{ "query": "what are sets and spaces? Indicate how many words is your response", "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? Indicate how many words is your response\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": "The provided context focuses on the geometry of numbers and doesn't explicitly define \"sets\" and \"spaces\" in the way you might find in a set theory or topology textbook. \n\nHowever, the context does use the concept of sets in a few ways:\n\n* **Sets of Lattice Points:** The text discusses lattice points, which are points with integer coordinates. It mentions \"every lattice point other than \\(O\\)\" and \"no lattice point can be in either semicircle.\" This implies a set of lattice points within a given region.\n* **Sets of Linear Inequalities:** The context mentions \"solving in integers a set of linear inequalities.\" This refers to finding integer solutions that satisfy a collection of inequalities, which can be viewed as a set of conditions.\n\nThe context also uses the concept of \"space\" in the context of geometry:\n\n* **n-dimensional Space:** The text mentions generalizing the fundamental theorem of Minkowski to \"n-dimensional space.\" This refers to a geometric space with n dimensions.\n\nWhile the context doesn't provide a formal definition of \"sets\" and \"spaces,\" it uses these concepts implicitly within the discussion of geometric and number-theoretic problems. \n\nWould you like me to elaborate on any of these concepts or provide examples from the context?" }, { "role": "assistant", "content": "The context provided does not explicitly define \"sets\" and \"spaces\" in a formal mathematical sense, but it does utilize these concepts in relation to geometry and number theory. Here\u2019s a brief overview based on the context:\n\n1. **Sets:**\n - The context refers to sets of lattice points, which are points in a coordinate system where both coordinates are integers. For example, the mention of \"every lattice point other than \\(O\\)\" indicates a specific set of points excluding the origin.\n - It also discusses sets of linear inequalities, which involve finding integer solutions that satisfy multiple conditions. This can be viewed as a collection of constraints that define a set of possible solutions.\n\n2. **Spaces:**\n - The term \"n-dimensional space\" is used, which refers to a geometric space that extends into n dimensions. This concept is fundamental in various areas of mathematics, including algebraic topology and geometry.\n\nIf you would like more detailed explanations or examples related to these concepts, please let me know!" }, { "role": "assistant", "content": "The context provided does not explicitly define \"sets\" and \"spaces\" in a formal mathematical sense, but it does utilize these concepts in relation to geometry and number theory. Here\u2019s a brief overview based on the context:\n\n1. **Sets:**\n - The context refers to sets of lattice points, which are points in a coordinate system where both coordinates are integers. For example, the mention of \"every lattice point other than \\(O\\)\" indicates a specific set of points excluding the origin.\n - It also discusses sets of linear inequalities, which involve finding integer solutions that satisfy multiple conditions. This can be viewed as a collection of constraints that define a set of possible solutions.\n\n2. **Spaces:**\n - The term \"n-dimensional space\" is used, which refers to a geometric space that extends into n dimensions. This concept is fundamental in various areas of mathematics, including algebraic topology and geometry.\n\nIf you would like more detailed explanations or examples related to these concepts, please let me know!" }, { "role": "user", "content": "what are sets and spaces? in 250 words" }, { "role": "assistant", "content": "The provided context is from a book on number theory, focusing on concepts like compositions, partitions, congruences, and combinatorial number theory. It doesn't explicitly define \"sets\" and \"spaces\" in the way you might find in a set theory or topology textbook. \n\nHowever, the context does use the concept of sets in a few ways:\n\n* **Sets of Integers:** The text frequently discusses sets of integers, such as the set of positive integers less than \\(2^n\\), the set of consecutive integers \\(1, 2, \\ldots, n\\), and the set of integers representable as sums of two squares.\n* **Sets of Classes:** The text discusses separating integers into classes, which can be viewed as sets of integers with specific properties. For example, it mentions separating integers into \\(r\\) classes and analyzing the properties of these classes.\n\nThe context also uses the concept of \"space\" in the context of geometry:\n\n* **n-dimensional Space:** The text mentions generalizing the fundamental theorem of Minkowski to \"n-dimensional space.\" This refers to a geometric space with \\(n\\) dimensions.\n\nWhile the context doesn't provide a formal definition of \"sets\" and \"spaces,\" it uses these concepts implicitly within the discussion of number-theoretic and geometric problems. \n\nWould you like me to elaborate on any of these concepts or provide examples from the context?" }, { "role": "assistant", "content": "The context provided primarily revolves around number theory and its various concepts, such as compositions, partitions, and congruences. While it does not explicitly define \"sets\" and \"spaces\" in the formal sense typically found in set theory or topology, it does utilize these concepts in a practical manner.\n\n1. **Sets**: The text refers to sets of integers, such as:\n - The set of positive integers less than \\(2^n\\).\n - The set of consecutive integers \\(1, 2, \\ldots, n\\).\n - Sets of integers that can be expressed in certain forms, like sums of two squares.\n\n2. **Spaces**: The term \"space\" is used in a geometric context, particularly:\n - The mention of \"n-dimensional space\" relates to geometric considerations and generalizations of theorems, such as Minkowski's theorem.\n\nIf you would like further clarification on any specific aspect of these concepts or examples from the text, please let me know!" }, { "role": "user", "content": "what are sets and spaces? Indicate how many words is your response" }, { "role": "assistant", "content": "" } ], "stream": true, "language_level": "", "chat_channel": ":4457:1727871997978", "language": "German", "tone": "neutral", "writing_style": "standard", "model": "gemini-1.5-flash", "knowledgebase": "ki-dev-large", "seed": 1055, "client_id": 1727871997978, "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? Indicate how many words is your response Model: gemini-1.5-flash **Elapsed Time: 0.00 seconds** ROUTING Query type: research **Elapsed Time: 1.60 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: 0.66 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: 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: 15 Context: # 1.2 Positional number systems is a function \( h \) such that \( h = o(h) \) and \( h = \Theta(g) \). Give an explicit construction for the function \( h \) in terms of \( g \). 6. (This exercise is a warmup for exercise 7.) Below there appear several mathematical propositions. In each case, write a proposition that is the negation of the given one. Furthermore, in the negation, do not use the word 'not' or any negation symbols. In each case the question is, “If this isn’t true, then what is true?” (a) \( \forall x > 0, f(x) \neq 0 \) (b) \( \forall y > 0, f(y) > 0 \) (c) \( \forall z \in \mathbb{R}, \, f(z) < f(x) \) (d) \( \exists y > 3 \, \forall z < f(y) \) (e) \( \forall x \exists z \, z < f(y) \) Can you formulate a general method for negating such propositions? Given a proposition that contains \( \forall, \exists \), what rule would apply in order to negate the proposition and leave the result in positive form (containing no negation symbols or ‘not’s)? 7. In this exercise we will work out the definition of the \( \Omega \): (a) Write out the precise definition of the statement \( \lim_{n \to \infty} h(n) = 0 \) (use `\varepsilon`). (b) Write out the negation of your answer to part (a) as a positive assertion. (c) Use your answer to part (b) to give a positive definition of the assertion \( f(n) \neq o(g(n)) \), and thereby justify the definition of the \( \Omega \) symbol that was given in the text. 8. Arrange the following functions in increasing order of their rates of growth, for large \( n \). List them so that each one is ‘little o’ of its successor: \[ 2^{\sqrt{n}}, n^{3.01}, n^{2}, n^{1.6}, \log^{3} n + 1, \sqrt{n}, n \log n, n^{3} \log(\log n), n^{2}, (n + 4)^{12} \] 9. Find a function \( f(x) \) such that \( f(x) = O(2^{x}) \) is true for every \( \varepsilon > 0 \), but for which it is not true that \( f(x) = O(1) \). 10. Prove that the statement \( f(n) = O(2^{\varepsilon}) \) for every \( \varepsilon > 0 \) is equivalent to the statement \( f(n) = \Omega(2^{\varepsilon}) \) for every \( \varepsilon > 0 \). ## 1.2 Positional number systems This section will provide a brief review of the representation of numbers in different bases. The usual decimal system represents numbers by using the digits \( 0, 1, \ldots, 9 \). For the purpose of representing whole numbers we can imagine that the powers of 10 are displayed below like this: \[ \cdots, 100000, 10000, 1000, 100, 10, 1. \] Then, to represent an integer we can specify how many copies of each power of 10 we would like to have. If we write 237, for example, then that means that we want 2 copies of \( 10^2 \) and 3 copies of \( 10^1 \) and 7 copies of \( 10^0 \). In general, if we write out the string of digits that represents a number in the decimal system, as \( d_m d_{m-1} \ldots d_1 d_0 \), then the number that is being represented by that string of digits is \[ n = \sum_{i=0}^{m} d_i \cdot 10^i. \] Now let’s try the binary system. Instead of using \( 10 \)’s we’re going to use \( 2 \)’s. So we imagine that the powers of \( 2 \) are displayed below as: \[ \cdots, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1. \] Image Analysis: ### Comprehensive Examination of the Visual Content #### 1. Localization and Attribution: - The entire content is one cohesive image displaying a single document page. - The page includes both text and mathematical notation, interspersed with explanations. #### 2. Object Detection and Classification: - **Detected Objects:** - Text: The page is predominantly filled with textual content. - Mathematical Notations: Various mathematical expressions and equations are present. #### 3. Scene and Activity Analysis: - **Scene:** The image represents a textbook page. - **Activities:** - The text appears to be an explanation of mathematical concepts, problems, and theorems. - The page is broken down into several exercises or sections (labelled 6, 7, 8, 9, etc.). #### 4. Text Analysis: - **Extracted Text:** The text includes exercises and explanations regarding functions, propositions in mathematics, the definition and application of the 'Ω' (big-Omega) notation, and a brief discussion on positional number systems. - **Key Text Components:** - **Exercise 6:** Deals with understanding the concepts of functions with respect to 'o(h)' and 'o(g)'. - **Exercise 7:** Focuses on the definition of the 'Ω' notation and its implications. - **Exercise 8:** Sorting functions based on their growth rates. - **Exercise 9:** Finding a function \( f(n) \) relative to growth orders. - **Section 1.2:** Discusses positional number systems and their representation. #### 5. Diagram and Chart Analysis: - **No diagrams or charts are present in the content.** #### 7. Anomaly Detection: - **None detected.** The page content appears consistent with typical textbook formatting. #### 8. Color Analysis: - **Color Composition:** The page is in grayscale, predominantly black text on a white background. #### 9. Perspective and Composition: - **Perspective:** Standard head-on view of the document page. - **Composition:** The text is well-organized with structured sections and numbered exercises, typical of textbook formatting. #### 10. Contextual Significance: - **Context:** The image is part of a mathematical or computer science textbook. - **Contributions:** It serves an educational purpose, likely aimed at undergraduate students studying algorithms or theoretical computer science. #### 13. Graph Numbers: - **Data Points:** - **Functions arranged by growth rates:** \( 2^{\sqrt{\log n}}, e^{(\log n)^2}, n^{3\log \log n}, 2^n, n^2, n^{2^n}, \log^2 n + \sqrt{n}, n^{1.6}, \log^2 n + 1, \sqrt{n}, n^{\log \log n}, n^3 \log n, (\log \log n)^3, n^{1.52}, (\pi + 4)^{\log n}, n^3 log n, n (log \log n)^{3/2}, (\pi + 4)^2 \). - These descriptions categorize various functions by their growth rates, providing a foundational understanding of algorithmic efficiency and complexity. #### Textual Breakdown and Analysis: - **Mathematical Propositions:** Exercise 6 gives multiple scenarios of logical and mathematical propositions, exploring negations and defining related terms. - **Growth Rates:** Exercise 8 asks to organize various functions by their growth rates, critical for algorithm analysis. - **Positional Number Systems:** The section 1.2 provides an introduction to positional number systems, a fundamental concept in number theory and computer science. #### Tables: - **No explicit tables are present.** This analysis helps to understand the content of the page thoroughly, laying out detailed insights into the educational content it presents and its structural components. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 24 Context: ### Typen Bezeichnung (Type Designations): - **Types or Categories**: - **Theorem 1.4.1**: Describes a mathematical type concerned with sequences and inequalities. - **Exercises**: Various types as each explores different aspects or variations of sequence relations. ### Trend and Interpretation: - **Trends**: The theorem and proof suggest a trend towards formalizing and generalizing inequalities in sequences. ### Anomaly Detection: - **Unusual Elements**: The small black square near the bottom-left of the page stands out. It might be a formatting error or a placeholder without textual significance. ### Color Analysis: - **Dominant Colors**: The page is primarily black and white, emphasizing clarity and readability typical of academic texts. This comprehensive analysis covers the key requested aspects. If further detailed breakdown or additional aspects not mentioned here are required, please specify. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 25 Context: # 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 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: ```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: 28 Context: # Chapter 1: Mathematical Preliminaries where we have used Stirling's formula (1.1.10). Equation (1.5) shows that the single biggest binomial coefficient accounts for a very healthy fraction of the sum of all of the coefficients of order n. Indeed, the sum of all of them is \( 2^n \), and the biggest one is approximately \( \sim \sqrt{n/2m} \). When n is large, therefore, the largest coefficient contributes a fraction \( \sim \sqrt{n/2m} \) of the total. If we think in terms of the subsets that these coefficients count, what we will see is that a large fraction of all of the subsets of an n-set have cardinality \( n/2 \); in fact \( |B(n, k)| \) of them do. This kind of probabilistic thinking can be very useful in the design and analysis of algorithms. If we are designing an algorithm that deals with subsets of \( x \), for instance, we should recognize that a large percentage of the customers for that algorithm will have cardinalities near \( n/2 \), and make every effort to see that the algorithm is fast for such subsets, even at the expense of possibly slowing it down on subsets whose cardinalities are very small or very large. ## Exercises for section 1.5 1. How many subsets of even cardinality does \([n]\) have? 2. By observing that \( (1+x)^{n}(1+x^n) = (1+x)^{n+1} \), prove that the sum of the squares of all binomial coefficients of order n is \( 2^n \). 3. Evaluate the following sums in simple form. 1. \( \sum_{j=0}^{n} \binom{n}{j} \) 2. \( \sum_{j=0}^{n} \binom{n}{j}^2 \) 3. \( \sum_{j=0}^{n} \binom{n}{j} \cdot 1^{j+1} \) 4. Find, by direct application of Taylor's theorem, the power series expansion of \( f(x) = 1/(1-x)^{n+1} \) about the origin. Express the coefficients as certain binomial coefficients. 5. Complete the following twiddles. 1. \( \left( \binom{n}{k} \right) \sim ? \) 2. \( \left( \binom{n}{k} \right) \sim ? \) 3. \( \left( \binom{n}{k} \right) \sim ? \) 4. \( \left( \binom{n}{k} \right) \sim ? \) 6. How many ordered pairs of unequal elements of \([n]\) are there? 7. Which one of the numbers \( \{ 2^{k} \}_{k=0}^{n} \) is the biggest? ## 1.6 Graphs A graph is a collection of vertices, certain unordered pairs of which are called its edges. To describe a particular graph we first say what its vertices are, and then we say how pairs of vertices are its edges. The set of vertices of a graph \( G \) is denoted by \( V(G) \), and its set of edges is \( E(G) \). If \( u \) and \( v \) are vertices of a graph \( G \), and if \( (u, v) \) is an edge of \( G \), then we say that vertices \( u \) and \( v \) are adjacent in \( G \). Consider the graph \( G \) whose vertex set is \( \{ 1, 2, 3, 4, 5 \} \) and whose edges are the set of pairs \( (1,2), (2,3), (3,4), (4,5) \). This is a graph of 5 vertices and 4 edges. A nice way to present a graph to an audience is to draw a picture of it, instead of just listing the pairs of vertices that are its edges. To draw a picture of a graph we would first make a point for each vertex, and then we would draw an arc between two vertices \( u \) and \( v \) if and only if \( (u, v) \) is an edge of the graph we are talking about. The graph \( G \) of 5 vertices and 4 edges that is lived above can be drawn as shown in Fig. 1.6.1(a). It could also be drawn as shown in Fig. 1.6.1(b). They’re both the same graph. Only the pictures are different, but the pictures aren't "really" the graph; the graph is the vertex list and the edge list. The pictures are helpful to us in visualizing and remembering the graph, but that's all. The number of edges that contain (are incident with) a particular vertex \( v \) of a graph \( G \) is called the degree of that vertex, and is usually denoted by \( p(v) \). If we add up the degrees of every vertex \( v \) of \( G \) we will have counted exactly two contributions from each edge of \( G \), one at each of its endpoints. Hence, for every Image Analysis: ### Analysis of the Attached Visual Content #### 1. Localization and Attribution - **Image Position:** - The visual content depicted is a single page from a book or document. - The main images (diagrams) appear within the text in the '1.6 Graphs' section. - **Numbering of Images:** - **Image 1:** Diagram labeled as Fig 1.6.1(a) - **Image 2:** Diagram labeled as Fig 1.6.1(b) #### 2. Object Detection and Classification - **Image 1: Diagram (Fig 1.6.1(a))** - Objects: Vertices (Vertex, Points), Edges (Line segments connecting vertices). - Key Features: - Vertices are labeled with numbers (1 to 5). - Edges are represented as lines connecting the vertices, indicating pairs like (1,2), (2,3), (3,4), etc. - **Image 2: Diagram (Fig 1.6.1(b))** - Objects: Same vertices and edges as in Image 1. - Key Features: - Similar representation but could be shown with different emphasis or layout. #### 3. Scene and Activity Analysis - **Image 1:** - Scene: A diagram representing a graph G with vertices and edges. - Activities: Visual aid for understanding the structure of graph G, focusing on vertices (1,2,3,4,5) and their connecting edges. - **Image 2:** - Scene: A slightly different depiction of the same graph to illustrate the same vertices and edges. #### 4. Text Analysis - **Detected Text:** - Title: "Chapter 1: Mathematical Preliminaries" - Subsections: "Exercises for section 1.5", "1.6 Graphs". - Questions and formulae pertaining to combinatorial mathematics and graph theory: - Exercises related to subsets, binomials, Taylor's theorem, and twiddles. - Graph section explaining vertices, edges, and the degree of a vertex. - **Analysis of Text:** - The text is educational, providing exercises for understanding binomial coefficients and properties of graph theory. - Key definitions: - **Graph:** Collection of vertices and edges connecting pairs of vertices. - **Degree of a vertex:** Number of edges incident to the vertex. #### 6. Product Analysis - **Adjective Questions for Exercises:** - Typesetting involves mathematical equations and formulas for binomial coefficients, subset problems, Taylor’s theorem, etc. - No physical products are depicted. #### 7. Anomaly Detection - No apparent anomalies within the images or text. Everything seems to be intentionally placed for educational purposes. #### 8. Color Analysis - **Color Composition:** - The page is primarily black and white typical of a textbook. - Impact: Monochrome color scheme ensures clarity and focuses on the content without distractions. #### 9. Perspective and Composition - **Perspective:** - Bird’s eye view of the diagrams ensuring all vertices and edges are visible. - **Composition:** - Balanced composition of text and diagrams. Diagrams are located centrally within the “Graphs” section to aid understanding of the descriptions. #### 10. Contextual Significance - **Significance:** - The overall document seems to be a mathematical textbook or study guide. - The images and text contribute to teaching advanced mathematical concepts relating to combinatorics and graph theory. #### 13. Table Analysis - **Tables:** - The text contains lists rather than tables, presenting exercises and definitions in a structured format. #################### 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: 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: 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: 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 nonsolated 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 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 - {1})` and `maxset1(G - {1} - Nbhd(1))` 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. Now 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 these distinctions alter the classification of problems into "polynomial time-doable" versus "hard"? Take the graph problems, for instance. How many kinds of input data does it take to describe a graph? Well, certainly we can think about the entire list of \( n(n - 1)/2 \) 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: 45 Context: # 2.3 Recursive Graph Algorithms An algorithm that one might think of for this problem, which is to examine every single subset of the vertices of \( G \) and ask if it is an independent set or not. That algorithm would take \( O(2^n) \) time since there are \( 2^n \) subsets of vertices to look at. Hence we have traded in a \( 2^n \) for a \( 1.619^n \) by being a little bit cagey about the algorithm. Can we do still better? There have not been a number of improvements of the basic maxset algorithm worked out. Of these, the most successful is perhaps the one of Tarjan and Trojanowski that is cited in the bibliography at the end of this chapter. We are not going to work out all of those ideas here, but instead we will show what kind of improvements on the basic idea will help us to do better in the time estimate. So, as our next thought, we might replace the instruction "choose some vertex \( v \) of \( G \)" in any instruction "choose some vertex \( v \) of \( G \) that has at least two neighbors." Then we could be quite certain that \( G - \{v\} - Nbd(v) \) would have at most \( n - 3 \) vertices. What if there isn't any such vertex in the graph \( G \)? Then \( G \) could contain only vertices with 0 or 1 neighbors. Such a graph \( G \) would be a collection of \( E \) disjoint edges together with a number of isolated vertices. The size of the largest independent set of vertices in such a graph is easy to find. An maximum independent set contains one vertex from each of the \( E \) edges and it contains all of the isolated vertices. Hence in this case, maxset \( = E + m = |V(G)| - |E(G)| \), and we obtain a second try at a good algorithm in the following form: ``` procedure marsel2(G); returns the size of the largest independent set of vertices of \( G \) if \( G \) has no vertex of degree \( > 2 \) then marsel2 := \( |V(G)| - |E(G)| \) else choose a vertex \( v \) of degree \( \geq 2 \); \( n_1 := marsel2(G - \{v\}); \) \( n_2 := marsel2(G - \{v\} - Nbd(v)); \) maxset2 := max(n_1, n_2) end(marsel2) ``` How much have we improved the complexity estimate? If we apply to marsel2 the reasoning that led to (2.3.2) we find: \[ f(n) \leq cm^2 + f(n - 1) + f(n - 3) \] where \( f(n) \) is once more the worst-case time bound for graphs of \( n \) vertices. Just as before, (2.3.3) is a recurrent inequality of the form that was studied at the end of section 1.4, in theorem 1.4.1. Using the conclusion of that theorem, we find from (2.3.3) that \( f(n) = O(c^n) \) where \( c = 1.46557 \). It is the positive root of the equation \( c^2 = c^2 + 1 \). The next result of our effort to improve marsel2 has been to reduce the running-time bound from \( O(1.619^n) \) to \( O(1.477^n) \), which isn’t a bad day’s work. In the exercises below we will develop marsel3, whose runtime will be \( O(1.391^n) \). The idea will be that since in marsel2 we were able to ensure that \( v \) had at least two neighbors, why not try to insure that \( v \) has at least 3 neighbors! As long as we have been able to reduce the time bound more and more by ensuring that the selected vertex has lots of neighbors, why don’t we keep it up, and insist that \( v \) should have 4 or more neighbors. Regrettably the intended route ran out of steam precisely at that moment. So why ask what the "trivial case" would look like. We would be working on a graph \( G \) in which vertices are more than 3 neighbors. Well, what "trivializing" shall we do in this "trivial case"? The fact is that there isn’t any way of finding the maximum independent set in a graph where all vertices have \( \leq 3 \) neighbors that’s any better than the general methods that we’ve already discussed. In fact, if one could find a fast method for that restricted problem it would have extremely important consequences because we would then be able to do all graphs rapidly, not just those special ones. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 51 Context: # Exercises for section 2.3 1. Let \( G \) be a cycle of \( n \) vertices. What is the size of the largest independent set of vertices in \( V(G) \)? 2. Let \( G \) be a path of \( n \) vertices. What is the size of the largest independent set of vertices in \( V(G) \)? 3. Let \( G \) be a connected graph in which every vertex has degree 2. What must such a graph consist of? 4. Let \( G \) be a connected graph in which every vertex has degree \( \leq 2 \). What must such a graph look like? What is the size of the largest independent set of vertices in such a graph? How long would it take you to calculate that number for such a graph \( G \)? How would you do it? 5. Write out algorithm `maxset3`, which finds the size of the largest independent set of vertices in a graph. Its trivial case will occur if \( G \) has no vertices of degree \( \geq 3 \). Otherwise, it will choose a vertex \( v \) of degree \( \geq 3 \) and proceed as in master2. 6. Analyze the complexity of your algorithm `maxset3` from exercise 6 above. 7. Use (2.3.4) to prove by induction that \( P(K_i;G) \) is a polynomial in \( K \) of degree \( |V(G)| \). Then show that if \( G \) is a tree then \( P(K;G) = K(K - 1)^{|V(G)| - 1} \). 8. Write out an algorithm that will change the vertex adjacency matrix of a graph \( G \) to the vertex adjacency matrix of the graph \( G' \) (where \( e \) is a given edge of \( G \)). 9. How many edges must \( G \) have before the second quantity inside the ‘O’ in (2.3.12) is the smaller of the two? 10. Let \( \alpha(G) \) be the size of the largest independent set of vertices of a graph \( G \), let \( \chi(G) \) be its chromatic number, and let \( n = |V(G)| \). Show that, for every \( G \), \( \alpha(G) \geq n/\chi(G) \). ## 2.4 Fast matrix multiplication Everybody knows how to multiply two \( 2 \times 2 \) matrices. If we want to calculate \[ \begin{pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{pmatrix} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} \] (2.1) then, "of course," \[ c_{ij} = \sum_{k=1}^{2} a_{ik} b_{kj} \quad (i,j=1,2). \] (2.2) Now look at (2.4.2) a little more closely. In order to calculate each one of the \( c_{ij} \)'s we have to do 2 multiplications of numbers. The cost of multiplying two \( 2 \times 2 \) matrices is therefore 8 multiplications of numbers. If we measure the cost in units of additions of numbers, the cost is 4 such additions. Hence, the matrix multiplication method that is shown in (2.4.1) has a complexity of \( 8 \) multiplications of numbers and \( 4 \) additions of numbers. This may seem rather unsettling, but the best ideas often have humble origins. Suppose you could find another way of multiplying two \( 2 \times 2 \) matrices in which the cost was only 7 multiplications of numbers, together with more than 4 additions of numbers. What would be a cause for dancing in the streets, or would it be just a curiosity, of little importance? In fact, it would be extremely important, and the consequences of such a step were fully appreciated only in 1965 by V. Strassen, to whom the ideas that we are now discussing are due. What we’re going to do next in this section is the following: (a) Describe another way of multiplying two \( 2 \times 2 \) matrices in which the cost will be only 7 multiplications of numbers plus a bunch of additions of numbers, and (b) Convince you that it was worth the trouble. The usefulness of the ideas stems from the following amazing fact: if two \( 2 \times 2 \) matrices can be multiplied with only \( 7 \) multiplications of numbers, then two \( N \times N \) matrices can be multiplied using only \( O(N^{2.81}) \). * V. Strassen, Gaussian elimination is not optimal, Numerische Math. 13 (1969), 354-6. Image Analysis: ### Localization and Attribution 1. **Image 1:** - **Location:** Single page, top-to-bottom position. - **Image Number:** Image 1. ### Object Detection and Classification **Image 1:** - **Objects Detected:** - Mathematical notations and text. - Formulae and equations enclosed in parentheses. ### Text Analysis **Image 1:** - **Extracted Text:** - The text contains multiple mathematical exercises, specifically for section 2.3, discussing various graph theory problems. - There's also an excerpt for section 2.4 discussing "Fast matrix multiplication." - **Content Analysis:** - **Exercises for section 2.3:** - The questions revolve around determining sizes of the largest independent sets in different types of graphs like cycles, paths, and connected graphs. - Questions concerning algorithms and complexity analysis are also present. - Inductive proof and graph vertex adjacency matrix modification algorithms are mentioned. - **Section 2.4 Fast Matrix Multiplication:** - The calculation of matrix products is introduced with a basic example using 2x2 matrices. - Formulas \( c_{11}, c_{12}, c_{21}, c_{22} \) are elaborated upon. - A hypothesis about reducing the number of multiplications from 8 to 7 is discussed, referencing V. Strassen's work. ### Diagram and Chart Analysis **Image 1:** - **Diagrams Detected:** - A mathematical matrix multiplication diagram is present. - Matrix elements and multiplication processes are illustrated. ### Scene and Activity Analysis **Image 1:** - **Scene Description:** - The image illustrates a textbook-like page, with text-based exercises and explanations about graph theory and matrix multiplication. - **Activities:** - Reading mathematical exercises, solving graph theory problems, and understanding matrix multiplication concepts. ### Contextual Significance **Image 1:** - **Contribution to Overall Document:** - The image seems to be part of a mathematics textbook or academic publication, focusing on graph theory and matrix algorithms. - It aids in the understanding of complex mathematical concepts and provides practical problems for the reader to solve. ### Process Descriptions **Image 1:** - **Description:** - The process of calculating the size of the largest independent set for various types of graphs is broken down. - Basic matrix multiplication processes are described, highlighting both traditional methods and potential optimizations. ### Typen Bezeichnung (Type Designations) **Image 1:** - **Types Described:** - Different types of graphs (cycles, paths, connected graphs, and non-necessarily-connected graphs) and their properties are identified. - Types of vertices (degrees, vertex adjacency matrix) are also designated. ### Trend and Interpretation **Image 1:** - **Identified Trends:** - A focus on enhancing computational efficiency, particularly in matrix multiplication, is emphasized. - Analyzing the size of independent sets in various graph structures is a recurring theme. ### Tables **Image 1:** - **Detected Content:** - The page does not contain any structured tables but includes mathematical equations which are presented in a structured format resembling tables. By addressing different aspects, the analysis provides a comprehensive understanding of the mathematical concepts and exercises illustrated in the image. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 109 Context: # What is a language? by a polynomial in B if T = n^K and B ∼ n^2? It would seem so; in fact obviously T = O(B^K), and that’s a polynomial, isn’t it? The key point resides in the order of the qualifiers. We must give the polynomial that works for every instance of the problem first. Then that one single polynomial must work on every instance. If the 'polynomial' that we give is B(K), well that’s a different polynomial in B for different instances of the problem, because K is different for different instances. Therefore if we say that a certain program for 'Independent Set' will always get an answer before B^K/2 minutes, where B is the length of the input bit string, then we would not have provided a polynomial-time guarantee in the form of a single polynomial in B that applies uniformly to all problem instances. The distinction is a little thorny, but is worthy of careful study because it’s of fundamental importance. What we are discussing is usually called a worst-case time bound, meaning a bound on the running time that applies to every instance of the problem. Worst-case time bounds aren’t the only possible interesting ones. Sometimes we might not care if an algorithm is occasionally very slow as long as it is almost always fast. In other situations we might be satisfied with an algorithm that is fast on average. For the present, however, we will stick to the worst-case time bounds and study some of the theory that applies to that situation. In sections 5.6 and 5.7 we will study some average time bounds. Now let’s return to the properties of the NP-complete family of problems. Here are some of them: 1. The problems all seem to be computationally very difficult, and no polynomial time algorithms have been found for any of them. 2. It has been proved that polynomial time algorithms for these problems do not exist. 3. But this is not just a random list of hard problems. If a fast algorithm could be found for one NP-complete problem then there would be fast algorithms for all of them. 4. Conversely, if it could be proved that no fast algorithm exists for one of the NP-complete problems, then there could not be a fast algorithm for any other of those problems. These properties are not intended to be a definition of the concept of NP-completeness. We’ll get that later on in this section. They are intended as a list of some of the interesting features of these problems, which, when coupled with their theoretical and practical importance, accounts for the intense worldwide research effort that has gone into understanding them in recent years. The question of the existence or nonexistence of polynomial-time algorithms for the NP-complete problems learns probably falls as the principal unsolved problem that faces theoretical computer science today. Our next task will be to develop the formal machinery that will permit us to give precise definitions of all of the concepts that are needed. In the remainder of this section we will discuss the additional ideas informally, and then in section 5.2 we’ll state them quite precisely. ## What is a decision problem? First, the idea of a decision problem. A decision problem is one that asks only for a yes-or-no answer: Can this graph be 5-colored? Is there a tour of length ≤ 15 miles? Is there a set of 67 independent vertices? Many of them problems that we are studying can be phrased as decision problems or as optimization problems: What is the smallest number of colors with which G can be colored? What is the length of the shortest tour of the cities? What is the size of the largest independent set of vertices in G? Usually if we find a fast algorithm for a decision problem then just a little more work will be able to solve the corresponding optimization problem. For instance, suppose we have an algorithm that solves the decision problem for graph coloring, and what we want is the solution of the optimization problem (the chromatic number). Let G be a graph G given, say of 100 vertices. Ask: can the graph be 50-colored? If so, then the chromatic number lies between 1 and 50. Then ask if it can be colored in 25 colors. If not, then the chromatic number lies between 26 and 50. Continue in this way, using bisection of the interval that is known to contain the chromatic number. After O(log n) steps we will have found the chromatic number of a graph of n vertices. The extra multiplicative factor of log n will not alter the polynomial vs. nonpolynomial runing time distinction. Hence if there is a fast way to do the decision problem then there is a fast way to do the optimization problem. The converse is obvious. Hence we will restrict our discussion to decision problems. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 126 Context: # Chapter 5: NP-completeness What kind of graph has such a ‘good chance’? A great deal of research has gone into the study of how many edges a graph has to have before one can assure it must contain certain given structures. For instance, how many edges must a graph of \( n \) vertices have before we can be almost certain that it will contain a complete graph of \( n \) vertices? To say that graphs have a property ‘almost certainly’ is to say that the ratio of the number of graphs on \( n \) vertices that have the property to the number of graphs on \( n \) vertices approaches 1 as \( n \) grows without bound. For the Hamilton path problem, an important dividing line, or threshold, turns out to be at the level of \( \log n \) edges. That is to say, a graph of \( n \) vertices that has \( o(\log n) \) edges has relatively little chance of being even connected, whereas a graph with \( \geq \log n \) edges is almost certainly connected, and almost certainly has a Hamilton path. We now state the theorem of Angluin and Valiant, which asserts that the algorithm above will almost surely succeed if the graph \( G \) has enough edges. ## Theorem 5.5.1 Fix a positive real number \( \epsilon \). There exist numbers \( M \) and \( c \) such that if we choose a graph \( G \) at random from among those of \( n \) vertices and at least \( cn \) edges, and we choose arbitrary vertices \( s, t \) in \( G \), then the probability that algorithm UHC returns 'success' before making a total of \( M \log n \) attempts to extend partly constructed paths is \( 1 - O(n^{-\epsilon}) \). ## 5.6 Backtracking (I): independent sets In this section, we are going to describe an algorithm that is capable of solving some NP-complete problems fast, on the average, while at the same time guaranteeing that a solution will always be found, be it quickly or slowly. The method is called backtracking, and it has long been a standard method in computer search problems when all else fails. It has been common to think of backtracking as a very long process, and indeed can be. But recently it has been shown that the method can be very fast on average, and that in the graph coloring problem, for instance, it functions in an average of constant time, i.e., the time is independent of the number of vertices, although to be sure, the worst-case behavior is very exponential. We first illustrate the backtrack method in the context of a search for the largest independent set of vertices (a set of vertices too two of which are joined by an edge) in a given graph \( G \), an NP-complete problem. In this case the average time behavior of the method is not constant, or even polynomial, but is subexponential. The method is also easy to analyze and describe in this case. Hence consider a graph \( G \) of \( n \) vertices, in which the vertices have been numbered \( 1, 2, \ldots, n \). We want to find, in \( G \), the size of the largest independent set of vertices. In Fig. 5.6.1 below, the graph \( G \) has 6 vertices. ![Fig. 5.6.1: Find the largest independent set](#) Begin by searching for an independent set \( S \) that contains vertex 1, so let \( S = \{ 1 \} \). Now attempt to enlarge \( S \). We cannot enlarge \( S \) by adjoining vertex 2 to it, but we can add vertex 3. Our set \( S \) is now \( \{ 1, 3 \} \). Now we cannot add vertex 4 (joined to 1) or vertex 5 (joined to 3) or vertex 6 (joined to 3), so we backtrack. Therefore we backtrack, by replacing the most recently added member of \( S \) by the next choice that we might have made for it. In this case, we delete vertex 3 from \( S \), and the next choice would be vertex 6. The set \( S \) is \( \{ 1, 6 \} \). Again we have a dead end. If we backtrack again, there are no further choices with which to replace vertex 6, so we backtrack even further, and not only delete 6 from \( S \) but also replace vertex 1 by the next possible choice for it, namely vertex 2. 122 Image Analysis: ### Analysis of Attached Visual Content #### 1. Localization and Attribution - **Number of Images:** - There is a single image embedded in the text, which will be referred to as Image 1. - **Position:** - Image 1 is located centrally on the page at the bottom part. #### 2. Object Detection and Classification - **Image 1:** - **Objects Detected:** - A labeled graph with vertices. - Text annotations and numerical notations. - Arrows indicating directions and connections. - **Classification:** - The object is a graph typically used in mathematical and computational contexts to illustrate the concept of independent sets. #### 3. Scene and Activity Analysis - **Image 1:** - **Scene Description:** - The image depicts a graphical representation of a graph, where nodes (vertices) connected by edges (lines) are showcased. The vertices are numbered, and connections between them indicate edges. - **Activities Taking Place:** - The activity involves finding the largest independent set, which is a set of vertices in a graph, none of which are adjacent to each other. #### 4. Text Analysis - **Text in Image 1:** - Text below the image: "Fig. 5.6.1: Find the largest independent set" - Intended to describe the process depicted in the illustration above it. - The numerical notations and annotations on the graph: - Vertices numbered 1 to 6. - An arrowed sequence illustrating connections and potential selections for the independent set. - **Content Significance:** - The text and numerical annotations are crucial for understanding the step-by-step process of identifying an independent set in graph theory. #### 5. Diagram and Chart Analysis - **Image 1:** - **Data and Trends Presented:** - The diagram represents a graph used for backtracking to find the largest independent set of vertices. - Shows a step-by-step process highlighting the vertices chosen and discarded. - **Axes, Scales, and Legends:** - Vertices are marked by numbers (1 to 6). - Lines (edges) connecting these vertices indicate relationships. - **Key Insights:** - Shows how backtracking can help identify an independent set in the graph. - Demonstrates the complexity and decision points in the algorithm. #### 7. Anomaly Detection - **No noticeable anomalies** in the context of the image and supporting text. #### 8. Color Analysis - **Predominant Colors:** - Black and white. - Impact: The monochromatic scheme is typical for scientific and mathematical diagrams, focusing attention on the structure and details without color distraction. #### 9. Perspective and Composition - **Image 1:** - **Perspective:** - The image is presented from a front view, ensuring clarity of all the nodes and connections. - **Composition:** - Balanced layout, central focus on the graph. - Clear and labeled vertices with connecting edges, illustrating the step-by-step process of finding the independent set. #### 10. Contextual Significance - **Image 1:** - **Context in the Document:** - The image supports the text explaining backtracking and its application in finding the largest independent set in NP-complete problems. - **Contribution to the Overall Message:** - Helps in visualizing and understanding the theoretical explanations provided in the text. #### 11. Metadata Analysis - **No metadata available** directly from the image. #### 12. Graph and Trend Analysis - **Image 1:** - **Trend Identification:** - Steps in selecting vertices aiming to maximize the independent set while avoiding adjacency. - **Data Interpretation:** - The graph and steps illustrate trials and errors in the selection process, indicating potential complexities and requiring backtracking. ### Conclusion: The attached content includes a detailed analysis of a single image illustrating the backtracking process to find an independent set in graph theory. The analysis comprehensively covers the visual elements, context, and significance of the image within the document. This aids in understanding the theoretical underpinnings of the text and provides clarity on the subject matter discussed. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 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: - \( \{1\} \) - \( \{3\} \) - \( \{13\} \) - \( \{16\} \) - \( \{2\} \) - \( \{24\} \) - \( \{245\} \) - \( \{25\} \) - \( \{34\} \) - \( \{345\} \) - \( \{4\} \) - \( \{45\} \) - \( \{6\} \) 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: 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: An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf Page: 35 Context: # Chapter 2: Combinatorial Number Theory Thus Schur’s theorem can be restated as follows: If \(1, 2, \ldots, r\) are separated into \(r\) classes in any manner whatever, at least one class will contain a solution to \(a + b = c\). We will prove this by assuming that the numbers \(1, 2, \ldots, r\) have been classified in ways with no class containing a solution of \(a + b = c\) and from this obtain a contradiction. Note that the condition of \(a + b = c\) means no class can contain the difference of two of its elements. Suppose that some class, say \(A_4\), contains elements \(a_1 < a_2 < \ldots\). The differences of these in the following manner: - \(b_1 = a_2 - a_1, \quad b_2 = a_3 - a_2, \ldots\) - \(c_1 = b_2 - b_1, \quad c_2 = b_3 - b_2, \ldots\) - \(d_1 = c_2 - c_1, \quad d_2 = c_3 - c_2, \ldots\) and so on. We note that all the \(b\)s, \(c\)s, \(d\)s, etc., are differences of \(a\)s and hence cannot lie in \(A_4\). Now, we start with \(n\) elements. At least \[ \left\lfloor \frac{n}{r - 1} \right\rfloor + 1 \] of these must lie in a single class, say \(A_4\). We then form \(T_n\)’s. These do not lie in \(A_4\) and hence lie in the remaining \(n - 1\) classes. At least \[ \left\lfloor \frac{T_{n-1}}{r - 1} \right\rfloor + 1 = T_{n-1} + 1 \] of them must lie in a single class, say \(A_4\). Forming their \(T_n\) differences, the class yields \(T_{n-3}\) numbers either in \(A_1, A_2, A_3\). In this manner we eventually produce a \(T_1\) number not belonging to \(A_1, A_2, \ldots, A_4\). But all numbers from among the numbers \(1, 2, \ldots, T_n\) so we have a contradiction, which proves the theorem. We state, without proof, the connection with Fermat’s last theorem. A natural approach to Fermat’s theorem would be to show that \(x^n + y^n = z^n\) (mod \(p\)) is insolvable modulo some \(p\), provided \(p\) does not divide \(x+y\). However, Schur’s theorem can be used to show that this indeed must fail if \(p > 2\) and \(x^m + y^m \equiv p\) has a solution with not a factor of \(p\). Somewhat related to Schur’s theorem is a famous theorem of Van der Waerden, which we briefly investigate. In the early 1920’s the following problem arose in connection with the theory of distributing consecutive residues. Imagine the set of all integers to be divided in any manner into \(r\) classes. Can one assert that arithmetic progressions of arbitrary length can belong to at least one of these classes? The problem remained unsolved for several years in spite of concentrated efforts by many outstanding mathematicians. It was finally solved in 1928 by Van der Waerden. As is not uncommon with such problems, Van der Waerden’s first step was to tackle the more general case. Van der Waerden proved the following: Given integers \(r\) and \(k\), there exists an integer \(N = N(r, k)\) such that if the numbers \(1, 2, \ldots, N\) are separated into \(r\) classes in any manner, then at least one class will contain \(k\) terms in arithmetic progression. We will not give Van der Waerden’s proof here. It is extremely difficult to see strong, and leads only to fantastically large bounds for \(W(k, r)\). For this reason the reader might consider the very worthwhile unsolved problem of finding an alternative simpler proof that \(W(k, r)\) exists and finding reasonable bounds for it. We will have a little more to say about the function \(W(k, r)\) a little later. Our next problem of combinatorial number theory deals with “non-averaging” sequences. We call a sequence \(A = \{a_1 < a_2 < a_3 < \ldots\}\) non-averaging if it does not contain the average of two of its elements, i.e., if \(a_i + a_j \in A\) for \(i \neq j\). Let \(A(n)\) denote the number of elements in \(A\) not averaging \(n\). The main problem is to estimate how large \(A(n)\) can be if it is nonaveraging. We can form a nonaveraging sequence by starting with \(1, 2, \ldots\) and then always taking the smallest number that does not violate the conditions for nonaveraging sets. In this way we obtain \(1, 2, 4, 5, 10, 11, 14, 15, 28, 29, \ldots\). It is an interesting fact that this sequence is related to the famous Cantor ternary set. Indeed, we have an easier sequence to prove that this sequence can be obtained by adding \(1\) to each integer representation in base \(3\) containing only \(0\)s and \(1\)s. This sequence is visual in the sense that there cannot be integers without the successive digit violating its nonaveraging character. This, as well as other facts, led Schur (about 1930) to conjecture that this set was dens as any focusing on \(r\). The trick, then, is that the counting function can easily be estimated by Spencer and Spencer (1992) proved that one could have a nonaveraging set of integers of size at least \(2^{-\frac{1}{3}}n^{2/3}\). Given \(a\) numbers in base ten, we describe \(x\) as \(R\) in the vector form in which it is the first digit in the first bracket, the next digit in the second bracket, the next digit in the third bracket, and so on. If the last incorrectly bracketed the bracket in the left does not exist or does not strictly order of digits, it will overflow with zeros. For instance, the numbers \[ a = (000)(265)(320)(0)(20)(0). \] ########## """QUERY: what are sets and spaces? Indicate how many words is your response""" 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 9, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 15, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 24, 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 28, 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 45, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 51, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 109, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 126, 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 35 ================================================== **Elapsed Time: 0.00 seconds** ================================================== FINAL ANSWER Answer: [Answer was streamed to client] ================================================== **Elapsed Time: 2.85 seconds** ==================================================