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You are Simon, an intelligent personal assistant within the KIOS system. You can access knowledge bases provided in the user's "CONTEXT" and should expertly interpret this information to deliver the most relevant responses. Your role is to act as an expert writer. Prioritize clarity, trustworthiness, and appropriate formality when communicating with enterprise users. If a topic is outside your knowledge scope, admit it honestly and suggest alternative ways to obtain the information. Utilize chat history effectively to avoid redundancy and enhance relevance, continuously integrating necessary details. Use as much tokens as possible to provide a detailed response. **Elapsed Time: 0.18 seconds** FINAL QUERY Final Query: CONTEXT: ########## File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 1 Context: # Algorithms and Complexity **Herbert S. Wilf** University of Pennsylvania Philadelphia, PA 19104-6395 ## Copyright Notice Copyright 1994 by Herbert S. Wilf. This material may be reproduced for any educational purpose; multiple copies may be made for classes, etc. Charges, if any, for reproduced copies must be just enough to recover reasonable costs of reproduction. Reproduction for commercial purposes is prohibited. This cover page must be included in all distributed copies. --- ## Internet Edition, Summer, 1994 This edition of *Algorithms and Complexity* is available at the web site [http://www.cis.upenn.edu/~wilf](http://www.cis.upenn.edu/~wilf). It may be taken at no charge by all interested persons. Comments and corrections are welcome and should be sent to [wilf@math.upenn.edu](mailto:wilf@math.upenn.edu). A Second Edition of this book was published in 2003 and can be purchased now. The Second Edition contains solutions to most of the exercises. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 3 Context: # Chapter 5: NP-completeness ## 5.1 Introduction ................................................ 104 ## 5.2 Turing machines ............................................. 109 ## 5.3 Cook's theorem .............................................. 112 ## 5.4 Some other NP-complete problems ............................. 116 ## 5.5 Half a loaf .................................................. 119 ## 5.6 Backtracking (I): independent sets ........................... 122 ## 5.7 Backtracking (II): graph coloring ............................ 124 ## 5.8 Approximate algorithms for hard problems ..................... 128 #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 4 Context: # Preface For the past several years, mathematics majors in the computing track at the University of Pennsylvania have taken a course in continuous algorithms (numerical analysis) in the junior year, and in discrete algorithms in the senior year. This book has grown out of the senior course as I have been teaching it recently. It has also been tried out on a large class of computer science and mathematics majors, including seniors and graduate students, with good results. Selection by the instructor of topics of interest will be very important, because normally I've found that I can't cover anywhere near all of this material in a semester. A reasonable choice for a first try might be to begin with Chapter 2 (recursive algorithms) which contains lots of motivation. Then, as new ideas are needed in Chapter 2, one might delve into the appropriate sections of Chapter 1 to get the concepts and techniques well in hand. After Chapter 2, Chapter 4, on number theory, discusses material that is extremely attractive, and surprisingly pure and applicable at the same time. Chapter 5 would be next, since the foundations would then all be in place. Finally, material from Chapter 3, which is rather independent of the rest of the book, but is strongly connected to combinatorial algorithms in general, might be studied as time permits. Throughout the book, there are opportunities to ask students to write programs and get them running. These are not mentioned explicitly, with a few exceptions, but will be obvious when encountered. Students should all have the experience of writing, debugging, and using a program that is nontrivially recursive, for example. The concept of recursion is subtle and powerful, and is helped a lot by hands-on practice. Any of the algorithms of Chapter 2 would be suitable for this purpose. The recursive graph algorithms are particularly recommended since they are usually quite foreign to students’ previous experience and therefore have great learning value. In addition to the exercises that appear in this book, student assignments might consist of writing occasional programs, as well as delivering reports in class on assigned readings. The latter might be found among the references cited in the bibliographies in each chapter. I am indebted first of all to the students on whom I worked out these ideas, and to a number of colleagues for their helpful advice and friendly criticism. Among the latter I will mention Richard Brault, Daniel Kleitman, Albert Nijenhuis, Robert Tarjan, and Alan Tucker. For the no-doubt-numerous shortcomings that remain, I accept full responsibility. This book was typeset in TeX. To the extent that it’s a delight to look at, thank TeX. For the deficiencies in its appearance, thank my limitations as a typesetter. It was, however, a pleasure for me to have had the chance to typeset my own book. My thanks to the Computer Science department of the University of Pennsylvania, and particularly to Aravind Joshi, for generously allowing me the use of TeX facilities. Herbert S. Wilf #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 5 Context: # Chapter 0: What This Book Is About ## 0.1 Background An algorithm is a method for solving a class of problems on a computer. The complexity of an algorithm is the cost, measured in running time, or storage, or whatever units are relevant, of using the algorithm to solve one of those problems. This book is about algorithms and complexity, and so it is about methods for solving problems on computers and the costs (usually the running time) of using those methods. Computing takes time. Some problems take a very long time, others can be done quickly. Some problems seem to take a long time, and then someone discovers a faster way to do them (a ‘faster algorithm’). The study of the amount of computational effort that is needed in order to perform certain kinds of computations is the study of computational complexity. Naturally, we would expect that a computing problem for which millions of bits of input data are required would probably take longer than another problem that needs only a few items of input. So the time complexity of a calculation is measured by expressing the running time of the calculation as a function of some measure of the amount of data that is needed to describe the problem to the computer. For instance, think about this statement: “I just bought a matrix inversion program, and it can invert an \( n \times n \) matrix in just \( 1.2n^3 \) minutes.” We see here a typical description of the complexity of a certain algorithm. The running time of the program is being given as a function of the size of the input matrix. A faster program for the same job might run in \( 0.8n^3 \) minutes for an \( n \times n \) matrix. If someone were to make a really important discovery (see section 2.4), then maybe we could actually lower the exponent, instead of merely shaving the multiplicative constant. Thus, a program that would invert an \( n \times n \) matrix in only \( n^2 \log n \) minutes would represent a striking improvement of the state of the art. For the purposes of this book, a computation that is guaranteed to take at most \( c n^3 \) time for input of size \( n \) will be thought of as 'easy' computation. One that needs at most \( n^{10} \) time is also easy. If a certain calculation on an \( n \times n \) matrix were to require \( 2^n \) minutes, then that would be a 'hard' problem. Naturally some of the computations that we are calling 'easy' may take a very long time to run, but still, from our present point of view the important distinction to maintain will be the polynomial time guarantee or lack of it. The general rule is that if the running time is not a polynomial function of the amount of input data, then the calculation is an easy one; otherwise it’s hard. Many problems in computer science are known to be easy. To convince someone that a problem is easy, it is enough to describe a fast method for solving that problem. To convince someone that a problem is hard is hard, because you will have to prove to them that it is impossible to find a fast way of doing the calculation. It will not be enough to point to a particular algorithm and to lament its slowness. After all, that algorithm may be slow, but maybe there’s a faster way. Matrix inversion is easy. The familiar Gaussian elimination method can invert an \( n \times n \) matrix in time of at most \( O(n^3) \). To give an example of a hard computational problem we have to go far afield. One interesting one is called the 'tiling problem'. Suppose we are given infinitely many identical floor tiles, each shaped like a regular hexagon. Then we can tile the whole plane with them, i.e., we can cover the plane with no empty spaces left over. This can also be done if the tiles are identical rectangles, but not if they are regular hexagons. In Fig. 0.1 we show a tiling of the plane by identical rectangles, and in Fig. 0.2 is a tiling by regular hexagons. --- 1. See, for instance, Martin Gardner’s article in *Scientific American*, January 1977, pp. 110-121. 2. R. Berger, “The undecidability of the domino problem,” *Amer. Math. Soc.* 66 (1966), Amer. Image Analysis: ## Analysis of the Attached Visual Content ### 1. Localization and Attribution - **Image**: There is only one image present on the page. ### 2. Scene and Activity Analysis - **Scene Description**: The entire scene appears to be a page from a book or document. - **Activity**: There is no specific activity depicted. The main focus is on the textual content provided on this page. ### 4. Text Analysis - **Extracted Text**: - **Title**: "Chapter 0: What This Book Is About" - **Section**: "0.1 Background" - **Content**: The text describes algorithms, computational complexity, and introduces the purpose and scope of the book. It provides definitions and examples to explain how computational problems can be classified as easy or difficult based on the amount of computational effort required. - **Examples and Explanations**: - Discusses matrix inversion problems. - Mention of "tiling problem" with illustrations (Fig. 0.1 and Fig. 0.2) referred to but not shown in the extracted image. - **References**: - Martin Gardner's article in *Scientific American*. - R. Berger's work on the undecidability of the domino problem. ### 7. Anomaly Detection - **Anomalies**: There are no noticeable anomalies or unusual elements in the image. The layout and content appear to be standard for an academic or technical document. ### 8. Color Analysis - **Color Composition**: The image is black and white with text and some footnotes in a standard format. - **Dominant Colors**: Black text on a white background, which is typical for printed academic material. ### 9. Perspective and Composition - **Perspective**: The perspective is a direct, flat view of the page, typical of a scanned document or a digital book page. - **Composition**: The text is structured with a title, section heading, main body text, and footnotes. There is clear demarcation between sections with proper use of paragraphs and indentations. ### 12. Graph and Trend Analysis - **Existing Graphs**: There are references to figures (Fig. 0.1 and Fig. 0.2) related to tiling problems, but these figures are not part of the extracted image. ### Additional Aspects: #### Prozessbeschreibungen (Process Descriptions) - **Description**: The text provides a process description of computational complexity, explaining how various problems can be classified based on the time they take to compute. #### Typen Bezeichnung (Type Designations) - **Types/Categories**: - Problems are classified based on computational effort. - Examples given involve matrix inversion and tiling problems. - Terms like "easy", "hard", and "polynomial time" are used to categorize problems. ### Contextual Significance - **Overall Document Context**: This page serves as an introductory section to an academic or technical book on algorithms and computational complexity. It provides the reader with foundational knowledge necessary to understand subsequent chapters. - **Contribution to Overall Message**: The image sets the stage for comprehensive discourse on algorithms by explaining fundamental concepts and laying the groundwork for more detailed exploration of computational problems. ### Conclusion The visual content analyzed is an informative introductory page of a book or document centered on computational complexity and algorithms. It introduces key concepts, definitions, and examples, preparing readers for detailed discussions in subsequent sections. The layout, text structure, and references are typical of technical or academic literature. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 6 Context: # Chapter 0: What This Book Is About ![Fig. 0.1: Tiling with rectangles](path/to/rectangles-image) ![Fig. 0.2: Tiling with hexagons](path/to/hexagons-image) The way to do it, so even looking for an algorithm would be fruitless. That doesn’t mean that the question is hard for every polygon. Hard problems can have easy instances. What has been proved is that no single method exists that can guarantee that it will decide this question for every polygon. The fact that a computational problem is hard doesn’t mean that every instance of it has to be hard. The problem is hard because we cannot devise an algorithm for which we can give a guarantee of fast performance for all instances. Notice that the amount of input data to the computer in this example is quite small. All we need to input is the shape of the basic polygon. Yet not only is it impossible to devise a fast algorithm for this problem, it has been proved impossible to devise any algorithm at all that is guaranteed to terminate with a Yes/No answer after finitely many steps. That’s really hard! ## 0.2 Hard vs. Easy Problems Let’s take a moment more to say in another way exactly what we mean by an ‘easy’ computation vs. a ‘hard’ one. Think of an algorithm as being a little box that can solve a certain class of computational problems. Into the box goes a description of a particular problem in that class, and then, after a certain amount of time, or of computational effort, the answer appears. A **fast** algorithm is one that carries a guarantee of fast performance. Here are some examples. **Example 1.** It is guaranteed that if the input problem is described with B bits of data, then an answer will be output after at most \(6B^3\) minutes. **Example 2.** It is guaranteed that every problem that can be input with B bits of data will be solved in at most \(O(B^{1.5})\) seconds. A performance guarantee, like the two above, is sometimes called a ‘worst-case complexity estimate,’ and it’s easy to see why. If we have an algorithm that will, for example, sort any given sequence of numbers into ascending order of size (see section 2.1) it may find that some sequences are easier to sort than others. For instance, the sequence `1, 2, 7, 11, 10, 15, 20` is nearly in order already, so our algorithm might, if it takes advantage of the near-order, sort it very rapidly. Other sequences might be a lot harder for it to handle, and might therefore take more time. Math. Soc., Providence, RI 2 Image Analysis: ### Image Analysis #### 1. Localization and Attribution - **Image 1** is at the top center of the page. - **Image 2** is below **Image 1**, also at the center of the page. #### 2. Object Detection and Classification - **Image 1:** - **Object:** Grid pattern - **Category:** Geometric figure - **Key Features:** Consists of several intersecting horizontal and vertical lines forming rectangles. - **Image 2:** - **Object:** Hexagonal tiling - **Category:** Geometric figure - **Key Features:** Composed of multiple hexagons arranged in a regular pattern. #### 3. Scene and Activity Analysis - **Image 1:** - **Scene Description:** The image shows a simple grid pattern created by perpendicular lines intersecting to form rectangles. - **Main Actors and Actions:** None. - **Image 2:** - **Scene Description:** The image depicts a pattern formed by multiple hexagons joined together. - **Main Actors and Actions:** None. #### 4. Text Analysis - **Detected Text:** - "Chapter 0: What This Book Is About" - "Fig. 0.1: Tiling with rectangles" - "Fig. 0.2: Tiling with hexagons" - Additional body text discusses the difference between hard and easy computational problems and examples of each. - **Text Content Analysis:** - The text explains computational complexity, differentiating between easy and hard problems. It provides a brief introduction to the chapter's content and uses figures to illustrate examples of tiling patterns, which likely relate to computational problem examples. - Significant extracts: - "Fig. 0.1: Tiling with rectangles" and "Fig. 0.2: Tiling with hexagons" are captions that explain the images of tiling patterns. - The body text refers to the problem of tiling patterns, implying some connection to algorithmic complexity and computational problems. #### 10. Contextual Significance - The images and text are part of a chapter introduction about computational problems, specifically focusing on examples that illustrate differences between easy and hard problems. The tiling patterns with rectangles and hexagons visually represent examples of problems that an algorithm might need to solve. #### 12. Graph and Trend Analysis - There are no traditional graphs with axes and data points, but the tiling patterns might be interpreted as visual "graphs" in the sense of theoretical computer science problems (e.g., tiling and coverage problems). #### 13. Graph Numbers - Not applicable as there are no numerically labeled graphs. #### Additional Aspects - **Ablaufprozesse (Process Flows):** - None depicted. - **Prozessbeschreibungen (Process Descriptions):** - Descriptions refer to computational problem definitions and characteristics in the text, not shown visually as processes. - **Typen Bezeichnung (Type Designations):** - Types of problems (hard vs. easy) defined and exemplified in the text. - **Trend and Interpretation:** - Trend towards explaining complexity through visual and textual examples, emphasizing how different problem types demonstrate computational difficulty. - **Tables:** - None present in the document. ### Summary The document's main focus is to introduce computational problems and distinguish between those that are considered hard and easy. The images are geometric figures showing tiling with rectangles and hexagons, correlating to examples mentioned in the text. The text provides context on the computational complexity of problems using these visual examples to help illustrate different types of computational challenges. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 7 Context: The problem is this. Let \(n\) be a given integer. We want to find out if \(n\) is prime. The method that we choose is the following. For each integer \(m = 2, 3, \ldots, \sqrt{n}\)| we ask if \(m\) divides (evenly into) \(n\). If all of the answers are ‘No,’ then we declare \( n \) to be a prime number, else it is composite. First, we look at the computational complexity of this algorithm. That means that we are going to find out how much work is involved in doing the test. For a given integer \( n \) the work that we have to do can be measured in units of divisions of a whole number by another whole number. In those units, we obviously will do about \(\sqrt{n}\) units of work. It seems as though this is a tractable problem, because, after all, \(\sqrt{n}\) is of polynomial growth in \( n \). For instance, we do less than \( n \) units of work, and that’s certainly a polynomial in \( n \), isn’t it? So, according to our definition of fast and slow algorithms, the distinction was made on the basis of polynomial vs. faster-than-polynomial growth of the work done with the problem size, and therefore this problem must be easy. Right? Well, no not really. References to the distinction between fast and slow methods will show that we have to measure the amount of work done as a function of the number of bits of input to the problem. In this example, \( n \) is not the number of bits of input. For instance, if \( n = 59 \), we don’t need 59 bits to describe it, but only 6. In general, the number of binary digits in the bit string of an integer \( n \) is close to \(\log_2 n\). So in the problem of this example, testing the primality of a given integer \( n \), the length of the input bit string \( B \) is about \(\log_2n\). Seen in this light, the calculation suddenly seems very long. A string consisting of mere \(\log_2 n\) 0’s and 1’s has caused our mighty computer to do about \(\sqrt{n}\) units of work. If we express the amount of work done as a function of \( B \), we find that the complexity of this calculation is approximately \(2^{B/2}\), and that grows much faster than any polynomial function of \( B \). Therefore, the method that we have just discussed for testing the primality of a given integer is slow. See Chapter 4 for further discussions of this problem. At the present time no one has found a fast way to test for primality, nor has anyone proved that there isn’t a fast way. Primality testing belongs to the (well-populated) class of seemingly, but not provably, intractable problems. In this book we will deal with some easy problems and some seemingly hard ones. It’s the ‘seemingly’ that makes things very interesting. These are problems for which no one has found a fast computer algorithm, --- ### Scene and Activity Analysis: #### Described Scene and Activities: - The scene described in the image is a text-heavy document discussing computational problems, particularly focusing on the complexity and computational time of algorithms. - There are discussions of worst-case bounds, average case bounds, and the difference between fast and slow algorithms, followed by an example problem regarding the determination of whether a given integer is prime. #### Main Actors: - The main actors in the scene are algorithms and computational problems. - The text describes the actions of measuring computational complexity, evaluating polynomial vs. non-polynomial time, and testing algorithm efficiency. ### Contextual Significance: #### Contribution to Overall Message: - This text excerpt appears to be from a textbook or academic paper focusing on computational complexity, likely aimed at explaining the difference between easy and hard problems in computer science. - The example provided (primality testing) is used to illustrate these concepts and highlight the challenges associated with algorithm efficiency. ### Perspective and Composition: #### Perspective: - The perspective is that of a reader or student studying the material presented on the page. #### Composition: - The text is organized in a traditional scholarly format, with clear headings, paragraphs, and a structured presentation of concepts and examples. ### Anomaly Detection: - No anomalies were detected in the image; the text appears to be consistent with a standard academic document discussing computational complexity. ### Color Analysis: #### Color Composition: - The image consists entirely of black text on a white background. #### Dominant Colors: - The dominant colors are black and white. ### Graph and Trend Analysis: - There are no graphs present in this image. ### Diagram and Chart Analysis: - There are no diagrams or charts present in this image. ### Metadata Analysis: - There is no visible metadata within the image. ### Additional Aspects: #### Prozessbeschreibungen (Process Descriptions): - The described process is related to evaluating the computational complexity of algorithms, specifically mentioning worst-case and average-case scenarios and providing an example with testing for primality. #### Typen Bezeichnung (Type Designations): - Type designations include polynomial-time algorithms and non-polynomial-time algorithms, as well as tractable and intractable problems. ### Conclusion: This image contains a detailed section of text from what appears to be a textbook or academic paper on computational complexity, focusing on the difference between easy and hard computational problems, with a specific example relating to primality testing. The text touches upon polynomial-time guarantees, worst-case and average-case bounds, and the conceptual distinction between fast and slow algorithms. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 7 Context: So in some problems whose input bit string has \( B \) bits, the algorithm might operate in time \( 6B \) and on others it might need, say, \( 100 \log B \) time units, and for still other problems instances of length \( B \) the algorithm might need \( 5B^2 \) time units to get the job done. Well then, what would the warranty card say? I would have to pick out the worst possibility; otherwise the guarantee wouldn’t be valid. It would assure me that if the input problem instance can be described by \( B \) bits, then an answer will appear after at most \( 5B^2 \) time units. Hence, a performance guarantee is equivalent to an estimation of the worst possible scenario: the longest possible calculation that might ensue if \( B \) were input to the program. ### Worst-case bounds Worst-case bounds are the most common kind, but there are other kinds of bounds for running time. We might give an average case bound instead (see section 5.7). That wouldn’t guarantee performance no worse than \( \varepsilon \); it would state that if the performance is averaged over all possible input bit strings of \( B \) bits, then the average amount of computing time will be so-and-so (as a function of \( B \)). Now let’s talk about the difference between easy and hard computational problems and between fast and slow algorithms. A warranty that would not guarantee ‘fast’ performance would contain some function of \( B \) that grows faster than any polynomial, like \( 2^B \), for instance, or like \( n^{\log n} \), etc. It is the polynomial time vs. necessarily polynomial time guarantee that makes the difference between the easy and hard classes of problems, or between the fast and the slow algorithms. It is highly desirable to work with algorithms such that we can give a performance guarantee for running time that is at most a polynomial function of the number of bits of input. An algorithm is slow if, whatever polynomial \( P \) we think of, there exists arbitrary large values of \( B \), such that the input data strings of \( B \) bits, that cause the algorithm to do more than \( P(B) \) units of work. A computational problem is **intractable** if it can be proved that there is no fast algorithm for it. ### Example 3 Here is a familiar computational problem and a method, or algorithm, for solving it. Let’s see if the method has a polynomial time guarantee or not. The problem is this. Let \( n \) be a given integer. We want to find out if \( n \) is prime. The method that we choose is the following. For each integer \( n = 2, 3, \ldots, \sqrt{n} \) we ask if \( n \) divides (evenly into) \( n \). If all the answers are ‘No,’ then we declare \( n \) to be a prime number; else it is composite. In this way, we look at the computational complexity of this algorithm. Therefore, we are going to find out how much work is involved in doing the test. For a given integer \( n \) that we have to do can be measured in units of divisions of a whole number by another whole number. In those units, we obviously will do about \( \sqrt{n} \) units of work. It seems as though this is a tractable problem, because, after all, \( \sqrt{n} \) is of polynomial growth in \( n \). For instance, we do less than \( n \) units of work, and that’s certainly a polynomial in \( n \), isn’t it? So, according to our definition of fast and slow algorithms, this distinction was made on the basis of polynomial vs. faster-than-polynomial growth of the work done with the problem size, and therefore this problem must be easy. Right? Well, not really. Reference to the distinction between fast and slow methods will show that we have to measure the amount of work done as a function of the number of bits of input to the problem. In this example, \( n \) is not the number of bits of input. For instance, if \( n = 59 \), we don’t need 59 bits to describe it, but only 6. In general, the number of binary digits in the bit string of an integer \( n \) is close to \( \log_2 n \). So in the problem of this example, testing the primality of a given integer \( n \), the length of the input bit string \( B \) is about \( 10 \log_2 n \). Seen in this light, the calculation suddenly seems very long, a string consisting of mere log \( 0 \)’s and 1’s has caused our mighty computer to do about \( \sqrt{n} \) units of work. If we express the amount of work done as a function of \( B \), we find that the complexity of this calculation is approximately \( 2^{B/2} \), and that grows much faster than any polynomial function of \( B \). Therefore, note that we have just discussed for testing the primality of a given integer is slow. See Chapter 6 for further discussion of this problem. At the present time no one has found a fast way to test for primality, nor has anyone proved that there isn’t a fast way. Primality testing belongs to the (well-publicized) class of seemingly, but not provably, intractable problems. In this book, we will deal with some easy problems and some seemingly hard ones. It’s the ‘seemingly’ that makes things very interesting. These are problems for which no one has found a fast computer algorithm. Image Analysis: ### Localization and Attribution - The image consists of a single page of a text document. - This can be referred to as **Image 1**. ### Text Analysis #### Extracted Text The main section of the image contains printed text. Below is the complete text extracted from the image: --- **0.2 Hard vs. easy problems** So in some problems whose input bit string has \(B\) bits the algorithm might operate in time \(6B\), and on others its input need, say, \(120\log B\) time units, and for still other problem instances of length \(B\) bits the algorithm might need \(5B^2\) time units to get the job done. Well then, what would the warranty card say? It would have to pick out the worst possibility, otherwise the guarantee wouldn’t be valid. It would assure a user that if the input problem instance can be described by \(B\) bits, then an answer will appear after at most \(5B^2\) time units. Hence a performance guarantee is equivalent to an estimation of the worst possible scenario: the longest possible calculation that might ensue if \(B\) bits are input to the program. Worst-case bounds are the most common kind, but there are other kinds of bounds for running time. We might give an average case bound instead (see section 5.7). That wouldn’t guarantee performance no worse than so-and-so; it would state that if the performance is averaged over all possible input bit strings of \(B\) bits, then the average amount of computing time will be so-and-so (as a function of \(B\)). Now let’s talk about the difference between easy and hard computational problems and between fast and slow algorithms. A warranty that would not guarantee ‘fast’ performance would contain some function of \(B\) that grows faster than any polynomial. Like \(B^B\), for instance, or like \(2^{2^B}\), etc. It is the polynomial time vs. not necessarily polynomial time guarantee that makes the difference between the easy and the hard classes of problems, or between the fast and the slow algorithms. It is highly desirable to work with algorithms such that we can give a performance guarantee for running time that is at most a polynomial \(P(B)\) of the number of bits of input. An algorithm is slow if, whatever polynomial \( P \) we think of, there exist arbitrarily large values of \( B \), and input data strings of \( B \) bits, that cause the algorithm to do more than \( P(B) \) units of work. A computational problem is trackable if there is a fast algorithm that will find all its instances of it. A computational problem is intractable if it can be proved that there is no fast algorithm for it. **Example 3.** Here is a familiar computational problem and a method, or algorithm, for solving it. Let’s see if the method has a polynomial time guarantee or not. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 9 Context: # Chapter 1: Mathematical Preliminaries ## 1.1 Orders of Magnitude In this section, we're going to discuss the rates of growth of different functions and to introduce the five symbols of asymptotics that are used to describe those rates of growth. In the context of algorithms, the reason for this discussion is that we need a good language for the purpose of comparing the speeds with which different algorithms do the same job, or the amounts of memory that they use, or whatever other measure of the complexity of the algorithm we happen to be using. Suppose we have a method of inverting square nonsingular matrices. How might we measure its speed? Most commonly, we would say something like: "If the matrix is n × n, then the method will run in time 16.8n³." Then we would know that if a 100 × 100 matrix can be inverted, with this method, in 1 minute of computer time, then a 200 × 200 matrix would require 2³ = 8 times as long, or about 8 minutes. The constant 16.8 wasn't used at all in this example; only the fact that the labor grows as the third power of the matrix size was relevant. Hence, we need a language that will allow us to say that the computing time, as a function of n, grows "in the order of n³", or not as fast as n², or at least as fast as n×log n, etc. The key symbols that are used in the language of comparing the rates of growth of functions are the following five: - `o` (read as "little o" of) - `O` (read as "big O" of) - `Θ` (read as "theta of") - `ω` (read as "little omega of") - `Ω` (read as "big omega of") Now let's explain what each of them means. Let \( f(n) \) and \( g(n) \) be two functions of \( n \). Each of the five symbols above is intended to compare the rapidity of growth of \( f \) and \( g \). If we say that \( f(n) = o(g(n)) \), then informally we are saying that \( f \) grows more slowly than \( g \) does when \( n \) is very large. Formally, we state the following: **Definition.** We say that \( f(n) = o(g(n)) \) if \( \lim_{n \to \infty} f(n)/g(n) \) exists and is equal to 0. Here are some examples: 1. \( x^2 = o(x^3) \) 2. \( \sin x = o(x) \) 3. \( 1.4\log n = o(n/2 + 7 \cos x) \) 4. \( 1/n = o(1) \) 5. \( 23\log^2 n = o(n^2) \) We can see from these few examples that sometimes it might be easy to prove that a relationship is true and sometimes it might be rather difficult. Example (4), for instance, requires the use of L'Hôpital's rule. If we leave two computer programs, and if one of them inverts a n × n matrices in time \( c n^3 \) and the other does so in time \( o(n^3) \), then we know that for all sufficiently large values of \( n \), the performance guarantee of the second program will be superior to that of the first program. Of course, the first program might run faster on small matrices, say up to size 10,000 x 10,000. If a certain program runs in time \( n^{2.8} \) and someone were to produce another program for the same problem that runs in \( n^{2} \log n \), then that second program would be an improvement, at least in the theoretical sense. The reason for the "theoretical" qualification, once more, is that the second program would be known to be superior only if \( n \) were sufficiently large. The second symbol of the asymptotics vocabulary is the `O`. When we say that \( f(n) = O(g(n)) \), we mean, informally, that \( f \) certainly doesn't grow at a faster rate than \( g \). It might grow at the same rate or it might grow more slowly; both are possibilities that the "O" permits. Formally, we have the set: **Definition.** We say that \( f(n) = O(g(n)) \) if there exists \( c > 0 \) such that \( |f(x)| \leq c|g(x)| \) for \( x \) large enough. The qualifier \( n \to \infty \) will usually be omitted, since it will be understood that we will most often be interested in large values of the variables that are involved. For example, it is certainly true that \( n = O(n) \), but even more can be said, namely that \( n = O(1) \). Also \( 2x^2 + 7x = O(x^2) \) and \( 1/(1 + x^2) = O(1) \). Now we can sharpen the last example by saying that \( 1/(1 + x^2) = o(1) \). This is Image Analysis: ### Comprehensive Examination of the Visual Content #### Localization and Attribution - **Image 1**: There is only one image on the page. #### Text Analysis - **Image 1**: - **Heading**: - "Chapter 1: Mathematical Preliminaries" - "1.1 Orders of magnitude" - **Main Body**: - The text discusses the rates of growth of different functions and the asymptotic notations used to describe these rates. It focuses on the context of algorithms and their complexity. - Introduces the commonly used symbols of asymptotics: - 'ω' (read 'is little oh of') - 'O' (read 'is big oh of') - 'θ' (read 'is theta of') - '~' (read 'is asymptotically equal to') - 'Ω' (read 'is omega of') - Defines the formal mathematical notation and theorems for asymptotic behavior: - If there exist two functions \( f(x) \) and \( g(x) \), then \( f(x) = o(g(x)) \) means that \( \lim_{x \to \infty} \frac{f(x)}{g(x)} = 0 \). - Examples and explanations: - \( x^2 = o(x^3) \) - \( sin x = o(x) \) - \( 14.7ln_{10}x = o(x/2 + 7 cos x) \) - \( 1/x = θ(1) \) - \( 2^3logx = θ(log x) \) - Mention of error rates, comparing two algorithms' performance, one with order \( n^3 \) and another optimized to \( n^2 log n \). - Qualifiers and sharper classifications in complex cases beyond just 'big O' notation. #### Scene and Activity Analysis - **Image 1**: - The image contains text from an academic book or lecture notes on mathematical preliminaries, particularly focused on orders of magnitude and asymptotic analysis. There are no interactive activities; the image is purely educational. #### Contextual Significance - **Image 1**: - This image appears to be from an educational or reference book in mathematics or computer science, especially useful for students or professionals in understanding algorithm complexities. It contributes significantly by laying the groundwork for more complex discussions on algorithm analysis. The content is foundational for those learning about computational efficiency and problem-solving within computer science. #### Perspective and Composition - **Image 1**: - The image is a full-page scan from a book, with a top-down perspective showing clear text layout. The heading and subheadings are bolded for emphasis, and examples within the text are indented and numbered for clarity. No objects, diagrams, or charts are depicted in the image presented, focusing solely on textual content describing mathematical concepts. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 11 Context: ### Contextual Significance: - **Overall Message**: This page aims to educate the reader on the different rates at which functions grow and to establish a foundational understanding of comparative growth rates in mathematical analysis. - **Contribution to Theme**: It likely contributes to a larger section or chapter on mathematical analysis, specifically focusing on asymptotic behavior of functions. This foundational knowledge is essential for understanding more advanced concepts in mathematical and computational analysis. #### 12. Graph and Trend Analysis: - The page does not contain graphs or visual data plots but rather focuses on textual descriptions and mathematical expressions. #### 13. Graph Numbers: - Since no graphs are present on this page, no numerical data points for graphs can be provided. ### Summary: - The provided visual content is an excerpt from a mathematical text discussing the concept of orders of magnitude in function growth. It provides formal definitions, examples using limits, and explanations of different growth rates (polynomial, exponential, and beyond). The analysis here helps in understanding the structure and significance of the content in a broader educational context. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 11 Context: **Perspective and Composition:** 1. **Perspective:** - Standard straight-on view of the page, as commonly found in scanned documents or electronic texts. 2. **Composition:** - Text is organized into sections with headings for definitions, examples, and specific types of function growth. - Each paragraph builds on the previous, contributing to the overall understanding of function growth rates. **Contextual Significance:** 1. **Overall Document Context:** - Likely part of a mathematical textbook or academic paper. - The content plays a crucial role in teaching or explaining the complexity related to function growth rates, which is relevant in fields like computer science, mathematics, and related disciplines focusing on algorithm complexity. **Tables:** - No tables are included in the image. **Metadata Analysis:** - No metadata can be analyzed from the visual. **Graph and Trend Analysis:** - No graphs are included in the image. **Graph Numbers:** - No graphs to provide data points for. **Ablaufprozesse (Process Flows):** - No process flows depicted. **Prozessbeschreibungen (Process Descriptions):** - No detailed processes depicted beyond the instructional explanations about different function growth rates. **Typen Bezeichnung (Type Designations):** - Discusses types of function growth (e.g., polynomial, exponential). **Trend and Interpretation:** 1. **Trend Identification:** - The trend focuses on the increasing complexity and rates of growth in mathematical functions. 2. **Interpretation:** - Understanding these trends is crucial for comparing algorithms and their efficiencies. In summarizing the visual content, the focus is academic, dissecting different types of functions and their growth rates, benefiting professionals and students in mathematics and computer science. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 11 Context: # 1.1 Orders of magnitude **Definition.** We say that \( f(x) = \Omega(g(x)) \) if there is an \( \epsilon > 0 \) and a sequence \( x_1, x_2, \ldots \) such that \( \forall j: f(x_j) > \epsilon g(x_j) \). Now let's introduce a hierarchy of functions according to their rates of growth when \( x \) is large. Among commonly occurring functions of \( x \) that grow without bound as \( x \to \infty \), perhaps the slowest growing ones are functions like \( \log x \) or maybe \( (\log x)^3 \) or things of that sort. It is certainly true that \( \log \log x \to \infty \), but it takes its time about it. When \( x = 1{,}000{,}000 \), for example, \( \log_2 x \) has the value 19.3. Just a bit faster growing than the ‘snails’ above is \( x^{\frac{1}{10}} \). After all, for \( x = 1{,}000 \), we have \( 1{,}000^{\frac{1}{10}} \approx 2.5 \). If we had a computer algorithm that could do \( n \) things in time \( \log n \) and someone found another method that could do the same job in time \( O(\log \log n) \), then the second method, other things being equal, would indeed be an improvement, but it might have to be extremely large before you would notice the improvement. Next on the scale of rapidity of growth we might mention the powers of \( x \). For instance, think about \( x^{0.1} \). It grows faster than \( \log x \), although you wouldn’t believe it if you tried to substitute a few values of \( x \) and to compare the answers (see exercise 1 at the end of this section). **How could we prove that \( x^{0.1} \) grows faster than \( \log x \)?** By using L'Hôpital's rule. **Example.** Consider the limit \( \lim_{x \to \infty} \frac{x^{0.1}}{\log x} \) as \( x \to \infty \). As \( x \to \infty \) assumes the indeterminate form \( \infty/\infty \), and it is therefore a candidate for L'Hôpital's rule, which tells us that if we want to find the limit then we can differentiate the numerator, differentiate the denominator, and try again to let \( x \to \infty \). If we do this, then instead of the original ratio, we find the ratio \[ \frac{(-0.1)x^{-0.9}}{1/x} = -0.1x^{0.1} \] which obviously grows without bound as \( x \to \infty \). Therefore the original ratio \( \frac{x^{0.1}}{\log x} \) also grows without bound. What we have proved, precisely, is that \( x^{0.1} \) grows faster than \( \log x \). To continue up the scale of rates of growth, we meet \( x^2, x^3, x^{15} \log^2 x \), etc., and then encounter functions that grow faster than every fixed power of \( x \) just as \( x^2 \) grows faster than every fixed power of \( x \). Consider \( e^{x^2} \). Since this is the same as \( e^{x \cdot x} \), it will obviously grow faster than \( x^{1000} \), in fact it will be larger than \( x^{1000} \) as soon as \( x > 1000 \), i.e., as soon as \( x > e^{1000} \) (don’t hold your breath!). Hence \( e^{x^2} \) is an example of a function that grows faster than every fixed power of \( x \). Another such example is \( e^{\sqrt{x}} \). **Definition.** A function that grows faster than \( x^n \) for every constant \( n \), but grows slower than \( e^c \) for every constant \( c \) is said to be of moderately exponential growth if for every \( \epsilon > 0 \) we have \( f(x) = \Omega(x^n) \) and for every \( c > 0 \) we have \( f(x) = O(e^c) \). Beyond the range of moderately exponential growth are the functions that grow exponentially fast. Typical of such functions are \( 10^{n} \), \( 2^{n^2} \), and so forth. Formally, we have the: **Definition.** A function \( f \) is of exponential growth if there exists \( c > 1 \) such that \( f(x) = \Omega(c^x) \) and there exists \( d \) such that \( f(x) = O(d^x) \). If we truncate up a function of exponential growth with smaller functions then we will not change the fact that it is of exponential growth. The series \( e^{x^2/(n^2 + 37)} \) remains of exponential growth, because \( e^{x^2} \) is by itself, and it resists the efforts of the smaller functions to change its mind. Beyond the exponentially growing functions there are functions that grow as fast as you might please. Like \( n! \), for instance, which grows faster than \( c^n \) for every fixed constant \( c \) and like \( 2^{n!} \), which grows much faster than \( n! \). The growth ranges that are the most common to computer scientists are ‘between’ the very slowly, logarithmically growing functions and the functions that are of exponential growth. The reason is simple: if a computer algorithm requires more than an exponential amount of time to do its job, then it will probably not be used, or at any rate it will be used only in highly unusual circumstances. In this book, the algorithms that we will deal with fall in this range. Now we have discussed the various symmetries and asymptotes that are used to compare the rates of growth of pairs of functions, and we have dismissed the pecking order of rapidity of growth, so that we have a small catalogue of functions that grow slowly, medium-fast, fast, and super-fast. Next, let’s look at the growth of sums that involve elementary functions, with a view toward discovering the rates at which the sums grow. Image Analysis: **Text Analysis:** 1. **Text Detection and Extraction:** - The page contains a significant amount of text discussing mathematical functions and their rates of growth. 2. **Content Analysis and Significance:** - **Definitions and Explanations:** - The text defines the notation \(f(x) = \Omega(g(x))\) for functions that grow without bound. - It discusses functions growing faster than linear, such as logarithmic, polynomial, and exponential functions. - Examples illustrate the concepts further using L'Hôpital's rule to compare growth rates. - **Function Hierarchy:** - It introduces a hierarchy of functions according to their growth rates when \(x \to \infty\). - Examples include \(\log \log x\), \(\log x\), \(x^c\), and various polynomial and exponential functions. - **Moderately Exponential and Exponential Growth:** - Definitions around moderately exponential growth (\(f(x) = \Omega(a^{(\gamma x)}\))) - Explanation involving exponential growth (\(f(x) = \Theta(e^{cx})\)) and functions of faster growth beyond exponential. - **General Discussion:** - The importance of comparing growth rates is emphasized, particularly for understanding computational complexity. - Techniques such as L'Hôpital's rule are presented for proving and understanding these contrasts in growth rates. **Localization and Attribution:** 1. **Page Layout:** - Single page visual, holding multiple sections of text with definitions, explanations, and examples all centered around mathematical functions and growth rates. **Scene and Activity Analysis:** 1. **Scene Description:** - The entire scene is text-based with mathematical formulas and theoretical explanations filling the content. - The activity involves explanations, definitions, and examples intended to educate readers on the rates of growth for functions in mathematical terms. There are no human actors; the main component is the academic content. **Diagram and Chart Analysis:** - There are no diagrams or charts in this section of text. **Product Analysis:** - No products are depicted. **Anomaly Detection:** - No anomalies are apparent. **Color Analysis:** 1. **Color Composition:** - The text is black on a white background, creating high readability and a standard academic format. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 15 Context: # 1.2 Positional number systems is a function \( h \) such that \( h = o(h) \) and \( h = \Theta(g) \). Give an explicit construction for the function \( h \) in terms of \( g \). 6. (This exercise is a warmup for exercise 7.) Below there appear several mathematical propositions. In each case, write a proposition that is the negation of the given one. Furthermore, in the negation, do not use the word 'not' or any negation symbols. In each case the question is, “If this isn’t true, then what is true?” (a) \( \forall x > 0, f(x) \neq 0 \) (b) \( \forall y > 0, f(y) > 0 \) (c) \( \forall z \in \mathbb{R}, \, f(z) < f(x) \) (d) \( \exists y > 3 \, \forall z < f(y) \) (e) \( \forall x \exists z \, z < f(y) \) Can you formulate a general method for negating such propositions? Given a proposition that contains \( \forall, \exists \), what rule would apply in order to negate the proposition and leave the result in positive form (containing no negation symbols or ‘not’s)? 7. In this exercise we will work out the definition of the \( \Omega \): (a) Write out the precise definition of the statement \( \lim_{n \to \infty} h(n) = 0 \) (use `\varepsilon`). (b) Write out the negation of your answer to part (a) as a positive assertion. (c) Use your answer to part (b) to give a positive definition of the assertion \( f(n) \neq o(g(n)) \), and thereby justify the definition of the \( \Omega \) symbol that was given in the text. 8. Arrange the following functions in increasing order of their rates of growth, for large \( n \). List them so that each one is ‘little o’ of its successor: \[ 2^{\sqrt{n}}, n^{3.01}, n^{2}, n^{1.6}, \log^{3} n + 1, \sqrt{n}, n \log n, n^{3} \log(\log n), n^{2}, (n + 4)^{12} \] 9. Find a function \( f(x) \) such that \( f(x) = O(2^{x}) \) is true for every \( \varepsilon > 0 \), but for which it is not true that \( f(x) = O(1) \). 10. Prove that the statement \( f(n) = O(2^{\varepsilon}) \) for every \( \varepsilon > 0 \) is equivalent to the statement \( f(n) = \Omega(2^{\varepsilon}) \) for every \( \varepsilon > 0 \). ## 1.2 Positional number systems This section will provide a brief review of the representation of numbers in different bases. The usual decimal system represents numbers by using the digits \( 0, 1, \ldots, 9 \). For the purpose of representing whole numbers we can imagine that the powers of 10 are displayed below like this: \[ \cdots, 100000, 10000, 1000, 100, 10, 1. \] Then, to represent an integer we can specify how many copies of each power of 10 we would like to have. If we write 237, for example, then that means that we want 2 copies of \( 10^2 \) and 3 copies of \( 10^1 \) and 7 copies of \( 10^0 \). In general, if we write out the string of digits that represents a number in the decimal system, as \( d_m d_{m-1} \ldots d_1 d_0 \), then the number that is being represented by that string of digits is \[ n = \sum_{i=0}^{m} d_i \cdot 10^i. \] Now let’s try the binary system. Instead of using \( 10 \)’s we’re going to use \( 2 \)’s. So we imagine that the powers of \( 2 \) are displayed below as: \[ \cdots, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1. \] Image Analysis: ### Comprehensive Examination of the Visual Content #### 1. Localization and Attribution: - The entire content is one cohesive image displaying a single document page. - The page includes both text and mathematical notation, interspersed with explanations. #### 2. Object Detection and Classification: - **Detected Objects:** - Text: The page is predominantly filled with textual content. - Mathematical Notations: Various mathematical expressions and equations are present. #### 3. Scene and Activity Analysis: - **Scene:** The image represents a textbook page. - **Activities:** - The text appears to be an explanation of mathematical concepts, problems, and theorems. - The page is broken down into several exercises or sections (labelled 6, 7, 8, 9, etc.). #### 4. Text Analysis: - **Extracted Text:** The text includes exercises and explanations regarding functions, propositions in mathematics, the definition and application of the 'Ω' (big-Omega) notation, and a brief discussion on positional number systems. - **Key Text Components:** - **Exercise 6:** Deals with understanding the concepts of functions with respect to 'o(h)' and 'o(g)'. - **Exercise 7:** Focuses on the definition of the 'Ω' notation and its implications. - **Exercise 8:** Sorting functions based on their growth rates. - **Exercise 9:** Finding a function \( f(n) \) relative to growth orders. - **Section 1.2:** Discusses positional number systems and their representation. #### 5. Diagram and Chart Analysis: - **No diagrams or charts are present in the content.** #### 7. Anomaly Detection: - **None detected.** The page content appears consistent with typical textbook formatting. #### 8. Color Analysis: - **Color Composition:** The page is in grayscale, predominantly black text on a white background. #### 9. Perspective and Composition: - **Perspective:** Standard head-on view of the document page. - **Composition:** The text is well-organized with structured sections and numbered exercises, typical of textbook formatting. #### 10. Contextual Significance: - **Context:** The image is part of a mathematical or computer science textbook. - **Contributions:** It serves an educational purpose, likely aimed at undergraduate students studying algorithms or theoretical computer science. #### 13. Graph Numbers: - **Data Points:** - **Functions arranged by growth rates:** \( 2^{\sqrt{\log n}}, e^{(\log n)^2}, n^{3\log \log n}, 2^n, n^2, n^{2^n}, \log^2 n + \sqrt{n}, n^{1.6}, \log^2 n + 1, \sqrt{n}, n^{\log \log n}, n^3 \log n, (\log \log n)^3, n^{1.52}, (\pi + 4)^{\log n}, n^3 log n, n (log \log n)^{3/2}, (\pi + 4)^2 \). - These descriptions categorize various functions by their growth rates, providing a foundational understanding of algorithmic efficiency and complexity. #### Textual Breakdown and Analysis: - **Mathematical Propositions:** Exercise 6 gives multiple scenarios of logical and mathematical propositions, exploring negations and defining related terms. - **Growth Rates:** Exercise 8 asks to organize various functions by their growth rates, critical for algorithm analysis. - **Positional Number Systems:** The section 1.2 provides an introduction to positional number systems, a fundamental concept in number theory and computer science. #### Tables: - **No explicit tables are present.** This analysis helps to understand the content of the page thoroughly, laying out detailed insights into the educational content it presents and its structural components. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 16 Context: # Chapter 1: Mathematical Preliminaries To represent a number we will specify how many copies of each power of 2 we would like to have. For instance, if we write \(1101\), then we want an 8, a 4 and a 1, so this must be the decimal number 13. We will write \( (13)_{10} = (1101)_2 \) to mean that the number 13, in the base 10, is the same as the number 1101, in the base 2. In the binary system (base 2) the only digits we will ever need are 0 and 1. What that means is that if we use only 0’s and 1’s then we can represent every number in exactly one way. The unique representation of every number is, after all, what we must expect and demand of any proposed system. Let’s elaborate on this last point. If we were allowed to use more digits than just 0’s and 1’s then we would be able to represent the number \( (13)_{10} \) as a binary number in a whole lot of ways. For instance, we might make the mistake of allowing digits 0, 1, 2, 3, etc. Then 13 would be representable as \( 3 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0 \) etc. So if we were to allow too many different digits, then numbers would be representable in more than one way as a string of digits. If we were to allow too few different digits then we would find that some numbers have no representation at all. For instance, if we were to use the decimal system with only the digits 0, 1, ..., 9, then infinitely many numbers would not be able to be represented, so we had better keep the 9's. The general proposition is this. ## Theorem 1.2.1 Let \( b > 1 \) be a positive integer (the ‘base’). Then every positive integer \( n \) can be written in one and only one way in the form \[ n = d_k b^k + d_{k-1} b^{k-1} + d_{k-2} b^{k-2} + \ldots \] if the digits \( d_0, d_1, \ldots \) lie in the range \( 0 \leq d_i < b - 1 \) for all \( i \). **Remark:** The theorem says, for instance, that in the base 10 we need the digits 0, 1, 2, ..., 9; in the base 2 we need only 0 and 1, in the base 16 we need sixteen digits, etc. **Proof of the theorem:** If \( b \) is fixed, the proof is by induction on \( n \), the number being represented. Clearly, the number 1 can be represented in one and only one way with the available digits (why?). Suppose, inductively, that every integer \( 1, 2, \ldots, n - 1 \) is uniquely representable. Now consider the integer \( n \). Define \( n \mod b \). Then \( d \) is one of the permissible digits. By induction, the number \( n - d \) is uniquely representable, say \[ n - d = \frac{n - d}{b} \] Then clearly, \[ n = d + b \cdot \frac{n - d}{b} \] \[ = d + d_0 b + d_1 b^2 + d_2 b^3 + \ldots \] is a representation of \( n \) that uses only the allowed digits. Finally, suppose that \( n \) has some other representation in this form also. Then we would have \[ n = a_0 + a_1 b + a_2 b^2 + \ldots \] \[ = c_0 + c_1 b + c_2 b^2 + \ldots \] Since \( a_0 \) and \( c_0 \) are both equal to \( n \mod b \), they are equal to each other. Hence the number \( n' = (n - a_0)/b \) has two different representations, which contradicts the inductive assumption, since we have assumed the truth of the result for all \( n < n \). The bases \( b \) that are the most widely used are, aside from 10, 2 (‘binary system’), 8 (‘octal system’) and 16 (‘hexadecimal system’). The binary system is extremely simple because it uses only two digits. This is very convenient if you’re a computer or a computer designer, because the digits can be determined by some component being either ‘on’ (digit 1) or ‘off’ (digit 0). The binary digits of a number are called its bits or its bit string. Image Analysis: ### Analysis of the Provided Visual Content **Localization and Attribution:** - **Single Image Identified**: The content appears to be a single-page excerpt from a mathematical text. - **Page Number**: The image corresponds to page number 12. **Text Analysis:** - **Text Extraction**: 1. **Title**: "Chapter 1: Mathematical Preliminaries" 2. **Paragraphs**: - Discusses the representation of numbers, specifically focusing on binary (base 2) and its comparison with other bases such as base 10 and base 16. - Highlights the importance of representation in binary digits (0 and 1) and the unique representation of each number. - Provides an in-depth explanation with examples, e.g., the number 13 represented in binary as (1101)₂. 3. **Theorem 1.2.1**: "Let b > 1 be a positive integer (the ‘base’). Then every positive integer n can be written in one and only one way in the form..." 4. **Remark**: Discusses the number of digits needed for different bases. 5. **Proof of the theorem**: Provides a proof by induction on n and explains the uniqueness of number representation in any given base b. 6. **Additional Explanation**: Explains the contradiction proof for unique representation and uses the concept of congruences. - **Content Significance**: - The text is significant in an educational or academic context as it explains fundamental concepts in number representation and bases, a foundational topic in mathematics and computer science. **Object Detection and Classification:** - **Objects**: The detected objects are: - Text blocks - Mathematical equations - Page elements such as titles and section headings. - **Classification**: - **Text Blocks**: Academic mathematical content. - **Equations**: Mathematical notations and variables. **Scene and Activity Analysis:** - **Scene Description**: The page presents a focused educational layout with the main activity being the explanation and teaching of mathematical concepts related to number bases. - **Main Actors**: While no human figures are present, the 'actors' can be considered the mathematical numbers, variables, and equations being utilized to explain concepts. **Color Analysis:** - **Color Composition**: Predominantly black text on a white background, typical for academic or textbook-style documents. - **Impact on Perception**: The high-contrast colors (black on white) ensure readability and focus on the textual content. **Diagram and Chart Analysis:** - **Absence of Diagrams/Charts**: Since there are no diagrams or charts present, this aspect is not applicable. **Perspective and Composition:** - **Perspective**: The perspective is a flat, bird’s eye view typical of scanned or digital textbook pages. - **Composition**: Structured in a classical textbook format with sections, subsections, and logical flow of mathematical explanations. **Contextual Significance:** - **Overall Document Context**: The image is extracted from an academic publication focused on mathematics, essential for teaching and understanding mathematical theories. - **Contribution to Theme**: This page contributes foundational knowledge crucial for anyone studying mathematical preliminaries, especially number representations in different bases. **Anomaly Detection:** - **No Anomalies Detected**: The page content follows a standard format with no noticeable anomalies. **Metadata Analysis:** - **Absence of Metadata**: Since the image appears to be a page from a text document, no explicit metadata details (like date or camera settings) are visible or applicable. **Additional Aspects:** - **Procesbeschreibungen (Process Descriptions)**: The page thoroughly describes the process of representing numbers in various bases through progressive explanation and proofs. - **Trend and Interpretation**: The text does not present trends but focuses on the theoretical and proof-based interpretation of number representations. - **Typen Bezeichnung (Type Designations)**: The types/categories specified are numerical bases – binary (base 2), decimal (base 10), and hexadecimal (base 16). Overall, this analysis comprehensively examines and categorizes the textual content of an academic page explaining mathematical preliminaries in number representation. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 22 Context: #### Additional Aspects: - **Ablaufprozesse (Process Flows)**: - Recurrence relation solving methods, from first-order equations to Fibonacci sequence solutions. - **Prozessbeschreibungen (Process Descriptions)**: - Detailed description of deriving general solutions for recurrence relations and solving quadratic characteristic equations. - **Typen Bezeichnung (Type Designations)**: - Various types of recurrence relations and quadratic characteristic equations addressed (first-order, second-order, homogeneous equations). ### Summary The image is a page from an academic text on mathematical preliminaries, focusing on solving first and second-order recurrence relations. It includes several mathematical notations and equations, as well as detailed explanations of solving techniques, particularly relating to the Fibonacci sequence. The text is well-organized with numbered equations and contexts clearly defined for educational purposes. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 22 Context: 5. **Contextual Significance:** - The provided content seems to be part of a mathematical textbook or an academic paper focused on recurrence relations and their solutions. - This section (from Chapter 1) introduces fundamental concepts and methods that are likely necessary for understanding more complex topics later in the document. - The example of the Fibonacci sequence serves to illustrate the practical application of the theoretical methods discussed. The provided analysis covers the text and context of the content in depth. Specific numerical and algebraic expressions have been noted to capture the mathematical rigor presented in the image. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 23 Context: # 1.4 Recurrence Relations The Fibonacci numbers are in fact 1, 1, 2, 3, 5, 8, 13, 21, 34, ... It isn't even obvious that the formula (1.4.15) gives integer values for the \( F_n \). The reader should check that the formula indeed gives the first few \( F_n \) correctly. Just to exercise our newly acquired skills in asymptotics, let's observe that since \( (1 + \sqrt{5})/2 > 1 \) and \( |(1 - \sqrt{5})/2| < 1 \), it follows that when \( n \) is large we have: \[ F_n \sim \left( \frac{(1 + \sqrt{5})}{2} \right)^{n+1} \frac{1}{\sqrt{5}} \] The process of looking for a solution in a certain form, namely in the form \( \alpha^n \), is subject to the same kind of special treatment, in the case of repeated roots, that we find in differential equations. Corresponding to a double root of the associated quadratic equation \( a^2 - \alpha a + \beta = 0 \), we would find two independent solutions \( \alpha^n \) and \( n\alpha^n \), so the general solution would be in the form \( \alpha^n (c_1 + c_2 n) \). ## Example Consider the recurrence: \[ x_{n+1} = 2x_n - x_{n-1} \quad (n \geq 1; \, x_0 = 5; \, x_1 = 5) \tag{1.4.16} \] If we try a solution of the type \( x_n = \alpha^n \), then we find that it satisfies the quadratic equation \( a^2 - 2a + 1 = 0 \). Hence the two roots are \( a = 1 \). The general solution is \( x_n = c_1(\alpha^n) + c_2(n\alpha^n) \). After inserting the given initial conditions, we find that: \[ x_0 = 1 = c_1; \quad x_1 = 5 = c_1 + c_2 \] If we solve for \( c_1 \) and \( c_2 \), we obtain \( c_1 = 1, c_2 = 4 \), and therefore the complete solution of the recurrence (1.4.16) is given by: \[ x_n = 4n + 1 \] Now let's look at recurrent inequalities, like this one: \[ x_{n+1} \leq x_n + x_{n-1} + n^2 \quad (n \geq 1; \, x_0 = 0; \, x_1 = 0) \tag{1.4.17} \] The question is, what restriction is placed on the growth of the sequence \( \{x_n\} \) by (1.4.17)? By analogy with the case of difference equations with constant coefficients, the thing to try here is to set \( x_n \leq K \cdot n^t \). So suppose it is true that \( x_n \leq K n^t \) for \( n = 0, 1, 2, \ldots, N \). Then from (1.4.17) with \( n = N \), we find: \[ x_{N+1} \leq K N^t + K N^{t-1} + N^2 \] Let \( c \) be the positive real root of the equation \( \alpha^2 - c = 1 \) and suppose that \( \alpha > c \). Then \( \alpha^2 > c + 1 \); say, where \( t > 0 \). Hence: \[ x_{N+1} \leq K N^{t-1}(1 + t) + N^2 = K n^{t-1} \left( K n^{t-1} \right) \tag{1.4.18} \] In order to ensure that \( x_{N+1} < K n^{t+1} \), we need to fix \( K < \max_{n \geq 2} \left( \frac{N^2}{N^{t-1}} \right) \). As long as we choose: \[ K < \max_{n \geq 2} \left( \frac{N^2}{N^{t-1}} \right) \tag{1.4.19} \] In which the right member is clearly finite, the inductive step will go through. The conclusion is that (1.4.17) implies that for every fixed \( t > 0 \), \( x_n = O(n^{t+ε}) \), where \( ε = (1 + \sqrt{5})/2 \). The same argument applies to the general situation that is expressed in: Image Analysis: ### Analysis of the Attached Visual Content #### 1. Localization and Attribution - **Image Positioning:** The document contains a single image occupying the entire visible area of the page. It will be referred to as **Image 1**. #### 2. Object Detection and Classification - **Categories Detected:** - **Text Sections:** The image contains various sections of text including paragraphs and mathematical equations. - **Mathematical Equations:** Displayed prominently within the text. #### 3. Scene and Activity Analysis - **Description:** Image 1 depicts a page from an academic or educational document likely related to mathematics or computer science. - **Main Activities:** The main activity includes the explanation of mathematical recurrence relations and the exploration of the Fibonacci numbers. #### 4. Text Analysis - **Text Extraction:** - Title/Header: "1.4 Recurrence relations" - Paragraphs explaining the concept of recurrence relations and their properties. - Example recurrence relation shown as \( x_{n+1} = 2x_n - x_{n-1} \) and subsequent mathematical steps. - Additional explanations about the growth of sequences and inferences from specific equations. - Concluding text about general cases and implications of mathematical findings. - **Content Significance:** - **Educational Value:** The text is highly educational, explaining sophisticated mathematical concepts and recurrence relations. It is written to enhance the reader's understanding of solving recurrence relations using mathematical examples and proofs. - **Instructional Examples:** Provides clear mathematical examples and step-by-step solutions to illustrate important points. #### 5. Diagram and Chart Analysis - **Mathematical Equations:** Several equations are included throughout the text to explain and demonstrate the properties of recurrence relations. #### 7. Anomaly Detection - **Unusual Elements:** No anomalies detected in the page content; the text appears to follow logical and expected formatting for an academic document. #### 8. Color Analysis - **Color Scheme:** The page is predominantly black and white, typical of printed or digital academic documents. This maintains clarity and legibility. #### 9. Perspective and Composition - **Perspective:** The image is shown from a top-down, direct view, allowing full visibility of the page. - **Composition:** The page is well-organized, with clear sections for titles, text, and equations. The use of bold sections and mathematical notation contributes to a structured layout. #### 10. Contextual Significance - **Overall Contribution:** This page contributes significantly to a larger educational text by laying the groundwork for understanding recurrence relations. It positions itself as an informational and foundational piece within the studied subject. #### 13. Graph Numbers - **Data Points:** Not applicable for this image as it does not contain graph data points but rather textual and mathematical content. ### Additional Aspects #### Ablaufprozesse (Process Flows) - **Described Processes:** The text describes the process of solving recurrence relations and analyzing their properties. #### Prozessbeschreibungen (Process Descriptions) - **Detailed Descriptions:** The document elaborates on how to derive solutions for recurrence relations and the resulting sequence behavior. ### Conclusion Image 1 is an academic document page focused on recurrence relations in mathematics. It provides comprehensive text and mathematical examples to explain the concept, making it an educational resource for students or readers interested in advanced mathematical topics. The page layout is clear and well-structured to facilitate understanding. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 24 Context: ### Typen Bezeichnung (Type Designations): - **Types or Categories**: - **Theorem 1.4.1**: Describes a mathematical type concerned with sequences and inequalities. - **Exercises**: Various types as each explores different aspects or variations of sequence relations. ### Trend and Interpretation: - **Trends**: The theorem and proof suggest a trend towards formalizing and generalizing inequalities in sequences. ### Anomaly Detection: - **Unusual Elements**: The small black square near the bottom-left of the page stands out. It might be a formatting error or a placeholder without textual significance. ### Color Analysis: - **Dominant Colors**: The page is primarily black and white, emphasizing clarity and readability typical of academic texts. This comprehensive analysis covers the key requested aspects. If further detailed breakdown or additional aspects not mentioned here are required, please specify. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 25 Context: ```markdown 8. Write out a complete proof of theorem 1.4.1. 9. Show by an example that the conclusion of theorem 1.4.1 may be false if the phrase “for every fixed \( \varepsilon > 0 \)” were replaced by “for every fixed \( \varepsilon \geq 0 \) . . .” 10. In theorem 1.4.1, if we find the phrase “the positive real root of . . . ” Prove that this phrase is justified, in that the equation shown always has exactly one positive real root. Exactly what special properties of that equation did you use in your proof? # 1.5 Counting For a given positive integer \( n \), consider the set \( \{ 1, 2, \ldots, n \} \). We will denote this set by the symbol \( [n] \), and we want to discuss the number of subsets of various kinds that it has. Here is a list of all the subsets of \( [n] \): \( \emptyset, \{1\}, \{2\}, \{1, 2\} \). There are \( 2^n \) subsets. We claim that the set \( [n] \) has exactly \( 2^n \) subsets. To see why, notice that we can construct the subsets of \( [n] \) in the following way. Either choose, or don’t choose, the element \( 1 \); then either choose, or don’t choose, the element \( 2 \); etc., finally choosing, or not choosing, the element \( n \). Each of the \( n \) choices that you encountered could have been made in either of 2 ways. The totality of choices, therefore, might have been made in \( 2^n \) ways, so that is the number of subsets that are set of \( n \) objects has. Next, suppose we have \( n \) distinct objects, and we want to arrange them in a sequence. In how many ways can we do that? For the first object in our sequence we may choose any one of the \( n \) objects. The second element of the sequence can be any of the remaining \( n - 1 \) possible ways to make the first two decisions. Then there are \( n - 2 \) choices for the third element, and so we have \( n(n - 1)(n - 2) \cdots 2 \cdot 1 = n! \) ways to form the whole sequence. One of the subsets of \( [n] \), how many have exactly \( k \) objects in them? The number of elements in a set is called its cardinality. The cardinality of a set \( S \) is denoted by \( |S| \), so for example, \( |[6]| = 6 \). A set of cardinality \( k \) is called a \( k \)-set, and a subset of cardinality \( k \) is naturally enough, a \( k \)-subset. The question is, for how many subsets of \( [n] \) is it true that \( |S| = k \)? We can construct \( k \)-subsets \( S \) of \( [n] \) (written \( S \subseteq [n] \)) as follows. Choose an element \( a_1 \) (in possible choices). Of the remaining \( n - 1 \) elements, choose one (in \( n - 1 \) possible choices), etc., until a sequence of \( k \) different elements have been chosen. Obviously there were \( n(n - 1)(n - 2) \cdots (n - k + 1) \) ways in which you might have chosen that sequence, so the number of ways to choose an (ordered) sequence of \( k \) elements from \( [n] \) is \[ n(n - 1)(n - 2) \cdots (n - k + 1) = \frac{n!}{(n - k)!} \quad (n \geq 0, 0 \leq k \leq n). \] But there are more sequences of \( k \) elements than there are \( k \)-sets, because any particular \( k \)-subset will correspond to \( k! \) different ordered sequences, namely all possible rearrangements of the elements in the subset. Hence the number of \( k \)-subsets of \( [n] \) is equal to the number of \( k \)-sequences divided by \( k! \). In other words, there are exactly \( \frac{n!}{k!(n - k)!} \) \( k \)-subsets of a set of \( n \) objects. The quantities \( \frac{n!}{k!(n - k)!} \) are the famous binomial coefficients, and they are denoted by \[ \binom{n}{k} = \frac{n!}{k!(n - k)!}, \quad (n \geq 0; 0 \leq k \leq n) \tag{1.5.1} \] Some of their special values are \[ \binom{n}{0} = 1 \quad (\text{for } n \geq 0); \quad \binom{n}{1} = n \quad (\text{for } n \geq 0); \quad \binom{n}{2} = \frac{n(n - 1)}{2} \quad (\text{for } n \geq 2); \quad \binom{n}{n} = 1 \quad (\text{for } n \geq 0). \] It is convenient to define \( \binom{n}{k} \) to be \( 0 \) if \( k < 0 \) or if \( k > n \). We can summarize the developments so far with ``` Image Analysis: ### Comprehensive Examination of the Attached Visual Content #### 1. Localization and Attribution: 1. **Text and Image Position:** - The visual content is a single page divided into several text blocks. - The text is located at the center of the page, structured in paragraphs and numbered sections. #### 2. Object Detection and Classification: 1. **Objects Detected:** - Primary Object: A digitally typed text page - Secondary Objects: Mathematical expressions and formulas within the text #### 3. Scene and Activity Analysis: - **Scene Description:** - The page presents a scholarly text focusing on mathematical concepts, particularly set theory and combinatorics. - **Activities:** - The text describes the process of counting subsets of a given set and explains binomial coefficients. #### 4. Text Analysis: 1. **Extracted Text:** #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 25 Context: #### 4. Text Analysis: 1. **Extracted Text:** ``` 8. Write out a complete proof of theorem 1.4.1. 9. Show by an example that the conclusion of theorem 1.4.1 may be false if the phrase ‘for every fixed ϵ0 > 0...’ were replaced by ‘for every fixed ϵ > 0 ...’. 10. In theorem 1.4.1 we find the phrase: ‘…the positive real root of …’. Prove that this phrase is justified, in that the equation shown always has exactly one positive real root. Exactly what special properties of that equation did you use in your proof? 1.5 Counting For a given positive integer n, consider the set {1,2,...,n}. We will denote this set by the symbol [n], and we want to discuss the number of subsets of various kinds that it has. Here is a list of all of the subsets of {2}, {1}, {2}, {1, 2}. There are 4 of them. We claim that the set [n] has exactly 2^n subsets. To see why, notice that we can construct the subsets of [n] in the following way. Either choose, or don’t choose, the element ‘1’; then either choose, or don’t choose, the element ‘2’; etc., finally choosing, or not choosing, the element ‘n’. Each of the n choices that you encountered could have been made in either 2 of ways. The totality of n choices, therefore, might have been made in 2^n different ways, so that is the number of subsets that a set of n objects has. Next, suppose we have n distinct objects, and we want to arrange them in a sequence. In how many ways can we do that? For the first object in our sequence we may choose any one of the n objects. The second element of the sequence can be any of the remaining n − 1 objects, so there are n(n − 1) possible ways to make these first two decisions. Then there are n − 2 choices for the third element, and so we have n(n − 1)(n − 2) ways to arrange the first three elements of the sequence. It is no doubt clear now that there are exactly n(n − 1)(n − 2)∙∙∙3∙2∙1 = n! ways to form the whole sequence. Of the n subsets of [n], how many have exactly k objects in them? The number of elements in a set is called its cardinality. The cardinality of a set S is denoted by |S|, so, for example, |{6}| = 6. A set whose cardinality is k is called a ‘k-set’, and a subset of cardinality k is, naturally enough, a ‘k-subset’. The question is, for how many subsets S of [n] is it true that |S| = k? We can construct k-subsets S of [n] (written S ⸦ [n] ∖) as follows. Choose an element a_1 (in possible choices). Of the remaining n − 1 elements, choose one (in n − 1 possible choices), etc., until a sequence of k different elements have been chosen. Obviously there were n(n − 1)(n − 2) ⸦⸦⸦ (n − k + 1) ways in which we might have chosen that sequence, so the number of ways to choose an (ordered) sequence of k elements from [n] is n(n − 1)(n − 2) ∙∙∙ (n − k + 1) = n!/(n − k)! But there are more sequences of k elements than there are k-subsets, because any particular k-subset S will correspond to k! different ordered sequences, namely all possible rearrangements of the elements of that subset. Hence the number of k-subsets of [n] is equal to the number of k-sequences divided by k!. In other words, there are exactly n!/[k!(n − k)!] k-subsets of a set of n objects. The quantities n!/[k!(n − k)!] are the famous binomial coefficients, and they are denoted by ( ) = n!/[k!(n − k)!] (n ≥ 0; 0 ≤ k ≤ n). Some of their special values are (n) = 1 ( ∀n ≥ 0); (n) = n ( ∀n ≥ 0); (k) = n(n − 1)/2 ( ∀n ≥ 0); (n) = 1 ( ∀n ≥ 0). ( 0 ) ( 1 ) ( 2 ) ( n ) It is convenient to define (2) to be 0 if k < 0 or if k > n. We can summarize the developments so far with ``` 2. **Content Analysis:** - **Mathematical Tasks:** The text includes exercises (tasks 8, 9, and 10) asking the reader to provide mathematical proofs and to critique theorem revisions. - **Main Text:** The main body of text discusses mathematical counting principles, such as the number of subsets of a set, permutations, and binomial coefficients. - **Significance:** This text is instructional, aiming to teach the concepts of combinatorics, specifically counting subsets and calculating binomial coefficients. #### 6. Product Analysis: - **Products Depicted:** - Not applicable as there are no products shown in the image. #### 8. Color Analysis: - **Color Composition:** - The image primarily consists of black text on a white background. - The monochrome effect aids in readability and focuses the reader's attention on the content without distraction. #### 9. Perspective and Composition: - **Perspective:** - The image is presented from a straight, eye-level perspective, typical for readable documents. - **Composition:** - The content is organized into numbered sections and paragraphs, which helps in structuring the reading flow. - Mathematical formulas are interspersed with explanatory text, making the complex content manageable. #### 10. Contextual Significance: - **Contribution to Overall Message:** - The image, being a page from an academic document or textbook, contributes to teaching mathematical concepts on counting subsets and permutations. - It is part of a larger text aimed at undergraduate or graduate students studying mathematics, particularly combinatorics and set theory. ### Conclusion: The analysis reveals that the visual content is an educational text aimed at explaining combinatorial mathematics. It provides exercises and thorough explanations of concepts such as subsets, permutations, and binomial coefficients. The monochromatic color scheme is typical for academic texts, ensuring clarity and focus on the content. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 25 Context: - **Contribution to Overall Document:** - The content on this page likely contributes to a larger section on counting principles in mathematics, providing foundational knowledge and detailed examples to support learning and understanding of subsets, sequences, and combinations. ### Summary: The image is an excerpt from an academic textbook's combinatorics section, focusing on counting principles involving subsets, sequences, and binomial coefficients. The clean, black-and-white text layout ensures readability and structured presentation of mathematical concepts and examples. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 27 Context: # 1.5 Counting The binomial theorem is the statement that \( \forall n \geq 0 \) we have \[ (1+x)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^{k} \tag{1.5.4} \] **Proof:** By induction on \( n \). Eq. (1.5.4) is clearly true when \( n = 0 \), and if it is true for some \( n \) then multiply both sides of (1.5.4) by \( (1+x) \) to obtain \[ (1+x)^{n+1} = \sum_{k=0}^{n} \binom{n}{k} x^{k} + \sum_{k=0}^{n} \binom{n}{k} x^{k+1} \] \[ = \sum_{k=0}^{n} \binom{n}{k} x^{k} + \sum_{k=1}^{n+1} \binom{n}{k-1} x^{k} \] \[ = \sum_{k=0}^{n} \binom{n}{k} x^{k} + \sum_{k=0}^{n} \binom{n}{k} x^{k+1} \] \[ = \sum_{k=0}^{n} \left( \binom{n}{k} + \binom{n}{k-1} \right) x^{k} = \sum_{k=0}^{n+1} \binom{n+1}{k} x^{k} \] which completes the proof. Now let's ask how big the binomial coefficients are, as an exercise in asymptotics. We will refer to the coefficients in row \( n \) of Pascal's triangle, that is, \[ \begin{pmatrix} n \\ 0 \end{pmatrix},\ \begin{pmatrix} n \\ 1 \end{pmatrix},\ \ldots,\ \begin{pmatrix} n \\ n \end{pmatrix} \] as the coefficients of order \( n \). Then, by (1.5.2) (or by (1.5.4) with \( x = 1 \)), the sum of all the coefficients of order \( n \) is also fairly apparent, from an inspection of Table 1.5.1, that the largest one(s) of the coefficients of order \( n \) is (are) the one(s) in the middle. More precisely, if \( n \) is odd, then the largest coefficients of order \( n \) are \( \binom{n}{(n-1)/2} \) and \( \binom{n}{(n+1)/2} \); whereas if \( n \) is even, the largest one is uniquely \( \binom{n}{n/2} \). It will be important, in some of the applications to algorithms later on in this book, for us to be able to pick out the largest term in a sequence of this kind, so let's see how we could prove that the biggest coefficients are the ones cited above. For \( k \) fixed, we will compute the ratio of the \( (k+1)^{th} \) coefficient of order \( n \) to the \( k^{th} \) coefficient of order \( n \): \[ \text{Ratio} = \frac{\binom{n}{k+1}}{\binom{n}{k}} = \frac{n!/(k+1)!(n-k-1)!}{n!/(k)!(n-k)!} = \frac{(k+1)(n-k)}{(n-k-1)(k+1)} = \frac{n-k}{k+1} \] and is \( > 1 \) if \( k < (n - 1)/2 \), as claimed. OK, the biggest coefficients are in the middle, but how big are they? Let’s suppose that \( n \) is even, just to keep things simple. Then the biggest binomial coefficient of order \( n \) is \[ \binom{n}{n/2} \sim \frac{n!}{(n/2)^{n/2}(n/2)!} \sim \frac{n^{n/2}}{(n/2)^{n/2}\sqrt{2\pi(n/2)}} \sim \frac{2^{n/2}}{\sqrt{2\pi n/2}} \tag{1.5.5} \] 23 Image Analysis: ### 1. **Localization and Attribution:** - There is only one image on the page. ### 2. **Object Detection and Classification:** - The objects in the image are: - Mathematical text and equations. - A rectangular bold box used to signify the end of a proof. ### 3. **Scene and Activity Analysis:** - The scene is of a textbook or academic paper discussing mathematical concepts. - The primary activity is explaining and proving the binomial theorem using algebraic manipulations and asymptotic analysis. ### 4. **Text Analysis:** - **Detected Text:** - The binomial theorem is the statement that ∀ n ≥ 0 we have: (1+x)^n = Σ(k=0 to n) (n choose k) x^k. (1.5.4) - Proof: By induction on n. Eq. (1.5.4) is clearly true when n = 0, and if it is true for some n then multiply both sides of (1.5.4) by (1 + x) to obtain... - The text proceeds to elaborate on the binomial coefficients, sums, largest coefficients in rows, and an asymptotic analysis of coefficients. - Equations and binomial coefficients such as n choose k, n choose k + 1, (n + 1)/2, etc. - Ends with the equation: ∼ sqrt(2/π) 2^n. (1.5.5) - Page number: 23. ### 6. **Product Analysis:** - Not applicable; no products are depicted. ### 8. **Color Analysis:** - The image is in black and white, which is typical for printed academic materials. The dominant color is black text on a white background, facilitating easy readability and focus on the content. ### 9. **Perspective and Composition:** - The perspective is a straight-on view typical of a scanned or photographed page from a book. - The composition is standard for academic texts with logical flow from the theorem, proof, and further explanations broken into paragraphs and sections for clarity. ### 12. **Graph and Trend Analysis:** - The equations and text provide an analytical breakdown of mathematical trends, particularly around binomial coefficients. - It discusses the largest coefficients in rows of Pascal’s triangle, their positions, and asymptotic behaviors. ### 13. **Graph Numbers:** - Specific numeric data points are provided in the equations and analysis, such as n, k, and asymptotic ratios involving n choose k. ### **Ablaufprozesse (Process Flows):** - The process of proving the binomial theorem: - Initial statement of the theorem. - Proof by induction: base case and inductive step. - Deriving properties from Pascal's triangle. - Asymptotic analysis of binomial coefficients. ### **Prozessbeschreibungen (Process Descriptions):** - Detailed steps of the mathematical proof, asymptotic analysis, and properties of binomial coefficients. ### **Trend and Interpretation:** - The primary trend highlighted is the position and size of the largest binomial coefficients in any given row of Pascal’s triangle and their asymptotic growth. - Interpretation involves understanding the central binomial coefficient and its relation to n. Overall, the image provides a clear, academic explanation of the binomial theorem and its properties, with a particular focus on coefficients and asymptotics. The page serves as a typical example of mathematical exposition. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 28 Context: # Chapter 1: Mathematical Preliminaries where we have used Stirling's formula (1.1.10). Equation (1.5) shows that the single biggest binomial coefficient accounts for a very healthy fraction of the sum of all of the coefficients of order n. Indeed, the sum of all of them is \(2^n\), and the biggest one is \(\sim \sqrt{\frac{2}{\pi n}}\). When n is large, therefore, the largest coefficient contributes a fraction \(\sim \sqrt{\frac{1}{n}}\) of the total. If we think in terms of the subsets that these coefficients count, what we will see is that a large fraction of all of the subsets of an n-set have cardinality \(\frac{n}{2}\); in fact \(\binom{n}{n/2}\) of them do. This kind of probabilistic thinking can be very useful in the design and analysis of algorithms. If we are designing an algorithm that deals with subsets of \([n]\), for instance, we should recognize that a large percentage of the customers for that algorithm will have cardinalities near \(n/2\), and make every effort to see that the algorithm is fast for such subsets, even at the expense of possibly slowing it down on subsets whose cardinalities are very small or very large. ## Exercises for section 1.5 1. How many subsets of even cardinality does \([n]\) have? 2. By observing that \((1 + x)^n(1 + x^2) = (1 + x)^{n+1}\), prove that the sum of the squares of all binomial coefficients of order n is \(2^n\). 3. Evaluate the following sums in simple form. - (i) \(\sum_{k=0}^{n} \binom{n}{k}\) - (ii) \(\sum_{k=0}^{n} \binom{n}{k}^2\) - (iii) \(\sum_{k=0}^{n} \binom{n}{k} k\) 4. Find, by direct application of Taylor’s theorem, the power series expansion of \(f(x) = \frac{1}{1 - x}\) about the origin. Express the coefficients as certain binomial coefficients. 5. Complete the following twiddles. - (i) \(\binom{n}{r} \sim ?\) - (ii) \(\left( \binom{n}{r} \right)^{1/n} \sim ?\) - (iii) \(\left( \binom{n}{r} \right)^{1/r} \sim ?\) - (iv) \(\left( \binom{n}{r} \right)^{r/n} \sim ?\) 6. How many ordered pairs of unequal elements of \([n]\) are there? 7. Which one of the numbers \(\{2^{\binom{n}{2}}\}_{n \in \mathbb{N}}\) is the biggest? ## 1.6 Graphs A graph is a collection of vertices, certain unordered pairs of which are called its edges. To describe a particular graph we first say what its vertices are, and then we say how pairs of vertices are its edges. The set of vertices of a graph \(G\) is denoted by \(V(G)\), and its set of edges is \(E(G)\). If \(u\) and \(v\) are vertices of a graph \(G\), and if \((u,v)\) is an edge of \(G\), then we say that vertices \(u\) and \(v\) are adjacent in \(G\). Consider the graph \(G\) whose vertex set is \(\{1, 2, 3, 4, 5\}\) and whose edges are the set of pairs \((1,2)\), \((2,3)\), \((4,5)\). This is a graph of 5 vertices and 5 edges. A nice way to present a graph to an audience is to draw a picture of it, instead of just listing the pairs of vertices that are its edges. To draw a picture of a graph we would first make a point for each vertex, and then we would draw arcs between two vertices \(u\) and \(v\) if and only if \((u,v)\) is an edge of the graph we are talking about. The graph \(G\) of 5 vertices and 5 edges that is listed above can be drawn as shown in Fig. 1.6.1(a). It could also be drawn as shown in Fig. 1.6.1(b). They're both the same graph. Only the pictures are different, but the pictures aren't "really" the graph; the graph is the vertex list and the edge list. The pictures are helpful to us in visualizing and remembering the graph, but that's all. The number of edges that contain (are incident with) a particular vertex \(v\) of a graph \(G\) is called the degree of that vertex, and is usually denoted by \(p(v)\). If we add up the degrees of every vertex of \(G\) we will have counted exactly two contributions from each edge of \(G\), one at each of its endpoints. Hence, for every Image Analysis: ### Comprehensive Examination of the Visual Content **Localization and Attribution:** - Single image on the page. - The image includes different sections such as text, mathematical formulas, and diagrams. - Assigned number: **Image 1**. **Text Analysis:** - **Image 1** contains extensive textual content divided into multiple sections: - **Title:** Chapter 1: Mathematical Preliminaries. - **Main Text:** Discusses Stirling's formula, binomial coefficients, subsets of n-set, probability thinking in algorithms, etc. - **Exercises for section 1.5:** Consists of seven questions that involve mathematical proofs, sums, and applications of Taylor’s theorem. - **Section 1.6 Graphs:** Describes the components and properties of graphs with an example graph G. **Object Detection and Classification:** - **Text blocks:** Several paragraphs of text, exercises, and explanations. - **Mathematical Equations and Formulas:** Various equations related to binomial coefficients, sums, Taylor's theorem, and twiddles. - **Graph Diagram:** Depicts a graph G with vertices {1, 2, 3, 4, 5} and edges {(1,2), (2,3), (3,4), (4,5), (1,5)} in Figure 1.6.1(a). **Scene and Activity Analysis:** - **Entire scene:** Academic content focusing on mathematical preliminaries, specifically: - Description of mathematical concepts. - Problem-solving exercises. - Explanation of graph theory along with a visual example. **Diagram and Chart Analysis:** - **Graph Diagram (Figure 1.6.1(a)):** - **Vertices:** {1, 2, 3, 4, 5} - **Edges:** {(1,2), (2,3), (3,4), (4,5), (1,5)} - **Explanation:** Different ways to present a graph, the figure helps in visualizing and remembering the graph structure. **Color Analysis:** - The content is in black and white, typical for textbook pages. - Dominant color: Black text on a white background. **Perspective and Composition:** - **Perspective:** Eye-level view, standard for reading academic content. - **Composition:** Well-structured layout with headings, sections, exercises, and diagrams organized logically. **Contextual Significance:** - **Overall context:** Academic document or textbook on Mathematical Preliminaries. - **Contribution:** The image breaks down complex mathematical concepts, provides exercises for practice, and visual aids (graph) for better understanding. **Ablaufprozesse (Process Flows):** - **Problem-solving process in exercises:** How to derive subsets, evaluate sums, apply formulas, and understand graph theory step-by-step. **Prozessbeschreibungen (Process Descriptions):** - **Mathematical explanations:** Show the processes involved in mathematical problem-solving. - **Graph creation and interpretation:** Process of drawing and understanding a graph’s components. **Typen Bezeichnung (Type Designations):** - **Types of mathematical problems:** Binomial coefficients sums, Taylor series, twiddles, and graph properties analysis. **Trend and Interpretation:** - **Trend:** Progressive learning and detailed comprehension of mathematical concepts and graph theory. - **Interpretation:** Structured to build foundational knowledge in mathematics through description, problem-solving, and visual examples. **Tables:** - No tables are present in this image. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 33 Context: ``` ## 1.6 Graphs 13. Let \( n \) be a multiple of 3. Consider a labeled graph \( G \) that consists of \( n/3 \) connected components, each of them a \( K_3 \). How many maximal independent sets does \( G \) have? 14. Describe the complement of the graph \( G \) in exercise 13 above. How many cliques does it have? 15. In how many labeled graphs of \( n \) vertices is the subgraph that is induced by vertices \( \{1, 2, 3\} \) a triangle? 16. Let \( H \) be a labeled graph of \( L \) vertices. In how many labeled graphs of \( n \) vertices is the subgraph that is induced by vertices \( \{1, \ldots, L\} \) equal to \( H \)? 17. Devise an algorithm that will decide if a given graph, of \( n \) vertices and \( m \) edges, does or does not contain a triangle, in time \( O(m \max(n^{2}, m)) \). 18. Prove that the number of labeled graphs of \( n \) vertices all of whose vertices have even degree is equal to the number of all labeled graphs of \( n - 1 \) vertices. ``` #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 35 Context: ```markdown ## 2.2 Quicksort **Procedure:** ```plaintext procedure calculate(list of variables); if (trivialcase) then do (triviallything); else do {call calculate(smaller values of the variables)}; {maybe do a few more things} end. ``` In this chapter, we’re going to work out a number of examples of recursive algorithms, of varying sophistication. We will see how the recursive structure helps us to analyze the running time, or complexity, of the algorithms. We will also find that there is a bit of art involved in choosing the list of variables that a recursive procedure operates on. Sometimes the first list we think of doesn’t work because the recursive call seems to need more detailed information than we have provided for it. So we try a larger list, and then perhaps it works, or maybe we need a still larger list ..., but more of this later. ### Exercises for Section 2.1 1. Write a recursive routine that will find the digits of a given integer n in the base b. There should be no visible loops in your program. ## 2.2 Quicksort Suppose that we are given an array `x[1], ..., x[n]` of n numbers. We would like to rearrange these numbers as necessary so that they end up in nondecreasing order of size. This operation is called **sorting** the numbers. For instance, if we are given `{9, 4, 7, 2, 1}`, then we want our program to output the sorted array `{1, 2, 4, 7, 9}`. There are many methods of sorting, but we are going to concentrate on methods that rely on only two kinds of basic operations, called **comparisons** and **interchanges**. This means that our sorting routine is allowed to: - (a) pick up two numbers (‘keys’) from the array, compare them, and decide which is larger. - (b) interchange the positions of two selected keys. Here is an example of a rather primitive sorting algorithm: 1. (i) find, by successive comparisons, the smallest key 2. (ii) interchange it with the first key 3. (iii) find the second smallest key 4. (iv) interchange it with the second key, etc. Here is a more formal algorithm that does the job above: ```plaintext procedure slowsort(X: array[1..n]); {sorts a given array into nondecreasing order} for i = 1 to n - 1 do for j = i + 1 to n do if X[i] > X[j] then swap(X[i], X[j]) end. slowsort ``` If you were wondering why we called this method ‘primitive’, ‘slowsort’, and other pejorative names, the reason will be clearer after we look at its complexity. What is the cost of sorting n numbers by this method? We will look at two ways to measure that cost. First let’s choose our unit of cost to be one comparison of two numbers, and then we will choose a different unit of cost, namely one interchange of position (‘swap’) of two numbers. ``` #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 40 Context: # Chapter 2: Recursive Algorithms is done first of all over all n! of the possible input orderings of the array elements, and second, for each such input ordering, we average also over all sequences of choices of the splitting elements. Now let's consider the behavior of the function \( F(n) \). What we are going to show is that \( F(n) = O(n \log n) \). The label that \( F(n) \) estimates has two components. First there are the pairwise comparisons involved in choosing a splitting element and rearranging the array about the chosen splitting value. Second there are the comparisons that are done in the two recursive calls that follow the creation of a splitter. As we have seen, the number of comparisons involved in splitting the array is \( n - 1 \). Hence it remains to estimate the number of comparisons in the recursive calls. For this purpose, suppose we have rearranged the array about the splitting element, and that it has turned out that the splitting entry now occupies the \( i \)th position in the array. Our next remark is that each value of \( i = 1, 2, \ldots, n \) is equally likely to occur. The reason for this is that we chose the splitter originally by choosing a random array entry. Since all orderings of the array entries are equally likely, the one that we happened to have chosen was just as likely to have been the largest entry as to have been the smallest, or the 17th-from-largest, or whatever. Since each value of \( i \) is equally likely, each \( i \) has probability \( \frac{1}{n} \) of being chosen as the residence of the splitter. If the splitting element lives in the \( i \)th array position, the two recursive calls to Quicksort will be on two subarrays, one of which has length \( i - 1 \) and the other of which has length \( n - i \). The average numbers of pairwise comparisons that are involved in such recursive calls are \( F(i - 1) \) and \( F(n - i) \), respectively. It follows that our average complexity function \( F \) satisfies the relation: \[ F(n) = n - 1 + \frac{1}{n} \sum_{i=1}^{n} \left[ F(i - 1) + F(n - i) \right] \quad (n \geq 2) \tag{2.2.2} \] together with the initial value \( F(0) = 0 \). How can we find the solution of the recurrence relation (2.2.2)? First let's simplify it a little by noticing that \[ \sum_{i=1}^{n} F(n - i) = F(n - 1) + F(n - 2) + \ldots + F(0) \tag{2.2.3} \] and so (2.2.2) can be written as: \[ F(n) = n - 1 + \frac{2}{n} \sum_{i=1}^{n} F(i - 1) \tag{2.4} \] We can simplify (2.4) a lot by getting rid of the summation sign. This next step may seem like a trick at first (and it is!), but it's a trick that is used in so many different ways that now we call it a 'method.' What we do is first to multiply (2.4) by \( n \), to get: \[ n F(n) = n(n - 1) + 2 \sum_{i=1}^{n} F(i - 1) \tag{2.5} \] Next, in (2.5), we replace \( n \) by \( n - 1 \), yielding: \[ (n - 1)F(n - 1) = (n - 1)(n - 2) + 2 \sum_{i=1}^{n - 1} F(i - 1) \tag{2.6} \] Finally, we subtract (2.6) from (2.5), and the summation sign obliquely disappears, leaving behind just: \[ n F(n) - (n - 1)F(n - 1) = n(n - 1) - (n - 1)(n - 2) + 2F(n - 1) \tag{2.7} \] Image Analysis: ### Analysis of the Attached Visual Content: --- #### 1. Localization and Attribution: - The image contains a single page, analyzed as **Image 1**. --- #### 2. Object Detection and Classification: - **Image 1**: - Objects detected: - Text blocks. - Mathematical equations/forms. - Page number indicator. - Categories: - The text paragraphs are categorized as educational content on computer science/mathematics. - The equations are identified as mathematical formulas. --- #### 3. Scene and Activity Analysis: - **Image 1**: - Scene Description: This page appears to be from a textbook or academic paper discussing recursive algorithms. - Activities: It shows an explanation of a recursive function \(F(n)\), involving mathematical derivations and steps to solve a recurrence relation. --- #### 4. Text Analysis: - **Image 1**: - Extracted Text: - Title: "Chapter 2: Recursive Algorithms" - Main content: Detailed explanation of a computational complexity function \(F(n)\), which is discussed through several steps and equations to show its behavior. - Specifically, steps discuss the estimation of recursive steps, the average complexity function, and simplification of summation. - Significance in Context: - The text is critical for understanding recursion in algorithms, particularly in analyzing the time complexity of recursive procedures. --- #### 5. Diagram and Chart Analysis: - Not applicable as there are no diagrams or charts present. --- #### 6. Product Analysis: - Not applicable as there are no products depicted. --- #### 7. Anomaly Detection: - No unusual elements or anomalies detected. --- #### 8. Color Analysis: - **Image 1**: - Dominant Colors: The page predominantly uses black text on a white background. This contrast facilitates readability and is typical in academic documents. --- #### 9. Perspective and Composition: - **Image 1**: - Perspective: The image is taken from a straight-on (eye-level) perspective, common for document scanning. - Composition: The page layout is structured in a standard academic format, with the title at the top, followed by coherent paragraphs and mathematical derivations down the page. --- #### 10. Contextual Significance: - **Image 1**: - The image functions as a page in an academic resource aimed at explaining recursive algorithms, which is essential for computer science students and professionals working with algorithm analysis. --- #### 11. Metadata Analysis: - Metadata not provided or visible. --- #### 12. Graph and Trend Analysis: - Not applicable as there are no graphs presented. --- #### 13. Graph Numbers: - Not applicable as there are no numerical graphs provided. --- #### Additional Aspects: - **Ablaufprozesse (Process Flows):** - **Image 1** details the process flow of estimating and simplifying the complexity of a recursive function \(F(n)\). - **Prozessbeschreibungen (Process Descriptions):** - **Image 1** provides a detailed mathematical process of breaking down and simplifying the recurrence relation to understand the behavior of \(F(n)\). - **Typen Bezeichnung (Type Designations):** - **Image 1** categorizes the function \(F(n)\) as a measure of average computational complexity. - **Trend and Interpretation:** - Analyzing the recurrence \(F(n)\), it reveals how simplification techniques can reduce computational complexity analysis. - **Tables:** - Not applicable as there are no tables present. --- This comprehensive analysis captures the essential aspects of the attached visual content, maintaining focus on relevant attributes while ignoring immaterial details. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 41 Context: # 2.2 Quicksort After some tidying up, (2.2.7) becomes \[ F(n) = \left( 1 + \frac{1}{n} \right) F(n - 1) + \left( 2 - \frac{2}{n} \right), \tag{2.2.8} \] which is exactly in the form of the general first-order recurrence relation that we discussed in section 1.4. In section 1.4 we saw that to solve (2.2.8) the winning tactic is to change to a new variable, that is defined, in this case, by \[ F(n) = \frac{n + 1}{n - 1} - \frac{2}{n + 1} y_n. \tag{2.2.9} \] If we make the change of variable \( F(n) = (1 + y_n) \), in (2.2.8), then it takes the form \[ y_n = y_{n - 1} + 2\left( n - 1 \right)/n\left( n + 1 \right) \quad (n \geq 1) \tag{2.2.10} \] as an equation for the \( y_n \)s (\( y_0 = 0 \)). The solution of (2.2.10) is obviously \[ y_n = 2 \sum_{j=1}^{n} \frac{j - 1}{j(y_j + 1)} = 2 \sum_{j=1}^{n} \frac{2}{j + 1} - \frac{1}{j} = 2 \sum_{j=1}^{n} \frac{1}{j} - 4n/(n + 1). \] Hence from (2.2.9), \[ F(n) = 2(n + 1) \sum_{j=1}^{n} \frac{1}{j} - 4n \tag{2.2.11} \] is the average number of pairwise comparisons that we do if we Quicksort an array of length \( n \). Evidently \( F(n) \sim 2n \log(n - \infty) \) (see (1.1) with \( g(n) = 1/n \)), and we have proved ## Theorem 2.2.2. The average number of pairwise comparisons of array entries that Quicksort makes when it sorts arrays of \( n \) elements is exactly as shown in (2.2.11), and is \( \sim 2n \log(n - \infty) \). Quicksort is, on average, a very quick sorting method, even though its worst case requires a quadratic amount of labor. --- ## Exercises for section 2.2 1. Write out an array of 10 numbers that contains no splitter. Write out an array of 10 numbers that contains 10 splitters. 2. Write a program that does the following. Given a positive integer \( n \). Choose 100 random permutations of \([1, 2, \ldots, n!]\) and count how many of the 100 had at least one splitter. Execute your program for \( n = 5, 6, \ldots, 12 \) and tabulate the results. 3. Think of some method of sorting \( n \) numbers that isn't in the text. In the worst case, how many comparisons might your method do? How many swaps? * For a fast and easy way to do this see A. Nijhuis and H. S. Wilf, *Combinatorial Algorithms*, 2nd ed. (New York: Academic Press, 1978), chap. 6. Image Analysis: ### Analysis of the Visual Content #### Image 1 ##### Text Analysis: 1. **Mathematical Content and Equations**: - The image contains various mathematical equations and explanations related to "Quicksort" in section 2.2. Noteworthy equations include: - Equation (2.2.7): \( F(n) = \left(1 + \frac{2}{n}\right)F(n-1) + \left(2 - \frac{2}{n}\right) \) - Equation (2.2.9): \( F(n) = \frac{n+1}{n} + \frac{n-1}{n-1} - \frac{2}{n} \cdot \frac{1}{n_{h}} \) - Equation (2.2.10) provides \( y_{n} \) calculation. - Final Equation (2.2.11): \( F(n) = 2(n+1) \left(\sum_{j=1}^{n}\frac{1}{j}\right) - 4n \) 2. **Theorem and Explanation**: - Theorem 2.2.2: It states the average number of pairwise comparisons of array entries that Quicksort makes is exactly as shown in Equation 2.2.11 and is approximately \( 2n \log_{e} (n) \) as \( n \) approaches infinity. 3. **Exercises**: - This section provides three exercises: 1. Write out arrays with specific conditions regarding splitters. 2. Write a program to determine the number of splitters in randomly generated arrays. 3. Invent a sorting method and calculate the number of swaps in the worst case. 4. **Reference Note**: - A footnote suggests referring to "A. Nijenhuis and H. S. Wilf, Combinatorial Algorithms, 2nd ed." ##### Diagram and Chart Analysis: 1. **Absence of Diagrams/Charts**: - The content does not include any diagrams or charts. Equations and text are the primary focus. #### Scene and Activity Analysis: 1. **Educational Context**: - The scene is a textbook or academic document explaining the Quicksort algorithm and its analysis. The main activity includes walking through the mathematical foundations, theorems, and practical programming exercises. ##### Perspective and Composition: 1. **Textbook Layout**: - The perspective is a straightforward, eye-level view typical of textbook pages. - The composition is standard, with headings, equations, theoretical explanations, exercises, and additional reference notes. #### Conclusion: - This image comprises an analytical deep dive into the Quicksort algorithm within a textbook context. The explanations are thorough, emphasizing the mathematical derivation and providing exercises for practical understanding. The lack of visual elements like diagrams suggests a focus on rigorous mathematical content. The reference provides additional reading to support the topic discussed. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 42 Context: # Chapter 2: Recursive Algorithms ## 4. Consider the array ``` x = {2, 4, 1, 10, 5, 3, 9, 7, 8, 6} ``` with `left = 1` and `right = 10`. Suppose that the procedure `split` is called, and suppose the random integer `L` in `split` happens to be 5. Carry out the complete split algorithm (not on a computer; use pencil and paper). Particularly, record the condition of the array `x` after each value of `j` is processed in the for `j ...` loop. 5. Suppose `H(0) = 1` and `H(n) ≤ 1 + Σ_{i=1}^{n-1} H(i - 1)` for `n ≥ 1`. How might `H(n)` be? 6. If `Q(0) = 0` and `Q(n) ≤ n^2 + Σ_{i=1}^{Q(i - 1)}` for `n ≥ 1`, how big might `Q(n)` be? 7. (Research problem) Find the asymptotic behavior, for large `n`, of the probability that a randomly chosen permutation of `n` letters has a splitter. ## 2.3 Recursive Graph Algorithms Algorithms on graphs are another rich area of applications of recursive thinking. Some of the problems are quite different from the ones that we have so far been studying in that they seem to need exponential amounts of computing time, rather than the polynomial times that were required for sorting problems. We will illustrate the dramatically increased complexity with a recursive algorithm for the ‘maximum independent set problem’, one which has received a great deal of attention in recent years. Suppose a graph `G` is given. An **independent set** of vertices of `G` means a set of vertices no two of which are connected by an edge of `G`. In the graph in Fig. 2.3.1 the set `{1, 2, 6}` is an independent set and so is the set `{1, 3}`. The largest independent set in the graph shown there is the set `{1, 2, 3, 6}`. The problem of finding the size of the largest independent set in a given graph is computationally very difficult. All algorithms known to date require exponential amounts of time, in their worst cases, although no one has proved the nonexistence of fast (polynomial time) algorithms. If the problem itself seems unusual, and maybe not deserving of a lot of attention, be advised that in Chapter 5 we will see that it is a member in good standing of a large family of very important computational problems (the ‘NP-complete’ problems) that are tightly bound together, in that if we can figure out better ways to compute any one of them, then we will be able to do all of them faster. Here is an algorithm for the independent set problem that is easy to understand and to program, although, of course, it may take a long time to run on a large graph `G`. We are looking for the size of the largest independent set of vertices of `G`. Suppose we denote that number by `maxSet(G)`. Fix some vertex of the graph, say vertex `v`. Let’s distinguish two kinds of independent sets of vertices of `G`. There are those that contain vertex `v` and those that don’t contain vertex `v`. If an independent set `S` of vertices contains vertex `v`, then what does the rest of the set `S` consist of? The remaining vertices of `S` are an independent set in a smaller graph, namely the graph that is obtained from `G` by deleting vertex `v` as well as all vertices that are connected to vertex `v` by an edge. This latter set of vertices is called the **neighborhood** of vertex `v`, and is written `Nbd(v)`. The set `S` consists, therefore, of vertex `v` together with an independent set of vertices from the graph `G - {v} - Nbd(v)`. Now consider an independent set `S` that doesn’t contain vertex `v`. In that case the set `S` is simply an independent set in the smaller graph `G - {v}`. Image Analysis: ### Comprehensive Examination of the Visual Content #### 1. **Localization and Attribution** - **Image 1** is located at the bottom center of the page and is labeled as "Fig. 2.3.1". #### 2. **Object Detection and Classification** - **Image 1**: The diagram depicts a graph with vertices and edges. - **Objects Detected:** - Vertices: Represented by the circles in the diagram. - Edges: Represented by the lines connecting the vertices. - **Classification:** - Graph Theory Elements (Vertices and Edges). #### 3. **Scene and Activity Analysis** - **Image 1**: The scene involves a graph, which is part of an explanation regarding recursive graph algorithms focusing on the 'maximum independent set problem'. - **Main Actors:** - Vertices labeled as 1, 2, and 3. - Edges connecting these vertices. #### 4. **Text Analysis** - **Image 1**: The text related to the image is a part of the main content describing recursive algorithms, specifically about the maximum independent set problem. - **Extracted Text**: "Fig. 2.3.1" - **Contextual Significance**: Fig. 2.3.1 is used to illustrate an example within the section discussing an independent set of vertices in a graph. #### 5. **Diagram and Chart Analysis** - **Image 1**: The diagram shows a simple graph with three vertices connected by edges. The vertices are part of an independent set example. - **Axes, Scales, and Legends**: Not applicable. - **Key Insights**: - Demonstrates the concept of an independent set in graph theory. - Vertices {1, 2, 6} form an independent set as no two are connected by an edge. #### 10. **Contextual Significance** - **Image 1**: Fig. 2.3.1 is used to illustrate the discussion on recursive graph algorithms and the independent set problem. - **Contribution to Overall Message**: The image helps elucidate the practical application of recursive algorithms in solving complex problems in graph theory. #### 13. **Graph Numbers** - **Image 1**: Vertices are labeled as 1, 2, and 3. #### **Additional Aspect:** ##### Procesbeschreibungen (Process Descriptions): - **Image 1**: Describes the process of finding an independent set in a graph. - **Process**: 1. Identify the independent set of vertices. 2. Distinguish between vertices connected to a specified vertex and those not connected. 3. Formulate a smaller graph by removing a specified vertex and its connections. 4. Identify if the given set remains an independent set within the smaller graph. ##### Typen Bezeichnung (Type Designations): - **Image 1**: Type designation involves the 'independent set' in the context of graph theory. ### Summary The document highlights recursive algorithms with an emphasis on recursive graph algorithms. A specific problem, the maximum independent set problem, is illustrated with a diagram (Fig 2.3.1) showing a simple graph. The text explains the recursive approach to find the largest independent set, aiding the understanding of complex computation versus polynomial time approaches. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 45 Context: # 2.3 Recursive Graph Algorithms An algorithm that one might think of for this problem, which is to examine every single subset of the vertices of \( G \) and ask if it is an independent set or not. That algorithm would take \( O(2^n) \) time since there are \( 2^n \) subsets of vertices to look at. Hence we have traded in a \( 2^n \) for a \( 1.619^n \) by being a little bit cagey about the algorithm. Can we do still better? There have not been a number of improvements of the basic maxset algorithm worked out. Of these, the most successful is perhaps the one of Tarjan and Trojanowski that is cited in the bibliography at the end of this chapter. We are not going to work out all of those ideas here, but instead we will show what kind of improvements on the basic idea will help us to do better in the time estimate. So, as our next thought, we might replace the instruction "choose some vertex \( v \) of \( G \)" in any instruction "choose some vertex \( v \) of \( G \) that has at least two neighbors." Then we could be quite certain that \( G - \{v\} - Nbd(v) \) would have at most \( n - 3 \) vertices. What if there isn't any such vertex in the graph \( G \)? Then \( G \) could contain only vertices with 0 or 1 neighbors. Such a graph \( G \) would be a collection of \( E \) disjoint edges together with a number of isolated vertices. The size of the largest independent set of vertices in such a graph is easy to find. An maximum independent set contains one vertex from each of the \( E \) edges and it contains all of the isolated vertices. Hence in this case, maxset \( = E + m = |V(G)| - |E(G)| \), and we obtain a second try at a good algorithm in the following form: ``` procedure marsel2(G); returns the size of the largest independent set of vertices of \( G \) if \( G \) has no vertex of degree \( > 2 \) then marsel2 := \( |V(G)| - |E(G)| \) else choose a vertex \( v \) of degree \( \geq 2 \); \( n_1 := marsel2(G - \{v\}); \) \( n_2 := marsel2(G - \{v\} - Nbd(v)); \) maxset2 := max(n_1, n_2) end(marsel2) ``` How much have we improved the complexity estimate? If we apply to marsel2 the reasoning that led to (2.3.2) we find: \[ f(n) \leq cm^2 + f(n - 1) + f(n - 3) \] where \( f(n) \) is once more the worst-case time bound for graphs of \( n \) vertices. Just as before, (2.3.3) is a recurrent inequality of the form that was studied at the end of section 1.4, in theorem 1.4.1. Using the conclusion of that theorem, we find from (2.3.3) that \( f(n) = O(c^n) \) where \( c = 1.46557 \). It is the positive root of the equation \( c^2 = c^2 + 1 \). The next result of our effort to improve marsel2 has been to reduce the running-time bound from \( O(1.619^n) \) to \( O(1.477^n) \), which isn’t a bad day’s work. In the exercises below we will develop marsel3, whose runtime will be \( O(1.391^n) \). The idea will be that since in marsel2 we were able to ensure that \( v \) had at least two neighbors, why not try to insure that \( v \) has at least 3 neighbors! As long as we have been able to reduce the time bound more and more by ensuring that the selected vertex has lots of neighbors, why don’t we keep it up, and insist that \( v \) should have 4 or more neighbors. Regrettably the intended route ran out of steam precisely at that moment. So why ask what the "trivial case" would look like. We would be working on a graph \( G \) in which vertices are more than 3 neighbors. Well, what "trivializing" shall we do in this "trivial case"? The fact is that there isn’t any way of finding the maximum independent set in a graph where all vertices have \( \leq 3 \) neighbors that’s any better than the general methods that we’ve already discussed. In fact, if one could find a fast method for that restricted problem it would have extremely important consequences because we would then be able to do all graphs rapidly, not just those special ones. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 46 Context: # Chapter 2: Recursive Algorithms We will learn more about this phenomenon in Chapter 5, but for the moment let's leave just the observation that the general problem of **master** turns out to be no harder than the special case of **master** in which no vertex has more than 3 neighbors. Aside from the complexity issue, the algorithm **master** has shown how recursive ideas can be used to transform questions about graphs to questions about smaller graphs. Here's another example of such a situation. Suppose \( G \) is a graph, and that we have a certain supply of colors available. To be exact, suppose we have \( k \) colors. We can then attempt to color the vertices of \( G \) properly in \( k \) colors (see section 1.6). If we don't have enough colors, and \( G \) has lots of edges, this will not be possible. For example, suppose \( G \) is the graph of Fig. 2.3.4, and suppose we have just 3 colors available. Then there is no way to color the vertices without ever finding that both endpoints of some edge have the same color. On the other hand, if we have four colors available then we can do the job. \begin{figure}[h] \centering \includegraphics[width=0.5\textwidth]{path_to_image} \caption{Fig. 2.3.4} \end{figure} There are many interesting computational and theoretical problems in the area of coloring of graphs. Just for its general interest, we are going to mention the four-color theorem, and then we will turn to a study of some of the computational aspects of graph coloring. First, just for general cultural reasons, let's slow down for a while and discuss the relationship between graph colorings in general and the four-color problem, even though it isn't directly relevant to what we’re doing. The original question was this. Suppose that a delegation of Earthlings were to visit a distant planet and find there a society of human beings. Since that race is well known for its squabbling habits, you can be sure that the planet will have been carved up into millions of little countries, each with its own ruling class, system of government, etc., and of course, all at war with each other. The delegation wants to escape quickly, just before doing so it draws a careful map of the 5,000,000 countries into which the planet has been divided. To make the map easier to read, the countries are then colored in such a way that whenever two countries share a stretch of border they are of two different colors. Surprisingly, it was found that the coloring could be done using only red, blue, yellow, and green. It was noticed over 100 years ago that no matter how complicated a map is drawn, and no matter how many countries are involved, it seems to be possible to color the countries in such a way that: - (a) every pair of countries that have a common stretch of border have different colors and - (b) no more than four colors are used in the entire map. It was then conjectured that four colors are always sufficient for the proper coloring of the countries of any map at all. Settling this conjecture turned out to be a very hard problem. It was finally solved in 1976 by K. Appel and W. Haken* by means of an extraordinary proof with two main ingredients. First they showed how to reduce the general problem to only a finite number of cases, by a mathematical argument. Then, since the 'finite' number was quite large, they settled all of those cases with a lengthy computer calculation. So now we have the **Four Color Theorem**, which asserts that no matter how we carve up the plane or the sphere into countries, we will always be able to color those countries with not more than four colors so that countries with a common frontier are colored differently. We can change the map coloring problem into a graph coloring problem as follows. Given a map, from the map we will construct a graph \( G \). There will be a vertex of \( G \) corresponding to each country on the map. Two of these vertices will be connected by an edge of the graph \( G \) if the two countries that two countries have just a single point in common are allowed to have the same color. As an illustration: * Every planar map is four colorable, **Bull. Amer. Math. Soc.**, 82 (1976), 711-712. 42 Image Analysis: ### Analysis of Attached Visual Content #### Image Information and Localization 1. **Localization and Attribution:** - **Text-Based Paragraphs:** - The main body of the page is comprised of text arranged in paragraphs. - There are three main sections: an introduction discussing the algorithm "marset," an illustration labeled as Fig. 2.3.4, and a text block discussing the "four-color problem." - **Diagram:** - The diagram is centrally located and labeled Fig. 2.3.4. 2. **Object Detection and Classification:** - **Image 1:** Diagram - The content of the drawing depicts a simplified graph with nodes and edges. - The key features include several nodes connected by lines representing edges. 3. **Scene and Activity Analysis:** - **Image 1:** - Scene: This is a mathematical illustration showing an example graph with nodes and edges. - Activity: The graph is likely part of an explanatory section in a textbook about graph coloring (four-color theorem). 4. **Text Analysis:** - **General Text:** - This page discusses recursive algorithms, specifically focusing on the complexity of problems, introducing the algorithm "marset," and illustrating a concept by reference to Fig. 2.3.4. - This example illustrates that with a certain number of available colors, a graph's vertices can be colored appropriately without two adjacent vertices sharing the same color. - The text mentions the "four-color problem," explaining it theoretically and mentioning its historical context and significance. - **Text in Diagram (Fig. 2.3.4):** - The diagram is labeled Fig. 2.3.4 and shows the structure of a simple graph to explain a theoretical point about graph coloring. - The nodes are represented as vertices v, and the figure demonstrates an alternative view of a map coloring problem as a graph coloring problem. 8. **Color Analysis:** - **Image 1:** - The diagram does not utilize colors but rather grayscale for nodes and edges, focusing purely on the schematic depiction of the graph. 10. **Contextual Significance:** - **Overall Document:** - The provided image is from a chapter on recursive algorithms in a textbook. The focal topic of this segment is about applying recursive algorithms to the graph coloring problem (four-color problem) and how the problem can be simplified and addressed using theoretical computational approaches. - **Contributions:** - The diagram and associated explanation contribute to a deeper understanding of the four-color problem, demonstrating its complexity and providing historical context to underline its significance in computational theory. Summary: - The page is a part of a textbook on recursive algorithms. - It discusses the four-color problem utilizing an illustrative graph (Fig. 2.3.4) to conceptualize vertex coloring. - The four-color problem is explained, with emphasis on its historical computation and solutions that transformed map-coloring problems into graph-theoretical problems. - There is a detailed introduction of the problem, theoretical aspects, historical significance, solution approaches, and illustrative examples. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 47 Context: ### Conclusion: The visual content includes detailed diagrams and descriptive text related to graph theory, specifically planar graphs. The figures provide visual explanations, while the text offers theoretical context and practical insights into graph planarity and recursive algorithms. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 49 Context: 2.3 Recursive graph algorithms ============================== or if we solve for \( P(K; G) \), then we have \[ P(K; G) = P(K; G - \{e\}) - P(K; G / \{e\}) \tag{2.3.4} \] The quantity \( P(K; G) \), the number of ways of properly coloring the vertices of a graph \( G \) in \( K \) colors, is called the chromatic polynomial of \( G \). We claim that it is, in fact, a polynomial in \( K \) of degree \( |V(G)| \). For instance, if \( G \) is the complete graph on \( n \) vertices then obviously \( P(K; G) = K(K - 1) \cdots (K - n + 1) \), and that is indeed a polynomial in \( K \) of degree \( n \). **Proof of claim:** The claim is certainly true if \( G \) has just one vertex. Next suppose the assertion is true for graphs of \( < V \) vertices, then we claim it is true for graphs of \( V \) vertices also. This is surely true if \( G \) has \( V \) vertices and no edges at all. Hence, suppose it is true for all graphs of \( V \) vertices and fewer than \( E \) edges. Then (2.3.4) implies that \( P(K; G) \) is a polynomial of the required degree \( E \) because \( G - \{e\} \) has fewer edges than \( G \) does, so its chromatic polynomial is a polynomial of degree \( V \). \( G / \{e\} \) has fewer vertices than \( G \) has, and so \( P(K; G / \{e\}) \) is a polynomial of lower degree. The claim is proved, by induction. Equation (2.3.4) gives a recursive algorithm for computing the chromatic polynomial of a graph \( G \), since the two graphs that appear on the right are both ‘smaller’ than \( G \), one in the sense that it has fewer edges than \( G \) has, and the other in that it has fewer vertices. The algorithm is the following. ```plaintext function chromopoly(G: graph): polynomial; // computes the chromatic polynomial of a graph G if G has no edges then chromopoly = K^{|V(G)|} else choose an edge e of G; chromopoly = chromopoly(G - {e}) - chromopoly(G / {e}) end (chromopoly) ``` Next we are going to look at the complexity of the algorithm chromopoly (we will also refer to it as ‘the delete-and-identify’ algorithm). The graph \( G \) can be input in any one of a number of ways. For example, we might input the list of edges of \( G \), as a list of pairs of vertices. The first step of the computation is to choose the edge \( e \) and to create the edge list of the graph \( G - \{e\} \). The latter operation is trivial, since all we have to do is to ignore one edge in the list. Next we call chromopoly on the graph \( G - \{e\} \). The third step is to create the edge list of the collapsed graph \( G / \{e\} \) from the edge list of \( G \) itself. That involves some work, but it is rather routine, and the cost is linear in the number of edges of \( G \), say \( |E(G)| \). Finally we call chromopoly on the graph \( G / \{e\} \). Let \( F(V, E) \) denote the maximum cost of calling chromopoly on any graph of at most \( V \) vertices and at most \( E \) edges. Then we see at once that \[ F(V, E) \leq F(V, E - 1) + cE + F(V - 1, E - 1) \tag{2.3.5} \] together with \( F(V, 0) = 0 \). If we put, successively, \( E = 1, 2, 3, \) we find that \( F(V, 1) < c, F(V, 2) < 2c, \) and \( F(V, 3) \leq 3c \). Hence we seek a solution of (2.3.5) in the form \( F(V, E) \leq E(c) \), and we quickly find that if \[ f(E) = 2Ef(E - 1) + E \tag{2.3.6} \] then we will have such a solution. Since (2.3.6) is a first-order difference equation of the form (1.4.5), we find that \[ f(E) = 2E \sum_{j=0}^{E-1} j^2 \sim \frac{2E^3}{3} \tag{2.3.7} \] Image Analysis: ### Analysis of the Visual Content --- **Localization and Attribution:** - **Document Page Location:** This is a single page of the document shown in the image. - **Image Numbering:** This is Image 1 of the document. **Text Analysis:** - **Text Detected and Extracted:** - Section Title: "2.3 Recursive graph algorithms" - Mathematical Expressions: Present throughout the text (e.g., P(K;G) = P(K;G - {e}) - P(K;G/{e}), chrompoly(G), etc.) - Paragraph Descriptions: The text provides detailed explanations of recursive graph algorithms, focusing heavily on the chromatic polynomial of a graph and various procedures to compute it. - Pseudocode: A function named `chrompoly(G: graph): polynomial` described with conditional logic. - Equations: References to various equations such as (2.3.4), (2.3.5), (2.3.6), and (2.3.7). **Diagram and Chart Analysis:** - **No diagrams or charts present:** **Product Analysis:** - **No products present:** **Scene and Activity Analysis:** - **No scenes or activities present:** **Graph and Trend Analysis:** - **No graphs present:** **Tables:** - **No tables present:** **Anomaly Detection:** - **No anomalies or unusual elements detected:** **Color Analysis:** - **Color Composition:** The page primarily features black text on a white background. This standard color scheme provides clear readability and is commonly used for mathematical and academic documents. **Perspective and Composition:** - **Perspective:** The image is captured directly perpendicular to the page, providing a clear and straightforward view of the text. - **Composition:** The layout includes a mix of text, equations, and pseudocode. The text appears to be organized in a typical academic format with sections, definitions, and proofs arranged sequentially. **Contextual Significance:** - **Overall Document Context:** The image appears to be part of a textbook or academic paper related to graph theory and algorithms. The section is dealing specifically with recursive algorithms related to graph coloring problems. **Metadata Analysis:** - **No metadata available:** ### Additional Aspects --- **Ablaufprozesse (Process Flows):** - **Described Process Flows:** - The pseudocode for the function `chrompoly(G: graph): polynomial` describes the process of computing the chromatic polynomial. - Detailed steps are provided for computing chrompoly on different subsections of the graph through edge deletions. **Prozessbeschreibungen (Process Descriptions):** - **Detailed Process Descriptions:** - The process involves checking for edges in the graph and recursively computing the chromatic polynomial of subgraphs obtained by deleting edges. - Steps include choosing an edge, creating edge lists, and recursively calling the function. **Typen Bezeichnung (Type Designations):** - **Type Designations in Text:** - References to graph types and polynomial degrees (e.g., G, G - {e}, G/{e}, etc.). - Use of mathematical symbols to denote various graph structures and operations. **Trend and Interpretation:** - **No specific trend analysis required:** --- This comprehensive examination provides a detailed analysis based on the detected aspects of the provided image. The focus on graph algorithms, particularly the chromatic polynomial, is evident through the extracted text and process descriptions. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 51 Context: # Exercises for section 2.3 1. Let \( G \) be a cycle of \( n \) vertices. What is the size of the largest independent set of vertices in \( V(G) \)? 2. Let \( G \) be a path of \( n \) vertices. What is the size of the largest independent set of vertices in \( V(G) \)? 3. Let \( G \) be a connected graph in which every vertex has degree \( 2 \). What must such a graph consist of? Prove. 4. Let \( G \) be a connected graph in which every vertex has degree \( \leq 2 \). What must such a graph look like? What is the size of the largest independent set of vertices in such a graph? How long would it take you to calculate that number for such a graph \( G \)? How would you do it? 5. Write out algorithm `maxIS`, which finds the size of the largest independent set of vertices in a graph. Its trivial case will occur if \( G \) has no vertices of degree \( \geq 3 \). Otherwise, it will choose a vertex of degree \( \geq 3 \) and proceed as in master 2. 6. Analyze the complexity of your algorithm `maxIS` from exercise 6 above. 7. Use (2.3.4) to prove by induction that \( P(K;G) \) is a polynomial in \( K \) of degree \( |V(G)| \). Then show that if \( G \) is a tree then \( P(K;G) = K(K - 1)^{|V| - 1} \). 8. Write out an algorithm that will change the vertex adjacency matrix of a graph \( G \) to the vertex adjacency matrix of the graph \( G/{e} \), where \( e \) is a given edge of \( G \). 9. How many edges must \( G \) have before the second quantity inside the ‘O’ in (2.3.12) is the smaller of the two? 10. Let \( \alpha(G) \) be the size of the largest independent set of vertices of a graph \( G \), let \( \chi(G) \) be its chromatic number, and let \( n = |V(G)| \). Show that, for every \( G \), \( \alpha(G) \geq n/\chi(G) \). ## 2.4 Fast Matrix Multiplication Everybody knows how to multiply two \( 2 \times 2 \) matrices. If we want to calculate \[ \begin{pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{pmatrix} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} \] then, ‘of course,’ \[ c_{ij} = \sum_{k=1}^{2} a_{ik} b_{kj} \quad (i,j = 1,2). \] Now look at (2.4.2) a little more closely. In order to calculate each one of the \( c_{ij} \)’s we have to do 2 multiplications of numbers. The cost of multiplying two \( 2 \times 2 \) matrices is therefore 8 multiplications of numbers. If we measure the cost in units of additions of numbers, the cost is 4 such additions. Hence, the matrix multiplication method that is shown in (2.4.1) has a complexity of 8 multiplications of numbers and 4 additions of numbers. This may seem rather unsettling, but the best ideas often have humble origins. Suppose you could find another way of multiplying two \( 2 \times 2 \) matrices in which the cost was only 7 multiplications of numbers, together with more than 4 additions of numbers. What would be a cause for dancing in the streets, or would it be just a curiosity of little importance? In fact, it would be extremely important, and the consequences of such a step were fully appreciated only in 1969 by V. Strassen, to whom the ideas that we are now discussing are due.* What we’re going to do next in this section is the following: - (a) Describe another way of multiplying two \( 2 \times 2 \) matrices in which the cost will be only 7 multiplications of numbers plus a bunch of additions of numbers, and - (b) convince you that it was worth the trouble. The usefulness of the ideas stems from the following amazing fact: if two \( 2 \times 2 \) matrices can be multiplied with only 7 multiplications of numbers, then two \( N \times N \) matrices can be multiplied using only \( O(N^{2.81}) \). * V. Strassen, Gaussian elimination is not optimal, Numerische Math. 13 (1969), 354-6. Image Analysis: ### Comprehensive Analysis of the Attached Visual Content #### 1. Localization and Attribution - **Image Positioning:** Entire content displayed is on a single page. - **Numbering:** Image 1 (whole page content). #### 2. Object Detection and Classification - **Object 1:** Text (Instructions and Mathematical Content) - **Object 2:** Diagram (Mathematical Matrix Equation) #### 3. Scene and Activity Analysis - **Scene Description:** The scene depicts a page from a mathematical or computer science textbook, specifically focusing on exercises related to graph theory and an explanation of matrix multiplication. - **Activities:** - Solving graph theory exercises. - Learning about fast matrix multiplication. #### 4. Text Analysis - **Text Detected:** - Titles and Sections: - "Exercises for section 2.3" - "2.4 Fast matrix multiplication" - Exercises: 1. Let \( G \) be a cycle of \( n \) vertices. What is the size of the largest independent set of vertices in \( V(G) \)? 2. Let \( G \) be a path of \( n \) vertices. What is the size of the largest independent set of vertices in \( V(G) \)? 3. Let \( G \) be a connected graph in which every vertex has degree 2. What must such a graph consist of? Prove it. 4. Let \( G \) be a connected graph in which every vertex has degree \( \leq 2 \). What must such a graph look like? 5. Let \( G \) be a not-necessarily-connected graph in which every vertex has degree \( \leq 2 \). What must such a graph look like? What is the size of the largest independent set of vertices in such a graph? How long would it take you to calculate that number for such a graph \( G \)? How would you do it? 6. Write out algorithm maxset3, which finds the size of the largest independent set of vertices in a graph. Its trivial case will occur if \( G \) has no vertices of degree \( \geq 3 \). Otherwise, it will choose a vertex \( v \) of degree \( \geq 3 \) and proceed as in maxset2. 7. Analyze the complexity of your algorithm maxset3 from exercise 6 above. 8. Use (2.3.4) to prove by induction that \( P(K;G) \) is a polynomial in \( K \) of degree \( |V(G)| \). Then show that if \( G \) is a tree then \( P(K;G)= K(K-1)^{|V(G)|-1} \). 9. Write out an algorithm that will change the vertex adjacency matrix of a graph \( G \) to the vertex adjacency matrix of the graph \( G+e \), where \( e \) is a given edge of \( G \). 10. How many edges \( G \) must have before the second quantity inside the ‘O’ in (2.3.12) is the smaller of the two? 11. Let \( \alpha(G) \) be the size of the largest independent set of vertices of a graph \( G \), let \( \chi(G) \) be its chromatic number, and let \( n=|V(G)| \). Show that, for every \( G \), \( \alpha(G) \geq n/\chi(G) \). - **Fast Matrix Multiplication Paragraph:** - Explains how to multiply 2x2 matrices - Describes the logic and cost calculation of 2x2 matrix multiplication - Mentions the contribution of V. Strassen #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 53 Context: ```markdown ## 2.4 Fast Matrix Multiplication ```plaintext function MatProd(A, B: matrix: N: integer): matrix; { MatProd is AB, where A and B are N x N } uses Strassen method; if N is not a power of 2 then border A and B by zeros and columns of O’s until their size is the next power of 2 and change N; if N = 1 then MatProd := AB else partition A and B as shown in (2.4.5): I := MatProd(A11, A22, B21 + B23, N/2); II := MatProd(A11 + A12, B21, B22 + B23, N/2); ... etc., through all of the formulas shown in (2.4.3), (2.4.4), ending with ... C22 := II + I1 + V - V1 end; { MatProd } Note that this procedure calls itself recursively 7 times. The plus and minus signs in the program each represent an addition or subtraction of two matrices and, therefore, each one of them involves a call to a matrix addition or subtraction procedure (just the usual method of adding, nothing fancy!). Therefore the function MatProd makes 25 calls, 7 of which are recursively to itself, and 18 of which are to a matrix addition/subtraction routine. We will study the complexity of the routine in two ways. We will count the number of multiplications of numbers that are needed to multiply two \(2^n \times 2^n\) matrices using MatProd (call that number \(f(n)\)), and then we will count the number of additions of numbers (call that number \(g(n)\)) that MatProd needs in order to multiply two \(2^n \times 2^n\) matrices. The multiplications of numbers are easy to count. MatProd calls itself 7 times, in each of which it does exactly \(f(n - 1)\) multiplications of numbers, hence \(f(n) = 7f(n - 1) + f(0) = 1\). Therefore we see that \(f(n) = 7^n\) for all \(n \geq 0\). Hence MatProd does \(7^n\) multiplications of numbers in order to do the multiplication of \(2^n \times 2^n\) matrices. Let’s take the last sentence in the above paragraph and replace \(2^n\) by \(n\) throughout. It then tells us that MatProd does \((\log_2 N) \times N^2\) multiplications of numbers in order to do one multiplication \(N \times N\) matrices. Since \(n \log_2 n \approx n^{\log_2 3} \approx n^{2.8}\), we see that Strassen's method uses only \(O(N^{\log_2 7})\) multiplications of numbers, in place of the \(N^3\) multiplications that are required by the usual formulas. It remains to count the additions/subtractions of numbers that are needed by MatProd. In each call to it, it recurses calls to itself MatProd does \(g(n - 1)\) additions of numbers. In each of its 18 calls to the procedure that adds or subtracts matrices it does a number of additions of numbers that is equal to the square of the size of the matrices that are being added or subtracted. That size is \(2^{n - 1}\), so it follows that \(g(0) = 0\) and for \(n \geq 1\) we have \[ g(n) = 7g(n - 1) + 18 \cdot 2^{n - 2} \] We follow the method of section 1.4 on this first-order linear difference equation. Hence we make the change of variable \(h(n) = g(n) - 18 \cdot 2^{(n - 2)}\) (where \(n \geq 0\) and we find that \(h(0) = 0\) and for \(n \geq 1\), \[ h_n = \frac{9}{2} \sum_{i=1}^{n} 7^{i - 1} = \frac{9}{2} \cdot \frac{7^n - 1}{7 - 1} \] \[ \leq \frac{9}{2} \cdot \frac{7^n}{6} \] = \(21/2\) ``` Image Analysis: ### **Analysis of the Provided Page** --- **Localization and Attribution:** - The page does not contain multiple images; instead, it appears to be a single image of a written text page, likely from a book or academic paper. - The page number is indicated at the bottom as "49." --- **Text Analysis:** 1. **Page Header:** - **Section Title:** "2.4 Fast matrix multiplication" - This indicates that the content discussed in this section focuses on advanced techniques for performing matrix multiplication efficiently. 2. **Main Text Content:** - **Function Definition:** - A function named `MatrProd(A, B: matrix; N: integer): matrix` is described, where `A` and `B` are `N x N` matrices. - The function uses Strassen’s method for matrix multiplication, which is efficient for large matrices. - **Procedure Breakdown:** - Conditions are given for different cases: - If `N` is not a power of 2, matrices `A` and `B` are resized to fit the next power of 2. - If `N = 1`, the multiplication is straightforward: `A * B = AB`. - Otherwise, matrices `A` and `B` are partitioned into submatrices and further recursive calls are made. 3. **Mathematical Equations and Formulas:** - Several references are made to other sections such as (2.4.3), (2.4.4). - The significant mathematical representations in this function include the handling of matrix partitions, operations on submatrices, and recursion. 4. **Complexity Analysis:** - A detailed complexity analysis is provided, explaining: - The procedure calls itself recursively 7 times. - The calculation of the function's execution in terms of multiplications (`f(n)`) and additions (`g(n)`). 5. **Recursive Equations:** - Equations used to calculate the number of operations are provided: - \(g(n) = 7g(n - 1) + 18 \cdot 4^{n - 1}\) - The method for solving these equations is based on first-order linear difference equations. 6. **Final Calculations:** - Summation formulas: - Describing the series sums and leading to the final results for `gm` and `ym`. --- **Perspective and Composition:** - **Perspective:** - The image is taken from a bird’s-eye view or top view, focusing on the entire page with clear visibility of the text. - **Composition:** - The image is composed in a manner that centers the text, maintaining the entire page within the frame. Mathematical notations, functions, and explanations are neatly aligned and provide structured information. --- **Contextual Significance:** - **Subject Matter:** - The content appears to be from a mathematical or computer science textbook or research paper. It is dealing with topics relevant to numerical methods and algorithms, specifically how to efficiently perform matrix multiplication through Strassen’s algorithm. - **Contribution to Overall Document:** - It contributes to the reader’s understanding of efficient algorithms for matrix operations, essential in areas like linear algebra, computer graphics, and scientific computing. --- **Anomaly Detection:** - No apparent anomalies or unusual elements detected in the image. The text and mathematical content appear standard for educational or academic material. --- **Color Analysis:** - **Color Composition:** - The document is primarily monochromatic with black text on a white background, which is typical for printed academic content. - **Impact on Perception:** - The clear contrast between the text and the background ensures readability and focus on the content. --- This analysis covers the detailed examination of the aspects provided for the given visual content. The context indicates a focus on computational algorithms and their theoretical underpinnings, relevant to students and professionals in the field. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 53 Context: We follow the method of section 1.4 on this first-order linear difference equation. Hence we make the change of variable g(n) = 7 ^n y(n) (n ≥ 0) and we find that y0 = 0 and for n ≥ 1, y_n - 7y_n–1 + 2 ^(4/n^ n )= 9. If we sum over n we obtain y_n = 9 ∑_(i=1)^n (4/7)^i ≤ 9 ∑_(i=1)^∞ (4/7)^n = 9/((1 – 4/7)) = 21/2. ``` - **Analysis of Content:** The text is an excerpt from a mathematical or computer science document discussing the Strassen method for fast matrix multiplication. It includes a pseudocode function `MatrProd` for multiplying matrices and explains the recursive nature of the algorithm. The complexity analysis includes counting the number of multiplications and additions/subtractions necessary for the algorithm. Detailed recursive calls and complexity calculations are provided, illustrating the efficiency improvement from O(N^3) to O(N^2.81) for matrix multiplication. #### 8. Color Analysis - **Color Composition:** - The page has a predominantly white background with black text, typical of a printed or digital document. - There are no other significant colors present, which focuses the reader's attention on the textual content. #### 9. Perspective and Composition - **Perspective:** - The image is a top-down view of a text document, likely appearing in a book or a digitally scanned paper. - **Composition:** - The text is arranged in typical paragraph format. - The pseudocode and equations are indented and formatted distinctly to differentiate them from the main body of text, aiding readability. #### 14. Trend and Interpretation - **Trend in Visual Content:** - The visual content suggests a trend toward presenting complex mathematical and algorithmic concepts in a step-by-step manner, providing both theoretical explanations and practical pseudocode. - **Interpretation:** - The use of pseudocode and complexity analysis serves to educate readers on efficient matrix multiplication methods, specifically emphasizing the Strassen algorithm's recursive approach and reduced computational complexity. #### Additional Observations - **Prozessbeschreibungen (Process Descriptions):** - The text describes the recursive process of Strassen's matrix multiplication algorithm in detail, including its base case, recursive case, and the operations needed to combine intermediate results. - **Typen Bezeichnung (Type Designations):** - The algorithm categorizes its steps into multiplication and addition/subtraction processes, describing the type of operations performed at each recursive level. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 54 Context: # Chapter 2: Recursive Algorithms Finally, \( g(n) = 7^{n} \leq (10.5)^{n} = O(n^{m}) \), and this is \( O(N^{2.81}) \) as before. This completes the proof of ## Theorem 2.4.1 In Strassen's method of fast matrix multiplication the number of multiplications of numbers, of additions of numbers and of subtractions of numbers that are needed to multiply together two \( N \times N \) matrices are each \( O(N^{2.81}) \) (in contrast to the \( O(N^{3}) \) of the conventional method). In the years that have elapsed since Strassen's original paper many researchers have been whittling away at the exponent of \( N \) in the complexity bounds. Several new, and more elaborate algorithms have been developed, and the exponent, which was originally \( 3 \), has progressed downward through \( 2.81 \) to values below \( 2.5 \). It is widely believed that the true minimum exponent is \( 2 \), i.e., that two \( N \times N \) matrices can be multiplied in time \( O(N^{2}) \), but there seems to be a good deal of work to be done before that result can be achieved. ## Exercises for section 2.4 1. Suppose we could multiply together two \( 3 \times 3 \) matrices with only \( 22 \) multiplications of numbers. How fast, recursively, would we then be able to multiply two \( N \times N \) matrices? 2. (cont.) With what would the ‘22’ in problem 1 above have to be replaced in order to achieve an improvement over Strassen's algorithm given in the text? 3. (cont.) Still more generally, with how few multiplications would we have to be able to multiply two \( M \times N \) matrices in order to ensure that recursively we would then be able to multiply two \( N \times N \) matrices faster than the method given in this section? 4. We showed in the text that if \( N \) is a power of \( 2 \) then two \( N \times N \) matrices can be multiplied in at most time \( C N^{2} \), where \( C \) is a suitable constant. Prove that if \( N \) is not a power of \( 2 \) then two \( N \times N \) matrices can be multiplied in time at most \( T(N) \). ## 2.5 The Discrete Fourier Transform It is a lot easier to multiply two numbers than to multiply two polynomials. If you should want to multiply two polynomials \( f \) and \( g \) of degrees \( 77 \) and \( 94 \), respectively, you are in for a lot of work. To calculate just one coefficient of the product is already a lot of work. Think about the calculation of the coefficient of \( x^{60} \) in the product, for instance, and you will see that \( 50 \) numbers must be multiplied together and added in order to calculate just that one coefficient of \( fg \), and there are \( 171 \) other coefficients to calculate! Instead of calculating the coefficients of the product \( fg \), it would be much easier just to calculate the values of the product at, say, \( 172 \) points. To do that we could just multiply the values of \( f \) and \( g \) at each of those points, and after a total cost of \( 172 \) multiplications we would have the values of the product. The values of the product polynomial at \( 172 \) distinct points determine that polynomial completely, so that sequence of values is the answer. It’s just that we humans prefer to see polynomials given by means of their coefficients instead of by their values. The Fourier transform, that is the subject of this section, is a method of converting from one representation of a polynomial to another. More exactly, it converts from the sequence of coefficients of the polynomial to the sequence of values of that polynomial at a certain set of points. Ease of converting between these two representations of a polynomial is vitally important for many reasons, including multiplication of polynomials, high precision integer arithmetic in computers, creation of medical images in CAT scanners, etc. Hence, in this section we will study the discrete Fourier transform of a finite sequence of numbers, methods of calculating it, and some applications. This is a computational problem which at first glance seems very simple. What we’re asked to do, basically, is to evaluate a polynomial of degree \( n - 1 \) at \( n \) different points. So what could be so difficult about that? If we just calculate the \( n \) values by brute force, we certainly wouldn’t need to do more than \( O(n^{2}) \) multiplications altogether. Image Analysis: Based on the provided visual content, here is a comprehensive examination of the various aspects: 1. **Localization and Attribution:** - This document contains one single page with multiple sections of text. - The sections are as follows based on their positioning from top to bottom: 1. Section Title: "Chapter 2: Recursive Algorithms" 2. Theorem and Explanation: Theorem 2.4.1 and its explanation about matrix multiplication. 3. Exercises: A list of exercises related to section 2.4. 4. Section Title: "2.5 The discrete Fourier transform" 5. Explanation: An explanation on the discrete Fourier transform. 2. **Object Detection and Classification:** - There are no images containing objects to be classified in this document. 4. **Text Analysis:** - **Section Title**: "Chapter 2: Recursive Algorithms" - This section header indicates the main topic for this chapter, focusing on recursive algorithms. - **Theorem and Explanation**: "Theorem 2.4.1" - This section describes a theorem related to fast matrix multiplication, specifically highlighting the number of multiplications needed for \(N \times N\) matrices. - The text includes mathematical expressions, such as \(O(N^{2.81})\), and terms like "complexity bounds," indicating the advanced nature of the mathematical concepts being discussed. - **Exercises for section 2.4**: - A set of numbered exercises from 1 to 4 related to matrix multiplication and recursive algorithms. - Exercises involve scenarios and questions that challenge the understanding of the concepts presented in the previous section. - **Section Title**: "2.5 The discrete Fourier transform" - This title indicates the beginning of a new topic focusing on the discrete Fourier transform, which is another mathematical concept. - **Explanation**: - An in-depth explanation about calculating the discrete Fourier transform of polynomials. - This section discusses the process and significance of the discrete Fourier transform in computational mathematics. 5. **Diagram and Chart Analysis:** - There are no diagrams or charts present in this document. 6. **Product Analysis:** - There are no products depicted in the provided visual content. 7. **Anomaly Detection:** - There are no anomalies or unusual elements detected in this document. 8. **Color Analysis:** - The document is in grayscale, with no particular use of color except for varying shades of black and white. - Dominant colors are black text on a white background, which is standard for printed or digital documents. 9. **Perspective and Composition:** - The perspective is a standard eye-level view suitable for reading. - The composition is typical of a text document, structured with headings, paragraphs, and sections to organize the content logically and sequentially. Note: The metadata, contextual significance, graph and trend analysis, ablaufprozesse, prozessbeschreibungen, typen bezeichnung, and tables aspects do not apply to the provided image as no such elements are present. This covers all observable information from the provided visual content. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 57 Context: # 2.5 The Discrete Fourier Transform There is one small problem, though. In (2.5.8) we want to compute \( f(j) \) for \( j = 0, \ldots, 2^n - 1 \). However, the Fourier transform of the shorter sequence (2.5.9) is defined for only \( 2^{r-1} \) values of \( j \) namely for \( j = 0, \ldots, 2^{r-1} - 1 \). So if we calculate the first sum by a recursive call, then we will need its values for \( j \) that are outside the range for which it was computed. This problem is no sooner recognized than solved. Let \( Q(j) \) denote the first sum in (2.5.8). Then we claim that \( Q(j) \) is a periodic function of \( j \), of period \( 2^{r-1} \): \[ Q(j + 2^{r-1}) = \sum_{m=0}^{2^{r-1}-1} x_m \exp(2\pi i (j + 2^{r-1})/2^r) \] \[ = \sum_{m=0}^{2^{r-1}-1} x_m \exp(2\pi i j / 2^r) \exp(2\pi i m/2^{r-1}) = \sum_{m=0}^{2^{r-1}-1} x_m \exp(2\pi i j / 2^r) \cdot \exp(2\pi i m/2^{r-1}) = Q(j) \quad \text{(2.5.11)} \] for all integers \( j \). If \( Q(j) \) has been computed only for \( 0 \leq j \leq 2^{r-1} - 1 \) and if we should want its value for some \( j \geq 2^{r} \), then we can get that value by asking for \( Q(j \mod 2^{r-1}) \). Now we can state the recursive form of the Fast Fourier Transform algorithm in the most important case where \( n \) is a power of 2. In the algorithm we will use the type `complexarray` to denote an array of complex numbers. ```matlab function FFT(x: complexarray): complexarray; % computes Fast Fourier transform of n = 2^k numbers x if n == 1 then FFT[0] := x[0] end if; evenarray := {x[0], x[2], ..., x[n - 2]}; oddarray := {x[1], x[3], ..., x[n - 1]}; {u[0], u[1], ..., u[n - 1]} := FFT(n/2, evenarray); {v[0], v[1], ..., v[n - 1]} := FFT(n/2, oddarray); for j = 0 to n/2 - 1 do x[j] := u[j] + exp(2\pi i * j/n) * v[j]; x[j + n/2] := u[j] - exp(2\pi i * j/n) * v[j]; end for; end function; ``` Let \( y(k) \) denote the number of multiplications of complex numbers that will be done if we call FFT on an array whose length is \( n = 2^k \). The call to FFT(n/2, evenarray) costs \( y(k - 1) \) multiplications as does the call to FFT(n/2, oddarray). The for loop \( j = 0 \) to \( n/2 - 1 \) requires \( n \) more multiplications. Hence, \[ y(k) = 2y(k - 1) + 2^k \quad (k \geq 1); \quad y(0) = 0. \quad (2.5.12) \] If we change variables by writing \( y(k) = 2^{k+1} - 2 \), then that is \( y(k) = 2(k + 1) \), together with \( y(0) = 0 \), and therefore that \( y(k) = k2^{k+1} \). This proves **Theorem 2.5.1.** The Fourier transform of a sequence of \( n \) complex numbers is computed using \( O(n \log n) \) multiplications of complex numbers by means of the procedure FFT, if \( n \) is a power of 2. Next* we will discuss the situation when \( n \) is not a power of 2. The reader may observe that by 'padding out' the input array with additional 0's we can extend the length of the array until it becomes a power of 2, and then call the FFT procedure that we have already. Image Analysis: ### Detailed Examination #### Localization and Attribution: - **Image 1** is the only image on the page. #### Text Analysis: 1. **Detected Text:** - Title: “2.5 The discrete Fourier transform” - Main Content: - The page describes concepts related to the Fourier transform, specifically focusing on the recursive calculation for certain values of \(j\) and \(Q(j)\) and the Fast Fourier Transform (FFT) algorithm. - Mathematical formulae and equations are present, such as: - \( Q(j + 2^r-1) = Q(j) \) - \( y(k) = 2y(k - 1) + 2^k \) - Explanation of the FFT algorithm, including the function definition: - ```Text function FFT(integer n; x: complexarray):complexarray; ``` - Important Points: - Fourier transform challenges, recursive solution: - Calculations for \( j = 0, 1, \ldots, 2^r - 1 \) - Recursive formula adaptation: leveraging \( Q(j + 2^r-1) = Q(j) \) - Expansion and calculations involving complex exponentials - Computing \( y(k) = 2y(k - 1) + 2^k \) and the application in FFT - Summation symmetry and logic behind FFT implementations 2. **Significance:** - The text provides thorough explanations and derivations for understanding and implementing the Fast Fourier Transform, a key algorithm in computational mathematics and signal processing. It offers valuable insight into the recursive approach to computing Fourier transforms efficiently. #### Process Descriptions: - The text describes two main computational processes: - The computation of function \(f_c(j)\) for \( 2^r \) values of \( j \). - The process of the Fast Fourier Transform (FFT), from basic definition to a more advanced recursive calculation involving even and odd indexed arrays, aiming to reduce multiplications of complex numbers. #### Typen Bezeichnung (Type Designations): - **Mathematical Notations**: - Typical notations used: \(j\), \(Q(j)\), \(2^r\), \(x\), “mod” arithmetic, and \(y(k)\). - **Function Definition**: - ```Text function FFT(integer n; x: complexarray):complexarray; ``` - Variables in the function: \( j, k, u, v, y \), indicating the iterative nature of the algorithm. #### Diagram and Chart Analysis: - **Equations**: - Equations present intricate mathematical relationships fundamental to discrete Fourier transform and FFT algorithms. - Key insights involve recognizing the periodic nature of \( Q(j) \), efficiency gains with recursive functions, and explicit calculations within FFT. #### Perspective and Composition: - **Perspective**: - The page is laid out in a standard mathematical textbook format with detailed text, equations, and algorithmic functions. - **Composition**: - Structured: Title at the top, dense textual explanation, embedded equations, clear logical flow, followed by defined programs or algorithms. #### Contextual Significance: - Within a broader context, this page is part of a textbook or comprehensive guide on Fourier transforms and their computational techniques, critical for students and professionals in fields such as applied mathematics, engineering, and computer science. #### Color Analysis: - The page mainly consists of black text on a white background, standard for textbook material, ensuring clarity and readability. This detailed examination covers text analysis, process descriptions, type designations, and other essential details from the attached image. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 59 Context: # 2.5 The discrete Fourier transform The computation, hence all of the necessary values of \( a_j \) can be found with \( r_1 \) complex multiplications. Once the \( a_j \) are all in hand, then the computation of the one value of the transform from (2.5.13) will require an additional \( r_1 \) complex multiplications. Since \( n = r_1 r_2 \) values of the transform have to be computed, we will need \( r_1 r_2 (n-1) \) complex multiplications. The complete computation needs \( r_1 r_2 + r_1 r_2 + r_1 r_2(n-1) \) multiplications if we choose a particular factorization in terms of \( r_1 r_2 \). The factorization that should be chosen is the one that minimizes the labor, so we have the recurrence \[ g(n) = \min_{r_1, r_2 \mid n} \{ g(r_1) + g(r_2) + r_1 r_2 - n \} \tag{2.5.16} \] If \( n = p \) is a prime number, then there are no factorizations to choose from and our algorithm is of no help at all. There is no recourse but to calculate the \( p \) values of the transform directly from the definition (2.5.6), at that will require \( p - 1 \) complex multiplications to be done in order to get each of those \( p \) values. Hence we have, in addition to the recurrence formula (2.5.16), the special values \[ g(p) = p - 1 \quad \text{(if \( p \) is prime)} \tag{2.5.17} \] The recurrence formula (2.5.16) together with the starting values that are shown in (2.5.17) completely determine the function \( g(n) \). Before proceeding, the reader is invited to calculate (2.12) and (2.18). We are going to work out the exact solution of the interesting recurrence (2.5.16), (2.5.17), and when we are finished we will see which factorization of \( n \) is the best one to choose. If we leave that question in abeyance for a while, though, we can summarize by stating the (otherwise) complete algorithm for the fast Fourier transform. ```plaintext function FFT(x:complexarray; n:integer):complexarray; {computes Fourier transform of a sequence x of length n} if n is prime then for j:0 to n-1 do FFT[j] := ∑_{k=0}^{n-1} a[k] x^{jk} else let n = r_1 r_2 be some factorization of n; {see below for best choice of r_1, r_2} for k:0 to r_1 - 1 do {a[0], a[1], ..., a[r_2 - 1]} {x := FFT[1]{k][r_1], ..., k[r_2 - 1]}; r_2} for j:0 to n - 1 do FFT[j] := ∑_{k=0}^{n-1} a[k] mod r_2^{sk} end; ``` Our next task will be to solve the recurrences (2.5.16), (2.5.17), and thereby to learn the best choice of the factorization of \( n \). Let \( g(n) = n h(n) \), where \( h \) is a new unknown function. Then the recurrence that we have to solve takes the form \[ h(n) = \begin{cases} \min\{ h(d) \mid d \mid n \text{ and } d \neq 1 \text{ and } e \neq n \} - 1 & \text{if } n \text{ is composite} \\ 1 & \text{if } n \text{ is prime.} \end{cases} \tag{2.5.18} \] In (2.5.18), the "min" is taken over all \( d \) that divide one other than \( d = 1 \) and \( e = n \). The above relation determines the value of \( h \) for all positive integers \( n \). For example, \[ h(15) = \min(h(15/3) + 4, h(15/5) + 3) - 1 = \min(h(5) + 3, h(3) + 5) - 1 = \min(7, 6) = 6 \] #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 60 Context: ### Summary The visual content is a page from an academic or technical text focused on recursive algorithms, specifically the Fast Fourier Transform (FFT). The text includes mathematical notations, theorems, proofs, and practical applications, emphasizing the efficiency of FFT in computational tasks. The included table illustrates the reduction in complexity provided by using the FFT. The document's formal structure and detailed mathematical content suggest it is intended for readers with a strong background in mathematics or computer science. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 60 Context: **Text Analysis:** **Image 1:** - **Detected and extracted text:** ``` Chapter 2: Recursive Algorithms ... and so forth. To find the solution in a pleasant form, let n = p^aqi^a ... p_ka^ak (2.5.19) be the canonical factorization of n into primes. We claim that the function H(n) = a1 (p − 1) + a(p − 1) + acp2 − 1) + ...+ a (p − 1) (2.5.20) is the solution of (2.5.18) (this claim is obviously (x) correct if n is prime). To prove the claim in general, suppose it to be true for 1... 1, and suppose that n is not prime. Then every divisor d of n must be of the form d = p^i, pzp^, .. . Heat where the primes are the same as those that appear in (2.5.19) and each h, is 4; < a,. Hence from (2.5.18) we get H(n) = min((a − b)(p − 1) + ...to + ... (a − b)(p − 1) + p^i . pAl ... pa ^ 1) − 1 where now the 'min' extends over all admissible choices of the 's, namely exponents 4,..., such that 0 bi s ai (v i = 1, s) and not all bi are 0 and not all al) b = cl. One such admissible choice would be to take, say, b, = 1 and all other a, 0. If we let H(hi...b) denote the quantity in braces in (2.5.21), then with this choice the value of H would be a,(p, – 1) + ... (a(p, – 1) + exactly what we need to prove our claim (2.5.20). Hence what we have to show is that the above choice of the 's is the best one. We will show that if one of the b, is larger than 1 then we can reduce it without increasing the value of H. To prove this, observe that for each i = 1, s we have H(d, ...b, b + 1, ...b) – H(hi..., b) = – p^b + 41(a _ 1) = d – 1)(p –1). Since the divisor d 2 and the prime 'i' 2, the last difference is nonnegative. Hence H doesn’t increase if we decrease one of the bi's by 1 unit, as long as not all bi = 0. It follows that the minimum of H occurs among the prime divisors of d n. Further, if dis prime, then we can easily check from (2.5.21) that it doesn't matter which prime divisor of 'n that we choose to be + d, the function H(n) is always given by (2.5.20). If we recall the change of variable g(y) = nli(y) we find that we have proved Theorem 2.5.2. (Complexity of the Fast Fourier Transform) The best choice of the factorization 'n to nodes in algorithm FFT is to take r, to be a prime divisor of n. If that is done, then algorithm FFT requires g(n) = a(p_ (p, – 1) + a(p – 1) + ...+ a(p, 1)) complex multiplications in order to do its job, where n = p^i . pa^ ... ... pg, f is the canonical factorization of the integer 'n. Table 2.5.1 shows the number g(y) of complex multiplications required by FFT as a function of 'n. The saving over the straightforward algorithm that uses n(n – 1) multiplications for each is is apparent. If it is a power of 2, Say n = 2', then the formula of theorem 2.5.2 reduces to g(y) In log/n/g log 2, i agreement with theorem 2.5.1. What does the formula say if n is a power of 3? It if is a product of distinct primes? 2.6 Applications of the FFT Finally, we will discuss some applications of the FFT. A family of such applications begins with the observation that the FFT provides the fastest game in town for multiplying two polynomials together. Consider a multiplication like (1 + 2x + 2x - 2x x³) - (4 -5x - x - 2 - x³ + 114x + x³). ``` - **Analysis:** - The excerpt is from "Chapter 2: Recursive Algorithms." - The focus is on the factorization of integers into their prime components and the related function \( H(n) \). - Mathematical notations such as factorization, algorithms, and theorems are present. - The text delves into the details of finding the minimum cost of complex multiplications using the Fast Fourier Transform (FFT). **Diagram and Chart Analysis:** **Image 1:** - **Detected Table:** - "Table 2.5.1 shows the number g(n) of complex multiplications required by FFT as a function of n." - **Analysis:** - The table likely includes data about complex multiplications for different values of \( n \). - Exact content and numbers from the table cannot be extracted from the visual, but it is referred to in the explanatory text. **Perspective and Composition:** **Image 1:** - **Perspective:** - The image is a straight-on, centered view of a single-page mathematical text. - **Composition:** - The page is arranged in a structured format, typically found in academic textbooks, with sections, sub-sections, and mathematical notation. **Contextual Significance:** **Image 1:** - The text appears in the context of a textbook or academic paper focused on recursive algorithms. - The page elaborates on the theory and practical applications of prime factorization and FFT, emphasizing mathematical proofs and theorems. - This page contributes to the overall theme of computational efficiency and algorithm optimization within the wider context of the document. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 62 Context: - **Tables:** - No tables are included in the content. ### Conclusion This page presents a comprehensive discussion and procedural guide on using recursive algorithms and FFT for multiplying polynomials. It is academically oriented and contributes to the broader field of computer science and mathematics by offering an efficient methodology for polynomial multiplication. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 62 Context: #### Contextual Significance - **Contribution to Overall Document**: - This page is part of a larger document focused on recursive algorithms. - The detailed algorithmic approach and theoretical explanations provide foundational knowledge crucial for advanced understanding in computational mathematics and computer science. #### Summary The image is a page from an educational text on recursive algorithms, providing a deep dive into polynomial multiplication using the FFT and roots of unity. Key features include detailed steps, equations, and explanatory text that elucidate both the forward and inverse FFT processes. The layout, color scheme, and textual organization all contribute to the goal of delivering complex mathematical concepts in a clear and comprehensible manner. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 63 Context: ```markdown ## 2.7 A review An outgrowth of the rapidity with which we can now multiply polynomials is a rethinking of the methods by which we do ultrahigh-precision arithmetic. How fast can we multiply two integers, each of which has ten million bits? By using ideas that developed directly (though not at all trivially) from the ones that we have been discussing, Schönhage and Strassen found the fastest known method for doing such large-scale multiplications of integers. The method relies heavily on the FFT, which may not be too surprising since an integer \( n \) is given in terms of its bits \( b_0, b_1, \ldots, b_m \) by the relation \[ n = \sum_{i=0}^{m} b_i 2^i. \] However, the sum in (2.5) is seen at once to be the value of a certain polynomial at \( z = 2 \). Hence in asking for the bits of the product of two such integers we are asking for something very similar to the coefficients of the product of two polynomials, and indeed the fastest known algorithms for this problem depend upon the Fast Fourier Transform. ### Exercises for section 2.6 1. Let \( \omega \) be an \( n \)-th root of unity, and let \( k \) be a fixed integer. Evaluate \[ 1 + \omega^k + \omega^{2k} + \cdots + \omega^{(n-1)k}. \] 2. Verify that the relations (2.6.3) and (2.6.4) indeed are inverses of each other. 3. Let \( f = \sum_{j=0}^{n-1} a_j \omega^j \). Show that \[ \frac{1}{n} \sum_{n=1}^{\omega} |f(\omega)|^2 = |a_0|^2 + \cdots + |a_{n-1}|^2. \] 4. The values of a certain cubic polynomial at \( i, -1, -i \) are \( 1, 2, 3, 4 \), respectively. Find its value at 2. 5. Write a program that will do the FFT in the case where the number of data points is a power of 2. Organize your program so as to minimize additional array storage beyond the input and output arrays. 6. Prove that a polynomial of degree \( n \) is uniquely determined by its values at \( n + 1 \) distinct points. ## 2.7 A review Here is a quick review of the algorithms that we studied in this chapter. Sorting is an easy computational problem. The most obvious way to sort an array elements takes time \( \Theta(n^2) \). We discussed a recursive algorithm that sorts in an average time of \( \Theta(n \log n) \). Finding a maximum independent set in a graph is a hard computational problem. The most obvious way to do it might take time \( \Omega(2^{|V|}) \) if the graph \( G \) has \( V \) vertices. We discussed a recursive method that runs in time \( O(1.39^n) \). The best known methods run in time \( O(2^{|V|/3}) \). Finding out if a graph is \( k \)-colorable is a hard computational problem. The most obvious way to do it takes time \( O(k^n) \), if \( G \) has \( n \) vertices. We also discussed methods that run in time \( O(1.5^n) \) if \( G \) has \( m \) edges. One recently developed method *runs in time* \( O(1.5^n) \). We will see in section 5.7 that this problem can be done in an average time that is \( O(1) \) for fixed \( k \). Multiplying two matrices is an easy computational problem. The most obvious way to do it takes time \( O(n^3) \) if the matrices are \( n \times n \). We discussed a recursive method that runs in time \( O(n^{2.81}) \). A recent method **runs in time \( O(n^{2.5}) \)** for some \( \epsilon < 2.5 \). - E. Lawler, A note on the complexity of the chromatic number problem, Information Processing Letters 5 (1976), 66-7. - D. Coppersmith and S. Winograd, On the asymptotic complexity of matrix multiplication, SIAM J. Comp. 11 (1980), 472-492. ``` Image Analysis: ### Image Analysis #### 1. **Localization and Attribution:** - **Image Position**: The entire provided content appears as a single page document. - **Image Number**: Image 1 #### 4. **Text Analysis:** - **Detected Text**: ``` 2.7 A review An outgrowth of the rapidity with which we can now multiply polynomials is a rethinking of the methods by which we do ultrahigh-precision arithmetic. How fast can we multiply two integers, each of which has ten million bits? By using ideas that developed directly (though not at all trivially) from the ones that we have been discussing, Schönhage and Strassen found the fastest known method for doing such large-scale multiplications of integers. The method relies heavily on the FFT, which may not be too surprising since an integer n is given in terms of its bits by the relation n = ∑_(i=0)^k b_i 2^i. (2.6.5) However, the sum in (2.6.5) is seen at once to be the value of a certain polynomial at x = 2. Hence in asking for the bits of the product of two such integers we are asking for something very similar to the coefficients of the product of two polynomials, and indeed the fastest known algorithms for this problem depend upon the Fast Fourier Transform. Exercises for section 2.6 1. Let ω be an nth root of unity, and let k be a fixed integer. Evaluate 1 + ω^k + ω^(2k) +···+ ω^[(n−1)k] . 2. Verify that the relations (2.6.3) and (2.6.4) indeed are inverses of each other. 3. Let f = ∑(n−1)_(j=0) a_jω^j. Show that 1/n ∑(n)_(ω=1) |f(ω^1)|^2 = |a_0|^2 + ···+ |a_(n−1)|^2 4. The values of a certain cubic polynomial at 1, i, -1, -i are 1, 2, 3, 4, respectively. Find its value at ω. 5. Write a program that will do the FFT in the case where the number of data points is a power of 2. Organize your program so as to minimize additional array storage beyond the input and output arrays. 6. Prove that a polynomial of degree n is uniquely determined by its values at n + 1 distinct points. 2.7 A review Here is a quick review of the algorithms that we studied in this chapter. Sorting is an easy computational problem. The most obvious way to sort an array elements takes time Θ(n^2). We discussed a recursive algorithm that sorts in an average time of Θ(n log n). Finding a maximum independent set in a graph is a hard computational problem. The most obvious way to do it might take time Θ(2^n) if the graph G has n vertices. We discussed a recursive method that runs in time O((1.3^n)). The best known methods run in time O((2.2^n)/3). Finding out if a graph is k-colorable is a hard computational problem. The most obvious way to do it takes time Θ(k^n), if G has n vertices. We discussed a recursive method that runs in time O((1.62^n) if G has n vertices and E edges. One recently developed method ** runs in time O(((4/3)^n)). We will see in section 5.7 that this problem can be done in an average time that is O(1) for fixed k**. Multiplying two matrices is an easy computational problem. The most obvious way to do it takes time Θ(n^3) if the matrices are n × n. We discussed a recursive method that runs in time O((n^(2.8)). A recent method ** runs in time O(n) for some γ < 2.5. ** E. Lawler, A note on the complexity of the chromatic number problem, Information Processing Letters 5 (1976), 66-7. ** D. Coppersmith and S. Winograd, On the asymptotic complexity of matrix multiplication, SIAM J. Comp. 11 (1980), 472-492. ``` #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 63 Context: - **Content Analysis**: 1. **Title and Sections**: - The document includes discussions on advanced computational arithmetic, primarily focusing on fast multiplication methods for large integers and the Fast Fourier Transform (FFT). - There is a review section summarizing algorithms discussed in the chapter. 2. **Mathematical Context**: - The text is heavily mathematical, dealing with concepts like polynomial multiplication, FFT, and computational complexity. - Specific problems and exercises are given that involve roots of unity, polynomial evaluation, and proof-writing. 3. **Algorithmic Overview**: - The review section covers various computational problems and their complexities, including sorting, finding a maximum independent set in a graph, graph coloring, and matrix multiplication. - It mentions both the naive (brute-force) and advanced (recursive or optimized) methods for each problem. 4. **References**: - References to academic works by E. Lawler and D. Coppersmith and S. Winograd indicate the scholarly nature of the document related to computational complexity. #### 9. **Perspective and Composition**: - **Perspective**: The image appears to present the content from a medium or standard top-down view, typical of scanning or photographing a document page. - **Composition**: The elements consist primarily of text sections, equations, and a list of exercises. The content is structured into paragraphs with equations and exercise items appropriately spaced. ### Conclusions The provided image represents a page from an academic textbook or document related to computational mathematics, specifically dealing with polynomial arithmetic, FFT, and algorithmic complexity. The text includes detailed exercises aimed at deepening the understanding of the discussed methods and algorithms, providing both theoretical insights and practical programming challenges. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 73 Context: - **Significance**: - The text provides a detailed explanation of the Ford-Fulkerson algorithm and the max-flow min-cut theorem, crucial for understanding network flow optimization. - Ford-Fulkerson is an iterative approach used to find the maximum flow in a network, whereas the max-flow min-cut theorem establishes a relationship between maximum flow and minimum cut in a network. #### 9. Perspective and Composition - **Perspective**: The image is taken from a direct, straight-on perspective, typical of a scanned or photographed document page. - **Composition**: The text is organized in typical document format with headings, code blocks, and explanatory paragraphs, structured to guide readers through the algorithm and theorem step-by-step. #### 10. Contextual Significance - **Overall Document/Website Context**: This image is likely part of an academic or technical document on algorithms, specifically focusing on network flow problems. - **Contribution to Overall Message/Theme**: This part of the document contributes essential details about the Ford-Fulkerson algorithm and the max-flow min-cut theorem, foundational concepts in network flow theory used in operations research and computer science. #### 11. Metadata Analysis - **Capture Date and Camera Settings**: No metadata is available to analyze from the given visual content. ### Conclusion The provided image details critical algorithmic procedures and theoretical foundations essential in understanding network optimization, emphasizing the Ford-Fulkerson method and the max-flow min-cut theorem. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 74 Context: This proves the \( = \) part of (3.4.1), and the \( \leq \) part is obvious, completing the proof of lemma 3.4.1. We now know that the maximum value of the flow in a network cannot exceed the minimum of the capacities of the cuts in the network. To complete the proof of the theorem we will show that a flow of maximum value, which surely exists, must saturate the edges of some cut. **3.5 The Complexity of the Ford-Fulkerson Algorithm** The algorithm of Ford and Fulkerson terminates if and when it arrives at a stage where the sink is not labeled but no more vertices can be labeled. If at that time we let \( W \) be the set of vertices that have been labeled, then we have seen that \( (W, \overline{W}) \) is a minimum cut of the network, and the present value of the flow is the desired maximum for the network. The question now is, how long does it take to arrive at that stage, and indeed, is it guaranteed that we will ever get there? We are asking if the algorithm is finite, surely the most primitive complexity question imaginable. First consider the case where every edge of the given network \( X \) has integer capacity. Then during the labeling and flow augmentation algorithms, various additions and subtractions are done, but there is no way that any nonintegral flows can be produced. It follows that the augmented flow is still integral. The value of the flow is increased by an integer amount during each augmentation. On the other hand if, say, \( C^* \) denotes the combined capacity of all edges that are outbound from the source, then it is eminently clear that the value of the flow can never exceed \( C^* \). Since the value of the flow increases by at least 1 unit per augmentation, we see that no more than \( C^* \) flow augmentations will be needed before a maximum flow is reached. This yields ... **Significance:** The text describes procedural and theoretical aspects of network flow problems, specifically dealing with the Ford-Fulkerson algorithm and its properties. This excerpt likely serves an educational purpose in teaching the mathematical reasoning and algorithms used in network flow theory. #### Color Analysis: - The page background is primarily white. - The text color is black, with some mathematical expressions clearly marked by specific notation styles in italics and bold. #### Perspective and Composition: - **Perspective:** The image appears to be a direct screenshot of a page, with a clear bird's-eye view. - **Composition:** The page consists of multiple paragraphs of continuous text, segmented into sections with different topics. Mathematical expressions and proofs are embedded within the text, providing visual contrast and separating different types of content. ### Conclusion The image is of a textbook page detailing lemma 3.4.1 and a proof, along with a discussion on the complexity of the Ford-Fulkerson algorithm. The page aims to provide a theoretical foundation and algorithmic understanding within the context of network flow problems. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 75 Context: # 3.5 Complexity of the Ford-Fulkerson algorithm ## Theorem 3.5.1 In a network with integer capacities on all edges, the Ford-Fulkerson algorithm terminates after a finite number of steps with a flow of maximum value. This is good news and bad news. The good news is that the algorithm is finite. The bad news is that the complexity estimate that we have proved depends not only on the numbers of edges and vertices in \( X \) but on the edge capacities. If the bound \( C^* \) represents the true behavior of the algorithm, rather than some weakness in our analysis of the algorithm, then even on very small networks it will be possible to assign edge capacities so that the algorithm takes a very long time to run. And it is possible to do that. We will show below an example due to Ford and Fulkerson in which the situation is even worse than the one envisaged above: not only will the algorithm take a very long time to run; it won't converge at all! Consider the network \( X \) that is shown in Fig. 3.5.1. It has 10 vertices \( s, t, v_1, \ldots, v_4 \). There are directed edges \( (x_i, x_j) \neq (y_i, y_j) \) for \( i, j = 1, 2, 3, 4 \). In this network, the four edges \( A_i = (v_i, y_i) \) (for \( i = 1, 4 \)) will be called the special edges. Next we will give the capacities of the edges of \( X \). Write \( r = (-1 + \sqrt{5})/2 \), and let \[ S = (3 + \sqrt{5})/2 = \sum_{n=0}^{\infty} r^n. \] Then to every edge of \( X \) except the four special edges, we assign the capacity \( S \). The special edges \( A_1, A_2, A_3, A_4 \) are given capacities \( 1, r^1, r^2 \) respectively (you can see that this is going to be interesting). Suppose, for our first augmentation step, we find the flow augmenting path \( s \rightarrow v_1 \rightarrow t \), and that we augment the flow by 1 unit along that path. The four special edges will then have residual capacities (excesses of capacity over flow) of \( 0, r^2, r^3, r^4 \), respectively. Inductively, suppose we have arrived at a stage of the algorithm where the four special edges, taken in some rearrangement \( A_1', A_2', A_3', A_4' \), have residual capacities \( 0, r^{n-1}, r^n \). We will now show that the algorithm might next do two flow augmentation steps the next result of which would be that the inductive state of affairs would again hold, with \( n \) replaced by \( n + 1 \). Indeed, choose the flow augmenting path \[ s \rightarrow v_2 \rightarrow v_3 \rightarrow v_4 \rightarrow t. \] Image Analysis: ### Comprehensive Examination of the Visual Content: #### Image 1: 1. **Localization and Attribution:** - The visual content consists of a single page with both textual and graphical elements. The image contains some mathematical descriptions and an illustrative diagram. Let's refer to the diagram as "Image 1." 2. **Object Detection and Classification:** - Diagrams: A single network flow diagram with nodes and directed edges. - Text: Theorem description, mathematical formulas, and an explanation pertaining to the Ford-Fulkerson algorithm. 3. **Scene and Activity Analysis:** - The image depicts a scene from a mathematical or algorithmic textbook explaining the complexity of the Ford-Fulkerson algorithm. The diagram specifically illustrates a network flow problem, demonstrating how to give the algorithm a hard time by using specific edge weights and capacities. 4. **Text Analysis:** - Theorem 3.5.1: "In a network with integer capacities on all edges, the Ford-Fulkerson algorithm terminates after a finite number of steps with a flow of maximum value." - A description follows, explaining the good and bad news about the algorithm and providing an example to show the potential issues that could cause long run times or failure to converge. - The network X is described with vertices and edges, specifically identifying the capacities and the augmentation paths used. 5. **Diagram and Chart Analysis:** - The diagram (Figure 3.5.1) illustrates a network with vertices (labeled 𝓋₁ to 𝓋₁₀) and directed edges. - The diagram shows a complex interconnected network with nodes and edges, each edge having a capacity. - The special edges A₁, A₂, A₃, A₄ are highlighted, showing specific flows and capacities. 6. **Contextual Significance:** - The image is likely a page from an academic textbook or research paper on algorithms, specifically focusing on network flow algorithms. The explanation and diagram aim to provide insight into the complexity and potential pitfalls of using the Ford-Fulkerson algorithm under certain conditions. - The example given suggests an in-depth discussion on algorithm performance and edge cases important for understanding theoretical and practical implications. 7. **Color Analysis:** - The image is in black and white, typical for academic prints. Contrast helps in distinguishing between text and the graphical components. 8. **Perspective and Composition:** - Perspective: Straight-on view, typical for textbook pages to facilitate reading and comprehension. - Composition: The top part of the image consists of text explaining the theorem and its implications. The central part contains the diagram (Figure 3.5.1), which visually supports the textual explanation. The bottom part provides a step-by-step explanation of the process outlined in the diagram. Given the detailed textual explanations and the accompanying network diagram, the reader is provided with a comprehensive understanding of the limitations and potential challenges faced by the Ford-Fulkerson algorithm in specific scenarios involving edge capacities and augmentation steps. The exercise is intended to elucidate the complexity and theoretical depth of maximum flow problems in network flow algorithms. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 87 Context: # 4.2 The Greatest Common Divisor If we use the customary abbreviation \( a \mod m \) for \( r \), the remainder in the division of \( a \) by \( m \), then what we have shown is that \[ \text{gcd}(m, n) = \text{gcd}(n, m \mod n) \] This leads to the following recursive procedure for computing the g.c.d. ```plaintext function gcd(n, m); {finds gcd of given nonnegative integers n and m} if m = 0 then gcd := n else gcd := gcd(m, n mod m) end. ``` The above is the famous "Euclidean algorithm" for the g.c.d. It is one of the oldest algorithms known. The reader is invited to write the Euclidean algorithm as a recursive program, and get it working on some computer. Use a recursive language, write the program more or less as above, and try it out with some large, healthy integers \( n \) and \( m \). The gcd program exhibits all of the symptoms of recursion. It calls itself with smaller values of its variable list. It begins with 'if trivially' then to 'trivially' (if \( m = 0 \)), and this case is all-important because it’s the only way the procedure can stop itself. If, for example, we want the g.c.d. of 13 and 21, we call the program with \( n = 13 \) and \( m = 21 \), and then recursively calls itself with the following arguments: ``` (21, 13), (13, 8), (8, 5), (5, 3), (3, 2), (2, 1), (1, 0) ``` When it arrives at a call in which the 'm' is \( 0 \), then the 'n', namely \( 1 \) in this case, is the desired g.c.d. What is the input to the problem? The two integers \( n, m \) whose g.c.d. we want are the input, and the number of bits that are needed to input those two integers is \( \Theta(\log n) + \Theta(\log m) \). Hence \( \log m \) then \( \log n \) is the length of the input bit string. Now let’s see how long the algorithm might run with an input string of that length. To measure the running time of the algorithm we need first to choose a unit of cost or work. Let’s agree that one unit of labor is the execution of a single \( \text{mod} \) operation. In this problem, an equivalent measure of cost would be the number of times the algorithm calls itself recursively. In the example (4.22) the cost was 7 units. ## Lemma 4.2.1 If \( 1 \leq b \leq a \) then \( a \mod b \leq (a - 1)/2 \). **Proof**: Clearly \( a \mod b \leq b - 1 \). Further, \[ a \mod b = a - \left\lfloor \frac{a}{b} \right\rfloor b \] Thus \( a \mod b < \min(a - b, b - 1) \). Now we distinguish two cases. First suppose \( b \geq (a + 1)/2 \). Then \( b - 1 \leq a - b \) so \[ a \mod b \leq b - 1 \leq \frac{a + 1}{2} - 1 = \frac{a - 1}{2} \] in this case. Next, suppose \( b < (a + 1)/2 \). Then \( a - b \leq a - \frac{a + 1}{2} = \frac{a - 1}{2} \), \[ a \mod b \leq a - b < a - \frac{a + 1}{2} = \frac{a - 1}{2} \] so the result holds in either case. *In Historia Mathematica 21 (1994), 401-419, Jeffrey Shallit traces this analysis back to Pierre-Joseph-Étienne Picard, in 1841.* Image Analysis: ### Comprehensive Examination of the Attached Visual Content #### 1. Localization and Attribution: - **Image 1**: The entire page is occupied by a single image. #### 4. Text Analysis: - **Detected Text**: - **Section Header**: "4.2 The greatest common divisor" - **Main Body Text**: - "If we use the customary abbreviation 'a mod m' for r, the remainder in the division of a by m, then what we have shown is that gcd(m, n) = gcd(n, m mod n)..." - Detailed algorithm and explanation on how to compute the greatest common divisor (g.c.d) using the Euclidean algorithm. - "**Lemma 4.2.1.** If 1 ≤ b ≤ a then a mod b ≤ (a − 1)/2." - "In Historia Mathematica 21 (1994), 401-419, Jeffrey Shallit traces this analysis back to Pierre-Joseph-Étienne Finck, in 1841." - **Analysis of Text Content**: - The text describes the Euclidean algorithm for finding the greatest common divisor (g.c.d) of two integers. It indicates the historical significance of the algorithm, recognizes its origin, and provides a detailed recursive procedural description along with an example. - There is a proof provided for a lemma regarding properties of the modulo operation. #### 9. Perspective and Composition: - **Perspective**: The image of the text appears to be a high-angle shot that captures the full page of a printed or digital document. - **Composition**: The text is structured in a clear academic format with a header, large body of text, algorithms in a different font or boxed, and mathematical notations and formulas presented clearly for readability. #### 13. Graph Numbers: - **Mathematical Algorithm**: - The given sequence for computing g.c.d.: - Initial sequence: (21,13), (13,8), (8,5), (5,3), (3,2), (2,1), (1,0) #### Additional Aspects: ##### Ablaufprozesse (Process Flows): - **Described Process**: The Euclidean algorithm for finding the g.c.d, which is recursively structured and continues until it reaches a base case where one of the integers becomes zero. ##### Prozessbeschreibungen (Process Descriptions): - **Detailed Description**: The Euclidean algorithm described involves a recursive function that repeatedly applies the modulus operation until the remainder is zero, thereby identifying the greatest common divisor. It begins by comparing inputs m and n, if m is not zero, assigns remainder of m divided by n to a recursive call, otherwise returns n as g.c.d. ##### Typen Bezeichnung (Type Designations): - **Types/Categories**: - **Mathematical Notations**: Symbols and operations related to the Euclidean algorithm. - **Mathematical Proofs**: Including steps and conditional checks provided in case scenarios. ### Summary The document is highly academic, detailing a historical and algorithmic approach to finding the greatest common divisor using the Euclidean algorithm. It provides both the procedural method and theoretical underpinnings, complete with example calculations and proofs. The page captures the structured logic and mathematical rigor typical of mathematical textbooks or research papers. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 88 Context: # Chapter 4: Algorithms in the Theory of Numbers ## Theorem 4.2.1 (A worst-case complexity bound for the Euclidean algorithm) Given two positive integers \(a, b\), the Euclidean algorithm will find their greatest common divisor after a cost of at most \(2 \log_2 M + 1\) integer divisions, where \(M = \max(a, b)\). Before we prove the theorem, let’s return to the example \((a, b) = (13, 21)\) of the display (4.2.2). In that case \(M = 21\) and \(20 \log_2 M + 1 = 9.78\). The theorem asserts that the g.c.d. will be found after about 9 operations. In fact, it was found after 7 operations in that case. ### Proof of theorem: Suppose first that \(a \geq b\). The algorithm generates a sequence \(a_0, a_1, \ldots\) where \(a_0 = a\), \(a_1 = b\), and \(a_{j+1} = a_j \mod a_{j-1}\) \((j \geq 1)\). By lemma 4.2.1, \[ a_{j+1} \leq \frac{a_{j-1} - 1}{2} \leq \frac{a_j}{2}. \] Then, by induction on \(j\), it follows that \[ a_{2j} \leq \frac{4n}{2^j} \quad (j \geq 0) \] \[ a_{2j+1} \leq \frac{a_j}{2} \quad (j \geq 0) \] and so, \[ a_j \leq 2^{-|r/2|}M \quad (r = 0, 1, 2, \ldots). \] Obviously, the algorithm has terminated if \(a < 1\), and this will have happened when \(r\) is large enough so that \(2^{-|r/2|}M < 1\), i.e., if \(r > 2 \log_2 M\). If \(r < 4\) then after 1 operation we will be in the case \(a < b\) that we have just discussed, and the proof is complete. The upper bound in the statement of theorem 4.2.1 can be visualized as follows. The number \(log_2 M\) is almost exactly the number of bits in the binary representation of \(M\) (what is ‘exactly’ that number of bits?). Theorem 4.2.1 therefore asserts that we can find the g.c.d. of two integers in a number of operations that is at most a linear function of the number of bits that it takes to represent the two numbers. In brief, we might say that \( \text{Time} = O(\text{bits})\); in the case of Euclid’s algorithm. ## Exercises for section 4.2 1. Write a nonrecursive program, in Basic or Fortran, for the g.c.d. Write a recursive program, in Pascal or a recursive language of your choice, for the g.c.d. 2. Choose 1000 pairs of integers \((n, m)\), at random between 1 and 1000. For each pair, compute the g.c.d. using a recursive program and a nonrecursive program. (a) Compare the execution times of the two programs. (b) There is a theorem to the effect that the probability that two random integers \(\geq 1\) is \((6/\pi^2)\). What, precisely, do you think that this theorem means by 'the probability that two'? What percentage of the 1000 pairs that you chose had g.c.d. = 1? Compare your observed percentage with \(100 - (6/\pi^2)\). 3. Find out when Euclid lived, and with exactly what words he described his algorithm. 4. Write a program that will light up a pixel in row \(r\) and column \(c\) of your CRT display if and only if \(gcd(n, m) = 1\). Run the program with enough values of \(n\) and \(m\) to fill your screen. If you see any interesting visual patterns, try to explain them mathematically. 5. Show that if \(m\) and \(n\) have a total of \(b\) bits, then Euclid's algorithm will not need more than \(2B + 3\) operations before reaching termination. 84 Image Analysis: 1. **Localization and Attribution:** - This document is a single-page text. - The text is divided into a theorem, its proof, and exercises related to Euclid's algorithm. 2. **Text Analysis:** - **Title and Section:** "Chapter 4: Algorithms in the Theory of Numbers" - **Theorem 4.2.1:** Discusses the worst-case complexity bound for the Euclidean algorithm, defining the number of steps needed to find the greatest common divisor (g.c.d) of two integers \( a \) and \( b \). - **Proof of Theorem 4.2.1:** Provides a mathematical proof, including lemmas and inductive reasoning. - **Exercises for section 4.2**: Lists exercises intended to deepen understanding of the Euclidean algorithm through different programming languages, mathematical proofs, and historical context. 3. **Diagram and Chart Analysis:** - No diagrams or charts are presented in this text. 4. **Anomaly Detection:** - No anomalies or unusual elements are identified in this text. 5. **Color Analysis:** - The document is in black and white, typical for mathematical proofs and educational content. The absence of color emphasizes the text. 6. **Perspective and Composition:** - Perspective is a standard viewer's perspective of reading a document. - The composition is well-organized with clear sections for the theorem, proof, and exercises. 7. **Process Descriptions:** - Euclidean Algorithm: The steps for finding the g.c.d. are systematically proved, showcasing the sequence and induction to establish the theorem's validity. - Steps involve iterative division and comparison to reduce the problem space until the g.c.d is found. 8. **Typen Bezeichnung (Type Designations):** - Euclidean algorithm – a specific type of algorithm for computing the greatest common divisor. 9. **Trend and Interpretation:** - The exercises suggest a trend towards understanding the efficiency and implementation of the Euclidean algorithm in both recursive and non-recursive programming methods. There is also an emphasis on exploring patterns and probabilities related to the algorithm. 10. **Tables:** - No tables are included in the text. **Summary:** This document is a mathematical exposition on the efficiency of the Euclidean algorithm, complete with a theorem, its detailed proof, and practical exercises for enhanced understanding. The text is methodically structured, adhering to educative norms tailored to students or professionals in the field of number theory and algorithm design. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 89 Context: ``` ## 4.3 The extended Euclidean algorithm 6. Suppose we have two positive integers \( m, n \), and we have factored them completely into primes, in the form \[ m = \prod_{i=1}^{k} p_i^{\alpha_i}; \quad n = \prod_{j=1}^{l} q_j^{\beta_j} \] How would you calculate \( \gcd(m,n) \) from the above information? How would you calculate the least common multiple (\(\mathrm{lcm}\)) of \( m \) and \( n \) from the above information? Prove that \( \gcd(m,n) = \frac{m}{\mathrm{lcm}(m,n)} \). 7. Calculate \( \gcd(102431, 56129) \) in two ways: use the method of exercise 6 above, then use the Euclidean algorithm. In each case count the total number of arithmetic operations that you had to do to get the answer. 8. Let \( F_n \) be the \( n \)th Fibonacci number. How many operations will be needed to compute \( \gcd(F_n, F_{n-1}) \) by the Euclidean algorithm? What is \( \gcd(F_n, F_{n-1})? \) ### 4.3 The extended Euclidean algorithm Again suppose \( n, m \) are two positive integers whose g.c.d. is \( g \). Then we can always write \( g \) in the form \[ g = t n + u m \quad (4.3.1) \] where \( t \) and \( u \) are integers. For instance, \( \gcd(14,11) = 1 \), so we can write \( 1 = 14(-1) + 11(2) \) for integers \( t, u \). Can you spot integers \( t, u \) that will work? One pair that does the job is \( (-4, 5) \), and there are others (can you find all of them?). The extended Euclidean algorithm finds not only the g.c.d. of \( n \) and \( m \), it also finds a pair of integers \( t \) that exists (4.3.1). One ‘application’ of the extended algorithm is that we will obtain an inductive proof of the existence of \( t \), that is not immediately obvious from (4.3.1) (see exercise 1 below). While this hardly rates as a ‘practical’ application, it represents a very important feature of recursive algorithms. We might say, rather generally, that the following items go hand-in-hand: - Recursive algorithms - Inductive proofs - Complexity analyses by recurrence formulas If we have a recursive algorithm, then it is natural to prove the validity of the algorithm by mathematical induction. Conversely, inductive proofs of theorems often (not always, alas!) yield recursive algorithms for the construction of the objects that are being studied. The complexity analysis of a recursive algorithm will use recurrence formulas, in a natural way. We saw that already in the analysis that proved theorem 4.2.1. Now let’s discuss the extended algorithm. Input it will be two integers \( a \) and \( b \). Output from it will be \( g = \gcd(a,m) \) and two integers \( t \) and \( u \) for which (4.3.1) is true. Let \( m \) be one step of the original Euclidean algorithm look to us from the problem of finding \( \gcd(n, m) \): \[ \gcd(n \mod m) \] Suppose, inductively, that we not only know \( g = \gcd(n,m) \), but we also know the coefficients \( t', u' \) for the equation \[ g = t' n + u' m \quad (4.3.2) \] Can we get out, at the next step, the corresponding coefficients \( t, u \) for (4.3.1)? Indeed we can, by substituting (4.3.2) the fact that \[ n \mod m = n - \left\lfloor \frac{n}{m} \right\rfloor m \quad (4.3.3) \] we find that \[ g = t' m + t' \left( n - \left\lfloor \frac{n}{m} \right\rfloor m \right) \quad (4.3.4) \] \[ = v'n + (t' - v')\left\lfloor \frac{n}{m} \right\rfloor m \] Hence the rule by which \( t', u' \) for equation (4.3.2) transform into \( t, u \) for equation (4.3.1) is that \[ t = u' \quad u = t' - \left\lfloor \frac{n}{m} \right\rfloor u' \] ``` Image Analysis: **1. Localization and Attribution:** - There is only one image to analyze. - Image Number: Image 1 **4. Text Analysis:** - The text in the image is extracted in the following sections: ``` 4.3 The extended Euclidean algorithm 6. Suppose we have two positive integers m, n, and we have factored them completely into primes, in the form m = ∏ pᵢᵃⁱ ; n = ∏ pᵢᵇᵢ. How would you calculate gcd(m,n) from the above information? How would you calculate the least common multiple (lcm) of m and n from the above information? Prove that gcd(m,n) = mn/lcm(m, n). 7. Calculate gcd(102131, 56129) in two ways: use the method of exercise 6 above, then use the Euclidean algorithm. In each case count the total number of arithmetic operations that you had to do to get the answer. 8. Let Fₙ be the nᵗʰ Fibonacci number. How many operations will be needed to compute gcd(Fₙ, Fₙ₋₁) by the Euclidean algorithm? What is gcd(Fₙ, Fₙ₋₁)? 4.3 The extended Euclidean algorithm Again suppose m, n are two positive integers whose g.c.d. is g. Then we can always write g in the form g = tn + um (4.3.1) where t and u are integers. For instance, gcd(14,11) = 1, so we can write 1 = 1(4) + 1(1)u for integers t, u. Can you spot integers t, u that will work? One pair that does the job is (4, -5) and there are others (can you find all of them?). The extended Euclidean algorithm finds not only the g.c.d. of m and n, it also finds a pair of integers t, u that exists (4.3.1). One ‘application’ of the extended algorithm is that we will obtain an inductive proof of the existence of t, u that is not immediately obvious from (4.3.1) (see exercise 1 below). While this hardly rates as a ‘practical’ application, it represents a very important feature of recursive algorithms. We might say, rather generally, that the following items go hand-in-hand: Recursive algorithms Inductive proofs Complexity analyses by recurrence formulas If we have a recursive algorithm, then it is natural to prove the validity of the algorithm by mathematical induction. Conversely, inductive proofs of theorems often (not always, alas!) yield recursive algorithms for the construction of the objects that are being studied. The complexity analysis of a recursive algorithm will use recurrence formulas, in a natural way. We saw that already in the analysis that proved theorem 4.2.1. Now let’s discuss the extended algorithm. Input to it will be two integers n and m. Output from it will be g = gcd(n,m) and two integers t and u for which (4.3.1) is true. g₀ is a simple step of the original Euclidean algorithm that looks from the problem of finding gcd(n,m) to finding gcd(n, n mod m). Suppose, inductively, that we not only know g = gcd(n,m, n mod m) but we also know the coefficients t, uʹ for the equation g = tʹn + uʹ(n mod m). (4.3.2) #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 89 Context: ``` ## 4.3 The extended Euclidean algorithm 6. Suppose we have two positive integers \( m \) and \( n \), and we have factored them completely into primes, in the form \[ m = \prod p_i^{\alpha_i}; \quad n = \prod p_i^{\beta_i} \] How would you calculate \( \gcd(m,n) \) from the above information? How would you calculate the least common multiple \( \mathrm{lcm}(m) \) of \( m \) and \( n \) from the above information? Prove that \( \gcd(m,n) = \prod p_i^{\min(\alpha_i, \beta_i)} \). 7. Calculate \( \gcd(1021, 56129) \) in two ways: use the method of exercise 6 above, then use the Euclidean algorithm. In each case count the total number of arithmetic operations that you had to do to get the answer. 8. Let \( F_n \) be the \( n \)-th Fibonacci number. How many operations will be needed to compute \( \gcd(F_n, F_{n-1}) \) by the Euclidean algorithm? What is \( \gcd(F_n, F_{n-1})? \) ### 4.3 The extended Euclidean algorithm Again suppose \( m, n \) are two positive integers whose \( \gcd \) is \( g \). Then we can always write \( g \) in the form \[ g = t n + u m \tag{4.3.1} \] where \( t \) and \( u \) are integers. For instance, \( \gcd(14, 11) = 1 \), so we can write \( 1 = 14(-1) + 11(2) \) for integers \( t, u \) that will work. One pair that does the job is \((-1, 2)\) and there are others (can you find all of them?). The extended Euclidean algorithm finds not only the \( \gcd \) of \( m \) and \( n \), it also finds a pair of integers \( t \) that exists (4.3.1). One 'application' of the extended algorithm is that we will obtain an inductive proof of the existence of \( t \) that is not immediately obvious from (4.3.1) (see exercise 1 below). While this hardly rates as a 'practical' application, it represents a very important feature of recursive algorithms. We might say, rather generally, that the following items go hand-in-hand: #### Recursive algorithms #### Inductive proofs #### Complexity analyses by recurrence formulas If we have a recursive algorithm, then it is natural to prove the validity of the algorithm by mathematical induction. Conversely, inductive proofs of theorems often (not always, alas!) yield recursive algorithms for the construction of the objects that are being studied. The complexity analysis of a recursive algorithm will use recurrence formulas, in a natural way. We saw that already in the analysis that proved theorem 4.2.1. Now let's discuss the extended algorithm. Input to it will be two integers \( m \) and \( n \). Output from it will be \( g = \gcd(m,n) \) and two integers \( t \) and \( u \) for which (4.3.1) is true. To get a single step of the original Euclidean algorithm to look like from the problem of finding \( \gcd(m,n) = \gcd(n, m \mod n) \). Suppose, inductively, that we not only know \( g = \gcd(m,n) \) but we also know the coefficients \( t', u' \) for the equation \[ g = l'm + u'(n \mod m). \tag{4.3.2} \] Can we get out, at the next step, the corresponding coefficients \( t, u \) for (4.3.1)? Indeed we can, by substituting \( n \mod m \): \[ n \mod m = n - \left\lfloor \frac{n}{m} \right\rfloor m \tag{4.3.3} \] we find that \[ g = l'm + u'(n - \left\lfloor \frac{n}{m} \right\rfloor m) \tag{4.3.4} \] \[ = u' n + (l' - u' \left\lfloor \frac{n}{m} \right\rfloor) m. \tag{4.3.5} \] Hence the rule by which \( t', u' \) for equation (4.3.2) transform into \( t, u \) for equation (4.3.1) is that \[ t = u' \] \[ u = l' - u' \left\lfloor \frac{n}{m} \right\rfloor. \] ``` Image Analysis: ### Analysis of the Visual Content #### 1. Localization and Attribution: - The document page contains a single image, hence it is identified as **Image 1**. #### 2. Object Detection and Classification: - **Objects detected in Image 1**: - Text: Various paragraphs and equations. #### 3. Scene and Activity Analysis: - **Scene Description**: - The page depicts a mathematical text discussing various concepts related to the Extended Euclidean Algorithm. - The section of the page is numbered 4.3 and is titled "The extended Euclidean algorithm." - **Activities**: - Presentation of mathematical problems and solutions. - Introduction and explanation of recursive algorithms, inductive proofs, and complexity analyses through recurrence formulas. #### 4. Text Analysis: - **Text Extracted**: - The extracted text includes theoretical explanations, mathematical formulas, and instructive problems related to number theory and algorithms. - **Content Analysis**: - Problems 6, 7, and 8 discuss calculating mathematical properties like gcd (greatest common divisor) using factorization and other methods. - Section 4.3 titled "The extended Euclidean algorithm," explains how two positive integers’ gcd can be expressed using integers t and u. - It includes example cases (like gcd(14,11) = 1), and the transformation of equations to find these integers. - Recursive algorithms, inductive proofs, and complexity analyses are linked together with inductive methodologies discussed. - Equations (4.3.1) to (4.3.5) are used to explain and transform the problem of solving gcd into integers related problems. #### 8. Color Analysis: - **Color Composition**: - The image is primarily black and white, indicative of typical textbook formatting. - The background is white, and the text is black, ensuring high contrast for readability. #### 9. Perspective and Composition: - **Perspective**: - The image is taken from a straightforward, head-on perspective typical of scanned or photographed pages ensuring all content is displayed flat and uniformly. - **Composition**: - The composition follows a structured, linear format with clear headings, numbered sections, and consistent typography, making it easy to follow and comprehend. #### 10. Contextual Significance: - **Overall Document Context**: - The page appears to be from an academic textbook or a lecture note focused on mathematical algorithms, particularly number theory and algorithm analysis. - This page likely contributes to the overall educational theme by providing detailed discussions on the extended Euclidean algorithm and related mathematical concepts. - **Contribution to the Theme**: - It educates readers on advanced concepts of mathematical algorithms, particularly how the Euclidean algorithm extends and applies to various problems in number theory. #### 12. Graph and Trend Analysis: - **Graph Numbers and Trends**: - No graphs are present in the image, so this aspect does not apply. #### Additional Aspects: - **Prozessbeschreibungen (Process Descriptions)**: - The page describes processes such as calculating gcd through different methods and the step-by-step transformation of equations within the extended Euclidean algorithm context. Overall, the image is a rich mathematical text providing detailed explanations and problems related to the extended Euclidean algorithm, aiming to educate readers on advanced algorithmic concepts through clear examples and step-by-step guides. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 94 Context: #### **Image 1:** - **Perspective:** - The image is a direct scan of a textbook page, taken from a straight, birds-eye view. - **Composition:** - Organized in a standard scientific literature format. - Includes clear section headers, theorem statements in bold or italic, and detailed explanations. ### Contextual Significance: #### **Image 1:** - **Context:** - This image is likely part of a mathematical textbook aimed at discussing algorithms in the field of number theory. - The theorems and corollaries presented contribute crucially to the understanding of primitive roots and cyclic groups. - **Contribution to Overall Message:** - The image aims to provide foundational mathematical theory (primitive roots, cylic groups) with rigor and clarity. - Example calculations and theorems ensure a comprehensive understanding of the topics. ### Conclusion: This page from the textbook focuses on advanced number theory concepts. It introduces important theorems related to primitive roots and cyclic groups and illustrates these concepts with examples in \( Z_6 \). The clear, structured presentation of information fosters a deep understanding of these mathematical principles. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 95 Context: # 4.5 Interlude: the ring of integers modulo n ## Theorem 4.5.4 Let \( \pi = \prod_{j=1}^r p_j^{\alpha_j} \). The mapping which associates with each \( z \in \mathbb{Z} \), the r-tuple \( (z_1, z_2, \ldots, z_r) \), where \( z_i \equiv z \mod p_i^{\alpha_i} \) (for \( i = 1, \ldots, r \)), in which - \( A) \ z_i \in \mathbb{Z}_{p_i^{\alpha_i}} \) (for \( i = 1 \) and) - \( B) \ (z_1, z_2, \ldots, z_r) \) = \( (z_1 + y_1, z_2 + y_2) \) and - \( C) \ (z_1, z_2, \ldots, z_r) \) = \( (x_1, x_2, \ldots, x_r) \) - \( D) \ In (b), the \( i \)-th sign on the right side is the addition operation of \( \mathbb{Z}_{p_i^{\alpha_i}} \), for each \( i = 1, \ldots, r \). The proof of theorem 4.5.4 follows at once from the following: ## Theorem 4.5.5 (‘The Chinese Remainder Theorem’) Let \( m_i \) (for \( i = 1, \ldots, r \)) be pairwise relatively prime positive integers, and let \[ M = m_1 m_2 \cdots m_r. \] Then the mapping that associates with each integer \( x \) (where \( 0 \leq s < M - 1 \)) the r-tuple \( (b_1, b_2, \ldots, b_r) \), where \( b_i \equiv x \mod m_i \) (for \( i = 1, \ldots, r \)), is a bijection between \( \mathbb{Z}/M\mathbb{Z} \) and \( \mathbb{Z}/m_1\mathbb{Z} \times \cdots \times \mathbb{Z}/m_r\mathbb{Z} \). A good theorem deserves a good proof. An outstanding theorem deserves two proofs, at least, one existential, and one constructive. So here are one of each for the Chinese Remainder Theorem. ### Proof 1 We must show that each r-tuple \( (b_1, \ldots, b_r) \), such that \( 0 \leq k < m_i \) (for \( i = 1, \ldots, r \)) occurs exactly once. There are obviously \( M \) such vectors, and so it will be sufficient to show that each of them occurs at most once as the image of some \( x \). In the contrary case we would have \( x \) and \( x' \) both corresponding to \( (b_1, b_2, \ldots, b_r) \), say. But then \( x - x' \equiv 0 \mod m_i \) for the \( m_i \). Here \( x - x' < M \), hence \( x = x' \). ### Proof 2 Here’s how to compute a number \( x \) that satisfies the simultaneous congruences \( x \equiv b_i \mod m_i \) (for \( i = 1, \ldots, r \)). First, by the extended Euclidean algorithm we can quickly find \( u_i, v_i, \ldots, u_r \), such that \( t_k(M/m_j) + u_j m_j \equiv 1 \) for \( j = 1, \ldots, r \). Then we claim that the number \[ x = \sum_{j=1}^{r} b_j t_j(M/m_j) \] satisfies all of the given congruences. Indeed, for each \( k = 1, \ldots, r \) we have \[ x \equiv \sum_{j=1}^{r} b_j t_j(M/m_j) \equiv b_k \ (mod \ m_k) \] where the first congruence holds because each \( M/m_j \) (for \( j \neq k \)) is divisible by \( m_k \), and the second congruence follows since \( t_k(M/m_k) = 1 - u_k m_k \equiv 1 \mod m_k \), completing the second proof of the Chinese Remainder Theorem. Now the proof of theorem 4.5.4 follows easily, and is left as an exercise for the reader. The factorization that is described in detail in theorem 4.5.4 will be written symbolically as \[ \mathbb{Z}_n \cong \bigoplus_{i=1}^r \mathbb{Z}_{p_i^{\alpha_i}} \tag{4.5.2} \] The factorization (4.5.2) of the ring \( \mathbb{Z} \) induces a factorization \[ U_n \cong \bigoplus_{i=1}^r U_{p_i^{\alpha_i}} \tag{4.5.3} \] Image Analysis: ### Analysis of the Attached Visual Content #### 1. Localization and Attribution - The page consists of a single image with mathematical text. - This image will be referred to as **Image 1**. #### 4. Text Analysis - The page contains advanced mathematical text, likely from a textbook on number theory or abstract algebra. Here is the extracted text: **Theorem 4.5.4:** Let \( n = p_1^{r_1} p_2^{r_2} \dots p_k^{r_k} \). The mapping which associates with each \( z \in \mathbb{Z} \), the \( r \)-tuple \((z_1, z_2, \dots, z_r)\), where \( z_i \equiv z \mod p_i^{r_i} \) (\( i = 1, \dots, r \)), is a ring isomorphism of \( \mathbb{Z}_n \) with the ring of \( r \)-tuples \((z_1, z_2, \dots, z_r) \) in which (a) \( z_i \in \mathbb{Z}_{p_i^{r_i}} \) (\( i = 1, \ldots, r \)) and (b) \((z_1 + x_1, \dots, z_r + y_r) = (z_1 + y_1, \dots, z_r + y_r)\) and (c) \((z_1 \dots z_r) \cdot (y_1 \dots y_r) = (z_1 \cdot y_1, \dots, z_r \cdot y_r) \) (d) In (b), the ith + sign on the right side is the additive operation of \( Z_{p_i^{r_i}} \) and in (e) the ith * sign is the multiplicative operation of \( Z_{p_i^{r_i}} \), for each \( i = 1, \dots, r \). **Theorem 4.5.5 (‘The Chinese Remainder Theorem’):** Let \( m_i (\equiv 1, \ldots, r) \) be pairwise relatively prime positive integers, and let: \( M = m_1 m_2 \ldots m_r \) Then the mapping that associates with each integer \( x (0 \leq x \leq M-1) \), the r-tuple \((b_1, b_2, \dots, b_r)\), where \( b_i \equiv x \mod m_i)(i = 1, \ldots, r \), is a bijection between \( \mathbb{Z}_M \) and \( \mathbb{Z}_{m_1} \times \cdots \times \mathbb{Z}_{m_r} \). A good theorem deserves a good proof. An outstanding theorem deserves two proofs, at least one existential, and one constructive. So here are one of each for the Chinese Remainder Theorem. **Proof 1:** We must show that each r-tuple \((b_1, \ldots, b_r)\) such that \( 0 \leq b_i < m_i (i = 1, \dots, r)\) occurs exactly once. There are obviously M such vectors and so it will be sufficient to show that each of them occurs at most once as the image of some x. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 95 Context: In the contrary case we would have x and x' both corresponding to (b1, b2, ..., br) say. But then \( x - x'\) ≡ 0 (mod each of the mi). Hence \( x - x'\) is divisible by \( M = m_1 m_2 \cdots m_r \). But \( | x' | ≤ M \), hence \( x = x' \). **Proof 2:** Here’s how to compute a number x that satisfies the simultaneous congruences \( x ≡ b_i \mod m_i (i = 1, r)\). First, by the extended Euclidean algorithm we can quickly find \( t_1, t_2, ..., t_r\) all \( x_i t_i mod m_i | by(i) = 1\) for j - 1 per t. For each \( x = b_1 + ... + I^(t_1, ..., t_n)\). we always have, \( x = \sum_{j=1}^r b_j (M/m_j)y_j (\mod M) \) where the first congruence holds because each \( M / m_j (j \neq k) \) is divisible by \( m_j \) and the second congruence follows since \( t_k (M/ m_k) + 1 = t_u, t_i \equiv m_i (\mod m_k)\) completing the second proof of the Chinese Remainder Theorem. Now the proof of theorem 4.5.4 follows easily, and is left as an exercise for the reader. The factorization that is described in detail in theorem 4.5.4 will be written symbolicially as: \( \mathbb{Z}_n = \bigotimes \\ \mathbb{Z}_{p_i^{r_i}}.\\) (4.5.2) The factorization (4.5.2) of the ring \( \mathbb{Z}_2 \) induces a factorization \( U, \bigotimes \\ U\\U \\ r^{-1}.) #### 9. Perspective and Composition - **Perspective:** The perspective is straightforward, presenting a flat, text-heavy image, typical of scanned book pages. - **Composition:** The composition includes standard formatting for a mathematical text, with theorems, proofs, and equations clearly divided into sections. #### 12. Graph and Trend Analysis - There are no graphs, charts, or data trends presented in the image. #### Conclusion - The content of the image is mathematical, focused on ring theory, specifically addressing isomorphisms and the Chinese Remainder Theorem. It includes theorems, proofs, and equations typical of an advanced mathematical text. - The context suggests an educational or reference document, likely part of a mathematical textbook, aiding in understanding complex algebraic structures. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 97 Context: **Typen Bezeichnung (Type Designations):** - Carmichael numbers: Specific composite numbers that behave like primes under certain tests are highlighted. ### Summary The document analyzed focuses on pseudoprimality tests used to determine if a number is composite or prime. The presented tests involve modular arithmetic and mathematical proofs, making it clear and comprehensive for those studying number theory. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 97 Context: # 4.6 Pseudorandomity tests ## Test 2 Given \( n \). Output \( n \) is composite if \( \gcd(b, n) \neq 1 \), else output ‘inconclusive.’ This one is a little better, but not yet good. If it is composite, the number of bases \( b \leq k \) for which Test 2 will produce the result ‘composite’ is \( n - \phi(n) \), where \(\phi\) is the Euler totient function, of (4.1.5). This number of useful bases will be large if \( n \) has some small prime factors, but in that case it’s easy to find out that \( n \) is composite by other methods. If \( n \) has only a few large prime factors, say if \( n = p^k \), then the proportion of useful bases is very small, and we have the same kind of inefficiency as in Test 1 above. Now we can state the third pseudorandomity test. ## Test 3 Given \( n \). (If \( b \) and \( n \) are not relatively prime or) if \( n \equiv 1 \,(\text{mod}\, n) \) then output \( n \) is composite; else output ‘inconclusive.’ Regrettably, the test is still not ‘good,’ but it’s a lot better than its predecessors. To cite an extreme case of its un-goodness, there exist composite numbers \( n \), called Carmichael numbers, with the property that the pair \( (b, n) \) produces the output ‘inconclusive’ for every integer \( b \) that is relatively prime to \( n \). An example of such a number is \( n = 1729 \), which is composite (\( 1729 = 17 \cdot 101 \)), but for which Test 3 gives the result ‘inconclusive’ on every integer \( b < 1729 \) that is relatively prime to \( 1729 \) (i.e., that is not divisible by \( 7 \) or \( 13 \) or \( 19 \)). Despite such misbehavior, the test usually seems to perform quite well. When \( n = 169 \) (a difficult integer for Tests 1 and 2) it turns out that there are \( 158 \) different \( b \in [1, 168] \) that produce the ‘composite’ outcome from Test 3, namely every such \( b \) except for \( 19, 22, 23, 70, 80, 89, 96, 147, 150, 168 \). Finally, we will describe a good pseudorandomity test. The familial resemblance to Test 3 will be apparent. ## Test 4 (the strong pseudorandomity test) Given \( (n, k) \). Let \( n = 1 \cdot 2^m \), where \( m \) is an odd integer. If either - (a) \( b^m \equiv 1 \,(\text{mod}\, n) \) or - (b) there is an integer \( i \in [0, q - 1] \) such that \( b^{n^2} \equiv -1 \,(\text{mod}\, n) \) then return ‘inconclusive’; else return ‘is composite.’ First we validate the test by proving the Proposition. ### Proposition If the test returns the message \( n \) is composite, then \( n \) is composite. **Proof:** Suppose not. Then \( n \) is an odd prime. We claim that \[ b^{m^2} \equiv 1 \,(\text{mod}\, n) \] for all \( i = q, q - 1, \ldots, 0 \). If so then the case \( i = 0 \) will contradict the outcome of the test, and thereby complete the proof. To establish the claim, it is clearly true when \( i = q \) by Fermat’s theorem. If true for \( i \), then it is true for \( i - 1 \), because \[ (b^{m^{i-1}})^2 \equiv b^{m^2} \equiv 1 \,(\text{mod}\, n) \] implies that the quantity being squared is \( +1 \) or \( -1 \). Since \( n \) is an odd prime, by corollary 4.5.3 \( U_n \) is cyclic, and so the equation \( x^2 \equiv 1 \,(\text{mod}\, n) \) has only the solutions \( x \equiv \pm 1 \). But \( -1 \) is ruled out by the outcome of the test, and the proof of the claim is complete. What is the computational complexity of the test? Consider first the computational problem of raising a number to a power. We can calculate, for example, \( b^2, b^4, b^8, \ldots \), by squaring, and reducing modulus immediately after each squaring operation, rather than waiting until the final exponent is reached. Then we use the binary expansion of the exponent to tell us which of these powers of \( b \) we should multiply together in order to compute \( b^n \). For instance, \[ b^{37} = b^{32} \cdot b^{4} \cdot b^{1}. \] Image Analysis: ## Image Analysis ### Image Localization and Attribution: - **Image 1**: The only image present on the page. ### Object Detection and Classification: - **Image 1**: - Objects Detected: - Multiple pieces of text. - Mathematical formulas and expressions. ### Scene and Activity Analysis: - **Image 1**: - Scene: Academic or scientific publication page, specifically focusing on pseudoprimality tests in number theory. - Activities: Detailed explanation of various pseudoprimality tests, including complete descriptions and examples. ### Text Analysis: - **Image 1**: - Text Extracted: ``` 4.6 Pseudoprimality tests Test 2. Given b, n. Output ‘n is composite’ if gcd(b, n) ≠ 1, else output ‘inconclusive.’ This one is a little better, but not yet good. If n is composite, the number of bases b ≤ n for which Test 2 will produce the result ‘composite’ is tn - 6(n)), where t() is the Euler totient function, of (4.1.5). This number of useful bases will be large if n has some small prime factors, but in that case it’s easy to find out that n is composite by other methods. if n has only a few large prime factors, say if n=ps, then the proportion of useful bases is very small, and we have the same kind of inefficiency as in Test 1 above. Now we can state the third pseudoprimality test. Test 3. Given b, n. (If b and n are not relatively prime or if bn - 7 ≠ 1 (mod n)) then output 'n is composite’; else output 'inconclusive.’ Regrettably, the test is still not ‘good,’ but it’s a lot better than its predecessors. To cite an extreme case of its un-goodness, there exist composite numbers n, called Carmichael numbers, with the property that the pair (b, n) produces the output ‘inconclusive’ for every integer b in [1, n - 1] that is relatively prime to n. An example of such a number is n = 1729, which is composite (1729 = 7 · 13 · 19), but for which Test 3 gives the result ‘inconclusive’ on every integer b € 1729 that is relatively prime to 1729 (i.e., that is not divisible by 7 or 13 or 19). Despite such misbehavior, the test usually seems to perform quite well. When n=169 (a difficult integer for Tests 1 and 2) it turns out that there are 158 different b’s in [1, 168] that produce the ‘composite’ outcome from Test 3, namely every such b except for 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168. Finally, we will describe a good pseudoprimality test. The familial resemblance to Test 3 will be apparent. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 97 Context: Test 4 (the strong pseudoprimality test): Given (b, n). Let n =1 - 2^q, where m is an old integer. If either (a) b^m ≠ 1 (modn) or (b) there is an integer i in [0, q- 1] such that b^mi ≠ -1 (modn) then return ‘inconclusive’ else return ‘n is composite.’ First we validate the test by proving the Proposition. If the test returns the message ‘n is composite,’ then n is composite. Proof: Suppose not. Then n is an odd prime. We claim that b^m ≡ 1 (modn) for all i = q, q — 1,..,, 0. If so then the case i = 0 will contradict the outcome of the test, and thereby complete the proof. To establish the claim, it is clearly true when i = q by Fermat’s theorem. If true for I, then it is true for i – 1 also, because (bm+1)2 == bm+2≡ 1 (mod n) implies that the quantity being squared is +1 or -1. Since n is an odd prime, by corollary 4.5.3 Un is cyclic, and so the equation z^2 = 1 in Un has only the solutions z = ±1. But -1 is ruled out by the outcome of the test, and the proof of the claim is complete. (q.e.d.) What is the computational complexity of the test? Consider first the computational problem of raising a number to a power. We can calculate, for example, bn mod n with O(log n) integer multiplications, by successive squaring. More precisely, we compute b, b^2, b^4, b^8,… by squaring, and reducing modulo n immediately after each squaring operation, rather than waiting until the final exponent is reached. Then we use the binary expansion of the exponent to tell us which of these powers of b we should multiply together in order to compute bp. For instance, b337 = b256 . b64 . b16 . b. 93 ``` - Significance: The extracted text details four pseudoprimality tests, explaining their effectiveness, limitations, examples, and mathematical proofs. This is essential for understanding advancements in determining the primality of numbers, a fundamental topic in number theory and cryptographic applications. ### Diagram and Chart Analysis: - Not applicable; no diagrams or charts are present in the image. ### Product Analysis: - Not applicable; no products are depicted in the image. ### Anomaly Detection: - No anomalies detected in the image. Everything appears coherent and well-structured. ### Color Analysis: - **Image 1**: - Dominant Colors: Black and white text on a white background. - Impact: The high contrast between the black text and the white background ensures readability. ### Perspective and Composition: - **Image 1**: - Perspective: Front-facing, as typical for a page of text in a book or article. - Composition: The text is structured in a standard, double-column format common in academic publications. ### Contextual Significance: - The image, being a page from a mathematical text, contributes to the overall theme of pseudoprimality tests within the broader document. The detailed explanations and proofs underscore the scientific rigor and depth of the subject matter. ### Metadata Analysis: - Not applicable; no metadata information is available. ### Graph and Trend Analysis: - Not applicable; no graphs are included in the image. ### Graph Numbers: - Not applicable; no graphs are present in the image. ### Ablaufprozesse (Process Flows): - Not explicitly depicted; the text describes step-by-step tests in a linear, explanatory format. ### Prozessbeschreibungen (Process Descriptions): - Details descriptions of four different pseudoprimality tests, explaining procedural steps and mathematical justifications. ### Typen Bezeichnung (Type Designations): - Identifies tests by labels (Test 2, Test 3, Test 4). ### Trend and Interpretation: - Identifies a progression in the effectiveness and reliability of pseudoprimality tests, indicating a trend towards more robust and comprehensive methods. ### Tables: - Not applicable; no tables are included in the image. The detailed extraction and analysis provide a comprehensive understanding of the pseudoprimality tests and their significance within the mathematical context. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 99 Context: **Prozessbeschreibungen (Process Descriptions):** - The proofs describe iterative and logical processes related to mathematical properties and how they lead to conclusions about the group \(U_n\) and primality. **Typen Bezeichnung (Type Designations):** - Types and categories include: - **Groups:** \(U_n\) - **Subgroups:** \(C(x)\), \(B\), and \(B'\) - **Mathematical Functions:** Euler's totient function \(\phi\) - **Mathematical Concepts:** Order, cyclic groups, prime power **Trend and Interpretation:** - The lemmas and theorem develop a trend in understanding how the structure and properties of \(U_n\) can influence and be used in primality testing. Specifically, it interprets the order and subgroup generation properties to conclude about the composite or prime nature of the number 'n.' The content maintains a scientific and educational tone, focusing on theoretical aspects crucial in advanced mathematics and computer science, particularly in fields like cryptography. Note: The remaining aspects such as metadata analysis, color analysis, perspective and composition, contextual significance are not applicable for analysis as no related information is provided in the visual content and it is a black and white text-focused document. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 99 Context: φ(x)2 = φ(z)2=1. Since (the square is 1,φ(z) has order 1 or 2. Hence ifψ(z)^ 1 it is of order 2. If the cyclic group CU) is of odd order, then it contains no element of even order. Hence Cu) is of even order and contains -1. Then it can contain no other element of order 2, so φ(z)*1 in this case. Hence for every x£ B,y(z)=±1.Die Suppose B generates the full group Un. Then not only for every x€ B but for every x E Un is true that φ(x)* Suppose n is not a prime power. Then s > 1 in the factorization (4.5.2) of Un. Consider the element y of Un, which, when written out as an s-tuple according to that factorization, is of the form v =(1,1,1,...1,w,w2, 1,...,1.....yw.) where the ‘v is in the ith component, y E Un, (recall that j as described above, in the second sentence of this proof). We can suppose yiy to be an element of order exactly 2' in Un since Un is cycic Consider y(z)=1. Clearly ψ(z) is not 1, for otherwise the order y'.%a would divide 2l'-lm, his impossible because m iso. Also,φ(z) is not 41 because the element -1 of Un is represented uniquely by the s-tuple all of whose entries are –1. Thus φ(s”) is neither 1 nor–1 in Un, which contradicts the italicized assertion above. Hence s= 1 nd n is a prime power, completing the proof. Now we can prove the main result of Solovay, Strassen and Rabin, which asserts that Test 4 is good. Theorem 4.7.1. LetIFn the set of integers to m3a such that In, (n)y returns ‘ inconclusive’ in Test 4. (a) If B generates Un, thenn is prime. (b) Ifnis composite then B' consists of at most half of the integers in In 1|| :| Proof: Suppose 6 €B and let m be the odd part of n - 1. Then either3m^1 1 or y=1 forsome s€ {0,4 - 1). In the former case the cyclic subgroup (C)b) has odd order, since m is odd, and in the latter case (Ch contains -1. ``` - **Analysis:** - This page contains advanced mathematical content typical of academic literature, specifically focusing on pseudoprimality tests. - **Lemma 4.7.1** and **Lemma 4.7.2** state mathematical properties related to the order of elements in certain cyclic groups. - **Theorem 4.7.1** establishes a criterion for determining the primality of an integer based on generating sets, related to Solovay, Strassen, and Rabin's work on primality tests. #### 8. **Color Analysis:** - **Dominant Colors:** - The image predominantly features black text on a white background. #### 9. **Perspective and Composition:** - **Perspective:** - The perspective is that of a flat, face-on view, typical of a page of text. - **Composition:** - The page consists of structured text with headings, sub-headings, numbered lemmas, and their proofs. The text is aligned evenly with consistent spacing, which is typical for mathematical or academic documents. #### 10. **Contextual Significance:** - **Overall Context:** - The image is part of an academic document or textbook on pseudoprimality tests in number theory or a related field. - The text on the page contributes significantly to the understanding and application of pseudoprimality tests, a crucial concept in computational number theory and cryptography. ### Summary: The analyzed image is a page from a mathematical text dealing with pseudoprimality tests. It includes detailed lemmas and a theorem related to the order of elements in cyclic groups and their application in pseudoprimality testing, with proofs providing rigorous validation of these lemmas and theorem. The page is composed primarily of structured text, black on white, and is geared towards readers with an advanced understanding of mathematics, specifically in number theory. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 100 Context: # Chapter 4: Algorithms in the Theory of Numbers Hence in either case \( B' \subseteq B \), where \( B \) is the set defined in the statement of lemma 4.7.2 above. If \( B' \) generates the full group \( U_n \), then \( B \) does too, and by lemma 4.7.2, if \( n \) is a prime power, \( n = p^k \). Also, in either of the above cases we have \( b^{\phi(n)} \equiv 1 \), so the same holds for all \( b \in B' \), and so for all \( x \in U_n \), we have \( x^{n-1} \equiv 1 \), since \( B' \) generates \( U_n \). Now, \( n \) is cyclic or direct. By theorem 4.5.3 there are primitive roots modulo \( n = p^k \). Let \( g \) be one of these. The order of \( g \) is, on the one hand, \( p^{k-1}(p-1) \) since the set of all its powers is identical with \( U_n \), and on the other hand is a divisor of \( n - 1 = i^{k} - 1 \) for all \( a \), and in particular for \( a = g \). Hence \( p^{k-1}(p - 1) \) (which, if \( k > 1 \), is a multiple of \( p \)) divides \( p \) (which is one less than a multiple of \( p \)), and so \( k = 1 \), which completes the proof of part (a) of the theorem. In part (b), \( n \) is composite and so \( B' \) cannot generate all of \( U_n \) by part (a). Hence \( B' \) generates a proper subgroup of \( U_n \), and so can contain at most half as many elements as \( U_n \) contains, and the proof is complete. Another application of the same circle of ideas to compute elements occurs in the generation of random numbers on a computer. A good way to do this is to choose a primitive root modulo the word size of your computer, and then, each time the user asks for a random number, output the next higher power of the primitive root. The fact that you started with a primitive root insures that the number of "random numbers" generated before repetition sets in will be as large as possible. Now we'll summarize the way in which the primality test is used. Suppose there is a large integer \( n \), and we would like to determine if it is prime. We can do this with: ``` function test(n, outcome); times = 0; repeat choose an integer \( k \) uniformly at random in \( [2, n-1] \); apply the strong pseudoprimality test (Test 4) to the pair \( (n, k) \); times = times + 1 until \( (result = 'is composite') \text{ or times = 100} \; if times = 100 then outcome := 'is probably prime' else outcome := 'is composite' end{test} ``` If the procedure exits with \( n \) is composite, then we can be certain that \( n \) is not prime. If we want to see the factors of \( n \) then it will be necessary to use some factorization algorithm, such as the one described below in section 4.9. On the other hand, if the procedure halts because it has been through 100 trials without a conclusive result, then the integer \( n \) is very probably prime. More precisely, the chance that a composite integer \( n \) would have behaved like that is less than \( 2^{-100} \). If we want certainty, however, it will be necessary to apply a test whose outcome will prove primality, such as the algorithm of Adleman, Rumely and Pomerance, referred to earlier. In section 4.9 we will discuss a probabilistic factoring algorithm. Before doing so, in the next section we will present a remarkable application of the complexity of the factoring problem to cryptography. Such applications remind us that primality and factorization algorithms have important applications beyond pure mathematics, in areas of vital public concern. ## Exercises for section 4.7 1. For \( n = 9 \) and \( r = 15 \), find all of the cyclic groups \( C_u \) of lemma 4.7.2, and find the set \( B \). 2. For \( n = 9 \) and \( r = 15 \) find the set \( B' \) of theorem 4.1. Image Analysis: ### Image Analysis 1. **Localization and Attribution:** - **Image Number:** 1 4. **Text Analysis:** - **Detected Text:** The text primarily consists of explanations on algorithms in the theory of numbers, specifically discussing primitive roots, power modulo, orders of elements, and a probabilistic algorithm for testing primitivity. It includes a pseudo-code for the function `testN` to determine the primality of an integer. - **Content and Significance:** - **Chapter Title:** "Chapter 4: Algorithms in the Theory of Numbers" - **Main Points:** - Definition and properties of sets related to primitive roots modulo. - Discussion on generating full groups \(U_n\) and conditions for \(B'\). - Explanation of the order of elements. - Introduction to a probabilistic test for small primes using a test function. - **Code:** - The pseudo-code outlined is for `testN`, a loop that repeats a certain number of times to determine if a number is composite or probably prime. - **Exercises:** - There are two exercises related to finding cyclic groups and set \( B' \). 8. **Color Analysis:** - **Color Composition:** - The text is in black font on a white background, typical of a textbook or academic paper, ensuring readability and focus on the content. 9. **Perspective and Composition:** - **Perspective:** - The text is viewed from a straightforward, bird's eye perspective. - **Composition:** - The format is typical of academic texts, with a clear hierarchy of sections, subsections, and code blocks. 10. **Contextual Significance:** - **Contribution:** - This page contributes to an overall educational or instructional document on number theory and algorithms. It provides theoretical insights as well as practical methods for determining the properties of numbers, useful in both academic and practical applications. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 104 Context: # Chapter 4: Algorithms in the Theory of Numbers learned nothing. However if neither \( u \equiv (mod \, n) \) nor \( u \equiv -v \, (mod \, n) \) is true then we will have found a nontrivial factor of \( n \), namely \( \gcd(u - v, n) \) or \( \gcd(u + v, n) \). ## Example: Take as a factor base \( B = \{-2, 5\} \), and let it be required to find a factor of \( n = 1729 \). Then we claim that 186 and 267 are B-numbers. To see that 186 is a B-number, note that \( 186^2 = 20 - 1729 + (-2)^2 \); and similarly, since \( 267^2 = 41 - 1729 + (-2)^2 \), we see that 267 is a B-number, for this factor base \( B \). The exponent vectors of 186 and 267 are \( (4, 0) \) and \( (4, 2) \) respectively, and these sum to \( (0, 1) \) mod \( 2 \), hence we find that \[ u = 186 \times 267 = 1250 \, \quad (mod \, 1729) \] \[ r_1 = 4 \, \quad r_2 = 1 \\ r_1 = 2^{(-2)}(5^1) = 80 \\ \gcd(u - v, n) = \gcd(1170, 1729) = 13 \] There might have seemed to be some legerdemain involved in plugging the B-numbers 186 and 267 out of the air, in the example above. In fact, as the algorithm has been implemented by its author, J. D. Dixon, one simply chooses integers uniformly at random from \([1, n-1]\) until enough B-numbers have been found, so their exponent vectors are linearly dependent modulo 2. In Dixon’s implementation the factor base that is used consists of \( -1 \) together with the first \( k \) prime numbers. It can then be proved that if \( n \) is not a prime power then with a correct choice of \( b \) relative to \( n \), if we repeat the random choices until a factor of \( n \) is found, the average running time will be: \[ \mathrm{exp}((2 + o(1))(\log \log n)^3). \] This is not polynomial time, but it is moderately exponential only. Nevertheless, it is close to being about the best that we know how to do on the elusive problem of factoring a large integer. ## 4.10 Proving primality In this section we will consider a problem that sounds a lot like primality testing, but is really a little different because the rules of the game are different. Basically the problem is to convince a skeptical audience that a certain integer is prime, requiring them to do only a small amount of computation in order to be so persuaded. First, though, suppose you were writing a 100-decimal-digit integer \( n \) on the blackboard in front of a large audience and you wanted to prove to them that \( n \) was not a prime. If you simply wrote down two smaller integers whose product was \( n \), the job would be done. Anyone who wished to be certain could spend a few minutes multiplying the factors together and verifying the product was indeed \( n \), and all doubts would be dispelled. Indeed*, a speak ear at a mathematical convention in 1903 announced the result that \( 2^{67} - 1 \) is not a prime number, and to be utterly convincing all he had to do was to write: \[ 2^{67} - 1 = 193707721 \times 761838257287. \] We note that the speaker probably had to work very hard to find those factors, but having found them it became quite easy to convince others of the truth of the claimed result. A pair of integers \( r, s \) for which \( r \, | \, s \, | \, n \) and \( n = r s \) constitute a certificate attesting to the compositeness of \( n \). With this certificate \( C(n) \) and an auxiliary checking algorithm, viz. 1. Verify that \( r \neq 1 \) and that \( r \neq s \). 2. Verify that \( s \neq n \). we can prove, in polynomial time, that \( n \) is not a prime number. *We follow the account given in V. Pratt, Every prime has a succinct certificate, SIAM J. Computing, 4 (1975), 214-220. Image Analysis: ### Analysis of the Visual Content #### Image Details - **Localization and Attribution:** - Only one image is present in the provided visual content, hence it is designated as **Image 1**. #### Text Analysis: - The page consists of a substantial amount of text and mathematical notations. Here is a breakdown of notable components: - The heading is "**Chapter 4: Algorithms in the Theory of Numbers**." - The text under the heading discusses methods for finding factors of integers and various properties and theorems relevant to number theory. - **Example Section:** - Discusses an example involving the B-numbers method to find a factor of n = 1729. - Detailed calculations are given to find the factor using given base B and subsequent exponential properties. - **Mathematical Functions & Variables:** - gcd(u, v, n), exponential factors, verification methods, and calculations are prominent. - Specifically outlined steps like \(186 \times 267 \equiv 1250 \) (mod 1729). - **Paragraph on Legerdemain in B-number Selection:** - Describes historical context and the implementation specifics of the algorithm. - Explains that the numbers 186 and 267 may seem arbitrary but were likely chosen for practical reasons in a computational algorithm setup. - **Mathematical Formula:** - Exponential Time Complexity given by \(\exp[2(\theta(1))(log \log log n)^3]\). - **Section 4.10: Proving Primality:** - Discusses methods to prove a number is not a prime. - Use of historical anecdote involving a 100-digit integer to demonstrate the math behind proving non-primality. - Verification method using \(2^{67} - 1 = 193707721 \times 761838257287\). - **Pair of integers theorem:** - Considers a pair (r, s) where r, s ≠ 1 and the condition n = r * s. - The verification process is concise but laid out step by step. - **Reference:** - Mentions V. Pratt’s succinct certificate method found in "SIAM J. Computing, 4 (1975), 214-220". #### Scene and Activity Analysis: - **Book/Page Context:** - The page appears to be from an academic book or paper on number theory, specifically within a chapter on algorithms. - No visual illustrations or charts are present – the content is purely textual with mathematical notations. #### Anomalies: - There are no apparent anomalies in the textual content of this image. The mathematical notations and examples follow a coherent line of reasoning consistent with academic literature. #### Perspective and Composition: - **Perspective:** - Straight-on view of a single page from a book. - **Composition:** - The text is methodically arranged, with headings, paragraphs, mathematical notations, and references structured in a logical flow. #### Contextual Significance: - This page forms a part of a detailed exposition on algorithms in the theory of numbers, likely used in higher education or for research purposes. - It contributes significantly by integrating practical examples, theoretical explorations, and historical anecdotes to deepen understanding of number theory algorithms. ### Conclusion: The provided page is rich with detailed explanations, mathematical derivations, and historical contexts relating to algorithms in number theory. It serves as a comprehensive resource for understanding the computational methods to identify factors of large integers and offers techniques for primality testing. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 105 Context: # 4.10 Proving primality Now comes the hard part. How might we convince an audience that a certain integer \( n \) is a prime number? The rules are that we are allowed to do any immense amount of calculation beforehand, and the results of that calculation can be written on a certificate \( C(n) \) that accompanies the integer \( n \). The audience, however, will need to see only a polynomial amount of further computation in order to convince themselves that \( n \) is prime. We will describe a primality-checking algorithm \( A \) with the following properties: 1. If \( n \) is the integer and \( A \) a certain certificate \( C(n) \). 2. If \( n \) is prime then the action of \( A \) on the inputs \( (n, C(n)) \) results in the output `it is prime`. 3. If \( n \) is not prime then for every possible certificate \( C(n) \) the action of \( A \) on the inputs \( (n, C(n)) \) results in the output `primality of \( n \) is not verified`. 4. Algorithm \( A \) runs in polynomial time. Now the question is, does such a procedure exist for primality verification? The answer is affirmative, and we will now describe one. The fact that primality can be quickly verified, if not quickly discovered, is of great importance for the developments of Chapter 5. In the language of section 5.1, what we are about to do is show that the problem “Is \( n \) prime?” belongs to the class NP. The next lemma is a kind of converse to Fermat’s little theorem (theorem 4.5.2). ## Lemma 4.10.1 Let \( p \) be a positive integer. Suppose there is an integer \( x \) such that \( x^{p-1} \equiv 1 \,(\text{mod } p) \) and such that for all divisors of \( p-1 \), \( d < p-1 \), we have \( x^{d} \not\equiv 1 \,(\text{mod } p) \). Then \( p \) is prime. **Proof:** First we claim that \( \gcd(d, x) = 1 \), for \( d = \text{gcd}(p, x) \). Then \( x = g^{r} \) and \( p = g^{s} \). Since \( x^{p-1} \equiv 1 \), it follows that \( x^{p-1} = 1 + t + x^{p-1} - t = (gs)^{p-1} - 1 \). The left side is a multiple of \( g \). The right side is not, unless \( g = 1 \). It follows that \( x \in U_{p} \), the group of units \( Z_{p} \). Thus \( x \) is an element of order \( p - 1 \) in a group of order \( \varphi(p) \). Hence \( \varphi(p) \leq p - 1 \). Hence \( \varphi(p) = p - 1 \) and \( p \) is prime. Lemma 4.10.1 is the basis for V. Pratt’s method of constructing certificates of primality. The construction of the certificate is actually recursive since step 3b allows for certificates of smaller primes. We suppose that the certificate of the prime \( 2 \) is the trivial case, and that it can be verified at no cost. Here is a complete list of the information that is on the certificate \( C(p) \) that accompanies an integer \( p \) whose primality is to be attested to: 1. \( P \): a list of the primes \( p_i \) and the exponents \( a_i \) for the canonical factorization \( p - 1 = \prod_{i=1}^{k} p_i^{a_i} \). 2. \( C_{p} \): the certificates \( C_{p_i} \) of each of the primes \( p_1, \ldots, p_k \). 3. \( x \): a positive integer \( x \). To verify that \( p \) is prime we could execute the following algorithm \( B \): **(B1)** Check that \( p - 1 = \prod_{i=1}^{k} p_i^{a_i} \). **(B2)** Check that each \( p_i \) is prime, using the certificates \( C_{p_i} \) \( (i = 1, r) \). **(B3)** For each divisor \( d \) of \( p - 1 \), \( d < p - 1 \), check that \( x^{d} \not\equiv 1 \,(\text{mod } p) \). **(B4)** Check that \( x^{p-1} \equiv 1 \,(\text{mod } p) \). This algorithm \( B \) is correct, but it might not operate in polynomial time. In step \( B3 \) we are looking at every divisor of \( p - 1 \) and there may be a lot of them. Fortunately, it isn’t necessary to check every divisor of \( p-1 \). The reader will have no trouble proving that there is a divisor of \( p - 1 \) (or \( < d \) such that \( x^{d} \equiv 1 \,(\text{mod } p) \) and only if there is such a divisor that has the special form \( d = (p - 1)/t \). The primality checking algorithm \( A \) now reads as follows: **(A1)** Check that \( p - 1 = \prod_{i=1}^{k} p_i^{a_i} \). **(A2)** Check that each \( p_i \) is prime, using the certificates \( C_{p_i} \) \( (i = 1, r) \). **(A3)** For each \( i = 1 \) to \( k \), check that \( x^{(p-1)/p_i} \not\equiv 1 \,(\text{mod } p) \). Image Analysis: ### Image Analysis #### 1. **Localization and Attribution** - **Image Number**: 1 - **Location on the Page**: Entire page has a single image consisting of text. #### 4. **Text Analysis** - **Detected Text and Content**: This image consists entirely of text discussing a mathematical proof related to prime number verification. The image includes definitions, lemmas, proofs, and algorithms related to the topic. - **Analyzed Content and Significance**: - **Header**: "4.10 Proving primality" indicates the section discussing methods to prove whether a number is prime. - **Main Text**: - **Algorithm Description**: Describes a polynomial-time algorithm \( A \) to verify the primality of an integer \( n \). - **Verification Conditions**: Includes conditions (1) through (4) that algorithm \( A \) must satisfy to verify primality. - **Significant Definitions and Lemmas**: - **Lemma 4.10.1**: States a lemma crucial for understanding the proof method. - **Proof Section**: Detailed proof of Lemma 4.10.1, with algebraic and number-theoretic reasoning. - **Process Description**: Outlines the process and steps to verify primality using the proposed algorithm, referred to as Algorithm \( B \) and modifications for Algorithm \( A \). - **Steps for Primality Checking**: Detailed step-by-step methodology for verifying prime numbers, including checks on divisors and certificates of primality. #### 6. **Product Analysis** - **Textual Product**: The page is composed solely of text discussing a mathematical proof; there are no physical products depicted. #### 10. **Contextual Significance** - **Document Context**: The document is evidently a part of a detailed mathematical text or textbook on number theory or algorithms. - **Contribution to Overall Theme**: The image contributes to the academic and educational theme by outlining theoretical and practical aspects of primality proving, key for students or enthusiasts in mathematics or computer science. #### 12. **Graph and Trend Analysis** - **Logical Flow**: Demonstrates the logical progression of mathematical theorem proving, starting from definitions to stepwise verification algorithms. #### 13. **Graph Numbers** - **No Graphs**: The page does not contain any graphical elements or numerical data points. #### Prozessbeschreibungen (Process Descriptions) - **Process Flow for Verifying Primality**: - **Steps**: - Check specific properties of \( p-1 \) - Verify primality of the prime divisors of \( p-1 \) - Use certificates to confirm divisors and primality - Ensure the calculation follows polynomial time to ensure efficiency ### Summary This page from a mathematical textbook outlines a comprehensive method for verifying the primality of numbers using an algorithmic approach. It includes detailed theoretical explanations, step-by-step proofs, and algorithmic descriptions necessary for academic understanding and practical application in mathematical computations or related fields. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 108 Context: # Chapter 5: NP-completeness ## 5.1 Introduction In the previous chapter we met two computational problems for which fast algorithms have never been found, but neither have such algorithms been proved to be unattainable. Those were the polynomial-time problem, for which the best-known algorithm is delicately poised on the brink of polynomial time, and the integer-factoring problem, for which the known algorithms are in a more primitive condition. In this chapter we will meet a large family of such problems (hundreds of them now!). This family is not just a list of seemingly difficult computational problems. It is in fact bound together by strong ties. The collection of problems, called the NP-complete problems, includes many well-known and important questions in discrete mathematics, such as the following: - **The travelling salesman problem (TSP)**: Given n points in the plane (the "cities"), and a distance D, is there a tour that visits all of the cities, returns to its starting point, and has total length ≤ D? - **Graph coloring**: Given a graph G and an integer K. Can the vertices of G be properly colored in K or fewer colors? - **Independent set**: Given a graph G and an integer K. Does V(G) contain an independent set of K vertices? - **Bin packing**: Given a finite set S of positive integers, and an integer N (the number of bins). Does there exist a partition of S into N or fewer subsets such that the sum of the integers in each subset is ≤ K? In other words, can we "pack" the integers of S into at most N "bins", where the "capacity" of each bin is K? These are very difficult computational problems. Take the graph coloring problem, for instance. We could try every possible way of coloring the vertices of G in K colors to see if any of them work. There are \( K^n \) such possibilities, if G has n vertices. Here, if there are a large amount of colorings to be done, enough so that if G has 50 vertices and we have 10 colors at our disposal, the problem would lie far beyond the capabilities of the fastest computers that are now available. Hard problems can have easy instances. If the graph G happens to have no edges at all, or very few of them, then it will be very easy to find out if a coloring is possible, or if an independent set of K vertices is present. The real question is this (let's use "Independent Set" as an illustration): Is it possible to design an algorithm that will come packaged with a performance guarantee of the following kind: > **The seller warrants that if a graph G, of n vertices, and a positive integer K are input to this program, then it will correctly determine if there is an independent set of K or more vertices in V(G), and it will do so in an amount of time that is at most 1000n^k.** Hence there is no contradiction between the facts that the problem is hard and that there are easy cases. The hardness of the problem stems from the seeming impossibility of producing such an algorithm accompanied by such a manufacturer's warranty card. Of course the "1000" didn’t have to be exactly that. But some quite specific polynomial in the length of the input string must appear in the performance guarantee. Hence "357n²" might have appeared in the guarantee, and so might "23n", but \( n^k \) would not be allowed. Let's look carefully at why \( n^k \) would not be an acceptable worst-case polynomial time performance bound. In the "Independent Set" problem the input must describe the graph G and the integer K. How many bits are needed to do that? The graph can be specified, for example, by its vertex adjacency matrix A. This is an \( n \times n \) matrix in which the entry in row i and column j is 1 if (i, j) ∈ E(G) and is 0 else. Evidently, \( n^2 \) bits will describe the matrix A. The integers K and n can be entered with just \( O(\log n) \) bits, so the entire input bit string for the "Independent Set" problem is \( n^2 \) bits long, let D denote the number of bits in the input string. Suppose that on the warranty card the program was guaranteed to run in a time that is ≤ \( n^k \). Is this a guarantee of polynomial time performance? That question raises: Is there a polynomial P such that for every instance of "Independent Set" the running time T will be at most P(D)? Well, is it bounded? Image Analysis: ### Analysis of the Attached Visual Content ### 1. Localization and Attribution - The entire content appears to be a single page of text from a document. - It is identified as being on page 104. ### 2. Object Detection and Classification - The object is a single page of text formatted with sections and paragraphs. ### 3. Scene and Activity Analysis - The scene shows documentation of a specific chapter from a larger document or book. It focuses on explaining concepts related to NP-completeness. ### 4. Text Analysis #### Chapter Title and Sections: - **Chapter Title**: Chapter 5: NP-completeness - **Sections**: The document has the following sections: - **5.1 Introduction**: This section introduces the concept of NP-complete problems and mentions the traveling salesman problem (TSP), Graph coloring problem, Independent set problem, and Bin packing problem. #### Content: - **Introduction and Complexity Problems**: - Describes how this chapter will introduce NP-complete problems. - Mentions that these problems are difficult and includes families of these problems. - Provides examples such as TSP, graph coloring, independent set, and bin packing. - **Independent Set Problem**: - Given a graph \( G \) and an integer \( K \). - Seeks to determine if there is an independent set of \( K \) vertices. - **Warranty Example**: - An example warranty statement is given to illustrate the difficulty of finding polynomial-time algorithms for NP-complete problems. - "The seller warrants that if a graph \( G \), integer \( K \), and a positive integer \( K \) are input to this program, then it will correctly determine if there is an independent set of \( K \) vertices... in an amount of time that is at most 1000(*)\( V^{1000} \)." This highlights that current understandings of polynomial-time algorithms are inadequate for such problems. ### 8. Color Analysis - The page is primarily composed of black text on a white background, typical of printed or digital text documents. ### 9. Perspective and Composition - The perspective is a straight-on view typically used for reading text. - Composition follows a conventional document layout with headings and paragraphs arranged in a standard, readable format. ### 12. Graph and Trend Analysis - Although no graphs are present, discussions of algorithm complexity, such as "1000(*)\( V^{1000} \)" and Big-O notation hint at more complex hypothetical graphs depicting algorithmic performance metrics. ### Additional Aspects: #### Prozessbeschreibungen (Process Descriptions): - Description of the Independent Set problem and how it needs polynomial time algorithms. - Challenges in computational complexity and the specific hypothetical warranty example for illustrating polynomial-bound guarantee challenges. #### Trend and Interpretation: - Trends in computational problems around NP-completeness. - Complexity boundaries and performance issues of algorithms in finding solutions to NP-complete problems. ### Conclusion This document page is primarily an explanatory text focusing on computational complexity, specifically regarding NP-complete problems and polynomial-time algorithms. It introduces key concepts, provides examples, and uses a hypothetical scenario to illustrate points related to computational difficulty and performance guarantees. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 109 Context: # What is a language? by a polynomial in B if T = n^K and B ∼ n^2? It would seem so; in fact obviously T = O(B^K), and that’s a polynomial, isn’t it? The key point resides in the order of the qualifiers. We must give the polynomial that works for every instance of the problem first. Then that one single polynomial must work on every instance. If the 'polynomial' that we give is B(K), well that’s a different polynomial in B for different instances of the problem, because K is different for different instances. Therefore if we say that a certain program for 'Independent Set' will always get an answer before B^K/2 minutes, where B is the length of the input bit string, then we would not have provided a polynomial-time guarantee in the form of a single polynomial in B that applies uniformly to all problem instances. The distinction is a little thorny, but is worthy of careful study because it’s of fundamental importance. What we are discussing is usually called a worst-case time bound, meaning a bound on the running time that applies to every instance of the problem. Worst-case time bounds aren’t the only possible interesting ones. Sometimes we might not care if an algorithm is occasionally very slow as long as it is almost always fast. In other situations we might be satisfied with an algorithm that is fast on average. For the present, however, we will stick to the worst-case time bounds and study some of the theory that applies to that situation. In sections 5.6 and 5.7 we will study some average time bounds. Now let’s return to the properties of the NP-complete family of problems. Here are some of them: 1. The problems all seem to be computationally very difficult, and no polynomial time algorithms have been found for any of them. 2. It has been proved that polynomial time algorithms for these problems do not exist. 3. But this is not just a random list of hard problems. If a fast algorithm could be found for one NP-complete problem then there would be fast algorithms for all of them. 4. Conversely, if it could be proved that no fast algorithm exists for one of the NP-complete problems, then there could not be a fast algorithm for any other of those problems. These properties are not intended to be a definition of the concept of NP-completeness. We’ll get that later on in this section. They are intended as a list of some of the interesting features of these problems, which, when coupled with their theoretical and practical importance, accounts for the intense worldwide research effort that has gone into understanding them in recent years. The question of the existence or nonexistence of polynomial-time algorithms for the NP-complete problems learns probably falls as the principal unsolved problem that faces theoretical computer science today. Our next task will be to develop the formal machinery that will permit us to give precise definitions of all of the concepts that are needed. In the remainder of this section we will discuss the additional ideas informally, and then in section 5.2 we’ll state them quite precisely. ## What is a decision problem? First, the idea of a decision problem. A decision problem is one that asks only for a yes-or-no answer: Can this graph be 5-colored? Is there a tour of length ≤ 15 miles? Is there a set of 67 independent vertices? Many of them problems that we are studying can be phrased as decision problems or as optimization problems: What is the smallest number of colors with which G can be colored? What is the length of the shortest tour of the cities? What is the size of the largest independent set of vertices in G? Usually if we find a fast algorithm for a decision problem then just a little more work will be able to solve the corresponding optimization problem. For instance, suppose we have an algorithm that solves the decision problem for graph coloring, and what we want is the solution of the optimization problem (the chromatic number). Let G be a graph G given, say of 100 vertices. Ask: can the graph be 50-colored? If so, then the chromatic number lies between 1 and 50. Then ask if it can be colored in 25 colors. If not, then the chromatic number lies between 26 and 50. Continue in this way, using bisection of the interval that is known to contain the chromatic number. After O(log n) steps we will have found the chromatic number of a graph of n vertices. The extra multiplicative factor of log n will not alter the polynomial vs. nonpolynomial runing time distinction. Hence if there is a fast way to do the decision problem then there is a fast way to do the optimization problem. The converse is obvious. Hence we will restrict our discussion to decision problems. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 110 Context: # Chapter 5: NP-completeness ## What is a language? Since every decision problem can have only the two answers 'Y/N', we can think of a decision problem as asking if a given word (the input string) does or does not belong to a certain language. The language is the totality of words for which the answer is 'Y'. The graph 3-coloring language, for instance, is the set of all symmetric, square matrices of 0s and 1s, with zeros on the main diagonal (these are the vertex adjacency matrices of graphs) such that the graph that the matrix represents is 3-colorable. We can imagine that somewhere there is a vast dictionary of all the words in this language. A 3-colorability computation is therefore nothing but an attempt to discover whether a given word belongs to the dictionary. ## What is the class P? We say that a decision problem belongs to the class P if there is an algorithm \( A \) and a number \( s \) such that for every instance \( I \) of the problem the algorithm \( A \) will provide a solution in time \( O(B) \), where \( B \) is the number of bits in the input string that represents \( I \). To put it more briefly, P is the set of easy decision problems. Examples of problems in P are most of the ones that we have already met in this book: Are these two integers relatively prime? Is this integer divisible by two? Is this graph 2-colorable? Is there a flow of value greater than \( K \) in this network? Can this graph be disconnected by the removal of \( K \) or fewer edges? Is there a matching of more than \( K \) edges in this bipartite graph? For each of these problems, there is a fast (polynomial time) algorithm. ## What is the class NP? The class NP is a little more subtle. A decision problem \( Q \) belongs to NP if there is an algorithm \( A \) that does the following: 1. Associated with each word of the language \( Q \) (i.e., with each instance for which the answer is 'Yes') there is a certificate \( C(I) \) such that when the pair \( (I, C(I)) \) are input to algorithm \( A \), it recognizes that \( I \) belongs to the language \( Q \). 2. If there is some word that does not belong to the language \( Q \) then there is no choice of certificate \( C(I) \) that will cause \( A \) to recognize \( I \) as a member of \( Q \). 3. Algorithm \( A \) operates in polynomial time. To put this one more briefly, NP is the class of decision problems for which it is easy to check the correctness of a claimed answer, with the aid of a little extra information. So we aren't asking for a way to find a solution, but only to verify that an alleged solution really is correct. Here is an analogy that may help to clarify the distinction between the classes P and NP. We have all had the experience of reading through a tiny ingenious and difficult proof of some mathematical theorem, and wondering how the person who found the proof in the first place ever did it. Our task, as a reader, was only to verify the proof, and that is a much easier job than the mathematician who invented the proof had. To pursue the analogy a bit farther, some proofs are extremely time consuming even to check (see the proof of the Four-Color Theorem!), and similarly, some computational problems are not even known to belong to NP, let alone to P. In P are the problems where it's easy to find a solution, and in NP are the problems where it's easy to check a solution that may have been very tedious to find. Here's another example. Consider the graph coloring problem to be the decision problem Q. Certainly this problem is not known to be in P. It is, however, in NP, and here is an algorithm, and a method of constructing certificates that proves it. Suppose \( G \) is some graph that is K-colorable. The certificate of \( G \) might be a list of the colors that get assigned to each vertex to ensure proper \( K \)-coloring of the vertices of \( G \). Where do we find that list, you ask? Well, we've said it was easy to construct a certificate. If you actually want to find one then you will have to solve a hard problem. But we're really only talking about checking the correctness of an alleged answer. To check that a certain graph \( G \) really is \( K \)-colorable we can be convinced if you will show us the color of each vertex in a proper \( K \)-coloring. If you do provide that certificate, then our checking algorithm \( A \) is very simple. It checks first that every vertex has a color and only one color. It then checks that no more than \( K \) colors have been used altogether. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 111 Context: It finally shows that for each edge \( e \in G \) it is true that the two endpoints of \( e \) have different colors. Hence the graph coloring problem belongs to NP. The traveling salesman problem would provide a certificate that contains a tour, whose total length is \( K \), of all of the cities. The checking algorithm \( A \) would then verify that the tour really does visit all of the cities and really does have total length \( \leq K \). The traveling salesman problem, therefore, also belongs to NP. "Well," you might reply, "if we’re allowed to look at the answers, how could a problem fail to belong to NP?" Try this decision problem: an instance of the problem consists of a set of cities in the plane and a positive number \( K \). The question is "Is it true that there is not a tour of all of these cities whose total length is less than \( K \)?" Clearly this is a kind of a negation of the traveling salesman problem. Does it belong to NP? If so, there must be an algorithm \( A \) and a way of making a certificate \( C \) for each instance \( I \) that we can quickly verify that no such tour exists of the given cities. Any suggestions for the certificate? The algorithm? No one else knows how to do this either. It is not known if this negation of the traveling salesman problem belongs to NP. Are there problems that do belong to NP but for which it isn't immediately obvious that this is so? Yes. In fact that's one of the main reasons that we studied the algorithm of Pratt, in section 4.10. Pratt's algorithm is exactly a method of producing a certificate with the aid of which we can quickly check whether a given integer is prime. The decision problem "Given \( n \), is it prime?" is thereby revealed to belong to NP, although that fact wasn't obvious at a glance. It is very clear that \( P = NP \). Indeed if \( Q \in P \) is some decision problem then we can verify membership in the language \( Q \) with the empty certificate. That is, we don't even need a certificate in order to do a quick calculation that checks membership in the language because the problem itself can be quickly solved. It seems natural to suppose that NP is larger than P. That is, one might presume that there are problems whose solutions can be quickly checked with the aid of a certificate even though they can’t be quickly found in the first place. No example of such a problem has ever been produced (and proved), nor has it been proved that no such problem exists. The question of whether or not NP = P is the one that we talked earlier as being perhaps the most important open question in the subject area today. It is fairly obvious that the class P is called the class P: \( P \) is the first letter of "Polynomial Time". But what does "NP" stand for? Stay tuned. The answer will appear in section 5.2. ## What is reducibility? Suppose that we want to solve a system of 100 simultaneous linear equations in 100 unknowns, of the form \( Ax = b \). We run down to the local software emporium and quickly purchase a program for $49.95 that solves such systems. When we get home and read the fine print on the label we discover, to our chagrin, that the system works only on systems where the matrix \( A \) is symmetric, and the coefficient matrix in the system that we want to solve is, of course, not symmetric. One possible response to this predicament would be to look for the solution to the system \( A^Tx = A^b \), in which the coefficient matrix \( A^T A \) is now symmetric. What we should have done would be to have reduced the problem that we really are interested in to an instance of a problem for which we have an algorithm. More generally, let \( Q \) and \( Q' \) be two decision problems. We will say that \( Q \) is quickly reducible to \( Q' \) whenever we are given an instance \( I \) of the problem \( Q \) we can convert it, with only a polynomial amount of labor, into an instance \( I' \) of \( Q' \) in such a way that \( I' \) and both have the same answer ("Yes" or "No"). If we buy a program to solve \( Q \), then we can use it to solve \( Q' \), with just a small amount of extra work. ## What is NP-completeness? How would you like to buy one program, for $49.95, that can solve 500 different kinds of problems? That’s what NP-completeness is about. To state it a little more carefully, a decision problem is NP-complete if it belongs to NP and every problem in NP is quickly reducible to it. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 112 Context: # Chapter 5: NP-completeness The implications of NP-completeness are numerous. Suppose we could prove that a certain decision problem \( Q \) is NP-complete. Then we could concentrate our efforts to find polynomial-time algorithms on just that one problem \( Q \). Indeed if we were to succeed in finding a polynomial-time algorithm to the instances of \( Q \) then we would automatically have found a fast algorithm for solving every problem in NP. How does that work? Take an instance \( I' \) of some problem \( Q' \in NP \). Since \( Q' \) is quickly reducible to \( Q \), we could transform the instance \( I' \) into an instance \( I \) of \( Q \). Then use the super algorithm that we found for problems in \( Q \) to decide \( I \). Altogether only a polynomial amount of time will have been used from start to finish. Let’s be more specific. Suppose that tomorrow morning we prove that the graph coloring problem is NP-complete, and that on the next morning you find a fast algorithm for solving it. Then consider some instance of the bin packing problem. Since graph coloring is NP-complete, the instance of bin packing can be quickly converted into an instance of graph coloring for which the “Yes/No” answer is the same. Now use the fast graph coloring algorithm that you found (congratulations, by the way!) on the converted problem. The answer you get is the correct answer for the original bin packing problem. So, a fast algorithm for some NP-complete problem implies a fast algorithm for every problem in NP. Conversely, suppose we can prove that it is impossible to find a fast algorithm for some particular problem \( Q \) in NP. Then we can’t find a fast algorithm for any NP-complete problem \( Q' \) either. For if we could then we would be able to solve instances of \( Q \) by quickly reducing them to instances of \( Q' \) and solving them. If we could prove that there is no fast way to test the primality of a given integer then we would have proved that it is fast to decide if a graph is \( k \)-colorable, because, as we will see, every graph problem \( P \) is NP-complete and primality testing is in NP. Think about that for a few moments, and the extraordinary beauty and structural unity of these computational problems will begin to reveal itself. To summarize, quick for one NP-complete problem implies quick for all NP-complete problems. There’s just one small detail to attend to. We’ve been discussing the economic advantages of keeping flocks of unison instead of sheep. If there aren’t any unicorns then the discussion is a little silly. NP-complete problems have all sorts of marvelous properties. It’s lovely that every problem in NP can be quickly reduced to just one NP-complete problem. But are there any NP-complete problems? After all, should there be a single computational problem with the property that every one of the diverse creatures that inhabit NP should be quickly reducible to it? Well, we have NP-complete problems, hordes of them, and proving that will occupy our attention for the next two sections. Here’s the plan. In section 5.2 we are going to talk about a simple computer, called a Turing machine. It is an idealized computer, and its purpose is to standardize ideas of compatibility and time of computation by referring all problems to the one standard machine. A Turing machine is an extremely simple finite-state computer, and when it performs a computation, a unit of computational labor will be very clearly and unambiguously describable. It turns out that the important aspects of polynomial time computability do not depend on the particular computer that is chosen as the model. The beauty of the Turing machine is that it is one such a strong concept that it can in principle perform any calculation that any other finite state machine can do, while at the same time it is logically equivalent to the machine being used for proving theorems about complexity. The microcomputer on your desktop might have been chosen as the standard against which polynomial time compatibility is measured. If that had been done then the class of \( P \) quickly solvable problems would scarcely have changed at all (the polynomials would be different but would still be polynomials), but the proofs that we humans would have to give in order to establish the relevant theorems would have gotten much more complicated because of the variety of different kinds of states modern computers have. Next, in section 5.3 we will prove that there is an NP-complete problem; in fact, it is called the satisfiability problem. Its status as an NP-complete problem was established by S. Cook in 1971, and from that work all later progress in the field has flowed. The proof uses the theory of Turing machines. The first NP-complete problem was the hardest one to find. We will find, in section 5.4, a few more NP-complete problems, so the reader will get some idea of the methods that are used in identifying them. Since nobody knows a fast way to solve these problems, various methods have been developed that give approximate solutions quickly, or that give exact solutions in fast average time, and so forth. The beautiful #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 113 Context: # 5.2 Turing Machines ## Exercises for section 5.1 1. Prove that the following decision problem belongs to P: Given integers \( K \) and \( a_1, \ldots, a_n \). Is the median of the \( a_i \)'s smaller than \( K \)? 2. Prove that the following decision problem is in NP: Given an \( n \times n \) matrix \( A \) of integer entries, is \( \det A = 0 \)? 3. For which of the following problems can you prove membership in P? - (a) Given a graph \( G \). Does \( G \) contain a circuit of length 4? - (b) Given a graph \( G \). Is \( G \) bipartite? - (c) Given integers. Is there a subset of them whose sum is an even number? - (d) Given \( n \) integers. Is there a subset of them whose sum is divisible by 3? - (e) Given a graph \( G \). Does \( G \) contain an Euler circuit? 4. For which of the following problems can you prove membership in NP? - (a) Given a set of integers and another integer \( K \). Is there a subset of the given integers whose sum is \( K \)? - (b) Given a graph \( G \) and an integer \( K \). Does \( G \) contain a path of length \( \geq K \)? - (c) Given a set of \( K \) integers. Is it true that not all of them are prime? - (d) Given a set of \( K \) integers. Is it true that all of them are prime? ## 5.2 Turing Machines A Turing machine consists of: - A doubly infinite tape, that is marked off into squares that are numbered as shown in Fig. 5.2.1 below. Each square can contain a single character from the character set that the machine recognizes. For simplicity, we can assume that the character set consists of just three symbols: '0', '1', and ' ' (blank). - A tape head that is capable of either reading a single character from a square on the tape or writing a single character on a square, or moving its position relative to the tape by an increment of one square in either direction. - A finite list of states such that at every instant the machine is in exactly one of those states. The possible states of the machine are, first of all, the regular states \( q_1, \ldots, q_n \), and second, three special states: - \( q_0 \): the initial state - \( q_y \): the final state in a problem to which the answer is ‘Yes’ - \( q_n \): the final state in a problem to which the answer is ‘No’ - A program (or program module, if we think of it as a pluggable component) that directs the machine through the steps of a particular task. ``` ... ... --- --- --- -2 ─→ 0 1 2 3 ... ... ``` Fig. 5.2.1: A Turing machine tape Image Analysis: ### Localization and Attribution: - **Image 1**: The page consists of one primary image—a diagram located centrally within the text, designated as Image 1. ### Object Detection and Classification: - **Image 1**: - **Objects Detected**: The image contains multiple squares arranged in a sequential manner, each square containing symbols or being blank. There are also labels underneath the squares. - **Classification**: This image can be categorized as a diagram depicting a Turing machine tape. ### Scene and Activity Analysis: - **Image 1**: - **Scene Description**: The image depicts a Turing machine tape, an abstract computational device used in theoretical computer science to understand the limits of what can be computed. - **Main Actors and Actions**: The main elements include labeled squares with symbols which depict the action of a tape head reading or writing a symbol. ### Text Analysis: - **Image Throughout the Page**: - **Extracted Text**: Title "5.2 Turing Machines", text describing "A Turing Machine..." and Fig. 5.2.1 caption: "A Turing machine tape". - **Content Analysis**: The text in this section describes the concept of a Turing machine, including its components: an infinite tape, a tape head, and a finite set of states. The figure caption highlights that the diagram is an illustration of the tape component of a Turing machine. ### Diagram and Chart Analysis: - **Image 1**: - **Diagram Description**: Image 1 is a diagram (Fig. 5.2.1) showing a sequence of squares that mimic a Turing machine tape. The squares are labeled “...”, “-2”, “-1”, “0”, “1”, “2”, “...” from left to right. - **Axes and Legends**: There are no typical axes or legends as this is a conceptual diagram, not a graph. - **Key Insights**: This diagram visualizes the tape of the Turing machine, indicating positions and possible contents (blank or symbols ‘0’ and ‘1’). ### Contextual Significance: - **Text Context**: The image is a part of a theoretical explanation of Turing machines within the larger context of computational theory and problem-solving approaches related to NP-completeness. - **Contribute to Overall Message**: This visual aids in the comprehension of how a Turing machine operates by illustrating its main component, facilitating a better understanding of the theoretical concepts discussed in the text. ### Perspective and Composition: - **Perspective**: The diagram is presented from a straightforward, head-on perspective for clarity, emphasizing the linear, sequential nature of the Turing machine tape. - **Composition**: Elements are arranged in a left-to-right sequence which mirrors the way the tape would be read or written by the Turing machine. ### Process Descriptions: - **Process Shown**: The process of how a Turing machine reads and writes symbols on an infinite tape is described. When a symbol is read, the machine transitions to a new state and performs an action based on the symbol and state. ### Trend and Interpretation: - **Trends Identified**: The visual shows a repeating and linear pattern of squares which represents the infinite nature of the Turing machine tape. - **Interpretation**: It reinforces the idea that theoretical machines like the Turing machine operate in a deterministic and procedural manner, following specified rules. ### Tables: - **Page Content**: The page intersperses table-like structures with exercises listing problems related to computational complexity (sections 5.1-5.2). - **Description**: Though not traditional tables, these organized lists are used to systematically challenge understanding of P, NP, and related problems in computer science theory. The combined text and diagrams on this page offer a comprehensive tutorial on Turing machines and exercises to practice understanding computational complexity. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 114 Context: # Chapter 5: NP-completeness One step of the program, therefore, goes from ``` (state, symbol) ⇒ (newstate, newsymbol, increment). (5.21) ``` If and when the state reaches ϕ or ψ the computation is over and the machine halts. The machine should be thought of as part hardware and part software. The programmer's job is, as usual, to write the software. To write a program for a Turing machine, what we have to do is to tell it how to make each and every one of the transitions (5.21). A Turing machine program looks like a table, in which, for every possible pair (state, symbol) that the machine might find itself in, the programmer has specified what the newstate, the newsymbol and the increment shall be. To begin a computation with a Turing machine we take the input string x of length B, say, that describes the problem that we want to solve, and we write x in squares 1, 2, ..., B of the tape. The tape head is then positioned over square 1, the machine is put into state ϕ, the program module that the programmer prepared is plugged into its slot, and the computation begins. The machine reads the symbol in square 1. It now is in state ϕ and has read symbol s, so it can consult the program module to find out what to do. The program instructs it to write at square 1 a newsymbol, to move the head either to square 0 or to square 2, and to enter a certain newstate, say α. The whole process is then repeated, possibly forever, but hopefully after finitely many steps the machine will enter the state ϕ or state ψ, at which moment the computation will halt with the decision having been made. If we want to watch a Turing machine in operation, we don’t have to build it. We can simulate one. Here is a pidgin-Pascal simulation of a Turing machine that can easily be turned into a functioning program. It is in two principal parts. The procedure `turnback` has for input a string x of length B, and for output it sets the Boolean variable accept to True or False, depending on whether the outcome of the computation is that the machine halted in state ϕ or ψ respectively. This procedure is the “hardware” part of the Turing machine. It doesn’t vary from one job to the next. ## Procedure `getnext` The procedure `getnext` is the program module of the machine, and it will be different for each task. Its inputs are the present state of the machine and the symbol that was just read from the tape. Its outputs are the newstate into which the machine goes next, the newsymbol that the tape head now writes on the current square, and the increment (±1) by which the tape head will now move. ```pascal procedure turnback(B: integer; x: array[1:B]: accept: Boolean); {simulates Turing machine action on input string x of length B} {write input string on tape in first B squares} for square := 1 to B do tape[square] := x[square]; {record boundaries of written-on part of tape} leftmost := 1; rightmost := B; {initialize tape head and state} state := 0; square := 1; while state ≠ 'Y' and state ≠ 'N' do read symbol at current tape square; if square ≤ leftmost or square ≥ rightmost then symbol := '∅' else symbol := tape[square]; {ask program module for state transition} getnext(state, symbol, newstate, newsymbol, increment); state := newstate; {update boundaries and write new symbol}; if square ≤ rightmost then leftmost := square; tape[square] := newsymbol; {move tape head} square := square + increment; end; while (state ≠ 'Y') accept := (state = 'Y'); end; ``` #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 115 Context: Now let's try to write a particular program module `gonetto`. Consider the following problem: given an input string \( x \), consisting of 0s and 1s, of length \( B \). Find out if it is true that the string contains an odd number of 1s. We will write a program that will scan the input string from left to right, and at each moment the machine will be in state 0 if it has so far scanned an even number of 1s, in state 1 otherwise. In Fig. 5.2.2 we show a program that will get the job done. | state | symbol | newstate | newsymbol | increment | |-------|--------|----------|-----------|-----------| | 0 | 0 | 0 | 0 | +1 | | 0 | 1 | 1 | 1 | +1 | | 0 | blank | \( q_v \) | blank | -1 | | 1 | 0 | 1 | 0 | +1 | | 1 | 1 | 0 | 1 | +1 | | 1 | blank | \( q_v \) | blank | -1 | **Fig. 5.2.2:** A Turing machine program for bit parity **Exercise.** Program the above as procedure `gonetto`, run it for some input string, and print out the state of the machine, the contents of the tape, and the position of the tape head after each step of the computation. In the next section we are going to use the Turing machine concept to prove Cook's theorem, which is the assertion that a certain problem is NP-complete. Right now let’s review some of the ideas that have already been introduced from the point of view of Turing machines. We might immediately notice that some terms were just a little bit fuzzy before are now much more sharply in focus. Take the notion of polynomial time, for example. To make that idea precise one needs a careful definition of what 'the length of the input bit string' means, and what one means by the number of ‘steps’ in a computation. But on a Turing machine both of these ideas come through with crystal clarity. The input bit string \( x \) is what we write on the tape to get things started, and its length is the number of tape squares it occupies. A ‘step’ in a Turing machine calculation is obviously a single call to the program module. A Turing machine calculation was done in ‘time \( P(B) \)’ if the input string occupied \( B \) tape squares and the calculation took \( P(B) \) steps. Another word that we have been using without real precision is ‘algorithm’. We all understand informally what an algorithm is. But now we understand formally how. An algorithm for a problem is a program module for a Turing machine that will cause the machine to halt after finitely many steps in state ‘Y’ for every instance whose answer is ‘Yes’, and after finitely many steps in state ‘N’ for every instance whose answer is ‘No’. Now let’s see how the idea of a Turing machine can clarify the description of the class NP. This is the class of problems for which the decisions can be made quickly if the input strings are accompanied by suitable certificates. By a certificate, we mean a finite string of Turing machine tape, consisting of 0 or more squares, each of which contains a symbol from the character set of the machine. A certificate can be loaded into a Turing machine as follows. If the certificate contains \( m > 0 \) tape squares, then replace the segment from square number \( -m \) to square number \( -1 \) of the tape. We then write a verifying program for the program module in which the program verifies that the string \( x \) is indeed a word in the language of the machine, and in the course of the verification the program is quite free to examine the certificate as well as the problem instance. A Turing machine that is being used as a verifying computer is called a nondeterministic machine. The hardware is the same, but the manner of input and the question that is being asked are different from the #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 116 Context: If we choose the truth values (T, T, F) for the variables, respectively, then the four clauses would acquire the truth values (T, T, T, F), and so this would be a satisfying truth assignment for the set of clauses. There are only eight possible ways to assign truth values to three variables, and after a little. more experimentation we might find out that these clauses would in fact be satisfied if we were to make the assignment (T,T,I) (how can we recognize a set of clauses that is satisfied by assigning to every variable the value ‘T T ?). The example already leaves one with the feeling that SAT might be a tough computational problem, because there are Zn possible sets of truth values that we might have to explore ff we were to do an exhaustive search. It is quite clear, however, that this problem belongs to NP. Indeed, it is a decision problem. Furthermore we can easily assign a certificate to every set of clauses for which the answer to SAT is ‘Yes, the clauses ``` - **Text Analysis:** - The text discusses the concept of NP-completeness and introduces Cook's Theorem. - It details the satisfiability problem (SAT) and provides an example to explain the concept of satisfiability. - The text is significant as it explains foundational concepts in computational complexity theory, particularly the classification of problems and the importance of NP-complete problems. ### 9. Perspective and Composition - **Perspective:** - The image is a direct scan or shot of a textbook page, taken from a straight-on perspective, ensuring all text is clearly visible and readable. - **Composition:** - The composition consists of structured text with sections, paragraphs, and an example. The page layout follows a typical textbook style with chapter headings, section numbers, and definitions. ### 13. Graph Numbers - No graphs are present in the image, thus no numerical data points to list. ### Additional Aspects #### Ablaufprozesse (Process Flows): - The image explains the process of determining the satisfiability of a set of clauses by assigning truth values to variables and evaluating the clauses based on those assignments. #### Prozessbeschreibungen (Process Descriptions): - The process description involves assigning truth values to Boolean variables, determining the truth value of literals, and subsequently the truth value of clauses to check for satisfiability. #### Typen Bezeichnung (Type Designations): - Types or categories specified include: - Boolean variables - Literals - Clauses #### Trend and Interpretation: - The text discusses a trend in computational problem classification, particularly focusing on the complexity class NP and NP-complete problems. It interprets the difficulty of satisfiability problems and their importance in computational theory. #### Tables: - No tables are present in the image. Overall, the image contributes significantly to the understanding of NP-completeness and the satisfiability problem, explaining key concepts and processes in computational complexity theory. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 116 Context: # Chapter 5: NP-completeness The class NP (Nondeterministic Polynomial) consists of these decision problems for which there exists a fast (polynomial time) algorithm that will verify, given a problem instance string \( s \) and a suitable certificate \( C(s) \), that \( s \) belongs to the language recognized by the machine, and for which, if \( s \) does not belong to the language, no certificate would cause an accepting computation to ensue. ## 5.3 Cook's Theorem The NP-complete problems are the hardest problems in NP, in the sense that if \( Q' \) is any decision problem in NP and \( Q \) is NP-complete problem, then every instance of \( Q' \) is polynomially reducible to an instance of \( Q \). As we have already remarked, the surprising thing is that there is an NP-complete problem at all, since it is not immediately clear why any single problem should hold the key to the polynomial time solvability of every problem in the class NP. But there is one. As soon as we see why there is one, then we'll be able to see more easily why there are hundreds of them, including many computational questions about discrete structures such as graphs, networks and games and about optimization problems, about algebraic structures, formal logic, and so forth. ### Here is the satisfiability problem, the first problem that was proved to be NP-complete, by Stephen Cook in 1971. We begin with a list of (Boolean) variables \( x_1, \ldots, x_n \). A literal is either one of the variables \( x_i \) or the negation of one of the variables, as \( \neg x_i \). There are \( 2n \) possible literals. A clause is a set of literals. The rules of the game are these. We assign the value ‘True’ (T) or ‘False’ (F) to each one of the variables. Having done that, each one of the literals inherits a truth value, namely a literal \( l \) has the same truth or falsity as the corresponding variable \( x_i \), and a literal \( \neg x_i \) has the opposite truth value from that of the variable \( x_i \). Finally each of the clauses also inherits a truth value from this process, and it is determined as follows. A clause has the value ‘True’ if and only if at least one of the literals in that clause has the value ‘T’; and otherwise it has the value ‘F’. Hence starting with an assignment of truth values to the variables, some true and some false, we end up with a determination of the truth values of each of the clauses, some true and some false. #### Definition A set of clauses is satisfiable if there exists an assignment of truth values to the variables that makes all the clauses true. Think of the word ‘or’ as being between each of the literals in a clause, and the word ‘and’ as being between the clauses. ### The satisfiability problem (SAT). Given a set of clauses. Does there exist a set of truth values (T or F), one for each variable, such that every clause contains at least one literal whose value is T (i.e., such that every clause is satisfied)? **Example:** Consider the set \( x_1, x_2, x_3 \) of variables. From these we might manufacture the following list of four clauses: - \( \{x_1, x_2\} \) - \( \{x_1, \neg x_3\} \) - \( \{x_2, x_3\} \) - \( \{\neg x_1, x_3\} \) If we choose the truth values (T, T, F) for the variables, respectively, then the four clauses would assign the truth values (T, T, F), and so this would not be a satisfying truth assignment for the set of clauses. There are only eight possible ways to assign truth values to three variables, and after a little more experimentation we might find out that these clauses would in fact be satisfied if we were to make the assignments (T, T, T) (how can we recognize a set of clauses that is satisfied by assigning to every variable the value ‘T’?) The example already leaves one with the feeling that SAT might be a tough computational problem, because there are \( 2^n \) possible sets of truth values that we might have to explore if we were to do an exhaustive search. It is quite clear, however, that this problem belongs to NP. Indeed, it is a decision problem. Furthermore we can easily assign a certificate to every set of clauses for which the answer to SAT is ‘Yes’, the clauses. Image Analysis: Based on the provided visual content, here is a detailed analysis according to the specified aspects: ### 1. Localization and Attribution - **Image Location:** The image is a single page from a textbook, located within a single view. - **Image Number:** Image 1 ### 4. Text Analysis - **Detected Text:** ``` Chapter 5: NP-completeness situation with a deterministic Turing machine, in which we decide whether or not the input string is in the language, without using any certificates. The class NP (Nondeterministic Polynomial) consists of those decision problems for which there exists a fast (polynomial time) algorithm that will verify, given a problem instance string and a suitable certificate C(x), that x belongs to the language recognized by the machine, and for which, if x does not belong to the language, no certificate would cause an accepting computation to ensue. 5.3 Cook’s Theorem The NP-complete problems are the hardest problems in NP, in the sense that if Q is any decision problem in NP and Q is an NP-complete problem, then every instance of Q is polynomially reducible to any instance of Q. As we have already remarked, the surprising thing is that there is an NP-complete problem at all, since it is not immediately clear why any single problem should hold the key to the polynomial time solvability of every problem in the class NP. But there is one. As soon as we see why there is one, then we’ll be able to see more easily why there are hundreds of them, including many computational questions about discrete structures such as graphs, networks and games and about optimization problems, about algebraic structures, formal logic, and so forth. Here is the satisfiability problem, the first problem that was proved to be NP-complete by Stephen Cook in 1971. We begin with a list of (Boolean) variables x1, ..., xn. A literal is either one of the variables xi or the negation of one of the variables, as x1. There are 2n possible literals. A clause is a set of literals. The rules of the game are these. We assign the value ‘True’ (T) or ‘False’ (F), to each one of the variables. Having done that, each one of the literals inherits a truth value, namely a literal xi has the same truth or falsity as the corresponding variable xi, and a literal xi has the opposite truth value from that of the variable xi. Finally each of the clauses also inherits a truth value from this process, and it is determined as follows. A clause has the value ‘T’ if and only if at least one of the literals in that clause has the value ‘T’; and otherwise it has the value ‘F’. Hence starting with an assignment of truth values to the variables, some true and some false, we end up with a determination of the truth values of each of the clauses, some true and some false. Definition. A set of clauses is satisfiable if there exists an assignment of truth values to the variables that makes all of the clauses true. Think of the word ‘or’ as being between each of the literals in the clause, and the word ‘and’ as being between the clauses. The satisfiability problem (SAT). Given a set of clauses. Does there exist a set of truth values (= T or F), one for each variable, such that every clause contains at least one literal whose value is T (i.e., such that every clause is satisfied)? Example: Consider the set x1, x2, x3 of variables. From these we might manufacture the following list of four clauses: {x1, x2}, {x1, x3}, {x2, x3}, {x1, x3}. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 116 Context: # Chapter 5: NP-completeness The situation with a deterministic Turing machine, in which we decide whether or not the input string is in the language, without using any certificates. The class NP (Nondeterministic Polynomial) consists of those decision problems for which there exists a fast (polynomial time) algorithm that will verify, given a problem instance string and a suitable certificate \(C\), that \(x\) belongs to the language recognized by the machine, and for which, if \(x\) does not belong to the language, no certificate would cause an accepting computation to ensue. ## 5.3 Cook's Theorem The NP-complete problems are the hardest problems in NP, in the sense that if \(Q'\) is any decision problem in NP and \(Q\) is NP-complete, then every instance of \(Q'\) is polynomially reducible to an instance of \(Q\). As we have already remarked, the surprising thing is that there is an NP-complete problem at all, since it is not immediately clear why any single problem should hold the key to the polynomial time solvability of every problem in the class NP. But there is one. As soon as we see why there is one, then we'll be able to see more easily why there are hundreds of them, including many computational questions about discrete structures such as graphs, networks and games and about optimization problems, about algebraic structures, formal logic, and so forth. ### Here is the satisfiability problem, the first problem that was proved to be NP-complete, by Stephen Cook in 1971. We begin with a list of (Boolean) variables \(x_1, \ldots, x_n\). A literal is either one of the variables \(x_i\) or the negation of one of the variables, as \( \neg x_i\). There are 2n possible literals. A clause is a set of literals. The rules of the game are these: We assign the value 'True' (T) or 'False' (F), to each one of the variables. Having done that, each one of the literals inherits a truth value; namely, a literal \(x_i\) has the same truth or falsity as the corresponding variable \(x_i\), and a literal \(\neg x_i\) has the opposite truth value from that of the variable \(x_i\). Finally each of the clauses also inherits a truth value from this process, and it is determined as follows. A clause has the value 'T' if and only if at least one of the literals in that clause has the value 'T'; and otherwise it has the value 'F'. Hence starting with an assignment of truth values to the variables, some true and some false, we end up with a determination of the truth values of each of the clauses, some true and some false. **Definition**: A set of clauses is satisfiable if there exists an assignment of truth values to the variables that makes all the clauses true. Think of the word 'or' as being between each of the literals in a clause, and the word 'and' as being between the clauses. ### The satisfiability problem (SAT). Given a set of clauses, Does there exist a set of truth values (T or F), one for each variable, such that every clause contains at least one literal whose value is T (i.e., such that every clause is satisfied)? **Example**: Consider the set \(x_1, x_2, x_3\) of variables. From these we might manufacture the following list of four clauses: 1. \( \{x_1, x_2\} \) 2. \( \{x_1, x_3\} \) 3. \( \{x_2, x_3\} \) 4. \( \{ \neg x_1, x_3\} \) If we choose the truth values (T, T, F) for the variables, respectively, then the four clauses would assume the truth values (T, T, F), and so this would not be a satisfying truth assignment for the set of clauses. There are only eight possible ways to assign truth values to three variables, and after a little more experimentation we might find out that these clauses would in fact be satisfied if we were to make the assignments \(T, T, F\) (how can we recognize a set of clauses that is satisfied by assigning to every variable the value 'T'?). The example already leaves one with the feeling that SAT might be a tough computational problem, because there are \(2^n\) possible sets of truth values that we might have to explore if we were to do an exhaustive search. It is quite clear, however, that this problem belongs to NP. Indeed, it is a decision problem. Furthermore we can easily assign a certificate to every set of clauses for which the answer to SAT is 'Yes', the clauses. Image Analysis: ### Localization and Attribution: - **Image 1**: Entire page. ### Text Analysis: - **Image 1**: - The image contains a single page of text extracted from a document. The headline is "Chapter 5: NP-completeness". - The text comprehensively discusses the complexities related to NP problems, defining the class NP, and introducing **Cook's Theorem**. - Important terms include: "deterministic Turing machine", "Nondeterministic Polynomial", and "Cook's Theorem". - Key concepts explained are: - NP-complete problems and their significance. - The "satisfiability problem (SAT)". - The rules for assigning truth values to literals (both variables and their negations). - An example is provided to illustrate the assignment of truth values to clauses. ### Scene and Activity Analysis: - **Image 1**: - This is a static scene comprising a dense page of academic text. - There are no people or clear activities depicted beyond textual content. ### Diagram and Chart Analysis: - None identified. ### Product Analysis: - None identified. ### Anomaly Detection: - None identified. ### Color Analysis: - **Image 1**: - This is a black-and-white image, with black text on a white background. The lack of color can impact reading ease and emphasizes the scholarly nature of the document. ### Perspective and Composition: - **Image 1**: - The perspective is a straightforward, head-on view typical in scanned documents or digital texts. - The composition is uniformly structured with a heading and paragraph format, facilitating academic reading. ### Contextual Significance: - **Image 1**: - This page appears to be a part of an academic textbook or scholarly document discussing theoretical computer science. - It significantly contributes to the overall understanding of NP-completeness by explaining foundational concepts and their implications. ### Diagram and Trend Analysis: - None identified. ### Graph Numbers: - None identified. ### Ablaufprozesse (Process Flows): - **Image 1**: - The process described involves determining the satisfiability of Boolean variables by methodically assigning truth values and analyzing the results. ### Prozessbeschreibungen (Process Descriptions): - **Image 1**: - The text details the step-by-step process of evaluating the satisfiability of Boolean variables, emphasizing logical clauses and truth value assignments. ### Typen Bezeichnung (Type Designations): - **Image 1**: - Different types defined are variables, literals, clauses, and the satisfiability problem (SAT). ### Trend and Interpretation: - **Image 1**: - The trend reflects an academic approach towards explaining and validating complex computational problems. - The logical flow from definitions to examples enhances comprehension. ### Tables: - None identified. Overall, the page offers a thorough introduction and explanation of NP-completeness and Cook's Theorem, essential for understanding computational complexity. The logical structure and step-by-step exposition serve as a didactic method to facilitate the reader's grasp of theoretical concepts. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 117 Context: # 5.3 Cook’s Theorem **Theorem 5.3.1.** (S. Cook, 1971): SAT is NP-complete. Before we carry out the proof, it may be helpful to give a small example of the reducibility ideas that we are going to use. ## Example: Reducing graph-coloring to SAT Consider the graph \( G \) of four vertices that is shown in Fig. 5.3.1, and the decision problem “Can the vertices of \( G \) be properly colored in 3 colors?” Let’s see how that decision problem can be reduced to an instance of SAT. We will use 12 Boolean variables: the variable \( x_{ij} \) corresponds to the assertion that “vertex \( i \) has been colored in color \( j \)” (i = 1, 2, 3; j = 1, 2, 3). The instance of SAT that we construct has 31 clauses. The first 16 of these are: - \( C(i) := \{x_{i1}, x_{i2}, x_{i3}\} \quad (i = 1, 2, 3, 4) \) - \( T(i) := \{x_{i1}, x_{i2}\} \quad (i = 1, 2, 3) \) - \( U(i) := \{x_{i2}, x_{i3}\} \quad (i = 1, 2, 3, 4) \) (5.3.1) - \( V(i) := \{x_{i1}, x_{i3}\} \quad (i = 1, 2, 3, 4) \) In the above, the four clauses \( C(i) \) assert that each vertex has been colored in at least one color. The clauses \( T(i) \) say that no vertex has both color 1 and color 2. Similarly, the clauses \( U(i) \) guarantee that no vertex has been colored 1 and 3 (resp. 2 and 3). Next we have to construct the clauses that will ensure us that the two endpoints of an edge of the graph are never the same color. For this purpose we define, for each edge \( e \) of the graph \( G \) and a clause \( D(e) \) as follows. Let \( a \) and \( b \) be the two endpoints of the edge \( e \) of \( G \): \( D(e) := \neg(x_{ai} \land x_{bi}) \), which asserts that both endpoints of the edge have the same color of \( i \). The original instance of the graph coloring problem has now been reduced to an instance of SAT. In more detail, there exists an assignment of values for the 12 Boolean variables \( x_{11}, x_{12}, \ldots, x_{34} \) such that each of the 31 clauses contains at least one literal that subcase is true if and only if the vertices of the graph \( G \) can be properly colored in three colors. The graph is 3-colorable if and only if the clauses are satisfiable. It is clear that if we have an algorithm that will solve SAT, then we can also solve graph coloring problems. A few remedies of thought will convince the reader that the transformation of one problem to the other that was carried out above involves only a polynomial amount of computation, despite the seemingly large number of variables and clauses. Hence graph coloring is quickly reducible to SAT. ![Fig. 5.3.1: A 3-coloring problem](attachment:image) Image Analysis: ### Comprehensive Image Analysis: #### 1. Localization and Attribution: - **Image 1**: Positioned centrally on the page, labeled as "Fig. 5.3.1." #### 2. Object Detection and Classification: - **Image 1**: Shows a graph with four vertices and edges connecting them in a specific pattern. The vertices are labeled and connected by edges forming a square with a diagonal. #### 3. Scene and Activity Analysis: - **Image 1**: The entire scene involves a diagram illustrating a 3-coloring problem in graph theory. The vertices are numerically labeled (1, 2, 3, 4), and the graph represents how these vertices are connected. - The main activity shown is the interaction of vertices and edges, demonstrating the 3-coloring problem. #### 4. Text Analysis: - The image is accompanied by detailed text explaining Theorem 5.3.1 (S. Cook, 1971: SAT is NP-complete) and providing an example of reducing graph-coloring to SAT. - Key points from the text: - **Fig. 5.3.1: A 3-coloring problem**: Describes the graph of four vertices that need to be properly colored in three colors. - The text explains the construction of SAT that has 31 clauses with specific assignments to ensure proper coloring. - The instance of SAT is broken down into clauses \(C(i)\), \(T(i)\), \(U(i)\), and \(V(i)\) ensuring that each vertex is colored in a specific way, and that adjacent vertices are assigned different colors. - Finally, it connects the clause construction back to solving the original graph coloring problem by reiterating SAT reduction. #### 10. Contextual Significance: - The image and accompanying text are part of a larger discussion on Cook's Theorem within the domain of computer science and graph theory. - The graph (Fig. 5.3.1) serves as a visual guide to help readers understand the step-by-step reduction process from a 3-coloring problem to SAT, assisting in grasping the NP-completeness of the problem. #### 11. Metadata Analysis: - No metadata is available from the provided image for additional analysis. #### 12. Graph and Trend Analysis: - **Graph Trend**: - The graph illustrates vertex connections that are essential for understanding the 3-coloring problem. - The main trend is the connectivity pattern of the vertices (pairs: (1,2), (2,3), (3,4), (4,1), (1,3)) which guides the formulation of clauses in SAT. ### Additional Aspects: - **Prozessbeschreibungen (Process Descriptions)**: - The process described involves reducing a graph-coloring problem to an SAT instance, detailing the specific clauses and their derivations. - Each described clause ensures the correct application of graph-coloring rules in the context of SAT. ### Conclusion: - The image and text collectively illustrate a segment of computational theory focusing on the 3-coloring problem and its reduction to SAT, providing a visual and narrative guide to understand the NP-complete classification. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 118 Context: # Chapter 5: NP-completeness ## Proof of Cook's theorem We want to prove that SAT is NP-complete, i.e., that every problem in NP is polynomially reducible to an instance of SAT. Hence let \( Q \) be some problem in NP and let \( I \) be an instance of problem \( Q \). Since \( Q \) is in NP there exists a Turing machine that recognizes encoded instances of problem \( Q \) accompanied by a suitable certificate, in polynomial time. Let \( TMQ \) be such a Turing machine, and let \( P(n) \) be a polynomial in its argument with the property that \( TMQ \) recognizes every pair \( (x, C(y)) \), where \( x \) is a word in the language \( Q \) and \( C(y) \) is its certificate, in time \( S(P(n)) \), where \( n \) is the length of \( x \). We intend to construct, corresponding to each word \( i \) in the language \( Q \), an instance \( f(i) \) of SAT for which the answer is "Yes, the clauses are all simultaneously satisfiable." Conversely, if the word \( i \) is not in the language \( Q \), the clauses will not be satisfiable. The idea can be summarized like this: the instance of SAT that will be constructed will be a collection of clauses that together express the fact that there exists a certificate that causes Turing machine \( TMQ \) to an accepting calculation. Therefore, in order to test whether or not the word \( i \) belongs to the language, it suffices to check that the collection of clauses is satisfiable. To construct an instance of SAT means that we are going to define a number of variables, of literals, and of clauses, in such a way that the clauses are satisfiable if and only if \( i \) is in the language \( Q \), i.e., the machine \( TMQ \) accepts \( x \) and its certificate. What we must do, then, is to express the accepting computation of the Turing machine as the simultaneous satisfaction of a number of logical propositions. It is precisely here that the relative simplicity of a Turing machine allows us to enumerate all of the possible paths to an accepting computation in a way that would be quite unthinkable with a 'real' computer. Now we will describe the Boolean variables that will be used in the clauses under construction. - Variable \( Q_i \): is true if step \( i \) (after step \( i \), symbol \( a \) is in tape square \( j \)). - Variable \( T_{j,k} \): (after step \( i \), the tape head is positioned over square \( j \)). Let’s count the variables that we’ve just introduced. Since the Turing machine \( TMQ \) does its calculation in time \( P(n) \), it follows that the tape head will never venture more than \( 2 + P(n) \) squares away from its starting position. Therefore the subscript \( j \), which runs through the tape squares that are scanned during the computation, can assume only \( O(P(n)) \) different values. Index \( k \) runs over the letters in the alphabet that the machine can read, so it takes at most \( O(P(n)) \) values. Finally, \( i \) indexes the states of the Turing machine, and there is only some fixed finite number, \( K \), say, of states that \( TMQ \) might be in. Hence there are altogether \( O(P(n)^2) \) variables, a polynomial number of variables. Is it true that every random assignment of true or false values to each of these variables corresponds to an accepting computation on \( (x, C(y)) \)? Certainly not. For example, if we weren’t careful we might assign true values to \( T_{j,4} \) and \( T_{j,3} \), thereby giving the wrong bearings on the tape transport mechanism! (Why?) Our variables taken, then, will be to describe precisely the conditions under which a set of clauses assigned to the variables listed above actually defines a possible accepting calculation for \( (x, C(y)) \). Then we will be sure that whatever set of satisfying values of the variables might be found by solving the SAT problem, they will determine a real accepting calculation of the machine \( TMQ \). This will be done by requiring that a number of clauses be all true (“satisfied”) at once, where each clause will express one necessary condition. In the following, the bold face type will describe, in words, the condition that we want to express, and it will be followed by the formal set of clauses that actually expresses the condition on input to SAT. At each step, the machine is in at least one state. Here at least one of the \( K \) available state variables must be true. This leads to the first set of clauses: #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 119 Context: # At step P(n) the machine is in state qₓ. One for each step i of the computation: ``` {Q₁,₁, Q₂,₁, …, Qₖ,₁} ``` Since I assume O(P(n)) values, these are O(P(n)) clauses. ## At each step, the machine is not in more than one state Therefore, for each step i, and each pair j, j' of distinct states, the clause ``` {Qᵢ,₁, Qⱼ,ⁿ} ``` must be true. These are O(P(n)) additional clauses to add to the list, but still more are needed. ## At each step, each tape square contains exactly one symbol from the alphabet of the machine. This leads to two lists of clauses which require, first, that there is at least one symbol in each square at each step, and second, that there are not two symbols in each square at each step. The clauses that do this are: ``` {S₁,₁, S₂,₁, …, Sₐ,₁} ``` where A is the number of letters in the machine's alphabet, and ``` {Sᵢ,ₗ, Sᵢ,ₖ} ``` for each step i, square j, and pair k', k'' of distinct symbols in the alphabet of the machine. The reader will by now have gotten the idea of how to construct the clauses, so for the next three categories we will simply list the functions that must be performed by the corresponding lists of clauses, and leave the construction of the clauses as an exercise. ## At each step, the tape head is positioned over a single square. Initially the machine is in state 0, the head is over square 1, the input string z is in squares 1 to n, and C(z) (the input certificate of z) is in squares 0, -1, ... -P(n). ## At step P(n) the machine is in state qᵧ. The last set of restrictions is a little trickier: ### At each step the machine moves to its next configuration (state, symbol, head position) in accordance with the application of its program module to its previous (state, symbol). To find the clauses that will do this job, consider first the following condition: the symbol in square j of the tape cannot change during step i of the computation if the tape head isn't positioned there at that moment. This translates into the collection ``` {Tᵢ,j, Sᵢ,j, S₁,j, k} ``` of clauses, one for each triple (i, j, k) = (state, square, symbol). These clauses express the condition in the following way: either (at time t) the tape head is positioned over square j (Tᵢ is true) or else the head is not positioned there, in which case either symbol k is not in the jth square before the step or symbol k is [still] in the jth square after the step is executed. It remains to express the fact that the transitions from one configuration of the machine to the next are the direct results of the operation of the program module. The three sets of clauses that do this are: ``` {Fᵢ,j, Qₖ,ₗ, Sᵢ,j, Tᵢ,j+1, H₁,ₙC} ``` ``` {Tᵢ, Qₗ, Sᵢ,j, Qₖ,ₕ} ``` ``` {Tᵣ, Qₖ, Sᵢ,j, S₁,j+1, H} ``` #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 121 Context: # 5.4 Some other NP-complete problems To prove the claim, first suppose that (5.4.1) is satisfied by some assignment \( x^* \). Then one, at least, of the \( k \) literals \( z_1, \ldots, z_k, \) say \( z_r \), has the value \texttt{True}. Then we can satisfy all \( k - 2 \) of the transformed clauses (5.4.2) by assigning \( z_s = \texttt{True} \) for \( s < r - 2 \) and \( z_t = \texttt{False} \) for \( s > r - 2 \). It is easy to check that each one of the \( k - 2 \) new clauses is satisfied. Conversely, suppose that all of the new clauses are satisfied by some assignment of the \( x's \) and the \( z's \). We will show that at least one of the \( x's \) must be \texttt{True}, so that the original clause will be satisfied. Suppose, to the contrary, that all of the \( x's \) are false. Since, in the new clauses, none of the \( x's \) are negated, the fact that the new clauses are satisfied tells us that they would remain satisfied without any of the \( x's \). Hence the clauses \[ \{x_1\}, \{x_2\}, \{x_3\}, \ldots, \{x_{k-4}, \neg x_{k-3}\} \] are satisfied by the values of the \( x's \). If we scan the list from left to right we discover, in turn, that \( x_1 \) is true, \( x_2 \) is true, \ldots, and finally, much to our surprise, that \( x_{k-3} \) is true, and \( x_k \) is also false, a contradiction which establishes the truth of the claim made above. The observation that the transformations just discussed can be carried out in polynomial time completes the proof of theorem 5.4.1. We remark, in passing, that the problem \texttt{2SAT} is in P. Our collection of NP-complete problems is growing. Now we have two, and a third is on the way. We will show next how to reduce \texttt{3SAT} to a graph coloring problem, thereby proving ## Theorem 5.4.2. The graph vertex coloring problem is NP-complete. **Proof:** Given an instance of \texttt{SAT}, that is to say, given a collection of \( c \) clauses, involving \( n \) variables and having at most three literals per clause, we will construct, in polynomial time, a graph \( G \) with the property that its vertices can be properly colored in \( n + 1 \) colors if and only if the given clauses are satisfiable. We will assume that \( n > 4 \), the contrary case being trivial. The graph \( G \) will have \( n + k \) vertices: \[ \{x_1, \ldots, x_n\}, \{x_{n+1}, \ldots, x_{n+k}\}, \{y_1, \ldots, y_k\}, \{C_1, \ldots, C_c\} \] Now we will describe the set of edges of \( G \). First each vertex \( x_i \) is joined to \( x_j \) (for \( i, j = 1, \ldots, n \)). Next, every vertex \( x_j \) is joined to every other vertex \( x_j \) (for \( j \neq i \)), and to every vertex \( x_j \) (for \( j \neq j \)). Vertex \( x_i \) is connected to \( C_j \) if \( x_i \) is not one of the literals in \( C_j \). Finally, it is connected to \( C_j \) if \( x_i \) is not one of the literals in \( C_j \). May we interrupt the proceedings to say again why we’re doing all of this? You have just read the description of a certain graph \( G \). The graph is one that can be drawn as soon as someone hands us a \texttt{SAT} problem. We described the graph by listing its vertices and then listing its edges. What does the graph do for us? We will suppose that we have just bought a computer program that can decide if graphs are colorable in a given number of colors. We paid \$9.95 for it, and we’d like to use it. But the first problem that needs solving happens to be a \texttt{3SAT} problem, not a graph coloring problem. We aren’t so easily discouraged, though. We convert the \texttt{3SAT} problem into a graph \( G \) that is \( (n + 1) \)-colorable if and only if the original \texttt{3SAT} problem was satisfiable. Now we can get our money's worth by running the graph coloring program even though what we really wanted to do was solve a \texttt{3SAT} problem. Image Analysis: 1. **Localization and Attribution:** - **Page Layout:** The page contains a large block of text structured into several paragraphs and sections. - **Text Columns:** There’s a single column of text on the page. 2. **Object Detection and Classification:** - **Objects:** The image consists only of text and mathematical notations. There are no other objects or images present. 3. **Scene and Activity Analysis:** - **Entire Scene:** The scene is textual, featuring detailed explanations and proofs related to computational problems (specifically NP-complete problems) in computer science. - **Main Actors and Actions:** The main actors are mathematical variables (like `x_i, z_i`) and abstract computational problems (3SAT, 2SAT). The primary actions are proving claims and explaining logical connections in computational theory. 4. **Text Analysis:** - **Detected Text:** - Title: "5.4 Some other NP-complete problems" - Subsection: "Theorem 5.4.2. The graph vertex coloring problem is NP-complete." - Key Variables: `x_i, z_i, x_s, z_s` - Key Phrases: "prove the claim", "theorem", "graph vertex coloring", "NP-complete", "2SAT", "3SAT" - **Content Analysis:** - **Section 5.4:** Discusses various NP-complete problems. - **Proofs and Claims:** Provides proofs about NP-complete problems, specifically showing how 3SAT can be reduced to a graph coloring problem. - **Significance:** This text is significant in the context of computational complexity theory, demonstrating how different computational problems are related through reductions, proving the NP-completeness of certain problems. 9. **Perspective and Composition:** - **Perspective:** The image is viewed from a front-facing, flat angle typical for reading documents. - **Composition:** The page is composed of several blocks of text without images or graphics. The text includes titles, subsections, mathematical notations, and paragraphs structured to guide the reader logically through the arguments and proofs. Overall, the page is from a mathematical or computational theory book discussing advanced topics in computational complexity, outlining proofs to establish the NP-completeness of certain problems. The text is analytically dense and seeks to convey rigorous theoretical arguments to a knowledgeable audience. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 123 Context: # Maximum clique We are given a graph \( G \) and an integer \( K \). The question is to determine whether or not there is a set of \( K \) vertices in \( G \), each of which is joined by an edge to all of the others. ## Edge coloring Given a graph \( G \) and an integer \( K \), can we color the edges of \( G \) in \( K \) colors, so that whenever two edges meet at a vertex, they will have different colors? Let us refer to an edge coloring of this kind as a proper coloring of the edges of \( G \). A beautiful theorem of Vizing's deals with this question. If \( \Delta \) denotes the largest degree of any vertex in the given graph, the Vizing's theorem asserts that the edges of \( G \) can be properly colored in either \( \Delta \) or \( \Delta + 1 \) colors. Since it is obvious that at least \( \Delta \) colors will be needed, this means that the edge chromatic number is in doubt by only one unit, for every graph \( G \). Nevertheless, the decision as to whether the correct answer is \( \Delta \) or \( \Delta + 1 \) is NP-complete. ## Hamilton path In a given graph \( G \), is there a path that visits every vertex of \( G \) exactly once? ## Target sum Given a finite set of positive integers whose sum is \( S \)? The above list, together with SAT, 3SAT, Travelling Salesman and Graph Coloring, constitutes a modest sampling of the class of these seemingly intractable problems. Of course, it must not be assumed that every problem that "sounds like" an NP-complete problem is necessarily so hard. If for example we ask for an Euler path instead of a Hamilton path (i.e., if we want to traverse edges rather than vertices) the problem would no longer be NP-complete, and in fact it would be in P, thanks to Theorem 1.6.1. As another example, the fact that one can find the edge connectivity of a given graph in polynomial time (see section 3.8) is rather amazing considering the quite difficult appearance of the problem. One of our motivations for including the network flow algorithms in this book was, indeed, to show how very sophisticated algorithms can sometimes prove that seemingly hard problems are in fact computationally tractable. # Exercises for section 5.4 1. Is the claim that we made and proved above (just after (5.4.2)) identical with the statement that the clause (5.4.1) is satisfiable if and only if the clauses (5.4.2) are simultaneously satisfiable? Discuss. 2. Is the claim that we made and proved above (just after (5.4.2)) identical with the statement that the Boolean expression (5.4.1) is equal to the product of the Boolean expressions (5.4.2) in the sense that their truth values are identical on every set of inputs? Discuss. 3. Let it be desired to find out if a given graph \( G \) of \( V \) vertices can be vertex colored in \( K \) colors. If we transform the problem into an instance of SAT, exactly how many clauses will there be? # 5.5 Half a loaf ... If we simply want to solve an NP-complete problem, then we are faced with a very long computation. Is there anything that can be done to lighten the load? In a number of cases various kinds of probabilistic and approximate algorithms have been developed, some very ingenious, and these may often be quite serviceable, as we have already seen in the case of primality testing. Here are some of the strategies of "near" solutions that have been developed. ## Type I: "Almost surely ..." Suppose we have an NP-complete problem that asks if there is a certain kind of substructure embedded inside a given structure. Then we may be able to develop an algorithm with the following properties: 1. It always runs in polynomial time 2. When it finds a solution then that solution is always a correct one 3. It doesn't always find a solution, but it "almost always" does, in the sense that the ratio of successes to total cases approaches unity as the size of the input grows large. An example of such an algorithm is one that will find a Hamilton path in almost all graphs, failing to do so sometimes, but not often, and running always in polynomial time. We will describe such an algorithm below. * V. G. Vizing. On an estimate of the chromatic class of a \( p \)-graph (Russian), Diskret. Analiz. 3 (1964), 25-30. Image Analysis: ### Image Analysis **Localization and Attribution:** - **Image Position:** Single page. - **Image Number:** Image 1. **Text Analysis:** - **Detected Text Extract:** - **Section Headings and Subheadings:** - "Maximum clique" - "Edge coloring" - "Hamilton path" - "Target sum" - "5.5 Half a loaf ..." - "Type 1: ‘Almost surely ...’" - **Exercises Section:** - "Exercises for section 5.4" - **Body Text:** \[ \text{We are given a graph } G \text{ and an integer } K. \text{ The question is to determine whether or not there is a set of } K \text{ vertices in } G, \text{ each of which is joined, by an edge of } G, \text{ to all of the others.} \] \[ \text{If we simply have to solve an NP-complete problem, then we are faced with a very long computation.} \] - **Citation:** - Refers to works by V.G. Vizing, including "On an estimate of the chromatic class of a p-graph" (1964). **Contextual Significance:** - **Context and Contribution:** - The text is part of a mathematical or computer science document discussing various graph theory problems and complexities such as NP-complete problems, maximum cliques, edge coloring, and Hamilton paths. - The document also discusses theoretical aspects and exercises to help understand these problems. - The section "5.5 Half a loaf ..." provides insight into approaches for solving NP-complete problems and highlights algorithms with guaranteed properties. **Object Detection and Classification:** - **Objects Detected:** - Primarily includes text blocks and mathematical symbols/formulae. - The only 'object' categorically identified is the text content relevant to graph theory. **Scene and Activity Analysis:** - **Scene Description:** - A structured page or section from a book or research paper focusing on theoretical computer science or mathematics. - Contains text, possibly as part of a book chapter or a journal article, discussing specific problems in graph theory and computational complexity. **Perspective and Composition:** - **Perspective:** - The page is viewed directly from above, simulating a reading perspective. - **Composition:** - The layout is columnar, typical of a textbook or academic paper, with headings, subheadings, and bullet points for exercises. - The text is dense, requiring a deep understanding of mathematical concepts and terms in graph theory. **Conclusion:** The attached diagram/image appears to be a page from a textbook or academic journal in the field of mathematics or computer science. It focuses on complex problems of graph theory such as the maximum clique, edge coloring, and Hamilton paths. Exercises are provided to reinforce the theoretical concepts discussed, and the supplementary section "5.5 Half a loaf ..." explains algorithmic approaches and strategies to cope with NP-complete problems. The content is highly specialized, intended for an audience versed in advanced graph theory and computational complexity. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 124 Context: # Chapter 5: NP-completeness ## Type II: 'Usually fast ...' In this category of quasi-solution are algorithms in which the uncertainty lies not in whether a solution will be found, but in how long it will take to find one. An algorithm of this kind will 1. always find a solution and the solution will always be correct, and 2. operate in an average of subexponential time, although occasionally it may require exponential time. The averaging is over all input strings of a given size. An example of this sort is an algorithm that will surely find a maximum independent set in a graph, will on the average require 'only' \( O(n^{\log n}) \) time to do so, but will occasionally, i.e., for some graphs, require nearly \( 2^n \) time to get an answer. We will outline such an algorithm below, in section 5.6. Note that \( O(n^{\log n}) \) is not a polynomial time estimate, but it’s an improvement over \( 2^n \). ## Type II: 'Usually fast ...' In this kind of algorithm, we don’t even get the right answer, but it’s close. Since this means giving up quite a bit, people like these algorithms to be very fast. Of course, we are going to drop our insistence that the questions be posed as decision problems, and instead they will be asked as optimization problems: find the shortest tour through these cities, or find the size of the maximum clique in this graph, or find a coloring of this graph in the fewest possible colors, etc. In response, these algorithms will 1. run in polynomial time 2. always produce some output 3. provide a guarantee that the output will not deviate from the optimal solution by more than such-and-such. An example of this type is the approximate algorithm for the traveling salesman problem that is given below, in section 5.8. It quickly yields a tour of the cities that is guaranteed to be at most twice as long as the shortest possible tour. Now let’s look at examples of each of these kinds of approximation algorithms. An example of an algorithm of Type I is due to Angluin and Valiant. It tries to find a Hamilton path (or circuit) in a graph \( G \). It doesn’t always find such a path, but in theorem 5.5.1 below we will see that it usually does, at least if the graph is from a class of graphs that are likely to have Hamilton paths at all. Input to the algorithm are the graph \( G \) and two distinguished vertices \( s, t \). It looks for a Hamilton circuit between the vertices \( s, t \) (if \( s = t \) on input then we are looking for a Hamilton circuit in \( G \)). The procedure maintains a partially constructed Hamilton path \( P \), from \( s \) to some vertex \( v_{dp} \), and it attempts to extend \( P \) by adjoining an edge to a new, previously unsatisfied vertex. In the process of doing so it will delete from the graph \( G \), from time to time, an edge, so we will also maintain a variable graph \( G' \) that is initially set to \( G \), but which is acted upon by the program. To do this, the algorithm chooses at random an edge \( (u_{dp}, v) \) that is incident with the current endpoint of the partial path \( P \), and it deletes the edge \( (u_{dp}, v) \) from the graph \( G' \), so it will never be chosen again. If \( v \) is a vertex that is not on the path \( P \) then the path is extended by adjoining the new edge \( (u_{dp}, v) \). So much is fairly clear. However if the new vertex \( v \) is already on the path \( P \), then we short circuit the path by deleting an edge from it and drawing in a new edge, as is shown below in the formal statement of the algorithm, and in Fig. 5.5.1. In that case the path does not get longer, but it changes so that it now has... #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 125 Context: # 5.5 Half a loaf ... enhanced chances of ultimate completion. ![Fig. 5.5.1: The short circuit](path/to/image) Here is a formal statement of the algorithm of Angluin and Valiant for finding a Hamilton path or circuit in an undirected graph \( G \): ```plaintext procedure uh(G: graph; s: vertex); { finds a Hamilton path (if \( s \neq t \)) or a Hamilton circuit (if \( s = t \)) in an undirected graph \( G \) and returns `success`, or fails, and returns `failure` } G' := G; ndp := s; P := empty path; repeat if ndp is an isolated point of \( G' \) then return `failure` else choose uniformly at random an edge (ndp, v) from among the edges of \( G' \) that are incident with ndp and delete that edge from \( G' \); if \( v \neq t \) and \( v \in P \) then adjoint the edge (ndp, v) to \( P \); ndp := v else if \( v \neq t \) and \( v \in P \) then { This is the short-circuit of Fig. 5.5.1 } u := neighbor of \( t \) in \( P \) that is closer to ndp; delete edge (u, v) from \( P \); adjoint edge (ndp, u) to \( P \); ndp := u end; { then } end; else until \( P \) contains every vertex of \( G \) (except \( T \), if \( s \neq t \) and edge (ndp, t) is in \( G \) but not in \( G' \); adjoint edge (ndp, t) to \( P \) and return `success` end; { uh } ``` As stated above, the algorithm makes only a very modest claim: either it succeeds or it fails! Of course, what makes it valuable is the accompanying theorem, which asserts that in fact the procedure almost always succeeds, provided the graph \( G \) has a good chance of having a Hamilton path or circuit. Image Analysis: ### Analysis of the Attached Visual Content #### 1. **Localization and Attribution:** - **Image 1:** Positioned at the top of the page. - **Image 2:** Positioned below Image 1. #### 2. **Object Detection and Classification:** - **Image 1:** - Objects: Graph showing a path labeled "Path P (before)". - Key Features: The path includes points connected by edges and a specific point labeled "ndp". - **Image 2:** - Objects: Graph showing a path labeled "Path P (after)". - Key Features: This path also includes points connected by edges and a specific point labeled "ndp", but the layout of the path differs from the first graph. #### 3. **Scene and Activity Analysis:** - **Image 1:** Depicts the initial state of a path (Path P) in an undirected graph with a marked point "ndp". This suggests steps are to be taken to transform this path. - **Image 2:** Shows the final state of a path (Path P) after a transformation involving the point "ndp". This indicates a change or development towards achieving a goal, likely related to finding a Hamilton path or circuit. #### 4. **Text Analysis:** - **Extracted Text:** - Title: "Fig. 5.5.1: The short circuit" - Descriptive Text: "Here is a formal statement of the algorithm of Angluin and Valiant for finding a Hamilton path or circuit in an undirected graph G:" - Procedure (Algorithm Description): ``` procedure uhc(G: graph, s, t: vertex); {finds a Hamilton path (if s ≠ t) or a Hamilton circuit (if s = t) P in an undirected graph G and returns `success`, or fails, and returns `failure`} G' := G; ndp := s; P := empty path; repeat if ndp is an isolated point of G' then return `failure` else choose uniformly at random an edge (ndp, v) from among the edges of G' that are incident with ndp and delete that edge from G'; if v ≠ t and v ∉ P then adjoin the edge (ndp, v) to P; ndp := v else if v ≠ t and v ∈ P then {This is the short-circuit of Fig. 5.5.1} u := neighbor of v in P that is closer to ndp; delete edge (u, v) from P; adjoin edge (ndp, v) to P; ndp := u end; {then} end; {else} until P contains every vertex of G (except T, if s ≠ t) and edge (ndp, t) is in G but not in G'; adjoin edge (ndp, t) to P and return `success` end; {uhc} ``` - Context: The text provides a detailed algorithm for finding a Hamilton path or circuit, explaining logical steps and conditions. This algorithm is crucial for understanding the transformation depicted in the graphs (Images 1 and 2). #### 5. **Diagram and Chart Analysis:** - **Image 1:** Represents the initial layout of a graph (Path P before transformation) with specific vertices and an isolated point "ndp". - **Image 2:** Shows the same graph after a transformation process described in the algorithm, depicting how to potentially find a Hamilton path or circuit. - **Axes and Legends:** Not applicable to these visual representations, as they are pure network diagrams. #### 6. **Product Analysis:** Not applicable. #### 7. **Anomaly Detection:** No anomalies detected; all elements appear to be consistent with the algorithm's steps described in the text. #### 8. **Color Analysis:** - **Image 1 and Image 2:** Both diagrams are in black and white, focusing on lines and points. The lack of colors emphasizes the structure and connectivity of the graphs. #### 9. **Perspective and Composition:** - **Perspective:** Direct 2D view from above, presenting a clear view of each point and edge in the graphs. - **Composition:** Simple and functional, with points and edges arranged to clearly depict the connections and transformations of the paths. #### 10. **Contextual Significance:** - These images and the accompanying text form part of a section in a document or a book discussing graph algorithms, specifically the Hamilton path and circuit problem. The diagrams illustrate the intermediate and final stages of the algorithm applied to a graph. ### Summary: - This page illustrates and explains an algorithm (by Angluin and Valiant) for finding a Hamilton path or circuit in an undirected graph. - Two diagrams show the state of the path before and after the application of the algorithm. - The text provides a formal algorithmic statement, describing each step and the conditions involved in transforming the path. - The significance lies in the visual and textual explanation of a complex algorithm, aiding in understanding its practical application. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 126 Context: # Chapter 5: NP-completeness What kind of graph has such a ‘good chance’? A great deal of research has gone into the study of how many edges a graph has to have before one can assure it must contain certain given structures. For instance, how many edges must a graph of \( n \) vertices have before we can be almost certain that it will contain a complete graph of \( n \) vertices? To say that graphs have a property ‘almost certainly’ is to say that the ratio of the number of graphs on \( n \) vertices that have the property to the number of graphs on \( n \) vertices approaches 1 as \( n \) grows without bound. For the Hamilton path problem, an important dividing line, or threshold, turns out to be at the level of \( \log n \) edges. That is to say, a graph of \( n \) vertices that has \( o(\log n) \) edges has relatively little chance of being even connected, whereas a graph with \( \geq \log n \) edges is almost certainly connected, and almost certainly has a Hamilton path. We now state the theorem of Angluin and Valiant, which asserts that the algorithm above will almost surely succeed if the graph \( G \) has enough edges. ## Theorem 5.5.1 Fix a positive real number \( \epsilon \). There exist numbers \( M \) and \( c \) such that if we choose a graph \( G \) at random from among those of \( n \) vertices and at least \( cn \) edges, and we choose arbitrary vertices \( s, t \) in \( G \), then the probability that algorithm UHC returns 'success' before making a total of \( M \log n \) attempts to extend partly constructed paths is \( 1 - O(n^{-\epsilon}) \). ## 5.6 Backtracking (I): independent sets In this section, we are going to describe an algorithm that is capable of solving some NP-complete problems fast, on the average, while at the same time guaranteeing that a solution will always be found, be it quickly or slowly. The method is called backtracking, and it has long been a standard method in computer search problems when all else fails. It has been common to think of backtracking as a very long process, and indeed can be. But recently it has been shown that the method can be very fast on average, and that in the graph coloring problem, for instance, it functions in an average of constant time, i.e., the time is independent of the number of vertices, although to be sure, the worst-case behavior is very exponential. We first illustrate the backtrack method in the context of a search for the largest independent set of vertices (a set of vertices too two of which are joined by an edge) in a given graph \( G \), an NP-complete problem. In this case the average time behavior of the method is not constant, or even polynomial, but is subexponential. The method is also easy to analyze and describe in this case. Hence consider a graph \( G \) of \( n \) vertices, in which the vertices have been numbered \( 1, 2, \ldots, n \). We want to find, in \( G \), the size of the largest independent set of vertices. In Fig. 5.6.1 below, the graph \( G \) has 6 vertices. ![Fig. 5.6.1: Find the largest independent set](#) Begin by searching for an independent set \( S \) that contains vertex 1, so let \( S = \{ 1 \} \). Now attempt to enlarge \( S \). We cannot enlarge \( S \) by adjoining vertex 2 to it, but we can add vertex 3. Our set \( S \) is now \( \{ 1, 3 \} \). Now we cannot add vertex 4 (joined to 1) or vertex 5 (joined to 3) or vertex 6 (joined to 3), so we backtrack. Therefore we backtrack, by replacing the most recently added member of \( S \) by the next choice that we might have made for it. In this case, we delete vertex 3 from \( S \), and the next choice would be vertex 6. The set \( S \) is \( \{ 1, 6 \} \). Again we have a dead end. If we backtrack again, there are no further choices with which to replace vertex 6, so we backtrack even further, and not only delete 6 from \( S \) but also replace vertex 1 by the next possible choice for it, namely vertex 2. 122 Image Analysis: ### Analysis of Attached Visual Content #### 1. Localization and Attribution - **Number of Images:** - There is a single image embedded in the text, which will be referred to as Image 1. - **Position:** - Image 1 is located centrally on the page at the bottom part. #### 2. Object Detection and Classification - **Image 1:** - **Objects Detected:** - A labeled graph with vertices. - Text annotations and numerical notations. - Arrows indicating directions and connections. - **Classification:** - The object is a graph typically used in mathematical and computational contexts to illustrate the concept of independent sets. #### 3. Scene and Activity Analysis - **Image 1:** - **Scene Description:** - The image depicts a graphical representation of a graph, where nodes (vertices) connected by edges (lines) are showcased. The vertices are numbered, and connections between them indicate edges. - **Activities Taking Place:** - The activity involves finding the largest independent set, which is a set of vertices in a graph, none of which are adjacent to each other. #### 4. Text Analysis - **Text in Image 1:** - Text below the image: "Fig. 5.6.1: Find the largest independent set" - Intended to describe the process depicted in the illustration above it. - The numerical notations and annotations on the graph: - Vertices numbered 1 to 6. - An arrowed sequence illustrating connections and potential selections for the independent set. - **Content Significance:** - The text and numerical annotations are crucial for understanding the step-by-step process of identifying an independent set in graph theory. #### 5. Diagram and Chart Analysis - **Image 1:** - **Data and Trends Presented:** - The diagram represents a graph used for backtracking to find the largest independent set of vertices. - Shows a step-by-step process highlighting the vertices chosen and discarded. - **Axes, Scales, and Legends:** - Vertices are marked by numbers (1 to 6). - Lines (edges) connecting these vertices indicate relationships. - **Key Insights:** - Shows how backtracking can help identify an independent set in the graph. - Demonstrates the complexity and decision points in the algorithm. #### 7. Anomaly Detection - **No noticeable anomalies** in the context of the image and supporting text. #### 8. Color Analysis - **Predominant Colors:** - Black and white. - Impact: The monochromatic scheme is typical for scientific and mathematical diagrams, focusing attention on the structure and details without color distraction. #### 9. Perspective and Composition - **Image 1:** - **Perspective:** - The image is presented from a front view, ensuring clarity of all the nodes and connections. - **Composition:** - Balanced layout, central focus on the graph. - Clear and labeled vertices with connecting edges, illustrating the step-by-step process of finding the independent set. #### 10. Contextual Significance - **Image 1:** - **Context in the Document:** - The image supports the text explaining backtracking and its application in finding the largest independent set in NP-complete problems. - **Contribution to the Overall Message:** - Helps in visualizing and understanding the theoretical explanations provided in the text. #### 11. Metadata Analysis - **No metadata available** directly from the image. #### 12. Graph and Trend Analysis - **Image 1:** - **Trend Identification:** - Steps in selecting vertices aiming to maximize the independent set while avoiding adjacency. - **Data Interpretation:** - The graph and steps illustrate trials and errors in the selection process, indicating potential complexities and requiring backtracking. ### Conclusion: The attached content includes a detailed analysis of a single image illustrating the backtracking process to find an independent set in graph theory. The analysis comprehensively covers the visual elements, context, and significance of the image within the document. This aids in understanding the theoretical underpinnings of the text and provides clarity on the subject matter discussed. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 128 Context: # Chapter 5: NP-completeness Hence in (5.6.1) we have an exact formula for the average number of independent sets in a graph of \( n \) vertices. A short table of values of \( I_n \) is shown below, in Table 5.6.1, along with values of \( 2^n \) for comparison. Clearly the average number of independent sets in a graph is a lot smaller than the maximum number that graphs of that size might have. | \( n \) | \( I_n \) | \( 2^n \) | |---------|----------|----------| | 2 | 3.5 | 4 | | 3 | 5.6 | 8 | | 4 | 8.5 | 16 | | 5 | 12.3 | 32 | | 10 | 52 | 1024 | | 15 | 149.8 | 32768 | | 20 | 350.6 | 1048576 | | 30 | 1342.5 | 1073741824 | | 40 | 3862.9 | 1099511627776 | **Table 5.6.1: Independent sets and all sets** In the exercises, it will be seen that the rate of growth of \( I_n \) as \( n \) grows large is \( O(n \log n) \). Hence the average amount of labor in a backtrack search for the largest independent set in a graph grows subexponentially, although faster than polynomially. It is some indication of how hard this problem is that even on the average the amount of labor needed is not of polynomial growth. ## Exercises for section 5.6 1. What is the average number of independent sets of size \( k \) that are in graphs of \( V \) vertices and \( E \) edges? 2. Let \( t_k \) denote the \( k \)th term in the sum (5.6.1). 1. Show that \( t_k/t_{k-1} > 1 \) when \( k \) is small, then let \( k \) pass a certain critical value \( k_0 \). Hence show that the terms in the sum (5.6.1) increase in size until \( k = k_0 \) and then decrease. 3. Now we will estimate the size of \( k_0 \) in the previous problem. 1. Show that \( k_0 < n \) when \( k = \log_2 n \) and \( k_1 > 1 \) when \( k = \log_2 n - \log_2 \log_2 n \). Hence the index \( k_0 \) of the largest term in (5.6.1) satisfies \[ [\log_2 n - \log_2 k] < k_0 \leq [\log_2 n] \] The entire sum in (5.6.1) is at most \( n + 1 \) times as large as its largest single term. Use Stirling's formula (1.11.10) and (3A) above to show that the \( k \)th term is \( O(n^{\epsilon + k}) \) and therefore the same is true of the whole sum, i.e., of \( I_n \). ## 5.7 Backtracking (II): Graph coloring In another NP-complete problem, that of graph-coloring, the average amount of labor in a backtrack search is \( O(1) \) (bounded) as \( n \), the number of vertices in the graph, grows without bound. More precisely, for fixed \( k \), if we ask "Is the graph \( G \) of \( V \) vertices, properly vertex-colorable in \( k \) colors?", then the answer in a backtrack search for the answer is bounded. Hence not only is the average of polynomial growth, but the polynomial is of degree \( O(1) \). To be more concrete, consider the case of 3 colors. It is already NP-complete to ask if the vertices of a given graph can be colored in 3 colors. Nevertheless, the average number of nodes in the backtrack search tree for this problem is about 197, averaged over all graphs of all sizes. This means that if we input a random graph of \( 1,000,000 \) vertices, and ask if it is 3-colorable, then we can expect an answer (probably "No") after only about 197 steps of computation. To prove this we will need some preliminary lemmas. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 129 Context: # 5.7 Backtracking (II): Graph Coloring ## Lemma 5.7.1 Let \( s_1, \ldots, s_K \) be nonnegative numbers whose sum is \( L \). Then the sum of their squares is at least \( \frac{L^2}{K} \). **Proof:** We have \[ 0 \leq \sum_{i=1}^K s_i^2 = \sum_{i=1}^K \left( s_i - \frac{L}{K} \right)^2 + \frac{L^2}{K} \] \[ = \sum_{i=1}^K s_i^2 - \frac{L^2}{K^2} + \frac{L^2}{K} \] \[ = \sum_{i=1}^K s_i^2 - \frac{L^2}{K^2} + \frac{L^2}{K} \] \[ \Rightarrow \sum_{i=1}^K s_i^2 \geq \frac{L^2}{K}. \] The next lemma deals with a kind of inside-out chromatic polynomial question. Instead of asking “How many proper colorings can a given graph have?”, we ask “How many graphs can have a given proper coloring?” ## Lemma 5.7.2 Let \( C \) be one of the \( K^L \) possible ways to color in \( K \) colors a set of \( L \) abstract vertices \( 1, 2, \ldots, L \). Then the number of graphs \( G \) whose vertex set is that set of \( L \) colored vertices and for which \( C \) is a proper coloring of \( G \) is at most \( 2^{L(L - 1)/2} \). **Proof:** In the coloring \( C \), suppose \( s_1 \) vertices get color 1, \( \ldots, s_K \) get color \( K \), where, of course, \( s_1 + \ldots + s_K = L \). If a graph \( G \) is to admit \( C \) as a proper vertex coloring then its edges can be drawn only between vertices of different colors. The number of edges that \( G \) might have is therefore \[ s_1 s_2 + s_1 s_3 + \ldots + s_K s_{K-1} + \ldots + s_{K-1} s_K \] for which we have the following estimate: \[ \sum_{1 \leq j < k \leq K} s_j s_k \leq \frac{1}{2} \sum_{j=1}^K s_j^2 \] \[ = \frac{1}{2} \left( \sum_{j=1}^K s_j^2 - \sum_{j=1}^K s_j^2 \right) = \frac{1}{2} \sum_{j=1}^K s_j^2 \] \[ \Rightarrow \sum_{1 \leq j < k \leq K} s_j s_k \leq \frac{L^2}{2} - \frac{1}{2} \sum_{j=1}^K s_j^2 \quad \text{(by lemma 5.7.1)} \] The number of possible graphs \( G \) is therefore at most \( 2^{(L(L - 1)/2)} \). ## Lemma 5.7.3 The total number of proper colorings in \( K \) colors of all graphs of \( L \) vertices is at most \[ K \cdot L^{2^{(L-1)/K}}. \] **Proof:** We are counting the pairs \( (G, C) \), where the graph \( G \) has \( L \) vertices and \( C \) is a proper coloring of \( G \). If we keep \( C \) fixed and sum on \( G \), then by lemma 5.7.2 the sum is at most \( 2^{(L-1)/K} \). Since there are \( K \) such \( C \), the proof is finished. Now let's think about a backtrack search for a \( K \)-coloring of a graph. Begin by using color 1 on vertex 1. Then use color 2 on vertex 2 unless \( (1, 2) \) is an edge, in which case use color 2. As the coloring progresses through vertices \( 1, 2, \ldots, L \), we color each new vertex with the lowest available color number that does not cause a conflict with some vertex that has previously been colored. Image Analysis: ### Comprehensive Analysis of the Attached Visual Content #### 1. **Localization and Attribution** - The visual content consists of one page from a document. - The page includes multiple mathematical expressions, proofs, and text explanations. - There is no numbering of images as there is a single continuous page of text and equations. #### 2. **Object Detection and Classification** - **Text and equations**: The page is composed mainly of paragraphs containing text and mathematical expressions. - **Mathematical Symbols**: Various symbols including summation (∑), inequality signs (≤), and fractions are present. - **Rectangles with numbers**: There are black rectangles with white text inside, representing the end of lemmas and proofs. #### 3. **Scene and Activity Analysis** - The scene is academic and mathematical, with the content focused on graph coloring and related lemmas and proofs. - The activities involve the presentation and proof of lemmas relating to the sum of squares and proper graph coloring. #### 4. **Text Analysis** - **Lemma 5.7.1**: Discusses nonnegative numbers and the sum of their squares. - **Significance**: The lemma concludes that the sum of the squares of nonnegative numbers under certain conditions is at least \(L^2 / K\). - **Lemma 5.7.2**: Explores a coloring problem in graph theory, providing a bound on the number of possible graphs for a given proper coloring. - **Significance**: This is crucial for understanding the complexity and constraints in graph coloring. - **Lemma 5.7.3**: Calculates the total number of proper colorings in \(K\) colors of all graphs with \(L\) vertices. - **Significance**: Provides a formula (\( K^L \cdot 2^{(L^2 - L) \cdot (1 - 1/K) / 2} \)) for quantifying the proper colorings, advancing theoretical knowledge in graph theory. - **Proofs**: Detailed mathematical proofs are provided for each lemma, following a typical structure of definitions, logical deductions, and conclusions. #### 7. **Anomaly Detection** - There are no apparent anomalies or unusual elements in the visual content. All elements align with standard mathematical documentation practices. #### 8. **Color Analysis** - **Dominant Colors**: The document uses a standard black-and-white color scheme typical for academic papers. - **Black**: Used for all text and mathematical symbols, enhancing readability. - **White**: Background color, ensuring contrast and clarity. #### 9. **Perspective and Composition** - **Perspective**: The perspective is direct, as expected from a scanned or digitally created academic document. - **Composition**: The page is well-structured with clear sections for lemmas and proofs. Equations are centered and inline with text, following standard mathematical document formatting. #### 13. **Graph Numbers** - **Lemma 5.7.1**: - The sum of squares formula is \[ \sum_{i=1}^K s_i^2 = \frac{L^2}{K} \] - **Lemma 5.7.2**: - The number of graphs \(G\) possible is \[ 2^{(L^2 - L) \cdot (1 - 1/K) / 2} \] - **Lemma 5.7.3**: - The total number of proper colorings is \[ K^L \cdot 2^{(L^2 - L) \cdot (1 - 1/K) / 2} \] #### **Contextual Significance** - The page appears to be from a chapter on "Backtracking (II): Graph Coloring" in a mathematical or computer science text. - **Contribution**: This section contributes to the overall understanding of graph coloring by providing theoretical bounds and proofs that facilitate further study and algorithms development. Overall, the page is a detailed exploration of specific aspects of graph theory, primarily focused on proper coloring and combinatorial bounds, aimed at readers with a strong background in mathematics or computer science. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 132 Context: # Chapter 5: NP-completeness ## 5.8 Approximate algorithms for hard problems Finally, we come to Type III of the three kinds of 'half-a-loaf-is-better-than-none' algorithms that were described in section 5.5. In these algorithms, we don't find the exact solution of the problem, only an approximate one. As consolation, we have an algorithm that runs in polynomial time as well as a performance guarantee to the effect that while the answer is approximate, it can certainly deviate by no more than such-and-such from the exact answer. An elegant example of such a situation is the Travelling Salesman Problem, which we will now express as an optimization problem rather than as a decision problem. We are given \( n \) points ('cities') in the plane, as well as the distances between every pair of them, and we are asked to find a round-trip tour of all of these cities that has minimum length. We will assume throughout the following discussion that the distances satisfy the triangle inequality. This restriction of the TSP is often called the 'Euclidean' Travelling Salesman Problem. The algorithm that we will discuss for this problem has the properties: 1. It runs in polynomial time and 2. The round-trip tour that it finds will never be more than twice as long as the shortest possible tour. The first step in carrying out the algorithm is to find a minimum spanning tree (MST) for the \( n \) given cities. A MST is a tree whose nodes are the cities in question, and which, among all possible trees on the vertex set, has minimum possible length. It may seem that finding a MST is just as hard as solving the TSP, but NIN (No, It's Not). The MST problem is one of those all-too-rare computational situations in which it pays to be greedy. Generally speaking, in a greedy algorithm, 1. We are trying to construct some optimal structure by adding one piece at a time, and 2. At each step we make the decision about which piece will be added next by choosing, among all available pieces, the single one that will carry us as far as possible in the desirable direction (be greedy!). The reason that greedy algorithms are not usually the best possible ones is that it may be better not to take the single best piece at each step, but to take some other piece, in the hope that at a later step we will be able to improve things even more. In other words, the global problem of finding the best structure might not be solvable by the local procedure of being as greedy as possible at each step. In the MST problem, though, the greedy strategy works, as we see in the following algorithm. ``` procedure mst(x: array of n points in the plane); { constructs a spanning tree of minimal length on the vertices \( x_1, x_2, \ldots, x_n \) in the plane } let T consist of a single vertex \( x_1 \); while T has fewer than n vertices do for each vertex \( v \) that is not yet in T, find the distance \( d(v) \) from \( v \) to the nearest vertex of T; let \( u \) be a vertex of smallest \( d(v) \); add \( u \) to the vertex set of T; add to T the edge from \( v \) to the nearest vertex \( u \) of T; end; end(mst); ``` ### Proof of correctness of mst: Let \( T \) be the tree that is produced by running mst, and let \( e_1, \ldots, e_k \) be its edges, listed in the same order in which the algorithm produced them. Let \( T' \) be a minimum spanning tree for \( x \). Let \( e \) be the first edge of \( T \) that does not appear in \( T' \). In the minimum tree \( T' \), edges \( e_1, \ldots, e_{i-1} \) all appear, and we let \( S \) be the union of their vertices in \( T' \) plus the edge that joins the subtree on \( S \) to the subtree on the remaining vertices of \( x \). Suppose \( f \) is shorter than \( e \). Then \( f \) was one of the edges that was available to the algorithm and that it chose \( e_r \), and since \( e_r \) was the shortest edge available at that moment, we have a contradiction. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 133 Context: ```markdown # 5.7 Backtracking (II): Graph Coloring Suppose \( f \) is longer than \( \epsilon \). Then \( T' \) would not be minimal because the tree that we would obtain by exchanging \( f \) for \( e_i \) in \( T' \) (why is it still a tree if we do that exchange?) would be shorter, contradicting the minimality of \( T' \). Hence \( f \) and \( g \) have the same length. In \( T' \) exchange \( f \) for \( e_i \). Then \( T' \) is still a tree, and is still a minimum spanning tree. The index of the first edge of \( T \) that does not appear in \( T' \) is now at least \( r + 1 \), one unit larger than before. The process of replacing edges of \( T \) that do not appear in \( T' \) without affecting the minimality of \( T \) can be repeated until every edge of \( T \) appears in \( T' \), i.e., until \( T' = T \). Hence \( T \) was a minimum spanning tree. That fulfills one step of the process that leads to a polynomial time travelling salesman algorithm that finds a tour of at most twice the minimum length. The next step involves finding an Euler circuit. Way back in theorem 1.6.1 we learned that a connected graph has an Euler circuit if and only if every vertex has even degree. Recall that the proof was recursive in nature, and immediately implies a linear time algorithm for finding Euler circuits recursively. We also noted that the proof remains valid even if we are dealing with a multigraph, that is, with a graph in which several edges are permitted between single pairs of vertices. We will in fact need that extra flexibility for the purpose at hand. Now we have the ingredients for a quick near-optimal travelling salesman tour. **Theorem 5.8.1**. An algorithm that operates in polynomial time and will return a travelling salesman tour whose length is at most twice the length of a minimum tour. Here is the algorithm. Given \( n \) cities in the plane: 1. Find a minimum spanning tree \( T \) for the cities. 2. Double each edge of the tree, thereby obtaining a ‘multitree’ \( T' \) in which each pair of vertices there are \( 0 \) or \( 2 \) edges. 3. Since every vertex of the doubled tree has even degree, there is an Eulerian tour \( W \) of the edges of \( T' \); find one, as in the proof of theorem 1.6.1. 4. Now we construct the output tour of the cities. Begin at some city and follow the walk \( W \). However, instead of starting at some vertex \( v \), go from \( v \) directly (via a straight line) to the next vertex of the walk \( W \) that you haven't visited yet. This means that you will often short-circuit portions of the walk \( W \) by going directly from some vertex to another via several edges "down the road." The tour \( Z \) that results from (4) above is indeed a tour of all the cities in which each city is visited once and only once. We claim that its length is at most twice optimal. Let \( Z \) be an optimum tour, and let \( e \) be some edge of \( Z \). Then \( Z - e \) is a path that visits all of the cities. Since a path is a tree, \( Z - e \) is a spanning tree of the cities, hence \( Z - e \) is at least as long as \( T \), and so \( Z \) is surely at least as long as \( T \). Next consider the length of the tour \( Z \). A step of \( Z \) that walks along an edge of the walk \( W \) has length equal to the length of that edge of \( W \). A step of \( Z \) that short-circuits several edges of \( W \) has length at most equal to the sum of the lengths of the edges of \( W \) that were short-circuited. If we sum these inequalities over all steps of \( Z \) we find that the length of \( Z \) is at most equal to the length of \( W \), which is in turn twice the length of the tree \( T \). If we put all of this together we find that \[ \text{length}(Z) \leq \text{length}(Z - e) \leq \text{length}(T) = \frac{1}{2} \text{length}(W) \leq \frac{1}{2} \text{length}(Z') \] as claimed. More recently it has been proved (Christodes, 1976) that polynomial time we can find a TSP tour whose total length is at most \( 3/2 \) as long as the original tour. The algorithm makes use of Edmonds’s algorithm for maximum matching in a general graph (see the reference at the end of Chapter 3). It will be interesting to see if the factor \( 3/2 \) can be further refined. Polynomial time algorithms are known for other NP-complete problems that guarantee that the answer obtained will not exceed, by more than a constant factor, the optimum answer. In some cases this guarantees apply to the difference between the answer that the algorithm gives and the best one. See the references below for more information. ``` Image Analysis: ### Text Analysis: #### Image 1 - **Text Content:** - The text discusses a polynomial time approximation algorithm for finding a near-optimal traveling salesman tour. - It restates a theorem for the polynomial time working of this algorithm. - References to previous sections are included (e.g., Section 1.6.1). - Descriptions of the steps involved in the algorithm for n cities are provided. - Definitions and explanations about Euler circuits, multitreets, and process flows related to constructing and optimizing the traveling salesman path are mentioned. - A mathematical expression demonstrates the length relationship between different paths and circuits, including a proof. - **Significance in Context:** - Provides context for the algorithm's efficiency and optimality in solving the Traveling Salesman Problem under certain conditions. - Explains theoretical underpinnings (from earlier sections and external references) to give a potential context of research or course material on computational theory and graph algorithms. ### Color Analysis: - **Image 1:** - Dominated by black text on a white background, which is typical for textual academic or technical content. - The monochromatic color scheme enhances readability and is appropriate for the clear communication of complex mathematical recommendations and proofs. ### Perspective and Composition: - **Image 1:** - Straightforward, top-down perspective, typical of textbook pages. - The layout is columnar with margins and justified text, adhering to standard academic formatting. ### Contextual Significance: - **Image 1:** - Likely taken from an academic textbook or research paper dealing with graph theory, specifically focusing on algorithms related to the Euler circuits and Traveling Salesman Problem. - This page provides necessary theoretical explanations as part of broader material, indicating its role in teaching or scholarly examination of algorithms. ### Conclusion: This analysis indicates that the given image is a page filled with text from an advanced academic context, focusing on polynomial-time algorithms for the Traveling Salesman Problem. The explanations, definitions, and formal proof presented signify that this material is aimed at a reader with a solid understanding of theoretical computer science, likely providing coursework or supporting material for in-depth study of graph theory and its applications. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 133 Context: ### Color Analysis - **Color Composition:** - The image is primarily composed of black text on a white background, typical of printed documents. No additional color variations are present that contribute to the content. ### Perspective and Composition - **Perspective:** - The perspective is that of a flat, directly photographed or scanned page. - **Composition:** - The image is structured in columns of text with occasional numbered lists. Sections are clearly delineated with headings. ### Contextual Significance - **Contribution to Overall Message:** - The image contributes detailed theoretical content pertinent to the field of graph theory and algorithm design, specifically focusing on the Traveling Salesman Problem and Euler circuits. ### Graph and Trend Analysis - **Graphs and Trends:** - No graphs or explicit data trends are presented in this image. The focus is on theoretical descriptions and algorithmic steps. ### Prozessbeschreibungen (Process Descriptions) - **Processes Described:** - The image describes the step-by-step procedure of an algorithm for finding a near-optimal traveling salesman tour. ### Typen Bezeichnung (Type Designations) - **Type Designations:** - The text refers to types of mathematical structures such as trees, 'multitrees,' Eulerian tours, and spanning trees. ### Trend and Interpretation - **Identified Trends and Interpretations:** - The trend in the text shows a progressive build-up from theoretical background to practical algorithm description and mathematical proofs. ### Tables - **Content Description:** - No tables are present in this image. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 134 Context: # Chapter 5: NP-completeness ## Exercises for section 5.8 1. Consider the following algorithm: ``` procedure mst2(x: array of n points in the plane); { allegedly finds a tree of minimum total length that visits every one of the given points } if n = 1 then T := {x₁} else T := mst2(n - 1, x − xₙ); let u be the vertex of T that is nearest to xₙ; mst2 := T plus vertex xₙ plus edge (xₙ, u) end. ``` Is this algorithm a correct recursive formulation of the minimum spanning tree greedy algorithm? If so then prove it, and if not then give an example of a set of points where mst2 gets the wrong answer. ## Bibliography Before we list some books and journal articles, it should be mentioned that research in the area of NP-completeness is moving rapidly, and the state of the art is changing all the time. Readers who would like updates on the subject are referred to a series of articles that have appeared in issues of the Journal of Algorithms in recent years. These are called "NP-completeness: An ongoing guide." They are written by David S. Johnson, and each of them is a thorough survey of recent progress in one particular area of NP-completeness research. They are written as updates of the first reference below. Journals that contain a good deal of research on the areas of this chapter include the Journal of Algorithms, the Journal of the Association for Computing Machinery, the SIAM Journal of Computing, Information Processing Letters, and SIAM Journal of Discrete Mathematics. The most complete reference on NP-completeness is - M. Garey and D. S. Johnson, *Computers and Intractability: A guide to the theory of NP-completeness*, W. H. Freeman and Co., San Francisco, 1979. The above is highly recommended. It is readable, careful, and complete. The earliest ideas on the computational intractability of certain problems go back to - Alan Turing, "On computable numbers, with an application to the Entscheidungsproblem," Proc. London Math. Soc., Ser. 2, 42 (1936), 230-265. Cook's theorem, which originated the subject of NP-completeness, is in - S. A. Cook, "The complexity of theorem proving procedures," Proc., Third Annual ACM Symposium on the Theory of Computing, ACM, New York, 1971, 151-158. After Cook's work was done, a large number of NP-complete problems were found by - Richard M. Karp, "Reducibility among combinatorial problems," in R. E. Miller and J. W. Thatcher, eds., *Complexity of Computer Combinations*, Plenum, New York, 1972, 85-103. The above paper is recommended both for its content and its clarity of presentation. The approximate algorithm for the traveling salesman problem is in - D. J. Rothlauf, R. E. Stearns and P. M. Lewis, "An analysis of several heuristics for the travelling salesman problem," SIAM J. Comp. 6 (1977), 563-581. Another approximate algorithm for the Euclidean TSP which guarantees that the solution found is no more than 3/2 as long as the optimum tour, was found by - N. Christofides, "Worst case analysis of a new heuristic for the travelling salesman problem," Technical Report, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, 1976. The minimum spanning tree algorithm is due to - R. C. Prim, "Shortest connection networks and some generalizations," Bell System Tech. J. 36 (1957), 1389-1401. The probabilistic algorithm for the Hamilton path problem can be found in --- 130 #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 138 Context: ```markdown # Index - **k-subset** 35 - **language** 132 - **Lawler, E.** 99 - **layered network** 120f - **Lenstra, H. W., Jr.** 176 - **Levyque, W. J.** 175 - **Lewis, P. A. W.** 103 - **Lewis, P. M.** 227 - **L'Hôpital's rule** 12 - **Little, O.** 54 ## Network - **network** 105 - **flow** 105f - **dense** 107 - **layered** 108, 120f - **height** 125 - **Nijenhuis, A.** 60 - **nondeterministic** 193 - **NP** 182 - **NP-complete** 61, 180 - **NP-completeness** 178f - **octal system** 21 - **optimization problem** 181 - **orders of magnitude** 6f ## P - **P** 12 - **Palmer, E. M.** 228 - **Pav, V.** 103 - **Pascal’s triangle** 36 - **path** 41 - **predicate function** 87 - **polynomial time** 2, 179, 185 - **polynomials, multiplication of** 96 - **Ponomarev, C.** 143, 164, 176 - **positional number systems** 19f - **Pramodh-Kumar, M.** 108f, 135 - **Pratt, V.** 171, 172 - **Prim, R. C.** 227 - **primality testing** 148f, 186 - **routing** 179 - **prime number** 2 - **primitive root** 152 - **pseudorandomity test** 149, 156ff - **strong** 158 - **public key encryption** 150, 165 - **Quicksort** 50f - **Rabin, M. O.** 149, 162, 175 - **Ralston, A.** 103 ``` #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 139 Context: # Index - recurrence relations 26/f - recurrent inequality 31 - recursive algorithms 48/f - reducibility 185 - relative prime 138 - ring Z, 151 - Rivest, R. 165, 176 - roots of unity 86 - Rosenkrantz, D. 227 - RSA system 165, 168 - Rumyantsev, R. 149, 164, 176 - Runge, C. 103 ## SAT - SAT 195 - satisfiability 187, 195 - second vertex 111 - Schöning, A. 103 - Selbig, J. 176 - Shamir, A. 165, 176 - slowest 50 - Solvable, R. 149, 162, 176 - splitter 52 - Stearns, R. E. 227 - Stirling's formula 16, 216 - Strassen, V. 73, 103, 149, 162, 176 - synthetic division 86 ## 3SAT - target sum 206 - Tajima, R. E. 66, 70, 103, 135 - Θ (Theta of) 10 - tiling ? - tractable 5 - travelling salesman problem 178, 184, 221 - tree 45 - Trojanowski, A. 66, 103 - TSP 178, 201 - Tukey, J. W. 103 - Turing, A. 226 - Turing machine 187/f - Ullman, J. D. 103 - usable edge 111 - Valiant, L. 208-11, 227 - vertices 40 - Wizing, V. 206 - Wagstaff, S. 176 - Welch, P. D. 103 - Wilf, H. S. 60, 103, 227, 228 - Winograd, S. 99 - worst-case 4, 180 - Wright, E. M. 175 #################### File: Feedback Page: 1 Context: Feedback: Prompt: how much is a fried chicken in "Pakens"? Response: The provided context is from two books on mathematics, one on number theory and the other on algorithms and complexity. Neither book mentions anything about "Pakens" or fried chicken. To answer your question, I need more information. Could you please clarify what "Pakens" refers to? Is it a restaurant, a country, or something else? Note: Reference this document as 'previous feedback'. #################### File: An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf Page: 4 Context: # Preface These lectures are intended as an introduction to the elementary theory of numbers. I use the word “elementary” both in the technical sense—complex variable theory is to be avoided—and in the usual sense—that of being easy to understand, I hope. I shall not concern myself with questions of foundations and shall presuppose familiarity only with the most elementary concepts of arithmetic, i.e., elementary divisibility properties, e.g., (greatest common divisor), l.c.m. (least common multiple), essentially unique factorization into primes and the fundamental theorem of arithmetic: if \( p \mid ab \) then \( p \mid a \) or \( p \mid b \). I shall consider a number of rather distinct topics each of which could easily be the subject of 15 lectures. Hence, I shall not be able to penetrate deeply in any direction. On the other hand, it is well known that in number theory, more than in any other branch of mathematics, it is easy to reach the frontiers of knowledge. It is easy to propound problems in number theory that are unsolved. I shall mention many of these problems; but the trouble with the natural problems of number theory is that they are either too easy or much too difficult. I shall therefore try to expose some problems that are of interest and unsolved but for which there is at least a reasonable hope for a solution by you or me. The topics I hope to touch on are outlined in the Table of Contents, as are some of the main reference books. Most of the material I want to cover will consist of old theorems proved in old ways, but I also hope to produce some old theorems proved in new ways and some new theorems proved in old ways. Unfortunately I cannot produce many new theorems proved in really new ways. #################### File: An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf Page: 6 Context: ``` ## Chapter 1: Compositions and Partitions ### Examples As an exercise I would suggest using both the combinatorial method and the algebraic approach to prove the following results: 1. The number of compositions of \( n \) into exactly \( r \) parts is \( C(n-1, r-1) \) (Catalan); 2. The number of compositions of \( n \) into even parts is \( 2^{r-1} \) if \( n \) is even and \( 0 \) if \( n \) is odd; 3. The number of compositions of \( n \) into an even number of parts is equal to the number of compositions of \( n \) into an odd number of parts. Somewhat more interesting is the determination of the number of compositions \( c(n) \) of \( n \) into odd parts. Here the algebraic approach yields: \[ \sum_{n=1}^{\infty} c(n)x^n = \sum_{n=1}^{\infty} \frac{x^n}{1-x^n} = \frac{x}{1-x} \sum_{n=0}^{\infty} \left( \frac{x}{1-x} \right)^n. \] By cross multiplying the last two expressions we see that \[ F_{n+2} = F_n + F_{n+1}, \quad F_0 = 0, \, F_1 = 1. \] Thus the \( F \)'s are the so-called Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, ... The generating function yields two explicit expressions for these numbers. First, by "partial fractioning": \[ \frac{x}{1-x^2} = \frac{1}{\sqrt{5}} \left( \frac{\sqrt{5}}{2} - \left( \frac{-1+\sqrt{5}}{2} \right)^{n} \right). \] Another expression for \( F_n \) is obtained by observing that \[ \frac{x}{1-x^2} = x \sum_{n=1}^{\infty} (x^2)^{n-1} = x \left( 1 + (x^2) + (x^2)^2 + \cdots \right). \] Comparing the coefficients here we obtain (Lucas): \[ F_n = \binom{n-1}{1} + \binom{n-2}{1} + \cdots. \] You might consider the problem of deducing this formula by combinatorial arguments. ### Another Approach Suppose we denote by \( a(n) \) the number of compositions of \( n \) with summands at least 2. An interesting result is that \( a(n) = b(n-2) \). I will prove this result and suggest the problem of finding a reasonable generalization. First note that \( a(n) = a(n-1) + a(n-2) \). This follows from the fact that every admissible composition of \( n \) either ends in 2 (the last summand), we obtain an admissible composition of \( n-2 \) respectively. Since \( a(1) = 0 \), it follows that \( a(n) = F_{n-2} \). An interesting idea for compositions is that of weight of a composition. Suppose we associate with each composition a number called the weight, which is the product of the summands. We shall determine the sum \( w(n) \) of the weights of the compositions on \( n \). The generating function of \( a(n) \) is: \[ \sum_{n=1}^{\infty} u(n)x^n = \sum_{n=1}^{\infty} \frac{x(x^2 + 2x^3 + 3x^4 + \cdots)}{1-x^2} = \frac{z}{1-x^2} - z. \] From this we find that \( a(n) = 3a(n-1) - 1 \) for \( n \geq 2 \). I leave it as an exercise to prove from this that \( w(n) = F_n \). Now let’s turn to partitions. There is no simple explicit formula for \( p(n) \). Our main objective here will be to prove the recursion formula \[ p(n) = p(n-1) + p(n-2) + p(n-3) - p(n-5) - p(n-7) + p(n-12) + \cdots. \] This was discovered by Euler. The algebraic approach to partition theory depends on algebraic manipulations with the generating function \[ \sum_{n=0}^{\infty} p(n)x^n = \frac{1}{(1-x)(1-x^2)(1-x^3)\cdots}. \] The combinatorial approach depends on the use of partition (Ferrers) diagrams. For example, the Ferrers diagram of the partition \( 7 = 4 + 2 + 1 \) is: ``` ``` #################### File: An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf Page: 18 Context: ``` # Chapter 3: Distribution of Primes 1. \( r \) and interpreting \( \binom{2n}{n} \) as the number of ways of choosing \( r \) objects from \( 2n \), we conclude that the second expression is indeed smaller than the first. This contradiction proves the theorem when \( r > 5 \). The results 7, 29, 97, 389, and 1543 show that the theorem is also true for \( r \leq 5 \). The proof of Bertrand's Postulate by this method is left as an exercise. Bertrand's Postulate may be used to prove the following results. 1. \(\frac{1}{n} \sum_{p \leq n} \frac{1}{p} \text{ is never an integer.}\) 2. Every integer \( > 6 \) can be written as the sum of distinct primes. 3. Every prime \( p_n \) can be expressed in the form \[ p_n = 2 + 3 + 5 + \cdots + p_{n-1} \] with an error of most 1 (Sierk). 4. The equation \( \pi(n) = \psi(n) \) has exactly 8 solutions. About 1949 a sensation was created by the discovery by Erdős and Selberg of an elementary proof of the Prime Number Theorem. The main new result was the following estimate, proved in an elementary manner: \[ \sum_{p \leq x} \log p \sim 2x \log x + O(x). \] Although Selberg's inequality is still far from the Prime Number Theorem, the latter can be deduced from it in various ways without recourse to any further number theoretical results. Several proofs of this lemma have been given, perhaps the simplest being due to Tzwarzow and Szelk. Selberg's original proof depends on consideration of the functions \[ \lambda_n = \mu_n = \sum_{d \mid d} \mu(d) \frac{1}{d^2} \] and \[ T(r) = \sum_{n=1}^{r} \lambda^2. \] Some five years ago J. Landeck and L. Moser showed that one could prove Selberg's lemma in a completely elementary way, i.e., using properties of integers only. One of the main tools for doing this is the following rational analogue of the logarithmic function. Let \[ h(r) = \frac{1}{1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{r}}. \] We prove in an quite elementary way that \[ |h(e) - h(c)| < \frac{1}{k}. \] The results we have established are useful in the investigation of the magnitudes of the arithmetic functions \( \sigma(n) \), \( \phi(n) \) and \( g(n) \). Since these depend not only on the magnitude of \( n \) but also strongly on the arithmetic structure of \( n \), we cannot expect to approximate them by the elementary functions of analysis. Nevertheless, we shall will see that “on average” these functions have a rather simple behavior. Let \( f \) and \( g \) be functions for which \[ f(1) + f(2) + \cdots + f(n) + g(1) + g(2) + \cdots + g(n), \] we say that \( f \) and \( g \) have the same average order. We will see, for example, that the average order of \( \sigma(n) \) is long, that of \( \phi(n) \) is \(\frac{n}{\log n}\). Let us consider first a purely heuristic argument for obtaining the average value of \( a(n) \). The probability that \( r \mid n \) and \( r \leq 1 \) contributes \( \left( \frac{1}{k} \right)^r \) to \( a(n) \). Thus the expected value of \( a(n) \) is \[ \frac{1}{T(r)} \left( \sum_{k=1}^{1} \frac{1}{k} + 1 + \frac{1}{k} \right). \] For \( k = 0 \) this will be about \( \log n \). For \( n \geq 1 \) it will be about \( n^{k}(k + 1) \), e.g., for \( n = 1 \) it will be about \( n(k^2) = n^2 + \frac{1}{k^2} \). Before proceeding to the proof and refinement of some of these results we consider some applications of the inversion of order of summation in certain double sums. Let \( f \) be an arithmetic function and associate with it the function \[ F(n) = \sum_{d \mid n} f(d) \] and \( g(n) = \sum_{d \mid n} f(d). \] We will obtain two expressions for \[ F(n) = \sum_{n=1}^{r} f(d). \] ``` #################### File: An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf Page: 32 Context: # Chapter 6: Diophantine Equations We now go on to quite a different problem. Has the equation \[ 1^n + 2^n + \ldots + (m - 1)^n = m^n \] any solutions in integers other than \( n = 2 \)? Here are some near solutions: \[ 1^3 + 2^3 = 3^3 \] \[ 3^4 + 4^4 + 5^4 + 6^4 + 7^4 + 8^4 + 9^4 + 10^4 = 212^4 \] \[ 1^6 + 2^6 + 3^6 + 4^6 + 5^6 + 6^6 + 7^6 + 12^6 + 13^6 + 15^6 + 16^6 + 18^6 + 20^6 + 22^6 + 23^6 + 25^6 = 256^6 \] We now outline a proof that other solutions exist for \( n > 1000 \). The rest of this section appeared originally as the paper "On the Diophantine Equation \( 1^n + 2^n + \ldots + (m - 1)^n = m^n \)", *Rendiconti del Circolo Matematico di Palermo*, 19 (1953), pp. 84-85. A number of isolated equations expressing the sum of the \( n \)th powers of integers as an \( n \)th power of an integer have long been known. Some examples are: \[ 3^4 + 4^3 = 6^3 \] \[ \sum_{k=1}^{4} (-1)^k \cdot 2^k - 3^k - 10^k - 14^k - 72^k = 212^4 \] \[ 1^6 + 2^6 + 4^6 + 5^6 + 6^6 + 7^6 + 12^6 + 13^6 + 15^6 + 16^6 + 18^6 + 20^6 + 22^6 + 23^6 + 25^6 = 256^6 \] Further examples and references to such results are given in (1, 2). On the other hand, the only known solution in integers for the equation in the title is the trivial one \( 1 + 2 = 3 \). In a letter to the author, P. Erdos conjectured that this is the only solution. The object of this note is to show that if the equation has a solution with \( n \), then \( n > 1000 \). Let \( S_n(m) \) denote \[ S_n(m) = \sum_{j=1}^{m} j^n, \quad n > 1. \] It is possible to examine (1) with various modular and thereby obtain restrictions on \( m \) and \( n \). This is essentially our method, but the modulo are so chosen that we are able to combine the resulting congruences so as to obtain extremely large bounds for \( m \) without laborious computation. We use the following lemma. ## Lemma 1 If \( p \) is a prime and \( c(p) \) is defined by \( c(p) = \pm 1 \) when \( n \equiv 0 \mod p \) (1) does not divide \( n \) then \[ S_n(p - 1) \equiv 0 \mod p. \] A simple proof of (2) is given in [2, p. 90]. Now suppose \( p \equiv 1 \mod n \), then \[ S_n(m) \equiv \sum_{j=1}^{m} j^n \equiv \frac{m^n - 1}{p - 1} - c(p) \quad ( \text{mod } p). \] On the other hand, let \( m = 1 \quad ( \text{mod } p) \) so that by (1) \[ \frac{m - 1}{p} \equiv 1 \quad ( \text{mod } p). \] Hence \( c(p) \neq 0 \quad ( \text{mod } p) \) so that from the definition of \( c(p) \) it follows that \( c(p) = -1 \) and \[ p \equiv 1 \quad ( \text{mod } n). \] Thus (3) can be put in the form \[ \frac{m - 1}{p - 1} + 1 \equiv 0 \quad ( \text{mod } p) \] or \[ m - 1 \equiv 0 \quad ( \text{mod } p^2). \] From (6) it follows that \( m - 1 \) is squarefree. Further, since it is easily checked that \( m - 1 \) if \( 2 \) it follows that \( m - 1 \) must have at least one prime divisor, so by (4) \( n \) is even. We now multiply together all congruences of the type (5), that is one for each prime dividing \( m - 1 \). Since \( m - 1 \) is squarefree, the resulting modulus is \[ m - 1. \quad \text{Furthermore, products containing two or more distinct factors of the form (1 - j) will be divisible by \( m - 1 \). Thus we obtain} \] \[ (m - 1) \cdot \sum_{k=1}^{p} \frac{1}{k - 1} + 1 \equiv 0 \quad ( \text{mod } 1). \] The only roots of \( m < 1000 \) which satisfy (8) are \( 3, 7, 43 \). We proceeded to develop three more congruences, similar to (8), which when combined with (6) led us to the main result. Equation (12) can be written in the form \[ S_n(m + 2) = 2^{m + 1}. \] Suppose that \( p \equiv 1 \). Using (2) and the fact that \( n \) is even, we obtain \[ | p | (1 + 1) \implies \quad 0 \quad ( \text{mod } p). \] #################### File: An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf Page: 36 Context: ``` # Chapter 2: Combinatorial Number Theory Let $$ b = \frac{(100)(2)(6)(10)(0)}{(100)(150)(60)}, $$ $$ c = \frac{(1008)(600)(290)(50)(0)}{(1008)(600)(290)(50)(0)}. $$ respectively. Now suppose the $\text{n}$ bracket in $\text{R}$ contains nonzero digits, but all further brackets to the left are 0. Call the number represented by the digits in the $\text{n}$ bracket, $x_r$, for $r = 1, 2, \ldots, r - 2$. Further, denote by $y$ the number represented by the digit in the last two brackets taken together, but excluding the last digit. For $x_r$ to belong to $\text{R}$ we require 1. The last digit of $x_r$ must be 1. 2. $x_r$ must begin with 0 for $r = 1, 2, \ldots, r - 2$. 3. $x_1 + x_2 + \ldots + x_r = z$. In particular, note that 2 satisfies (2) but violates (1) and (3) so that $x$ is not in $\text{R}$, but $x$ cannot satisfy all three conditions and are in $\text{R}$. To check (3) we note that $b^6 + 1^6 = 2600$. We next prove that no three integers in $\text{R}$ are in an arithmetic progression. First note that if two elements of $\text{R}$ have a different number of nonempty brackets their average cannot satisfy (1). Thus we need only consider averages of elements of $\text{R}$ having the same number of nonempty brackets. From (1) and (3) it follows that the two elements of $\text{R}$ can be averaged by bracket for the first $r - 2$ brackets and also for the last two brackets taken together. Thus, in our example, $$ \frac{(60 + 50)}{5} = \frac{1}{2}(150 + 290) = 220, $$ $$ \frac{1}{2}(10020060 + 100856600) = 100635300, $$ $$ \frac{1}{2}(h + c) = \frac{(10065)(230)(5)(0)}. $$ This violates (3) and so is not in $\text{R}$. In general we will prove that if $x$ and $y$ are in $\text{R}$ then $x = \frac{1}{2}(x + y)$ and $x$ is not in $\text{R}$. Since $x$ and $y$ are in $\text{R}$. On the other hand, if $z \in \text{R}$ implies $$ z = \sum_{i=1}^{r} x_i^2 + \frac{y^2}{2} - \sum_{i=1}^{r} (x_i + y_i)^2, $$ thus $$ \left( \sum_{i=1}^{r} (x_i + y_i)^2 \right) = 0, $$ which implies $x_r = y_r$ for $r = 1, 2, \ldots, r - 2$. This together with (1) and (2) implies that $x$ and $y$ are not distinct. Seeker's sequence starts with 100000, 1001000, 1002000, ... Nevertheless, the terms of this sequence are eventually much smaller than the corresponding terms of Seeker's sequence. We now estimate how many integers in $\text{R}$ contain exactly $r$ brackets. Given $r$ brackets we can make the first digit in each of the $r - 2$ brackets 0. We can fill up the first $r - 2$ brackets in as arbitrary manner. This can be done in $$ 10^{(r+2)-(\text{len}(r)-2)} = 10^{r-2} $$ ways. The last two brackets can be filled in such a way as to satisfy (1) and (3). To see this we need only check that the last two brackets will not be overfilled, and that the last digit, which we shall set equal to 1, will not be interfered with. This follows from the inequality $$ (10)^r + (10)^{r-2} + \ldots + (10)^{r-2} < 10^{r-n}. $$ For a given $r$ let $b$ be the integer determined by $$ 10^{\text{len}(r)-2} < k < 10^{\text{len}(r)+2}. \quad (4) $$ Since all integers with at most $r$ brackets will not exceed $n$, and since $r$ brackets can be filled to specification in $10^{(r - 2)(r-2)}$ ways, we have $$ R(n) \ge 10^{(r-2)-2}. \quad (5) $$ From the right hand side of (4) we have $$ r + z \approx 2 \sqrt{\log n} $$ that implies that $$ R(n) \ge (r)(r - 2) - 3 \log_{10}(r - \sqrt{n}), $$ where all logs are to base 10. ``` #################### File: An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf Page: 43 Context: ```markdown 14. Show that \( \frac{n}{\sum_{i=1}^{n} \pi(i)} \geq 1 \). 15. Show that \( \sum_{d|n} d = |n| \). 16. Show that for \( n \geq 3 \), \( 1 + g^2 + g^3 + g^4 \) is not a square. 17. For an integer \( a \geq 0 \), prove that \( \sum_{k=0}^{a} \left| k + \frac{1}{n} \right| = \lfloor a + \frac{1}{n} \rfloor \). 18. Show that \( \frac{1}{\sqrt{n}} \) is asymptotically equivalent to \( \frac{n}{d(n)} \). 19. Prove that \( \sum_{k=1}^{n} \frac{1}{k^2} = x - 2x^2 \). 20. Prove that \( N(n) \geq \frac{1}{1 - x} \sum_{k=1}^{n} 2^k \). 21. Prove that \( F(n) = \prod_{d|n} f(d) \) if and only if \( f(n) = \prod_{d|n} \frac{F(d)}{d} \). 22. Show that the sum of the odd divisors of \( n \) is \( n - \sum_{d|n} (-1)^d \). 23. Prove that the product of the integers \( \leq n \) and relatively prime to \( n \) is \[ n^{\frac{n}{\pi(n)}} \prod_{d|n} \left( \frac{\pi(d)}{d} \right). \] 24. Show that every integer has a multiple of the form \( 11...00 \). 25. Prove that there are infinitely many square-free numbers of the form \( n^2 + 1 \). 26. Prove that \[ \left( \binom{m}{3} \right) \left( \binom{m}{6} \right) \neq 0 \text{ (mod 3)}. \] 27. Show that the number of representations of \( n \) as the sum of one or more consecutive positive integers is \( r(n) \) where \( n_1 \) is the largest odd divisor of \( n \). 28. Prove that \( r(n) = 1 \) is solvable for every \( n \). 29. Prove that \( r(x) = 2^k \) is not solvable for any \( k \). 30. Prove that 30 is the largest integer such that every integer less than it and relatively prime to it is 1 or a prime. 31. Let \( a, a_1, a_2, \ldots \) be ordered by not necessarily distinct nonzero residue classes (mod \( p \)). Prove that there exist integers \( 1 \leq j \leq S - 1 \) such that \( a_i + a_j \equiv 1 \text{ (mod } p \text{)} \). 32. Show that the only solutions of \( \varphi(n) = \varphi(n) \) are \( n = 2, 3, 4, 6, 12, 20, 90 \). 33. Show that \( \varphi(n + 1) = p + 1 \) is valid only for \( 1 \leq n \leq 5 \). 34. Show that \( 2a(n) + 2b(n) \) is an integer. 35. Show that if \( a(n) = 1 \), then \( \frac{(a - 1)(b - 1)}{c - 1} \) is an integer. 36. Show that an integral polynomial of at least the first degree cannot represent exactly many primes. 37. Show that if \( f(x) \) is an integral polynomial of degree \( 0 \), then \( f(x) \) for \( x = 1, 2, \ldots \) has an infinite number of distinct prime divisors. 38. Find the number of integers prime to \( n \) in the set \( \{1, 2, \ldots, m(n+1)\} \). 39. Prove that the Fermat primes are relatively prime in pairs. 40. Let \( T_1 = 2, T_n = T_{n-1} - T_{n-2} \). Prove that \( (T_i, T_j) = 1 \) if \( i \neq j \). 41. Prove that \[ 2(z) = \sum_{k=1}^{\infty} \frac{1}{k} \left( 1 + \frac{1}{2} + \frac{1}{3} + \ldots \right) = 0. \] 42. Prove that the density of numbers for which \( (n, \varphi(n)) = 1 \) is zero. 43. Show that for some \( n \), \( 2^n \) has 1000 consecutive 7s in its digital representation. 44. Show that infinitely many squares do not contain the digit 0. 45. Show that for some \( n \), \( n \) contains 1000 consecutive 7s in its digital representation. 46. Show that the density of the numbers for which \( \varphi(n) = n \) is solvable is zero. 47. Show that if \( \varphi(n) = n \) has exactly one solution then \( n > 10^{100} \). 48. Prove that \( \varphi(n) = n - \frac{\varphi(n)}{p - 1} \). 49. Let \( a_1, a_2, \ldots \) be ordered by not necessarily distinct nonzero residue classes \( (\text{mod } p) \). Prove that there exist integers \( 1 \leq j \leq S - 1 \) such that \( a_i + a_j \equiv 1 \text{ (mod } p \text{)} \). 50. Show that the \( n^{th} \) prime is the limit of the sequence \[ n_0 = n^k, n_k = n + \varphi(n + j) + \ldots. \] 51. Show that the \( n^{th} \) odd prime is the limit of the sequence \[ n, n + \varphi(n), n + \varphi(n + r(n)), \ldots. \] ``` #################### File: An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf Page: 44 Context: ``` 78 Miscellaneous Problems 52. Prove that every positive integer is either of the form n + φ(n) or of the form n + P_n, but not both. 53. Show that \( (3 + 2 \sqrt{2})^{n} + (3 - 2 \sqrt{2})^{n} - 2 \) is a square for every \( n \geq 1 \). 54. Prove that for every real \( x > 2 \) there exists a real \( r \) such that the fractional part of \( r \) is greater than \( 1 - \frac{1}{x} \) for every integer \( x \). 55. Show that if \( p \) and \( q \) are integers such that \( p \) then it is possible to arrange \( n \) or fewer unit resistances to give a combined resistance of \( \phi \). 56. Show that \( (a_n)_{n \geq 1} \) and \( x = a - 12 \sum_{k=1}^{n} \left\lfloor \frac{km}{n} \right\rfloor \) imply \( x \equiv 1 \, (\text{mod} \, n) \). 57. If \( (a, b) = 1 \) prove that \( \sum_{k=1}^{\infty} \frac{(-1)^{k}}{k^a} \) diverges. 58. Show that the sum of reciprocals of integers representable as sums of two squares is divergent. 59. Show that the sum of reciprocals of integers whose digital representation does not include 100 consecutive 7's is convergent. 60. Prove that every \( n > 1 \) can be expressed as the sum of two deficient numbers. 61. Prove that every \( n > 10^6 \) can be expressed as the sum of two abundant numbers. 62. Prove that every sufficiently large \( n \) can be expressed as the sum of \( k \)-abundant numbers. 63. Prove that the \( n \)th non-square is \( n + \sqrt{n} \) (where \( \{x\} \) denotes the integer closest to \( x \)). 64. Prove that the \( n \)th nonquasimir is \( n + \sqrt{2n} \). 65. Prove that the \( n \)th non-\( k \)-power is \[ n + \sqrt{n + 1} \] 66. Show that the binary operation \( \circ \) defined on nonnegative integers by \[ m \circ n = m + n + \sqrt{mn} \] is associative. 67. Prove the same for the operation \( m \times n = m + n + 2\sqrt{mn} \). 68. Show that for \( p > 5 \), \( (p - 1)! \) contains a prime factor \( p \). 69. Show that the only solutions of \( x + y = z - 1 \) are \( (n, k) = (2, 1), (3, 1), \) and \( (5, 2) \). 79 Miscellaneous Problems 70. Show that \( x^2 \equiv 2^{n} \, (\text{mod} \, p) \) has a solution for every prime \( p \geq 3 \). 71. Show that if \( f(x) \) is a polynomial with integer coefficients and \( f(a) \) is a square for each \( a \), then \( f(x) = g(x)f(y) \), where \( g(y) \) is a polynomial with integer coefficients. 72. Given integers \( a_1 < a_2 < ... < a_k < n \) with \( a_j \leq \frac{n}{j} \), prove that for some \( i < j \leq k, a_i \). 73. Show that two of the \( a_i's \) of Problem 72 are relatively prime. 74. With the \( c \)'s of Problem 72, show that \( a_j + c_j \) is solvable. 75. Show that the number of solutions of \( x + 2y + 3z = n \) in non-negative integers is \[ \frac{(n + 3 - 1)(n + 3 - 2)}{2} \] 76. Show that the number of solutions of \( x + 2y + 4z = n \) in non-negative integers is \[ \left\{ \begin{array}{ll} \frac{(1 + 2n + 5)(-1)^{n}}{16}, & \text{if } n \equiv 1 \, (\text{mod} \, 16) \\ 0, & \text{otherwise} \end{array} \right. \] 77. Show that \( n \) and \( n + 2 \) are simultaneously prime if and only if \[ \sum_{k=1}^{n} \left[ \frac{n + 2}{k} + \frac{-1}{-1 - n} \right] = 4. \] 78. Show that if \( m \) and \( n \) are simultaneously prime if and only if \[ 4(n - 1) + 1 = 0 \, (\text{mod} \, n + 2), \, (n > 1). \] 79. Show that for every \( n \), \( 6, 10^2, \) and \( 125 \cdot 10^{t} + 1 \) are Pythagorean triples. 80. Show that the number of ordered pairs of integers whose 1.c.m. is \( n \) is \( \sigma(n) \). 81. Show that \[ \frac{1}{2} + \frac{1}{3} + \cdots + \sqrt[n]{n} \] is never an integer. 82. Show that \( x^2 + 2x + y^2 \) is a square if and only if \( x = y \). 83. Show that \[ \sum_{k=1}^{n} \phi(k)x^{n + k} = x(1 + x)^{n - g(n - 1)} \] 84. Show that the number of regular \( p \)-gons with \( p \) edges is \( 2d \). 85. Prove that the \( n \)th order determinant with \( a_{ij} \) has the value \( \prod_{j=1}^{n} \phi(j) \). ``` #################### File: An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf Page: 45 Context: ``` 86. Prove that \[ \sum_{n=1}^\infty \sqrt{n} = \left( \sqrt{n} + 1 + \sqrt{n} - 1 \right). \] 87. Prove that if \( p = 4n + 3 \) and \( q = 7 \) are both prime then \( p \equiv -1 \mod 8 \). 88. Show how to split the positive integers into two classes so that neither class contains all the positive terms of any arithmetic progression with common difference exceeding 1. 89. Show that the reciprocal of every integer \( n > 1 \) can be expressed as the sum of a finite number of consecutive terms of the form \( \frac{1}{k} \). 90. In how many ways can this be done? (Answer: \( \Psi(\lfloor \sqrt{n} \rfloor) \)). 91. Show that every rational can be expressed as a sum of a finite number of distinct reciprocals of integers. 92. Show that the density of integers for which \( \lfloor \sqrt{n} \rfloor \) is 1 is \( \frac{1}{2} \). 93. Show that the expected value of \( \lfloor \sqrt{n} \rfloor \) is \( \frac{1}{2} \). 94. Prove that \( x^2 \equiv 0 \) (mod \( p \)) for every prime implies that \( a \) is a square. 95. Prove that \( f(a) = f(b) \) for all \( a, b \) and \( f(a + 1) = f(a) + 1 \) for every \( c \) imply that \( f(a) = a^2 \). 96. Find all primes in the sequence 101, 1001, 10101, 101001, ... 97. Find all primes in the sequence 101, 1001, 10101, 101001, ... 98. Show that if \( f(1) > 0 \) for all \( x \) and \( f(x) - f(2) > 0 \) as \( x \to \infty \) then there exists at most a finite number of solutions in integers of \( f(n) + f(p) = 1 \). 99. Prove that the last consecutive of every prime \( p > 2 \) is less than \( \sqrt{p} \). 100. Prove the existence of infinite sequences of \( 1's, 2's, \) and \( 3's \) so that finite part of which is immediately repeated. 101. Let \( d(n) \) denote the number of square divisors of \( n \). Prove that \[ \lim_{n \to \infty} \frac{1}{n} \sum_{m=1}^n d(m) = \frac{\pi^2}{6}. \] 102. Find all \( r \) such that \( r \) cannot end in zeros. 103. Let \( a_1, a_2, \ldots, a_k \) be integers with \( a_1 = 1 \) and \( a_{i+1} \leq a_i \). Prove that there exists a sequence \( (r_i) \) of \( k \) such that \( \sum_{i=1}^k r_i \to \infty \). 104. Show that for a prime \( p \), we can find \[ \sqrt{p} + \sqrt{2} + \ldots + \sqrt{r - 1} = \frac{2 - 1}{12}. \] 105. Prove that \( \pi^2 \) is irrational. 106. Prove that \( \cos \frac{1}{2} \) is irrational. 107. If \( \sum_{n=1}^{\infty} \frac{1}{n^2} \to \infty \) prove that \( \sum_{n=1}^{\infty} \frac{1}{n} \) is irrational. 108. Prove that \( a^2 + b^2 + c^2 \neq 0 \) if \( a, b, c \) are integers. 109. Prove that \[ \tau(n) = \left[ \sqrt{n} - \sqrt{-1} \right] + \frac{\sqrt{n}}{2} \sum_{k=1}^{\infty} \left( \frac{1}{|n - k|} \right). \] 110. Let \( n = a_0 + a_1p + a_2p^2 + ... + a_kp^k \) where \( p \) is a prime and \( 0 \leq a_k < p \). Show that the number of binomial coefficients of order \( k \) that are relatively prime to \( p \) is \( |P(1, k)| \). 111. Show that if \( f_1, f_2, \ldots, f_r \) form a complete residue system (mod \( p \)) then \( f_1, f_2, \ldots, f_{r-1} \) do not. 112. Show that 3 is a primitive root of every Fermat prime. 113. Show that the number of ways in which \( n \) can be represented as the product of two relatively prime factors is \( 2^{c - 1} \). 114. Prove that every even perfect number is of the form \( 2^{(p-1)}(2^p - 1) \). 115. Show that if \( f(x) \) is a polynomial with integer coefficients and there are \( v(n) \) integers relatively prime to \( n \) in the set \( f(1), f(2), \ldots, f(n) \) then \( v(n) \) is a weakly multiplicative function. 116. If \( p = 1 + 1 \) is a prime, show that \( (2n)^2 + 1 \equiv 0 \) (mod \( p \)). 117. Show that 128 is the largest integer not represented as the sum of distinct squares. 118. Show that \( x^3 + y^3 = z^5 \) has infinitely many solutions. 119. Show that \( x^3 + y^3 = z^5 \) has infinitely many solutions. 120. Show that for every \( x > 0 \) there exists a lattice point \( (x, y) \) such that for every lattice point \( (x, y) \) whose distance from \( (x, y) \) does not exceed \( k \), the gcd \( (x, y) > 1 \). 121. Prove that for distinct squares the arithmetic progression is impossible. 122. Prove that for \( n \) composite, \( \tau(n) < n \). 123. Prove that \( 2n \left( 1 + \sqrt{n} \right) \). ``` #################### File: An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf Page: 47 Context: ```markdown 84 Unsolved Problems and Conjectures 20. Conjecture: \( \frac{(2n)!}{(n!)^2} \) is an integer for infinitely many \( n \) (Erdős). 21. Is \( (2n - 1)! \) an integer for every \( k \) and infinitely many \( n \)? (Erdős). 22. Does there exist an \( n \) such that \( [A_n] \) is prime for every \( n \)? (Mills). 23. Does \( [e^n] \) represent infinitely many primes? 24. Does \( [e^n] \) represent infinitely many composite numbers? (Erdős). 25. The number 105 has the property that \( 105 - 2^k \) is prime whenever it is positive. Is 105 the largest number with this property? 26. Is \( p \) the largest number such that for all \( k \) (with \( n \) and \( k > 2 \), \( n - E_k \) is prime)? (Erdős). 27. Does there exist a prime \( p > 41 \) such that \( 2 - z^2 + p \) is prime for \( 1 \leq z \leq p - 17 \)? 28. Let \( a(n) \) denote the number of 1's in the binary representation of \( n \). Does there exist a \( k \) such that for infinitely many primes \( p \), \( a(p) < k \)? (Belman). 29. If \( f(x) \) is a polynomial with integer coefficients, \( f(0) = a \), and \( f(1) = b \), and \( f(a) \), can a sequence \( f(k), k = 1, 2, \ldots \) consist of primes? 30. For sufficiently large \( a \) and \( b \), does the polynomial \( x^n + ax + b \) assume more than \( k \) values (mod \( p \))? (Chowla). 31. Find pairs of integers \( m, n \) such that \( m \) have the same prime factors; \( n = 2^n - 2 \) and \( n = x^2 + 2 \). Are these the only cases? (Straszewicz). 32. What is the largest integer not representable as the sum of distinct cubes? Conjecture: \[ \limsup_{n \to \infty} \frac{a_n}{n^{1/2}} = 0. \] (Chowla and Derpoot) 34. Conjecture: \( \sum_{p} (−1)^{\phi(p)} = 2 \) (Pillai). 35. Can every prime \( p \equiv 3 \, (mod \, 8) \), \( p > 163 \), be written as the sum of three distinct squares? (Erdős). 36. Is \( (3) \) irrational? (Erdős) 37. Conjecture: The only solution of \( 1 + 2 + \ldots + m = (m - 1)1 + 1 + 2 = 2 \) (Bowen). 38. Conjecture: The solutions of \( x^{(k + 1)} + x^{(k + 2)} + \ldots + x^{(k + k)} = \) \( (x + 1 + 2)^6, 3^3 + 3^4 + 5^3 + 5^6 = 5 \). (Erdős). 39. Does the equation \( 1^2 + 2^2 + \ldots + n^2 = (n(n + 1))^2 \) have solutions? (Kelly). 40. Conjecture: If \( r > 0 \) is not an integer then the density of solutions of \( (n,m) \) is 16 for \( (k) \). (Lambek and Moser). 42. Conjecture: The only solutions of \[ \frac{1}{x_1} + \frac{1}{x_2} + \ldots + \frac{1}{x_n} = 1 \] are \[ \text{arc}^2 \left( \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{12} + \frac{1}{42} \right). \] (Erdős). 43. Is it true that for all primes \( p \), all sufficiently large numbers can be written as the sum of distinct numbers of the form \( p^k \)? (Erdős). Let \( c_k \) be the integers not exceeding \( n \) such that the sum of all integers is \( n - 1 \). What is the maximum of \( \sum_{k=1}^{n} c_k \)? (Conjecture: 31.0) (Erdős). Let \( 0 < a_1 < a_2 < \ldots < a_k \) be such that the sums of distinct \( a_i \) are distinct. Conjecture: \( k - \log_k 2 \) is bounded. 44. Give a relatively simple proof of Roth's theorem: any sequence that does not contain an arithmetic progression has zero density. 45. Give an elementary proof of Dirichlet's theorem on quadratic residues: \[ \sum_{p} \left( \frac{2}{p} \right) > 0 \text{ for } p \equiv 3 \, (mod \, 4). \] ``` #################### File: An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf Page: 48 Context: ``` 86 Let \( a_1 < a_2 < \cdots \) be a sequence of positive integers and let \( f(n) \) denote the number of solutions of \( a_i + a_j = n \). Conjecture: If \( f(n) > 0 \) for every \( n \) then \( f(n) \) is unbounded. [Erdős and Turán] If the \( f(n) \) of Problem 49 is \( > 0 \) for every \( n \) sufficiently large then it can be written as the sum of three distinct \( a_k \). Construct a sequence of \( a_k \) for which the \( f(n) \) becomes \( > 0 \) and for which \( f(n) \) is long for every \( n \). [Erdős has shown that such sequences exist.] 53. Does there exist a sequence with a counting function \( A(n) < c / \log n \) such that every integer can be represented in the form \( a_i + a_j \)? Improve the bound \( [*] \) in Schur's theorem in combinatorial number theory. 54. Conjecture: If \( a_i < a_j < a_k \) is a sequence of integers with \( a_i / a_j \to 1 \) and if for every \( d \) every residue (mod \( d \)) is representable as the sum of distinct \( a_k \), then at most a finite number of integers are not representable as the sum of distinct \( a_i \). [Erdős] 55. Is the sum of the reciprocals of those integers that are representable as the sum of \( k \) powered divergents? [Klamkin and Newman] 56. Conjecture: For every \( \epsilon > 0 \) there exists an \( N \in \mathbb{N} \) such that for \( n > N \) the 3-dimensional game of tic-tac-toe played on a \( 3 \times 3 \times 3 \) board must terminate before \( 3^n \) moves have been played. [Moer] 57. Same as Problem 56 with \( 3 \) replaced by \( k \). 58. Every integer belongs to one of the arithmetic progressions \( \{2n\}, \{3n\}, \{4n + 1\}, \{6n + 5\}, \{12n + 7\}, n = 1, 2, \ldots \). This is the simplest example of a finite set of arithmetic progressions, and with distinct common differences, all of those common differences are greater than, or contain all integers. Does there exist for every \( c > 0 \) a set of progressions, such that common difference being \( c \)? [End] 59. Give an explicit representation of \( n \) as the sum of super classes of arithmetic progression? 60. Do there exist for every \( n \) primes that are consecutive terms of an arithmetic progression? 61. \( \frac{1}{1 + 2^2} + \frac{1}{1 + 3^2} + \cdots + \frac{1}{n^2} = 1. \) 62. Are there infinitely many primes of the form \( 11 \cdots 1 \)? 63. Are there infinitely many primitive roots \( 2, 3, \ldots, p + 1 \)? 64. Conjecture: The least primitive root of a prime \( p \) is \( < p \). 65. Conjecture: The number of perfect numbers \( n \) is \( < \log n \). 66. Find good bounds for the density of the abundant numbers. 67. Prove that the ratio of residues to nonresidues in the range \( (1, \lfloor \sqrt{n} \rfloor) \) approaches \( 1 \) as \( n \to \infty \). 68. Give an elementary proof of \( \prod_{p > 3} 2^n \). 70. Conjecture: \( \lim_{n \to \infty} (a_{n+1} - a_n) \) implies \( \sum_{n}\frac{1}{n^2} \). [Erdős] 71. Find all solutions of \( x^4 + y^4 = z^4 + t^4 \). 72. Find all solutions of \( x^2 + y^2 + z^2 = t^2 \). 73. Find all solutions of \( x^n = y^n \). 74. Let \( (t_n) \) be the least \( n \) for which there exists a chain of integers \( a_0 = 1 < a_1 < a_2 < \cdots < a_k = n \), where for each \( i > 0, a_i = a_j + c \) for some \( j, k < i \) permitted). 75. Conjecture: \( t(n) < t(n) \) for all \( n > 0 \). [Utzi] 76. Let \( S(n) \) denote the number of solutions of \( t(n) = n \). Is it true that \( S(n) < S(n + 1) \) for all \( n > 0? \) [Utzi] 77. Polyá's conjecture: \( \sum_{n} \frac{1}{n} > 1 \) for \( n < 800,000 \). 78. Turán's conjecture: \( \sum_{n} \frac{M(n)}{n} \) = 1. [Checked for \( < 50,000 \).] 79. Pillar’s conjecture: \( |n - g| \neq N, m, n > 1 \) has for every \( n \) only a finite number of solutions. 80. \( \frac{1}{1 + 1/2^2} + \frac{1}{1 + 1/3^2} + \cdots + \frac{1}{1/n} = 1 \). ``` #################### File: Algebraic%20Topology%20AT-toc.pdf Page: 2 Context: I'm unable to view images. Please provide the text that you would like to be formatted in Markdown, and I'll be happy to help! Image Analysis: Based on the aspects provided, here is a comprehensive examination of the attached visual content: **1. Localization and Attribution:** - The image occupies the entirety of the provided space. It can be numbered as Image 1. **2. Object Detection and Classification:** - **Image 1:** - **Objects Detected:** - A person. - A large screen/projector behind the person. - A clock on the wall. - Three smaller screens or panels below the large screen. - A speaker or microphone setup on the podium. - A laptop on the podium. - **Classification by Category:** - Person: Human. - Screen/Projector: Electronic device. - Clock: Timekeeping device. - Screens/Panels: Electronic display devices. - Speaker/Microphone: Audio equipment. - Laptop: Computing device. **3. Scene and Activity Analysis:** - **Image 1:** - **Scene Description:** - The scene takes place in what appears to be a conference room or lecture hall. - The person is standing at a podium, presumably giving a presentation. - **Activities Taking Place:** - The main actor, a person, is engaged in speaking or presenting information to an audience. - The person appears to be using a laptop, possibly to control a presentation on the large screen behind them. **4. Text Analysis:** - **Image 1:** - There is no visible text in the image based on the provided resolution and image quality. **8. Color Analysis:** - **Image 1:** - **Dominant Colors:** - Predominantly neutral colors like white and gray for the walls and background. - The podium and the person's attire include darker shades like black and gray. - The large screen in the background is turned off, showing a dark color. - **Impact on Perception:** - The use of neutral and dark colors keeps the focus on the person presenting. **9. Perspective and Composition:** - **Image 1:** - **Perspective:** - The image is taken from a viewpoint that is at the audience level, facing toward the presenter. - This perspective gives the sense of being part of the audience. - **Composition:** - The composition places the presenter at the center. - The large screen behind the presenter acts as the background, with additional visual elements like a clock and smaller screens filling the side spaces. **14. Ablaufprozesse (Process Flows):** - **Image 1:** - No specific process flows are depicted directly in the image. **15. Prozessbeschreibungen (Process Descriptions):** - **Image 1:** - The main process described here involves a presentation setup, with the presenter utilizing electronic equipment to communicate information. **16. Typen Bezeichnung (Type Designations):** - **Image 1:** - The types of equipment include Audio-Visual equipment (screen, laptop, microphones) and timekeeping devices (clock). **17. Trend and Interpretation:** - **Image 1:** - The trend seems to indicate a professional or educational setting where technology is integrated to facilitate presentations and lectures. **19. Tables:** - **Image 1:** - There are no tables in the image to analyze. The image provides a focused look at a presentation scene in a formal setting. The arrangement of elements and the color palette underscore the importance of the presenter and the information being shared. ########## """QUERY: give me a 500 word essay""" Consider the chat history for relevant information. If query is already asked in the history double check the correctness of your answer and maybe correct your previous mistake. Use as much tokens as needed but at the same time be as efficient as possible. If you find information separated by a | in the context, it is a table formatted in Markdown - the whole context is formatted as md structure. Important: Take a look at the QUERY and only the QUERY. Please try always to answer the query question. If the client ask for a formatting structure follow his advise.But if the question is vague or unclear ask a follow-up question based on the context. Final Files Sources: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 1, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 3, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 4, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 5, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 6, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 7, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 9, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 11, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 15, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 16, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 22, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - 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