{ "query": "what are sets and spaces? quote information from your context", "namespace": "924dbc22-31b1-4285-ad02-9f10843c9937", "messages": [ { "role": "assistant", "content": "Hallo NF SepTest,\n\nich bin Simon, dein KI-Chatbot, der mit dem Large Language Model gemini-1.5-flash arbeitet. Ich bin hier, um all deine Fragen pr\u00e4zise und umfassend zu beantworten. \n\nUnser System KIOS bietet dir Zugriff auf eine Wissensdatenbank namens 10-2, die eine Vielzahl von Informationen enth\u00e4lt. In dieser Datenbank befinden sich folgende Dokumente:\n\n* **Algebraic Topology AT-toc.pdf:** Ein Inhaltsverzeichnis f\u00fcr ein Buch \u00fcber algebraische Topologie, das Themen wie Homotopie, Homologie, Kohomologie und Homotopietheorie behandelt.\n* **An Introduction to the Theory of Numbers - Leo Moser (PDF).pdf:** Vorlesungsunterlagen \u00fcber elementare Zahlentheorie, die Themen wie Kompositionen und Partitionen, arithmetische Funktionen, Verteilung von Primzahlen, irrationale Zahlen, Kongruenzen, diophantische Gleichungen, kombinatorische Zahlentheorie und Geometrie der Zahlen abdecken.\n\nM\u00f6chtest du mehr \u00fcber die Themen in diesen Dokumenten erfahren? Hier sind ein paar Vorschl\u00e4ge:\n\n* \ud83e\uddee **Grundlagen der algebraischen Topologie:** Was sind Homotopie und Homologie? Wie werden sie in der Topologie verwendet?\n* \ud83d\udd22 **Elementare Zahlentheorie:** Was sind Primzahlen und wie werden sie verteilt? Wie funktionieren arithmetische Funktionen?\n* \ud83d\udd0e **Ungel\u00f6ste Probleme in der Zahlentheorie:** Was sind die Goldbachsche Vermutung, die Euler-Vermutung und die Zwillings-Primzahl-Vermutung?\n* \ud83d\udcda **Vergleich der beiden Dokumente:** Welche Gemeinsamkeiten und Unterschiede gibt es zwischen den beiden Dokumenten?\n\nWas m\u00f6chtest du dir heute genauer ansehen? \ud83d\ude0a\n\n\nwhat are sets and spaces? quote information from your context\n\n FOR ADDITIONAL CONTEXT:\n\nKnowledgebase name: 10-2\nKnowledgebase files:\nAlgebraic Topology AT-toc.pdf\nAn Introduction to the Theory of Numbers - Leo Moser (PDF).pdf\nAlgorithms and Complexity - Herbert S. Wilf (PDF).pdf\nKnowledgebases crawlers:\n" }, { "role": "user", "content": "halo" }, { "role": "assistant", "content": "Hello! \ud83d\udc4b How can I help you today? \ud83d\ude0a" }, { "role": "user", "content": "can you write me 500 words on what is the contents of your context?" }, { "role": "assistant", "content": "The provided context is a Table of Contents (TOC) and a section titled \"Standard Notations\" from a book titled \"Algebraic Topology\" by Allen Hatcher. \n\n**Table of Contents:**\n\nThe TOC outlines the book's structure, dividing the content into chapters and sections. It covers fundamental geometric notions, the fundamental group, covering spaces, and additional topics. Each chapter is further divided into sections, providing a detailed overview of the book's content.\n\n**Standard Notations:**\n\nThis section defines various mathematical notations used throughout the book. It covers:\n\n- **Sets and Spaces:**\n - **Z, Q, R, C, H, O**: Integers, rationals, reals, complexes, quaternions, and octonions.\n - **Zn**: Integers modulo n.\n - **Rn**: n-dimensional Euclidean space.\n - **Cn**: Complex n-space.\n - **I**: Unit interval.\n - **Sn**: Unit sphere in \\(R^{n+1}\\).\n - **Dn**: Unit disk or ball in \\(R^n\\).\n - **\u2202Dn**: Boundary of the n-disk.\n - **en**: n-cell.\n - **|S|**: Disjoint union of sets or spaces.\n - **\u00d7, \u220f**: Product of sets, groups, or spaces.\n - **\u2245**: Isomorphism.\n - **A \u2282 B**: Set-theoretic containment.\n - **A \u21aa B**: Inclusion map.\n - **A - B**: Set-theoretic difference.\n\n- **Mathematical Symbols:**\n - **1**: Identity function.\n - **iff**: If and only if.\n\nThe section also clarifies that definitions of mathematical terms are provided within the text rather than in separate lists.\n\n**Overall, the context provides a glimpse into the structure and content of the book \"Algebraic Topology\" by Allen Hatcher. It outlines the book's organization, defines standard mathematical notations, and clarifies the book's approach to presenting definitions.**" }, { "role": "assistant", "content": "Thank you for providing the context. If you have any specific questions about the content of \"Algebraic Topology\" by Allen Hatcher, or if you need clarification on any of the standard notations or topics mentioned in the Table of Contents, please feel free to ask! I'm here to assist you with precise and accurate information." }, { "role": "assistant", "content": "It seems that I currently do not have access to the specific content of the files listed in the knowledgebase. However, if you have questions about algebraic topology or related topics, feel free to ask, and I will do my best to provide accurate and relevant information based on my training. If you need detailed information from the documents, I recommend reviewing them directly." }, { "role": "user", "content": "what are sets and spaces? quote information from your context" }, { "role": "assistant", "content": "" } ], "stream": true, "language_level": "", "chat_channel": ":4457:1727871687775", "language": "German", "tone": "neutral", "writing_style": "standard", "model": "gemini-1.5-flash", "knowledgebase": "ki-dev-large", "seed": 1055, "client_id": 1727871687775, "all_context": false, "follow_up_for": null, "knowledgebase_files_count": 3, "override_command": null, "disable_clarity_check": false, "custom_primer": "", "logging": true, "query_route": "" } INITIALIZATION Knowledgebase: ki-dev-large Base Query: what are sets and spaces? quote information from your context Model: gemini-1.5-flash **Elapsed Time: 0.00 seconds** ROUTING Query type: research **Elapsed Time: 1.57 seconds** RAG PARAMETERS Max Context To Include: 20 Lowest Score to Consider: 0.1 ================================================== **Elapsed Time: 0.14 seconds** ================================================== VECTOR SEARCH ALGORITHM TO USE Use MMR search?: False Use Similarity search?: True ================================================== **Elapsed Time: 0.10 seconds** ================================================== VECTOR SEARCH DONE ================================================== **Elapsed Time: 0.69 seconds** ================================================== PRIMER Primer: You are Simon, a highly intelligent personal assistant in a system called KIOS. You are a chatbot that can read knowledgebases through the "CONTEXT" that is included in the user's chat message. Your role is to act as an expert at reading the information provided by the user and giving the most relevant information. In your responses to enterprise users, prioritize clarity, trustworthiness, and appropriate formality. Be honest by admitting when a topic falls outside your scope of knowledge, and suggest alternative avenues for obtaining information when necessary. Make effective use of chat history to avoid redundancy and enhance response relevance, continuously adapting to integrate all necessary details in your interactions. Focus on precision and accuracy of your answers information. **Elapsed Time: 0.18 seconds** FINAL QUERY Final Query: CONTEXT: ########## File: Algebraic%20Topology%20AT-toc.pdf Page: 7 Context: # Preface The additional topics amount to nearly half the book, and they are included here both to make the book more comprehensive and to give the reader who takes the time to dive into them a more substantial sample of the true richness and beauty of the subject. There is also an Appendix dealing mainly with a number of matters of a point-set topological nature that arise in algebraic topology. Since this is a textbook on algebraic topology, details involving point-set topology are often treated lightly or skipped entirely in the body of the text. Not included in this book is the important but somewhat more sophisticated topic of spectral sequences. It was very tempting to include something about this marvelous tool here, but spectral sequences are such a big topic that it seemed best to start with them afresh in a new volume. This is tentatively titled *Spectral Sequences in Algebraic Topology* and is referred to herein as [SSAT]. There is also a third book in progress, on vector bundles, characteristic classes, and K-theory, which will be largely independent of [SSAT] and of much of the present book. This is referred to as [VBKT], its provisional title being *Vector Bundles and K-Theory*. In terms of prerequisites, the present book assumes the reader has some familiarity with the content of the standard undergraduate courses in algebra and point-set topology. In particular, the reader should know about quotient spaces, or identification spaces as they are sometimes called, which are quite important for algebraic topology. Good sources for this concept are the textbooks [Armstrong 1983] and [Jänich 1984] listed in the Bibliography. A book such as this one, whose aim is to present classical material from a rather classical viewpoint, is not the place to indulge in wild innovation. There is, however, one small novelty in the exposition that may be worth commenting upon, even though one might say that it plays a relatively minor role. This is the use of what we call Δ-complexes, which are a mild generalization of the classical notion of a simplicial complex. The idea is to decompose a space into simplices allowing different faces of a simplex to coincide and dropping the requirement that simplices are uniquely determined by their vertices. For example, if one takes the standard picture of the torus as a square with opposite edges identified and divides the square into two triangles by cutting along a diagonal, then the result is a Δ-complex structure on the torus having 2 triangles, 3 edges, and 1 vertex. By contrast, a simplicial complex structure on the torus must have at least 14 triangles, 21 edges, and 7 vertices. So Δ-complexes provide a significant improvement in efficiency, which is nice from a pedagogical viewpoint since it simplifies calculations in examples. A more fundamental reason for considering Δ-complexes is that they seem to be very natural objects from the viewpoint of algebraic topology. They are the natural domain of definition for simplicial homology, and a number of standard constructions produce Δ-complexes rather than simplicial complexes. Historically, Δ-complexes were first introduced by #################### File: Algebraic%20Topology%20AT-toc.pdf Page: 9 Context: # Standard Notations **Z, Q, R, C, H, O** : the integers, rationals, reals, complexes, quaternions, and octonions. - **Zn** : the integers mod n. - **Rn** : n-dimensional Euclidean space. - **Cn** : complex n-space. In particular, **R^0 = {0} = C^0**, zero-dimensional vector spaces. - **I = [0, 1]** : the unit interval. - **Sn** : the unit sphere in **R^(n+1)**, all points of distance 1 from the origin. - **Dn** : the unit disk or ball in **R^n**, all points of distance ≤ 1 from the origin. - **∂Dn = Sn-1** : the boundary of the n-disk. - **en** : an n-cell, homeomorphic to the open n-disk **Dn - ∂Dn**. In particular, **D^0** and **e^0** consist of a single point since **R^0 = {0}**. But **S^0** consists of two points since it is **∂D^1**. - **1** : the identity function from a set to itself. - **|S|** : disjoint union of sets or spaces. - **×, ∏** : product of sets, groups, or spaces. - **≅** : isomorphism. - **A ⊂ B** or **B ⊃ A** : set-theoretic containment, not necessarily proper. - **A ↪ B** : the inclusion map **A → B** when **A ⊂ B**. - **A - B** : set-theoretic difference, all points in **A** that are not in **B**. - **iff** : if and only if. There are also a few notations used in this book that are not completely standard. The union of a set **X** with a family of sets **Yi**, with **i** ranging over some index set, is usually written simply as **X ∪ Yi** rather than something more elaborate such as **X ∪ (∪i Yi)**. Intersections and other similar operations are treated in the same way. Definitions of mathematical terms are given within paragraphs of text, rather than displayed separately like theorems. These definitions are indicated by the use of **boldface** type for the more important terms, with italics being used for less important or simpler definitions as well as simple emphasis as in standard written prose. Terms defined using boldface appear in the Index, with the page number where the definition occurs listed first. #################### 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: 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: # 1.5 Counting 1. Write out a complete proof of theorem 1.4.1. 2. 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 \( \varepsilon \geq 0 \) ..." 3. In theorem 1.4, if we add 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 of 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 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 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 \[ n \cdot (n - 1) \cdot (n - 2) \cdots 2 \cdot 1 = n! \] for ways to form the whole sequence. Let \( S \) be a subset of \( [n] \), and let \( k \) be the number of elements in \( S \). The number of elements in a set is called its cardinality. The cardinality of a set \( S \) is denoted by \( |S| \), so for example, \( |S| = 6 \). A set with cardinality \( k \) is called a \( k \)-set; and a subset of cardinality \( k \) is naturally known as a \( k \)-subset. The question is, for how many subsets of \( S \) 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 we might have chosen that sequence, so the number of ways to choose an (ordered) sequence of \( k \) elements from \( [n] \) is \[ \frac{n!}{(n - k)!} \cdot k! \] But there are more sequences of \( k \) elements than there are \( k \)-subsets, because any particular \( k \)-subset corresponds to \( k! \) different ordered sequences, namely all possible rearrangements of the given \( k \)-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 \[ \frac{n!}{k!(n - k)!} \] subsets of a set of \( n \) objects. The quantities \( \binom{n}{k} = \frac{n!}{k!(n - k)!} \) are the famous binomial coefficients, and they are denoted by \[ \binom{n}{k} \quad (n \geq 0; 0 \leq k \leq n). \] Some of their special values are: \[ \binom{n}{0} = 1 \quad (\text{for } n \geq 0); \] \[ \binom{n}{1} = n \quad (\text{for } n \geq 0); \] \[ \binom{n}{2} = \frac{n(n - 1)}{2} \quad (\text{for } n \geq 2); \] \[ \binom{n}{n} = 1 \quad (\text{for } n \geq 0). \] 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: ### Detailed Analysis #### Image Localization and Attribution - **Image Number:** Image 1 - **Position:** This image is the only one on the page, centrally located. #### Text Analysis - **Text Extraction:** *Extracted Text (Partial):* ``` 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 ε > 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… ... 1.5 Counting For a given positive integer n, consider the set {1, 2,…, n}. We will denote this set by the symbol […] Example: List of all the subsets of {2}: { }, {1}, {2}, {1, 2}. There are […] We claim that the set […] has exactly […] Each of the n choices … -Sets ... We can construct k-subsets S of [n]… Therefore, the number of ways to choose an ordered sequence of k elements from [n] is … Some of their special values are... It is convenient to define (nk) to be 0 if k < 0 or if k > n. ``` - **Text Content and Significance:** - **1.5 Counting Section:** The text is an excerpt from a section titled "1.5 Counting", focusing on combinatorial mathematics. It discusses the counting principles around subsets and sequences from set {1, 2,...,n}. - **Examples:** The text includes examples of subsets and calculations involving combinations (\(\binom{n}{k}\)) and their properties. - **Special Values and Definitions:** Specific binomial coefficient values are noted, providing foundational knowledge in combinatorial analysis. #### Diagram and Chart Analysis - **Content:** The given page does not include diagrams or charts. #### Product Analysis - **Content:** There are no products depicted in this image. #### Anomaly Detection - **Content:** There are no noticeable anomalies or unusual elements within the image. #### Color Analysis - **Dominant Colors:** - The image consists primarily of black text on a white background, typical for a textbook or academic document page. - **Impact:** - The color scheme is standard for printed academic material, allowing for clear readability and minimal visual distraction. #### Perspective and Composition - **Perspective:** - The image appears to be a straight, directly overhead view, typical for a scanned or digitally-created page. - **Composition:** - The text is organized into paragraphs and lists. The section titled "1.5 Counting" is introduced mid-page with a clear hierarchical structure for easy navigation and comprehension. Subsections are denoted by new paragraphs, bullet points, and indentation, contributing to a structured and well-organized format. #### Contextual Significance - **Overall Context:** - The text appears to be part of an academic textbook or educational resource focusing on mathematical theories, specifically combinatorics and set theory. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 25 Context: ```markdown # 1.5 Counting 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 \( \epsilon > 0 \) . . . " were replaced by "for every fixed \( \epsilon \geq 0 \) . . . " 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 \( \{1, 2, \ldots, n\} \). 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 of the subsets of \( [2] \): \( \emptyset, \{1\}, \{2\}, \{1, 2\} \). 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 choices that you encountered could have been made in either 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 that 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 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 \( S \subseteq [n] \)) as follows. Choose an element \( a_1 \) (in possible choices). Of the remaining \( n - 1 \) elements, choose one \( a_2 \) (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 we might have chosen that sequence, so the number of ways to choose an (ordered) sequence of \( k \) elements from \( [n] \) is: \[ n(n - 1)(n - 2) \cdots (n - k + 1) = \frac{n!}{(n - k)!} \] 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 given 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: \[ \binom{n}{k} = \frac{n!}{k!(n - k)!} \quad (n \geq 0; 0 \leq k \leq n) \] Some of their special values are: \[ \binom{n}{0} = 1 \quad (n \geq 0); \] \[ \binom{n}{1} = n \quad (n \geq 0); \] \[ \binom{n}{2} = \frac{n(n - 1)}{2} \quad (n \geq 2); \] \[ \binom{n}{n} = 1 \quad (n \geq 0). \] 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: ### Analysis of the Provided Visual Content #### 1. Localization and Attribution: - **Image 1**: This is the only image present on the page. #### 4. Text Analysis: - The text seems to be part of a mathematical and theoretical discussion. The sections include references to specific theorems (e.g., Theorem 1.4.1) and exercises related to counting subsets. - **Section 1.5 Counting**: - This section discusses the counting of subsets for a given set. It uses mathematical notation and examples to illustrate various ways to arrange and count subsets. - **Detected Text**: - The detected text includes tasks for the reader such as writing a complete proof of a theorem, examining results under different conditions, and proving statements using certain properties and sequences. - There are also references to particular equations and combinatorial problems, like the computation of binomial coefficients. #### 8. Color Analysis: - The image is primarily in black and white, typical for a text document. #### 11. Metadata Analysis: - No metadata was visible in the context of this visual. #### 13. Graph Numbers: - The image yields the following relevant binomial coefficient values: - \(\binom{n}{0} = 1\) - \(\binom{n}{1} = n\) - \(\binom{n}{2} = n(n - 1)/2\) - \(\binom{n}{n} = 1\) #### Additional Aspects: - **Ablaufprozesse (Process Flows)**: - There is a sequence described for how to systematically construct subsets and k-subsets from a given set, presenting a logical and step-by-step process of counting them. - **Prozessbeschreibungen (Process Descriptions)**: - Detailed descriptions are included about how to choose elements of a set to form subsets and how to count them systematically. This involves combinatorial logic. - **Typen Bezeichnung (Type Designations)**: - k-sets and k-subsets, where k denotes the cardinality or number of elements in the subset. - **Tables**: - While there isn't a formal table, tabular information is provided in the context of listings (cardinality and binomial coefficients). ### Summary: The image provides a thorough analysis rooted in combinatorial mathematics, specifically focusing on counting subsets of a set using binomial coefficients. It demonstrates the logical processes for constructing subsets and k-subsets. The discussions and exercises prompt verification through proofs and the application of theorem results. #################### 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: 42 Context: # Chapter 2: Recursive Algorithms ## 4. Consider the array Let \( x = \{2, 4, 1, 10, 5, 3, 9, 7, 8, 6\} \) with left = 1 and right = 10. Suppose that the procedure `split` is called, and suppose the random integer \( L \) in step 1 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 \( j = \ldots \) loop. 1. Suppose \( H(0) = 1 \) and \( H(1) \leq 1 + \sum_{i=1}^{n-1} H(i - 1) \) (for \( n \geq 1 \)). How might \( H(n) \) be? 2. If \( Q(0) = 0 \) and \( Q(n) \leq n^2 + \sum_{i=1}^{n-1} Q(i - 1) \) (for \( n \geq 1 \)), how big might \( Q(n) \) be? 3. (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. By an independent set of vertices of \( G \) we mean 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 of vertices 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. ### Fig. 2.3.1 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 \( m(s, v) \). 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 of \( s \) 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, where the graph that is obtained from \( G \) by deleting vertex \( v \) as well as all vertices that are connected to vertex \( v \) are 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 \( G - \{v\} \). 38 Image Analysis: ### Comprehensive Analysis of the Visual Content #### 1. **Localization and Attribution:** - **Image 1:** Located in the middle of the page, below the text. Numbered as Image 1. #### 2. **Object Detection and Classification:** - **Image 1:** - **Objects:** - A 2D graph or diagram. - **Classification:** - The diagram illustrates a set of points and possibly a connection or relationship between them. #### 3. **Scene and Activity Analysis:** - **Image 1:** - **Scene Description:** - The diagram shows a simple graph with vertices labeled 1, 2, 3, and points connected by edges. - **Activities:** - The vertices and edges suggest it is demonstrating a concept in graph theory. #### 4. **Text Analysis:** - **Text Content:** - The text preceding the diagram discusses recursive algorithms, particularly focusing on an array and the maximum independent set problem in graphs. - Key excerpts: - "Consider the array x = {2, 4, 1, 10, 5, 3, 9, 7, 8, 6}" - "Recursive graph algorithms" - Description of maximum independent set problem. - **Context and Significance:** - The text explains the concept behind recursive algorithms and gives an example involving an array and a maximum independent set problem in a graph context. #### 5. **Diagram and Chart Analysis:** - **Image 1 (Fig. 2.3.1):** - **Description:** - The diagram shows points (vertices) {1, 2, 6} forming an independent set. - **Axes and Scales:** - No specific axes or scales since it's a conceptual illustration. - **Key Insights:** - The diagram is used to illustrate the independent set in a graph G. - Helps in understanding the complexity and structure of the independent set problem. #### 7. **Anomaly Detection:** - **Image 1:** - **Anomalies:** - No noticeable anomalies. Diagram is clear and aligns with the contextual explanation of the text. #### 8. **Color Analysis:** - **Image 1:** - **Color Composition:** - Black and white diagram. - **Impact:** - Color choice emphasizes clarity and simplicity, focusing on the conceptual understanding rather than visual appeal. #### 9. **Perspective and Composition:** - **Image 1:** - **Perspective:** - Front-on view, typical for illustrative diagrams. - **Composition:** - Vertices and edges are clearly labeled and arranged for easy comprehension. #### 10. **Contextual Significance:** - **Image 1:** - The diagram complements the text discussion on graph theory and recursive algorithms. - It serves to visualize the maximum independent set problem, enhancing the reader's understanding of the concept discussed in the text. #### 12. **Graph and Trend Analysis:** - **Image 1:** - **Graph Numbers:** - Vertices labeled {1, 2, 6}. - **Data Points and Trends:** - Independent set in a graph, conceptual rather than numerical data. #### 14. **Ablaufprozesse (Process Flows):** - **Text and Image Context:** - Explains the process of finding the largest independent set in a graph recursively. #### 16. **Prozessbeschreibungen (Process Descriptions):** - **Text Context:** - Provides a step-by-step description for solving the maximum independent set problem recursively. - Involves fixing a vertex, distinguishing sets containing or not containing the vertex, and recursively applying the problem to subproblems. ### Conclusion: The visual content and accompanying text provide a detailed explanation and illustration of recursive algorithms in the context of graph theory. Image 1 (Fig. 2.3.1) effectively visualizes an independent set in a graph, aiding in the comprehension of the discussed algorithms and their computational complexities. #################### 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: 43 Context: # 2.3 Recursive Graph Algorithms We now have all of the ingredients of a recursive algorithm. Suppose we have found the two numbers `maxSet(G = (V, E))` and `maxSet(G = (V - {v*}, Nbhd(v*))`. Then, from the discussion above, we have the relation: ``` maxSet(G) = max{maxSet(G - {v*}) + 1, maxSet(G - {v*} - Nbhd(v*))}. ``` We obtain the following recursive algorithm. ```plaintext function maxSet1(G); {returns the size of the largest independent set of vertices of G} if G has no edges then maxSet1 := |V(G)|. else choose some nonisolated vertex v* of G; n1 := maxSet1(G - {v*}); n2 := maxSet1(G - {v*} - Nbhd(v*)); maxSet1 := max(n1, 1 + n2); end(maxSet1); ``` ## Example: Here is an example of a graph \( G \) and the result of applying the `maxSet1` algorithm to it. Let the graph \( G \) be a 5-cycle. That is, it has 5 vertices and its edges are (1,2), (2,3), (3,4), (4,5), (1,5). What are the two graphs on which the algorithm calls itself recursively? Suppose we select vertex number 1 as the chosen vertex \( v* \) in the algorithm. Then \( G - \{1\} \) and \( G - \{1\} - Nbhd(1) \) are respectively the two graphs shown in Fig. 2.3.2. ``` 2 3 | | 5--1---4 ``` **Fig. 2.3.2:** \( G - \{1\} \) | \( G - \{1\} - Nbhd(1) \) The reader should now check that the size of the largest independent set of \( G \) is equal to the larger of the two numbers `maxSet1(G - {v*}) + 1` and `maxSet1(G - {v*} - Nbhd(v*))` in this example. Of course the creation of these two graphs from the original input graph is just the beginning of the story, as far as the computation is concerned. Unbeknownst to the programmer, who innocently wrote the recursive routine `maxSet1` and then sat back to watch, the compiler will go ahead with the computation by generating a tree full of graphs. In Fig. 2.3.3 we show the collection of all of the graphs that the compiler might generate while executing a single call to `maxSet1` on the input graph of this example. In each case, the graph that is below and to the left of a given one is the one obtained by deleting a single vertex, and the one below and to the right of each graph is obtained by deleting a vertex and its entire neighborhood. Next, we are going to study the complexity of `maxSet1`. The results will be sufficiently depressing that we will then think about how to speed up the algorithm, and we will succeed in doing that to some extent. To open the discussion, let's recall that in Chapter 1 it was pointed out that the complexity of a calculation is usefully expressed as a function of the number of bits of input data. In problems about graphs, however, it is more natural to think of the number of labor as a function of \( n \), the number of vertices of the graph. In problems about matrices it is more natural to use \( m \), the size of the matrix, and so forth. Do those distinctions alter the classification of problems into "polynomial time-doable" vs. "hard"? Take the graph problems, for instance. How many bits of input data does it take to describe a graph? Well, certainly we can enumerate through the entire list of \( n \times (n - 1) / 2 \) pairs of vertices and the edges that are actually edges in the input graph to the problem. Hence we can describe a graph to a computer by making #################### 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 **marset** turns out to be no harder than the special case of **marset** in which no vertex has more than 3 neighbors. Aside from the complexity issue, the algorithm **marset** 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. ![Fig. 2.3.4](path_to_image) 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 cease 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 1. every pair of countries that have a common stretch of border have different colors and 2. 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 \( v \) for each country on the map. Two of these vertices will be connected by an edge of the graph \( G \) if the two countries that correspond to them share a common stretch of border (we keep saying "stretch of border" to emphasize that if two countries have just a single point in common they are allowed to have the same color). As an illustration: * Every planar map is colorable, **Bull. Amer. Math. Soc.**, 82 (1976), 711-712. 42 Image Analysis: ### Image Analysis #### 1. **Localization and Attribution**: - The page contains a single image centrally placed. - Numbered as: **Image 1** #### 2. **Object Detection and Classification**: - **Image 1**: The image depicts a geometric illustration, likely a graph from graph theory. - **Detected Objects**: Vertices and edges of a graph. - **Classification**: Mathematical diagram, specifically an undirected graph. #### 3. **Scene and Activity Analysis**: - **Image 1**: The scene consists of a diagram showing interconnected points (vertices) and lines (edges). - **Main Activity**: Demonstration of a concept in graph theory, presumably related to coloring vertices. #### 4. **Text Analysis**: - **Text Extraction**: - The page has multiple paragraphs discussing: 1. The general problem of `maxset` and its complexity. 2. An algorithm showing recursive ideas for transforming questions about graphs. 3. A discussion of a graph `G` with `K` colors. 4. Details about the Four-Color Theorem and its proof. 5. Explanation of how to represent a map as a graph and the ability to color it using only four colors. - **Text in Image 1**: "Fig. 2.3.4" - **Text Content**: - Discusses theoretical problems in graph coloring, specifically the four-color problem. - The significance of the Four-Color Theorem, which states any map can be colored with no more than four colors, ensuring no two adjacent regions share the same color. #### 5. **Diagram and Chart Analysis**: - **Diagram Analysis (Fig. 2.3.4)**: - **Data and Trends**: Illustrates a graph with vertices and edges used to explain a concept in graph theory. - **Axes, Scales, and Legends**: Not applicable as it is a categorical illustration. - **Key Insights**: Visual representation of a theorem or concept in graph theory, probably linked to vertex coloring. #### 9. **Perspective and Composition**: - **Perspective**: The diagram is presented from a top-down view. - **Composition**: The elements (vertices and edges) are symmetrically arranged, highlighting the connections between different points in the graph. #### 10. **Contextual Significance**: - The image and accompanying text elaborate on complex concepts in graph theory, providing a visual aid (Fig. 2.3.4) to demonstrate how graphs can be colored under certain constraints. - Contributes to the overall message of discussing recursive algorithms and computational problems in graph theory, culminating in the Four-Color Theorem's practical explanation. Overall, this page serves as an educational piece aimed at explaining and proving concepts in graph theory, specifically focused on graph coloring algorithms and the famous Four-Color Theorem. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 47 Context: # 2.3 Recursive Graph Algorithms In this section, we show in Fig. 2.3.5(a) a map of a distant planet, and in Fig. 2.3.5(b) the graph that results from the construction that we have just described. By a **planar graph** we mean a graph \( G \) that can be drawn in the plane in such a way that two edges never cross (except that two edges at the same vertex have that vertex in common). The graph that results from changing a map of countries into a graph as described above is always a planar graph. In Fig. 2.3.6(a) we show a planar graph \( G \). This graph doesn't look planar because two of its edges cross. However, that isn’t the graph’s fault, because with a little more care we might have drawn the same graph as in Fig. 2.3.6(b), in which its planarity is obvious. Don’t blame the graph if it doesn’t look planar. It might be planar anyway! The question of recognizing whether a given graph is planar is itself a formidable problem, although the solution, due to J. Hopcroft and R. E. Tarjan,* is an algorithm that makes the decision in linear time, i.e., in \( O(V) \) time for a graph of \( V \) vertices. Although every planar graph can be properly colored in four colors, there are still all of those other graphs that are not planar to deal with. For any of those graphs we can ask if a positive integer \( K \) is given, whether or not its vertices can be \( K \)-colored properly. As if that question weren’t hard enough, we might ask for even more detail, namely about the number of ways of properly coloring the vertices of a graph. For instance, if we have \( K \) colors to work with, suppose \( G \) is the empty graph \( R_n^K \). That is, the graph of \( n \) vertices that has no edges at all. Then \( G \) has quite a large number of proper colorings, \( K^n \), to be exact. Other graphs of \( n \) vertices have fewer proper colorings than that, and an interesting computational question is to count the proper colorings of a given graph. We will now find a recursive algorithm that will answer this question. Again, the complexity of the algorithm will be exponential, but as a small consolation we note that no polynomial time algorithm for this problem is known. Choose an edge \( e \) of the graph, and let its endpoints be \( u \) and \( v \). Now delete the edge from the graph, and let the resulting graph be called \( G - \{ e \} \). Then we will distinguish two kinds of proper colorings of \( G - \{ e \} \: \): those in which vertices \( u \) and \( v \) have the same color and those in which \( u \) and \( v \) have different colors. Obviously the number of proper colorings of \( G - \{ e \} \) if \( u \) and \( v \) are colored the same is the sum of the numbers of colorings of each of these two kinds. * Efficient planarity testing, J. Assoc. Comp. Mach. 21 (1974), 549-568. 43 Image Analysis: ### Comprehensive Examination of the Visual Content #### 1. Localization and Attribution - **Image 1**: Located at the top left (labeled as Fig. 2.3.5(a)). - **Image 2**: Located at the top right (labeled as Fig. 2.3.5(b)). - **Image 3**: Located at the bottom left (labeled as Fig. 2.3.6(a)). - **Image 4**: Located at the bottom right (labeled as Fig. 2.3.6(b)). #### 2. Object Detection and Classification - **Image 1**: - Objects: Nodes (labeled A-F), Edges connecting nodes. - Features: Circular shape with labeled sections. - Category: Graph/Diagram. - **Image 2**: - Objects: Nodes (labeled A-F), Edges connecting nodes. - Features: Pentagon and liner connections indicating a planar graph. - Category: Graph/Diagram. - **Image 3**: - Objects: Nodes (labeled 1-6), Edges connecting nodes. - Features: Hexagonal layout with all nodes interconnected. - Category: Graph/Diagram. - **Image 4**: - Objects: Nodes (labeled 1-6), Edges connecting nodes. - Features: Similar structure to Image 3 but arranged differently. - Category: Graph/Diagram. #### 3. Scene and Activity Analysis - **Image 1**: - Scene: Schematic map indicating regions A-F. - Activity: Representation of spatial relationships in a graph form. - **Image 2**: - Scene: Planar graph corresponding to the map. - Activity: Visualization of non-intersecting edges in a graph. - **Image 3 and Image 4**: - Scene: Graph representations analyzing planar properties. - Activity: Illustrations of node connections and planar graph structures. #### 4. Text Analysis - **Text Detected**: - Fig. 2.3.5(a) - 2.3.6(b): Figure labels providing references. - Additional text describing graphs and their properties: - "A map of a distant planet..." - "Efficient planarity testing" - **Significance**: These descriptions explain the graphs and their context, providing insight into the construction and properties of planar graphs. #### 5. Diagram and Chart Analysis - **Diagrams** (Fig. 2.3.5(a) - 2.3.6(b)): - Purpose: Illustrate planar graphs and show the transformation from maps to graph representations. - Key Insights: Demonstrate graph properties and planar conditions without intersecting edges. #### 8. Color Analysis - **General Observation**: All images are primarily black and white, focusing on nodes and edges for clarity. #### 9. Perspective and Composition - **Perspective**: All images are top-down schematic views for clarity and ease of understanding of graph properties. - **Composition**: - Nodes and edges are arranged to show relationships. - Clear labeling to identify nodes and connections. #### 12. Graphs and Trend Analysis - **Graphs**: - Detailed comparisons showing how altering edges affect planar properties. #### 13. Graph Numbers: - **Fig. 2.3.6(a) and 2.3.6(b)**: Nodes labeled from 1 to 6, showing different interconnections. ### Contextual Summary The images and corresponding text form part of a discussion on recursive graph algorithms, focusing on planar graphs. By illustrating the planar maps and their graph representations, the figures aid in visualizing complex graph properties and transformations. This contributes significantly to understanding the underlying concepts explored in this section of the document. The planar graphs and explanations tie back to theoretical computer science and efficient planarity testing, highlighting significant research and problem-solving strategies in the field. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 47 Context: # 2.3 Recursive Graph Algorithms of this construction, we show in Fig. 2.3.5(a) a map of a distant planet, and in Fig. 2.3.5(b) the graph that results from the construction that we have just described. By a ‘planar graph’ we mean a graph \( G \) that can be drawn in the plane in such a way that two edges never cross (except that two edges at the same vertex have that vertex in common). The graph that results from changing a map of countries into a graph as described above is always a planar graph. In Fig. 2.3.6(a) we show a planar graph \( G \). This graph doesn’t look planar because two of its edges cross. However, that isn’t the graph’s fault, because with a little more care we might have drawn the same graph as in Fig. 2.3.6(b), in which its planarity is obvious. Don’t blame the graph if it doesn’t look planar. It might be planar anyway! ## Fig. 2.3.5(a) ![Fig 2.3.5(a)](path_to_image_a) ## Fig. 2.3.5(b) ![Fig 2.3.5(b)](path_to_image_b) ## Fig. 2.3.6(a) ![Fig 2.3.6(a)](path_to_image_c) ## Fig. 2.3.6(b) ![Fig 2.3.6(b)](path_to_image_d) The question of recognizing whether a given graph is planar is itself a formidable problem, although the solution, due to J. Hopcroft and R. E. Tarjan*, is an algorithm that takes linear time, i.e., in \( O(V) \) time for a graph of \( V \) vertices. Although every planar graph can be properly colored in four colors, there are still all of those graphs that are not planar to deal with. For any one of those graphs we can ask if a positive integer \( K \) is given, whether or not its vertices can be \( K \)-colored properly. As if that question weren’t hard enough, we might ask for even more detail, namely about the number of ways of properly coloring the vertices of a graph. For instance, if we have \( K \) colors to work with, \( G \) is the empty graph \( K_n \). That is, the graph of \( n \) vertices that has no edges at all. Then \( G \) has quite a large number of proper colorings, \( K^n \), to be exact. Other graphs of \( n \) vertices have fewer proper colorings than that, and an interesting computational question is to count the proper colorings of a given graph. We will now find a recursive algorithm that will answer this question. Again, the complexity of the algorithm will be exponential, but as a small consolation we note that no polynomial time algorithm for this problem is known. Choose an edge of the graph, and let its endpoints be \( u \) and \( v \). Now delete the edge from the graph, and the resulting graph be called \( G - \{e\} \). Then we will distinguish two kinds of proper colorings of \( G - \{e\} \): those in which vertices \( u \) and \( v \) have the same color and those in which \( u \) and \( v \) have different colors. Obviously the number of proper colorings of \( G - \{e\} \) if \( k \) colors is the sum of the numbers of colorings of each of those two kinds. * Efficient planarity testing, J. Assoc. Comp. Mach. 21 (1974), 549-568. Image Analysis: ### Image Analysis #### 1. Localization and Attribution: - **Image 1**: Located at the top left of the page. - **Image 2**: Located at the top right of the page. - **Image 3**: Located at the bottom left of the page. - **Image 4**: Located at the bottom right of the page. #### 2. Object Detection and Classification: - **Image 1**: - **Objects**: Venn diagram. - **Key Features**: Letters 'A', 'B', 'C', 'D', and 'F' indicating regions within the diagram. - **Image 2**: - **Objects**: Labeled vertices 'A', 'B', 'C', 'D', and 'E' connected by edges forming a graph. - **Key Features**: The vertices and edges layout indicative of a graph. - **Image 3**: - **Objects**: Graph with vertices labeled numerically (1-5) and edges forming a network. - **Key Features**: Planarity depicted as the numbers 5 and 6 are arranged without crossing lines. - **Image 4**: - **Objects**: Similar graph to Image 3, the vertices labeled numerically (1-5) and edges forming a network. - **Key Features**: Planarity demonstrated with a different arrangement compared to Image 3. #### 3. Scene and Activity Analysis: - **Image 1**: - **Scene**: A Venn diagram illustrating the relations between different regions labeled A, B, C, D, and F. - **Main Actors and Actions**: The regions represent different subsets that overlap and interact. - **Image 2**: - **Scene**: Graph theory representation of vertices and edges. - **Main Actors and Actions**: Vertices connected by edges, illustrating a planar graph. - **Image 3**: - **Scene**: Planar graph with vertices numbered 1-5. - **Main Actors and Actions**: Vertices and edges arranged to show that no two edges cross. - **Image 4**: - **Scene**: Another planar graph similar to Image 3 but with a slightly different arrangement. - **Main Actors and Actions**: Similar to Image 3, vertices and edges arranged to depict a planar graph. #### 4. Text Analysis: - **Text Detected**: - Fig. 2.3.5(a) - Fig. 2.3.5(b) - Fig. 2.3.6(a) - Fig. 2.3.6(b) - Paragraphs below each figure explaining the concept of planar graphs, vertex coloring, and algorithmic solutions for graph problems. - Text excerpt mentioning "Efficient Planarity Testing" from sources. - **Content and Significance**: - The text explains the theoretical aspects of planar graphs, the importance of edges not crossing, and algorithms for recognizing planar graphs. - It describes the significance of proper vertex coloring, especially when K colors are involved. #### 11. Metadata Analysis: - **Metadata Information**: Not available from the visual content provided. #### 13. Graph Numbers: - **Image 3**: Vertices labeled as 1, 2, 3, 4, and 5. - **Image 4**: Vertices labeled similarly as 1, 2, 3, 4, and 5. ### Additional Aspects #### Prozesbeschreibungen (Process Descriptions): - The figures illustrate the step-by-step evolution from a map to a graph (Fig. 2.3.5(a) to Fig. 2.3.5(b)) and the different representations of planar graphs (Fig. 2.3.6(a) and Fig. 2.3.6(b)). **Conclusion**: The visual content provides a detailed exploration of planar graphs, their properties, and the theoretical underpinnings of graph theory. The diagrams and supporting text explain how sets and vertices interact and contribute to understanding the planar graphs and related algorithms. #################### 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: 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: 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 primality-testing 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. 1. **The travelling salesman problem (TSP)**: Given n points in the plane (`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? 2. **Graph coloring**: Given a graph G and an integer K. Can the vertices of G be properly colored in K or fewer colors? 3. **Independent set**: Given a graph G and an integer K. Does V(G) contain an independent set of K vertices? 4. **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 a very large amount of coloring will 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 r 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 \( 100n^k \). Hence there is no contradiction between the fact 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 `100n^k` 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 `357n^7` might have appeared in the guarantee, and so might `23n^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 \( (i, j) \in E(G) \) and is 0 else. Evidently \( n^2 \) bits of input 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, but 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 tests: Is there a polynomial P such that for every instance of `Independent Set` the running time T will be at most \( P(|B|) \)? Well, is T bounded? Image Analysis: ### Comprehensive Analysis of the Attached Visual Content #### 1. Localization and Attribution - The document contains one main image which is a single page of text. - The image will be referred to as Image 1. #### 4. Text Analysis - **Detected Text and Content** - The text is a section from a mathematical or computer science book focusing on the concept of NP-completeness. - The specific chapter title is "Chapter 5: NP-completeness" covering section "5.1 Introduction". - **Content Analysis and Significance** - **Chapter Title and Section** - The chapter deals with NP-completeness, a foundational topic in computational theory. - The introduction explains computational problems where fast algorithms have not been found and mentions the collection of problems known as NP-complete problems. - **Key Topics Discussed** - **Travelling Salesman Problem (TSP)** - A problem where a tour must visit all given points with minimal total distance. - **Graph Coloring Problem** - Determining if vertices of a graph can be colored with a given number of colors such that no two adjacent vertices share the same color. - **Independent Set Problem** - Finding if a graph contains an independent set of certain size. - **Bin Packing Problem** - Partitioning objects into bins with capacity constraints. - **Example Guarantee Statement** - The text provides an example of a "polynomial-time performance guarantee" for an algorithm solving the Independent Set problem. - The guarantee ensures the algorithm completes in time no greater than a polynomial function of the input size. - Illustrates with a hypothetical warranty card example, showing the complexity assurance (e.g., \(100n^2\), \(357n^9\)). - **Polynomial-Time Algorithms** - The guarantee statement revolves around whether a polynomial-time algorithm can exist for every instance of the 'Independent Set' problem. #### 9. Perspective and Composition - **Perspective** - The image is taken from a standard reading viewpoint, providing a straightforward view of the text. - **Composition** - The text is neatly formatted with distinct sections and subsections. - The chapter and section titles are bold and larger than the body text. - The body text is well-aligned and organized into paragraphs, making it accessible for detailed reading. - The inclusion of a boxed example in the middle of the text highlights an important concept distinctly within the flowing text. #### 13. Graph Numbers (Referencing Hypothetical Examples) - **Polynomial-Time Warranty** - Example values of polynomials used for guarantees include: - \( 100n^2 \) - \( 357n^9 \) ### Summary and Context - This page is an introductory segment on NP-completeness from a textbook likely used in computer science education. - The text breaks down complex theoretical concepts into understandable examples, aiding in student comprehension. - The detailed descriptions of various problems within the NP-complete class serve as foundational knowledge crucial for advanced studies in algorithm design and computational complexity. This document section provides a critical introduction to computational problems that define the boundaries of what can be efficiently solved using current algorithmic techniques. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 127 Context: # 5.6 Backtracking (I): Independent Sets To speed up the discussion, we will now show the list of all sets \( S \) that turn up from start to finish of the algorithm: 1. \(\{1\}\) 2. \(\{13\}\) 3. \(\{16\}\) 4. \(\{2\}\) 5. \(\{24\}\) 6. \(\{245\}\) 7. \(\{25\}\) 8. \(\{3\}\) 9. \(\{34\}\) 10. \(\{343\}\) 11. \(\{35\}\) 12. \(\{4\}\) 13. \(\{45\}\) 14. \(\{5\}\) 15. \(\{6\}\) A convenient way to represent the search process is by means of the backtrack search tree \( T \). This is a tree whose vertices are arranged on levels \( L = 0, 1, 2, \ldots \) for a graph of \( n \) vertices. Each vertex of \( T \) corresponds to an independent set of vertices in \( G \). Two vertices of \( T \) corresponding to independent sets \( S_1, S_2 \) of vertices of \( G \) are joined by an edge in \( T \) if \( S_1 \) and \( S_2 \) consist of a single element: the highest-numbered vertex in \( S_1 \). On level \( L \), we find a vertex \( S \) for every independent set of exactly \( L \) vertices of \( G \). Level 0 consists of a single root vertex, corresponding to the empty set of vertices of \( G \). The complete backtrack search tree for the problem of finding a maximum independent set in the graph \( G \) of Fig. 5.6.1 is shown in Fig. 5.6.2 below. ## Fig. 5.6.2: The Backtrack Search Tree The backtrack algorithm amounts just to visiting every vertex of the search tree \( T \), without actually having to write down the tree explicitly in advance. Observe that the list of sets \( S \) above, or equivalently, the list of nodes of the tree \( T \), consists of exactly every independent set in the graph \( G \). A reasonable measure of the complexity of the searching job, therefore, is the number of independent sets that \( G \) has. In the example above, the graph \( G \) had 19 independent sets of vertices, including the empty set. The question of the complexity of backtrack search is therefore the same as the question of determining the number of independent sets of the graph \( G \). Some graphs have an enormous number of independent sets. The graph \( T_n \), of \( n \) vertices and no edges whatever has \( 2^n \) independent sets of vertices. The backtrack tree will have \( 2^n \) nodes, and the search will be a long one indeed. The complete graph \( K_n \) of \( n \) vertices and every possible edge, \( n(n-1)/2 \) in all, has just \( n+1 \) independent sets of vertices. Any other graph \( G \) of \( n \) vertices will have a number of independent sets that lies between these two extremes of \( 2^n \) and \( 2^{n-1} \). Sometimes backtracking will take an exponentially long time, and sometimes it will be fairly quick. Now the question is, on the average how fast is the backtrack method for this problem? What we are asking for is the average number of independent sets that a graph \( G \) of \( n \) vertices has. But that is the sum, over all vertex subsets \( S \subseteq \{1, \ldots, n\} \), of the probability that \( S \) is independent. If \( S \) has \( k \) vertices, then the probability that \( S \) is independent is the probability that, among the \( k(k - 1)/2 \) possible edges that might join a pair of vertices in \( S \), exactly zero of these edges actually lie in the random graph \( G \). Since each of these \( \binom{k}{2} \) edges has a probability \( 1/2 \) of appearing in \( G \), the probability that none of them appear is \( 2^{-k(k-1)/2} \). Hence the average number of independent sets in a graph of \( n \) vertices is \[ I_n = \sum_{k=0}^{n} \binom{n}{k} 2^{-k(k-1)/2}. \tag{5.6.1} \] Image Analysis: ### Comprehensive Examination #### 1. Localization and Attribution - **Image 1:** - Located centrally in the middle of the page. - Identified as Fig. 5.6.2. #### 2. Object Detection and Classification - **Image 1:** - **Objects Detected:** - Different nodes represented by circles containing numbers. - Lines connecting these nodes, forming a tree structure. - **Classification by Category:** - Nodes: Represent different states in the backtracking algorithm. - Lines: Indicate relationships or transitions between the states. #### 3. Scene and Activity Analysis - **Image 1:** - **Scene Description:** - A hierarchical tree diagram with nodes labeled by numbers and letters. - **Activities Taking Place:** - The nodes represent vertices of a graph, and the connections indicate the exploration of possible sets without conflicts. #### 4. Text Analysis - **Text Extracted:** - "Fig. 5.6.2: The backtrack search tree" - Main text includes explanations about the backtracking algorithm to find independent sets, organization of sets on different levels, and explanation about backtrack search complexity and applications to other graph types. - **Content Significance:** - Describes the process, complexity, and practical implications of the backtrack search method to find independent sets in a graph. #### 5. Diagram and Chart Analysis - **Diagram Analysis:** - **Axes, Scales, and Legends:** - No explicit axes or scales. - Levels are labeled from Level 0 to Level 3. - **Key Insights:** - Representation of all possible sets expands exponentially as depicted by the increasing number of nodes at each level. - Demonstrates the practical depth-first search steps of a backtracking algorithm. #### 6. Product Analysis - **Not Applicable** #### 7. Anomaly Detection - **No noticeable anomalies or unusual elements identified.** #### 8. Color Analysis - **Color Composition:** - Black and white, typical for academic or technical diagrams. - Dominant colors: Black text and lines, white background. - **Impact on Perception:** - Clear contrast ensures easy readability and focus on details. #### 9. Perspective and Composition - **Perspective:** - Straight-on view, which is standard for tree diagrams. - **Composition:** - Centered, hierarchical structure with clear connections between nodes. - Nodes are well-spaced and aligned for readability. #### 10. Contextual Significance - **Context within Document:** - Supports the explanatory text about backtrack search strategy in finding independent sets in graphs. - Visual representation aligns with the theoretical descriptions and facilitates understanding. #### 11. Metadata Analysis - **Not Applicable** #### 12. Graph and Trend Analysis - **Not Applicable** #### 13. Graph Numbers - **Not Applicable** ### Additional Aspects #### Ablaufprozesse (Process Flows) - **Depicted Process Flows:** - The backtracking search process to find independent sets in graphs. - Explores all vertices and edge combinations systematically. #### Prozessbeschreibungen (Process Descriptions) - **Description:** - Backtrack search starts from the root node. - Each subset of vertices explored and expanded level by level. - Branches represent possible combinations extending in a depth-first manner. - Process detailed in the accompanying text for complete context. #### Typen Bezeichnung (Type Designations) - **Identified Types:** - Different independent sets of vertices. - Separated into levels indicating stages of the search. #### Trend and Interpretation - **Identification and Interpretation:** - Highlights exponential growth of potential combinations in larger graphs. - Illustrates complexity and efficiency considerations. #### Tables - **Not Applicable** #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 127 Context: ``` # 5.6 Backtracking (I): Independent Sets To speed up the discussion, we will now show the list of all sets S that turn up from start to finish of the algorithm: - {1}, {13}, {16}, {2}, {24}, {245}, {25}, {3}, - {34}, {345}, {35}, {4}, {45}, {5}, {6} A convenient way to represent the search process is by means of the backtrack search tree T. This is a tree whose vertices are arranged on levels \(L = 0, 1, 2, \ldots\) for a graph of \(n\) vertices. Each vertex of T corresponds to an independent set of vertices in G. Two vertices of T, corresponding to independent sets \(S', S''\) of vertices of G, are joined by an edge in T if \(S' \subset S''\) and \(S''\) consists of a single element: the highest-numbered vertex in \(S''\). On level \(L\) we find a vertex S of T for every independent set of exactly L vertices of G. Level 0 consists of a single root vertex, corresponding to the empty set of vertices of G. The complete backtrack search tree for the problem of finding a maximum independent set in the graph G of Fig. 5.6.1 is shown in Fig. 5.6.2 below. ## Fig. 5.6.2: The backtrack search tree The backtrack algorithm amounts just to visiting every vertex of the search tree T, without actually having to write down the tree explicitly, in advance. Observe that the list of sets S above, or equivalently, the list of nodes of the tree T, consists of exactly every independent set in the graph G. A reasonable measure of the complexity of the searching job, therefore, is the number of independent sets that G has. In the example above, the graph G had 19 independent sets of vertices, including the empty set. The question of the complexity of backtrack search is therefore the same as the question of determining the number of independent sets of the graph G. Some graphs have an enormous number of independent sets. The graph \(T_n\), of n vertices and no edges whatever has \(2^n\) independent sets of vertices. The backtrack tree will have \(2^n\) nodes, and the search will be a long one indeed. The complete graph \(K_n\) of n vertices and every possible edge, \(n(n-1)/2\), has just \(n+1\) independent sets of vertices. Any other graph G of n vertices will have a number of independent sets that lies between these two extremes of \(2^n\) and \(2^{n}\). Sometimes backtracking will take an exponentially long time, and sometimes it will be fairly quick. Now the question is, on the average how fast is the backtrack method for this problem? What we are asking for is the average number of independent sets that a graph G of \(n\) vertices has. But that is the sum, over all vertex subsets \(S \subseteq \{1, \ldots, n\}\), of the probability that S is an independent set. If S has k vertices, then the probability that S is independent is the probability that, among \(k(k - 1)/2\) possible edges that might join a pair of vertices in S, exactly zero of these edges actually live in the random graph G. Since each of these \(\binom{k}{2}\) edges has a probability \(1/2\) of appearing in G, the probability that none of them appear is \(2^{-k(k-1)/2}\). Hence the average number of independent sets in a graph of n vertices is: \[ I_n = \sum_{k=0}^{n} \binom{n}{k} 2^{-k(k-1)/2}. \quad (5.6.1) \] ``` Image Analysis: ### 1. Localization and Attribution: - **Image 1**: The only image on the page is a diagram labeled "Fig. 5.6.2: The backtrack search tree" positioned centrally on the page and located below a text paragraph. ### 2. Object Detection and Classification: - **Image 1**: - **Objects Identified**: - Nodes/Vertices: Contain labels like {}, {1}, {2}, {3}, {1,2}, etc. - Edges/Connections: Lines connecting these nodes. - **Key Features**: - Nodes are arranged in different levels (Level 0, Level 1, etc.). - Labels inside the nodes signify different sets of vertices. ### 3. Scene and Activity Analysis: - **Image 1**: - **Scene Description**: - The scene depicts a hierarchical structure or tree, representing the backtrack search tree. - **Activities Taking Place**: - The tree shows the process of exploring different vertex sets in a systematic manner, specifically for finding independent sets in a graph. ### 4. Text Analysis: - **Text in the Image**: - Detected text: - "Level: 0", "Level: 1", "Level: 2", "{}", "{1}", "{2}", "{3}", etc. - **Analysis**: - The text indicates different levels of the search tree and the sets of vertices at each level. ### 5. Diagram and Chart Analysis: - **Image 1**: - **Axes, Scales, and Legends**: - No axes are present; instead, the diagram is labeled with levels indicating depth in the tree. - **Key Insights**: - The diagram shows how the backtrack algorithm systematically searches through possible independent sets of a graph by breaking down the problem into smaller subproblems (represented by levels). ### 6. Product Analysis: - Not applicable to this image. ### 7. Anomaly Detection: - No anomalies detected in the image. ### 8. Color Analysis: - **Image 1**: - **Color Composition**: - Mainly black and white with lines and text in black, and background in white. - **Impact**: - Clear and easy to read, emphasizing the logical structure without unnecessary distractions. ### 9. Perspective and Composition: - **Image 1**: - **Perspective**: - Standard, orthogonal view; designed for clarity in understanding the diagram. - **Composition**: - Nodes are symmetrically arranged, making the hierarchical relationships clear. - The tree starts with a single root node and branches out to wider levels. ### 10. Contextual Significance: - **Image 1**: - **Overall Document**: - The image is part of a discussion on backtracking algorithms, specifically focusing on independent sets in graphs. - **Contribution**: - The image visually explains the concept described in the text, aiding in the understanding of the search process by depicting the backtrack tree. ### 11. Metadata Analysis: - No metadata available from the image. ### 12. Graph and Trend Analysis: - No specific graphs related to trends are visible in the image. ### 13. Graph Numbers: - Not applicable to this image. ### Additional Aspects: - **Ablaufprozesse (Process Flows)**: - **Image 1**: - Depicts a flow from the root to various levels, illustrating the branching process of the algorithm. - **Prozessbeschreibungen (Process Descriptions)**: - The text and image collectively describe the backtracking process used to find independent sets in a graph. - **Typen Bezeichnung (Type Designations)**: - Levels in the tree represent different stages/types in the search process for independent sets. - **Tables**: - No tables are present in the image. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 127 Context: ``` # 5.6 Backtracking (I): Independent Sets To speed up the discussion, we will now show the list of all sets \( S \) that turn up from start to finish of the algorithm: - \( \{1\} \) - \( \{3\} \) - \( \{13\} \) - \( \{16\} \) - \( \{2\} \) - \( \{24\} \) - \( \{245\} \) - \( \{25\} \) - \( \{34\} \) - \( \{345\} \) - \( \{4\} \) - \( \{45\} \) - \( \{6\} \) A convenient way to represent the search process is by means of the backtrack search tree \( T \). This is a tree whose vertices are arranged on levels \( L = 0, 1, 2, \ldots \) for a graph of \( n \) vertices. Each vertex of \( T \) corresponds to an independent set of vertices in \( G \). Two vertices of \( T \) corresponding to independent sets \( S_1 \) and \( S_2 \) of vertices of \( G \) are joined by an edge in \( T \) if \( S_1 \) and \( S_2 \) consist of a single element: the highest-numbered vertex in \( S_2 \). On level \( L \), we find a vertex \( S \) for every independent set of exactly \( L \) vertices of \( G \). Level 0 consists of a single root vertex, corresponding to the empty set of vertices of \( G \). The complete backtrack search tree for the problem of finding a maximum independent set in the graph \( G \) of Fig. 5.6.1 is shown in Fig. 5.6.2 below. ## Fig. 5.6.2: The Backtrack Search Tree The backtrack algorithm amounts just to visiting every vertex of the search tree \( T \), without actually having to write down the tree explicitly, in advance. Observe that the list of sets \( S \) above, or equivalently, the list of nodes of the tree \( T \), consists of exactly every independent set in the graph \( G \). A reasonable measure of the complexity of the searching job, therefore, is the number of independent sets that \( G \) has. In the example above, the graph \( G \) had 19 independent sets of vertices, including the empty set. The question of the complexity of backtrack search is therefore the same as the question of determining the number of independent sets of the graph \( G \). Some graphs have an enormous number of independent sets. The graph \( T_n \), of \( n \) vertices and no edges whatever has \( 2^n \) independent sets of vertices. The backtrack tree will have \( 2^n \) nodes, and the search will be a long one indeed. The complete graph \( K_n \) of \( n \) vertices and every possible edge, \( n(n-1)/2 \) in all, has just \( n+1 \) independent sets of vertices. Any other graph \( G \) of \( n \) vertices will have a number of independent sets that lies between these two extremes of \( n + 1 \) and \( 2^n \). Sometimes backtracking will take an exponentially long time, and sometimes it will be fairly quick. Now the question is, on the average how fast is the backtrack method for this problem? What we are asking for is the average number of independent sets that a graph \( G \) of \( n \) vertices has. But that is the sum, over all vertex subsets \( S \subseteq \{1, \ldots, n\} \), of the probability that \( S \) is an independent set. If \( S \) has \( k \) vertices, then the probability that \( S \) is independent is the probability that, among \( k(k - 1)/2 \) possible edges that might join a pair of vertices in \( S \), exactly zero of these edges actually live in the random graph \( G \). Since each of these \( \binom{k}{2} \) edges has a probability \( 1/2 \) of appearing in \( G \), the probability that none of them appear is \( 2^{-k(k-1)/2} \). Hence the average number of independent sets in a graph of \( n \) vertices is \[ I_n = \sum_{k=0}^{n} \binom{n}{k} 2^{-k(k-1)/2}. \] ``` Image Analysis: ### Comprehensive Examination **1. Localization and Attribution:** - **Image:** One image is present; it is a diagram in the middle of the page. - **Image Position:** The image is situated centrally below the text paragraphs. - **Identification:** Image 1. **2. Object Detection and Classification:** - **Detected Objects in Image 1:** Nodes, edges, rectangles. - **Classification:** - **Nodes** represent vertices or elements of a set. - **Edges** represent connections or relations between nodes. - **Rectangles** indicate groupings or levels in a hierarchical structure. **3. Scene and Activity Analysis:** - **Scene:** A hierarchical tree structure depicting levels and nodes connected by lines. - **Activities:** Visualization of the backtrack search process for independent sets in graph G. - **Main Actors:** Nodes (labeled with number sets), and vertices. **4. Text Analysis:** - **Detected Text:** - **Headers:** - "5.6 Backtracking (I): independent sets" - "A convenient way to represent the search process..." - "Fig. 5.6.2: The backtrack search tree" - **Body Text:** - Descriptions explaining the diagram, process flow of the backtrack algorithm, and independent sets in the context of tree structures. - **Symbols and Formulas:** - Set representations (e.g., {1}, {13}, {16}, ...) - Mathematical notations and equations (e.g., \(I_n = \sum_{k=0} X_k \binom{n}{k} 2^{-(k-1)/2}\), etc.) - **Significance:** Text provides detailed explanation and context for the depicted backtrack search tree, explaining the algorithm’s operation, efficiency, and theoretical foundations. **5. Diagram and Chart Analysis:** - **Image 1 Analysis:** - **Title:** "Fig. 5.6.2: The backtrack search tree" - **Components:** Nodes labeled with sets and edges connecting these nodes represent possible transitions in levels during the search for independent sets. - **Key Insights:** - Illustrated hierarchical levels (0 to 3). - Nodes at different levels represent different stages of the backtrack algorithm. - The root node contains the empty set and the branching visualizes the progression of the algorithm towards finding the independent sets. **7. Anomaly Detection:** - **Detected Anomalies:** No anomalies noted; the diagram and text are coherent and logically structured. **8. Color Analysis:** - **Color Composition:** Black and white (grayscale) coloring, typical for printed mathematical text and diagrams. - **Dominant Colors:** Black text and lines on a white background. - **Impact on Perception:** The high contrast ensures clarity and readability, aiding in the understanding of the mathematical concepts. **9. Perspective and Composition:** - **Perspective:** Straight-on view as it is a diagram. - **Composition:** - Balanced layout with the tree diagram centrally positioned. - Nodes are evenly spaced, and levels are clearly demarcated. - Text explanation is well-structured, providing context before and after the diagram. **10. Contextual Significance:** - **Context:** Part of a mathematical or computer science text dealing with algorithms and graph theory. - **Contribution:** The image elucidates the concept of the backtrack algorithm for independent sets, making it easier to conceptualize the theoretical explanation provided in the surrounding text. **13. Graph Numbers:** - **Node Data:** - **Level 0:** {} - **Level 1:** {1}, {2}, {3} - **Level 2:** {13}, {16}, (others implied but not shown in full) - **Level 3:** {134}, {135}, etc. This detailed analysis covers the given aspects thoroughly, providing insights into the algorithm and its visual representation within the document. #################### File: An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf Page: 40 Context: # Chapter 8. Geometry of Numbers ## Figure 4 The long side of the rectangle is \( 2R \), the short \( 2r \). This segment. Hence, no matter how large \( R \) is, we can construct a rectangle containing this segment, which contains no lattice point other than \( O \). By the fundamental theorem of Minkowski on the area of this rectangle does not exceed \( 4 \). Thus \( 2R < 4 \). Note that if \( (p, q) \) is a lattice point on the border of the rectangle then \( \frac{p}{q} \) so that the fundamental theorem of Minkowski will give some information about how closely an irrational number can be approximated by rationals. Let us now return to Hajos proof of the fundamental theorem of Minkowski. Consider the \( x-y \) plane cut up into an infinite chessboard with the base square of area \( 2 \) determined by \( |x| \leq 1, |y| \leq 1 \). We now cut up the chessboard along the edges of the squares and superimpose all the squares that contain parts of the region \( R \). We have now compressed an area \( A > 4 \) into a region of area \( 2 \). This implies that there will be some overlapping, i.e., we can stick a pin through the square so as to pierce \( R \) into two points \( V_1 \) and \( V_2 \). Now reassemble the region and let the points \( V_1 \) and \( V_2 \) be the vectors \( \vec{V_1} \) and \( \vec{V_2} \). Consider the fact that the \( x \) and \( y \) coordinates of \( V_1 \) and \( V_2 \) differ by a multiple of \( 2 \). We write \( V_1 = (x_1, y_1) \) and \( V_2 = (x_2, y_2) \). Thus \( |y_1 - y_2| \) is a lattice point different from \( 0 \) (since \( y_1 \neq y_2 \)) in \( R \). The fundamental theorem of Minkowski can easily be generalized to \( n \)-dimensional space. Indeed we need only replace it in the fundamental theorem of Minkowski by \( n^2 \) and Hajos' proof goes through. Many extensions and refinements of the fundamental theorem of Minkowski have been made. I shall return to some of them later. One of Polya's earliest papers has the long and curious title "Zahlentheorie: this includes Wohlstandsreichtheorektheoretisches in der Sichtweite in Wade und durch Schweball". A proof of Polya's main result in this paper can be significantly simplified and somewhat refined using the fundamental theorem of Minkowski. The problem is this. ## Figure 5 Suppose every lattice point other than \( O \) is surrounded by a circle of radius \( r \leq 4 \) (a tree in a forest). A man stands at \( O \). In direction \( \theta \) he can see a distance \( F(r) \). What is the furthest he can see in any direction? That is, determine: \[ F(r) = \max f(\theta, r) \] By looking past the circle centered at \( (1, 0) \) (Figure 5), we can see almost a distance \( \phi \). On the other hand we can prove that \( F(r) \leq 5 \). For suppose that we can see a distance \( F(r) \) in direction \( O \) about this line of vision construct a rectangle with side \( 2r \). This rectangle contains no lattice point, for otherwise the tree centered at such a lattice point would obstruct our line of vision; see Figure 6. ## Figure 6 Hence, by the fundamental theorem of Minkowski \( F(r) < 4 \) and \( F(r) \leq \frac{1}{2} \) is required. Note that no lattice point can be in either semicircle in the diagram. This enables us to improve slightly on Polya's result. I shall leave the details as an exercise. A more significant application of the fundamental theorem of Minkowski concerns the possibility of solving in integers a set of linear inequalities. ########## """QUERY: what are sets and spaces? quote information from your context""" Consider the chat history for relevant information. Use all information included. If query is already asked in the history double check the correctness of your answer and maybe correct your previous mistake. 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. If too many questions were asked already based on the history, just answer the question Final Files Sources: Algebraic%20Topology%20AT-toc.pdf - Page 7, Algebraic%20Topology%20AT-toc.pdf - Page 9, 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 25, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 42, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 43, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 46, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 47, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 94, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 95, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 108, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 127, An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf - Page 40 ================================================== **Elapsed Time: 0.00 seconds** ================================================== FINAL ANSWER Answer: [Answer was streamed to client] ================================================== **Elapsed Time: 3.19 seconds** ==================================================