{ "query": "can you give me a 2 sentence summary of your contents?", "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\ncan you give me a 2 sentence summary of your contents?\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": "what are the lyrics of the song never gonna give you up" }, { "role": "assistant", "content": "TheThe provided text is about the Fast Fourier Transform (FFT) algorithm and doesn't provided text is about the Fast Fourier Transform (FFT) algorithm and doesn't contain any information about the song \"Never Gonna Give You Up\". \n\nTo contain any information about the song \"Never Gonna Give You Up\". \n\nTo find the lyrics of that song, I recommend searching online for \"Never Gonna Give You Up lyrics\". \n find the lyrics of that song, I recommend searching online for \"Never Gonna Give You Up lyrics\"." }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "give me a 200 word summary of your context" }, { "role": "assistant", "content": "The provided context is from a book on Algebraic Topology by Allen Hatcher. The book was published in 2001 and allows for noncommercial use of its content. The book cover features a geometric design with intersecting loops in blue and red hues, visually representing the complexity of the subject matter. \n\nThe second page of the book's table of contents shows a picture of a person giving a presentation in a conference room or lecture hall. The person is using a laptop and a large screen, suggesting a technologically-driven presentation." }, { "role": "assistant", "content": "The last response provided by the chatbot contains a summary of the book \"Algebraic Topology\" by Allen Hatcher, including details about its publication, content, and visual elements. However, it does not accurately reflect the context provided in the query, which includes a detailed analysis of various pages from the book \"Algorithms and Complexity\" by Herbert S. Wilf.\n\nThe response incorrectly attributes the content to Hatcher's work on Algebraic Topology instead of focusing on the algorithms and complexity topics discussed in Wilf's book. Additionally, the response does not address the specific exercises or theoretical concepts related to algorithms, Turing machines, or graph theory that were present in the context.\n\nIn conclusion, I do not agree with the original response as it fails to accurately represent the context and information provided. The response should have focused on the content related to algorithms, complexity, and Turing machines, as outlined in the context, rather than misattributing it to a different book. \n\nTo improve the response, it should include a summary of the relevant topics from Wilf's book, such as the exercises on decision problems, the description of Turing machines, and the discussions on graph theory and algorithms. This would provide a more accurate and comprehensive answer based on the context given." }, { "role": "assistant", "content": "It seems there may have been some confusion regarding the content of the previous responses. To clarify, the analysis provided was focused on the page from \"Algorithms and Complexity\" by Herbert S. Wilf, specifically discussing matrix multiplication and related concepts. \n\nIf you have specific questions about the content on that page or need further details about matrix multiplication, graph theory, or any other topic covered in Wilf's book, please let me know!" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "give a summary of yourr context" }, { "role": "assistant", "content": "The provided context is from a book on Algorithms and Complexity by Herbert S. Wilf. The book discusses various algorithms and their computational complexity, including sorting, graph algorithms, matrix multiplication, and number theory. It also explores the concept of NP-completeness and provides examples of NP-complete problems. The book is written in a clear and concise style, with numerous examples and exercises to help readers understand the concepts.\n\nThe book is divided into chapters, each focusing on a specific topic. Chapter 1 covers mathematical preliminaries, including orders of magnitude, positional number systems, and recurrence relations. Chapter 2 discusses recursive algorithms, including Quicksort, the maximum independent set problem, and the Fast Fourier Transform. Chapter 3 focuses on the network flow problem, including the Ford-Fulkerson algorithm and the max-flow min-cut theorem. Chapter 4 explores algorithms in number theory, including the Euclidean algorithm, primitive roots, and pseudoprimality tests. Chapter 5 introduces the concept of NP-completeness and provides examples of NP-complete problems.\n\nThe book is a valuable resource for anyone interested in learning about algorithms and their complexity. It is written in a clear and accessible style, making it suitable for both students and professionals." }, { "role": "user", "content": "give a summary of yourr context" }, { "role": "assistant", "content": "The provided context is from a book on Algorithms and Complexity by Herbert S. Wilf. The book discusses various algorithms and their computational complexity, including sorting, graph algorithms, matrix multiplication, and number theory. It also explores the concept of NP-completeness and provides examples of NP-complete problems. The book is written in a clear and concise style, with numerous examples and exercises to help readers understand the concepts.\n\nThe book is divided into chapters, each focusing on a specific topic. Chapter 1 covers mathematical preliminaries, including orders of magnitude, positional number systems, and recurrence relations. Chapter 2 discusses recursive algorithms, including Quicksort, the maximum independent set problem, and the Fast Fourier Transform. Chapter 3 focuses on the network flow problem, including the Ford-Fulkerson algorithm and the max-flow min-cut theorem. Chapter 4 explores algorithms in number theory, including the Euclidean algorithm, primitive roots, and pseudoprimality tests. Chapter 5 introduces the concept of NP-completeness and provides examples of NP-complete problems.\n\nThe book is a valuable resource for anyone interested in learning about algorithms and their complexity. It is written in a clear and accessible style, making it suitable for both students and professionals." }, { "role": "user", "content": "give a summary of yourr context" }, { "role": "assistant", "content": "The provided context is from a book on Algorithms and Complexity by Herbert S. Wilf. The book discusses various algorithms and their computational complexity, including sorting, graph algorithms, matrix multiplication, and number theory. It also explores the concept of NP-completeness and provides examples of NP-complete problems. The book is written in a clear and concise style, with numerous examples and exercises to help readers understand the concepts.\n\nThe book is divided into chapters, each focusing on a specific topic. Chapter 1 covers mathematical preliminaries, including orders of magnitude, positional number systems, and recurrence relations. Chapter 2 discusses recursive algorithms, including Quicksort, the maximum independent set problem, and the Fast Fourier Transform. Chapter 3 focuses on the network flow problem, including the Ford-Fulkerson algorithm and the max-flow min-cut theorem. Chapter 4 explores algorithms in number theory, including the Euclidean algorithm, primitive roots, and pseudoprimality tests. Chapter 5 introduces the concept of NP-completeness and provides examples of NP-complete problems.\n\nThe book is a valuable resource for anyone interested in learning about algorithms and their complexity. It is written in a clear and accessible style, making it suitable for both students and professionals." }, { "role": "user", "content": "give a summary of your context" }, { "role": "assistant", "content": "" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "can you give a 500 word summary of your context?" }, { "role": "assistant", "content": "The provided context is from a book on Algorithms and Complexity by Herbert S. Wilf. The book discusses various algorithms and their computational complexity, including sorting, graph algorithms, matrix multiplication, and number theory. It also explores the concept of NP-completeness and provides examples of NP-complete problems. The book is written in a clear and concise style, with numerous examples and exercises to help readers understand the concepts.\n\nThe book is divided into chapters, each focusing on a specific topic. Chapter 1 covers mathematical preliminaries, including orders of magnitude, positional number systems, and recurrence relations. Chapter 2 discusses recursive algorithms, including Quicksort, the maximum independent set problem, and the Fast Fourier Transform. Chapter 3 focuses on the network flow problem, including the Ford-Fulkerson algorithm and the max-flow min-cut theorem. Chapter 4 explores algorithms in number theory, including the Euclidean algorithm, primitive roots, and pseudoprimality tests. Chapter 5 introduces the concept of NP-completeness and provides examples of NP-complete problems.\n\nThe book is a valuable resource for anyone interested in learning about algorithms and their complexity. It is written in a clear and accessible style, making it suitable for both students and professionals." }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "can you give me 200 words lorem ipsum?" }, { "role": "assistant", "content": "The provided text is about the Fast Fourier Transform (FFT) algorithm and doesn't provided text is about the Fast Fourier Transform (FFT) algorithm and doesn't contain any information about lorem ipsum. \n\nTo contain any information about lorem ipsum. \n\nTo find lorem ipsum, I recommend searching online for \"lorem ipsum generator\". \n find lorem ipsum, I recommend searching online for \"lorem ipsum generator\"." }, { "role": "user", "content": "can you give me 300 words lorem ipsum?" }, { "role": "assistant", "content": "The provided text is about the Fast Fourier Transform (FFT) algorithm and doesn't provided text is about the Fast Fourier Transform (FFT) algorithm and doesn't contain any information about lorem ipsum. \n\nTo contain any information about lorem ipsum. \n\nTo find lorem ipsum, I recommend searching online for \"lorem ipsum generator\". \n find lorem ipsum, I recommend searching online for \"lorem ipsum generator\"." }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "can you list all documents in your knowledgebase?" }, { "role": "assistant", "content": "The documents in my knowledge base are:\n\n1. Algorithms and Complexity - Herbert S. Wilf (PDF).pdf\n2. Algebraic Topology AT-toc.pdf\n3. An Introduction to the Theory of Numbers - Leo Moser (PDF).pdf\n\nIs there anything else I can help you with?" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "can you give a 500 word summary of your context?" }, { "role": "assistant", "content": "The provided context is from two books on mathematics: \"Algebraic Topology\" by Allen Hatcher and \"Algorithms and Complexity\" by Herbert S. Wilf. \n\n**\"Algebraic Topology\"** is a textbook on the subject of algebraic topology, a branch of mathematics that uses algebraic methods to study topological spaces. The book cover features a geometric design with intersecting loops in blue and red hues, visually representing the complexity of the subject matter. The second page of the book's table of contents shows a picture of a person giving a presentation in a conference room or lecture hall. The person is using a laptop and a large screen, suggesting a technologically-driven presentation.\n\n**\"Algorithms and Complexity\"** is a book that discusses various algorithms and their computational complexity, including sorting, graph algorithms, matrix multiplication, and number theory. It also explores the concept of NP-completeness and provides examples of NP-complete problems. The book is written in a clear and concise style, with numerous examples and exercises to help readers understand the concepts.\n\nThe book is divided into chapters, each focusing on a specific topic. Chapter 1 covers mathematical preliminaries, including orders of magnitude, positional number systems, and recurrence relations. Chapter 2 discusses recursive algorithms, including Quicksort, the maximum independent set problem, and the Fast Fourier Transform. Chapter 3 focuses on the network flow problem, including the Ford-Fulkerson algorithm and the max-flow min-cut theorem. Chapter 4 explores algorithms in number theory, including the Euclidean algorithm, primitive roots, and pseudoprimality tests. Chapter 5 introduces the concept of NP-completeness and provides examples of NP-complete problems.\n\nThe book is a valuable resource for anyone interested in learning about algorithms and their complexity. It is written in a clear and accessible style, making it suitable for both students and professionals." }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "can you give a 500 word summary of your context?" }, { "role": "assistant", "content": "The provided context is from two books on mathematics: \"Algebraic Topology\" by Allen Hatcher and \"Algorithms and Complexity\" by Herbert S. Wilf. \n\n\"Algebraic Topology\" is a textbook on the subject of algebraic topology, a branch of mathematics that uses algebraic methods to study topological spaces. The book cover features a geometric design with intersecting loops in blue and red hues, visually representing the complexity of the subject matter. The second page of the book's table of contents shows a picture of a person giving a presentation in a conference room or lecture hall. The person is using a laptop and a large screen, suggesting a technologically-driven presentation.\n\n\"Algorithms and Complexity\" is a book that discusses various algorithms and their computational complexity, including sorting, graph algorithms, matrix multiplication, and number theory. It also explores the concept of NP-completeness and provides examples of NP-complete problems. The book is written in a clear and concise style, with numerous examples and exercises to help readers understand the concepts.\n\nThe book is divided into chapters, each focusing on a specific topic. Chapter 1 covers mathematical preliminaries, including orders of magnitude, positional number systems, and recurrence relations. Chapter 2 discusses recursive algorithms, including Quicksort, the maximum independent set problem, and the Fast Fourier Transform. Chapter 3 focuses on the network flow problem, including the Ford-Fulkerson algorithm and the max-flow min-cut theorem. Chapter 4 explores algorithms in number theory, including the Euclidean algorithm, primitive roots, and pseudoprimality tests. Chapter 5 introduces the concept of NP-completeness and provides examples of NP-complete problems.\n\nThe book is a valuable resource for anyone interested in learning about algorithms and their complexity. It is written in a clear and accessible style, making it suitable for both students and professionals." }, { "role": "user", "content": "can you give a 500 word summary of your context?" }, { "role": "assistant", "content": "" }, { "role": "user", "content": "can you give a 500 word summary of your context?" }, { "role": "assistant", "content": "" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "tes" }, { "role": "user", "content": "test" }, { "role": "user", "content": "test" }, { "role": "user", "content": "test" }, { "role": "user", "content": "test" }, { "role": "user", "content": "test" }, { "role": "user", "content": "test" }, { "role": "user", "content": "test" }, { "role": "user", "content": "test" }, { "role": "user", "content": "test" }, { "role": "user", "content": "test" }, { "role": "user", "content": "test" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Okay, I understand. I will focus on answering your questions directly and follow any formatting instructions you provide. If your question is unclear, I will ask for clarification based on the context. \n\nPlease tell me what you would like to test. I'm ready!" }, { "role": "user", "content": "test" }, { "role": "assistant", "content": "Hello! It seems like you're interested in testing something related to the knowledgebase. How can I assist you with that? Is there any specific information or topic you would like to explore?" }, { "role": "user", "content": "can you give me a 2 sentence summary of your contents?" }, { "role": "assistant", "content": "My knowledge base contains two books on mathematics: \"Algorithms and Complexity\" by Herbert S. Wilf and \"Algebraic Topology\" by Allen Hatcher. The first book focuses on algorithms and their computational complexity, while the second explores the field of algebraic topology." }, { "role": "assistant", "content": "Thank you for providing the context regarding the knowledgebases available. If you have specific questions or topics you would like to explore related to \"Algorithms and Complexity\" by Herbert S. Wilf or \"Algebraic Topology\" by Allen Hatcher, please let me know, and I will do my best to assist you with precise and relevant information." }, { "role": "assistant", "content": "Thank you for providing the context regarding the knowledgebases available. If you have specific questions or topics you would like to explore within \"Algorithms and Complexity\" by Herbert S. Wilf or \"Algebraic Topology\" by Allen Hatcher, please let me know, and I will do my best to assist you with precise and relevant information." }, { "role": "assistant", "content": "Thank you for providing the context regarding the knowledgebases available. If you have specific questions or topics you would like to explore within \"Algorithms and Complexity\" by Herbert S. Wilf or \"Algebraic Topology\" by Allen Hatcher, please let me know, and I will do my best to assist you with precise and relevant information." }, { "role": "assistant", "content": "Thank you for providing the context regarding the knowledgebases available. If you have specific questions or topics you would like to explore related to \"Algorithms and Complexity\" by Herbert S. Wilf or \"Algebraic Topology\" by Allen Hatcher, please let me know, and I will do my best to assist you with precise and relevant information." }, { "role": "assistant", "content": "" } ], "stream": true, "language_level": "", "chat_channel": ":4660:1728479571675", "language": "German", "tone": "neutral", "writing_style": "standard", "model": "gemini-1.5-flash", "knowledgebase": "ki-dev-large", "seed": 1055, "client_id": 1728479571675, "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: can you give me a 2 sentence summary of your contents? Model: gemini-1.5-flash **Elapsed Time: 0.00 seconds** ROUTING Query type: summary **Elapsed Time: 1.77 seconds** RAG PARAMETERS Max Context To Include: 100 Lowest Score to Consider: 0.1 ================================================== **Elapsed Time: 0.14 seconds** ================================================== VECTOR SEARCH ALGORITHM TO USE Use MMR search?: True Use Similarity search?: False ================================================== **Elapsed Time: 0.10 seconds** ================================================== VECTOR SEARCH DONE ================================================== **Elapsed Time: 6.55 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. In the "CONTEXT", prioritize information from data tagged "FEEDBACK:". Your role is to act as an expert at summarization and analysis. 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. Use as much tokens as possible to provide a detailed response. **Elapsed Time: 0.17 seconds** FINAL QUERY Final Query: CONTEXT: ########## File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 60 Context: ### Summary The visual content is a page from an academic or technical text focused on recursive algorithms, specifically the Fast Fourier Transform (FFT). The text includes mathematical notations, theorems, proofs, and practical applications, emphasizing the efficiency of FFT in computational tasks. The included table illustrates the reduction in complexity provided by using the FFT. The document's formal structure and detailed mathematical content suggest it is intended for readers with a strong background in mathematics or computer science. #################### File: Algebraic%20Topology%20AT-toc.pdf Page: 4 Context: # Chapter 2. Homology . . . . . . . . . . . . . . . . . . . . 97 ## 2.1. Simplicial and Singular Homology . . . . . . . . . 102 - Δ-Complexes 102 - Simplicial Homology 104 - Singular Homology 108 - Homotopy Invariance 110 - Exact Sequences and Excision 113 - The Equivalence of Simplicial and Singular Homology 128 ## 2.2. Computations and Applications . . . . . . . . . . 134 - Degree 134 - Cellular Homology 137 - Mayer-Vietoris Sequences 149 - Homology with Coefficients 153 ## 2.3. The Formal Viewpoint . . . . . . . . . . . . . . . 160 - Axioms for Homology 160 - Categories and Functors 162 ### Additional Topics - **2.A**. Homology and Fundamental Group 166 - **2.B**. Classical Applications 169 - **2.C**. Simplicial Approximation 177 # Chapter 3. Cohomology . . . . . . . . . . . . . . . . . . 185 ## 3.1. Cohomology Groups . . . . . . . . . . . . . . . . 190 - The Universal Coefficient Theorem 190 - Cohomology of Spaces 197 ## 3.2. Cup Product . . . . . . . . . . . . . . . . . . . 206 - The Cohomology Ring 212 - A Künneth Formula 214 - Spaces with Polynomial Cohomology 220 ## 3.3. Poincaré Duality . . . . . . . . . . . . . . . . 230 - Orientations and Homology 233 - The Duality Theorem 239 - Connection with Cup Product 249 - Other Forms of Duality 252 ### Additional Topics - **3.A**. Coefficients for Homology 261 - **3.B**. The General Künneth Formula 268 - **3.C**. H-Spaces and Hopf Algebras 281 - **3.D**. The Cohomology of \( S^0(n) \) 292 - **3.E**. Bockstein Homomorphisms 321 - **3.F**. Limits and Ext 311 - **3.G**. Local Coefficients 327 #################### File: An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf Page: 4 Context: # Preface These lectures are intended as an introduction to the elementary theory of numbers. I use the word “elementary” both in the technical sense—complex variable theory is to be avoided—and in the usual sense—that of being easy to understand, I hope. I shall not concern myself with questions of foundations and shall presuppose familiarity only with the most elementary concepts of arithmetic, i.e., elementary divisibility properties, e.g., (greatest common divisor), l.c.m. (least common multiple), essentially unique factorization into primes and the fundamental theorem of arithmetic: if \( p \mid ab \) then \( p \mid a \) or \( p \mid b \). I shall consider a number of rather distinct topics each of which could easily be the subject of 15 lectures. Hence, I shall not be able to penetrate deeply in any direction. On the other hand, it is well known that in number theory, more than in any other branch of mathematics, it is easy to reach the frontiers of knowledge. It is easy to propound problems in number theory that are unsolved. I shall mention many of these problems; but the trouble with the natural problems of number theory is that they are either too easy or much too difficult. I shall therefore try to expose some problems that are of interest and unsolved but for which there is at least a reasonable hope for a solution by you or me. The topics I hope to touch on are outlined in the Table of Contents, as are some of the main reference books. Most of the material I want to cover will consist of old theorems proved in old ways, but I also hope to produce some old theorems proved in new ways and some new theorems proved in old ways. Unfortunately I cannot produce many new theorems proved in really new ways. #################### File: Algebraic%20Topology%20AT-toc.pdf Page: 2 Context: I'm unable to view images. Please provide the text that you would like to be formatted in Markdown, and I'll be happy to help! Image Analysis: Based on the aspects provided, here is a comprehensive examination of the attached visual content: **1. Localization and Attribution:** - The image occupies the entirety of the provided space. It can be numbered as Image 1. **2. Object Detection and Classification:** - **Image 1:** - **Objects Detected:** - A person. - A large screen/projector behind the person. - A clock on the wall. - Three smaller screens or panels below the large screen. - A speaker or microphone setup on the podium. - A laptop on the podium. - **Classification by Category:** - Person: Human. - Screen/Projector: Electronic device. - Clock: Timekeeping device. - Screens/Panels: Electronic display devices. - Speaker/Microphone: Audio equipment. - Laptop: Computing device. **3. Scene and Activity Analysis:** - **Image 1:** - **Scene Description:** - The scene takes place in what appears to be a conference room or lecture hall. - The person is standing at a podium, presumably giving a presentation. - **Activities Taking Place:** - The main actor, a person, is engaged in speaking or presenting information to an audience. - The person appears to be using a laptop, possibly to control a presentation on the large screen behind them. **4. Text Analysis:** - **Image 1:** - There is no visible text in the image based on the provided resolution and image quality. **8. Color Analysis:** - **Image 1:** - **Dominant Colors:** - Predominantly neutral colors like white and gray for the walls and background. - The podium and the person's attire include darker shades like black and gray. - The large screen in the background is turned off, showing a dark color. - **Impact on Perception:** - The use of neutral and dark colors keeps the focus on the person presenting. **9. Perspective and Composition:** - **Image 1:** - **Perspective:** - The image is taken from a viewpoint that is at the audience level, facing toward the presenter. - This perspective gives the sense of being part of the audience. - **Composition:** - The composition places the presenter at the center. - The large screen behind the presenter acts as the background, with additional visual elements like a clock and smaller screens filling the side spaces. **14. Ablaufprozesse (Process Flows):** - **Image 1:** - No specific process flows are depicted directly in the image. **15. Prozessbeschreibungen (Process Descriptions):** - **Image 1:** - The main process described here involves a presentation setup, with the presenter utilizing electronic equipment to communicate information. **16. Typen Bezeichnung (Type Designations):** - **Image 1:** - The types of equipment include Audio-Visual equipment (screen, laptop, microphones) and timekeeping devices (clock). **17. Trend and Interpretation:** - **Image 1:** - The trend seems to indicate a professional or educational setting where technology is integrated to facilitate presentations and lectures. **19. Tables:** - **Image 1:** - There are no tables in the image to analyze. The image provides a focused look at a presentation scene in a formal setting. The arrangement of elements and the color palette underscore the importance of the presenter and the information being shared. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 113 Context: # 5.2 Turing Machines ## Exercises for section 5.1 1. Prove that the following decision problem belongs to P: Given integers \( K \) and \( a_1, \ldots, a_n \). Is the median of the \( a_i \)'s smaller than \( K \)? 2. Prove that the following decision problem is in NP: Given an \( n \times n \) matrix \( A \) of integer entries, is \( \det A = 0 \)? 3. For which of the following problems can you prove membership in P? - (a) Given a graph \( G \). Does \( G \) contain a circuit of length 4? - (b) Given a graph \( G \). Is \( G \) bipartite? - (c) Given integers. Is there a subset of them whose sum is an even number? - (d) Given \( n \) integers. Is there a subset of them whose sum is divisible by 3? - (e) Given a graph \( G \). Does \( G \) contain an Euler circuit? 4. For which of the following problems can you prove membership in NP? - (a) Given a set of integers and another integer \( K \). Is there a subset of the given integers whose sum is \( K \)? - (b) Given a graph \( G \) and an integer \( K \). Does \( G \) contain a path of length \( \geq K \)? - (c) Given a set of \( K \) integers. Is it true that not all of them are prime? - (d) Given a set of \( K \) integers. Is it true that all of them are prime? ## 5.2 Turing Machines A Turing machine consists of: - A doubly infinite tape, that is marked off into squares that are numbered as shown in Fig. 5.2.1 below. Each square can contain a single character from the character set that the machine recognizes. For simplicity, we can assume that the character set consists of just three symbols: '0', '1', and ' ' (blank). - A tape head that is capable of either reading a single character from a square on the tape or writing a single character on a square, or moving its position relative to the tape by an increment of one square in either direction. - A finite list of states such that at every instant the machine is in exactly one of those states. The possible states of the machine are, first of all, the regular states \( q_1, \ldots, q_n \), and second, three special states: - \( q_0 \): the initial state - \( q_y \): the final state in a problem to which the answer is ‘Yes’ - \( q_n \): the final state in a problem to which the answer is ‘No’ - A program (or program module, if we think of it as a pluggable component) that directs the machine through the steps of a particular task. ``` ... ... --- --- --- -2 ─→ 0 1 2 3 ... ... ``` Fig. 5.2.1: A Turing machine tape Image Analysis: ### Localization and Attribution: - **Image 1**: The page consists of one primary image—a diagram located centrally within the text, designated as Image 1. ### Object Detection and Classification: - **Image 1**: - **Objects Detected**: The image contains multiple squares arranged in a sequential manner, each square containing symbols or being blank. There are also labels underneath the squares. - **Classification**: This image can be categorized as a diagram depicting a Turing machine tape. ### Scene and Activity Analysis: - **Image 1**: - **Scene Description**: The image depicts a Turing machine tape, an abstract computational device used in theoretical computer science to understand the limits of what can be computed. - **Main Actors and Actions**: The main elements include labeled squares with symbols which depict the action of a tape head reading or writing a symbol. ### Text Analysis: - **Image Throughout the Page**: - **Extracted Text**: Title "5.2 Turing Machines", text describing "A Turing Machine..." and Fig. 5.2.1 caption: "A Turing machine tape". - **Content Analysis**: The text in this section describes the concept of a Turing machine, including its components: an infinite tape, a tape head, and a finite set of states. The figure caption highlights that the diagram is an illustration of the tape component of a Turing machine. ### Diagram and Chart Analysis: - **Image 1**: - **Diagram Description**: Image 1 is a diagram (Fig. 5.2.1) showing a sequence of squares that mimic a Turing machine tape. The squares are labeled “...”, “-2”, “-1”, “0”, “1”, “2”, “...” from left to right. - **Axes and Legends**: There are no typical axes or legends as this is a conceptual diagram, not a graph. - **Key Insights**: This diagram visualizes the tape of the Turing machine, indicating positions and possible contents (blank or symbols ‘0’ and ‘1’). ### Contextual Significance: - **Text Context**: The image is a part of a theoretical explanation of Turing machines within the larger context of computational theory and problem-solving approaches related to NP-completeness. - **Contribute to Overall Message**: This visual aids in the comprehension of how a Turing machine operates by illustrating its main component, facilitating a better understanding of the theoretical concepts discussed in the text. ### Perspective and Composition: - **Perspective**: The diagram is presented from a straightforward, head-on perspective for clarity, emphasizing the linear, sequential nature of the Turing machine tape. - **Composition**: Elements are arranged in a left-to-right sequence which mirrors the way the tape would be read or written by the Turing machine. ### Process Descriptions: - **Process Shown**: The process of how a Turing machine reads and writes symbols on an infinite tape is described. When a symbol is read, the machine transitions to a new state and performs an action based on the symbol and state. ### Trend and Interpretation: - **Trends Identified**: The visual shows a repeating and linear pattern of squares which represents the infinite nature of the Turing machine tape. - **Interpretation**: It reinforces the idea that theoretical machines like the Turing machine operate in a deterministic and procedural manner, following specified rules. ### Tables: - **Page Content**: The page intersperses table-like structures with exercises listing problems related to computational complexity (sections 5.1-5.2). - **Description**: Though not traditional tables, these organized lists are used to systematically challenge understanding of P, NP, and related problems in computer science theory. The combined text and diagrams on this page offer a comprehensive tutorial on Turing machines and exercises to practice understanding computational complexity. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 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: 18 Context: # Chapter 1: Mathematical Preliminaries ## 1.3 Manipulations with Series In this section, we will look at operations with power series, including multiplying them and finding their sums in simple form. We begin with a little catalogue of some power series that are good to know. First we have the finite geometric series: $$ \frac{(1 - x^n)}{(1 - x)} = 1 + x + x^2 + \ldots + x^{n-1}. \tag{1.3.1} $$ This equation is valid certainly for all \( x \neq 1 \), and it remains true when \( x = 1 \) also if we take the limit indicated on the left side. Why is \( 1 - x^n \) true? Just multiply both sides by \( 1 - x \) to clear fractions. The result is: $$ 1 - x^n = \left( 1 + x + x^2 + \ldots + x^{n-1} \right)(1 - x) = \left( 1 + x + x^2 + \ldots + x^{n-1} \right) - \left( x + x^2 + \ldots + x^n \right) = 1 - x^n $$ and the proof is finished. Now try this one. What is the value of the sum $$ \sum_{j=0}^{\infty} 3^j? $$ Observe that we are looking at the right side of (1.3.1) with \( x = 3 \). Therefore the answer is \( \frac{3}{(1 - 3)} \). Try to get used to the idea that a series in powers of \( x \) becomes a number if \( x \) is replaced by a number, and if we know a formula for the sum of the series then we know the number that it becomes. Here are some more series to keep in your zoo. A parenthetical remark like \( |x| < 1 \) shows the set of values of \( x \) for which the series converges. $$ \sum_{k=0}^{\infty} x^k = \frac{1}{(1 - x)} \quad (|x| < 1) \tag{1.3.2} $$ $$ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \tag{1.3.3} $$ $$ \sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n + 1)!} \tag{1.3.4} $$ $$ \cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} \tag{1.3.5} $$ $$ \log \left( \frac{1}{(1 - x)} \right) = \sum_{j=1}^{\infty} \frac{x^j}{j} \quad (|x| < 1) \tag{1.3.6} $$ Can you find a simple form for the sum (the logarithms are 'natural')? $$ 1 + \log 2 + \frac{(\log 2)^2}{2!} + \frac{(\log 2)^3}{3!} + \ldots $$ Hint: Look at (1.3.3), and replace \( x \) by \( \log 2 \). Aside from merely substituting values of \( x \) into known series, there are many other ways of using known series to express sums in simple form. Let's think about the sum $$ 1 + 2 + 2 + 3 + 4 + 4 + 5 + 5 + 6 + \ldots + N2^{N-1}. \tag{1.3.7} $$ #################### 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: 8 Context: # Chapter 0: What This Book Is About but also, no one has proved the impossibility of doing so. It should be added that the entire area is vigorously being researched because of the attractiveness and the importance of the many unanswered questions that remain. Thus, even though we just don't know many things that we'd like to know in this field, it isn't for lack of trying! ## 0.3 Preview Chapter 1 contains some of the mathematical background that will be needed for our study of algorithms. It is not intended that reading this book or using it as a text in a course must necessarily begin with Chapter 1. It’s probably a better idea to plunge into Chapter 2 directly, and then when particular skills or concepts are needed, to read the relevant portions of Chapter 1. Otherwise the definitions and ideas that are in that chapter may seem to be unmotivated, when in fact motivation in great quantity resides in the later chapters of the book. Chapter 2 deals with recursive algorithms and the analysis of their complexities. Chapter 3 is about a problem that seems as though it might be hard, but turns out to be easy, namely the network flow problem. Thanks to quite recent research, there are fast algorithms for network flow problems, and they have many important applications. In Chapter 4 we study algorithms in one of the oldest branches of mathematics, the theory of numbers. Remarkably, the connections between this ancient subject and the most modern research in computer methods are very strong. In Chapter 5 we will see that there is a large family of problems, including a number of very important computational questions, that are bound together by a good deal of structural unity. We don’t know if they’re hard or easy. We do know that we haven’t found a fast way to do them yet, and most people suspect that they’re hard. We also know that if any one of these problems is hard, then they all are, and if any one of them is easy, then they all are. We hope that, having found out something about what people know and what people don’t know, the reader will have enjoyed the trip through this subject and may be interested in helping to find out a little more. #################### 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: 48 Context: #### *Prozessbeschreibungen (Process Descriptions):* - **Description:** Detailed explanation of how the graph vertices are merged to form a new graph, maintaining the properties discussed in the lemmas. ### Conclusion: The visual content effectively illustrates the theoretical concepts discussed in the text, aiding in the understanding of graph colorings and vertex collapsing in a mathematical context. The diagrams are clear, labeled appropriately, and positioned to complement the textual explanations of the lemmas. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 126 Context: - **Tables**: None included in the provided content. This detailed analysis covers multiple aspects to provide a comprehensive understanding of the attached visual content. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 38 Context: # Chapter 2: Recursive Algorithms ## Procedure `split(x, left, right, t)` ```markdown {chooses at random an entry T of the subarray [x[left], x[right]], and splits the subarray around T} {the output integer i is the position of T in the output array x[1..n] = T:} 1. L := a random integer in [left, right]; 2. swap(x[left], x[L]); 3. {now the splitter is first in the subarray} 4. T := x[left]; 5. i := left; 6. for j := left + 1 to right do begin 7. if x[j] < T then 8. begin 9. i := i + 1; 10. swap(x[i], x[j]); 11. end; 12. end; 13. swap(x[left], x[i]); 14. end. ``` ## Theorem 2.2.1 Procedure `split` correctly splits the array x around the chosen value T. ### Proof We claim that as the loop in lines 7 and 8 is repeatedly executed for j := left + 1 to right, the following three assertions will always be true just after each execution of lines 7, 8: 1. \( x[left] = T \) and 2. \( x[k] < T \) for all \( left < k < i \) and 3. \( x[j] \geq T \) for all \( i < j \leq right \) Fig. 2.2.1 illustrates the claim. ### Fig. 2.2.1: Conditions (a), (b), (c) To see this, observe first that (a), (b), (c) are surely true at the beginning, when j = left + 1. Next, if for some j they are true, then the execution of lines 7, 8 guarantee that they will be true for the next value of j. Now look at (a), (b), (c) when j = right. It tells us that just prior to the execution of line 9 the condition of the array will be: 1. \( x[left] = T \) and 2. \( x[k] < T \) for all \( left < k < i \) and 3. \( x[j] \geq T \) for all \( i < j \leq right \) When line 9 executes, the array will be in the correctly split condition. Now we can state a ‘final’ version of `qsort` (and therefore of Quicksort too). Image Analysis: ### Analysis of Visual Content #### **1. Localization and Attribution:** - **Image 1:** Top left, the procedure and theorem text block. - **Image 2:** Bottom, the diagram labeled Fig. 2.2.1. #### **2. Object Detection and Classification:** - **Image 1:** Includes text, blocks, and coding elements. - Objects identified: Text blocks, code lines, and theorem description. - Key features: Text describing algorithmic procedures and mathematical proofs. - **Image 2:** Line diagram illustrating a sorting algorithm. - Objects identified: Rectangles containing numerical or symbolic values. - Key features: Arrangement of elements to show conditions of an array during a sorting algorithm. #### **4. Text Analysis:** - **Image 1:** - **Text Detected:** - "procedure split(x, left, right, t)" - "chooses at random an entry T of the subarray" - Theorem 2.2.1, Proof explanations - **Content Analysis:** The text describes a functional procedure called `split`, a theorem (2.2.1) detailing its correctness, and a proof of the theorem. This forms part of an explanation of the `quicksort` algorithm. - **Image 2:** - **Text Detected:** - "≤T", "≤T", "≤T", " 1\) be a positive integer (the 'base'). Then every positive integer \(n\) can be written in one and only one way in the form \[ n = d_k b^k + d_{k-1} b^{k-1} + d_{k-2} b^{k-2} + \ldots \] if the digits \(d_0, d_1, \ldots\) lie in the range \(0 \leq d_i < b - 1\) for all \(i\). **Remark:** The theorem says, for instance, that in the base \(10\) we need the digits \(0, 1, 2, \ldots, 9\); in the base \(2\) we need only \(0\) and \(1\), in the base \(16\) we need sixteen digits, etc. **Proof of the theorem:** If \(b\) is fixed, the proof is by induction on \(n\), the number being represented. Clearly, the number \(1\) can be represented in one and only one way with the available digits (why?). Suppose, inductively, that every integer \(1, 2, \ldots, n - 1\) is uniquely representable. Now consider the integer \(n\). Define \(d = n \mod b\). Then \(d\) is one of the permissible digits. By induction, the number \(n' = (n - d) / b\) is uniquely representable, say \[ n' = d_k b^k + d_{k-1} b^{k-1} + d_{k-2} b^{k-2} + \ldots \] Then clearly, \[ n = d + n' b = d + d_k b^{k+1} + d_{k-1} b^{k} + d_{k-2} b^{k-1} + \ldots \] is a representation of \(n\) that uses only the allowed digits. Finally, suppose that \(n\) has some other representation in this form also. Then we would have \[ n = a_0 + a_1 b + a_2 b^{2} + \ldots = c_0 + c_1 b + c_2 b^{2} + \ldots \] Since \(a_0\) and \(c_0\) are both equal to \(n \mod b\), they are equal to each other. Hence the number \(n' = (n - a_0) / b\) has two different representations, which contradicts the inductive assumption, since we have assumed the truth of the result for all \(n < m\). The bases \(b\) that are the most widely used are, aside from \(2\) (binary system), \(8\) (octal system) and \(16\) (hexadecimal system). The binary system is extremely simple because it uses only two digits. This is very convenient if you’re a computer or a computer designer because the digits can be determined by some component being either ‘on’ (digit \(1\)) or ‘off’ (digit \(0\)). The binary digits of a number are called its bits or its bit string. Image Analysis: ### Analysis of the Attached Visual Content #### 1. Localization and Attribution - **Image Position**: The provided image is positioned on a single page with continuous textual content. #### 2. Object Detection and Classification - **Detected Objects**: - Textual content with mathematical expressions and explanations. - Equations and formulas. #### 3. Scene and Activity Analysis - **Scene Description**: The scene is a page from a mathematical textbook or an academic paper. - **Activities**: Mathematical explanation and theorem proof demonstration. #### 4. Text Analysis - **Extracted Text**: - Chapter heading: "Chapter 1: Mathematical Preliminaries" - Explanation on binary number representation. - Theorem discussion: "Theorem 1.2.1. Let b > 1 be a positive integer (the ‘base’). ..." - Mathematical proof and associated equations. - Remarks on different base representations and their implications. - **Text Content Analysis**: - This text is a part of a mathematical discussion about number representation in different bases. - It starts with an introduction to binary numbers and extends to a general theorem concerning the representation of numbers in any base greater than 1. - Theorem 1.2.1 is stated and proved utilizing induction. - Remarks provide practical insights into the utility of different bases, particularly the binary system. #### 9. Perspective and Composition - **Perspective**: The image is captured from a straight-on perspective, typical of scanned or digitized documents. - **Composition**: The page is well-structured with headings, paragraphs, and equations clearly demarcated and logically arranged. #### 11. Metadata Analysis - **Captured Metadata**: Not available in the provided visual content. ### Additional Aspects: #### Prozessbeschreibungen (Process Descriptions): - **Description**: - **Binary Representation**: Explanation of how the number 13 is represented in binary as (1101)_2. - **General Theorem**: Step-by-step proof demonstrates that any number can be uniquely represented in a given base. - **Inductive Proof**: The proof involves demonstrating that any integer can be represented using a set of permissible digits from 0 to b-1. ### Summary The document is a detailed excerpt from a mathematical text that focuses on number representation in various bases, specifically emphasizing binary, decimal, and hexadecimal systems. It provides theoretical explanations, theorem proofs, and practical remarks on the applications and representations of numbers in different bases, particularly within computer science and numerical analysis. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 47 Context: ### Conclusion: The visual content includes detailed diagrams and descriptive text related to graph theory, specifically planar graphs. The figures provide visual explanations, while the text offers theoretical context and practical insights into graph planarity and recursive algorithms. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 63 Context: ``` 2.7 A review =========== An outgrowth of the rapidity with which we can now multiply polynomials is a rethinking of the methods by which we do ultrahigh-precision arithmetic. How fast can we multiply two integers, each of which has ten million bits? By using ideas that developed directly (though not at all trivially) from the ones that we have been discussing, Schönhage and Strassen found the fastest known method for doing such large-scale multiplications of integers. The method relies heavily on the FFT, which may not be too surprising since an integer n is given in terms of its bits b₀, b₁, ..., b_{m} by the relation n = ∑_{i=0}^{m} b_i 2^i. (2.5) However the sum in (2.5) is seen at once to be the value of a certain polynomial at x = 2. Hence in asking for the bits of the product of two such integers we are asking for something very similar to the coefficients of the product of two polynomials, and indeed the fastest known algorithms for this problem depend upon the Fast Fourier Transform. Exercises for section 2.6 --------------------------- 1. Let ω be an nth root of unity, and let k be a fixed integer. Evaluate \[ 1 + \omega^k + \omega^{2k} + \ldots + \omega^{(n-1)k}. \] 2. Verify that the relations (2.6.3) and (2.6.4) indeed are inverses of each other. 3. Let f = ∑_{j=0}^{n-1} a_j ω^j. Show that \[ \frac{1}{n} \sum_{ω^{1}}^{n} |f(ω)|^2 = |a_0|^2 + \ldots + |a_{n-1}|^2. \] 4. The values of a certain cubic polynomial at 1, -1, -i are 1, 2, 3, 4, respectively. Find its value at 2. 5. Write a program that will do the FFT in the case where the number of data points is a power of 2. Organize your program so as to minimize additional array storage beyond the input and output arrays. 6. Prove that a polynomial of degree n is uniquely determined by its values at n + 1 distinct points. 2.7 A review ------------- Here is a quick review of the algorithms that we studied in this chapter. Sorting is an easy computational problem. The most obvious way to sort an array elements takes time θ(n log n). Finding a maximum independent set in a graph is a hard computational problem. The most obvious way to do it might take time θ(2^n) if the graph G has n vertices. We described a recursive method that runs in time O(1.39^n). The best known methods run in time O(2^{n/3}). Finding out if a graph is k-colorable is a hard computational problem. The most obvious way to do it takes time O(k^n), if G has n vertices. We described a recursive method that runs in time O(1.62^n) if G has n vertices and E edges. One recently developed method * runs in time O(1.3^n). We will see in section 5.7 that this problem can be done in an average time that is O(1) for fixed k. Multiplying two matrices is an easy computational problem. The most obvious way to do it takes time θ(n^3) if the matrices are n x n. We discussed a recursive method that runs in time O(n^{2.376}). A recent method ** runs in time O(n^{2.5}) for some γ < 2.5. References ---------- 1. E. Lawler, A note on the complexity of the chromatic number problem, Information Processing Letters 5 (1976), 66-7. 2. D. Coppersmith and S. Winograd, On the asymptotic complexity of matrix multiplication, SIAM J. Comp. 11 (1980), 472-492. ``` Image Analysis: ### Comprehensive Analysis #### Text Analysis: ##### Section Title: - **"2.7 A review"**: This section provides a summary of previous content. ##### Paragraph Content: - **First Paragraph**: Describes improvements in algorithms for multiplying polynomials and numbers, especially focusing on the use of Fast Fourier Transform (FFT). - **Key Features**: - The sum \( n = \sum_{i=0}^{k} b_i 2^i \) indicates a polynomial calculation. - Denotes relation with polynomial coefficients and the FFT. ##### Exercises for Section 2.6: - **Exercise 1**: Focuses on evaluating a sum involving roots of unity. - **Exercise 2**: Discusses verifying mathematical relations. - **Exercise 3**: Deals with a complex function involving roots of unity. - **Exercise 4**: Involves finding values of a cubic polynomial. - **Exercise 5**: Write a program for FFT. - **Exercise 6**: Proves uniqueness of polynomial degree determination. ##### 2.7 Review Content: - **Summary of Algorithms**: - Sorting algorithms. - Maximum independent set. - Coloring graphs. - Matrix multiplication. - **Complexity Notation Used**: - \(Θ(n \log n)\) for sorting. - \(Θ(2^n)\) for independent set problem. - \(O(2^n)\) for matrix multiplication if \(n \times n\). ##### References: - Cites significant works related to the chromatic number problem and matrix multiplication methodologies. #### Diagram and Chart Analysis: - There are no diagrams or charts in the content provided. #### Tables: - No tables are present in the content. #### Process Descriptions: - The algorithms are discussed from a high-level approach with steps for sorting, maximum independent set, coloring, and matrix multiplication. #### Color Analysis: - The page mainly features black text on a white background, which is typical for academic documents. No additional color analysis is necessary. #### Localization and Attribution: - This appears to be a single page from a chapter in an academic or technical book, focusing on a review section. ### Notes: - This analysis is based on the detection and extraction of text content in the image. Other aspects such as diagrams, charts, and metadata are not present and hence not analyzed. Overall, the content provides a review of complex computational problems, highlighting polynomial multiplication improvements, and placing a strong focus on FFT. Exercises reinforce the concepts discussed in the text passage. Lastly, references strengthen the academic rigor of the document, directing towards further reading on computational complexity. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 102 Context: # Chapter 4: Algorithms in the Theory of Numbers Fig. 4.8.1: Who knows what ## (B) How to send a message The sender takes the message \( P \), looks at the public keys \( E \) and \( n \), computes \( C = P^E \mod n \), and transmits \( C \) over the public airwaves. Note that the sender has no private codebook or anything secret other than the message itself. ## (C) How to decode a message The receiver receives \( C \) and computes \( C^D \mod n \). Observe, however, that \( (p-1)(q-1) \) is \( \phi(n) \), and so we have: \[ C^D \equiv P^{DE} \equiv P^{(1+k\phi(n))} \quad (t \text{ is some integer}) \] \[ \equiv P \quad (\text{mod } n) \] where the last equality is by Fermat’s theorem (4.5.1). The receiver has now recovered the original message \( P \). If the receiver suspects that the code has been broken, i.e., that the adversaries have discovered the primes \( p \) and \( q \), then the sender can change them without having to send any secret messages to anyone else. Only the public numbers \( n \) and \( E \) would change. The sender would not need to be informed of any other changes. Before proceeding, the reader is urged to construct a little scenario. Make up a short (very short!) message. Choose values for the other parameters that are needed to complete the picture. Send the message as the sender would, and decode it as the receiver would. Then try to intercept the message, as an eavesdropper would, and see what the difficulties are. ## (D) How to intercept the message An eavesdropper who receives the message \( C \) would be unable to decode it without (inventing some entirely new decoding scheme or) knowing the inverse \( D \) of \( E \) (mod \( (p-1)(q-1) \)). The eavesdropper, however, does not even know the modulus \( n = (p-1)(q-1) \) because \( p \) and \( q \) are unknown (only the receiver knows them), and knowing the product \( p \times q = n \) alone is insufficient. The eavesdropper is thereby compelled to derive a polynomial-time factoring algorithm for large integers. May success attend those efforts! The reader might well remind her that the receiver has a substantial computational problem in creating two large primes \( p \) and \( q \). To a certain extent this is so, but two factors make the task a good deal easier. First, \( p \) and \( q \) will need to have only half as many bits as \( n \) has, so the job is of smaller size. Second, there Image Analysis: ### Comprehensive Image Analysis #### 1. **Localization and Attribution:** - **Image 1:** - **Location:** Upper-middle section of the page. - **Content:** Diagram illustrating "Who knows what" in a cryptographic context. - **Number:** Image 1 #### 2. **Object Detection and Classification:** - **Image 1:** - **Objects Detected:** - Two cartoon heads labeled "Sender" and "Receiver." - Text bubbles indicating what each person knows ("n E" for Sender and "p, q, d, E⁻¹" for Receiver). - **Classification:** - Heads: Representations of human characters. - Text bubbles: Cryptographic keys and information. #### 3. **Scene and Activity Analysis:** - **Image 1:** - **Scene Description:** - A cartoon of two individuals, one labeled "Sender" and the other "Receiver," highlighting what cryptographic information each person holds. - Indicates the public and private keys held by the sender and receiver. - **Activities:** - The image is a visual aid to explain cryptographic key distribution in a communication system. #### 4. **Text Analysis:** - **Text in Image 1:** - **Extracted Text:** - "n E" (Public keys known to the sender) - "p, q, d, E⁻¹" (Keys known to the receiver including the private key components) - **Significance:** - Illustrates the concept of public-key cryptography, showing the distribution of public and private keys between the sender and receiver. - **Extracted Text from Surrounding Text:** - **How to send a message:** - Details the process of using public keys to encrypt a message without a private codebook. - **How to decode a message:** - Explains the decoding process using the receiver's private key. - **How to interpret the message:** - Discusses the computational complexity for eavesdroppers to decrypt the message without private keys. - **How to intercept the message:** - Highlights the challenges for eavesdroppers in breaking the encryption code. #### 9. **Perspective and Composition:** - **Image 1:** - **Perspective:** - Straight-on, illustrative perspective meant to clearly show concepts. - **Composition:** - Symmetrical arrangement with two characters side by side, each annotated with the keys they know. #### 11. **Metadata Analysis:** - **Metadata:** - Not available in the analyzed context; no additional stylistic or technical metadata provided. #### 13. **Graph Numbers:** - **Textual Context Analysis:** - Specific numeric data is not presented as a graph but rather through textual mathematical expressions relevant to public-key cryptography. - Example: \( C ≡ P^E \ (\text{mod} \ n) \) #### **Additional Aspects:** - **Prozessbeschreibungen (Process Descriptions):** - **How to send a message:** Encrypting a message using the receiver's public key. - **How to decode a message:** Decrypting a message using the receiver's private key. - **How to interpret the message:** Understanding the computational challenges for eavesdroppers. - **How to intercept the message:** Discussing the limitations for an eavesdropper trying to break the encryption. - **Typen Bezeichnung (Type Designations):** - Descriptions of cryptographic roles (Sender and Receiver) and key types (Public, Private, and their components). ### Summary: The document page helps convey complex cryptographic concepts through a combination of textual descriptions and informative diagrams. It explains the roles of public and private keys, the process of sending and decrypting messages, and the challenges faced by eavesdroppers. The visual elements support the textual content, ensuring a clearer understanding of cryptographic principles. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 23 Context: **Tables:** - No tables are present. This comprehensive analysis provides detailed insights into the instructional content depicted in the image, enhancing understanding of its academic focus on mathematical recurrence relations. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 99 Context: **Prozessbeschreibungen (Process Descriptions):** - The proofs describe iterative and logical processes related to mathematical properties and how they lead to conclusions about the group \(U_n\) and primality. **Typen Bezeichnung (Type Designations):** - Types and categories include: - **Groups:** \(U_n\) - **Subgroups:** \(C(x)\), \(B\), and \(B'\) - **Mathematical Functions:** Euler's totient function \(\phi\) - **Mathematical Concepts:** Order, cyclic groups, prime power **Trend and Interpretation:** - The lemmas and theorem develop a trend in understanding how the structure and properties of \(U_n\) can influence and be used in primality testing. Specifically, it interprets the order and subgroup generation properties to conclude about the composite or prime nature of the number 'n.' The content maintains a scientific and educational tone, focusing on theoretical aspects crucial in advanced mathematics and computer science, particularly in fields like cryptography. Note: The remaining aspects such as metadata analysis, color analysis, perspective and composition, contextual significance are not applicable for analysis as no related information is provided in the visual content and it is a black and white text-focused document. #################### File: An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf Page: 18 Context: ``` # Chapter 3: Distribution of Primes 1. \( r \) and interpreting \( \binom{2n}{n} \) as the number of ways of choosing \( r \) objects from \( 2n \), we conclude that the second expression is indeed smaller than the first. This contradiction proves the theorem when \( r > 5 \). The results 7, 29, 97, 389, and 1543 show that the theorem is also true for \( r \leq 5 \). The proof of Bertrand's Postulate by this method is left as an exercise. Bertrand's Postulate may be used to prove the following results. 1. \(\frac{1}{n} \sum_{p \leq n} \frac{1}{p} \text{ is never an integer.}\) 2. Every integer \( > 6 \) can be written as the sum of distinct primes. 3. Every prime \( p_n \) can be expressed in the form \[ p_n = 2 + 3 + 5 + \cdots + p_{n-1} \] with an error of most 1 (Sierk). 4. The equation \( \pi(n) = \psi(n) \) has exactly 8 solutions. About 1949 a sensation was created by the discovery by Erdős and Selberg of an elementary proof of the Prime Number Theorem. The main new result was the following estimate, proved in an elementary manner: \[ \sum_{p \leq x} \log p \sim 2x \log x + O(x). \] Although Selberg's inequality is still far from the Prime Number Theorem, the latter can be deduced from it in various ways without recourse to any further number theoretical results. Several proofs of this lemma have been given, perhaps the simplest being due to Tzwarzow and Szelk. Selberg's original proof depends on consideration of the functions \[ \lambda_n = \mu_n = \sum_{d \mid d} \mu(d) \frac{1}{d^2} \] and \[ T(r) = \sum_{n=1}^{r} \lambda^2. \] Some five years ago J. Landeck and L. Moser showed that one could prove Selberg's lemma in a completely elementary way, i.e., using properties of integers only. One of the main tools for doing this is the following rational analogue of the logarithmic function. Let \[ h(r) = \frac{1}{1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{r}}. \] We prove in an quite elementary way that \[ |h(e) - h(c)| < \frac{1}{k}. \] The results we have established are useful in the investigation of the magnitudes of the arithmetic functions \( \sigma(n) \), \( \phi(n) \) and \( g(n) \). Since these depend not only on the magnitude of \( n \) but also strongly on the arithmetic structure of \( n \), we cannot expect to approximate them by the elementary functions of analysis. Nevertheless, we shall will see that “on average” these functions have a rather simple behavior. Let \( f \) and \( g \) be functions for which \[ f(1) + f(2) + \cdots + f(n) + g(1) + g(2) + \cdots + g(n), \] we say that \( f \) and \( g \) have the same average order. We will see, for example, that the average order of \( \sigma(n) \) is long, that of \( \phi(n) \) is \(\frac{n}{\log n}\). Let us consider first a purely heuristic argument for obtaining the average value of \( a(n) \). The probability that \( r \mid n \) and \( r \leq 1 \) contributes \( \left( \frac{1}{k} \right)^r \) to \( a(n) \). Thus the expected value of \( a(n) \) is \[ \frac{1}{T(r)} \left( \sum_{k=1}^{1} \frac{1}{k} + 1 + \frac{1}{k} \right). \] For \( k = 0 \) this will be about \( \log n \). For \( n \geq 1 \) it will be about \( n^{k}(k + 1) \), e.g., for \( n = 1 \) it will be about \( n(k^2) = n^2 + \frac{1}{k^2} \). Before proceeding to the proof and refinement of some of these results we consider some applications of the inversion of order of summation in certain double sums. Let \( f \) be an arithmetic function and associate with it the function \[ F(n) = \sum_{d \mid n} f(d) \] and \( g(n) = \sum_{d \mid n} f(d). \] We will obtain two expressions for \[ F(n) = \sum_{n=1}^{r} f(d). \] ``` #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 20 Context: # Chapter 1: Mathematical Preliminaries Similarly, multiplying the \( n \)th coefficient of a power series by \( r^n \) will change the sum from \( f(x) \) to \((x-r)f(r)\), but that’s not all. What happens if we multiply the coefficient of \( x^n \) by, say, \( 3n^2 + 2n + 5 \)? If the sum previously was \( f(x) \), then it will be changed to \((3x^2 + 2x + 5)f(x)\). The sum \[ \sum_{j=0}^{\infty} (2j^2 + 5j) \] is therefore equal to \( (2\cdot\frac{d^2}{dx^2} + 5)\frac{1}{(1-x)} \), and after doing the differentiations we find the answer in the form \(\frac{1}{x^2 - 8x + 5}(1 - x^{-2})\). Here is the general rule: if \( P(x) \) is any polynomial then \[ \sum_{j=0}^{\infty} P(j)x^j = P\left(\frac{d}{dx}\right)\sum_{j=0}^{\infty} x^j. \] ## Exercises for section 1.3 1. Find simple, explicit formulas for the sums of each of the following series. - \( \sum_{j=1}^{\infty} \log(j)/j \) - \( \sum_{j=1}^{\infty} (2n + 7)/5^n \) - \( \sum_{j=0}^{\infty} j^2/(2^j) \) - \( \sum_{j=1}^{\infty} 1/(1 - x^j) \) - Explain why \( \sum_{n=0}^{\infty} (-1)^n x^{n+1}/(2n + 1) = 0 \). - (a) \( (1 + t^2)/t \) - (b) \( (3 + 2^2)/n^2 \) - (c) \( (1 + t^2)/(t - 1)^2 \) ## 4 Recurrence relations A recurrence relation is a formula that permits us to compute the members of a sequence once after another, starting with one or more given values. Here is a small example. Suppose we are to find an infinite sequence of numbers \( x_0, x_1, \ldots \) by means of: \[ x_{n+1} = Cx_n, \quad (n \geq 0; x_0 = 1). \] This relation tells us that \( x_1 = Cx_0 \), and \( x_2 = Cx_1 \), etc., and therefore \( x_n = C^n x_0 \). We say that the solution of the recurrence relation (e.g., ‘difference equation’) (1.41) is given by \( x_n = C^n \) for all \( n \geq 0 \). Equation (1.41) is a first-order recurrence relation because a new value of the sequence is computed from just one preceding value (i.e., \( x_{n+1} \) is obtained solely from \( x_n \) and does not involve \( x_{n-1} \) or any earlier values). Observe the format of the equation (1.41). The parenthetical remarks are essential. The first one \( x_2 > 0 \) tells us for what values of the recurrence formula is valid, and the second one ‘\( x_0 = 1 \)’ gives the starting value. If one of these is missing, the solution may not be uniquely determined. The recurrence relation \[ x_{n+1} = x_n + x_{n-1} \] needs two starting values in order to ‘get going’; but it is missing both of those starting values and the range of \( n \). Consequently, (1.42) (which is a second-order recurrence) does not uniquely determine the sequence. Image Analysis: ### Analysis of the Attached Visual Content #### Localization and Attribution: - **Page Layout:** - The visual content consists of a single page. - The page is divided into several sections with text and formulas. #### Text Analysis: - **Content:** - The text appears to be an excerpt from a mathematical textbook or academic paper, touching on mathematical preliminaries. - The main topics covered on the page include the manipulation of power series coefficients, exercises related to the section, and an introduction to recurrence relations. - **Sections and Content Description:** - **Title:** - "**Chapter 1: Mathematical Preliminaries**" - **Power Series Coefficient Manipulation:** - Theorem and explanation about how multiplying the \(n\)th coefficient of a power series by \(n^r\) alters the sum \[f(x)\]. - Specifically, multiplying the coefficient of \(x^n\) by \(3n^2 + 2n + 5\) transforms the sum to \[3\left( \frac{x^3}{(1 - x)^4} \right) + 2\left( \frac{x^2}{(1 - x)^3} \right) + 5 \left( \frac{x}{(1 - x)^2} \right).\] - Further simplification leads to the form \[(7x^2 - 8x + 5) / (1 - x)^3\]. - **General Rule:** - If \(P(x)\) is any polynomial, \[\sum_{j} P(j) a_j x^j = P \left(x \frac{d}{dx} \right) \sum_{j} a_j x^j.\] - **Exercises for Section 1.3:** - Explicit formulas for series sums. - Example exercises include: 1. \(\sum_{j \geq 0} \log^6 j / j!\) 2. \(\sum_{m \geq 0} (2m + 7)^5 / 5^m\) 3. etc. - **Recurrence Relations:** - Definition and examples of recurrence relations. - Introduction to both first-order and second-order recurrence relations. - General form of recurrence relations and necessary initial values for solutions. #### Diagram and Chart Analysis: - **Mathematical Formulas:** - Various mathematical formulas and series expansions are provided. - No graphical diagrams or charts are present. #### Contextual Significance: - **Relevance:** - The content serves as foundational material, likely intended for students or researchers in mathematics. - This page seems to be part of a larger chapter on mathematical preliminaries, contributing to a broader understanding of power series and recurrence relations. ### Conclusion: The examined visual content is a detailed segment from a mathematical textbook, focusing on the manipulation of power series coefficients, providing exercises for the reader, and introducing the concept of recurrence relations. The structured explanations and examples are designed to help develop a deeper understanding of these mathematical concepts. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 129 Context: # 5.7 Backtracking (II): Graph Coloring ## Lemma 5.7.1 Let \( s_1, \ldots, s_K \) be nonnegative numbers whose sum is \( L \). Then the sum of their squares is at least \( \frac{L^2}{K} \). **Proof:** We have \[ 0 \leq \sum_{i=1}^K s_i^2 = \sum_{i=1}^K \left( s_i - \frac{L}{K} \right)^2 + \frac{L^2}{K} \] \[ = \sum_{i=1}^K s_i^2 - \frac{L^2}{K^2} + \frac{L^2}{K} \] \[ = \sum_{i=1}^K s_i^2 - \frac{L^2}{K^2} + \frac{L^2}{K} \] \[ \Rightarrow \sum_{i=1}^K s_i^2 \geq \frac{L^2}{K}. \] The next lemma deals with a kind of inside-out chromatic polynomial question. Instead of asking “How many proper colorings can a given graph have?”, we ask “How many graphs can have a given proper coloring?” ## Lemma 5.7.2 Let \( C \) be one of the \( K^L \) possible ways to color in \( K \) colors a set of \( L \) abstract vertices \( 1, 2, \ldots, L \). Then the number of graphs \( G \) whose vertex set is that set of \( L \) colored vertices and for which \( C \) is a proper coloring of \( G \) is at most \( 2^{L(L - 1)/2} \). **Proof:** In the coloring \( C \), suppose \( s_1 \) vertices get color 1, \( \ldots, s_K \) get color \( K \), where, of course, \( s_1 + \ldots + s_K = L \). If a graph \( G \) is to admit \( C \) as a proper vertex coloring then its edges can be drawn only between vertices of different colors. The number of edges that \( G \) might have is therefore \[ s_1 s_2 + s_1 s_3 + \ldots + s_K s_{K-1} + \ldots + s_{K-1} s_K \] for which we have the following estimate: \[ \sum_{1 \leq j < k \leq K} s_j s_k \leq \frac{1}{2} \sum_{j=1}^K s_j^2 \] \[ = \frac{1}{2} \left( \sum_{j=1}^K s_j^2 - \sum_{j=1}^K s_j^2 \right) = \frac{1}{2} \sum_{j=1}^K s_j^2 \] \[ \Rightarrow \sum_{1 \leq j < k \leq K} s_j s_k \leq \frac{L^2}{2} - \frac{1}{2} \sum_{j=1}^K s_j^2 \quad \text{(by lemma 5.7.1)} \] The number of possible graphs \( G \) is therefore at most \( 2^{(L(L - 1)/2)} \). ## Lemma 5.7.3 The total number of proper colorings in \( K \) colors of all graphs of \( L \) vertices is at most \[ K \cdot L^{2^{(L-1)/K}}. \] **Proof:** We are counting the pairs \( (G, C) \), where the graph \( G \) has \( L \) vertices and \( C \) is a proper coloring of \( G \). If we keep \( C \) fixed and sum on \( G \), then by lemma 5.7.2 the sum is at most \( 2^{(L-1)/K} \). Since there are \( K \) such \( C \), the proof is finished. Now let's think about a backtrack search for a \( K \)-coloring of a graph. Begin by using color 1 on vertex 1. Then use color 2 on vertex 2 unless \( (1, 2) \) is an edge, in which case use color 2. As the coloring progresses through vertices \( 1, 2, \ldots, L \), we color each new vertex with the lowest available color number that does not cause a conflict with some vertex that has previously been colored. Image Analysis: ### Comprehensive Analysis of the Attached Visual Content #### 1. **Localization and Attribution** - The visual content consists of one page from a document. - The page includes multiple mathematical expressions, proofs, and text explanations. - There is no numbering of images as there is a single continuous page of text and equations. #### 2. **Object Detection and Classification** - **Text and equations**: The page is composed mainly of paragraphs containing text and mathematical expressions. - **Mathematical Symbols**: Various symbols including summation (∑), inequality signs (≤), and fractions are present. - **Rectangles with numbers**: There are black rectangles with white text inside, representing the end of lemmas and proofs. #### 3. **Scene and Activity Analysis** - The scene is academic and mathematical, with the content focused on graph coloring and related lemmas and proofs. - The activities involve the presentation and proof of lemmas relating to the sum of squares and proper graph coloring. #### 4. **Text Analysis** - **Lemma 5.7.1**: Discusses nonnegative numbers and the sum of their squares. - **Significance**: The lemma concludes that the sum of the squares of nonnegative numbers under certain conditions is at least \(L^2 / K\). - **Lemma 5.7.2**: Explores a coloring problem in graph theory, providing a bound on the number of possible graphs for a given proper coloring. - **Significance**: This is crucial for understanding the complexity and constraints in graph coloring. - **Lemma 5.7.3**: Calculates the total number of proper colorings in \(K\) colors of all graphs with \(L\) vertices. - **Significance**: Provides a formula (\( K^L \cdot 2^{(L^2 - L) \cdot (1 - 1/K) / 2} \)) for quantifying the proper colorings, advancing theoretical knowledge in graph theory. - **Proofs**: Detailed mathematical proofs are provided for each lemma, following a typical structure of definitions, logical deductions, and conclusions. #### 7. **Anomaly Detection** - There are no apparent anomalies or unusual elements in the visual content. All elements align with standard mathematical documentation practices. #### 8. **Color Analysis** - **Dominant Colors**: The document uses a standard black-and-white color scheme typical for academic papers. - **Black**: Used for all text and mathematical symbols, enhancing readability. - **White**: Background color, ensuring contrast and clarity. #### 9. **Perspective and Composition** - **Perspective**: The perspective is direct, as expected from a scanned or digitally created academic document. - **Composition**: The page is well-structured with clear sections for lemmas and proofs. Equations are centered and inline with text, following standard mathematical document formatting. #### 13. **Graph Numbers** - **Lemma 5.7.1**: - The sum of squares formula is \[ \sum_{i=1}^K s_i^2 = \frac{L^2}{K} \] - **Lemma 5.7.2**: - The number of graphs \(G\) possible is \[ 2^{(L^2 - L) \cdot (1 - 1/K) / 2} \] - **Lemma 5.7.3**: - The total number of proper colorings is \[ K^L \cdot 2^{(L^2 - L) \cdot (1 - 1/K) / 2} \] #### **Contextual Significance** - The page appears to be from a chapter on "Backtracking (II): Graph Coloring" in a mathematical or computer science text. - **Contribution**: This section contributes to the overall understanding of graph coloring by providing theoretical bounds and proofs that facilitate further study and algorithms development. Overall, the page is a detailed exploration of specific aspects of graph theory, primarily focused on proper coloring and combinatorial bounds, aimed at readers with a strong background in mathematics or computer science. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 22 Context: 5. **Contextual Significance:** - The provided content seems to be part of a mathematical textbook or an academic paper focused on recurrence relations and their solutions. - This section (from Chapter 1) introduces fundamental concepts and methods that are likely necessary for understanding more complex topics later in the document. - The example of the Fibonacci sequence serves to illustrate the practical application of the theoretical methods discussed. The provided analysis covers the text and context of the content in depth. Specific numerical and algebraic expressions have been noted to capture the mathematical rigor presented in the image. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 53 Context: We follow the method of section 1.4 on this first-order linear difference equation. Hence we make the change of variable g(n) = 7 ^n y(n) (n ≥ 0) and we find that y0 = 0 and for n ≥ 1, y_n - 7y_n–1 + 2 ^(4/n^ n )= 9. If we sum over n we obtain y_n = 9 ∑_(i=1)^n (4/7)^i ≤ 9 ∑_(i=1)^∞ (4/7)^n = 9/((1 – 4/7)) = 21/2. ``` - **Analysis of Content:** The text is an excerpt from a mathematical or computer science document discussing the Strassen method for fast matrix multiplication. It includes a pseudocode function `MatrProd` for multiplying matrices and explains the recursive nature of the algorithm. The complexity analysis includes counting the number of multiplications and additions/subtractions necessary for the algorithm. Detailed recursive calls and complexity calculations are provided, illustrating the efficiency improvement from O(N^3) to O(N^2.81) for matrix multiplication. #### 8. Color Analysis - **Color Composition:** - The page has a predominantly white background with black text, typical of a printed or digital document. - There are no other significant colors present, which focuses the reader's attention on the textual content. #### 9. Perspective and Composition - **Perspective:** - The image is a top-down view of a text document, likely appearing in a book or a digitally scanned paper. - **Composition:** - The text is arranged in typical paragraph format. - The pseudocode and equations are indented and formatted distinctly to differentiate them from the main body of text, aiding readability. #### 14. Trend and Interpretation - **Trend in Visual Content:** - The visual content suggests a trend toward presenting complex mathematical and algorithmic concepts in a step-by-step manner, providing both theoretical explanations and practical pseudocode. - **Interpretation:** - The use of pseudocode and complexity analysis serves to educate readers on efficient matrix multiplication methods, specifically emphasizing the Strassen algorithm's recursive approach and reduced computational complexity. #### Additional Observations - **Prozessbeschreibungen (Process Descriptions):** - The text describes the recursive process of Strassen's matrix multiplication algorithm in detail, including its base case, recursive case, and the operations needed to combine intermediate results. - **Typen Bezeichnung (Type Designations):** - The algorithm categorizes its steps into multiplication and addition/subtraction processes, describing the type of operations performed at each recursive level. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 62 Context: - **Tables:** - No tables are included in the content. ### Conclusion This page presents a comprehensive discussion and procedural guide on using recursive algorithms and FFT for multiplying polynomials. It is academically oriented and contributes to the broader field of computer science and mathematics by offering an efficient methodology for polynomial multiplication. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 3 Context: # Chapter 5: NP-completeness ## 5.1 Introduction ................................................ 104 ## 5.2 Turing machines ............................................. 109 ## 5.3 Cook's theorem .............................................. 112 ## 5.4 Some other NP-complete problems ............................. 116 ## 5.5 Half a loaf .................................................. 119 ## 5.6 Backtracking (I): independent sets ........................... 122 ## 5.7 Backtracking (II): graph coloring ............................ 124 ## 5.8 Approximate algorithms for hard problems ..................... 128 #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 7 Context: The problem is this. Let \(n\) be a given integer. We want to find out if \(n\) is prime. The method that we choose is the following. For each integer \(m = 2, 3, \ldots, \sqrt{n}\)| we ask if \(m\) divides (evenly into) \(n\). If all of the answers are ‘No,’ then we declare \( n \) to be a prime number, else it is composite. First, we look at the computational complexity of this algorithm. That means that we are going to find out how much work is involved in doing the test. For a given integer \( n \) the work that we have to do can be measured in units of divisions of a whole number by another whole number. In those units, we obviously will do about \(\sqrt{n}\) units of work. It seems as though this is a tractable problem, because, after all, \(\sqrt{n}\) is of polynomial growth in \( n \). For instance, we do less than \( n \) units of work, and that’s certainly a polynomial in \( n \), isn’t it? So, according to our definition of fast and slow algorithms, the distinction was made on the basis of polynomial vs. faster-than-polynomial growth of the work done with the problem size, and therefore this problem must be easy. Right? Well, no not really. References to the distinction between fast and slow methods will show that we have to measure the amount of work done as a function of the number of bits of input to the problem. In this example, \( n \) is not the number of bits of input. For instance, if \( n = 59 \), we don’t need 59 bits to describe it, but only 6. In general, the number of binary digits in the bit string of an integer \( n \) is close to \(\log_2 n\). So in the problem of this example, testing the primality of a given integer \( n \), the length of the input bit string \( B \) is about \(\log_2n\). Seen in this light, the calculation suddenly seems very long. A string consisting of mere \(\log_2 n\) 0’s and 1’s has caused our mighty computer to do about \(\sqrt{n}\) units of work. If we express the amount of work done as a function of \( B \), we find that the complexity of this calculation is approximately \(2^{B/2}\), and that grows much faster than any polynomial function of \( B \). Therefore, the method that we have just discussed for testing the primality of a given integer is slow. See Chapter 4 for further discussions of this problem. At the present time no one has found a fast way to test for primality, nor has anyone proved that there isn’t a fast way. Primality testing belongs to the (well-populated) class of seemingly, but not provably, intractable problems. In this book we will deal with some easy problems and some seemingly hard ones. It’s the ‘seemingly’ that makes things very interesting. These are problems for which no one has found a fast computer algorithm, --- ### Scene and Activity Analysis: #### Described Scene and Activities: - The scene described in the image is a text-heavy document discussing computational problems, particularly focusing on the complexity and computational time of algorithms. - There are discussions of worst-case bounds, average case bounds, and the difference between fast and slow algorithms, followed by an example problem regarding the determination of whether a given integer is prime. #### Main Actors: - The main actors in the scene are algorithms and computational problems. - The text describes the actions of measuring computational complexity, evaluating polynomial vs. non-polynomial time, and testing algorithm efficiency. ### Contextual Significance: #### Contribution to Overall Message: - This text excerpt appears to be from a textbook or academic paper focusing on computational complexity, likely aimed at explaining the difference between easy and hard problems in computer science. - The example provided (primality testing) is used to illustrate these concepts and highlight the challenges associated with algorithm efficiency. ### Perspective and Composition: #### Perspective: - The perspective is that of a reader or student studying the material presented on the page. #### Composition: - The text is organized in a traditional scholarly format, with clear headings, paragraphs, and a structured presentation of concepts and examples. ### Anomaly Detection: - No anomalies were detected in the image; the text appears to be consistent with a standard academic document discussing computational complexity. ### Color Analysis: #### Color Composition: - The image consists entirely of black text on a white background. #### Dominant Colors: - The dominant colors are black and white. ### Graph and Trend Analysis: - There are no graphs present in this image. ### Diagram and Chart Analysis: - There are no diagrams or charts present in this image. ### Metadata Analysis: - There is no visible metadata within the image. ### Additional Aspects: #### Prozessbeschreibungen (Process Descriptions): - The described process is related to evaluating the computational complexity of algorithms, specifically mentioning worst-case and average-case scenarios and providing an example with testing for primality. #### Typen Bezeichnung (Type Designations): - Type designations include polynomial-time algorithms and non-polynomial-time algorithms, as well as tractable and intractable problems. ### Conclusion: This image contains a detailed section of text from what appears to be a textbook or academic paper on computational complexity, focusing on the difference between easy and hard computational problems, with a specific example relating to primality testing. The text touches upon polynomial-time guarantees, worst-case and average-case bounds, and the conceptual distinction between fast and slow algorithms. #################### File: Algebraic%20Topology%20AT-toc.pdf Page: 6 Context: # Preface This book was written to be a readable introduction to algebraic topology with rather broad coverage of the subject. The viewpoint is quite classical in spirit, and stays well within the confines of pure algebraic topology. In a sense, the book could have been written thirty or forty years ago since virtually everything in it is at least that old. However, the passage of the intervening years has helped clarify what are the most important results and techniques. For example, CW complexes have proved over time to be the most natural class of spaces for algebraic topology, so they are emphasized here much more than in the books of an earlier generation. This emphasis also illustrates the book's general slant towards geometric, rather than algebraic, aspects of the subject. The geometry of algebraic topology is so pretty, it would seem a pity to slight it and to miss all the intuition it provides. At the elementary level, algebraic topology separates naturally into the two broad channels of homology and homotopy. This material is here divided into four chapters, roughly according to increasing sophistication, with homotopy split between Chapters 1 and 4, and homology and its mirror variant cohomology in Chapters 2 and 3. These four chapters do not have to be read in this order, however. One could begin with homology and perhaps continue with cohomology before turning to homotopy. In the other direction, one could postpone homology and cohomology until after parts of Chapter 4. If this latter strategy is pushed to its natural limit, homology and cohomology can be developed just as branches of homotopy theory. Appealing as this approach is from a strictly logical point of view, it places more demands on the reader, and since readability is one of the first priorities of the book, this homotopic interpretation of homology and cohomology is described only after the latter theories have been developed independently of homotopy theory. Preceding the four main chapters there is a preliminary Chapter 0 introducing some of the basic geometric concepts and constructions that play a central role in both the homological and homotopical sides of the subject. This can either be read before the other chapters or skipped and referred back to later for specific topics as they become needed in the subsequent chapters. Each of the four main chapters concludes with a selection of additional topics that the reader can sample at will, independent of the basic core of the book contained in the earlier parts of the chapters. Many of these extra topics are in fact rather important in the overall scheme of algebraic topology, though they might not fit into the time #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 25 Context: #### 4. Text Analysis: 1. **Extracted Text:** ``` 8. Write out a complete proof of theorem 1.4.1. 9. Show by an example that the conclusion of theorem 1.4.1 may be false if the phrase ‘for every fixed ϵ0 > 0...’ were replaced by ‘for every fixed ϵ > 0 ...’. 10. In theorem 1.4.1 we find the phrase: ‘…the positive real root of …’. Prove that this phrase is justified, in that the equation shown always has exactly one positive real root. Exactly what special properties of that equation did you use in your proof? 1.5 Counting For a given positive integer n, consider the set {1,2,...,n}. We will denote this set by the symbol [n], and we want to discuss the number of subsets of various kinds that it has. Here is a list of all of the subsets of {2}, {1}, {2}, {1, 2}. There are 4 of them. We claim that the set [n] has exactly 2^n subsets. To see why, notice that we can construct the subsets of [n] in the following way. Either choose, or don’t choose, the element ‘1’; then either choose, or don’t choose, the element ‘2’; etc., finally choosing, or not choosing, the element ‘n’. Each of the n choices that you encountered could have been made in either 2 of ways. The totality of n choices, therefore, might have been made in 2^n different ways, so that is the number of subsets that a set of n objects has. Next, suppose we have n distinct objects, and we want to arrange them in a sequence. In how many ways can we do that? For the first object in our sequence we may choose any one of the n objects. The second element of the sequence can be any of the remaining n − 1 objects, so there are n(n − 1) possible ways to make these first two decisions. Then there are n − 2 choices for the third element, and so we have n(n − 1)(n − 2) ways to arrange the first three elements of the sequence. It is no doubt clear now that there are exactly n(n − 1)(n − 2)∙∙∙3∙2∙1 = n! ways to form the whole sequence. Of the n subsets of [n], how many have exactly k objects in them? The number of elements in a set is called its cardinality. The cardinality of a set S is denoted by |S|, so, for example, |{6}| = 6. A set whose cardinality is k is called a ‘k-set’, and a subset of cardinality k is, naturally enough, a ‘k-subset’. The question is, for how many subsets S of [n] is it true that |S| = k? We can construct k-subsets S of [n] (written S ⸦ [n] ∖) as follows. Choose an element a_1 (in possible choices). Of the remaining n − 1 elements, choose one (in n − 1 possible choices), etc., until a sequence of k different elements have been chosen. Obviously there were n(n − 1)(n − 2) ⸦⸦⸦ (n − k + 1) ways in which we might have chosen that sequence, so the number of ways to choose an (ordered) sequence of k elements from [n] is n(n − 1)(n − 2) ∙∙∙ (n − k + 1) = n!/(n − k)! But there are more sequences of k elements than there are k-subsets, because any particular k-subset S will correspond to k! different ordered sequences, namely all possible rearrangements of the elements of that subset. Hence the number of k-subsets of [n] is equal to the number of k-sequences divided by k!. In other words, there are exactly n!/[k!(n − k)!] k-subsets of a set of n objects. The quantities n!/[k!(n − k)!] are the famous binomial coefficients, and they are denoted by ( ) = n!/[k!(n − k)!] (n ≥ 0; 0 ≤ k ≤ n). Some of their special values are (n) = 1 ( ∀n ≥ 0); (n) = n ( ∀n ≥ 0); (k) = n(n − 1)/2 ( ∀n ≥ 0); (n) = 1 ( ∀n ≥ 0). ( 0 ) ( 1 ) ( 2 ) ( n ) It is convenient to define (2) to be 0 if k < 0 or if k > n. We can summarize the developments so far with ``` 2. **Content Analysis:** - **Mathematical Tasks:** The text includes exercises (tasks 8, 9, and 10) asking the reader to provide mathematical proofs and to critique theorem revisions. - **Main Text:** The main body of text discusses mathematical counting principles, such as the number of subsets of a set, permutations, and binomial coefficients. - **Significance:** This text is instructional, aiming to teach the concepts of combinatorics, specifically counting subsets and calculating binomial coefficients. #### 6. Product Analysis: - **Products Depicted:** - Not applicable as there are no products shown in the image. #### 8. Color Analysis: - **Color Composition:** - The image primarily consists of black text on a white background. - The monochrome effect aids in readability and focuses the reader's attention on the content without distraction. #### 9. Perspective and Composition: - **Perspective:** - The image is presented from a straight, eye-level perspective, typical for readable documents. - **Composition:** - The content is organized into numbered sections and paragraphs, which helps in structuring the reading flow. - Mathematical formulas are interspersed with explanatory text, making the complex content manageable. #### 10. Contextual Significance: - **Contribution to Overall Message:** - The image, being a page from an academic document or textbook, contributes to teaching mathematical concepts on counting subsets and permutations. - It is part of a larger text aimed at undergraduate or graduate students studying mathematics, particularly combinatorics and set theory. ### Conclusion: The analysis reveals that the visual content is an educational text aimed at explaining combinatorial mathematics. It provides exercises and thorough explanations of concepts such as subsets, permutations, and binomial coefficients. The monochromatic color scheme is typical for academic texts, ensuring clarity and focus on the content. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 9 Context: # Chapter 1: Mathematical Preliminaries ## 1.1 Orders of Magnitude In this section, we're going to discuss the rates of growth of different functions and to introduce the five symbols of asymptotics that are used to describe those rates of growth. In the context of algorithms, the reason for this discussion is that we need a good language for the purpose of comparing the speeds with which different algorithms do the same job, or the amounts of memory that they use, or whatever other measure of the complexity of the algorithm we happen to be using. Suppose we have a method of inverting square nonsingular matrices. How might we measure its speed? Most commonly, we would say something like: "If the matrix is n × n, then the method will run in time 16.8n³." Then we would know that if a 100 × 100 matrix can be inverted, with this method, in 1 minute of computer time, then a 200 × 200 matrix would require 2³ = 8 times as long, or about 8 minutes. The constant 16.8 wasn't used at all in this example; only the fact that the labor grows as the third power of the matrix size was relevant. Hence, we need a language that will allow us to say that the computing time, as a function of n, grows "in the order of n³", or not as fast as n², or at least as fast as n×log n, etc. The key symbols that are used in the language of comparing the rates of growth of functions are the following five: - `o` (read as "little o" of) - `O` (read as "big O" of) - `Θ` (read as "theta of") - `ω` (read as "little omega of") - `Ω` (read as "big omega of") Now let's explain what each of them means. Let \( f(n) \) and \( g(n) \) be two functions of \( n \). Each of the five symbols above is intended to compare the rapidity of growth of \( f \) and \( g \). If we say that \( f(n) = o(g(n)) \), then informally we are saying that \( f \) grows more slowly than \( g \) does when \( n \) is very large. Formally, we state the following: **Definition.** We say that \( f(n) = o(g(n)) \) if \( \lim_{n \to \infty} f(n)/g(n) \) exists and is equal to 0. Here are some examples: 1. \( x^2 = o(x^3) \) 2. \( \sin x = o(x) \) 3. \( 1.4\log n = o(n/2 + 7 \cos x) \) 4. \( 1/n = o(1) \) 5. \( 23\log^2 n = o(n^2) \) We can see from these few examples that sometimes it might be easy to prove that a relationship is true and sometimes it might be rather difficult. Example (4), for instance, requires the use of L'Hôpital's rule. If we leave two computer programs, and if one of them inverts a n × n matrices in time \( c n^3 \) and the other does so in time \( o(n^3) \), then we know that for all sufficiently large values of \( n \), the performance guarantee of the second program will be superior to that of the first program. Of course, the first program might run faster on small matrices, say up to size 10,000 x 10,000. If a certain program runs in time \( n^{2.8} \) and someone were to produce another program for the same problem that runs in \( n^{2} \log n \), then that second program would be an improvement, at least in the theoretical sense. The reason for the "theoretical" qualification, once more, is that the second program would be known to be superior only if \( n \) were sufficiently large. The second symbol of the asymptotics vocabulary is the `O`. When we say that \( f(n) = O(g(n)) \), we mean, informally, that \( f \) certainly doesn't grow at a faster rate than \( g \). It might grow at the same rate or it might grow more slowly; both are possibilities that the "O" permits. Formally, we have the set: **Definition.** We say that \( f(n) = O(g(n)) \) if there exists \( c > 0 \) such that \( |f(x)| \leq c|g(x)| \) for \( x \) large enough. The qualifier \( n \to \infty \) will usually be omitted, since it will be understood that we will most often be interested in large values of the variables that are involved. For example, it is certainly true that \( n = O(n) \), but even more can be said, namely that \( n = O(1) \). Also \( 2x^2 + 7x = O(x^2) \) and \( 1/(1 + x^2) = O(1) \). Now we can sharpen the last example by saying that \( 1/(1 + x^2) = o(1) \). This is Image Analysis: ### Comprehensive Examination of the Visual Content #### Localization and Attribution - **Image 1**: There is only one image on the page. #### Text Analysis - **Image 1**: - **Heading**: - "Chapter 1: Mathematical Preliminaries" - "1.1 Orders of magnitude" - **Main Body**: - The text discusses the rates of growth of different functions and the asymptotic notations used to describe these rates. It focuses on the context of algorithms and their complexity. - Introduces the commonly used symbols of asymptotics: - 'ω' (read 'is little oh of') - 'O' (read 'is big oh of') - 'θ' (read 'is theta of') - '~' (read 'is asymptotically equal to') - 'Ω' (read 'is omega of') - Defines the formal mathematical notation and theorems for asymptotic behavior: - If there exist two functions \( f(x) \) and \( g(x) \), then \( f(x) = o(g(x)) \) means that \( \lim_{x \to \infty} \frac{f(x)}{g(x)} = 0 \). - Examples and explanations: - \( x^2 = o(x^3) \) - \( sin x = o(x) \) - \( 14.7ln_{10}x = o(x/2 + 7 cos x) \) - \( 1/x = θ(1) \) - \( 2^3logx = θ(log x) \) - Mention of error rates, comparing two algorithms' performance, one with order \( n^3 \) and another optimized to \( n^2 log n \). - Qualifiers and sharper classifications in complex cases beyond just 'big O' notation. #### Scene and Activity Analysis - **Image 1**: - The image contains text from an academic book or lecture notes on mathematical preliminaries, particularly focused on orders of magnitude and asymptotic analysis. There are no interactive activities; the image is purely educational. #### Contextual Significance - **Image 1**: - This image appears to be from an educational or reference book in mathematics or computer science, especially useful for students or professionals in understanding algorithm complexities. It contributes significantly by laying the groundwork for more complex discussions on algorithm analysis. The content is foundational for those learning about computational efficiency and problem-solving within computer science. #### Perspective and Composition - **Image 1**: - The image is a full-page scan from a book, with a top-down perspective showing clear text layout. The heading and subheadings are bolded for emphasis, and examples within the text are indented and numbered for clarity. No objects, diagrams, or charts are depicted in the image presented, focusing solely on textual content describing mathematical concepts. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 94 Context: #### Additional Aspects: - **Ablaufprozesse (Process Flows):** - **Addition Process:** Describes how to add ordered pairs in \( Z_6 \). - **Multiplication Process:** Describes how to multiply ordered pairs in \( Z_6 \). - **Prozessbeschreibungen (Process Descriptions):** - Detailed steps on arithmetic operations in modular arithmetic contexts. - **Typen Bezeichnung (Type Designations):** - Identification and classification of mathematical objects like primitive roots, cyclic groups, and \( Z_n \) sets. ### Overall Observations: This page from the document aims to educate readers on specific number theory topics, explaining the theoretical foundations and providing practical arithmetic examples. The content is structured in a logical and sequential manner to build comprehension, with emphasis on key theorems and their implications in the field of number theory. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 88 Context: # Chapter 4: Algorithms in the Theory of Numbers ## Theorem 4.2.1 (A worst-case complexity bound for the Euclidean algorithm) Given two positive integers \(a, b\), the Euclidean algorithm will find their greatest common divisor after a cost of at most \(2 \log_2 M + 1\) integer divisions, where \(M = \max(a, b)\). Before we prove the theorem, let’s return to the example \((a, b) = (13, 21)\) of the display (4.2.2). In that case \(M = 21\) and \(20 \log_2 M + 1 = 9.78\). The theorem asserts that the g.c.d. will be found after about 9 operations. In fact, it was found after 7 operations in that case. ### Proof of theorem: Suppose first that \(a \geq b\). The algorithm generates a sequence \(a_0, a_1, \ldots\) where \(a_0 = a\), \(a_1 = b\), and \(a_{j+1} = a_j \mod a_{j-1}\) \((j \geq 1)\). By lemma 4.2.1, \[ a_{j+1} \leq \frac{a_{j-1} - 1}{2} \leq \frac{a_j}{2}. \] Then, by induction on \(j\), it follows that \[ a_{2j} \leq \frac{4n}{2^j} \quad (j \geq 0) \] \[ a_{2j+1} \leq \frac{a_j}{2} \quad (j \geq 0) \] and so, \[ a_j \leq 2^{-|r/2|}M \quad (r = 0, 1, 2, \ldots). \] Obviously, the algorithm has terminated if \(a < 1\), and this will have happened when \(r\) is large enough so that \(2^{-|r/2|}M < 1\), i.e., if \(r > 2 \log_2 M\). If \(r < 4\) then after 1 operation we will be in the case \(a < b\) that we have just discussed, and the proof is complete. The upper bound in the statement of theorem 4.2.1 can be visualized as follows. The number \(log_2 M\) is almost exactly the number of bits in the binary representation of \(M\) (what is ‘exactly’ that number of bits?). Theorem 4.2.1 therefore asserts that we can find the g.c.d. of two integers in a number of operations that is at most a linear function of the number of bits that it takes to represent the two numbers. In brief, we might say that \( \text{Time} = O(\text{bits})\); in the case of Euclid’s algorithm. ## Exercises for section 4.2 1. Write a nonrecursive program, in Basic or Fortran, for the g.c.d. Write a recursive program, in Pascal or a recursive language of your choice, for the g.c.d. 2. Choose 1000 pairs of integers \((n, m)\), at random between 1 and 1000. For each pair, compute the g.c.d. using a recursive program and a nonrecursive program. (a) Compare the execution times of the two programs. (b) There is a theorem to the effect that the probability that two random integers \(\geq 1\) is \((6/\pi^2)\). What, precisely, do you think that this theorem means by 'the probability that two'? What percentage of the 1000 pairs that you chose had g.c.d. = 1? Compare your observed percentage with \(100 - (6/\pi^2)\). 3. Find out when Euclid lived, and with exactly what words he described his algorithm. 4. Write a program that will light up a pixel in row \(r\) and column \(c\) of your CRT display if and only if \(gcd(n, m) = 1\). Run the program with enough values of \(n\) and \(m\) to fill your screen. If you see any interesting visual patterns, try to explain them mathematically. 5. Show that if \(m\) and \(n\) have a total of \(b\) bits, then Euclid's algorithm will not need more than \(2B + 3\) operations before reaching termination. 84 Image Analysis: 1. **Localization and Attribution:** - This document is a single-page text. - The text is divided into a theorem, its proof, and exercises related to Euclid's algorithm. 2. **Text Analysis:** - **Title and Section:** "Chapter 4: Algorithms in the Theory of Numbers" - **Theorem 4.2.1:** Discusses the worst-case complexity bound for the Euclidean algorithm, defining the number of steps needed to find the greatest common divisor (g.c.d) of two integers \( a \) and \( b \). - **Proof of Theorem 4.2.1:** Provides a mathematical proof, including lemmas and inductive reasoning. - **Exercises for section 4.2**: Lists exercises intended to deepen understanding of the Euclidean algorithm through different programming languages, mathematical proofs, and historical context. 3. **Diagram and Chart Analysis:** - No diagrams or charts are presented in this text. 4. **Anomaly Detection:** - No anomalies or unusual elements are identified in this text. 5. **Color Analysis:** - The document is in black and white, typical for mathematical proofs and educational content. The absence of color emphasizes the text. 6. **Perspective and Composition:** - Perspective is a standard viewer's perspective of reading a document. - The composition is well-organized with clear sections for the theorem, proof, and exercises. 7. **Process Descriptions:** - Euclidean Algorithm: The steps for finding the g.c.d. are systematically proved, showcasing the sequence and induction to establish the theorem's validity. - Steps involve iterative division and comparison to reduce the problem space until the g.c.d is found. 8. **Typen Bezeichnung (Type Designations):** - Euclidean algorithm – a specific type of algorithm for computing the greatest common divisor. 9. **Trend and Interpretation:** - The exercises suggest a trend towards understanding the efficiency and implementation of the Euclidean algorithm in both recursive and non-recursive programming methods. There is also an emphasis on exploring patterns and probabilities related to the algorithm. 10. **Tables:** - No tables are included in the text. **Summary:** This document is a mathematical exposition on the efficiency of the Euclidean algorithm, complete with a theorem, its detailed proof, and practical exercises for enhanced understanding. The text is methodically structured, adhering to educative norms tailored to students or professionals in the field of number theory and algorithm design. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 32 Context: # Chapter 1: Mathematical Preliminaries It is not hard to prove that the following are equivalent descriptions of a tree. 1. A tree is a graph that is connected and has no cycles. 2. A tree is a graph \( G \) that is connected and for which \( |E(G)| = |V(G)| - 1 \). 3. A tree is a graph \( G \) with the property that between every pair of distinct vertices there is a unique path. If \( G \) is a graph and \( S \subseteq V(G) \), then \( S \) is an independent set of vertices of \( G \) if no two of the vertices in \( S \) are adjacent in \( G \). An independent set \( S \) is maximal if it is not a proper subset of another independent set of vertices of \( G \). Dually, if a vertex subset \( S \) induces a complete graph, then we speak of a complete subgraph of \( G \). A maximal complete subgraph of \( G \) is called a clique. A graph might be labeled or unlabelled. The vertices of a labeled graph are numbered \( 1, 2, \ldots, n \). One difference that this makes is that there are a lot more labeled graphs than there are unlabelled graphs. There are, for example, 3 labeled graphs that have 3 vertices and 1 edge. They are shown in Fig. 1.6.7. ![Fig. 1.6.7: Three labeled graphs...](image_link_here) There is, however, only 1 unlabelled graph that has 3 vertices and 1 edge, as shown in Fig. 1.6.8. ![Fig. 1.6.8: ... but only one unlabelled graph](image_link_here) Most counting problems on graphs are much easier for labeled than for unlabelled graphs. Consider the following question: how many graphs are there that have exactly \( n \) vertices? Suppose first that we mean labeled graphs. A graph of \( n \) vertices has a maximum of \( \binom{n}{2} \) edges. To construct a graph we would decide which of these possible edges would be used. We can make each of these \( \binom{n}{2} \) decisions independently, and for every way of deciding where to put the edges we would get a different graph. Therefore the number of labeled graphs of \( n \) vertices is \( 2^{\binom{n}{2}} \). If we were to ask the corresponding question for unlabelled graphs we would find it to be very hard. The answer is known, but the derivation involves Burnside’s lemma about the action of a group on a set, and some fairly delicate counting arguments. We will state the approximate answer to this question, which is easy to write out, rather than the exact answer, which is not. If \( g_n \) is the number of unlabelled graphs on \( n \) vertices then \[ g_n \sim 2^{n^2/n}. \] ## Exercises for Section 1.6 1. Show that a tree is a bipartite graph. 2. Find the chromatic number of the n-cycle. 3. Describe how you would find out, on a computer, if a given graph \( G \) is bipartite. 4. Given a positive integer \( K \). Find two different graphs each of whose chromatic numbers is \( K \). 5. Exactly how many labeled graphs of \( n \) vertices and \( E \) edges are there? 6. How many labeled graphs of \( n \) vertices do vertices \( \{1, 2, 3\} \) form an independent set? 7. True or false: A Hamilton circuit is an induced cycle in a graph. 8. Which graph of \( n \) vertices has the largest number of independent sets? How many does it have? 9. Draw all of the connected, unlabelled graphs of 4 vertices. 10. Let \( G \) be a bipartite graph that has \( k \) connected components. Show that there are exactly \( 2^k \) ways to properly color the vertices of \( G \) in 2 colors. 11. Find a graph \( G \) of \( n \) vertices, other than the complete graph, whose chromatic number is equal to 1 plus the maximum degree of any vertex of \( G \). 28 Image Analysis: ### 1. Localization and Attribution - **Image 1:** Located at the middle of the page, directly under the text explaining labeled graphs. - **Image 2:** Located directly beneath Image 1, depicting an unlabeled graph. ### 2. Object Detection and Classification - **Image 1 (Three Labeled Graphs):** - Objects: Three labeled graphs, each having 3 vertices and 1 edge. - Classification: Simple graphs. - Key Features: - Each graph consists of three labeled nodes connected by one edge. - The positioning and connections of the nodes vary between each graph. - **Image 2 (One Unlabeled Graph):** - Object: One unlabeled graph. - Classification: Simple graph. - Key Features: - Consists of three nodes (vertices), but no labels or edges connecting them. ### 3. Scene and Activity Analysis - **Image 1 (Three Labeled Graphs):** - Scene: Depicts various labeled graph configurations. - Activities: Presentation of different ways to construct labeled graphs with 3 vertices and 1 edge. - **Image 2 (One Unlabeled Graph):** - Scene: Displays a single unlabeled graph. - Activities: Highlights the concept of an unlabeled graph with 3 vertices and 1 edge. ### 4. Text Analysis - **Image Captions:** - Image 1 caption: "Fig. 1.6.7: Three labeled graphs..." - Significance: Demonstrates different labeled graph configurations. - Image 2 caption: "Fig 1.6.8: ... but only one unlabeled graph" - Significance: Illustrates that there is only one distinct configuration for an unlabeled graph with 3 vertices and 1 edge. - **Main Text:** - Discusses the equivalence of certain graph properties and the complexity of counting graphs when labeled versus unlabeled. - Key excerpts discuss independent sets, cliques, and counting problems related to graph labeling. ### 10. Contextual Significance - **Image 1 and 2:** - Context within the Document: The images contribute to the section on mathematical preliminaries in graph theory, providing visual examples to elucidate theoretical concepts discussed in the text. - The images align with the text’s explanation of labeled and unlabeled graphs, clarifying distinctions in graph representation and structural diversity. ### 11. Metadata Analysis - Not applicable as there is no visible metadata provided in the image. ### 13. Graph Numbers - **Labeled Graphs (Image 1):** Each graph has 3 vertices and 1 edge. - **Unlabeled Graph (Image 2):** The graph also consists of 3 vertices and 1 edge, but it is notable for having only one configuration. ### Additional Aspects #### Tables - No tables are present in the visual content. #### Ablaufprozesse (Process Flows) - No process flows are depicted. #### Prozessbeschreibungen (Process Descriptions) - Not applicable. #### Typen Bezeichnung (Type Designations) - Types/categories identified: - Labeled graphs - Unlabeled graphs #### Trend and Interpretation - Labeled graphs exhibit more configuration diversity compared to unlabeled graphs, highlighting increased complexity in labeling. ### Summary The visual content effectively illustrates key graph theory concepts discussed in the text. The images provide tangible examples of the theoretical distinctions between labeled and unlabeled graphs, supporting the explanation of combinatorial problems in graph theory. Through this comprehensive examination, it is clear that the visual aids are crucial for reinforcing the textual content and facilitating a deeper understanding of the topic. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 28 Context: #### Additional Aspects - **Ablaufprozesse (Process Flows) & Prozessbeschreibungen (Process Descriptions):** - Processes involved include mathematical reasoning, steps for solving combinatorial problems, and interpreting graph properties. - **Typen Bezeichnung (Type Designations):** - Different types of combinatorial terms and graph properties (e.g., binomial coefficients, vertices, edges, degrees). ### Conclusion The visual content on the provided page delivers comprehensive educational material on mathematical preliminaries, specifically combinatorial mathematics and graph theory. The page effectively uses diagrams and structured text to convey advanced concepts and exercises aimed at students or readers engaged in higher mathematical studies. #################### File: An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf Page: 3 Context: # Copyright Notice An Introduction to the Theory of Numbers © 1967 Leo MosER Distributed under a Creative Commons Attribution 4.0 International (CC BY 4.0) license. Informally, this license allows you to: - Share: copy and redistribute the material in any medium or format - Adapt: remix, transform, and build upon the material for any purpose, even commercially, under the following conditions: - **Attribution**: You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. - **No additional restrictions**: You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Full license terms at: [http://creativecommons.org/licenses/by/4.0/legalcode](http://creativecommons.org/licenses/by/4.0/legalcode) Published by The Trilla Group, West Lafayette, Indiana, USA ISBN 1-58901-114-1 First published: March 1, 2001. This version revised: July 31, 2011. The publisher "The Trilla Group" and The Trilla Group logo are trademarks of The Trilla Group. This book was prepared by William MosER from a typescript by Leo MosER. We thank Siwan Gunturo, Joseph Lipman, and Mark Hudson for proofreading parts of the manuscript. We intend to correct and update the work as needed. If you notice any mistakes in this work, please send email to Bradley Lucier (lucier@math.purdue.edu) and they will be corrected in a later version. --- # Contents - Preface .............................................................................. v - Chapter 1. Compositions and Partitions ............................... 1 - Chapter 2. Arithmetic Functions ......................................... 7 - Chapter 3. Distribution of Primes ...................................... 17 - Chapter 4. Irrational Numbers ............................................ 37 - Chapter 5. Congruences ..................................................... 43 - Chapter 6. Diophantine Equations ...................................... 53 - Chapter 7. Combinatorial Number Theory ........................... 59 - Chapter 8. Geometry of Numbers ....................................... 69 - Classical Unsolved Problems ............................................. 73 - Miscellaneous Problems .................................................... 75 - Unsolved Problems and Conjectures .................................... 83 #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 65 Context: # 2.7 A review ![](path/to/image) *Fig. 2.7.3: The recursive call tree for FFT* Finally, if we call the ‘power of 2’ version of the FFT algorithm on the sequence \( \{1, -1, 1, -1\} \), then FFT will proceed to manufacture the tree shown in Fig. 2.7.3. It must be emphasized that the creation of the tree of recursions is done by the compiler without any further effort on the part of the programmer. As long as we’re here, how does a compiler go about making such a tree? It does it by using an auxiliary stack. It adopts the philosophy that if it is asked to do two things at once, well after all, it can’t do that, so it does one of those two things and drops the other request on top of a stack of unfinished business. When it finishes executing the first request, it goes to the top of the stack to find out what to do next. ## Example Let’s follow the compiler through its tribulations as it attempts to deal with our request for maximum independent set size that appears in Fig. 2.3.3. We begin by asking for the `maxset` of the 5-cycle. Our program immediately makes two recursive calls to `maxset`, one for each of the two graphs that appear on the second level of the tree in Fig. 2.3.3. The stack is initially empty. The compiler says to itself ‘I can’t do these both at once’, and it puts the right-hand graph (involving vertices \(3, 4\)) on the stack, and proceeds to call itself on the left hand graph (vertices \(2, 3, 4, 5\)). When it tries to do that one, of course, two more graphs are generated, of which the right-hand one (4,5) is dropped onto the stack, on top of the graph that previously lived there, so now two graphs are on the stack, awaiting processing, and the compiler is dealing with the graph \( (3, 4, 5) \). This time the graph of just one vertex \( (5) \) is dropped onto the stack, which now holds three graphs, as the compiler works on \( (4, 5) \). Next, that graph is broken up into \( (5) \), and an empty graph, which is dutifully dropped onto the stack, so the compiler can work on \( (5) \). Finally, something fruitful happens: the graph \( (5) \) has no edges, so the program `maxset` gives, in its trivial case, very specific instructions as to how to deal with this graph. We now know that the graph that consists of just the single vertex \( (5) \) has a `maxset` value of 1. The compiler next reacts for the graph on top of the stack, finds that it is the empty graph, which has no edges at all, and therefore its `maxset` size is 0. Now it knows the \( n_1 = 1 \) and \( n_2 = 0 \) values that appear in the algorithm `maxset`, and therefore it can execute the instruction `maxset(n1, 1 + n2)`, from which it finds that the value of `maxset` for the graph \( (4, 5) \) is 1, and it continues from there, to dig itself out of the stack of unfinished business. In general, if it is trying to execute `maxset` on a graph that has edges, it will drop the graph \( G - \{v^* - N(v^*)\} \) on the stack and try to do the graph \( G - \{v\} \). The reader should try to write out, as a formal algorithm, the procedure that we have been describing, whereby the compiler deals with a recursive computation that branches into two sub-computations until a trivial case is reached. Image Analysis: ### Comprehensive Examination of the Visual Content #### Localization and Attribution - **Image 1:** - Located at the top-center of the page in section 2.7. - Diagram with caption "Fig. 2.7.3: The recursive call tree for FFT." #### Object Detection and Classification - **Image 1:** - Objects Detected: - A tree diagram with rectangular nodes. - Text labels within rectangles. - Key Features of Detected Objects: - Nodes in the diagram represent calls in the recursive function. - The root node is labeled `{1, i, −i, 1}`. - The branching nodes below it represent subsequent recursive calls. #### Text Analysis - **Image 1:** - Detected Text: - Main diagram: `{1, i, −i, 1}`, `{1, -i}`, `{1, i}`, `1`, `1`, `i`, `-i` - Analysis of the Text: - The text inside the nodes represents an FFT decomposition process. The notations are complex numbers or sets that define each stage of the recursion. #### Diagram and Chart Analysis - **Image 1:** - Analyzed Data: - The diagram is a recursive call tree for the FFT algorithm. - Description of Axes, Scales, and Legends: - No explicit axes or scales; it's a hierarchical structure. - Key Insights: - Provides a visual breakdown of recursion involved in FFT, illustrating how the input `{1, i, −i, 1}` is divided into smaller subproblems until base cases (`1`, `i`, `-i`) are reached. #### Scene and Activity Analysis - **Image 1:** - Scene Description: - The scene shows a structured graphical representation of recursive function calls. - Main Actors and Actions: - Main actors: The recursive function calls represented as nodes. - Actions: Each node represents a function invocation subdividing its problem into smaller parts. #### Process Flows (Ablaufprozesse) and Descriptions (Prozessbeschreibungen) - **Image 1:** - Process Flows: - Depicted recursively breaking down the input list in the FFT algorithm. - Process Descriptions: - Each call subdivides its input until reaching base cases (`1`, `i`, `-i`). #### Perspective and Composition - **Image 1:** - Perspective: - Frontal view of a tree diagram; nodes are evenly spaced. - Composition: - The root is at the top, branching out downward, clearly illustrating recursive splitting. #### Contextual Significance - **Image 1:** - Contextual Analysis: - This image is part of a section explaining the recursive nature of the FFT algorithm. - It complements the textual explanation by providing a visual aid for understanding recursion. #### Trend and Interpretation - **Image 1:** - Trend and Interpretation: - Visual representation reinforces the divide-and-conquer strategy of FFT. - It highlights the systematic reduction of the problem size until trivial cases are reached. ### Additional Observations - **Color Analysis:** - The diagram is mostly in black and white, ensuring clarity and readability. - **Text Extraction Outside Image:** - The accompanying text provides further insights into how the recursion is managed using a stack-based approach, with a detailed example provided. This examination provides a comprehensive analysis of the visual content and relevant aspects, offering detailed insights into the diagram and its contextual significance in explaining the FFT algorithm. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 51 Context: #### 5. Diagram and Chart Analysis - **Diagram Analysis:** - **Equation Diagram:** Shows the multiplication of two 2x2 matrices. - **Equations**: - \( c_{11} = a_{11}a_{11} + a_{12}b_{21} \) - \( c_{12} = a_{11}b_{12} + a_{12}b_{22} \) - \( c_{21} = a_{21}b_{11} + a_{22}b_{21} \) - \( c_{22} = a_{21}b_{12} + a_{22}b_{22} \) - \( c_{i,j} = \sum_{k=1}^2 a_{ik}b_{kj} \) #### 6. Product Analysis - **Not applicable** since the content doesn't depict any products. #### 7. Anomaly Detection - **No anomalies** were detected in the image content. #### 8. Color Analysis - **Color Composition:** - Monochrome (Black text on white background) - Limited color scheme as typical for academic textbooks #### 9. Perspective and Composition - **Perspective:** Eye-level perspective typical for reading material. - **Composition:** - Structured in a typical textbook format - Exercises and explanations divided into sections - Mathematical equations and text are appropriately formatted and aligned for readability. #### 10. Contextual Significance - **Purpose and Contribution:** - The image contributes educational content related to graph theory and matrix multiplication. - It serves as exercises and explanations for students studying these topics. #### 11. Metadata Analysis - **Metadata Not Provided:** No metadata can be inferred directly from the visual content provided. #### 12. Graph and Trend Analysis - **No graphs or trend charts** are present within this content. #### 13. Graph Numbers - **Not applicable** as there are no data points or row numbers in the graph. #### Additional Aspects - **Ablaufprozesse (Process Flows):** - Described process of multiplying matrices. - Explanation of algorithm steps for graph theory exercises. - **Prozessbeschreibungen (Process Descriptions):** - Detailed steps for matrix multiplication. - Breakdown of how to approach and solve graph-related problems. - **Typen Bezeichnung (Type Designations):** - Types of graphs (cycle, path, connected, not-necessarily-connected). - Type of matrix multiplication. - **Trend and Interpretation:** - Modern approaches and historical methods in matrix multiplication. - Progression in the understanding of matrix multiplication efficiency (reference to V. Strassen's work). - **Tables:** - No tables present in the content. This comprehensive analysis offers detailed insights and interpretations based on the visual and textual content of the page. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 24 Context: # Chapter 1: Mathematical Preliminaries ## Theorem 1.4.1. Let a sequence \( \{x_n\} \) satisfy a recurrent inequality of the form \[ x_{n+1} \leq b_0 x_n + b_1 x_{n-1} + \ldots + b_{p-2} x_{n-p+1} + G(n) \quad (n \geq p) \] where \( b_i \geq 0 \) (\( \forall i, \sum_{i=0}^{p-1} b_i > 1 \)). Further, let \( c \) be the positive real root of the equation \( s^{p+1} = b_0 s^{p-1} + \ldots + b_{p-1} \). Finally, suppose \( G(n) = o(n^c) \). Then for every fixed \( \epsilon > 0 \) we have \( x_n = O(n^{c + \epsilon}) \). **Proof:** Fix \( \epsilon > 0 \), and let \( \alpha = c + \epsilon \), where \( c \) is the root of the equation shown in the statement of the theorem. Since \( \alpha > c \), if we let \[ t = \alpha^{p+1} - b_0 \alpha^p - \ldots - b_{p-1} \] then \[ K = \max \left\{ \frac{|x_1|}{a_0}, \frac{|x_2|}{a_1}, \max_{n \geq 2} \frac{G(n)}{K(n)^{\alpha - s}} \right\} \] Then \( K \) is finite, and clearly \( |x_j| \leq K a^j \) for \( j \leq p \). We claim that \( |x_n| \leq K a^n \) for all \( n \), which will complete the proof. Indeed, if the claim is true for \( 0, 1, 2, \ldots, n \), then \[ |x_{n+1}| \leq b_0 |x_n| + \ldots + b_{p-1} |x_{n-p+1}| + G(n) \] \[ \leq K a^n \left( b_0 K a^{n-1} + \ldots + b_{p-1} K a^{n-p} \right) + G(n) \] \[ = K a^n \left( K a^{p-1} \left( b_0 + \ldots + b_{p-1} \right) + G(n) \right) \] \[ = K a^{n+1} \left( t K^{\alpha - p} + G(n) \right) \leq K a^{n+1} \] ## Exercises for Section 1.4 1. Solve the following recurrence relations: 1. \( x_n = 2x_{n-3} + 3 \quad (n \geq 0; x_0 = 2) \) 2. \( x_{n+1} = 2x_n/3 + 2 \quad (n \geq 0; x_0 = 0) \) 3. \( x_{n+1} = 2x_n + 1 \quad (n \geq 0; x_0 = 0) \) 4. \( x_{n+1} = x_{n-1} + x_n + 1 \quad (n \geq 1; x_1 = 5) \) 5. \( x_{n+1} = x_n + x_{n-1} \quad (n \geq 1; x_0 = 0; x_1 = 3) \) 6. \( x_{n+1} = 4x_n - 4x_{n-1} \quad (n \geq 1; x_0 = 1; x_1 = 1) \) 2. Find \( d \) if the sequence \( x_n \) satisfies the Fibonacci recurrence relation and if furthermore \( x_0 = 1 \) and \( x_1 = 1 \) (\( n \to \infty \)). 3. Let \( L_n \) be the average number of trailing 0's in the binary expansions of all integers \( 0, 1, 2, \ldots, n-1 \). Find a recursive relation satisfied by the sequence \( \{x_n\} \), solve it, and evaluate \( \lim_{n \to \infty} x_n \). 4. For what values of \( a \) and \( b \) is it true that no matter what the initial values \( x_0, x_1 \) are, the solution of the recurrence relation \( x_{n+1} = a x_n + b x_{n-1} \) (\( n \geq 2 \)) is guaranteed to be \( O(1) \) (as \( n \to \infty \))? 5. Suppose \( x_0 = 1, x_1 = 1 \), and for all \( n \geq 2 \) it is true that \( x_{n+1} \leq F_n \). Prove your answer. 6. Generalize the result of exercise 5, as follows. Suppose \( x_0 = x_1 = y \), where \( y = 1 \) or \( y \geq 1 \). If furthermore, \( x_{n+1} \leq x_{n-1} \) (\( n \geq 2 \)), can we conclude that \( \forall n : x_n \leq y \)? If not, describe conditions on \( a \) and \( b \) under which that conclusion would follow. 7. Find the asymptotic behavior in the form \( x_n \sim f(n) \) of the right side of (1.4.10). * See exercise 10, below. Image Analysis: Here is a comprehensive analysis of the attached visual content based on the provided aspects: ### 1. Localization and Attribution: - **Page Layout**: The content is divided into two primary sections: a theoretical proof (positioned at the top and middle) and a set of exercises (located at the bottom). - **Image Numbering**: Since the entire content is a single page, it is denoted as Image 1. ### 2. Object Detection and Classification: - **Image 1**: - **Text**: Mathematical notation, theorems, proofs, and exercises. - **Graphical Elements**: There is a small black square near the bottom-left of the page. ### 3. Scene and Activity Analysis: - **Image 1**: - **Scene Description**: The page appears to be from a textbook or an academic paper focusing on mathematical preliminaries. - **Activity**: The main activities include the presentation of Theorem 1.4.1 along with its proof, followed by exercises related to the theorem. ### 4. Text Analysis: - **Text Extraction**: - **Title**: "Chapter 1: Mathematical Preliminaries" - **Theorem**: 1.4.1 and the corresponding proof. - **Exercises**: Series of exercises numbered 1 to 7. - **Significance**: - **Theorem 1.4.1**: Presents a specific result about sequences and recurrent inequalities. - **Proof**: Provided to validate Theorem 1.4.1. - **Exercises**: Intended for practice and deeper understanding of the theorem's applications. ### 9. Perspective and Composition: - **Perspective**: The content is presented from a standard upright viewpoint common to academic texts, facilitating easy reading. - **Composition**: The page is well-structured with clear demarcation between the theorem, its proof, and the exercises. Mathematical expressions are neatly formatted. ### 10. Contextual Significance: - **Context**: Likely from a mathematics textbook or academic paper, this content is part of a broader discussion on mathematical sequence and recursion. - **Contribution to Overall Message**: The theorem, proof, and exercises together serve to educate and help students understand and apply mathematical concepts of sequences and recurrence relations. ### 14. Tables: - **In-text Table(s)**: Not applicable as there is no table present in the content. ### Diagrams and Charts: - **In-text Diagrams and Charts**: None are present or were detected. ### Metadata Analysis: - **Metadata**: Not accessible through visual content. ### Graph and Trend Analysis: - **Graphs and Trends**: None were detected. ### Graph Numbers: - **Data Points**: Not applicable since there are no graphs. ### Prozessbeschreibungen (Process Descriptions): - **Descriptions**: The theorem proof process can be considered a process description. It contains detailed steps logical steps to validate the theorem. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 41 Context: After some tidying up, (2.2.7) becomes F(n) = \(\left(1 + \frac{1}{n-1}\right)F(n-1) + \left(2 - \frac{2}{n}\right)\). which is exactly in the form of the general first-order recurrence relation that we discussed in section 1.4. In section 1.4 we saw that to solve (2.2.8) the winning tactic is to change to a new variable, that is defined, in this case, by F(n) = \(\frac{n + 1}{n - 1} - \frac{2}{n + 1}y_n\). If we make the change of variable F(n) = \( (1 + y_n)\), in (2.2.8), then it takes the form \(y_n = y_{n-1} + 2(n - 1)/n(a_n + 1) \quad (n \geq 1)\) as an equation for the \(y_n\)s (\(y_0 = 0\)). The solution of (2.2.10) is obviously \(y_n = 2 \sum_{j=1}^{n} \frac{j - 1}{j(y_j + 1)}\) \(\ = 2 \sum_{j=1}^{n} \frac{2}{j + 1} - \frac{1}{j}\) \(\ = 2 \sum_{j=1}^{n} \frac{1}{j} - 4n/(n + 1)\). Hence from (2.2.9), F(n) = \(2(n + 1)\sum_{j=1}^{n} \frac{1}{j} - 4n\) is the average number of pairwise comparisons that we do if we Quicksort an array of length n. Evidently F(n) ∼ \(2n\log(n - \infty)\) see (1.1.7) with \(g(t) = 1/t\), and we have proved **Theorem 2.2.2.** The average number of pairwise comparisons of array entries that Quicksort makes when it sorts arrays of n elements is exactly as shown in (2.2.11), and is ∼ \(2n\log(n - \infty)\). Quicksort is, on average, a very quick sorting method, even though its worst case requires a quadratic amount of labor. ## Exercises for section 2.2 1. Write out an array of 10 numbers that contains no splitter. Write out an array of 10 numbers that contains 10 splitters. 2. Write a program that does the following. Given a positive integer n, choose 100 random permutations of \([1, 2, \ldots, n!]\) and count how many of the 100 had at least one splitter. Execute your program for \(n = 5, 6, \ldots, 12\) and tabulate the results. 3. Think of some method of sorting n numbers that isn't in the text. In the worst case, how many comparisons might your method do? How many swaps? * For a fast and easy way to do this see A. Nijhuis and H. S. Wilf, *Combinatorial Algorithms*, 2nd ed. (New York: Academic Press, 1978), chap. 6. Image Analysis: ### Comprehensive Examination of the Attached Visual Content #### Localization and Attribution 1. **Image 1** - **Location**: The entire content appears to be on a single page, containing textual and mathematical content. - **Page Number**: 37 is visible at the bottom-center of the page. #### Object Detection and Classification 1. **Image 1** - **Objects Detected**: - Text - Mathematical equations - Numbered list (exercises) #### Text Analysis 1. **Image 1** - **Extracted Text**: - Sections from a book discussing "Quicksort" and related mathematical formulas. - Exercises for section 2.2 providing tasks involving arrays and algorithms. - **Content Analysis**: - **Mathematical Content**: The page elaborates on the recurrence relation and the average number of pairwise comparisons made by the Quicksort algorithm. It includes equations (2.2.8) to (2.2.11) that help in deriving the average number of comparisons. - **Theorem Statement**: Theorem 2.2.2 proves the average number of comparisons for Quicksort, specifying that it's based on equation (2.2.11) and asymptotically forms \(2n \ln(n) - 4n\). - **Exercises**: Three exercises aimed at understanding and applying the concepts of array sorting provided in section 2.2. These exercises involve creating arrays, running sorting algorithms, and counting the operations performed. #### Diagram and Chart Analysis - No diagrams or charts are present in the provided image. #### Color Analysis - The image is primarily black and white, suitable for the academic or technical nature of the document. #### Perspective and Composition - **Perspective**: The image is a direct frontal view of a page from a book. - **Composition**: Text is arranged in paragraphs, mathematical equations, and a numbered list for exercises. The layout is structured to facilitate easy reading and reference. #### Contextual Significance - **Overall Document Context**: The image seems to come from a textbook on algorithms, specifically detailing aspects of the Quicksort algorithm. - **Contribution to Theme**: The detailed explanation of the Quicksort algorithm, including theoretical aspects and practical exercises, contributes to a deeper understanding of data sorting methods. #### Tables - No tables are present in the provided image. ### Summary The image is a textbook page focused on the Quicksort algorithm, presenting theoretical explanations, mathematical derivations, and practical exercises. The text is dense with technical content, suitable for a computer science or mathematics educational course. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 97 Context: # 4.6 Pseudoprimality tests ## Test 2 Given \( n \). Output 'n is composite' if \(\gcd(n, b) \neq 1\), else output 'inconclusive'. This one is a little better, but not yet good. If \( n \) is composite, the number of bases \( b \leq n \) for which Test 2 will produce the result 'composite' is \( n - \phi(n) \), where \(\phi\) is the Euler totient function, of (4.1.5). This number of useful bases will be large if \( n \) has some small prime factors, but in that case it’s easy to find out that \( n \) is composite by other methods. If \( n \) has only a few large prime factors, say if \( n = p^2 \), then the proportion of useful bases is very small, and we have the same kind of inefficiency as in Test 1 above. Now we can state the third pseudoprimality test. ## Test 3 Given \( n \). (If \( b \) and \( n \) are not relatively prime or) if \( n \equiv 1 \,(\text{mod } n) \) then output 'n is composite'; else output 'inconclusive'. Regrettably, the test is still not 'good', but it's a lot better than its predecessors. To cite an extreme case of its un-goodness, there exist composite numbers \( n \), called Carmichael numbers, with the property that the pair \((b, n)\) produces the output 'inconclusive' for every integer \( b \neq 1 \) that is relatively prime to \( n \). An example of such a number is \( n = 1729 \), which is composite (1729 = 17 ⋅ 101), but for which Test 3 gives the result 'inconclusive' on every integer \( b < 1729 \) that is relatively prime to 1729 (i.e., that is not divisible by 17 or 101). Despite such misbehavior, the test usually seems to perform quite well. When \( n = 169 \) (a difficult integer for tests 1 and 2) it turns out that there are 158 different \( b \in [1, 168] \) that produce the 'composite' outcome from Test 3, namely every such \( b \) except for 19, 22, 23, 70, 80, 89, 96, 147, 150, 168. Finally, we will describe a good pseudoprimality test. The familial resemblance to Test 3 will be apparent. ## Test 4 (the strong pseudoprimality test) Given \( (n, k) \). Let \( n - 1 = 2^m \cdot q \), where \( m \) is an odd integer. If either (a) \( b^m \equiv 1 \,(\text{mod } n) \) or (b) there is an integer \( i \in [0, q - 1] \) such that \[ b^{2^i} \equiv -1 \,(\text{mod } n) \] then return 'inconclusive'; else return 'n is composite'. First, we validate the test by proving the Proposition. If the test returns the message 'n is composite', then \( n \) is composite. ### Proof Suppose not. Then \( n \) is an odd prime. We claim that \[ b^{m} \equiv 1 \,(\text{mod } n) \] for all \( i = q, q - 1, \ldots, 0 \). If so then the case \( i = 0 \) will contradict the outcome of the test, and thereby complete the proof. To establish the claim, it is clearly true when \( i = q \) by Fermat’s theorem. If true for \( i \) then it is true for \( i - 1 \) also, because \[ (b^{m})^{2^i} \equiv b^{m} \equiv 1 \,(\text{mod } n) \] implies that the quantity being squared is \( +1 \) or \( -1 \). Since \( n \) is an odd prime, by corollary 4.5.3 \( U_n \) is cyclic, and so the equation \( x^2 \equiv 1 \,(\text{mod } n) \) has only the solutions \( x \equiv \pm 1 \). But \( -1 \) is ruled out by the outcome of the test, and the proof of the claim is complete. What is the computational complexity of the test? Consider first the computational problem of raising a number to a power. We can calculate, for example, \( b^2, b^4, b^8, \ldots \) by squaring, and reducing modulo \( n \) immediately after each squaring operation, rather than waiting until the final exponent is reached. Then we use the binary expansion of the exponent to tell us which of these powers of \( b \) should multiply together in order to compute \( b^n \). For instance, \[ b^{37} = b^{32} \cdot b^{4} \cdot b^{1}. \] Image Analysis: **1. Localization and Attribution:** - **Image Position**: Entire page - **Image Number**: Image 1 **2. Object Detection and Classification:** - **Detected Objects**: Text, formulas, mathematical symbols - **Categories**: Mathematical text, equations **3. Scene and Activity Analysis:** - **Scene Description**: The image contains a scanned page from a mathematical textbook or academic paper. - **Activities**: The page is presenting mathematical tests and proofs. **4. Text Analysis:** - **Extracted Text**: The content includes definitions, propositions, and proofs related to pseudoprimality tests. - **Content Significance**: The text outlines different tests (Test 2, Test 3, Test 4) for identifying pseudoprimes. It explains the procedures for each test and provides a proof for the strong pseudoprimality test (Test 4). **5. Diagram and Chart Analysis:** - **Present Data**: No diagrams or charts are present. **6. Product Analysis:** - **Products Depicted**: None **7. Anomaly Detection:** - **Unusual Elements**: No anomalies observed. Content appears consistent with typical mathematical textbook or academic paper format. **8. Color Analysis:** - **Dominant Colors**: Black text on a white background. - **Impact on Perception**: Enhances readability and focus on content, typical for academic publications. **9. Perspective and Composition:** - **Perspective**: Standard front-facing view typical of scanned documents. - **Composition**: The text is organized into sections, with tests and proofs clearly separated. There are numbered lists for the different tests. **11. Contextual Significance:** - **Overall Context**: The image is likely part of a textbook or research paper detailing mathematical theories and proofs related to pseudoprimality. - **Contribution to Overall Message**: The page contributes detailed explanations and proofs necessary for understanding pseudoprimality tests. ### Detailed Text Analysis (continued): **Test 2:** - **Definition**: Given integers \(b\) and \(n\), the output is 'composite' if \(gcd(b,n) \neq 1\), else output 'inconclusive'. - **Explanation**: Describes the criteria and limitations of Test 2, including the case when \(n\) is composite and the number of bases \(b \leq n\). **Test 3:** - **Definition**: Given \(b\) and \(n\); if \(n-1 \nequiv 1 \pmod n\), the output is 'composite', else 'inconclusive'. - **Explanation**: Discusses composite numbers known as Carmichael numbers that can produce 'inconclusive' outputs for every integer \(b\). Provides an example with \( n = 1729\). **Test 4 (Strong Pseudoprimality Test):** - **Definition**: Given \( (b, n)\). Consider \(1 \leq m \leq 2^{m_q}\) where \( m \) is an old integer. The output is 'composite' if \( b^{mn2} = -1\). - **Explanation**: Outlines the process and validation for Test 4, including a proof by contradiction for the pseudoprimality test. **Proposition:** - **Claim**: If the test returns 'composite', then \(n\) is composite. - **Proof**: Detailed proof provided using the properties described for the pseudoprimality test. **Example Calculation and Complexity Discussion:** - **Procedure**: Describes a computational method for raising a number to a power efficiently using binary exponentiation. ### Conclusion: The image is a detailed mathematical exposition on pseudoprimality tests, including definitions, explanations, and a proof. The text is structured to convey complex mathematical concepts clearly and sequentially. #################### 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: 133 Context: ```markdown # 5.7 Backtracking (II): Graph Coloring Suppose \( f \) is longer than \( \epsilon \). Then \( T' \) would not be minimal because the tree that we would obtain by exchanging \( f \) for \( e_i \) in \( T' \) (why is it still a tree if we do that exchange?) would be shorter, contradicting the minimality of \( T' \). Hence \( f \) and \( g \) have the same length. In \( T' \) exchange \( f \) for \( e_i \). Then \( T' \) is still a tree, and is still a minimum spanning tree. The index of the first edge of \( T \) that does not appear in \( T' \) is now at least \( r + 1 \), one unit larger than before. The process of replacing edges of \( T \) that do not appear in \( T' \) without affecting the minimality of \( T \) can be repeated until every edge of \( T \) appears in \( T' \), i.e., until \( T' = T \). Hence \( T \) was a minimum spanning tree. That fulfills one step of the process that leads to a polynomial time travelling salesman algorithm that finds a tour of at most twice the minimum length. The next step involves finding an Euler circuit. Way back in theorem 1.6.1 we learned that a connected graph has an Euler circuit if and only if every vertex has even degree. Recall that the proof was recursive in nature, and immediately implies a linear time algorithm for finding Euler circuits recursively. We also noted that the proof remains valid even if we are dealing with a multigraph, that is, with a graph in which several edges are permitted between single pairs of vertices. We will in fact need that extra flexibility for the purpose at hand. Now we have the ingredients for a quick near-optimal travelling salesman tour. **Theorem 5.8.1**. An algorithm that operates in polynomial time and will return a travelling salesman tour whose length is at most twice the length of a minimum tour. Here is the algorithm. Given \( n \) cities in the plane: 1. Find a minimum spanning tree \( T \) for the cities. 2. Double each edge of the tree, thereby obtaining a ‘multitree’ \( T' \) in which each pair of vertices there are \( 0 \) or \( 2 \) edges. 3. Since every vertex of the doubled tree has even degree, there is an Eulerian tour \( W \) of the edges of \( T' \); find one, as in the proof of theorem 1.6.1. 4. Now we construct the output tour of the cities. Begin at some city and follow the walk \( W \). However, instead of starting at some vertex \( v \), go from \( v \) directly (via a straight line) to the next vertex of the walk \( W \) that you haven't visited yet. This means that you will often short-circuit portions of the walk \( W \) by going directly from some vertex to another via several edges "down the road." The tour \( Z \) that results from (4) above is indeed a tour of all the cities in which each city is visited once and only once. We claim that its length is at most twice optimal. Let \( Z \) be an optimum tour, and let \( e \) be some edge of \( Z \). Then \( Z - e \) is a path that visits all of the cities. Since a path is a tree, \( Z - e \) is a spanning tree of the cities, hence \( Z - e \) is at least as long as \( T \), and so \( Z \) is surely at least as long as \( T \). Next consider the length of the tour \( Z \). A step of \( Z \) that walks along an edge of the walk \( W \) has length equal to the length of that edge of \( W \). A step of \( Z \) that short-circuits several edges of \( W \) has length at most equal to the sum of the lengths of the edges of \( W \) that were short-circuited. If we sum these inequalities over all steps of \( Z \) we find that the length of \( Z \) is at most equal to the length of \( W \), which is in turn twice the length of the tree \( T \). If we put all of this together we find that \[ \text{length}(Z) \leq \text{length}(Z - e) \leq \text{length}(T) = \frac{1}{2} \text{length}(W) \leq \frac{1}{2} \text{length}(Z') \] as claimed. More recently it has been proved (Christodes, 1976) that polynomial time we can find a TSP tour whose total length is at most \( 3/2 \) as long as the original tour. The algorithm makes use of Edmonds’s algorithm for maximum matching in a general graph (see the reference at the end of Chapter 3). It will be interesting to see if the factor \( 3/2 \) can be further refined. Polynomial time algorithms are known for other NP-complete problems that guarantee that the answer obtained will not exceed, by more than a constant factor, the optimum answer. In some cases this guarantees apply to the difference between the answer that the algorithm gives and the best one. See the references below for more information. ``` Image Analysis: ### Text Analysis: #### Image 1 - **Text Content:** - The text discusses a polynomial time approximation algorithm for finding a near-optimal traveling salesman tour. - It restates a theorem for the polynomial time working of this algorithm. - References to previous sections are included (e.g., Section 1.6.1). - Descriptions of the steps involved in the algorithm for n cities are provided. - Definitions and explanations about Euler circuits, multitreets, and process flows related to constructing and optimizing the traveling salesman path are mentioned. - A mathematical expression demonstrates the length relationship between different paths and circuits, including a proof. - **Significance in Context:** - Provides context for the algorithm's efficiency and optimality in solving the Traveling Salesman Problem under certain conditions. - Explains theoretical underpinnings (from earlier sections and external references) to give a potential context of research or course material on computational theory and graph algorithms. ### Color Analysis: - **Image 1:** - Dominated by black text on a white background, which is typical for textual academic or technical content. - The monochromatic color scheme enhances readability and is appropriate for the clear communication of complex mathematical recommendations and proofs. ### Perspective and Composition: - **Image 1:** - Straightforward, top-down perspective, typical of textbook pages. - The layout is columnar with margins and justified text, adhering to standard academic formatting. ### Contextual Significance: - **Image 1:** - Likely taken from an academic textbook or research paper dealing with graph theory, specifically focusing on algorithms related to the Euler circuits and Traveling Salesman Problem. - This page provides necessary theoretical explanations as part of broader material, indicating its role in teaching or scholarly examination of algorithms. ### Conclusion: This analysis indicates that the given image is a page filled with text from an advanced academic context, focusing on polynomial-time algorithms for the Traveling Salesman Problem. The explanations, definitions, and formal proof presented signify that this material is aimed at a reader with a solid understanding of theoretical computer science, likely providing coursework or supporting material for in-depth study of graph theory and its applications. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 4 Context: # Preface For the past several years, mathematics majors in the computing track at the University of Pennsylvania have taken a course in continuous algorithms (numerical analysis) in the junior year, and in discrete algorithms in the senior year. This book has grown out of the senior course as I have been teaching it recently. It has also been tried out on a large class of computer science and mathematics majors, including seniors and graduate students, with good results. Selection by the instructor of topics of interest will be very important, because normally I've found that I can't cover anywhere near all of this material in a semester. A reasonable choice for a first try might be to begin with Chapter 2 (recursive algorithms) which contains lots of motivation. Then, as new ideas are needed in Chapter 2, one might delve into the appropriate sections of Chapter 1 to get the concepts and techniques well in hand. After Chapter 2, Chapter 4, on number theory, discusses material that is extremely attractive, and surprisingly pure and applicable at the same time. Chapter 5 would be next, since the foundations would then all be in place. Finally, material from Chapter 3, which is rather independent of the rest of the book, but is strongly connected to combinatorial algorithms in general, might be studied as time permits. Throughout the book, there are opportunities to ask students to write programs and get them running. These are not mentioned explicitly, with a few exceptions, but will be obvious when encountered. Students should all have the experience of writing, debugging, and using a program that is nontrivially recursive, for example. The concept of recursion is subtle and powerful, and is helped a lot by hands-on practice. Any of the algorithms of Chapter 2 would be suitable for this purpose. The recursive graph algorithms are particularly recommended since they are usually quite foreign to students’ previous experience and therefore have great learning value. In addition to the exercises that appear in this book, student assignments might consist of writing occasional programs, as well as delivering reports in class on assigned readings. The latter might be found among the references cited in the bibliographies in each chapter. I am indebted first of all to the students on whom I worked out these ideas, and to a number of colleagues for their helpful advice and friendly criticism. Among the latter I will mention Richard Brault, Daniel Kleitman, Albert Nijenhuis, Robert Tarjan, and Alan Tucker. For the no-doubt-numerous shortcomings that remain, I accept full responsibility. This book was typeset in TeX. To the extent that it’s a delight to look at, thank TeX. For the deficiencies in its appearance, thank my limitations as a typesetter. It was, however, a pleasure for me to have had the chance to typeset my own book. My thanks to the Computer Science department of the University of Pennsylvania, and particularly to Aravind Joshi, for generously allowing me the use of TeX facilities. Herbert S. Wilf #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 2 Context: # CONTENTS ## Chapter 0: What This Book Is About 0.1 Background 0.2 Hard vs. easy problems 0.3 A preview ## Chapter 1: Mathematical Preliminaries 1.1 Orders of magnitude 1.2 Positional number systems 1.3 Manipulations with series 1.4 Recurrence relations 1.5 Counting 1.6 Graphs ## Chapter 2: Recursive Algorithms 2.1 Introduction 2.2 Quicksort 2.3 Recursive graph algorithms 2.4 Fast matrix multiplication 2.5 The discrete Fourier transform 2.6 Applications of the FFT 2.7 A review ## Chapter 3: The Network Flow Problem 3.1 Introduction 3.2 Algorithms for the network flow problem 3.3 The algorithm of Ford and Fulkerson 3.4 The max-flow min-cut theorem 3.5 The complexity of the Ford-Fulkerson algorithm 3.6 Layered networks 3.7 The MPM Algorithm 3.8 Applications of network flow ## Chapter 4: Algorithms in the Theory of Numbers 4.1 Preliminaries 4.2 The greatest common divisor 4.3 The extended Euclidean algorithm 4.4 Primality testing 4.5 Interlude: the ring of integers modulo n 4.6 Pseudoprimality tests 4.7 Proof of goodness of the strong pseudoprimality test 4.8 Factoring and cryptography 4.9 Factoring large integers 4.10 Proving primality #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 19 Context: ``` # 1.3 Manipulations with series We are reminded of the finite geometric series (1.3.1), but (1.3.7) is a little different because of the multipliers \( 1, 2, 3, 4, \ldots, N \). The trick is this. When confronted with a series that is similar to, but not identical with, a known series, write down the known series as an equation, with the series on one side and its sum on the other. Even though the unknown series involves a particular value of \( x \) in this case \( z = 2 \), keep the known series with its variable unrestricted. Then reach for an appropriate tool that will be applied to both sides of that equation, and whose result will be that the known series will have been changed into the one whose sum we need. In this case, since (1.3.7) reminds us of (1.3.1), we'll begin by writing down (1.3.1) again, \[ (1 - z)^{-n} = 1 + z + z^2 + \ldots + z^{n-1} \tag{1.3.8} \] Don’t replace \( z \) by \( 2 \) yet, just walk up to the equation (1.3.8) carrying your tool kit and ask what kind of surgery you could do to both sides of (1.3.8) that would be helpful in evaluating the unknown (1.3.7). We are going to reach into our tool kit and pull out \( \frac{d}{dz} \). In other words, we are going to differentiate (1.3.8). The reason for choosing differentiation is that it will put the missing multipliers \( 1, 2, 3, \ldots, N \) into (1.3.8). After differentiation, (1.3.8) becomes \[ 1 + 2z + 3z^2 + 4z^3 + \ldots + (n - 1)z^{n-2} = \frac{1 - n z^{n-1} + (n - 1)z^n}{(1-z)^2} \tag{1.3.9} \] Now it's easy. To evaluate the sum (1.3.7), all we have to do is to substitute \( z = 2 \), \( n = N + 1 \) in (1.3.9), to obtain, after simplifying the right-hand side, \[ 1 + 2 + 2^2 + 3 \cdot 2^3 + \ldots + N 2^{N-1} = 1 + (N - 1)2^N \tag{1.3.10} \] Next try this one: \[ \frac{1}{2^3} + \frac{1}{3^3} + \ldots \tag{1.3.11} \] If we rewrite the series using summation signs, it becomes \[ \sum_{j=1}^{\infty} \frac{1}{j^3} \] Comparison with the series \( 200 \) shows great resemblance to the species (1.3.6). In fact, if we put \( x = \frac{1}{3} \) in (1.3.6) it tells us that \[ \sum_{j=1}^{\infty} \frac{1}{j^3} = \frac{1}{3} \log(3/2) \tag{1.3.12} \] The desired sum (1.3.11) is the result of dropping the term with \( j = 1 \) from (1.3.12), which shows that the sum in (1.3.11) is equal to \( \log(3/2) - 1 \). In general, suppose that \( f(z) = \sum a_n z^n \) is some series that we know. Then \( \sum n a_n z^{n-1} = f'(z) \) and \( \sum n^2 a_n z^{n-2} = z \frac{d}{dz} f'(z) \). In other words, if the \( n \)-th coefficient is multiplied by \( n^2 \), then the function changes from \( f(z) \) to \( z f''(z) \). If we apply the rule again, we find that multiplying the \( n \)-th coefficient of a power series by \( n^2 \) changes the sum from \( f(z) \) to \( \left( z \frac{d}{dz} \right)^2 f(z) \): \[ \sum_{j=0}^{\infty} j^2 z^j / j! = \left( z \frac{d}{dz} \right)^2 e^z \] ``` Image Analysis: ## Image Analysis ### 1. Localization and Attribution - **Image 1**: The entire page shown is considered as one image, and therefore, will be referred to as Image 1. ### 2. Object Detection and Classification - The primary objects in Image 1 are textual content and mathematical equations. No graphical objects or distinct classifications are detected other than these. ### 3. Scene and Activity Analysis - The scene involves a detailed mathematical exposition, likely from a textbook or academic paper discussing manipulations with series. ### 4. Text Analysis - The text covers explanations and derivations involving finite geometric series. There are references to equations (1.3.1), (1.3.8), (1.3.9), (1.3.10), (1.3.11), and (1.3.12). The explanations serve to guide the reader through the steps needed to derive specific series manipulations. ### 5. Diagram and Chart Analysis - No diagrams or charts are present in the image. ### 6. Product Analysis - There are no products depicted in the image. ### 7. Anomaly Detection - No anomalies or unusual elements are observed in the image. ### 8. Color Analysis - The image is predominantly black and white, typical for textual documents. The lack of color elements suggests a focus on readability and clarity in presenting mathematical information. ### 9. Perspective and Composition - The image is taken from a straightforward, eye-level perspective focusing on the textual content. The layout is typical of a page from a mathematical textbook or academic paper, with numbered equations and dense text blocks. ### 10. Contextual Significance - The content appears to be part of a section in a mathematical treatment of series manipulations, guiding readers through the differentiation of series and summations with respect to particular variables. ### 12. Graph and Trend Analysis - No graphs are present in the image. ### 13. Graph Numbers - Not applicable as there are no graphs. ### Additional Aspects #### Ablaufprozesse (Process Flows) - The document sequentially guides the reader through the process of manipulating series, such as the differentiation of the series (1.3.8). #### Prozessbeschreibungen (Process Descriptions) - The processes described involve: 1. Differentiating a known series. 2. Evaluating the series at specific points. 3. Summing pseudo-geometric series. #### Typen Bezeichnung (Type Designations) - Types designations mentioned include finite geometric series, differentiable series, summations of series, and pseudo-geometric series. #### Trend and Interpretation - The trend involves finding methods to simplify and manipulate series by differentiation, then evaluating series at certain points to derive simpler summation forms. ### Tables - No tables are present in the image. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 15 Context: # 1.2 Positional number systems 1. Let \( f \) be a function such that \( f = o(h) \) and \( h = \Omega(g) \). Give an explicit construction for the function \( h \) in terms of \( f \). 2. (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) \( \forall x > 0, f(x) > 0 \) - (c) \( \forall x \geq 0, f(x) < f(t) \) - (d) \( \exists y > 3 \; \forall x < f(x) \) - (e) \( \forall y \exists z \; g(z) < f(f(t)) \) - (f) \( \forall x \; \exists y \; x \cdot f(y) < \epsilon \) 3. Can you formulate a general method for negating such propositions? Given a proposition that contains \( \neg \), \( \land \), \( \lor \); what rule would apply in order to negate the proposition and leave the result in positive form (containing no negation symbols or `not's)? 4. 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 `\epsilon`). - (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 \( f(x) \neq g(x) \); then briefly justify the definition of the \( \Omega \) symbol that was given in the text. 5. Arrange the following functions in increasing order of their rates of growth, for large \( n \). That is, list them so that each one is `little o` of its successor: \[ 2^{\sqrt{n}}, n^{0.301}, n^{2}, n^{1.6} \log^{3} n + 1, n^{\sqrt{n}}, n \log n, n^{3} \log(\log n), n^{2}, n^{(n + 4)^{12}} \] 6. Find a function \( f(x) \) such that \( f(x) = O(2^{\epsilon}) \) is true for every \( \epsilon > 0 \), but for which it is not true that \( f(x) = O(1) \). 7. Prove that the statement: \( f(n) = O(2^{\epsilon}) \) for every \( \epsilon > 0 \) is equivalent to the statement: \( f(n) = \Omega(2^{\epsilon}) \) for every \( \epsilon > 0 \). ## 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: \[ \ldots, 100000, 10000, 1000, 100, 10, 1. \] 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 \( 10^{2} \)s and 3 \( 10^{1} \)s and 7 \( 10^{0} \)s. In general, if we write out the string of digits that represents a number in the decimal system, as \( d_{m}d_{m-1} \ldots d_{0} \), 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 \[ \ldots, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1. \] Image Analysis: ### Comprehensive Examination 1. **Localization and Attribution:** - **Image 1:** - Located at the top part of the page. It includes a block of text containing mathematical exercises and explanations. - **Image 2:** - Located at the bottom part of the page. It contains a continuation of the explanation in Image 1, specifically focused on positional number systems. 2. **Object Detection and Classification:** - **Image 1:** - Object Detected: Text - Key Features: Mathematical notations, Greek letters, and textual explanations. - **Image 2:** - Object Detected: Text - Key Features: Numerical sequences, mathematical notations, and explanatory text. 3. **Scene and Activity Analysis:** - **Image 1:** - Scene: A mathematical textbook page with exercises and explanations related to function notation, propositions, negation, growth rates, and specific functions. - Activities: Explanation of mathematical exercises, detailing how to negate propositions, and discussing growth rates of functions. - **Image 2:** - Scene: Continuation of the mathematical discussion, specifically addressing positional number systems and conversions. - Activities: Explanation of positional number systems in decimal and binary bases, mathematical sequences. 4. **Text Analysis:** - **Image 1:** - Extracted Text: Contains exercises 6-10, discussing functions, propositions, negations, definitions of symbols (like the 'Ω' symbol), and rate of growth of functions. - Significance: Provides detailed instructions and questions for understanding complex mathematical concepts related to function growth rates and logical negation. - **Image 2:** - Extracted Text: Discusses positional number systems, starting with decimal and moving to binary representations. - Significance: Introduces foundational concepts of representing numbers in different bases, crucial for understanding computer science and mathematical applications. 7. **Anomaly Detection:** - No noticeable anomalies detected in either Image 1 or Image 2. The text follows logical and mathematical consistency based on the context provided. 8. **Color Analysis:** - **Image 1 and Image 2:** - Dominant Colors: Black text on a white background. - Impact on Perception: The high contrast between the black text and white background enhances readability, which is important for understanding complex mathematical content. 9. **Perspective and Composition:** - **Image 1 and Image 2:** - Perspective: Direct, frontal view typical of textbook pages. - Composition: Text is organized in a structured, columnar format with numbering and clear distinctions between different exercises and explanations. ### Specific Sections of Text (Summarized for Key Insights): - **Exercise 6:** Focuses on negating mathematical propositions without using negative symbols. - **Exercise 7:** Explores the definition of the 'Ω' symbol in relation to functions. - **Exercise 8:** Arranges functions by their growth rates. - **Exercise 9:** Finding a function \( f(x) \) that meets a specific condition involving big-O notation. - **Exercise 10:** Discusses the equivalence of two statements involving big-O notation. - **Positional Number Systems:** - Describes representation of numbers using digits in different bases. - Examples using decimal (base 10) and binary (base 2) systems. ### Contextual Significance: - The images together provide a comprehensive segment from a mathematical textbook likely intended for advanced study in computer science or mathematics. They offer detailed exercises and explanations to enhance the reader’s understanding of fundamental concepts such as function growth rates, logical propositions, and number representation in various bases. The logical progression from function notation to positional number systems shows a structured approach to mathematical learning, ensuring that students grasp foundational concepts before moving to more complex topics. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 63 Context: - **Content Analysis**: 1. **Title and Sections**: - The document includes discussions on advanced computational arithmetic, primarily focusing on fast multiplication methods for large integers and the Fast Fourier Transform (FFT). - There is a review section summarizing algorithms discussed in the chapter. 2. **Mathematical Context**: - The text is heavily mathematical, dealing with concepts like polynomial multiplication, FFT, and computational complexity. - Specific problems and exercises are given that involve roots of unity, polynomial evaluation, and proof-writing. 3. **Algorithmic Overview**: - The review section covers various computational problems and their complexities, including sorting, finding a maximum independent set in a graph, graph coloring, and matrix multiplication. - It mentions both the naive (brute-force) and advanced (recursive or optimized) methods for each problem. 4. **References**: - References to academic works by E. Lawler and D. Coppersmith and S. Winograd indicate the scholarly nature of the document related to computational complexity. #### 9. **Perspective and Composition**: - **Perspective**: The image appears to present the content from a medium or standard top-down view, typical of scanning or photographing a document page. - **Composition**: The elements consist primarily of text sections, equations, and a list of exercises. The content is structured into paragraphs with equations and exercise items appropriately spaced. ### Conclusions The provided image represents a page from an academic textbook or document related to computational mathematics, specifically dealing with polynomial arithmetic, FFT, and algorithmic complexity. The text includes detailed exercises aimed at deepening the understanding of the discussed methods and algorithms, providing both theoretical insights and practical programming challenges. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 5 Context: # Chapter 0: What This Book Is About ## 0.1 Background An algorithm is a method for solving a class of problems on a computer. The complexity of an algorithm is the cost, measured in running time, or storage, or whatever units are relevant, of using the algorithm to solve one of those problems. This book is about algorithms and complexity, and so it is about methods for solving problems on computers and the costs (usually the running time) of using those methods. Computing takes time. Some problems take a very long time, others can be done quickly. Some problems seem to take a long time, and then someone discovers a faster way to do them (a ‘faster algorithm’). The study of the amount of computational effort that is needed in order to perform certain kinds of computations is the study of computational complexity. Naturally, we would expect that a computing problem for which millions of bits of input data are required would probably take longer than another problem that needs only a few items of input. So the time complexity of a calculation is measured by expressing the running time of the calculation as a function of some measure of the amount of data that is needed to describe the problem to the computer. For instance, think about this statement: “I just bought a matrix inversion program, and it can invert an \( n \times n \) matrix in just \( 1.2n^3 \) minutes.” We see here a typical description of the complexity of a certain algorithm. The running time of the program is being given as a function of the size of the input matrix. A faster program for the same job might run in \( 0.8n^3 \) minutes for an \( n \times n \) matrix. If someone were to make a really important discovery (see section 2.4), then maybe we could actually lower the exponent, instead of merely shaving the multiplicative constant. Thus, a program that would invert an \( n \times n \) matrix in only \( n^2 \log n \) minutes would represent a striking improvement of the state of the art. For the purposes of this book, a computation that is guaranteed to take at most \( c n^3 \) time for input of size \( n \) will be thought of as 'easy' computation. One that needs at most \( n^{10} \) time is also easy. If a certain calculation on an \( n \times n \) matrix were to require \( 2^n \) minutes, then that would be a 'hard' problem. Naturally some of the computations that we are calling 'easy' may take a very long time to run, but still, from our present point of view the important distinction to maintain will be the polynomial time guarantee or lack of it. The general rule is that if the running time is not a polynomial function of the amount of input data, then the calculation is an easy one; otherwise it’s hard. Many problems in computer science are known to be easy. To convince someone that a problem is easy, it is enough to describe a fast method for solving that problem. To convince someone that a problem is hard is hard, because you will have to prove to them that it is impossible to find a fast way of doing the calculation. It will not be enough to point to a particular algorithm and to lament its slowness. After all, that algorithm may be slow, but maybe there’s a faster way. Matrix inversion is easy. The familiar Gaussian elimination method can invert an \( n \times n \) matrix in time of at most \( O(n^3) \). To give an example of a hard computational problem we have to go far afield. One interesting one is called the 'tiling problem'. Suppose we are given infinitely many identical floor tiles, each shaped like a regular hexagon. Then we can tile the whole plane with them, i.e., we can cover the plane with no empty spaces left over. This can also be done if the tiles are identical rectangles, but not if they are regular hexagons. In Fig. 0.1 we show a tiling of the plane by identical rectangles, and in Fig. 0.2 is a tiling by regular hexagons. --- 1. See, for instance, Martin Gardner’s article in *Scientific American*, January 1977, pp. 110-121. 2. R. Berger, “The undecidability of the domino problem,” *Amer. Math. Soc.* 66 (1966), Amer. Image Analysis: ## Analysis of the Attached Visual Content ### 1. Localization and Attribution - **Image**: There is only one image present on the page. ### 2. Scene and Activity Analysis - **Scene Description**: The entire scene appears to be a page from a book or document. - **Activity**: There is no specific activity depicted. The main focus is on the textual content provided on this page. ### 4. Text Analysis - **Extracted Text**: - **Title**: "Chapter 0: What This Book Is About" - **Section**: "0.1 Background" - **Content**: The text describes algorithms, computational complexity, and introduces the purpose and scope of the book. It provides definitions and examples to explain how computational problems can be classified as easy or difficult based on the amount of computational effort required. - **Examples and Explanations**: - Discusses matrix inversion problems. - Mention of "tiling problem" with illustrations (Fig. 0.1 and Fig. 0.2) referred to but not shown in the extracted image. - **References**: - Martin Gardner's article in *Scientific American*. - R. Berger's work on the undecidability of the domino problem. ### 7. Anomaly Detection - **Anomalies**: There are no noticeable anomalies or unusual elements in the image. The layout and content appear to be standard for an academic or technical document. ### 8. Color Analysis - **Color Composition**: The image is black and white with text and some footnotes in a standard format. - **Dominant Colors**: Black text on a white background, which is typical for printed academic material. ### 9. Perspective and Composition - **Perspective**: The perspective is a direct, flat view of the page, typical of a scanned document or a digital book page. - **Composition**: The text is structured with a title, section heading, main body text, and footnotes. There is clear demarcation between sections with proper use of paragraphs and indentations. ### 12. Graph and Trend Analysis - **Existing Graphs**: There are references to figures (Fig. 0.1 and Fig. 0.2) related to tiling problems, but these figures are not part of the extracted image. ### Additional Aspects: #### Prozessbeschreibungen (Process Descriptions) - **Description**: The text provides a process description of computational complexity, explaining how various problems can be classified based on the time they take to compute. #### Typen Bezeichnung (Type Designations) - **Types/Categories**: - Problems are classified based on computational effort. - Examples given involve matrix inversion and tiling problems. - Terms like "easy", "hard", and "polynomial time" are used to categorize problems. ### Contextual Significance - **Overall Document Context**: This page serves as an introductory section to an academic or technical book on algorithms and computational complexity. It provides the reader with foundational knowledge necessary to understand subsequent chapters. - **Contribution to Overall Message**: The image sets the stage for comprehensive discourse on algorithms by explaining fundamental concepts and laying the groundwork for more detailed exploration of computational problems. ### Conclusion The visual content analyzed is an informative introductory page of a book or document centered on computational complexity and algorithms. It introduces key concepts, definitions, and examples, preparing readers for detailed discussions in subsequent sections. The layout, text structure, and references are typical of technical or academic literature. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 11 Context: ### Contextual Significance: - **Overall Message**: This page aims to educate the reader on the different rates at which functions grow and to establish a foundational understanding of comparative growth rates in mathematical analysis. - **Contribution to Theme**: It likely contributes to a larger section or chapter on mathematical analysis, specifically focusing on asymptotic behavior of functions. This foundational knowledge is essential for understanding more advanced concepts in mathematical and computational analysis. #### 12. Graph and Trend Analysis: - The page does not contain graphs or visual data plots but rather focuses on textual descriptions and mathematical expressions. #### 13. Graph Numbers: - Since no graphs are present on this page, no numerical data points for graphs can be provided. ### Summary: - The provided visual content is an excerpt from a mathematical text discussing the concept of orders of magnitude in function growth. It provides formal definitions, examples using limits, and explanations of different growth rates (polynomial, exponential, and beyond). The analysis here helps in understanding the structure and significance of the content in a broader educational context. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 94 Context: #### **Image 1:** - **Perspective:** - The image is a direct scan of a textbook page, taken from a straight, birds-eye view. - **Composition:** - Organized in a standard scientific literature format. - Includes clear section headers, theorem statements in bold or italic, and detailed explanations. ### Contextual Significance: #### **Image 1:** - **Context:** - This image is likely part of a mathematical textbook aimed at discussing algorithms in the field of number theory. - The theorems and corollaries presented contribute crucially to the understanding of primitive roots and cyclic groups. - **Contribution to Overall Message:** - The image aims to provide foundational mathematical theory (primitive roots, cylic groups) with rigor and clarity. - Example calculations and theorems ensure a comprehensive understanding of the topics. ### Conclusion: This page from the textbook focuses on advanced number theory concepts. It introduces important theorems related to primitive roots and cyclic groups and illustrates these concepts with examples in \( Z_6 \). The clear, structured presentation of information fosters a deep understanding of these mathematical principles. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 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: 73 Context: ```markdown procedure fordFulkerson(X : network; f : flow; maxFlowValue : real); {finds maximum flow in a given network X} set f := 0 on every edge of X; maxFlowValue := 0; repeat labelAndScan(X, f, asPath); if asPath = 'it’s time to augment' then augment(flow[X, f], amount); maxFlowValue := maxFlowValue + amount until asPath = 'flow is maximum'; end. {fordFulkerson} Let’s look at what happens if we apply the labeling and scanning algorithm to the network and flow shown in Fig. 3.1.2. First vertex s gets the label (-∞, +∞). We then scan s. Vertex A gets the label (-, ∞, 8). B cannot be labeled, and C gets labeled with (s, +, 10), which completes the scan of s. Next we scan vertex A, during which D acquires the label (A, +, 8). Then C is scanned, which results in E getting the label (C, - , 10). Finally, the scan of D results in the label (D, +, 3) for the sink t. From the label of t we see that there is a flow augmenting path in the network along which we can push 3 more units of flow from s to t. We find the path as in procedure augment(flow, amount), following the labels backwards from t to D, A and s. The path in question will be seen to be exactly the one shown in Fig. 3.3.1, and further augmentation proceeds as we have discussed above. ## 3.4 The max-flow min-cut theorem Now we are going to look at the state of affairs that holds when the flow augmentation procedure terminates because it has not been able to label the sink. We want to show that then the flow will have a maximum possible value. Let W ⊆ V(X), and suppose that W contains the source and W does not contain the sink. Let W denote all other vertices of X, i.e., W = V(X) - W. **Definition.** By the cut (W, W̄) we mean the set of all edges of X whose initial vertex is in W and whose terminal vertex is in W̄. For example, one cut in a network consists of all edges whose initial vertex is the source. Now, every unit of flow that leaves the source and arrives at the sink must at some moment flow from a vertex of W to a vertex of W̄, i.e., must flow along some edge of the cut (W, W̄). We define the **capacity of a cut** to be the sum of the capacities of all edges in the cut; then it seems clear that the value of a flow can never exceed the capacity of any cut, and therefore the **minimum value of a flow** cannot exceed the minimum capacity of any cut. The main result of this section is the ‘max-flow min-cut’ theorem of Ford and Fulkerson, which we state as **Theorem 3.4.1.** The maximum possible value of any flow in a network is equal to the minimum capacity of any cut in that network. **Proof.** We will first do a little computation to show that the value of a flow can never exceed the capacity of a cut. Second, we will show that when the Ford-Fulkerson algorithm terminates it has been unable to label the sink, and that that moment there is a cut in the network whose edges are saturated with flow, i.e., such that the flow in each edge of the cut is equal to the capacity of that edge. Let U and V be two (not necessarily disjoint) sets of vertices of the network X, and let f be a flow function for X. By f(U | V) we mean the sum of the values of the flow function along all edges whose initial vertices lies in U and whose terminal vertices lies in V. Similarly, by cu(U | V) we mean the sum of the capacities of all those edges. Finally, by the net flow out of U we mean f(U | V) - f(V | U). ``` Image Analysis: ### Image Analysis #### 1. Localization and Attribution - **Image 1** is located at the upper part of the page and consists of a pseudocode and a procedural explanation. - **Image 2** is the lower part of the page containing a textual explanation and theorem proof. #### 2. Object Detection and Classification - **Image 1**: - Objects: Text content, pseudocode block. - Classification: Academic content, algorithm explanation. - **Image 2**: - Objects: Text content. - Classification: Theorem proof, explanatory text. #### 4. Text Analysis - **Image 1**: - **Pseudocode Content**: - Title: `procedure fordFulkerson(X: network; f: flow; maxflowvalue:real);` - Description of a maximum flow algorithm in a network. - Main steps involve initializing flow values, scanning vertices, and augmenting flow. - The psuedocode describes the Ford-Fulkerson method to find the maximum flow in a given network. - **Image 2**: - **Text Content**: - Discusses the application of the Ford-Fulkerson algorithm step by step. - Highlights labels and augmentation process for vertices spread across various steps of the algorithm. - Defines **The Max-Flow Min-Cut Theorem**: - Provides terms and definitions related to sets of vertexes. - Discusses the logical proof of the theorem. - Establishes the relationship between maximum flow and minimum cut in network flow problems. #### 8. Color Analysis - Both images are predominantly black and white, typical for academic or instructional text. There are no significant colors that impact the perception. #### 9. Perspective and Composition - **Image 1**: Composed with a top-down perspective focusing on the pseudocode block. - **Image 2**: Text is laid out in standard multi-paragraph, columnar format used in technical documents. ### Additional Aspects #### Ablaufprozesse (Process Flows) - **Image 1**: - Depicts steps in the Ford-Fulkerson algorithm to augment flow in a network. Identifies steps beginning with initialization, edge scanning, and augmentation until maximum flow is achieved. #### Prozessbeschreibungen (Process Descriptions) - **Image 2**: - Describes the application and logical steps of the Ford-Fulkerson method. - Discusses how labels are added to vertices and how augmenting paths are found, highlighting core algorithm operations. - The flow process continues with analyzing resulting flow from source to sink vertices. #### Typen Bezeichnung (Type Designations) - **Image 1**: Does not directly specify types, but implicitly categorizes operations such as initialization, scanning, and augmentation. - **Image 2**: Defines sets and functions relevant to network flow and capacity. #### Trend and Interpretation - The text demonstrates the relationship and trends between flow and cut in a network. The main insight is that the maximum flow in a network is equal to the capacity of the minimum cut. ### Summary The images collectively provide a comprehensive explanation of the Ford-Fulkerson algorithm and the Max-Flow Min-Cut Theorem, which are crucial concepts in network flow theory. The pseudocode and proof offer detailed steps and logical reasoning, highlighting important procedures and outcomes in network flow problems. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 57 Context: # 2.5 The Discrete Fourier Transform There is one small problem, though. In (2.5.8) we want to compute \( f(j) \) for \( j = 0, \ldots, 2^n - 1 \). However, the Fourier transform of the shorter sequence (2.5.9) is defined for only \( 2^{r-1} \) values of \( j \) namely for \( j = 0, \ldots, 2^{r-1} - 1 \). So if we calculate the first sum by a recursive call, then we will need its values for \( j \) that are outside the range for which it was computed. This problem is no sooner recognized than solved. Let \( Q(j) \) denote the first sum in (2.5.8). Then we claim that \( Q(j) \) is a periodic function of \( j \), of period \( 2^{r-1} \): \[ Q(j + 2^{r-1}) = \sum_{m=0}^{2^{r-1}-1} x_m \exp(2\pi i (j + 2^{r-1})/2^r) \] \[ = \sum_{m=0}^{2^{r-1}-1} x_m \exp(2\pi i j / 2^r) \exp(2\pi i m/2^{r-1}) = \sum_{m=0}^{2^{r-1}-1} x_m \exp(2\pi i j / 2^r) \cdot \exp(2\pi i m/2^{r-1}) = Q(j) \quad \text{(2.5.11)} \] for all integers \( j \). If \( Q(j) \) has been computed only for \( 0 \leq j \leq 2^{r-1} - 1 \) and if we should want its value for some \( j \geq 2^{r} \), then we can get that value by asking for \( Q(j \mod 2^{r-1}) \). Now we can state the recursive form of the Fast Fourier Transform algorithm in the most important case where \( n \) is a power of 2. In the algorithm we will use the type `complexarray` to denote an array of complex numbers. ```matlab function FFT(x: complexarray): complexarray; % computes Fast Fourier transform of n = 2^k numbers x if n == 1 then FFT[0] := x[0] end if; evenarray := {x[0], x[2], ..., x[n - 2]}; oddarray := {x[1], x[3], ..., x[n - 1]}; {u[0], u[1], ..., u[n - 1]} := FFT(n/2, evenarray); {v[0], v[1], ..., v[n - 1]} := FFT(n/2, oddarray); for j = 0 to n/2 - 1 do x[j] := u[j] + exp(2\pi i * j/n) * v[j]; x[j + n/2] := u[j] - exp(2\pi i * j/n) * v[j]; end for; end function; ``` Let \( y(k) \) denote the number of multiplications of complex numbers that will be done if we call FFT on an array whose length is \( n = 2^k \). The call to FFT(n/2, evenarray) costs \( y(k - 1) \) multiplications as does the call to FFT(n/2, oddarray). The for loop \( j = 0 \) to \( n/2 - 1 \) requires \( n \) more multiplications. Hence, \[ y(k) = 2y(k - 1) + 2^k \quad (k \geq 1); \quad y(0) = 0. \quad (2.5.12) \] If we change variables by writing \( y(k) = 2^{k+1} - 2 \), then that is \( y(k) = 2(k + 1) \), together with \( y(0) = 0 \), and therefore that \( y(k) = k2^{k+1} \). This proves **Theorem 2.5.1.** The Fourier transform of a sequence of \( n \) complex numbers is computed using \( O(n \log n) \) multiplications of complex numbers by means of the procedure FFT, if \( n \) is a power of 2. Next* we will discuss the situation when \( n \) is not a power of 2. The reader may observe that by 'padding out' the input array with additional 0's we can extend the length of the array until it becomes a power of 2, and then call the FFT procedure that we have already. Image Analysis: ### Detailed Examination #### Localization and Attribution: - **Image 1** is the only image on the page. #### Text Analysis: 1. **Detected Text:** - Title: “2.5 The discrete Fourier transform” - Main Content: - The page describes concepts related to the Fourier transform, specifically focusing on the recursive calculation for certain values of \(j\) and \(Q(j)\) and the Fast Fourier Transform (FFT) algorithm. - Mathematical formulae and equations are present, such as: - \( Q(j + 2^r-1) = Q(j) \) - \( y(k) = 2y(k - 1) + 2^k \) - Explanation of the FFT algorithm, including the function definition: - ```Text function FFT(integer n; x: complexarray):complexarray; ``` - Important Points: - Fourier transform challenges, recursive solution: - Calculations for \( j = 0, 1, \ldots, 2^r - 1 \) - Recursive formula adaptation: leveraging \( Q(j + 2^r-1) = Q(j) \) - Expansion and calculations involving complex exponentials - Computing \( y(k) = 2y(k - 1) + 2^k \) and the application in FFT - Summation symmetry and logic behind FFT implementations 2. **Significance:** - The text provides thorough explanations and derivations for understanding and implementing the Fast Fourier Transform, a key algorithm in computational mathematics and signal processing. It offers valuable insight into the recursive approach to computing Fourier transforms efficiently. #### Process Descriptions: - The text describes two main computational processes: - The computation of function \(f_c(j)\) for \( 2^r \) values of \( j \). - The process of the Fast Fourier Transform (FFT), from basic definition to a more advanced recursive calculation involving even and odd indexed arrays, aiming to reduce multiplications of complex numbers. #### Typen Bezeichnung (Type Designations): - **Mathematical Notations**: - Typical notations used: \(j\), \(Q(j)\), \(2^r\), \(x\), “mod” arithmetic, and \(y(k)\). - **Function Definition**: - ```Text function FFT(integer n; x: complexarray):complexarray; ``` - Variables in the function: \( j, k, u, v, y \), indicating the iterative nature of the algorithm. #### Diagram and Chart Analysis: - **Equations**: - Equations present intricate mathematical relationships fundamental to discrete Fourier transform and FFT algorithms. - Key insights involve recognizing the periodic nature of \( Q(j) \), efficiency gains with recursive functions, and explicit calculations within FFT. #### Perspective and Composition: - **Perspective**: - The page is laid out in a standard mathematical textbook format with detailed text, equations, and algorithmic functions. - **Composition**: - Structured: Title at the top, dense textual explanation, embedded equations, clear logical flow, followed by defined programs or algorithms. #### Contextual Significance: - Within a broader context, this page is part of a textbook or comprehensive guide on Fourier transforms and their computational techniques, critical for students and professionals in fields such as applied mathematics, engineering, and computer science. #### Color Analysis: - The page mainly consists of black text on a white background, standard for textbook material, ensuring clarity and readability. This detailed examination covers text analysis, process descriptions, type designations, and other essential details from the attached image. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 26 Context: # Chapter 1: Mathematical Preliminaries ## Theorem 1.5.1 For each \( n \geq 0 \), a set of \( n \) objects has exactly \( 2^n \) subsets, and of these, exactly \( \binom{n}{k} \) have cardinality \( k \) (for \( k = 0, 1, \ldots, n \)). There are exactly \( n! \) different sequences that can be formed from a set of \( n \) distinct objects. Since every subset of \( [n] \) has some cardinality, it follows that \[ \sum_{k=0}^{n} \binom{n}{k} = 2^n \quad (n = 0, 1, 2, \ldots). \] In view of the convention that we adopted, we might have written (1.5.2) as \( \sum_{k} \binom{n}{k} = 2^n \), with no restriction on the range of the summation index \( k \). It would then have been understood that the range of \( k \) is from \( -\infty \) to \( \infty \), and that the binomial coefficient \( \binom{n}{k} \) vanishes unless \( 0 \leq k \leq n \). In Table 1.5.1, we show the values of some of the binomial coefficients \( \binom{n}{k} \). The rows of the table are thought of as labelled ‘\( n = 0\), \( n = 1\), etc., and the entries within each row refer, successively, to \( k = 0, 1, \ldots, n \). The table is called "Pascal's triangle." | | 0 | 1 | 2 | 3 | 4 | |----|-----|-----|-----|-----|-----| | 0 | 1 | | | | | | 1 | 1 | 1 | | | | | 2 | 1 | 2 | 1 | | | | 3 | 1 | 3 | 3 | 1 | | | 4 | 1 | 4 | 6 | 4 | 1 | | 5 | 1 | 5 | 10 | 10 | 5 | | 6 | 1 | 6 | 15 | 20 | 15 | | 7 | 1 | 7 | 21 | 35 | 21 | | 8 | 1 | 8 | 28 | 56 | 70 | **Table 1.5.1:** Pascal's triangle Here are some facts about the binomial coefficients: (a) Each row of Pascal's triangle is symmetric about the middle. That is, \[ \binom{n}{k} = \binom{n}{n-k} \quad (0 \leq k \leq n). \] (b) The sum of the entries in the \( n \)th row of Pascal’s triangle is \( 2^n \). (c) Each entry is equal to the sum of the two entries that are immediately above it in the triangle. The proof of (c) above can be interesting. What it says about the binomial coefficients is that \[ \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k} \quad ((n,k) \neq (0,0)). \] There are (at least) two ways to prove (1.5.3). The hammer-and-tongs approach would consist of expanding each of the three binomial coefficients that appears in (1.5.3), using the definition (1.5.1) in terms of factorials, and then cancelling common factors to complete the proof. That would work (try it), but there’s another way. Contemplate (this proof is by contemplation) the totality of \( k \)-subsets of \( [n] \). The number of them is on the left side of (1.5.3). Sort them out into two piles: those \( k \)-subsets that contain ‘1’ and those that don’t. If a \( k \)-subset of \( [n] \) contains ‘1’, then \( k - 1 \) elements can be chosen in \( \binom{n-1}{k-1} \) ways, and that accounts for the first term on the right of (1.5.3). If a \( k \)-subset does not contain ‘1’, then its \( k \) elements are all chosen from \( [n-1] \), and that completes the proof of (1.5.3). Image Analysis: Certainly! Here's the comprehensive examination of the provided visual content: 1. **Localization and Attribution:** - **Image 1**: The entire content provided. - This image occupies a full page. 2. **Object Detection and Classification:** - Detected Objects: Text, Mathematical Formulae, Table. - Categories: Educational/Mathematical Content. 3. **Scene and Activity Analysis:** - Scene Description: The scene depicts a mathematical explanation from a textbook, specifically focusing on binomial coefficients and Pascal's triangle. It includes theorems, mathematical equations, explanatory text, and a table. - Main Actors: Mathematical symbols (e.g., Σ, binomial coefficients), textual explanations, and the table. 4. **Text Analysis:** - **Text Detected and Extracted:** **Theorem 1.5.1** - "For each \( n \geq 0 \), a set of \( n \) objects has exactly \( 2^n \) subsets, and of these, exactly \( \binom{n}{k} \) have cardinality \( k \) ( \( \forall k = 0, 1, \ldots, n ) \). There are exactly \( n! \) different sequences that can be formed from a set of \( n \) distinct objects." **Formula** - \( \sum_{k=0}^{n} \binom{n}{k} = 2^n \) **Table 1.5.1: Pascal's triangle** - A triangular arrangement of binomial coefficients. **Explanation of Binomial Coefficients:** - Symmetry, row sums equal \( 2^n \), sums of adjacent entries. **Proof Example:** - Two ways to prove the binomial coefficients formula. 5. **Diagram and Chart Analysis:** - **Table 1.5.1: Pascal's triangle** - The table visually represents the binomial coefficients arranged in a triangular form, where the sum of values in each row corresponds to powers of 2. - Key Features: - Symmetry: Each row is symmetric about its middle. - Summative Property: Each entry is the sum of the two entries directly above it. 6. **Anomaly Detection:** - There are no noticeable anomalies or unusual elements within the image. The content appears as standard mathematical text and tables. 8. **Color Analysis:** - The image is monochromatic (black and white), typical for printed or scanned textbook pages. The monochromatic scheme focuses the reader’s attention solely on the content without the distraction of colors. 9. **Perspective and Composition:** - Perspective: The image is seen from a direct front-on view, as if looking at an open book or a printed page. - Composition: The content is structured logically with a hierarchy: the theorem and explanations are at the top, followed by a centered table (Pascal's triangle) and further explanations beneath it. 10. **Contextual Significance:** - This page appears to be from a mathematics textbook, providing an understanding of binomial coefficients and Pascal's triangle. - Contribution to Overall Theme: The image contributes instructional value, aiding the reader’s comprehension of combinatorial mathematics by visually illustrating binomial coefficients through Pascal’s triangle. 13. **Graph Numbers:** - Listings of specific data points for the first seven rows of Pascal's triangle: - 1 - 1, 1 - 1, 2, 1 - 1, 3, 3, 1 - 1, 4, 6, 4, 1 - 1, 5, 10, 10, 5, 1 - 1, 6, 15, 20, 15, 6, 1 - And so on, continuing with the specified pattern. To summarize, the provided image is a detailed instructional page from a mathematical textbook, focusing on the theorem of binomial coefficients, visualized through Pascal's triangle, and providing related proofs and explanations. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 133 Context: ### Color Analysis - **Color Composition:** - The image is primarily composed of black text on a white background, typical of printed documents. No additional color variations are present that contribute to the content. ### Perspective and Composition - **Perspective:** - The perspective is that of a flat, directly photographed or scanned page. - **Composition:** - The image is structured in columns of text with occasional numbered lists. Sections are clearly delineated with headings. ### Contextual Significance - **Contribution to Overall Message:** - The image contributes detailed theoretical content pertinent to the field of graph theory and algorithm design, specifically focusing on the Traveling Salesman Problem and Euler circuits. ### Graph and Trend Analysis - **Graphs and Trends:** - No graphs or explicit data trends are presented in this image. The focus is on theoretical descriptions and algorithmic steps. ### Prozessbeschreibungen (Process Descriptions) - **Processes Described:** - The image describes the step-by-step procedure of an algorithm for finding a near-optimal traveling salesman tour. ### Typen Bezeichnung (Type Designations) - **Type Designations:** - The text refers to types of mathematical structures such as trees, 'multitrees,' Eulerian tours, and spanning trees. ### Trend and Interpretation - **Identified Trends and Interpretations:** - The trend in the text shows a progressive build-up from theoretical background to practical algorithm description and mathematical proofs. ### Tables - **Content Description:** - No tables are present in this image. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 123 Context: # Maximum clique We are given a graph \( G \) and an integer \( K \). The question is to determine whether or not there is a set of \( K \) vertices in \( G \), each of which is joined, by an edge of \( G \), to all of the others. ## Edge coloring Given a graph \( G \) and an integer \( K \), can we color the edges of \( G \) in \( K \) colors, so that whenever two edges meet at a vertex, they will have different colors? Let us refer to an edge coloring of this kind as a proper coloring of the edges of \( G \). A beautiful theorem of Vizing's deals with this question. If \( \Delta \) denotes the largest degree of any vertex in the given graph, the Vizing's theorem asserts that the edges of \( G \) can be properly colored in either \( \Delta \) or \( \Delta + 1 \) colors. Since it is obvious that at least \( \Delta \) colors will be needed, this means that the edge chromatic number is in doubt by only one unit, for every graph \( G \)! Nevertheless, the decision as to whether the correct answer is \( \Delta \) or \( \Delta + 1 \) is NP-complete. ## Hamilton path In a graph \( G \), is there a path that visits every vertex of \( G \) exactly once? ## Target sum Given a finite set of positive integers whose sum is \( S \)? The above list, together with SAT, 3SAT, Travelling Salesman, and Graph Coloring, constitutes a modest sampling of the class of these seemingly intractable problems. Of course it must not be assumed that every problem that "sounds like" an NP-complete problem is necessarily so hard. If for example we ask for an Euler path instead of a Hamilton path (i.e., if we want to traverse edges rather than vertices) the problem would no longer be NP-complete, and in fact it would be in P, thanks to theorem 1.6.1. As another example, the fact that one can find the edge connectivity of a given graph in polynomial time (see section 3.8) is rather amazing considering the quite difficult appearance of the problem. One of our motivations for including the network flow algorithms in this book was, indeed, to show how very sophisticated algorithms can sometimes prove that seemingly hard problems are in fact computationally tractable. ## Exercises for section 5.4 1. Is the claim that we made and proved above (just after (5.4.21) related with the statement that the clause (5.4.1) is satisfiable if and only if the clauses (5.4.2) are simultaneously satisfiable? Discuss. 2. Is the claim that we made and proved above (just after (5.4.21) related with the statement that the Boolean expression (5.4.1) is equal to the product of the Boolean expressions (5.4.2) in the sense that their truth values are identical on every set of inputs? Discuss. 3. Let it be desired to find out if a given graph \( G \) of vertices, can be vertex colored in \( K \) colors. If we transform the problem into an instance of SAT, exactly how many clauses will there be? # 5.5 Half a loaf ... If we simply groan to solve an NP-complete problem, then we are faced with a very long computation. Is there anything that can be done to lighten the load? In a number of cases various kinds of probabilistic and approximate algorithms have been developed, some very ingenious, and these may often be quite serviceable, as we have already seen in the case of primality testing. Here are some of the strategies of "near" solutions that have been developed. ## Type I: 'Almost surely ...' Suppose we have an NP-complete problem that asks if there is a certain kind of substructure embedded in a given structure. Then we may be able to develop an algorithm with the following properties: - (a) It always runs in polynomial time. - (b) When it finds a solution then that solution is always a correct one. - (c) It doesn't always find a solution, but it 'almost always' does, in the sense that the ratio of successes to total cases approaches unity as the size of the input string grows large. An example of such an algorithm is one that will find a Hamilton path in almost all graphs, failing to do so sometimes, but not often, and running always in polynomial time. We will describe such an algorithm below. **Reference:** - V. G. Vizing, On an estimate of the chromatic class of a \( p \)-graph (Russian), Diskret. Analiz. 3 (1964), 25-30. Image Analysis: Here is a detailed analysis according to the specified aspects: 1. **Localization and Attribution:** - The content consists of a single page of text. It does not contain multiple images but rather a continuous segment of textual information. 2. **Object Detection and Classification:** - This content is text-based and does not contain identifiable objects. 4. **Text Analysis:** - **Headings and Subheadings:** - "Maximum clique:" - "Edge coloring:" - "A beautiful theorem of Vizing deals with this question." - "Hamilton path:" - "Target sum:" - **Key Points:** - The text discusses various computational problems and theorems related to graph theory such as Maximum clique, Edge coloring, Hamilton path, and Target sum. - **Maximum clique**: Discusses determining if there is a set of K vertices in graph G, each of which is joined. - **Edge coloring**: Talks about coloring edges with K colors ensuring no two adjacent edges share the same color on a vertex. - Vizing's theorem: It provides conditions under which a graph can be properly edge colored. - **Hamilton path**: This involves finding a path that visits every vertex exactly once. - **Target sum**: Whether there is a subset whose sum is S. - **Complexity**: Highlights that many graph problems which seem NP-complete can sometimes be solved efficiently. - **Exercise for section 5.4**: - Exercises related to satisfiability of claims and problem-solving scenarios within the context of graph theory. - **Section 5.5 Half a loaf ...**: - Discusses approximate solutions for NP-complete problems. 5. **Diagram and Chart Analysis:** - There are no diagrams or charts present in this content. 6. **Product Analysis:** - No products are depicted in this content. 7. **Anomaly Detection:** - There are no anomalies detected within this textual document. 8. **Color Analysis:** - The document has a black and white text format, no notable colors are present. 9. **Perspective and Composition:** - The text is composed in a straightforward column format, typical for academic papers or textbooks. 10. **Contextual Significance:** - This page appears to be from a textbook or academic paper on computational theory, focusing on challenges in graph theory and related computational problems. - It supports the overall message of mathematical complexity in algorithms and graph theory. 11. **Metadata Analysis:** - There is no explicit metadata visible within the image. 13. **Graph Numbers:** - No graphs are present in this document. **Additional Aspects Included:** 14. **Ablaufprozesse (Process Flows):** - Not applicable as no process flows are depicted. 15. **Prozessbeschreibungen (Process Descriptions):** - Descriptions of solving various graph theory problems are given, focusing on algorithmic complexity and potential solutions. 16. **Typen Bezeichnung (Type Designations):** - NP-complete problems and related classifications are discussed. 17. **Trend and Interpretation:** - Trends toward finding efficient solutions for NP-complete problems using approximation algorithms are discussed. 18. **Tables:** - No tables are included in this content. This content’s primary focus is on explaining different complex graph problems, their theoretical implications, and approximate solutions in computational theory, with an academic approach designed for students or researchers in the field. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 50 Context: # Chapter 2: Recursive Algorithms The last `~` follows from the evaluation \( \sum_{j=2}^{2} j = 2 \) that we discussed in section 1.3. To summarize the developments so far, we have found out that the chromatic polynomial of a graph can be computed recursively by an algorithm whose cost is \( O(2^E) \) for graphs of \( E \) edges. This is exponential cost, and such computations are prohibitively expensive except for graphs of very modest numbers of edges. Of course the more fact that our provable time estimate is \( O(2^E) \) doesn't necessarily mean that the algorithm can be that slow, because maybe our complexity analysis wasn't as sharp as it might have been. However, consider the graph \( G = (G_{s,t}) \) that consists of \( s \) disjoint edges and \( t \) isolated vertices, for a total of \( 2s + t \) vertices altogether. If we choose an edge \( (G_{s,t}) \) and delete it, we get \( G(s-1, t + 1) \). Each of these two new graphs has \( s-1 \) edges. We might imagine arranging the computation so that the extra isolated vertices will be `free`, i.e., will not cost any additional labor. Then the work that we do on \( G(s,t) \) will depend only on \( s \) and will be twice as much as the work we do on \( G(s-1,t) \). Therefore \( G(s,t) \) will cost at least \( 2^s \) operations, and our complexity estimate wasn't a mirage; there really are graphs that make the algorithm do an amount \( 2^{|G|} \) of work. Considering the above remarks it may be surprising that there is a slightly different approach to the complexity analysis that leads to a time bound (for the same algorithm) that is a bit shorter than \( O(2^E) \) in many cases (the work of the complexity analyst is never finished!). Let's look at the algorithm chromoid in another way. For a graph \( G \) we can define a number \( \gamma(G) = |V(G)| - |E(G)| \), which is rather an odd kind of thing to define, but it has a nice property with respect to this algorithm, namely that whatever \( G \) we begin with, we will find that \[ \gamma(G - \{ e \}) = \gamma(G) - 1; \quad \gamma(G / \{ e \}) \leq \gamma(G) - 2. \] Indeed, if we delete the edge \( e \) then \( \gamma \) must drop by 1, and if we collapse the graph on the edge \( e \) then we will have lost one vertex and at least one edge, so we will drop by at least 2. Hence, if \( h(n) \) denotes the maximum amount of labor that chromoid does on any graph \( G \) with \[ |V(G)| \leq |E(G)| \leq n, \] then we claim that \[ h(n) \leq h(n-1) + h(n-2) \quad (n \geq 2). \] Indeed, if \( G \) is a graph for which (2.3.9) holds, then if \( G \) has any edges at all we can do the delete-and-identify step to prove that the labor involved in computing the chromatic polynomial of \( G \) is not at most the quantity on the right side of (2.3.10). Else, if \( G \) has no edges then the labor is 1 unit, which is again at most equal to the right side of (2.3.10), so the result (2.3.10) follows. With the initial conditions \( h(0) = h(1) = 1 \) the solution of the recurrent inequality (2.3.10) is obviously the relation \( h(n) \leq F_n \), where \( F_n \) is the Fibonacci number. We have thereby proved that the time complexity of the algorithm chromoid is \[ O(F_{|V(G)| + |E(G)|}) = O\left( 1 + \frac{\sqrt{5}}{2} |G|^{|G| + |E(G)|} \right) = O(0.1 \cdot 1.62^{|V(G)| + |E(G)|}). \] This analysis does not, of course, contradict the earlier estimate, but complements it. What we have shown is that the labor involved is always \[ O\left( \min(2^{|E(G)|}, 1.62^{|V(G)| + |E(G)|}) \right). \] On a graph with `few` edges relative to its number of vertices (how few?) the first quantity in the parentheses in (2.3.12) will be the smaller one, whereas if \( G \) has more edges, then the second term is the smaller one. In either case the overall judgment about the speed of the algorithm (it's slow!) remains. Image Analysis: ### Comprehensive Analysis of the Attached Visual Content #### 1. **Localization and Attribution:** - **Image Position**: The entire page constitutes one image, denoted as Image 1. #### 4. **Text Analysis:** - **Text Extraction:** - The page is a segment (likely a book or a lengthy document) discussing complexity analysis of recursive algorithms in graph theory. - The text involves intricate mathematical notations and formulas related to the chromatic polynomial of a graph \(G\). - **Key Text Content and Significance:** - **Subscript: Chapter 2: Recursive Algorithms** - This indicates that the text is part of Chapter 2, which focuses on recursive algorithms. - **Evaluation of \( \Sigma_j^2 = J^2 = 2 \)**: - Numerical and algebraic manipulations, indicating mathematical proof or theorem derivations. - **Complexity Analysis**: - Detailed complexity estimation such as \(O(2^{E(G)})\) for computing the chromatic polynomial of a graph, where \(E(G)\) denotes the edge set of \(G\). - **Graph Algorithms**: - Definitions related to graphs \(G\) including vertices \(V(G)\) and edges \(E(G)\). - Discussion on edge deletion and its impact on complexity, referring to graphs \(G - e\) and \(h(\gamma)\). - **Theorems and Proofs**: - Mathematical theorems and inequalities like \(\gamma(G - \{e\}) = \gamma(G) - 1\) (Equation 2.3.8), and further explanations using Fibonacci numbers \(F_{\gamma}\). - Equation \(h(\gamma) \leq h(\gamma - 1) + h(\gamma - 2)\) (Equation 2.3.10) represents a recurrence relation reminiscent of Fibonacci’s sequence applied to graph complexity. - **Time Complexity**: - It establishes bounds and complexity classes such as \(O((1.62)^{|V(G)|+|E(G)|})\) (Equation 2.3.11), which incorporate both vertices and edges. - Further refinements or contradictory analysis are mentioned in \(O (\min (2^{|E(G)|}, 1.62^{|V(G)|+|E(G)|}))\) (Equation 2.3.12). #### 10. **Contextual Significance:** - **Overall Document/Website Context:** - The image is likely a page from an advanced text on algorithms, possibly a computer science textbook or research paper. The document centers on mathematical rigor specific to recursive algorithms and complexity theory in graph analysis. - **Contribution to Overall Message/Theme:** - The detailed complexity analysis underscores the depth and intricacy of recursive algorithms in theoretical computer science. - The commentary about the efficiency and speed of algorithms complements the mathematical proofs, aiming to provide a comprehensive understanding for readers studying advanced algorithm design. #### Additional Aspects: - **Type Designations (Typen Bezeichnung):** - The text categorizes various algorithms and complexity bounds, using terms like chromatic polynomial, Fibonacci numbers, and various bounds of complexity (e.g., \(O\)-notation). ### Summary: The attached visual content is a densely-packed text page likely from an academic source dealing with recursive algorithms and graph theory. It includes complex mathematical notations, theorems, and proofs focused on the chromatic polynomial and complexity analysis. The content contributes notably to the overarching theme of algorithm efficiency and mathematical rigor in computation. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 94 Context: # Chapter 4: Algorithms in the Theory of Numbers A number (like 5 in the example) whose powers run through all elements of \( U_n \) is called a primitive root modulo \( n \). Thus 5 is a primitive root modulo 18. The reader should now find, from Table 4.5.2, all of the primitive roots modulo 18. Alternatively, since the order of a group element must always divide the order of the group, every element of \( U_n \) has an order that divides \( \phi(n) \). The primitive roots are exactly the elements, if they exist, of maximum possible order \( \phi(n) \). We pause to note two corollaries of these remarks, namely ## Theorem 4.5.1 (‘Fermat’s theorem’) For every integer \( b \) that is relatively prime to \( n \) we have \[ b^{\phi(n)} \equiv 1 \quad (\text{mod } n). \tag{4.5.1} \] In particular, if \( n \) is a prime number then \( \phi(n) = n - 1 \), and we have ## Theorem 4.5.2 (‘Fermat’s little theorem’) If \( n \) is prime, then for all \( a \neq 0 \) (mod \( n \)) we have \( a^{\phi(n)} \equiv 1 \) (mod \( n \)). It is important to know which groups \( U_n \) are cyclic, i.e., which integers \( n \) have primitive roots. The answer is given by ## Theorem 4.5.3 An integer \( n \) has a primitive root if and only if \( n = 2 \) or \( n = p^k \) (for an odd prime \( p \) and \( k \) a non-negative integer). Hence, the groups \( U_n \) are cyclic for precisely such values of \( n \). The proof of theorem 4.5.3 is a little lengthy and is omitted. It can be found, for example, in the book of LeVeque that is cited at the end of this chapter. According to theorem 4.5.3, for example, \( U_6 \) is cyclic, which we have already seen, and \( U_{12} \) is not cyclic, which the reader should check. Further, we state as an immediate consequence of theorem 4.5.3: ## Corollary 4.5.3 If \( n \) is an odd prime, then \( U_n \) is cyclic, and in particular the equation \( x^2 = 1 \) in \( U_n \) has only the solutions \( x = \pm 1 \). Next we will discuss the fact that if the integer \( n \) can be factored in the form \( n = p_1^{k_1} p_2^{k_2} \cdots p_m^{k_m} \), the full ring \( \mathbb{Z}_n \) can also be factored, in a certain sense, as a 'product' of \( \mathbb{Z}_{p_1^{k_1}}, \mathbb{Z}_{p_2^{k_2}}, \ldots, \mathbb{Z}_{p_m^{k_m}} \). Let’s take \( \mathbb{Z}_n \) as an example. Since \( 6 = 2 \cdot 3 \), we expect that somehow \( \mathbb{Z}_6 \cong \mathbb{Z}_2 \times \mathbb{Z}_3 \). What this means is that we consider ordered pairs \( (x_1, x_2) \), where \( x_1 \in \mathbb{Z}_2 \) and \( x_2 \in \mathbb{Z}_3 \). Here is how we do the arithmetic with the ordered pairs. First, \( (x_1, x_2) + (y_1, y_2) = (x_1 + y_1, x_2 + y_2) \), in which the two ‘+’ signs on the right are different: the \( x_1 + y_1 \) is done in \( \mathbb{Z}_2 \) while the \( x_2 + y_2 \) is done in \( \mathbb{Z}_3 \). Second, \( (x_1, y_1) \cdot (y_1, y_2) = (x_1 \cdot y_1, x_2 \cdot y_2) \), in which the two multiplications on the right side are different: the \( x_1 \cdot y_1 \) is done in \( \mathbb{Z}_2 \) and the \( x_2 \cdot y_2 \) is done in \( \mathbb{Z}_3 \). Therefore the 6 elements of \( \mathbb{Z}_6 \) are \[ (0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2). \] A sample of the addition process is \[ (0, 2) + (1, 1) = (0 + 1 + 2 + 1) = (1, 0). \] Where the addition of the first components was done modulo 2 and of the second components was done modulo 3. A sample of the multiplication process is \[ (1, 2) \cdot (1, 2) = (1 \cdot 1, 1 \cdot 2) = (1, 1). \] In which multiplication of the first components was done modulo 2 and of the second components was done modulo 3. In full generality, we can state the factorization of \( \mathbb{Z}_n \) as Image Analysis: ### Text Analysis: #### **Image 1:** - **Location on Page:** Top half and extending to the bottom half of the page. - **Text Detected:** 1. **Chapter Title:** - *Chapter 4: Algorithms in the Theory of Numbers* 2. **Main Content:** - The main text discusses the concepts of primitive roots modulo \( n \) with relevant theorems and corollaries stated clearly. 3. **Theorems and Corollaries:** - **Theorem 4.5.1** (Fermat’s theorem): "For every integer \( a \) that is relatively prime to \( n \) we have \( a^{\varphi(n)} \equiv 1 \) (mod \( n \))." - **Theorem 4.5.2** (Fermat’s little theorem): "If \( n \) is prime, then for all \( a \not\equiv 0 \) (mod n) we have \( a^{n-1} \equiv 1 \) (mod \( n \))." - **Theorem 4.5.3**: Discusses the conditions under which an integer \( n \) has a primitive root. - **Corollary 4.5.3.a**: States conditions for \( U_n \) (the multiplicative group modulo \( n \)) to be cyclic. 4. **Addition and Multiplication Processes:** - Describes operations within \( Z_6 \) using ordered pairs, specifically detailing summation and multiplication processes with examples. 5. **Example and General Factorization:** - Describes ordered pairs in \( Z_6 \) and provides specific examples of addition and multiplication processes for clarity. ### Object Detection and Classification: #### **Image 1:** - **Objects Detected:** - **Text Blocks:** Significant portions of text focused on theoretical formulations in number theory. - **Mathematical Symbols and Equations:** Various mathematical notations, equations, and symbols including summation and product notation, modular arithmetic symbols, and Greek letters like \( \varphi(n) \). ### Scene and Activity Analysis: #### **Image 1:** - **Scene Description:** - The image is a scanned page from a mathematical textbook. - The layout includes a structured presentation with headers, explanations, theorem statements, corollaries, and examples. - **Activities:** - The main activity involves conveying complex mathematical concepts related to number theory through the presentation of theorems and examples. ### Diagram and Chart Analysis: #### **Image 1:** - No diagrams or charts are present in this image. ### Color Analysis: #### **Image 1:** - **Color Composition:** - Predominantly black text on a white background. - The high contrast between text and background aids readability. ### Perspective and Composition: #################### File: Algebraic%20Topology%20AT-toc.pdf Page: 3 Context: # Table of Contents ## Preface - Standard Notations ix - Standard Notations xii ## Chapter 0. Some Underlying Geometric Notions 1. Homotopy and Homotopy Type 1. Cell Complexes 5 2. Operations on Spaces 8 3. Two Criteria for Homotopy Equivalence 10 4. The Homotopy Extension Property 14 ## Chapter 1. The Fundamental Group ### 1.1 Basic Constructions - Paths and Homotopy 25 - The Fundamental Group of the Circle 29 - Induced Homomorphisms 34 ### 1.2 Van Kampen’s Theorem - Free Products of Groups 41 - The van Kampen Theorem 43 - Applications to Cell Complexes 49 ### 1.3 Covering Spaces - Lifting Properties 60 - The Classification of Covering Spaces 63 - Deck Transformations and Group Actions 70 ## Additional Topics ### 1.A Graphs and Free Groups 83 ### 1.B KG(1) Spaces and Graphs of Groups 85 #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 99 Context: φ(x)2 = φ(z)2=1. Since (the square is 1,φ(z) has order 1 or 2. Hence ifψ(z)^ 1 it is of order 2. If the cyclic group CU) is of odd order, then it contains no element of even order. Hence Cu) is of even order and contains -1. Then it can contain no other element of order 2, so φ(z)*1 in this case. Hence for every x£ B,y(z)=±1.Die Suppose B generates the full group Un. Then not only for every x€ B but for every x E Un is true that φ(x)* Suppose n is not a prime power. Then s > 1 in the factorization (4.5.2) of Un. Consider the element y of Un, which, when written out as an s-tuple according to that factorization, is of the form v =(1,1,1,...1,w,w2, 1,...,1.....yw.) where the ‘v is in the ith component, y E Un, (recall that j as described above, in the second sentence of this proof). We can suppose yiy to be an element of order exactly 2' in Un since Un is cycic Consider y(z)=1. Clearly ψ(z) is not 1, for otherwise the order y'.%a would divide 2l'-lm, his impossible because m iso. Also,φ(z) is not 41 because the element -1 of Un is represented uniquely by the s-tuple all of whose entries are –1. Thus φ(s”) is neither 1 nor–1 in Un, which contradicts the italicized assertion above. Hence s= 1 nd n is a prime power, completing the proof. Now we can prove the main result of Solovay, Strassen and Rabin, which asserts that Test 4 is good. Theorem 4.7.1. LetIFn the set of integers to m3a such that In, (n)y returns ‘ inconclusive’ in Test 4. (a) If B generates Un, thenn is prime. (b) Ifnis composite then B' consists of at most half of the integers in In 1|| :| Proof: Suppose 6 €B and let m be the odd part of n - 1. Then either3m^1 1 or y=1 forsome s€ {0,4 - 1). In the former case the cyclic subgroup (C)b) has odd order, since m is odd, and in the latter case (Ch contains -1. ``` - **Analysis:** - This page contains advanced mathematical content typical of academic literature, specifically focusing on pseudoprimality tests. - **Lemma 4.7.1** and **Lemma 4.7.2** state mathematical properties related to the order of elements in certain cyclic groups. - **Theorem 4.7.1** establishes a criterion for determining the primality of an integer based on generating sets, related to Solovay, Strassen, and Rabin's work on primality tests. #### 8. **Color Analysis:** - **Dominant Colors:** - The image predominantly features black text on a white background. #### 9. **Perspective and Composition:** - **Perspective:** - The perspective is that of a flat, face-on view, typical of a page of text. - **Composition:** - The page consists of structured text with headings, sub-headings, numbered lemmas, and their proofs. The text is aligned evenly with consistent spacing, which is typical for mathematical or academic documents. #### 10. **Contextual Significance:** - **Overall Context:** - The image is part of an academic document or textbook on pseudoprimality tests in number theory or a related field. - The text on the page contributes significantly to the understanding and application of pseudoprimality tests, a crucial concept in computational number theory and cryptography. ### Summary: The analyzed image is a page from a mathematical text dealing with pseudoprimality tests. It includes detailed lemmas and a theorem related to the order of elements in cyclic groups and their application in pseudoprimality testing, with proofs providing rigorous validation of these lemmas and theorem. The page is composed primarily of structured text, black on white, and is geared towards readers with an advanced understanding of mathematics, specifically in number theory. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 53 Context: ```markdown ## 2.4 Fast Matrix Multiplication ```plaintext function MatProd(A, B: matrix: N: integer): matrix; { MatProd is AB, where A and B are N x N } uses Strassen method; if N is not a power of 2 then border A and B by zeros and columns of O’s until their size is the next power of 2 and change N; if N = 1 then MatProd := AB else partition A and B as shown in (2.4.5): I := MatProd(A11, A22, B21 + B23, N/2); II := MatProd(A11 + A12, B21, B22 + B23, N/2); ... etc., through all of the formulas shown in (2.4.3), (2.4.4), ending with ... C22 := II + I1 + V - V1 end; { MatProd } Note that this procedure calls itself recursively 7 times. The plus and minus signs in the program each represent an addition or subtraction of two matrices and, therefore, each one of them involves a call to a matrix addition or subtraction procedure (just the usual method of adding, nothing fancy!). Therefore the function MatProd makes 25 calls, 7 of which are recursively to itself, and 18 of which are to a matrix addition/subtraction routine. We will study the complexity of the routine in two ways. We will count the number of multiplications of numbers that are needed to multiply two \(2^n \times 2^n\) matrices using MatProd (call that number \(f(n)\)), and then we will count the number of additions of numbers (call that number \(g(n)\)) that MatProd needs in order to multiply two \(2^n \times 2^n\) matrices. The multiplications of numbers are easy to count. MatProd calls itself 7 times, in each of which it does exactly \(f(n - 1)\) multiplications of numbers, hence \(f(n) = 7f(n - 1) + f(0) = 1\). Therefore we see that \(f(n) = 7^n\) for all \(n \geq 0\). Hence MatProd does \(7^n\) multiplications of numbers in order to do the multiplication of \(2^n \times 2^n\) matrices. Let’s take the last sentence in the above paragraph and replace \(2^n\) by \(n\) throughout. It then tells us that MatProd does \((\log_2 N) \times N^2\) multiplications of numbers in order to do one multiplication \(N \times N\) matrices. Since \(n \log_2 n \approx n^{\log_2 3} \approx n^{2.8}\), we see that Strassen's method uses only \(O(N^{\log_2 7})\) multiplications of numbers, in place of the \(N^3\) multiplications that are required by the usual formulas. It remains to count the additions/subtractions of numbers that are needed by MatProd. In each call to it, it recurses calls to itself MatProd does \(g(n - 1)\) additions of numbers. In each of its 18 calls to the procedure that adds or subtracts matrices it does a number of additions of numbers that is equal to the square of the size of the matrices that are being added or subtracted. That size is \(2^{n - 1}\), so it follows that \(g(0) = 0\) and for \(n \geq 1\) we have \[ g(n) = 7g(n - 1) + 18 \cdot 2^{n - 2} \] We follow the method of section 1.4 on this first-order linear difference equation. Hence we make the change of variable \(h(n) = g(n) - 18 \cdot 2^{(n - 2)}\) (where \(n \geq 0\) and we find that \(h(0) = 0\) and for \(n \geq 1\), \[ h_n = \frac{9}{2} \sum_{i=1}^{n} 7^{i - 1} = \frac{9}{2} \cdot \frac{7^n - 1}{7 - 1} \] \[ \leq \frac{9}{2} \cdot \frac{7^n}{6} \] = \(21/2\) ``` Image Analysis: ### **Analysis of the Provided Page** --- **Localization and Attribution:** - The page does not contain multiple images; instead, it appears to be a single image of a written text page, likely from a book or academic paper. - The page number is indicated at the bottom as "49." --- **Text Analysis:** 1. **Page Header:** - **Section Title:** "2.4 Fast matrix multiplication" - This indicates that the content discussed in this section focuses on advanced techniques for performing matrix multiplication efficiently. 2. **Main Text Content:** - **Function Definition:** - A function named `MatrProd(A, B: matrix; N: integer): matrix` is described, where `A` and `B` are `N x N` matrices. - The function uses Strassen’s method for matrix multiplication, which is efficient for large matrices. - **Procedure Breakdown:** - Conditions are given for different cases: - If `N` is not a power of 2, matrices `A` and `B` are resized to fit the next power of 2. - If `N = 1`, the multiplication is straightforward: `A * B = AB`. - Otherwise, matrices `A` and `B` are partitioned into submatrices and further recursive calls are made. 3. **Mathematical Equations and Formulas:** - Several references are made to other sections such as (2.4.3), (2.4.4). - The significant mathematical representations in this function include the handling of matrix partitions, operations on submatrices, and recursion. 4. **Complexity Analysis:** - A detailed complexity analysis is provided, explaining: - The procedure calls itself recursively 7 times. - The calculation of the function's execution in terms of multiplications (`f(n)`) and additions (`g(n)`). 5. **Recursive Equations:** - Equations used to calculate the number of operations are provided: - \(g(n) = 7g(n - 1) + 18 \cdot 4^{n - 1}\) - The method for solving these equations is based on first-order linear difference equations. 6. **Final Calculations:** - Summation formulas: - Describing the series sums and leading to the final results for `gm` and `ym`. --- **Perspective and Composition:** - **Perspective:** - The image is taken from a bird’s-eye view or top view, focusing on the entire page with clear visibility of the text. - **Composition:** - The image is composed in a manner that centers the text, maintaining the entire page within the frame. Mathematical notations, functions, and explanations are neatly aligned and provide structured information. --- **Contextual Significance:** - **Subject Matter:** - The content appears to be from a mathematical or computer science textbook or research paper. It is dealing with topics relevant to numerical methods and algorithms, specifically how to efficiently perform matrix multiplication through Strassen’s algorithm. - **Contribution to Overall Document:** - It contributes to the reader’s understanding of efficient algorithms for matrix operations, essential in areas like linear algebra, computer graphics, and scientific computing. --- **Anomaly Detection:** - No apparent anomalies or unusual elements detected in the image. The text and mathematical content appear standard for educational or academic material. --- **Color Analysis:** - **Color Composition:** - The document is primarily monochromatic with black text on a white background, which is typical for printed academic content. - **Impact on Perception:** - The clear contrast between the text and the background ensures readability and focus on the content. --- This analysis covers the detailed examination of the aspects provided for the given visual content. The context indicates a focus on computational algorithms and their theoretical underpinnings, relevant to students and professionals in the field. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 11 Context: # 1.1 Orders of magnitude **Definition.** We say that \( f(x) = \Omega(g(x)) \) if there is an \( \epsilon > 0 \) and a sequence \( x_1, x_2, \ldots \) such that \( \forall j: f(x_j) > \epsilon g(x_j) \). Now let's introduce a hierarchy of functions according to their rates of growth when \( x \) is large. Among commonly occurring functions of \( x \) that grow without bound as \( x \to \infty \), perhaps the slowest growing ones are functions like \( \log x \) or maybe \( (\log x)^3 \) or things of that sort. It is certainly true that \( \log \log x \to \infty \), but it takes its time about it. When \( x = 1{,}000{,}000 \), for example, \( \log_2 x \) has the value 19.3. Just a bit faster growing than the ‘snails’ above is \( x^{\frac{1}{10}} \). After all, for \( x = 1{,}000 \), we have \( 1{,}000^{\frac{1}{10}} \approx 2.5 \). If we had a computer algorithm that could do \( n \) things in time \( \log n \) and someone found another method that could do the same job in time \( O(\log \log n) \), then the second method, other things being equal, would indeed be an improvement, but it might have to be extremely large before you would notice the improvement. Next on the scale of rapidity of growth we might mention the powers of \( x \). For instance, think about \( x^{0.1} \). It grows faster than \( \log x \), although you wouldn’t believe it if you tried to substitute a few values of \( x \) and to compare the answers (see exercise 1 at the end of this section). **How could we prove that \( x^{0.1} \) grows faster than \( \log x \)?** By using L'Hôpital's rule. **Example.** Consider the limit \( \lim_{x \to \infty} \frac{x^{0.1}}{\log x} \) as \( x \to \infty \). As \( x \to \infty \) assumes the indeterminate form \( \infty/\infty \), and it is therefore a candidate for L'Hôpital's rule, which tells us that if we want to find the limit then we can differentiate the numerator, differentiate the denominator, and try again to let \( x \to \infty \). If we do this, then instead of the original ratio, we find the ratio \[ \frac{(-0.1)x^{-0.9}}{1/x} = -0.1x^{0.1} \] which obviously grows without bound as \( x \to \infty \). Therefore the original ratio \( \frac{x^{0.1}}{\log x} \) also grows without bound. What we have proved, precisely, is that \( x^{0.1} \) grows faster than \( \log x \). To continue up the scale of rates of growth, we meet \( x^2, x^3, x^{15} \log^2 x \), etc., and then encounter functions that grow faster than every fixed power of \( x \) just as \( x^2 \) grows faster than every fixed power of \( x \). Consider \( e^{x^2} \). Since this is the same as \( e^{x \cdot x} \), it will obviously grow faster than \( x^{1000} \), in fact it will be larger than \( x^{1000} \) as soon as \( x > 1000 \), i.e., as soon as \( x > e^{1000} \) (don’t hold your breath!). Hence \( e^{x^2} \) is an example of a function that grows faster than every fixed power of \( x \). Another such example is \( e^{\sqrt{x}} \). **Definition.** A function that grows faster than \( x^n \) for every constant \( n \), but grows slower than \( e^c \) for every constant \( c \) is said to be of moderately exponential growth if for every \( \epsilon > 0 \) we have \( f(x) = \Omega(x^n) \) and for every \( c > 0 \) we have \( f(x) = O(e^c) \). Beyond the range of moderately exponential growth are the functions that grow exponentially fast. Typical of such functions are \( 10^{n} \), \( 2^{n^2} \), and so forth. Formally, we have the: **Definition.** A function \( f \) is of exponential growth if there exists \( c > 1 \) such that \( f(x) = \Omega(c^x) \) and there exists \( d \) such that \( f(x) = O(d^x) \). If we truncate up a function of exponential growth with smaller functions then we will not change the fact that it is of exponential growth. The series \( e^{x^2/(n^2 + 37)} \) remains of exponential growth, because \( e^{x^2} \) is by itself, and it resists the efforts of the smaller functions to change its mind. Beyond the exponentially growing functions there are functions that grow as fast as you might please. Like \( n! \), for instance, which grows faster than \( c^n \) for every fixed constant \( c \) and like \( 2^{n!} \), which grows much faster than \( n! \). The growth ranges that are the most common to computer scientists are ‘between’ the very slowly, logarithmically growing functions and the functions that are of exponential growth. The reason is simple: if a computer algorithm requires more than an exponential amount of time to do its job, then it will probably not be used, or at any rate it will be used only in highly unusual circumstances. In this book, the algorithms that we will deal with fall in this range. Now we have discussed the various symmetries and asymptotes that are used to compare the rates of growth of pairs of functions, and we have dismissed the pecking order of rapidity of growth, so that we have a small catalogue of functions that grow slowly, medium-fast, fast, and super-fast. Next, let’s look at the growth of sums that involve elementary functions, with a view toward discovering the rates at which the sums grow. Image Analysis: **Text Analysis:** 1. **Text Detection and Extraction:** - The page contains a significant amount of text discussing mathematical functions and their rates of growth. 2. **Content Analysis and Significance:** - **Definitions and Explanations:** - The text defines the notation \(f(x) = \Omega(g(x))\) for functions that grow without bound. - It discusses functions growing faster than linear, such as logarithmic, polynomial, and exponential functions. - Examples illustrate the concepts further using L'Hôpital's rule to compare growth rates. - **Function Hierarchy:** - It introduces a hierarchy of functions according to their growth rates when \(x \to \infty\). - Examples include \(\log \log x\), \(\log x\), \(x^c\), and various polynomial and exponential functions. - **Moderately Exponential and Exponential Growth:** - Definitions around moderately exponential growth (\(f(x) = \Omega(a^{(\gamma x)}\))) - Explanation involving exponential growth (\(f(x) = \Theta(e^{cx})\)) and functions of faster growth beyond exponential. - **General Discussion:** - The importance of comparing growth rates is emphasized, particularly for understanding computational complexity. - Techniques such as L'Hôpital's rule are presented for proving and understanding these contrasts in growth rates. **Localization and Attribution:** 1. **Page Layout:** - Single page visual, holding multiple sections of text with definitions, explanations, and examples all centered around mathematical functions and growth rates. **Scene and Activity Analysis:** 1. **Scene Description:** - The entire scene is text-based with mathematical formulas and theoretical explanations filling the content. - The activity involves explanations, definitions, and examples intended to educate readers on the rates of growth for functions in mathematical terms. There are no human actors; the main component is the academic content. **Diagram and Chart Analysis:** - There are no diagrams or charts in this section of text. **Product Analysis:** - No products are depicted. **Anomaly Detection:** - No anomalies are apparent. **Color Analysis:** 1. **Color Composition:** - The text is black on a white background, creating high readability and a standard academic format. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 22 Context: # Chapter 1: Mathematical Preliminaries (b) solve that one by summation and (c) go back to the original unknowns. As an example, consider the first-order equation \[ x_{n+1} = 3x_n + n \quad (n \geq 0; \, x_0 = 0). \tag{1.49} \] The winning change of variable, from (1.46), is to let \( x_n = 3^n y_n \). After substituting in (1.49) and simplifying, we find \[ y_{n+1} = y_n + \frac{1}{3^{n+1}} \quad (n \geq 0; \, y_0 = 0). \] Now by summation, \[ y_n = \sum_{j=0}^{n-1} \frac{1}{3^{j+1}} \quad (n \geq 0). \] Finally, since \( x_n = 3^n y_n \), we obtain the solution of (1.49) in the form \[ x_n = 3^n \sum_{j=0}^{n-1} \frac{1}{3^{j+1}} \quad (n \geq 0). \tag{1.40} \] This is quite an explicit answer, but the summation can, in fact, be completely removed by the same method that you used to solve exercise 1(c) of section 1.3 (try it). That pretty well takes care of first-order recurrence relations of the form \( x_{n+1} = b x_n + c_n \), and it’s time to move on to linear second-order (homogeneous) recurrence relations with constant coefficients. These are of the form \[ x_{n+1} = a x_n + b x_{n-1} \quad (n \geq 1; \, x_0 \text{ and } x_1 \text{ given}). \tag{1.11} \] If we think back to differential equations of second-order with constant coefficients, we recall that there are always solutions of the form \( y(t) = e^{rt} \) where \( r \) is constant. Hence the road to the solution of such differential equations begins by trying a solution of that form and seeing what the constant or constants \( r \) turn out to be. Analogously, equation (1.11) calls for a trial solution of the form \( x_n = r^n \). If we substitute \( x_n = r^n \) in (1.11) and cancel a common factor of \( r^{n-1} \) we obtain a quadratic equation for \( r \), namely \[ r^2 = a r + b. \tag{1.12} \] Usually this quadratic equation will have two distinct roots, say \( r_1 \) and \( r_2 \), and then the general solution of (1.11) will look like \[ x_n = c_1 r_1^n + c_2 r_2^n \quad (n = 0, 1, 2, \ldots). \tag{1.13} \] The constants \( c_1 \) and \( c_2 \) will be determined so that \( x_0 \) and \( x_1 \) have their assigned values. ## Example The recurrence for the Fibonacci numbers is \[ F_{n+1} = F_n + F_{n-1} \quad (n \geq 2; \, F_1 = 1). \tag{1.14} \] Following the recipe that was described above, we look for a solution in the form \( F_n = r^n \). After substituting in (1.14) and cancelling common factors we find that the quadratic equation for \( r \) is, in this case, \( r^2 = r + 1 \). If we denote the two roots by \( r_1 = \frac{1+\sqrt{5}}{2} \) and \( r_2 = \frac{1-\sqrt{5}}{2} \), then we find that the general solution to the Fibonacci recurrence has been obtained, and it has the form (1.13). It remains to determine the constants \( c_1 \) and \( c_2 \) from the initial conditions \( F_0 = 0 \) and \( F_1 = 1 \). From the form of the general solution we have \( F_0 = c_1 + c_2 \) and \( F_1 = c_1 r_1 + c_2 r_2 \). If we solve these two equations in the two unknowns \( c_1 \) and \( c_2 \) we find that \( c_1 = \frac{1}{\sqrt{5}} \) and \( c_2 = -\frac{1}{\sqrt{5}} \). Finally, we substitute these values of the constants into the form of the general solution, and obtain an explicit formula for the \( n \)th Fibonacci number, \[ F_n = \frac{1}{\sqrt{5}} \left( \left( \frac{1+\sqrt{5}}{2} \right)^{n+1} - \left( \frac{1-\sqrt{5}}{2} \right)^{n+1} \right) \quad (n = 0, 1, \ldots). \tag{1.15} \] Image Analysis: **Analysis based on the provided image:** 1. **Localization and Attribution:** - There is only one image in the provided content. This image occupies the entire space of the page. 2. **Text Analysis:** - **Header:** - "Chapter 1: Mathematical Preliminaries" - **Body Text:** - The text starts with a problem-solving narrative: (b) solve that one by summation and (c) go back to the original unknowns. - The text provides an example with a first-order equation: \(x_{n+1} = 3x_n + n\) (\(n \geq 0\); \(x_0 = 0\)). - A variable change is described leading to the equation \(x_n = 3^{n}y_n\). - This leads to finding \(y_{n+1} = y_n + n/3^{n+1}\) (\(n \geq 0\); \(y_0 = 0\)). - Summation methods are applied to find \( y_n = \sum_{j=1}^{n-1} j/3^{j+1} \) (\(n \geq 0\)). - Further simplification leads to \( x_n = \sum_{j=1}^{n-1} j/3^{j+1} \) (\(n \geq 0\)). - The text explains that the summation can indeed be evaluated using similar methods previously used. - Discussion moves to second-order homogeneous recurrence relations: \( x_{n+1} = a x_n + b x_{n-1}\) (\(n \geq 1\); \(x_0 \) and \( x_1\) given). - Differential equations with constant coefficients are discussed, relating them to solutions like \( y(t) = e^{\alpha t}\). - For trial solutions of the form \( x_n = \alpha^n \), the quadratic equation \( \alpha^2 = a \alpha + b \) is introduced. - This equation yields two distinct roots \( \alpha_1 \) and \( \alpha_2 \), leading to general solutions \( x_n = c_1 \alpha_1^n + c_2 \alpha_2^n \) (\(n = 0,1,2,...\)). - **Example:** - A specific example is provided for the recurrence of the Fibonacci numbers \( F_{n+1} = F_n + F_{n-1} \) (\(n \geq 1\); \(F_0 = F_1 = 1\)). - The solution follows equations in the previous sections and leads to the formula for the n-th Fibonacci number: \( F_n = \frac{1}{\sqrt{5}} \left( \left(\frac{1+\sqrt{5}}{2}\right)^n+1 - \left(\frac{1-\sqrt{5}}{2} \right)^n+1 \right) (\(n=0,1,2,...\)) (1.4.15). 3. **Diagrams and Charts:** - No diagrams or charts are provided in the image. 4. **Prozessbeschreibungen (Process Descriptions):** - The text describes a mathematical procedure for solving recurrence relations and differential equations. - Each step of the process is clearly outlined, including changing variables, solving the equation through summation, and determining coefficients for solutions. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 54 Context: # Chapter 2: Recursive Algorithms Finally, \( g(n) = 7^{n} \leq (10.5)^{n} = O(n^{m}) \), and this is \( O(N^{2.81}) \) as before. This completes the proof of ## Theorem 2.4.1 In Strassen's method of fast matrix multiplication the number of multiplications of numbers, of additions of numbers and of subtractions of numbers that are needed to multiply together two \( N \times N \) matrices are each \( O(N^{2.81}) \) (in contrast to the \( O(N^{3}) \) of the conventional method). In the years that have elapsed since Strassen's original paper many researchers have been whittling away at the exponent of \( N \) in the complexity bounds. Several new, and more elaborate algorithms have been developed, and the exponent, which was originally \( 3 \), has progressed downward through \( 2.81 \) to values below \( 2.5 \). It is widely believed that the true minimum exponent is \( 2 \), i.e., that two \( N \times N \) matrices can be multiplied in time \( O(N^{2}) \), but there seems to be a good deal of work to be done before that result can be achieved. ## Exercises for section 2.4 1. Suppose we could multiply together two \( 3 \times 3 \) matrices with only \( 22 \) multiplications of numbers. How fast, recursively, would we then be able to multiply two \( N \times N \) matrices? 2. (cont.) With what would the ‘22’ in problem 1 above have to be replaced in order to achieve an improvement over Strassen's algorithm given in the text? 3. (cont.) Still more generally, with how few multiplications would we have to be able to multiply two \( M \times N \) matrices in order to ensure that recursively we would then be able to multiply two \( N \times N \) matrices faster than the method given in this section? 4. We showed in the text that if \( N \) is a power of \( 2 \) then two \( N \times N \) matrices can be multiplied in at most time \( C N^{2} \), where \( C \) is a suitable constant. Prove that if \( N \) is not a power of \( 2 \) then two \( N \times N \) matrices can be multiplied in time at most \( T(N) \). ## 2.5 The Discrete Fourier Transform It is a lot easier to multiply two numbers than to multiply two polynomials. If you should want to multiply two polynomials \( f \) and \( g \) of degrees \( 77 \) and \( 94 \), respectively, you are in for a lot of work. To calculate just one coefficient of the product is already a lot of work. Think about the calculation of the coefficient of \( x^{60} \) in the product, for instance, and you will see that \( 50 \) numbers must be multiplied together and added in order to calculate just that one coefficient of \( fg \), and there are \( 171 \) other coefficients to calculate! Instead of calculating the coefficients of the product \( fg \), it would be much easier just to calculate the values of the product at, say, \( 172 \) points. To do that we could just multiply the values of \( f \) and \( g \) at each of those points, and after a total cost of \( 172 \) multiplications we would have the values of the product. The values of the product polynomial at \( 172 \) distinct points determine that polynomial completely, so that sequence of values is the answer. It’s just that we humans prefer to see polynomials given by means of their coefficients instead of by their values. The Fourier transform, that is the subject of this section, is a method of converting from one representation of a polynomial to another. More exactly, it converts from the sequence of coefficients of the polynomial to the sequence of values of that polynomial at a certain set of points. Ease of converting between these two representations of a polynomial is vitally important for many reasons, including multiplication of polynomials, high precision integer arithmetic in computers, creation of medical images in CAT scanners, etc. Hence, in this section we will study the discrete Fourier transform of a finite sequence of numbers, methods of calculating it, and some applications. This is a computational problem which at first glance seems very simple. What we’re asked to do, basically, is to evaluate a polynomial of degree \( n - 1 \) at \( n \) different points. So what could be so difficult about that? If we just calculate the \( n \) values by brute force, we certainly wouldn’t need to do more than \( O(n^{2}) \) multiplications altogether. Image Analysis: Based on the provided visual content, here is a comprehensive examination of the various aspects: 1. **Localization and Attribution:** - This document contains one single page with multiple sections of text. - The sections are as follows based on their positioning from top to bottom: 1. Section Title: "Chapter 2: Recursive Algorithms" 2. Theorem and Explanation: Theorem 2.4.1 and its explanation about matrix multiplication. 3. Exercises: A list of exercises related to section 2.4. 4. Section Title: "2.5 The discrete Fourier transform" 5. Explanation: An explanation on the discrete Fourier transform. 2. **Object Detection and Classification:** - There are no images containing objects to be classified in this document. 4. **Text Analysis:** - **Section Title**: "Chapter 2: Recursive Algorithms" - This section header indicates the main topic for this chapter, focusing on recursive algorithms. - **Theorem and Explanation**: "Theorem 2.4.1" - This section describes a theorem related to fast matrix multiplication, specifically highlighting the number of multiplications needed for \(N \times N\) matrices. - The text includes mathematical expressions, such as \(O(N^{2.81})\), and terms like "complexity bounds," indicating the advanced nature of the mathematical concepts being discussed. - **Exercises for section 2.4**: - A set of numbered exercises from 1 to 4 related to matrix multiplication and recursive algorithms. - Exercises involve scenarios and questions that challenge the understanding of the concepts presented in the previous section. - **Section Title**: "2.5 The discrete Fourier transform" - This title indicates the beginning of a new topic focusing on the discrete Fourier transform, which is another mathematical concept. - **Explanation**: - An in-depth explanation about calculating the discrete Fourier transform of polynomials. - This section discusses the process and significance of the discrete Fourier transform in computational mathematics. 5. **Diagram and Chart Analysis:** - There are no diagrams or charts present in this document. 6. **Product Analysis:** - There are no products depicted in the provided visual content. 7. **Anomaly Detection:** - There are no anomalies or unusual elements detected in this document. 8. **Color Analysis:** - The document is in grayscale, with no particular use of color except for varying shades of black and white. - Dominant colors are black text on a white background, which is standard for printed or digital documents. 9. **Perspective and Composition:** - The perspective is a standard eye-level view suitable for reading. - The composition is typical of a text document, structured with headings, paragraphs, and sections to organize the content logically and sequentially. Note: The metadata, contextual significance, graph and trend analysis, ablaufprozesse, prozessbeschreibungen, typen bezeichnung, and tables aspects do not apply to the provided image as no such elements are present. This covers all observable information from the provided visual content. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 40 Context: # Chapter 2: Recursive Algorithms is done first of all over all n! of the possible input orderings of the array elements, and second, for each such input ordering, we average also over all sequences of choices of the splitting elements. Now let's consider the behavior of the function \( F(n) \). What we are going to show is that \( F(n) = O(n \log n) \). The label that \( F(n) \) estimates has two components. First there are the pairwise comparisons involved in choosing a splitting element and rearranging the array about the chosen splitting value. Second there are the comparisons that are done in the two recursive calls that follow the creation of a splitter. As we have seen, the number of comparisons involved in splitting the array is \( n - 1 \). Hence it remains to estimate the number of comparisons in the recursive calls. For this purpose, suppose we have rearranged the array about the splitting element, and that it has turned out that the splitting entry now occupies the \( i \)th position in the array. Our next remark is that each value of \( i = 1, 2, \ldots, n \) is equally likely to occur. The reason for this is that we chose the splitter originally by choosing a random array entry. Since all orderings of the array entries are equally likely, the one that we happened to have chosen was just as likely to have been the largest entry as to have been the smallest, or the 17th-from-largest, or whatever. Since each value of \( i \) is equally likely, each \( i \) has probability \( \frac{1}{n} \) of being chosen as the residence of the splitter. If the splitting element lives in the \( i \)th array position, the two recursive calls to Quicksort will be on two subarrays, one of which has length \( i - 1 \) and the other of which has length \( n - i \). The average numbers of pairwise comparisons that are involved in such recursive calls are \( F(i - 1) \) and \( F(n - i) \), respectively. It follows that our average complexity function \( F \) satisfies the relation: \[ F(n) = n - 1 + \frac{1}{n} \sum_{i=1}^{n} \left[ F(i - 1) + F(n - i) \right] \quad (n \geq 2) \tag{2.2.2} \] together with the initial value \( F(0) = 0 \). How can we find the solution of the recurrence relation (2.2.2)? First let's simplify it a little by noticing that \[ \sum_{i=1}^{n} F(n - i) = F(n - 1) + F(n - 2) + \ldots + F(0) \tag{2.2.3} \] and so (2.2.2) can be written as: \[ F(n) = n - 1 + \frac{2}{n} \sum_{i=1}^{n} F(i - 1) \tag{2.4} \] We can simplify (2.4) a lot by getting rid of the summation sign. This next step may seem like a trick at first (and it is!), but it's a trick that is used in so many different ways that now we call it a 'method.' What we do is first to multiply (2.4) by \( n \), to get: \[ n F(n) = n(n - 1) + 2 \sum_{i=1}^{n} F(i - 1) \tag{2.5} \] Next, in (2.5), we replace \( n \) by \( n - 1 \), yielding: \[ (n - 1)F(n - 1) = (n - 1)(n - 2) + 2 \sum_{i=1}^{n - 1} F(i - 1) \tag{2.6} \] Finally, we subtract (2.6) from (2.5), and the summation sign obliquely disappears, leaving behind just: \[ n F(n) - (n - 1)F(n - 1) = n(n - 1) - (n - 1)(n - 2) + 2F(n - 1) \tag{2.7} \] Image Analysis: ### Analysis of the Attached Visual Content: --- #### 1. Localization and Attribution: - The image contains a single page, analyzed as **Image 1**. --- #### 2. Object Detection and Classification: - **Image 1**: - Objects detected: - Text blocks. - Mathematical equations/forms. - Page number indicator. - Categories: - The text paragraphs are categorized as educational content on computer science/mathematics. - The equations are identified as mathematical formulas. --- #### 3. Scene and Activity Analysis: - **Image 1**: - Scene Description: This page appears to be from a textbook or academic paper discussing recursive algorithms. - Activities: It shows an explanation of a recursive function \(F(n)\), involving mathematical derivations and steps to solve a recurrence relation. --- #### 4. Text Analysis: - **Image 1**: - Extracted Text: - Title: "Chapter 2: Recursive Algorithms" - Main content: Detailed explanation of a computational complexity function \(F(n)\), which is discussed through several steps and equations to show its behavior. - Specifically, steps discuss the estimation of recursive steps, the average complexity function, and simplification of summation. - Significance in Context: - The text is critical for understanding recursion in algorithms, particularly in analyzing the time complexity of recursive procedures. --- #### 5. Diagram and Chart Analysis: - Not applicable as there are no diagrams or charts present. --- #### 6. Product Analysis: - Not applicable as there are no products depicted. --- #### 7. Anomaly Detection: - No unusual elements or anomalies detected. --- #### 8. Color Analysis: - **Image 1**: - Dominant Colors: The page predominantly uses black text on a white background. This contrast facilitates readability and is typical in academic documents. --- #### 9. Perspective and Composition: - **Image 1**: - Perspective: The image is taken from a straight-on (eye-level) perspective, common for document scanning. - Composition: The page layout is structured in a standard academic format, with the title at the top, followed by coherent paragraphs and mathematical derivations down the page. --- #### 10. Contextual Significance: - **Image 1**: - The image functions as a page in an academic resource aimed at explaining recursive algorithms, which is essential for computer science students and professionals working with algorithm analysis. --- #### 11. Metadata Analysis: - Metadata not provided or visible. --- #### 12. Graph and Trend Analysis: - Not applicable as there are no graphs presented. --- #### 13. Graph Numbers: - Not applicable as there are no numerical graphs provided. --- #### Additional Aspects: - **Ablaufprozesse (Process Flows):** - **Image 1** details the process flow of estimating and simplifying the complexity of a recursive function \(F(n)\). - **Prozessbeschreibungen (Process Descriptions):** - **Image 1** provides a detailed mathematical process of breaking down and simplifying the recurrence relation to understand the behavior of \(F(n)\). - **Typen Bezeichnung (Type Designations):** - **Image 1** categorizes the function \(F(n)\) as a measure of average computational complexity. - **Trend and Interpretation:** - Analyzing the recurrence \(F(n)\), it reveals how simplification techniques can reduce computational complexity analysis. - **Tables:** - Not applicable as there are no tables present. --- This comprehensive analysis captures the essential aspects of the attached visual content, maintaining focus on relevant attributes while ignoring immaterial details. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 64 Context: # Chapter 2: Recursive Algorithms Finding the discrete Fourier transform of an array of \( n \) elements is an easy computational problem. The most obvious way to do it takes time \( O(n^2) \). We discussed a recursive method that runs in time \( O(n \log n) \) if \( n \) is a power of \( 2 \). When we write a program recursively, we are making life easier for ourselves and harder for the compiler and the computer. A single call to a recursive program can cause it to execute a tree-full of calls to itself before it is able to respond to our original request. For example, if we call Quicksort to sort the array \[ \{ 5, 8, 13, 9, 15, 29, 44, 71, 67 \} \] then the tree shown in Fig. 2.7.1 might be generated by the compiler. ![Fig. 2.7.1: A tree of calls to Quicksort](#) Again, if we call `maxsell` on the 5-cycle, the tree in Fig. 2.3.3 of calls may be created. A single invocation of `chrompoly`, where the input graph is a 4-cycle, might generate the tree of recursive calls that appears in Fig. 2.7.2. ![Fig. 2.7.2: A tree of calls to chrompoly](#) 60 Image Analysis: ### Image Analysis **Localization and Attribution:** - **Image 1**: - Located near the top of the page. - Caption: "Fig. 2.7.1: A tree of calls to Quicksort". - **Image 2**: - Positioned towards the bottom of the page. - Caption: "Fig. 2.7.2: A tree of calls to chrompoly". **Object Detection and Classification:** - **Image 1**: - Objects detected: Nodes and arrows forming a tree structure. - Key Features: Labeled nodes with numbers `{5, 8, 13, 9, 15, 29, 44, 71, 67}`, arrows indicating hierarchical recursive calls, each node splits further into sub-nodes. - **Image 2**: - Objects detected: Nodes with textual boxes and shape indicators (e.g., rectangle, triangle), arrows forming another tree structure. - Key Features: Nodes represented as shapes with labels inside boxes, a central tree structure with recursive calls. **Scene and Activity Analysis:** - **Image 1**: - Scene: A hierarchical tree structure representing a series of recursive calls in the Quicksort algorithm. - Activity: Each node corresponds to a recursive call, breaking down the array `{5, 8, 13, 9, 15, 29, 44, 71, 67}` into smaller segments until sorted. - **Image 2**: - Scene: Another hierarchical tree but for the `chrompoly` function. - Activity: Recursive breakdown of a chrompoly operation, representing an algorithm's execution trace for a 4-cycle input graph. **Text Analysis:** - Extracted from the page: - **Paragraph 1**: Discussion on the complexity of the Discrete Fourier Transform. - **Paragraph 2**: Benefits and challenges of writing recursive programs. - **Example**: Sorting array `{5, 8, 13, 9, 15, 29, 44, 71, 67}` with Quicksort. - **Caption for Fig. 2.7.1**: "A tree of calls to Quicksort". - **Further Example**: Using `maxreell` and `chrompoly` with recursive call trees. - **Caption for Fig. 2.7.2**: "A tree of calls to chrompoly". **Diagram and Chart Analysis:** - **Image 1 (Fig. 2.7.1)**: - Axis/Scale: Nodes and arrows depicting an algorithm's recursive structure, no numerical scales. - Insights: Visualization of Quicksort's recursion, showing breakdown steps of sorting an array. - **Image 2 (Fig. 2.7.2)**: - Axis/Scale: Nodes and arrows depicting recursive calls in `chrompoly`, with shapes indicating different stages of recursion. - Insights: Visualization of chrompoly recursion procedure, showing how calls propagate through the algorithm. **Prozessbeschreibungen (Process Descriptions):** - **Image 1**: - Process: Quicksort algorithm's recursive process. - Description: Recursively sorts an array by dividing it into sub-arrays based on pivoting. - **Image 2**: - Process: Chrompoly algorithm's recursive breakdown. - Description: Recursive handling of a 4-cycle graph, with various calls representing different stages of the algorithm. **Trend and Interpretation:** - **Image 1 (Quicksort)**: - Trend: Recursive division of array elements, reducing complexity at each level. - Interpretation: Visual representation emphasizes the depth of recursion. - **Image 2 (chrompoly)**: - Trend: Recursive expansion of calls as represented by different shapes. - Interpretation: Highlights the complexity and depth of recursive calls in chrompoly algorithm. ### Summary The visuals mainly illustrate concepts of recursive algorithms explained in Chapter 2. They serve to provide a visual understanding of how algorithms like Quicksort and chrompoly execute through recursive calls, breaking down complex problems into smaller, manageable ones, while maintaining the structure and flow of the process. The diagrams are integral for comprehending the recursion depth and procedural breakdown of these algorithms. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 108 Context: # Chapter 5: NP-completeness ## 5.1 Introduction In the previous chapter we met two computational problems for which fast algorithms have never been found, but neither have such algorithms been proved to be unattainable. Those were the polynomial-time problem, for which the best-known algorithm is delicately poised on the brink of polynomial time, and the integer-factoring problem, for which the known algorithms are in a more primitive condition. In this chapter we will meet a large family of such problems (hundreds of them now!). This family is not just a list of seemingly difficult computational problems. It is in fact bound together by strong ties. The collection of problems, called the NP-complete problems, includes many well-known and important questions in discrete mathematics, such as the following: - **The travelling salesman problem (TSP)**: Given n points in the plane (the "cities"), and a distance D, is there a tour that visits all of the cities, returns to its starting point, and has total length ≤ D? - **Graph coloring**: Given a graph G and an integer K. Can the vertices of G be properly colored in K or fewer colors? - **Independent set**: Given a graph G and an integer K. Does V(G) contain an independent set of K vertices? - **Bin packing**: Given a finite set S of positive integers, and an integer N (the number of bins). Does there exist a partition of S into N or fewer subsets such that the sum of the integers in each subset is ≤ K? In other words, can we "pack" the integers of S into at most N "bins", where the "capacity" of each bin is K? These are very difficult computational problems. Take the graph coloring problem, for instance. We could try every possible way of coloring the vertices of G in K colors to see if any of them work. There are \( K^n \) such possibilities, if G has n vertices. Here, if there are a large amount of colorings to be done, enough so that if G has 50 vertices and we have 10 colors at our disposal, the problem would lie far beyond the capabilities of the fastest computers that are now available. Hard problems can have easy instances. If the graph G happens to have no edges at all, or very few of them, then it will be very easy to find out if a coloring is possible, or if an independent set of K vertices is present. The real question is this (let's use "Independent Set" as an illustration): Is it possible to design an algorithm that will come packaged with a performance guarantee of the following kind: > **The seller warrants that if a graph G, of n vertices, and a positive integer K are input to this program, then it will correctly determine if there is an independent set of K or more vertices in V(G), and it will do so in an amount of time that is at most 1000n^k.** Hence there is no contradiction between the facts that the problem is hard and that there are easy cases. The hardness of the problem stems from the seeming impossibility of producing such an algorithm accompanied by such a manufacturer's warranty card. Of course the "1000" didn’t have to be exactly that. But some quite specific polynomial in the length of the input string must appear in the performance guarantee. Hence "357n²" might have appeared in the guarantee, and so might "23n", but \( n^k \) would not be allowed. Let's look carefully at why \( n^k \) would not be an acceptable worst-case polynomial time performance bound. In the "Independent Set" problem the input must describe the graph G and the integer K. How many bits are needed to do that? The graph can be specified, for example, by its vertex adjacency matrix A. This is an \( n \times n \) matrix in which the entry in row i and column j is 1 if (i, j) ∈ E(G) and is 0 else. Evidently, \( n^2 \) bits will describe the matrix A. The integers K and n can be entered with just \( O(\log n) \) bits, so the entire input bit string for the "Independent Set" problem is \( n^2 \) bits long, let D denote the number of bits in the input string. Suppose that on the warranty card the program was guaranteed to run in a time that is ≤ \( n^k \). Is this a guarantee of polynomial time performance? That question raises: Is there a polynomial P such that for every instance of "Independent Set" the running time T will be at most P(D)? Well, is it bounded? Image Analysis: ### Analysis of the Attached Visual Content ### 1. Localization and Attribution - The entire content appears to be a single page of text from a document. - It is identified as being on page 104. ### 2. Object Detection and Classification - The object is a single page of text formatted with sections and paragraphs. ### 3. Scene and Activity Analysis - The scene shows documentation of a specific chapter from a larger document or book. It focuses on explaining concepts related to NP-completeness. ### 4. Text Analysis #### Chapter Title and Sections: - **Chapter Title**: Chapter 5: NP-completeness - **Sections**: The document has the following sections: - **5.1 Introduction**: This section introduces the concept of NP-complete problems and mentions the traveling salesman problem (TSP), Graph coloring problem, Independent set problem, and Bin packing problem. #### Content: - **Introduction and Complexity Problems**: - Describes how this chapter will introduce NP-complete problems. - Mentions that these problems are difficult and includes families of these problems. - Provides examples such as TSP, graph coloring, independent set, and bin packing. - **Independent Set Problem**: - Given a graph \( G \) and an integer \( K \). - Seeks to determine if there is an independent set of \( K \) vertices. - **Warranty Example**: - An example warranty statement is given to illustrate the difficulty of finding polynomial-time algorithms for NP-complete problems. - "The seller warrants that if a graph \( G \), integer \( K \), and a positive integer \( K \) are input to this program, then it will correctly determine if there is an independent set of \( K \) vertices... in an amount of time that is at most 1000(*)\( V^{1000} \)." This highlights that current understandings of polynomial-time algorithms are inadequate for such problems. ### 8. Color Analysis - The page is primarily composed of black text on a white background, typical of printed or digital text documents. ### 9. Perspective and Composition - The perspective is a straight-on view typically used for reading text. - Composition follows a conventional document layout with headings and paragraphs arranged in a standard, readable format. ### 12. Graph and Trend Analysis - Although no graphs are present, discussions of algorithm complexity, such as "1000(*)\( V^{1000} \)" and Big-O notation hint at more complex hypothetical graphs depicting algorithmic performance metrics. ### Additional Aspects: #### Prozessbeschreibungen (Process Descriptions): - Description of the Independent Set problem and how it needs polynomial time algorithms. - Challenges in computational complexity and the specific hypothetical warranty example for illustrating polynomial-bound guarantee challenges. #### Trend and Interpretation: - Trends in computational problems around NP-completeness. - Complexity boundaries and performance issues of algorithms in finding solutions to NP-complete problems. ### Conclusion This document page is primarily an explanatory text focusing on computational complexity, specifically regarding NP-complete problems and polynomial-time algorithms. It introduces key concepts, provides examples, and uses a hypothetical scenario to illustrate points related to computational difficulty and performance guarantees. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 7 Context: So in some problems whose input bit string has \( B \) bits, the algorithm might operate in time \( 6B \) and on others it might need, say, \( 100 \log B \) time units, and for still other problems instances of length \( B \) the algorithm might need \( 5B^2 \) time units to get the job done. Well then, what would the warranty card say? I would have to pick out the worst possibility; otherwise the guarantee wouldn’t be valid. It would assure me that if the input problem instance can be described by \( B \) bits, then an answer will appear after at most \( 5B^2 \) time units. Hence, a performance guarantee is equivalent to an estimation of the worst possible scenario: the longest possible calculation that might ensue if \( B \) were input to the program. ### Worst-case bounds Worst-case bounds are the most common kind, but there are other kinds of bounds for running time. We might give an average case bound instead (see section 5.7). That wouldn’t guarantee performance no worse than \( \varepsilon \); it would state that if the performance is averaged over all possible input bit strings of \( B \) bits, then the average amount of computing time will be so-and-so (as a function of \( B \)). Now let’s talk about the difference between easy and hard computational problems and between fast and slow algorithms. A warranty that would not guarantee ‘fast’ performance would contain some function of \( B \) that grows faster than any polynomial, like \( 2^B \), for instance, or like \( n^{\log n} \), etc. It is the polynomial time vs. necessarily polynomial time guarantee that makes the difference between the easy and hard classes of problems, or between the fast and the slow algorithms. It is highly desirable to work with algorithms such that we can give a performance guarantee for running time that is at most a polynomial function of the number of bits of input. An algorithm is slow if, whatever polynomial \( P \) we think of, there exists arbitrary large values of \( B \), such that the input data strings of \( B \) bits, that cause the algorithm to do more than \( P(B) \) units of work. A computational problem is **intractable** if it can be proved that there is no fast algorithm for it. ### Example 3 Here is a familiar computational problem and a method, or algorithm, for solving it. Let’s see if the method has a polynomial time guarantee or not. The problem is this. Let \( n \) be a given integer. We want to find out if \( n \) is prime. The method that we choose is the following. For each integer \( n = 2, 3, \ldots, \sqrt{n} \) we ask if \( n \) divides (evenly into) \( n \). If all the answers are ‘No,’ then we declare \( n \) to be a prime number; else it is composite. In this way, we look at the computational complexity of this algorithm. Therefore, we are going to find out how much work is involved in doing the test. For a given integer \( n \) that we have to do can be measured in units of divisions of a whole number by another whole number. In those units, we obviously will do about \( \sqrt{n} \) units of work. It seems as though this is a tractable problem, because, after all, \( \sqrt{n} \) is of polynomial growth in \( n \). For instance, we do less than \( n \) units of work, and that’s certainly a polynomial in \( n \), isn’t it? So, according to our definition of fast and slow algorithms, this distinction was made on the basis of polynomial vs. faster-than-polynomial growth of the work done with the problem size, and therefore this problem must be easy. Right? Well, not really. Reference to the distinction between fast and slow methods will show that we have to measure the amount of work done as a function of the number of bits of input to the problem. In this example, \( n \) is not the number of bits of input. For instance, if \( n = 59 \), we don’t need 59 bits to describe it, but only 6. In general, the number of binary digits in the bit string of an integer \( n \) is close to \( \log_2 n \). So in the problem of this example, testing the primality of a given integer \( n \), the length of the input bit string \( B \) is about \( 10 \log_2 n \). Seen in this light, the calculation suddenly seems very long, a string consisting of mere log \( 0 \)’s and 1’s has caused our mighty computer to do about \( \sqrt{n} \) units of work. If we express the amount of work done as a function of \( B \), we find that the complexity of this calculation is approximately \( 2^{B/2} \), and that grows much faster than any polynomial function of \( B \). Therefore, note that we have just discussed for testing the primality of a given integer is slow. See Chapter 6 for further discussion of this problem. At the present time no one has found a fast way to test for primality, nor has anyone proved that there isn’t a fast way. Primality testing belongs to the (well-publicized) class of seemingly, but not provably, intractable problems. In this book, we will deal with some easy problems and some seemingly hard ones. It’s the ‘seemingly’ that makes things very interesting. These are problems for which no one has found a fast computer algorithm. Image Analysis: ### Localization and Attribution - The image consists of a single page of a text document. - This can be referred to as **Image 1**. ### Text Analysis #### Extracted Text The main section of the image contains printed text. Below is the complete text extracted from the image: --- **0.2 Hard vs. easy problems** So in some problems whose input bit string has \(B\) bits the algorithm might operate in time \(6B\), and on others its input need, say, \(120\log B\) time units, and for still other problem instances of length \(B\) bits the algorithm might need \(5B^2\) time units to get the job done. Well then, what would the warranty card say? It would have to pick out the worst possibility, otherwise the guarantee wouldn’t be valid. It would assure a user that if the input problem instance can be described by \(B\) bits, then an answer will appear after at most \(5B^2\) time units. Hence a performance guarantee is equivalent to an estimation of the worst possible scenario: the longest possible calculation that might ensue if \(B\) bits are input to the program. Worst-case bounds are the most common kind, but there are other kinds of bounds for running time. We might give an average case bound instead (see section 5.7). That wouldn’t guarantee performance no worse than so-and-so; it would state that if the performance is averaged over all possible input bit strings of \(B\) bits, then the average amount of computing time will be so-and-so (as a function of \(B\)). Now let’s talk about the difference between easy and hard computational problems and between fast and slow algorithms. A warranty that would not guarantee ‘fast’ performance would contain some function of \(B\) that grows faster than any polynomial. Like \(B^B\), for instance, or like \(2^{2^B}\), etc. It is the polynomial time vs. not necessarily polynomial time guarantee that makes the difference between the easy and the hard classes of problems, or between the fast and the slow algorithms. It is highly desirable to work with algorithms such that we can give a performance guarantee for running time that is at most a polynomial \(P(B)\) of the number of bits of input. An algorithm is slow if, whatever polynomial \( P \) we think of, there exist arbitrarily large values of \( B \), and input data strings of \( B \) bits, that cause the algorithm to do more than \( P(B) \) units of work. A computational problem is trackable if there is a fast algorithm that will find all its instances of it. A computational problem is intractable if it can be proved that there is no fast algorithm for it. **Example 3.** Here is a familiar computational problem and a method, or algorithm, for solving it. Let’s see if the method has a polynomial time guarantee or not. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 95 Context: # 4.5 Interlude: The ring of integers modulo n ## Theorem 4.5.4 Let \( n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r} \). The mapping which associates with each \( z \in \mathbb{Z} \), the r-tuple \( (z_1, z_2, \ldots, z_r) \), where \( z_i \equiv z \mod p_i^{e_i} \) (for \( i = 1, \ldots, r \)), and - (a) \( z_i \in \mathbb{Z}_{p_i^{e_i}} \) \( (i = 1) \) - (b) \( (z_1, z_2, \ldots, z_r) \) = \( (z_1 + y_1, \ldots, z_r + y_r) \) and - (c) \( (x_1, x_2, \ldots, x_r) \) = \( (x_1 y_1, \ldots, x_r y_r) \) - (d) In (b), the \( i \)-th sign on the right side is the addition operation of \( \mathbb{Z}_{p_i^{e_i}} \), and in (c) the \( i \)-th sign is the multiplication operation of \( \mathbb{Z}_{p_i^{e_i}} \) for each \( i = 1, \ldots, r \). The proof of Theorem 4.5.4 follows at once from the following: ## Theorem 4.5.5 (The Chinese Remainder Theorem) Let \( m_i \) \( (i = 1, \ldots, r) \) be pairwise relatively prime positive integers, and let \[ M = m_1 m_2 \cdots m_r. \] Then the mapping that associates with each integer \( z \) \( (0 \leq s < M - 1) \) the r-tuple \( (b_1, b_2, \ldots, b_r) \), where \( b_i \equiv z \mod m_i \) \( (i = 1, \ldots, r) \), is a bijection between \( \mathbb{Z}_M \) and \( \mathbb{Z}_{m_1} \times \cdots \times \mathbb{Z}_{m_r} \). A good theorem deserves a good proof. An outstanding theorem deserves two proofs, at least, one existential, and one constructive. So here are one of each for the Chinese Remainder Theorem. ### Proof 1 We must show that each r-tuple \( (b_1, \ldots, b_r) \) such that \( 0 \leq k < m_i \) \( (i = 1, \ldots, r) \) occurs exactly once. There are obviously \( M \) such vectors, and so it will be sufficient to show that each of them occurs at most once as the image of some \( x \). In the contrary case we would have \( x \) and \( x' \) both corresponding to \( (b_1, b_2, \ldots, b_r) \), say. But then \( x - x' \equiv 0 \mod \text{lcm}(m_1, \ldots, m_r) \). But \( x - x' < M \), hence \( x = x' \). ### Proof 2 Here's how to compute a number \( x \) that satisfies the simultaneous congruences \( x \equiv b_i \mod m_i \) \( (i = 1, \ldots, r) \). First, by the extended Euclidean algorithm we can quickly find \( u_1, u_2, \ldots, u_r \), such that \[ t \left(\frac{M}{m_j}\right) + u_j m_j = 1 \quad \text{for } j = 1, \ldots, r. \] Then we claim that the number \[ x = \sum_{j=1}^{r} b_j \frac{M}{m_j} t_j \mod M \] satisfies all the given congruences. Indeed, for each \( i = 1, \ldots, r \), we have \[ t_k(M / m_i) \equiv -u_i m_i \equiv 1 \mod m_i, \] where the first congruence holds because \( M/m_i \) \( (j \neq k) \) is divisible by \( m_i \), and the second congruence follows since \[ t_k(M/m_i) = u_i m_i \equiv 1 \mod m_i, \] completing the second proof of the Chinese Remainder Theorem. Now the proof of Theorem 4.5.4 follows easily, and is left as an exercise for the reader. The factorization that is described in detail in Theorem 4.5.4 will be written symbolically as \[ \mathbb{Z}_n \cong \bigoplus_{i=1}^{r} \mathbb{Z}_{p_i^{e_i}} \tag{4.5.2} \] The factorization (4.5.2) of the ring \( \mathbb{Z} \) induces a factorization \[ U_n \cong \bigoplus_{i=1}^{r} U_{p_i^{e_i}} \tag{4.5.3} \] Image Analysis: **Analysis of the Visual Content:** 1. **Localization and Attribution:** - The image contains a single page from a book or document. Thus, it's referred to as Image 1. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 106 Context: # Chapter 4: Algorithms in the Theory of Numbers (A4) Check that \( x^{p-1} \equiv 1 \, (\text{mod} \, p) \). Now let's look at the complexity of algorithm \( A \). We will measure its complexity by the number of times that we have to do a computation of either of the types (a) \( s \equiv \prod_{i} g^{s_i} \) or (b) \( (b) \, s^e \equiv 1 \, (\text{mod} \, p) \). Let \( f(p) \) be that number. Then we have (remembering that the algorithm calls itself \( t \) times) \[ f(p) = 1 + \sum_{i=2}^{r} f(p_i) + r + 1 \tag{4.10.1} \] in which the four terms, as written, correspond to the four steps in the checking algorithm. The sum begins with \( i = 2 \) because the prime 2, which is always a divisor of \( p - 1 \), is 'free.' Now (4.10.1) can be written as \[ g(p) = 1 + f(p). \tag{4.10.2} \] We claim that \( g(p) \leq 4 \log_2 p \) for all \( p \). This is surely true if \( p = 2 \). If true for primes less than \( p \) then from (4.10.2), \[ g(p) \leq \sum_{i=2}^{r} 4 \log_2 p_i + 4 \] \[ = 4 \log_2 \left( \prod_{i=2}^{r} p_i \right) + 4 \] \[ \leq 4 \log_2 \left( \frac{(p-1)}{2} \right) + 4 \] \[ = 4 \log_2 (p - 1) \leq 4 \log_2 p. \] Hence \( f(p) \leq 4 \log_2 p - 1 \) for all \( p \geq 2 \). Since the number of bits in \( p \) is \( \Theta(\log p) \), the number \( f(p) \) is a number of executions of steps that is a polynomial in the length of the input bit string. We leave to the exercises the verification that the steps that \( f(p) \) counts is also executed in polynomial time, so the entire primality-verification procedure operates in polynomial time. This yields **Theorem 4.10.1.** (W. Pratt, 1975) There exists a checking algorithm and a certificate such that primality can be verified in polynomial time. ## Exercises for section 4.10 1. Show that two positive integers of \( b \) bits each can be multiplied with at most \( O(b^2) \) bit operations (multiplications and carries). 2. Prove that step A1 of algorithm \( A \) can be executed in polynomial time, where time is now measured by the number of bit operations that are implied by the integer multiplications. 3. Same as exercise 2 above, for steps 4.3 and A4. 4. Write out the complete certificate that attests to the primality of \( p \). 5. Find an upper bound for the total number of bits that are in the certificate of the integer \( p \). 6. Carry out the complete checking algorithm on the certificate that you prepared in exercise 4 above. 7. Let \( p = 15 \). Show that there is no integer \( x \) as described in the hypotheses of lemma 4.10.1. 8. Let \( p = 17 \). Find all integers \( x \) that satisfy the hypotheses of lemma 4.10.1. Image Analysis: ### Image Analysis Report #### Localization and Attribution: - **Number of Images:** 1 image is present in the provided visual content. #### Text Analysis (Image 1): - **Detected Text:** - The title of the section is "Chapter 4: Algorithms in the Theory of Numbers." - Subsection titled (A4): "Check that x^p-1 ≡ 1 (mod p)." - Detailed text discussing the complexity of algorithm A: - Descriptions of terms like f(p), g(p), and mathematical notation. - Equations and inequalities regarding computational complexity (4.10.1, 4.10.2). - Reference to a theorem: "Theorem 4.10.1. (V. Pratt, 1975) There exist a checking algorithm and a certificate such that primality can be verified in polynomial time." - Exercises for section 4.10: 1. "Show that two positive integers of b bits each can be multiplied with at most O(b^2) bit operations (multiplications and carries)." 2. "Prove that steps A1 of algorithm A can be executed in polynomial time, where time is now measured by the number of bit operations that are implied by the integer multiplications." 3. "Same as exercise 2 above, for steps A3 and A4." 4. "Write out the complete certificate that attests to the primality of 19." 5. "Find an upper bound for the total number of bits that are in the certificate of the integer p." 6. "Carry out the complete checking algorithm on the certificate that you prepared in exercise 4 above." 7. "Let p = 15. Show that there is no integer x as described in the hypotheses of lemma 4.10.1." 8. "Let p = 17. Find all integers x that satisfy the hypotheses of lemma 4.10.1." - **Content Significance:** - The text appears to be an excerpt from a mathematical textbook or a scholarly article on algorithms used in number theory. - The exercises illustrate the practical application of theoretical concepts in algorithmic complexity and primality verification. #### Scene and Activity Analysis (Image 1): - **Scene Description:** - The image presents a text-rich content that is presumably from a mathematical text or academic material. - Mathematical formulas, algorithmic steps, and theoretical explanations dominate the scene. - **Activities Taking Place:** - The activities involved seem to be theoretical expositions, problem-solving exercises, and illustrations of algorithmic verification steps. #### Diagram and Chart Analysis (Image 1): - **Diagrams/Figures:** - Some mathematical notations and equations (e.g., 4.10.1, 4.10.2), but no explicit diagrams or charts are present. #### Contextual Significance (Image 1): - **Overall Theme/Message:** - Informative and educational content focusing on the complexity of algorithms in number theory and providing exercises to reinforce learning. ### Conclusion: This image is a detailed excerpt from an academic source on number theory and algorithm complexity, specifically discussing the computational steps and theoretical foundations required to check primality of numbers using algorithm A. The content is further enriched with exercises designed to engage readers in understanding and applying the concepts discussed. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 79 Context: # 3.6 Layered Networks After augmenting the flow in the original network \( X \), we construct a new layered network, from \( X \) and the newly augmented flow function of \( X \). The various activities that are now being described may sound like some kind of thinly disguised repackaging of the Ford-Fulkerson algorithm, but they aren’t just that, because here is what can be proved to happen: 1. First, if we start with zero flow in \( X \), make the layered network \( Y \), find a blocking flow in \( Y \), augment the flow in \( X \), make a new layered network \( Y' \), find a blocking flow, etc., then after at most \( V \) phases (“phase” = layer + block + augment) we will have found the maximum flow in \( X \) and the process will halt. 2. Second, each phase can be done very rapidly. The MPM algorithm, to be discussed in section 3.7, finds a blocking flow in a layered network in time \( O(V) \). By the height of a layered network \( Y \) we will mean the number of edges in any path from source to sink. The network of Fig. 3.6.1 has height 1. Let’s now show \[ \text{Theorem 3.6.1.} \] The heights of the layered networks that occur in the consecutive phases of the solution of a network flow problem form a strictly increasing sequence of positive integers. Hence, for a network \( X \) with \( V \) vertices, there can be at most \( V \) phases before a maximum flow is found. Let \( Y(p) \) denote the layered network that is constructed at the \( p^{th} \) phase of the computation and let \( H(p) \) denote the height of \( Y(p) \). We will first prove \[ \text{Lemma 3.6.1.} \] If \[ v_0 \to v_1 \to v_2 \to \ldots \to v_m \quad (v_0 = \text{source}) \] is a path in \( Y(p+1) \), and if every vertex \( v_i \) (for \( i = 1, \ldots, m \)) of that path also appears in \( Y(p) \), then for every \( a = 0, m \) it is true that if vertex \( v_a \) was in layer \( b \) of \( Y(p) \) then \( 2 \geq b \). **Proof of lemma:** The result is clearly true for \( a = 0 \). Suppose it is true for \( v_0, v_1, \ldots, v_a \), and suppose \( v_a, v_{a+1} \) was not present in network \( Y(p) \) since two vertices were not in the consecutive layers. Hence the flow in \( Y \) between \( v_a \) and \( v_{a+1} \) could not have been affected by the augmentation procedure of phase \( p \). But edge \( v_i \) is in \( Y(p) \). Therefore it represented an edge of \( Y \) that was helpful from \( v_a \) at the beginning of phase \( p+1 \) was unaffected by phase \( p \) but was not helpful at the beginning of phase \( p \). This contradiction establishes the lemma. Now we will prove the theorem. Let \[ s \to v_1 \to v_2 \to \ldots \to v_{H(p+1)} \to t \] be a path from source to sink in \( Y(p+1) \). Consider first the case where every vertex of the path also lies in \( Y(p) \), and let \( m = 1 \) (i.e. \( H(p + 1) = a = m \)). We conclude that \( H(p + 1) > H(p) \). Now we want to exclude the \( v_i \)’s from \( Y(p + 1) \) = \( H(p) \) and so all the edges in \( Y(p) \) that contain the sink, but that layer is special, and contains only \( t \). Hence, if \( v_a \) is in layer of \( Y(p) \), then \( b + 1 = H(p) \) by the lemma once more, i.e. \( 2 \geq b + 1 \geq H(p) \), and therefore \( H(p + 1) > H(p) \), completing the proof of theorem 3.6.1. Image Analysis: ### Analysis of the Provided Visual Content ### 1. **Localization and Attribution:** - **Image:** Single image present on the page. - **Position:** The entire page is used for text description and mathematical proof presentation. ### 2. **Object Detection and Classification:** - **Objects Detected:** - Text paragraphs. - Titles and subtitles. - Mathematical equations. - Bulleted and numbered points. ### 4. **Text Analysis:** - **Detected Text:** - **Title:** "3.6 Layered networks" - **Content:** - Description of augmenting flow in a network, construction of new layered networks, and the MPM algorithm. - Definitions of terms such as "height of layered network". - **Theorem 3.6.1:** Details heights of layered networks in relation to the sequence of vertices. - **Lemma 3.6.1:** Mathematical proof establishing conditions for paths in layered networks. - **Significance:** - The text describes a segment of a mathematical or computational theory, likely related to network flow algorithms. It methodically outlines the process of layered network construction, followed by a theorem and its proof. - **Text Sections:** - *Augmentation Process:* Describes the methods to improve flow within the network. - *Height Definition:* Establishes the concept of network height. - *Theorem 3.6.1:* Provides a specific theorem about network path lengths. - *Lemma 3.6.1:* Contains a formal proof to support the theorem. ### 9. **Perspective and Composition:** - **Perspective:** The image is a direct top-down view of a page from a document. - **Composition:** The page is densely packed with text, organized into titles, paragraphs, and equations. Key elements are: - Top-Centered Title: "3.6 Layered networks". - Text Paragraphs: Detailed theoretical description. - Subsections: Theorem and Lemma sections. - Mathematical Symbols and Equations: Supported proofs and examples. ### 12. **Graph and Trend Analysis:** - **Graph Elements:** No graphical elements present, only text and mathematical formulas. ### Additional Aspects: #### Ablaufprozesse (Process Flows): - Describes the sequential process of creating layered networks and augmenting flow in a network, tools and proofs associated with evaluating these networks. #### Prozessbeschreibungen (Process Descriptions): - Stepwise descriptions provided in the text: 1. Creation of a new layered network. 2. Finding blocking flows. 3. Repeating the process until a maximum flow is achieved. ### Conclusion: - This content is highly specialized, discussing algorithmic processes in network theory. It combines descriptive, theorematic, and lemma-focused writing to explain and verify network flow augmentation processes and properties. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 130 Context: # Chapter 5: NP-completeness At some stage, we may reach a dead end: out of colors, but not out of vertices to color. In the graph of Fig. 5.7.1, if we try to 2-color the vertices we can color vertex 1 in color 1, vertex 2 in color 2, vertex 3 in color 1, and then we'd be stuck because neither color would work on vertex 4. ![Fig. 5.7.1: Color this graph](path_to_image) When a dead end is reached, back up to the most recently colored vertex for which other color choices are available, replace its color with the next available choice, and try again to push forward to the next vertex. The (futile) attempt to color the graph in Fig. 5.7.1 with 2 colors by the backtrack method can be portrayed by the **backtrack search** tree in Fig. 5.7.2. The search is thought of as beginning at ‘Root’. The label at each node of the tree describes the colors of the vertices that have so far been colored. Thus ‘212’ means that vertices 1, 2, and 3 have been colored, respectively, in colors 2, 1, and 2. ![Fig. 5.7.2: A frustrated search tree](path_to_image) ![Fig. 5.7.3: A happy search tree](path_to_image) Image Analysis: ### Comprehensive Analysis of the Attached Visual Content #### 1. Localization and Attribution - **Image 1:** - Positioned at the top of the page. - Caption: Fig. 5.7.1: Color this graph - **Image 2:** - Positioned in the middle of the page. - Caption: Fig. 5.7.2: A frustrated search tree - **Image 3:** - Positioned at the bottom of the page. - Caption: Fig. 5.7.3: A happy search tree #### 2. Object Detection and Classification - **Image 1 (Fig. 5.7.1):** - Objects: Vertices and edges forming a graph. - Key Features: The graph consists of four vertices connected by edges, forming a shape visible as a triangle with an extended line. - **Image 2 (Fig. 5.7.2):** - Objects: Nodes and connecting lines forming a search tree. - Key Features: The nodes are labeled with numerical values indicating the coloring state of vertices. - **Image 3 (Fig. 5.7.3):** - Objects: Nodes and connecting lines forming another search tree. - Key Features: This tree is more extensive with detailed numerals representing vertex colorings. #### 3. Scene and Activity Analysis - **Image 1 (Fig. 5.7.1):** - Scene: A simple graph illustration. - Activity: An attempt to color the vertices of the graph. - **Image 2 (Fig. 5.7.2):** - Scene: A binary search tree. - Activity: Illustrates a search process that ends in frustration due to being unable to find a successful coloring of the graph. - **Image 3 (Fig. 5.7.3):** - Scene: Another binary search tree. - Activity: Shows a successful search process, resulting in heuristics that lead to a correctly colored graph. #### 4. Text Analysis - **Text Content:** - At some stage we may reach a dead end: out of colors, but not out of vertices to color... - When a dead end is reached, back up to the most recently colored vertex... - The (futile) attempt to color the graph in Fig. 5.7.1 with 2 colors by the backtrack method... - **Significance:** - The text explains the process of graph coloring and the backtracking algorithm, highlighting frustration when a dead end is reached, and satisfaction when a proper path is found. #### 7. Anomaly Detection - **Image 1:** - No unusual elements detected. - **Image 2:** - No noticeable anomalies; tree represents unsuccessful search attempts. - **Image 3:** - The volume of nodes suggests a more complex and refined search, but no anomalies. #### 8. Color Analysis - **Overall Color Composition:** - The images are in black and white, typical for textbook illustrations focusing on structure and logic rather than color impact. #### 9. Perspective and Composition - **Image 1:** - Perspective: Planar view of a graph. - Composition: Balanced with four vertices connected cleanly with edges. - **Image 2 & 3:** - Perspective: Hierarchical tree structures viewed from the root (top) downwards. - Composition: Nodes are symmetrical, illustrating the progression of search algorithms. #### 10. Contextual Significance - **Overall Contribution:** - The images visually support an explanation in a chapter about NP-completeness, emphasizing the difficulties in solving certain computational problems (e.g., graph coloring) and demonstrating both unsuccessful and successful search strategies. #### 13. Graph Numbers - **Image 2 (Fig. 5.7.2):** - Data Points: Root (1, 2), Level 1 (12, 21), Level 2 (121, 212) - **Image 3 (Fig. 5.7.3):** - Data Points: Root (1, 2), Level 1 (12, 13, 21, 23, 31, 32), Level 2 and beyond include more extensive numbering indicating paths and successful searches. #### Ablaufsprozesse (Process Flows) - **Images 2 and 3:** - These illustrate decision-making processes in search trees. #### Prozessbeschreibungen (Process Descriptions) - **Images 2 and 3:** - They depict the backtracking process in the search for valid graph colorings. #### Typen Bezeichnung (Type Designations) - **Images 2 and 3:** - Types are designated by numerical labels indicating vertex colorings at different nodes in the search process. ### Summary The content comprehensively demonstrates the concept of graph coloring in the context of NP-completeness through visual aids, including graphs and search trees, which effectively illustrate the problem-solving process via backtracking. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 24 Context: #### Prozessbeschreibungen (Process Descriptions): - The process involves defining a specific inequality, assuming bounds, and deriving a general solution that is verified at each step to ensure correctness. #### Typen Bezeichnung (Type Designations): - The notation `{x_n}` designates sequences, while `b_0, b_1, ..., b_p` and `G(n)` are specific functions and constants used in the theoretical framework. #### Trend and Interpretation: - The sequence analysis methodologies facilitate understanding the behavior of complex recurrence relations, offering insights on their long-term behavior and convergence properties. #### Tables: - No tables are present in the image. ### Summary: The image is from a mathematical textbook or academic paper detailing Theorem 1.4.1 about recurrent inequalities and its proof. It includes a range of exercises meant to deepen the reader's understanding through direct application of the theoretic concepts discussed. The main features include structured mathematical arguments, proofs, and problems presented to elucidate complex ideas systematically. The context is fundamentally educational, aimed at learners and researchers in mathematics. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 46 Context: # Chapter 2: Recursive Algorithms We will learn more about this phenomenon in Chapter 5, but for the moment let's leave just the observation that the general problem of **master** turns out to be no harder than the special case of **master** in which no vertex has more than 3 neighbors. Aside from the complexity issue, the algorithm **master** has shown how recursive ideas can be used to transform questions about graphs to questions about smaller graphs. Here's another example of such a situation. Suppose \( G \) is a graph, and that we have a certain supply of colors available. To be exact, suppose we have \( k \) colors. We can then attempt to color the vertices of \( G \) properly in \( k \) colors (see section 1.6). If we don't have enough colors, and \( G \) has lots of edges, this will not be possible. For example, suppose \( G \) is the graph of Fig. 2.3.4, and suppose we have just 3 colors available. Then there is no way to color the vertices without ever finding that both endpoints of some edge have the same color. On the other hand, if we have four colors available then we can do the job. \begin{figure}[h] \centering \includegraphics[width=0.5\textwidth]{path_to_image} \caption{Fig. 2.3.4} \end{figure} There are many interesting computational and theoretical problems in the area of coloring of graphs. Just for its general interest, we are going to mention the four-color theorem, and then we will turn to a study of some of the computational aspects of graph coloring. First, just for general cultural reasons, let's slow down for a while and discuss the relationship between graph colorings in general and the four-color problem, even though it isn't directly relevant to what we’re doing. The original question was this. Suppose that a delegation of Earthlings were to visit a distant planet and find there a society of human beings. Since that race is well known for its squabbling habits, you can be sure that the planet will have been carved up into millions of little countries, each with its own ruling class, system of government, etc., and of course, all at war with each other. The delegation wants to escape quickly, just before doing so it draws a careful map of the 5,000,000 countries into which the planet has been divided. To make the map easier to read, the countries are then colored in such a way that whenever two countries share a stretch of border they are of two different colors. Surprisingly, it was found that the coloring could be done using only red, blue, yellow, and green. It was noticed over 100 years ago that no matter how complicated a map is drawn, and no matter how many countries are involved, it seems to be possible to color the countries in such a way that: - (a) every pair of countries that have a common stretch of border have different colors and - (b) no more than four colors are used in the entire map. It was then conjectured that four colors are always sufficient for the proper coloring of the countries of any map at all. Settling this conjecture turned out to be a very hard problem. It was finally solved in 1976 by K. Appel and W. Haken* by means of an extraordinary proof with two main ingredients. First they showed how to reduce the general problem to only a finite number of cases, by a mathematical argument. Then, since the 'finite' number was quite large, they settled all of those cases with a lengthy computer calculation. So now we have the **Four Color Theorem**, which asserts that no matter how we carve up the plane or the sphere into countries, we will always be able to color those countries with not more than four colors so that countries with a common frontier are colored differently. We can change the map coloring problem into a graph coloring problem as follows. Given a map, from the map we will construct a graph \( G \). There will be a vertex of \( G \) corresponding to each country on the map. Two of these vertices will be connected by an edge of the graph \( G \) if the two countries that two countries have just a single point in common are allowed to have the same color. As an illustration: * Every planar map is four colorable, **Bull. Amer. Math. Soc.**, 82 (1976), 711-712. 42 Image Analysis: ### Analysis of Attached Visual Content #### Image Information and Localization 1. **Localization and Attribution:** - **Text-Based Paragraphs:** - The main body of the page is comprised of text arranged in paragraphs. - There are three main sections: an introduction discussing the algorithm "marset," an illustration labeled as Fig. 2.3.4, and a text block discussing the "four-color problem." - **Diagram:** - The diagram is centrally located and labeled Fig. 2.3.4. 2. **Object Detection and Classification:** - **Image 1:** Diagram - The content of the drawing depicts a simplified graph with nodes and edges. - The key features include several nodes connected by lines representing edges. 3. **Scene and Activity Analysis:** - **Image 1:** - Scene: This is a mathematical illustration showing an example graph with nodes and edges. - Activity: The graph is likely part of an explanatory section in a textbook about graph coloring (four-color theorem). 4. **Text Analysis:** - **General Text:** - This page discusses recursive algorithms, specifically focusing on the complexity of problems, introducing the algorithm "marset," and illustrating a concept by reference to Fig. 2.3.4. - This example illustrates that with a certain number of available colors, a graph's vertices can be colored appropriately without two adjacent vertices sharing the same color. - The text mentions the "four-color problem," explaining it theoretically and mentioning its historical context and significance. - **Text in Diagram (Fig. 2.3.4):** - The diagram is labeled Fig. 2.3.4 and shows the structure of a simple graph to explain a theoretical point about graph coloring. - The nodes are represented as vertices v, and the figure demonstrates an alternative view of a map coloring problem as a graph coloring problem. 8. **Color Analysis:** - **Image 1:** - The diagram does not utilize colors but rather grayscale for nodes and edges, focusing purely on the schematic depiction of the graph. 10. **Contextual Significance:** - **Overall Document:** - The provided image is from a chapter on recursive algorithms in a textbook. The focal topic of this segment is about applying recursive algorithms to the graph coloring problem (four-color problem) and how the problem can be simplified and addressed using theoretical computational approaches. - **Contributions:** - The diagram and associated explanation contribute to a deeper understanding of the four-color problem, demonstrating its complexity and providing historical context to underline its significance in computational theory. Summary: - The page is a part of a textbook on recursive algorithms. - It discusses the four-color problem utilizing an illustrative graph (Fig. 2.3.4) to conceptualize vertex coloring. - The four-color problem is explained, with emphasis on its historical computation and solutions that transformed map-coloring problems into graph-theoretical problems. - There is a detailed introduction of the problem, theoretical aspects, historical significance, solution approaches, and illustrative examples. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 82 Context: # Chapter 3: The Network Flow Problem ![Matching people to jobs](path_to_image_1) Fig. 3.8.1: Matching people to jobs ![The network for the matching problem](path_to_image_2) Fig. 3.8.2: The network for the matching problem Given capacity 1. The result for the graph of Fig. 3.8.1 is shown in Fig. 3.8.2. Consider a maximum integer-valued flow in this network, of value \( Q \). Since each edge has capacity 1, \( Q \) edges of the type \( (s,p) \) can contain a unit of flow. Out of each vertex \( p \) that receives some of this flow, there will come one unit of flow (since inflow equals outflow at such vertices), which will then cross to a vertex \( j \) of \( J \). No such \( j \) will receive more than one unit because at most one unit can leave it for the sink \( t \). Hence the flow defines a matching of \( Q \) edges of the graph \( G \). Conversely, any matching in \( G \) defines a flow, hence a maximum flow corresponds to a maximum matching. In Fig. 3.8.3 we show a maximum flow in the network of Fig. 3.8.2 and therefore a maximum matching in the graph of Fig. 3.8.1. ![A maximum flow](path_to_image_3) Fig. 3.8.3: A maximum flow 78 Image Analysis: ### Image Analysis of the Provided Visual Content #### 1. Localization and Attribution - **Image 1:** Positioned at the top of the page. - **Image 2:** Situated in the middle of the page. - **Image 3:** Located at the bottom of the page. #### 2. Object Detection and Classification - **Image 1** - **Objects Detected:** Nodes and edges. - **Key Features:** The diagram consists of two sets of nodes vertically aligned, with edges (lines) connecting nodes from the top set to those in the bottom set. Each line represents a connection linking nodes across the two groups. - **Image 2** - **Objects Detected:** Nodes, edges, and flow network details. - **Key Features:** This diagram features a more complex network of nodes and edges, with nodes grouped into two categories and edges directed between them. This depicts a network flow system. - **Image 3** - **Objects Detected:** Nodes, edges, flows, and directional capacities. - **Key Features:** Similar to the second image, this diagram illustrates a flow network. It involves multiple nodes and directed edges, representing a maximum flow scenario within a bipartite network. #### 3. Scene and Activity Analysis - **Image 1** - **Scene Description:** A bipartite graph illustrating the matching between two sets (possibly people and jobs). - **Activities:** Demonstration of possible matches between the entities in the two sets. - **Image 2** - **Scene Description:** A network flow diagram showing the network structure for solving the matching problem. - **Activities:** Explains the structure and connections within the flow network to address matching. - **Image 3** - **Scene Description:** A flow network presenting a possible maximum flow scenario. - **Activities:** Demonstrates a solution for the maximum matching problem, showing the maximum number of units flowing through the network. #### 4. Text Analysis - **Extracted Text:** - **Below Image 1:** "Fig. 3.8.1: Matching people to jobs" - **Below Image 2:** "Fig. 3.8.2: The network for the matching problem" - **Below Image 3:** "Fig. 3.8.3: A maximum flow" - **Content Significance:** - **Image 1:** This text indicates that the diagram is used to show how people can be matched to jobs, representing a simple bipartite matching problem. - **Image 2:** This description specifies that the diagram represents the network structure required for addressing the matching problem. - **Image 3:** This text implies that the diagram presents a network demonstrating the maximum flow solution in a matching problem scenario. #### 5. Diagram and Chart Analysis - **Image 1:** Represents a bipartite graph without flow annotations. It suggests potential pairings between two distinct sets. - **Image 2:** Illustrates a network with direction and flow properties needed for solving matching problems. Nodes are connected in a more detailed manner. - **Image 3:** Shows a maximum flow network, indicating the successful flow between nodes within the graph with clear connections from sources to sinks. #### 6. Product Analysis - Not applicable in this context. #### 7. Anomaly Detection - No anomalies or unusual elements detected in any of the diagrams. All elements appear standard and relevant to the context of network flow problems. #### 8. Color Analysis - All diagrams are in black and white. #### 9. Perspective and Composition - **Image 1:** Standard orthogonal top-down view depicting a simple bipartite matching. - **Image 2:** Classic flow network presentation. - **Image 3:** Orthogonal top-down view of a maximum flow scenario. #### 10. Contextual Significance - The images collectively contribute to explaining different aspects of the network flow problem, from simple matching (Image 1) to network configuration (Image 2), and finally to demonstrating a maximum flow problem (Image 3), all relevant to Chapter 3: The Network Flow Problem. #### 11. Metadata Analysis - No metadata is available for analysis in the provided images. #### 12. Graph and Trend Analysis - Not applicable as there are no additional graphs besides those already described. #### 13. Graph Numbers - Not applicable as the graphs do not contain numerical data points for tabulation. #### Ablaufprozesse (Process Flows) - The images depict the progression from simple bipartite matching to network configuration necessary for solving the matching problem and then to finding a maximum flow. #### Prozessbeschreibungen (Process Descriptions) - The images provide a visual process flow for understanding network problems: beginning with match making, followed by network configuration, and culminating in the depiction of a maximum flow scenario. #### Typen Bezeichnung (Type Designations) - The images classify different types of network flow problems and solutions pertinent to matching theory. #### Trend and Interpretation - The visual content suggests a pedagogical trend to enrich understanding of network flow problems, by moving from basic concepts to more intricate problem-solving illustrations. #### Tables - No tables are present in the images provided. Overall, the images and accompanying text offer a coherent and structured insight into the complexities and solutions associated with network flow problems in a mathematical context. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 90 Context: # Chapter 4: Algorithms in the Theory of Numbers We can now formulate recursively the extended Euclidean algorithm. ``` procedure gcdext(m, n, g, t, u); // computes g.c.d. of m and n, and finds // integers t, u that satisfy (4.3.1) if m = 0 then g := n; t := 1; u := 0 else gcdext(n mod m, g, t, u); s := u; u := t - [n/m] * u; t := s; end gcdext; ``` It is quite easy to use the algorithm above to make a proof of the main mathematical result of this section (see exercise 1), which is ## Theorem 4.3.1. Let m and n be given integers, and let g be their greatest common divisor. Then there exist integers t, u such that g = m * t + n * u. An immediate consequence of the algorithm and the theorem is the fact that finding inverses modulo a given integer is an easy computational problem. We will need to refer to that fact in the sequel, so we state it as ## Corollary 4.3.1. Let m and n be given positive integers, and let g be their g.c.d. Then there is a multiplicative inverse modulo n if and only if g = 1. In that case, the inverse can be computed in polynomial time. **Proof:** By the extended Euclidean algorithm we can find, in linear time, integers t and u such that g = m * t + n. But this last equation says that t ≡ g * (mod n). If g = 1 then it is obvious that t is the inverse modulo of m. If g > 1 then there exists no t such that m ≡ 1 (mod n) since m = 1 + n implies that the g.c.d. of m and n is 1. We will now trace the execution of `gcdext` if it is called with `(m, n) = (14, 11)`. The routine first replaces `(14, 11)` with `(11, 3)` and calls itself successively with `(3, 2)`, `(2, 1)` and `(1, 0)`. When it executes `(n, m) = (1, 0)` it encounters the `if m = 0` statement, so it sets g := 1, t := 1, u := 0. Now it can complete the execution of the call with `(m, n) = (2, 1)`, which has so far been pending. To do this it sets - u := -[n/m] * u = 1 - t := 0. The call with `(n, m) = (2, 1)` is now complete. The call to the routine with `(m, n) = (3, 2)` has been in limbo until just this moment. Now that the `(2, 1)` call is finished, the `(3, 2)` call executes and finds - u := 0 - [3/2] * 1 = 1 - t := 1. The call to the routine with `(n, m) = (11, 3)` has so far been languishing, but its turn has come. It computes - u := 1 - [11/3] * (-1) = 4 - t := -1. Finally, the original call to `gcdext` from the user, with `(m, n) = (14, 11)`, can be processed. We find - u := (-1) - [14/11] * 4 = -5 - t := -4. 86 Image Analysis: ### Image Analysis: #### Image 1: **1. Localization and Attribution:** - The image occupies the entire space on the page. - It is classified as Image 1 for the purpose of this analysis. **2. Object Detection and Classification:** - Objects detected include: - Text blocks - Mathematical notation - Code snippet in a box format - Categories: - Text: Explanatory paragraphs, titles, theorems, proofs, and corollaries. - Code: Pseudocode representation of an algorithm. **3. Scene and Activity Analysis:** - Scene Description: - The image consists of mathematical content from a book or document, explaining the extended Euclidean algorithm. - The activity involves a detailed walkthrough of an algorithm (procedures, theorems, proofs). - Main Actors: - Abstract entities representing mathematical values and procedures are described. **4. Text Analysis:** - Text Detected: - Chapter title: "Chapter 4: Algorithms in the Theory of Numbers" - Algorithm description: "We can now formulate recursively the extended Euclidean algorithm..." - Pseudocode snippet: procedure gcdext(n, m, g, t, u); computes g.c.d. of m and n, and finds integers t, u that satisfy (4.3.1) if m = 0 then g := n; t := 1; u := 0... - Theorem 4.3.1 and Corollary 4.3.1 - Proof and example execution of the algorithm. - Significance: - The chapter provides a comprehensive understanding of the extended Euclidean algorithm, crucial for number theory applications. - Theorems, corollaries, and proofs consolidate the theoretical and practical aspects of the algorithm. **9. Perspective and Composition:** - Perspective: - The image is a direct overhead view of a page, capturing all textual contents clearly. - Composition: - The composition follows a logical flow from algorithm introduction, followed by theorems, corollaries, and concrete examples. - Structured with distinct sections including pseudocode, theorems, proofs, and explanations, making it easy to follow. **13. Graph Numbers:** - Given that there are no graphs, rows, or numerical tables, this section is not applicable. #### Additional Aspects: **Ablaufprozesses (Process Flows):** - The image depicts the step-by-step process of computing the extended Euclidean algorithm: 1. Initialization and base case check. 2. Recursive calls with updated parameters. 3. Calculation of coefficients for the linear combination. 4. Procedure termination based on base case. **Prozessbeschreibungen (Process Descriptions):** - Detailed description of how the algorithm is executed, including the base case, recursive steps, and coefficient calculations. - Example tracing of the algorithm with specific inputs to illustrate the process clearly. **Typen Bezeichnung (Type Designations):** - Types defined include integers \( n \) and \( m \), and their greatest common divisor, \( g \). - Extended coefficients \( t \) and \( u \) are also detailed to satisfy the equation \( tn + um = g \). **Trend and Interpretation:** - The trend observed is the recursive nature of the algorithm that simplifies the problem step-by-step until a base case is reached. - Interprets the efficiency and effectiveness of the extended Euclidean algorithm for finding modular inverses, essential in cryptographic applications. **Tables:** - No tables detected in the image content. ### Contextual Significance: - The image, as a part of a textbook or academic paper, contributes significantly to the understanding of algorithms in number theory. - Crucial for readers looking to understand the mathematical foundations and applications of the extended Euclidean algorithm. **Conclusion:** This detailed analysis of the image highlights its focus on explaining a mathematical algorithm through text, pseudocode, and formal proofs, thus serving as an educational resource in the context of number theory. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 28 Context: # Chapter 1: Mathematical Preliminaries where we have used Stirling's formula (1.1.10). Equation (1.5) shows that the single biggest binomial coefficient accounts for a very healthy fraction of the sum of all of the coefficients of order n. Indeed, the sum of all of them is \(2^n\), and the biggest one is \(\sim \sqrt{\frac{2}{\pi n}}\). When n is large, therefore, the largest coefficient contributes a fraction \(\sim \sqrt{\frac{1}{n}}\) of the total. If we think in terms of the subsets that these coefficients count, what we will see is that a large fraction of all of the subsets of an n-set have cardinality \(\frac{n}{2}\); in fact \(\binom{n}{n/2}\) of them do. This kind of probabilistic thinking can be very useful in the design and analysis of algorithms. If we are designing an algorithm that deals with subsets of \([n]\), for instance, we should recognize that a large percentage of the customers for that algorithm will have cardinalities near \(n/2\), and make every effort to see that the algorithm is fast for such subsets, even at the expense of possibly slowing it down on subsets whose cardinalities are very small or very large. ## Exercises for section 1.5 1. How many subsets of even cardinality does \([n]\) have? 2. By observing that \((1 + x)^n(1 + x^2) = (1 + x)^{n+1}\), prove that the sum of the squares of all binomial coefficients of order n is \(2^n\). 3. Evaluate the following sums in simple form. - (i) \(\sum_{k=0}^{n} \binom{n}{k}\) - (ii) \(\sum_{k=0}^{n} \binom{n}{k}^2\) - (iii) \(\sum_{k=0}^{n} \binom{n}{k} k\) 4. Find, by direct application of Taylor’s theorem, the power series expansion of \(f(x) = \frac{1}{1 - x}\) about the origin. Express the coefficients as certain binomial coefficients. 5. Complete the following twiddles. - (i) \(\binom{n}{r} \sim ?\) - (ii) \(\left( \binom{n}{r} \right)^{1/n} \sim ?\) - (iii) \(\left( \binom{n}{r} \right)^{1/r} \sim ?\) - (iv) \(\left( \binom{n}{r} \right)^{r/n} \sim ?\) 6. How many ordered pairs of unequal elements of \([n]\) are there? 7. Which one of the numbers \(\{2^{\binom{n}{2}}\}_{n \in \mathbb{N}}\) is the biggest? ## 1.6 Graphs A graph is a collection of vertices, certain unordered pairs of which are called its edges. To describe a particular graph we first say what its vertices are, and then we say how pairs of vertices are its edges. The set of vertices of a graph \(G\) is denoted by \(V(G)\), and its set of edges is \(E(G)\). If \(u\) and \(v\) are vertices of a graph \(G\), and if \((u,v)\) is an edge of \(G\), then we say that vertices \(u\) and \(v\) are adjacent in \(G\). Consider the graph \(G\) whose vertex set is \(\{1, 2, 3, 4, 5\}\) and whose edges are the set of pairs \((1,2)\), \((2,3)\), \((4,5)\). This is a graph of 5 vertices and 5 edges. A nice way to present a graph to an audience is to draw a picture of it, instead of just listing the pairs of vertices that are its edges. To draw a picture of a graph we would first make a point for each vertex, and then we would draw arcs between two vertices \(u\) and \(v\) if and only if \((u,v)\) is an edge of the graph we are talking about. The graph \(G\) of 5 vertices and 5 edges that is listed above can be drawn as shown in Fig. 1.6.1(a). It could also be drawn as shown in Fig. 1.6.1(b). They're both the same graph. Only the pictures are different, but the pictures aren't "really" the graph; the graph is the vertex list and the edge list. The pictures are helpful to us in visualizing and remembering the graph, but that's all. The number of edges that contain (are incident with) a particular vertex \(v\) of a graph \(G\) is called the degree of that vertex, and is usually denoted by \(p(v)\). If we add up the degrees of every vertex of \(G\) we will have counted exactly two contributions from each edge of \(G\), one at each of its endpoints. Hence, for every Image Analysis: ### Comprehensive Examination of the Visual Content **Localization and Attribution:** - Single image on the page. - The image includes different sections such as text, mathematical formulas, and diagrams. - Assigned number: **Image 1**. **Text Analysis:** - **Image 1** contains extensive textual content divided into multiple sections: - **Title:** Chapter 1: Mathematical Preliminaries. - **Main Text:** Discusses Stirling's formula, binomial coefficients, subsets of n-set, probability thinking in algorithms, etc. - **Exercises for section 1.5:** Consists of seven questions that involve mathematical proofs, sums, and applications of Taylor’s theorem. - **Section 1.6 Graphs:** Describes the components and properties of graphs with an example graph G. **Object Detection and Classification:** - **Text blocks:** Several paragraphs of text, exercises, and explanations. - **Mathematical Equations and Formulas:** Various equations related to binomial coefficients, sums, Taylor's theorem, and twiddles. - **Graph Diagram:** Depicts a graph G with vertices {1, 2, 3, 4, 5} and edges {(1,2), (2,3), (3,4), (4,5), (1,5)} in Figure 1.6.1(a). **Scene and Activity Analysis:** - **Entire scene:** Academic content focusing on mathematical preliminaries, specifically: - Description of mathematical concepts. - Problem-solving exercises. - Explanation of graph theory along with a visual example. **Diagram and Chart Analysis:** - **Graph Diagram (Figure 1.6.1(a)):** - **Vertices:** {1, 2, 3, 4, 5} - **Edges:** {(1,2), (2,3), (3,4), (4,5), (1,5)} - **Explanation:** Different ways to present a graph, the figure helps in visualizing and remembering the graph structure. **Color Analysis:** - The content is in black and white, typical for textbook pages. - Dominant color: Black text on a white background. **Perspective and Composition:** - **Perspective:** Eye-level view, standard for reading academic content. - **Composition:** Well-structured layout with headings, sections, exercises, and diagrams organized logically. **Contextual Significance:** - **Overall context:** Academic document or textbook on Mathematical Preliminaries. - **Contribution:** The image breaks down complex mathematical concepts, provides exercises for practice, and visual aids (graph) for better understanding. **Ablaufprozesse (Process Flows):** - **Problem-solving process in exercises:** How to derive subsets, evaluate sums, apply formulas, and understand graph theory step-by-step. **Prozessbeschreibungen (Process Descriptions):** - **Mathematical explanations:** Show the processes involved in mathematical problem-solving. - **Graph creation and interpretation:** Process of drawing and understanding a graph’s components. **Typen Bezeichnung (Type Designations):** - **Types of mathematical problems:** Binomial coefficients sums, Taylor series, twiddles, and graph properties analysis. **Trend and Interpretation:** - **Trend:** Progressive learning and detailed comprehension of mathematical concepts and graph theory. - **Interpretation:** Structured to build foundational knowledge in mathematics through description, problem-solving, and visual examples. **Tables:** - No tables are present in this image. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 30 Context: # Chapter 1: Mathematical Preliminaries ![Fig. 1.6.3(a)](path/to/image1) ![Fig. 1.6.3(b)](path/to/image2) Likewise, not every graph has an Eulerian path. The graph in Fig. 1.6.3(a) has one and the graph in Fig. 1.6.3(b) doesn't. There is a world of difference between Eulerian and Hamiltonian paths, however. If a graph \( G \) is given, then thanks to the following elegant theorem of Euler, it is quite easy to decide whether or not \( G \) has an Eulerian path. In fact, the theorem applies also to multigraphs, which are graphs except that they are allowed to have several different edges joining the same pair of vertices. ## Theorem 1.6.1 A (multi-)graph has an Eulerian circuit (resp. path) if and only if it is connected and has no (resp. has exactly two) vertices of odd degree. **Proof:** Let \( G \) be a connected multigraph in which every vertex has even degree. We will find an Eulerian circuit in \( G \). The proof for Eulerian paths will be similar, and is omitted. The proof is by induction on the number of edges of \( G \), and the result is clearly true if \( G \) has just one edge. Hence suppose the theorem is true for all such multigraphs of fewer than \( m \) edges, and let \( G \) have \( m \) edges. We will construct an Eulerian circuit of \( G \). Begin at some vertex \( v \) and walk along some edge to a vertex \( v' \). Generically, having arrived at a vertex \( v \), depart from \( v \) along an edge that hasn't been used yet, arriving at a new vertex, etc. The process halts when we arrive for the first time at a vertex \( v' \) such that all edges incident with \( v' \) have previously been walked on, so there is no exit. We claim that \( v' = v \), i.e., we’re back where we started. Indeed, if not, then we arrived at \( v' \) one more time than we departed from it, each time using a new edge, and finding no edges remaining at the end. Thus there were an odd number of edges of \( G \) incident with \( v' \), a contradiction. Else there are edges of \( G \) that are not in \( W \). Let \( C_1, \ldots, C_k \) denote the connected components of \( G' \). Each of them has only vertices of even degree because that was true of \( G \) and of the walk \( W \) that we subtracted from \( G \). Since each of the \( C_i \) has fewer edges than \( G \) had, there is, by induction, an Eulerian circuit in each of the connected components of \( G' \). We will treat them all together to make such a circuit for \( G \) itself. Begin at the same \( v \) and walk along 0 or more edges of \( W \) until you arrive for the first time at a vertex of component \( C_i \). This will certainly happen because \( G \) is connected. Then follow the circuit of the edges of \( C_i \), which will return you to vertex \( v \). Then continue your momentarily interrupted walk \( W \) until you reach for the first time a vertex of \( C_2 \), which surely will happen because \( G \) is connected, etc., and the proof is complete. It is extremely difficult computationally to decide if a given graph has a Hamiltonian path or circuit. We will see in Chapter 5 that this question is typical of a breed of problems that are the main subject of that chapter, and are perhaps the most (in-)famous unsolved problems in theoretical computer science. Thanks to Euler's theorem (theorem 1.6.1) it is easy to decide if a graph has an Eulerian path or circuit. Next we’d like to discuss graph coloring, surely one of the prettier parts of graph theory. Suppose that there are \( k \) colors available to us, and that we are presented with a graph \( G \). A proper coloring of the vertices of \( G \) is an assignment of a color to each vertex of \( G \) in such a way that \( v_i \neq v_j \) if \( (v_i, v_j) \in E(G) \) (the colors of 26 Image Analysis: ### Analysis of Attached Visual Content #### 1. Localization and Attribution - The page contains two primary images, referred to as: - **Image 1:** Located on the left side of the page. - **Image 2:** Located on the right side of the page. #### 2. Object Detection and Classification - **Image 1**: - Objects: Graphical representation of a connected and Eulerian path graph. - Key Features: It shows vertices and edges forming a circuit that can be traced without lifting a pen off the paper. - **Image 2**: - Objects: Graphical representation of a graph that does not have an Eulerian path. - Key Features: It contains vertices and edges wherein a path covering all edges without revisiting any edge is not possible. #### 3. Scene and Activity Analysis - Each image depicts different types of graphs studied in graph theory; the first has an Eulerian path while the second does not. #### 4. Text Analysis - **Detected Text:** - Chapter 1: Mathematical Preliminaries - Fig. 1.6.3(a) to identify the graph on the left side (Image 1). - Fig. 1.6.3(b) to identify the graph on the right side (Image 2). - Explanation on Eulerian and Hamiltonian paths, highlighting Theorem 1.6.1 and its proof. - **Analysis:** - The text explains the context in which Eulerian and Hamiltonian paths operate. It presents Theorem 1.6.1, which provides conditions for a multi-graph to have an Eulerian circuit or path. - The proof walks through constructing an Eulerian circuit for graphs meeting specific conditions. #### 5. Diagram and Chart Analysis - The diagrams in Image 1 and Image 2 show different types of graphs. - Image 1: Eulerian path - the connectivity and traversal of all edges and vertices. - Image 2: Non-Eulerian path - a graph where such traversal is not possible due to uneven vertex connections. #### 8. Color Analysis - Both images use a grayscale scheme: - Dominant color is black (edges and vertices) against a white background. - The choice of black and white ensures clarity and avoids distractions in technical illustrations. #### 9. Perspective and Composition - The perspective of the illustrations is a straightforward 2D top-down view. - The composition is centered, focusing on clearly displaying the structure of the graph relevant to the theorem. #### 12. Graph and Trend Analysis - The diagrams illustrate the practical application of theoretical concepts discussed in the text. - Image 1 shows a conformance to Eulerian path properties, while Image 2 demonstrates a violation of those properties. #### 14. Contextual Significance - The images are crucial for explaining the theoretical content given in Chapter 1 of a mathematical preliminaries section, focusing on graph theory. - They serve to visually substantiate the written proof and explanation provided in the text. By analyzing these aspects, we gain a comprehensive understanding of the content in the document, extracting key mathematical principles related to Eulerian paths and circuits. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 83 Context: # 3.6 Layered Networks If we remove all the edges incident to a single vertex \( v \), we will disconnect the graph. Hence the edge connectivity cannot exceed the minimum degree of vertices in the graph. However, the edge connectivity could be a lot smaller than the minimum degree as the graph of Fig. 3.8.4 shows, in which the minimum is large, but the removal of just one edge will disconnect the graph. ![Fig. 3.8.4: Big degree, low connectivity](path_to_image) Finding the edge connectivity is quite an important combinatorial problem, and it is by no means obvious that network flow methods can be used on it, but they can, and here is how. Given \( G \), a graph of \( V \) vertices. We solve not just one, but \( V - 1 \) network flow problems, one for each vertex \( j = 2, \ldots, V \). Fix such a vertex \( j \). Then consider vertex \( 1 \) of \( G \) to be the source and vertex \( j \) to be the sink of a network \( X \). Replace each edge of \( G \) by two edges of \( X \), one in each direction, each with capacity \( 1 \). Now solve the network flow problem in \( X \), obtaining a maximum flow \( Q(j) \). Then the smallest of the numbers \( Q(j), j = 2, \ldots, V \) is the edge connectivity of \( G \). We will not prove this here. As a final application of network flow, we discuss the beautiful question of determining whether or not there is a matrix of \( 0 \)s and \( 1 \)s that has given row and column sums. For instance, there is a \( 6 \times 8 \) matrix whose row sums are respectively \( (5, 5, 4, 3, 5, 6) \) and whose column sums are \( (3, 4, 4, 4, 3, 3, 4, 3) \). Of course the phrase "row sums" means the same thing as "number of \( 1 \)s in each row" since we have said that the entries are only \( 0 \) or \( 1 \). Hence in general, let there be a given row sum vector \( (r_1, \ldots, r_m) \) and a column sum vector \( (s_1, \ldots, s_n) \). We ask if there exists an \( m \times n \) matrix \( A \) of \( 0 \)s and \( 1 \)s that has exactly \( r_j \) \( 1 \)s in the \( j \)th column, for each \( j = 1, \ldots, m \) and \( i = 1, \ldots, n \). The reader will not have noticed that for such a matrix to exist it must surely be true that \[ r_1 + \ldots + r_m = s_1 + \ldots + s_n \] since each side counts the total number of \( 1 \)s in the matrix. Hence we will suppose that (3.8.1) is true. Now we will construct a network \( Y \) of \( m + n + 2 \) vertices named \( s, t, v_1, \ldots, v_m, w_1, \ldots, w_n, \) and \( t \). Here \( s \) is an edge of capacity \( r_j \) drawn from the source \( s \) to vertex \( v_j \), for each \( j = 1, \ldots, m \), and an edge of capacity \( s_j \) drawn from vertex \( w_j \) to the sink \( t \), for each \( j = 1, \ldots, n \). Finally, there are \( mn \) edges of capacity \( 1 \) drawn from each edge to each vertex \( w_j \). Next fix a maximum flow in this network. There is a \( 0-1 \) matrix with the given row and column sum vectors if and only if a maximum flow saturates every edge outbound from the source, that is, if and only if a maximum flow has value equal to the right side of equation (3.8.1). If such a flow exists then a matrix \( A \) of the desired kind is constructed by putting \( a_{ij} = 1 \) equal to the flow in the edge from \( x_i \) to \( y_j \). --- * S. Even and R. E. Tarjan, Network flow and testing graph connectivity, SIAM J. Computing 4 (1975), 507-518. Image Analysis: ### Comprehensive Examination of the Visual Content #### 1. Localization and Attribution - **Image Position:** - **Image 1:** Positioned centrally on the page. #### 2. Object Detection and Classification - **Image 1:** - **Objects Detected:** - Two separate network graphs. - **Classification:** - Network graph components such as nodes (vertices) and edges (links). #### 3. Scene and Activity Analysis - **Image 1:** - **Scene Description:** - The image depicts two undirected graphs illustrating a concept in network theory. - **Activities:** - Visualization of network connectivity and degree concepts. #### 4. Text Analysis - **Extracted Text:** - "Fig. 3.8.4: Big degree, low connectivity" - Several paragraphs discussing network connectivity. - **Content Analysis:** - The text explains how removing all edges incident to a single vertex in a graph will disconnect the graph, comparing the minimum degree and edge connectivity. - It uses the image (Fig. 3.8.4) to exemplify a graph with high degree yet low connectivity. The mathematical approach to solving network flow problems and determining edge connectivity is also discussed. - References are made to matrices of 0's and 1's, column and row sums, and constructing networks with given properties. #### 5. Diagram and Chart Analysis - **Image 1:** - **Graphs Details:** - Two graphs show different network structures with nodes connected by edges. - **Insights:** - The first graph shows a highly connected structure where removing one edge disconnects the network. - The second graph shows a less connected network but illustrates the key concept of edge connectivity and degree within network graphs. #### 8. Color Analysis - **Image 1:** - **Color Composition:** - Black and white. - **Dominant Colors:** - Black lines and nodes on a white background. - **Impact:** - The monochromatic scheme ensures clarity and focus on the network structure without distraction. #### 9. Perspective and Composition - **Image 1:** - **Perspective:** - Straight-on view of graph illustrations. - **Composition:** - Two network graphs are centrally aligned, labeled as Fig. 3.8.4. - The graphs are placed adjacent to each other to facilitate comparison. #### 10. Contextual Significance - **Document Context:** - The image and accompanying text form a section of a technical document detailing network theory. - **Image Contribution:** - The image supports the textual description by providing a visual representation of theoretical concepts, aiding in understanding the explanation of network connectivity. #### 13. Graph Numbers - **Data Points:** - Not explicitly numbered, the image is illustrative of concepts rather than numerical data points per se. - **Numerical Analysis:** - The network graphs in the image do not present specific numerical data points but illustrate the theoretical concept with generic labeling of nodes and edges. The provided visual content is integral to understanding the theoretical framework discussed. The graph images combined with detailed textual explanation help to conceptualize the key point of high degree but low connectivity in network graphs. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 48 Context: # Chapter 2: Recursive Algorithms Consider the proper colorings in which vertices \(v\) and \(w\) have the same color. We claim that the number of such colorings is equal to the number of all colorings of a certain new graph \(G/{\sim}\); whose construction we now describe. The vertices of \(G/{\sim}\) consist of the vertices of \(G\) other than \(v\) or \(w\) and one new vertex that we will call ‘new’, so \(G/{\sim}\) will have one less vertex than \(G\) has. Now we describe the edges of \(G/{\sim}\). First, if \(a\) and \(b\) are two vertices of \(G/{\sim}\) neither of which is the new vertex ‘new’, then \((a, b)\) is an edge of \(G/{\sim}\) if and only if it is an edge of \(G\). Second, \((v, b)\) is an edge of \(G/{\sim}\) if and only if either \(b\) is not \(w\) (or both) is an edge of \(G\). We can think of this as ‘collapsing’ the graph \(G\) by imagining that the edges of \(G\) are elastic bands, and we squeeze vertices \(v\) and \(w\) together into a single vertex. The result is \(G/{\sim}\) (anyway, it is if we replace any resulting double bands by single ones!). In Fig. 2.3.7(a) we show a graph \(G\) of \(t\) vertices and a chosen edge \(e\). The two endpoints of \(e\) are \(v\) and \(w\). In Fig. 2.3.7(b) we show the graph \(G/{\sim}\) that is the result of the construction that we have just described. ![Fig. 2.3.7(a)](link_to_figure_a) ![Fig. 2.3.7(b)](link_to_figure_b) The point of the construction is the following: ## Lemma 2.3.1. Let \(v\) and \(w\) be two vertices of \(G\) such that \(e = (v, w) \in E(G)\). Then the number of proper \(K\)-colorings of \(G - \{e\}\) in which \(v\) and \(w\) have the same color is equal to the number of all proper colorings of the graph \(G/{\sim}\). ### Proof: Suppose \(G/{\sim}\) has a proper \(K\)-coloring. Color the vertices of \(G - \{e\}\) itself in \(K\) colors as follows. Every vertex of \(G - \{e\}\) other than \(v\) or \(w\) keeps the same color that it has in the coloring of \(G/{\sim}\). Vertex \(v\) and vertex \(w\) each receive the color that vertex \(uv\) has in the coloring of \(G/{\sim}\). Now we have a \(K\)-coloring of the vertices of \(G - \{e\}\). It is a proper coloring because if \(j\) is any edge of \(G - \{e\}\) then the two endpoints of \(j\) have different colors. Indeed, this is obviously true if either endpoint of \(j\) is \(v\) or \(w\) because the coloring of \(G/{\sim}\) was a proper one. There remains only the case where endpoint of \(j\) is say, \(u\) but there is some vertex \(z\) other than \(v\) or \(w\). But then the colors of \(u\) and \(z\) must be different because \(v\) and \(w\) were joined in \(G/{\sim}\) by an edge, and therefore must have gotten different colors. ## Lemma 2.3.2. Let \(e = (v, w)\) be an edge of \(G\). Then the number of proper \(K\)-colorings of \(G - \{e\}\) in which \(v\) and \(w\) have different colors is equal to the number of all proper \(K\)-colorings of \(G\) itself. ### Proof: Obvious (isn't it?). Now let’s put together the results of the two lemmas above. Let \(P(K; G)\) denote the number of ways of properly coloring the vertices of a given graph \(G\). Then lemmas 2.3.1 and 2.3.2 assert that: \[ P(K; G - \{e\}) = P(K; G/{\sim}) + P(K; G) \] 44 Image Analysis: ### 1. Localization and Attribution **Image 1:** Positioned on the left, labeled as Fig. 2.3.7(a). **Image 2:** Positioned on the right, labeled as Fig. 2.3.7(b). ### 2. Object Detection and Classification **Image 1 (Fig. 2.3.7(a)):** - Objects: Vertices and edges of a graph. - Key Features: A graph with 7 vertices labeled 1 through 7, and edges connecting them. **Image 2 (Fig. 2.3.7(b)):** - Objects: Collapsed version of the graph from Image 1. - Key Features: A graph with 6 vertices (one less vertex due to collapsing) and different connections. ### 3. Scene and Activity Analysis **Image 1:** - Scene: A representation of a graph with vertices and edges. - Activities: Display of vertices connected by edges, possibly illustrating relationships or paths within a graph. **Image 2:** - Scene: A modified version of the graph in Image 1, where two vertices have been 'collapsed' into one. - Activities: Demonstrates the process of collapsing or merging vertices within a graph. ### 4. Text Analysis - Extracted Text: - "Chapter 2: Recursive Algorithms" - Descriptions about graph coloring and collapsing a graph. - Lemma 2.3.1 and 2.3.2, along with their proofs. - Analysis: - The content provides theoretical explanations and proofs about graph coloring. - Defines the process of collapsing vertices in a graph and its impact on graph colorings. - Provides mathematical proofs to establish lemmas related to graph colorings. ### 9. Perspective and Composition **Image 1:** - Perspective: Standard front view. - Composition: Vertices and edges arranged to clearly depict the structure of the graph. **Image 2:** - Perspective: Standard front view. - Composition: Vertices and edges are arranged to show the result of collapsing vertices from the graph in Image 1. ### 10. Contextual Significance - The images complement the text, providing visual examples of the concepts discussed (graph transformations and their impact on colorings). - They enhance the reader’s understanding of theoretical content through visual representation of key ideas. ### 13. Graph Numbers **Image 1 (Fig. 2.3.7(a))**: - Vertices: 1, 2, 3, 4, 5, 6, 7 - Edges: Edges are depicted as lines connecting various pairs of these vertices, demonstrating relationships and paths within the graph. **Image 2 (Fig. 2.3.7(b))**: - Vertices: Modified set of vertices after collapsing, renumbered accordingly. - Edges: Updated edges reflecting the changes made in the graph structure post-collapsing. ### Additional Aspects **Prozessbeschreibungen (Process Descriptions):** - Description of collapsing vertices in a graph: - *New vertex "w*": Created by collapsing vertices \(v\) and \(u\). - Updated edges to redefine connections focusing on the collapsed vertex. ### Summary The images and text collectively explain the theoretical concept of graph transformation through vertex collapsing, and its effect on graph colorings. The visual representation aids in illustrating the transformation steps, while the accompanying text provides detailed proofs and explanations of the underlying principles. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 23 Context: The Fibonacci numbers are in fact 1, 1, 2, 3, 5, 8, 13, 21, 34, ... It isn't even obvious that the formula (1.4.15) gives integer values for the \( F_n \). The reader should check that the formula indeed gives the first few \( F_n \) correctly. Just to exercise our newly acquired skills in asymptotics, let’s observe that since \( (1 + \sqrt{5})/2 > 1 \) and \( |(1 - \sqrt{5})/2| < 1 \), it follows that when \( n \) is large we have \[ F_n \sim \left( \frac{(1 + \sqrt{5})}{2} \right)^{n+1}/\sqrt{5}. \] The process of looking for a solution in a certain form, namely in the form \( \alpha^n \), is subject to the same kind of special treatment, in the case of repeated roots, that we find in differential equations. Corresponding to a double root of the associated quadratic equation \( \alpha^2 - \alpha \omega - 1 = 0 \) would find two independent solutions \( \alpha^n \) and \( n \alpha^n \), so the general solution would be in the form \( \alpha^n(c_1 + c_2 n) \). ### Example Consider the recurrence \[ x_{n+1} = 2x_n - x_{n-1} \quad (n \geq 1; \quad x_0 = 5; \quad x_1 = 5). \] (1.4.16) If we try a solution of the type \( x_n = \alpha^n \), then we find that it satisfies the quadratic equation \( \alpha^2 = 2\alpha - 1 \). Hence the two roots are \( 1 \) and \( 1 \). The general solution is \( x_n = c_1 \alpha^n + c_2 n \alpha^n \). After inserting the given initial conditions, we find that \[ x_0 = 1 = c_1; \quad x_1 = 5 = c_1 + c_2. \] If we solve for \( c_1 \) and \( c_2 \) we obtain \( c_1 = 1, \quad c_2 = 4 \), and therefore the complete solution of the recurrence (1.4.16) is given by \( x_n = 4n + 1 \). Now let’s look at recurrent inequalities, like this one: \[ x_{n+1} \leq x_n + x_{n-1} + n^2 \quad (n \geq 1; \quad x_0 = 0; \quad x_1 = 0). \] (1.4.17) The question is, what restriction is placed on the growth of the sequence \( \{ x_n \} \) by (1.4.17)? By analogy with the case of difference equations with constant coefficients, the thing to try here is \( x_n \leq K n^r \). So suppose it is true that \( x_n \leq K n^r \) for all \( n = 0, 1, 2, ..., N \). Then from (1.4.17) with \( n = N \) we find \[ x_{N+1} \leq K N^r + K N^{r-1} + N^2. \] Let \( c \) be the positive real root of the equation \( z^2 = c + 1 \) and suppose that \( \alpha > c \). Then \( \alpha^2 > \alpha + 1 \), say, where \( t > 0 \). Hence, \[ x_{N+1} \leq K N^{r-1}(1 + \alpha) + N^2 = K N^{r-1}(\alpha - t) + N^2 = K^{N+1} \left( K^{r-1} - N^r \right). \] (1.4.18) In order to ensure that \( x_{N+1} < K^{N+1}N^2 \), we need to fix \( K \). \[ K > \max \left( \frac{N^2}{K^{r-1}} \right). \] (1.4.19) In which the right member is clearly finite; the inductive step will go through. The conclusion is that (1.4.17) implies that for every fixed \( t > 0 \), \( x_n = O(n^{c+\epsilon}) \), where \( c = (1 + \sqrt{5})/2 \). The same argument applies to the general situation that is expressed in Image Analysis: ### Comprehensive Examination of the Attached Visual Content #### 1. **Localization and Attribution** - **Image on Page:** - This content is located centrally on the page and spans the entire document width. - Only one image is present on the entire page. - **Image Number: 1** #### 2. **Object Detection and Classification** - **Detected Objects:** - **Text:** - The content is predominantly textual, with paragraphs of written information. - **Formulas and Equations:** - Several mathematical expressions and formulas are present throughout the content. - **Example Box:** - There is a boxed example with mathematical steps and logical explanations. #### 3. **Scene and Activity Analysis** - **Scene Description:** - The scene is an academic or instructional document, likely a textbook or a study guide on mathematics, particularly focusing on recurrence relations and asymptotics in mathematical sequences. - **Main Actors and Actions:** - **Main Actors:** - The document appears to address the reader directly, guiding them through mathematical concepts. - **Actions:** - Explaining mathematical theories. - Solving equations. - Providing proofs and examples. #### 4. **Text Analysis** - **Extracted Text:** - **Introduction:** - Discussion begins with the Fibonacci numbers and transitions into more complex topics involving asymptotic analysis and recurrence relations. - **Main Body:** - The text contains formulas (1.4.15), (1.4.16), (1.4.17), (1.4.18), and (1.4.19), discussing the growth of sequences defined by recurrence relations. - An example illustrates solving a basic recurrence relation. - **Significance:** - The text aims to educate the reader on solving recurrence relations, particularly those similar to differential equations with repeated roots. The example provides a practical demonstration of these concepts. #### 5. **Diagram and Chart Analysis** - **Contextual Insight:** - There's a lack of diagrams or charts. The image is strictly textual without visual data representations. #### 6. **Product Analysis** - **Note:** - There's no depiction of products in the content. #### 7. **Anomaly Detection** - **Noteworthy Elements:** - No anomalies or unusual elements detected. The text is consistent with academic mathematical content. #### 8. **Color Analysis** - **Color Composition:** - The entire image is in black and white, which is typical for academic and instructional materials, making it easy to read and print. #### 9. **Perspective and Composition** - **Perspective:** - Straightforward head-on view generally used for documents. - **Composition:** - The document is well-structured with: - Enlarged section title at the top ("1.4 Recurrence relations"). - Paragraphs of explanatory text. - Boxed sections with examples. - Mathematical formulas clearly separated from the main body text. #### 10. **Contextual Significance** - **Document Context:** - The image is an excerpt from a mathematical textbook or guide. - **Contribution:** - The content introduces and explains recurrence relations in sequences, providing theoretical foundations and practical examples. #### 11. **Metadata Analysis** - **Metadata Availability:** - No metadata information is available within the image itself. #### 12. **Graph and Trend Analysis** - **Analysis:** - No graphs are present. #### 13. **Graph Numbers** - **Data Points:** - No graph data points are present. #### **Additional Aspects** **Ablaufprozesse (Process Flows):** - The text describes a logical process for solving recurrence relations, demonstrating this process in a step-by-step solved example. **Prozessbeschreibungen (Process Descriptions):** - Detailed explanations of solving recurrence relations and finding solutions to related quadratic equations. **Typen Bezeichnung (Type Designations):** - Type designations involve specific mathematical terms like recurrence relations and quadratic equations. **Trend and Interpretation:** - The trend involves an increasing complexity from simple Fibonacci numbers to more complex sequence analysis techniques. **Tables:** - No tables are present. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 121 Context: # 5.4 Some other NP-complete problems To prove the claim, first suppose that \( (5.4.1) \) is satisfied by some assignment \( x^* \). Then one, at least, of the \( k \) literals \( z_1, \ldots, z_r, \text{ say } x_r \) has the value \( \text{True} \). Then we can satisfy all \( k - 2 \) of the transformed clauses \( (5.4.2) \) by assigning \( z_s = \text{True} \) for \( s \leq r - 2 \) and \( z_r = \text{False} \) for \( s > r - 2 \). It is easy to check that each one of the \( k - 2 \) new clauses is satisfied. Conversely, suppose that all of the new clauses are satisfied by some assignment of the \( x \)’s and the \( z \)’s. We will show that at least one of the \( x \)’s must be \( \text{True} \); so that the original clause will be satisfied. Suppose, to the contrary, that all of the \( x \)’s are false. Since, in the new clauses none of the \( x \)’s are negated, the fact that the new clauses are satisfied tells us that they would remain satisfied without any of the \( x \)’s. Hence the clauses \( \{x_1\}, \{x_2\}, \{x_3\}, \ldots, \{x_{k-4}, x_{k-3}\}, \{x_k\} \) are satisfied by the values of the \( x \)’s. If we scan the list from left to right we discover, in turn, that \( x_1 \) is true, \( x_2 \) is true, and finally, much to our surprise, that \( x_{k-3} \) is true, and \( x_k \) is also false, a contradiction which establishes the truth of the claim made above. The observation that the transformations just discussed can be carried out in polynomial time completes the proof of theorem \( 5.4.1 \). We remark, in passing, that the problem \( 2SAT \) is in \( P \). Our collection of NP-complete problems is growing. Now we have two, and a third is on the way. We will show next how to reduce \( 3SAT \) to a graph coloring problem, thereby proving ## Theorem 5.4.2. The graph vertex coloring problem is NP-complete. **Proof:** Given an instance of \( SAT \), that is to say, given a collection of \( C_j \) clauses, involving \( n \) variables and having at most three literals per clause, we will construct, in polynomial time, a graph \( G \) with the property that its vertices can be properly colored in \( k + 1 \) colors if and only if the given clauses are satisfiable. We will assume that \( n > 4 \), the contrary case being trivial. The graph \( G \) will have \( n + k \) vertices: \[ \{x_1, \ldots, x_n\}, \{z_1, \ldots, z_k\}, \{y_1, \ldots, y_m\}, \{C_1, \ldots, C_k\} \] Now we will describe the set of edges of \( G \). First each vertex \( x_i \) is joined to \( C_j \) for \( j = 1, \ldots, k \). Next, every vertex \( z_j \) is joined to every other vertex \( x_j (j \neq i) \), to every vertex \( z_j (j \neq i) \), and to every vertex \( x_j (j \neq i) \). Vertex \( z_j \) is connected to \( C_j \) if \( x_i \) is not one of the literals in \( C_j \). May we interrupt the proceedings to say again why we’re doing all of this? You have just read the description of a certain graph \( G \). The graph is one that can be drawn as soon as someone hands us a \( SAT \) problem. We described the graph by listing its vertices and then listing its edges. What does the graph do for us? We suppose that we have just bought a computer program that can decide if graphs are colorable in a given number of colors. We paid \$9.95 for it, and we’d like to use it. But the first problem that needs solving happens to be a \( 3SAT \) problem, not a graph coloring problem. We aren’t so easily discouraged, though. We convert the \( 3SAT \) problem into a graph (that is \( n + k \)-colorable) if and only if the original \( 3SAT \) problem was satisfiable. Now we can get our money’s worth by running the graph coloring program even though what we really wanted to do was solve a \( 3SAT \) problem. Image Analysis: ### Analysis of the Attached Visual Content #### 1. Localization and Attribution - The entire content represents a single page of text. - Since there are no distinct images other than graphs or diagrams, it's considered as one image. - Image number assigned: **Image 1** #### 4. Text Analysis - **Detected Text:** The text is primarily discussing various NP-complete problems in computational theory, specifically focusing on reducing cases to graph coloring problems and 3SAT (3-satisfiability) problems. - **Content Highlights:** - **Introduction:** The text begins by mentioning some transformations and proofs related to a problem, referencing certain formulae and conditions. - **Claim and Proof:** Discusses a claim (referencing a formula labeled \(5.4.1\)) and provides a proof that involves satisfying assignments and literals. - **Contradiction Argument:** An argument by contradiction shows that at least one of the literals must be true to satisfy the original claim. - **Transformation Observation:** Observes that certain transformations discussed can be done in polynomial time. - **Problem 2SAT:** Mentions in passing that the problem "2SAT" is in P (Polynomial time). - **Collection Growth:** Notes the growing collection of NP-complete problems, mentioning specifically the focus on reducing 3SAT to graph coloring problem. - **Theorem 5.4.2:** - States that the graph vertex coloring problem is NP-complete. - **Proof Steps:** - Given an instance of 3SAT, constructs a polynomial time graph \(G\) ensuring it is properly colored if and only if the clauses are satisfiable. - **Graph \(G\) Description:** Has \(k + n\) vertices, described with certain sets and connections. - **Vertices \(v_i, C_j:** Describes how vertices \(v_i\) and \(C_j\) are connected under specific conditions. - **Example and Application:** Discusses a practical scenario of buying a computer program to illustrate solving the graph coloring problem can incidentally solve a 3SAT problem. #### 9. Perspective and Composition - **Perspective:** The image is a direct capture of a page from a text, presumably a textbook or academic paper, shown from a top-down, orthogonal perspective. - **Composition:** - The page consists of structured paragraphs with theorem and proof formats. - Text is evenly aligned with a clear hierarchical structure (theorems, proofs, descriptions). #### 10. Contextual Significance - **Overall Document Context:** The text appears to be an excerpt from a chapter focused on computational theory, specifically dealing with NP-complete problems. - **Contribution to the Theme:** The image contributes significantly by showcasing a detailed approach to solving NP-complete problems, particularly through reducing problems to graph coloring which is a common method in computational complexity theory. #### 12. Graph and Trend Analysis - The presence of set notation and graph-related terminology indicates an underlying emphasis on showing how graph theoretical approaches can solve logical satisfiability problems. #### 13. Graph Numbers - **No Visual Graphs Detected:** This page does not contain visual graphs. Instead, it describes graphs in a theoretical context. ### Summary - The single image contains dense theoretical content related to NP-complete problems and their reduction to graph coloring problems. - Text structure includes paragraphs, theorems, proofs, and explanatory examples. - The main themes revolve around computational problem-solving techniques in polynomial time. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 44 Context: # Chapter 2: Recursive Algorithms ## Fig. 2.3.3: A tree-full of graphs is created a list of \( n(n-1)/2 \) 0's and 1's. Each 1 represents a pair that is an edge, each 0 represents one that isn't an edge. Thus \( \Theta(n^2) \) bits describe a graph. Since \( n^2 \) is a polynomial in \( n \), any function of the number of input data bits that can be bounded by a polynomial in \( n \) itself. Hence, in the case of graph algorithms, the 'easiness' vs. 'hardness' judgment is not altered if we base the distinction on polynomials in itself, rather than on polynomials in the number of bits of input data. Hence, with a clear conscience, we are going to estimate the running time or complexity of graph algorithms in terms of the number of vertices of the graph that is input. Now let’s do this for algorithm `maxest` above. The first step is to find out if \( G \) has any edges. To do this we simply have to look at the input data. In the worst case we might look at all of the input data, all \( \Theta(n^2) \) bits of it. Then, if \( G \) actually has some edges, the additional labor needed to process \( G \) consists of two recursive calls on smaller graphs and one computation of the larger of two numbers. If \( F(G) \) denotes the total amount of computational labor that we do in order to find \( \text{maxest}(G) \), then we see that \[ F(G) \leq cn^2 + F(G - \{v\}) + F(G - \{v\} - \text{Nbhd}(v)). \tag{2.3.1} \] Next, let \( f(n) = \max_{|V(G)|=n} F(G) \), and take the maximum of (2.3.1) over all graphs \( G \) of \( n \) vertices. The result is that \[ f(n) \leq cn^2 + f(n - 1) + f(n - 2). \tag{2.3.2} \] because the graph \( G - \{v\} - \text{Nbhd}(v) \) might have as many as \( n - 2 \) vertices, and would have that many if \( v \) had exactly one neighbor. Now it’s time to ‘solve’ the recurrent inequality (2.3.2). Fortunately the hard work has all been done, and the answer is in theorem 1.4.1. That theorem was designed expressly for the analysis of recursive algorithms, and in this case it tells us that \( f(n) = O(1.619^n) \). Indeed the number in that theorem is \( (1 + \sqrt{5})/2 = 1.61803 \ldots \) We chose the \( \epsilon \) that appears in the conclusion of the theorem simply by rounding \( c \) upwards. What have we learned? Algorithm `maxest` will find the answer in a time of no more than \( O(1.619^n) \) units if the input graph \( G \) has \( n \) vertices. This is a little improvement of the most simple-minded possible. Image Analysis: ### Image Analysis #### 1. Localization and Attribution - **Image 1**: The tree diagram representing a recursive algorithm for generating graphs. - **Image 2**: Located below Image 1, this depicts different stages or subgraphs in smaller boxes. #### 2. Object Detection and Classification - **Image 1**: - **Objects**: Nodes and edges of a tree diagram. - **Key Features**: Root node labeled '5', subnodes labeled '2', '3', '5' representing recursive steps. - **Image 2**: - **Objects**: Small boxes containing subgraphs. - **Key Features**: Each subgraph is labeled with numbers ('1', '2', '3', '4', '5') and connected by edges. #### 3. Scene and Activity Analysis - **Image 1**: - **Scene**: A tree diagram illustrating recursive steps in a graph algorithm. - **Activities**: Recursive partitioning of nodes in the graph. - **Image 2**: - **Scene**: Subgraphs illustrating different stages or configurations in a recursive process. - **Activities**: Visual representation of splitting and merging nodes in the graph. #### 4. Text Analysis - **Detected Text**: - **Fig. 2.3.3: A tree-full of graphs is created**: Caption explaining that the diagram shows a recursive tree of graphs. - **Paragraphs**: Explanation of the recursive algorithm, its complexities, and the computational steps involved in estimating the running time for graph algorithms. #### 7. Anomaly Detection - No anomalies detected in the images; everything seems to follow a logical recursive structure as outlined in the algorithm descriptions. #### 9. Perspective and Composition - **Perspective**: - **Image 1**: Standard top-down perspective, common in tree diagrams. - **Image 2**: Indent box layout illustrating various stages of the recursive process. - **Composition**: - **Image 1**: Root node at the top, branching out into different substeps, with clear labels. - **Image 2**: Series of smaller boxes aligned horizontally and vertically, showing different subgraph stages clearly. #### 10. Contextual Significance - **Overall Document/Website Context**: - This image is part of a chapter on recursive algorithms in a book discussing complex computational concepts. - **Contribution**: It visually supports the explanation of a recursive algorithm for graph generation, making the complex process easier to understand. #### 12. Graph and Trend Analysis - **Equation Analysis**: - **Equations**: Recursive relation \( F(G) \leq cn^2 + F(G - \{v^*\}) + F(G - \{v^*\} - Nbd(v^*)) \) and others representing the recursive splitting steps. - **Trends**: These equations model the computational complexity and show the reduction process in each step of the recursive partitioning of nodes. #### 14. Trend and Interpretation - **Trends**: - Recursive equations illustrate the decreasing complexity as the graph is partitioned into subgraphs. - Final trend interpretation shows the convergence to a final computational time of \( O(1.619^n) \). #### 15. Tables - **Implicit Table Data**: - Lists of nodes and their dependencies in the diagram provide clear hierarchical data for the recursive process. The combination of visuals, text, and equations work together to provide a comprehensive description and analysis of recursive algorithms for graphs, aiding in the understanding of their computational complexity and implementation steps. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 12 Context: # Chapter 1: Mathematical Preliminaries Think about this one: $$ f(n) = \sum_{j=0}^{n} j^2 = 1^2 + 2^2 + 3^2 + \cdots + n^2 $$ (1.1) Thus, \( f(n) \) is the sum of the squares of the first \( n \) positive integers. How fast does \( f(n) \) grow when \( n \) is large? Notice at once that among the \( n \) terms in the sum that defines \( f(n) \), the biggest one is the last one, namely \( n^2 \). Since there are \( n \) terms in the sum and the biggest one is only \( n^2 \), it is certainly true that \( f(n) = O(n^3) \), and even more, that \( f(n) \sim \frac{n^3}{3} \) for all \( n \geq 1 \). Suppose we wanted more precise information about the growth of \( f(n) \), such as a statement like \( f(n) \sim \frac{n^3}{3} \). How might we make such a better estimate? The best way to begin is to visualize the sum in (1.1) as shown in Fig. 1.1.1. ![Fig. 1.1.1: How to overestimate a sum](image_path) In that figure we see the graph of the curve \( y = x^2 \) in the \( x-y \) plane. Further, there is a rectangle drawn over every interval of unit length in the range from \( 1 \) to \( n \). The rectangles all lie under the curve. Consequently, the total area of all of the rectangles is smaller than the area under the curve, which is to say that $$ \sum_{j=1}^{n} j^2 \leq \int_{1}^{n} x^2 dx $$ (1.2) If we compare (1.2) and (1.1.1) we notice that we have proved that \( f(n) \leq ((n+1)^3 - 1)/3 \). Now we're going to get a lower bound on \( f(n) \) in the same way. This time we use the setup in Fig. 1.2, where we again show the curve \( y = x^2 \), but this time we have drawn the rectangles so they lie above the curve. From the picture we see immediately that $$ 1^2 + 2^2 + \cdots + n^2 \geq \int_{1}^{n} x^2 dx = \frac{n^3}{3}. $$ (1.3) Now our function \( f(n) \) has been bounded on both sides, rather tightly. What we know about it is that $$ \forall n \geq 1: \quad n^3/3 \leq f(n) \leq ((n+1)^3 - 1)/3. $$ From this we have immediately that \( f(n) \sim \frac{n^3}{3} \), which gives us quite a good idea of the rate of growth of \( f(n) \) when \( n \) is large. The reader will also have noticed that the \( \sim \) gives a much more satisfying estimate of growth than the \( O \) does. Image Analysis: ### Comprehensive Examination #### 1. **Localization and Attribution:** - **Image 1**: Located at the center of the page; designated as Figure 1.1.1. #### 2. **Object Detection and Classification:** - **Image 1**: - Objects: Graph of the curve \( y = x^2 \), rectangles beneath the curve. - Key Features: The curve is a parabola representing the function \( y = x^2 \). Rectangles are under the curve for intervals from \( x = 1 \) to \( x = n \), with the width of each rectangle equal to 1. #### 3. **Scene and Activity Analysis:** - **Image 1**: - **Scene**: Mathematical visualization intended to explain the relationship between a summation and an integral. - **Activity**: The graph depicts the process of overestimating a sum by using rectangles under the curve \( y = x^2 \). #### 4. **Text Analysis:** - **Visible Text**: - Equation (1.1.1): \( f(n) = \sum_{j=1}^n j^2 \) - The explanation of how to estimate \( f(n) \) - Equation (1.1.2): \( \sum_{j=1}^{n-1} j^2 \leq \int_1^n x^2 dx = (n^3 - 1)/3 \) - Equation (1.1.3): \( \sum_{j=1}^n j^2 \geq \int_0^n x^2 dx = n^3 /3 \) - These textual elements discuss methods to overestimate and underestimate the sum of squares of the first n positive integers using integrals. #### 5. **Diagram and Chart Analysis:** - **Image 1**: - The chart depicts the curve \( y = x^2 \) and rectangles under the curve to demonstrate the approximation of the sum by an integral. - Axes: x-axis (horizontal) and y-axis (vertical) represent the x and y coordinates respectively. #### 12. **Graph and Trend Analysis:** - **Image 1**: - **Trend**: The graph illustrates how the area under the curve \( y = x^2 \) can be approximated using a series of rectangles. - **Significance**: This provides an intuitive way to understand the summation \( \sum_{j=1}^n j^2 \) in terms of integrals, showing that the sum can be bounded by an integral from above and below. #### 13. **Graph Numbers:** - **Image 1**: - The sum of squares of integers is approximated by the integral of \( x^2 \). #### **Ablaufprozesse (Process Flows):** - **Image 1**: - The process of using the integral to overestimate the summation involves calculating the area under the curve \( y = x^2 \) by using rectangles. #### **Prozessbeschreibungen (Process Descriptions):** - **Image 1**: - The text discusses the approach to bound the summation \( f(n) \) using integrals both from below and above. #### **Typen Bezeichnung (Type Designations):** - **Image 1**: - Designations include the summation \( \sum_{j=1}^n j^2 \) and its approximation using the integral \( \int_1^n x^2 dx \). ### Summary: - The document discusses a mathematical method for estimating the growth of the function \( f(n) = \sum_{j=1}^n j^2 \). - Visual aids, including a graph (Image 1, Figure 1.1.1), help demonstrate how to overestimate and underestimate this function using integral approximations. - Equations and textual explanations provide detailed insights into the bounding process. - This analysis emphasizes the significant concept of approximating sums via integral bounds. The page is clearly structured to teach the mathematical concept efficiently, using visual aids and step-by-step explanations. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 60 Context: **Text Analysis:** **Image 1:** - **Detected and extracted text:** ``` Chapter 2: Recursive Algorithms ... and so forth. To find the solution in a pleasant form, let n = p^aqi^a ... p_ka^ak (2.5.19) be the canonical factorization of n into primes. We claim that the function H(n) = a1 (p − 1) + a(p − 1) + acp2 − 1) + ...+ a (p − 1) (2.5.20) is the solution of (2.5.18) (this claim is obviously (x) correct if n is prime). To prove the claim in general, suppose it to be true for 1... 1, and suppose that n is not prime. Then every divisor d of n must be of the form d = p^i, pzp^, .. . Heat where the primes are the same as those that appear in (2.5.19) and each h, is 4; < a,. Hence from (2.5.18) we get H(n) = min((a − b)(p − 1) + ...to + ... (a − b)(p − 1) + p^i . pAl ... pa ^ 1) − 1 where now the 'min' extends over all admissible choices of the 's, namely exponents 4,..., such that 0 bi s ai (v i = 1, s) and not all bi are 0 and not all al) b = cl. One such admissible choice would be to take, say, b, = 1 and all other a, 0. If we let H(hi...b) denote the quantity in braces in (2.5.21), then with this choice the value of H would be a,(p, – 1) + ... (a(p, – 1) + exactly what we need to prove our claim (2.5.20). Hence what we have to show is that the above choice of the 's is the best one. We will show that if one of the b, is larger than 1 then we can reduce it without increasing the value of H. To prove this, observe that for each i = 1, s we have H(d, ...b, b + 1, ...b) – H(hi..., b) = – p^b + 41(a _ 1) = d – 1)(p –1). Since the divisor d 2 and the prime 'i' 2, the last difference is nonnegative. Hence H doesn’t increase if we decrease one of the bi's by 1 unit, as long as not all bi = 0. It follows that the minimum of H occurs among the prime divisors of d n. Further, if dis prime, then we can easily check from (2.5.21) that it doesn't matter which prime divisor of 'n that we choose to be + d, the function H(n) is always given by (2.5.20). If we recall the change of variable g(y) = nli(y) we find that we have proved Theorem 2.5.2. (Complexity of the Fast Fourier Transform) The best choice of the factorization 'n to nodes in algorithm FFT is to take r, to be a prime divisor of n. If that is done, then algorithm FFT requires g(n) = a(p_ (p, – 1) + a(p – 1) + ...+ a(p, 1)) complex multiplications in order to do its job, where n = p^i . pa^ ... ... pg, f is the canonical factorization of the integer 'n. Table 2.5.1 shows the number g(y) of complex multiplications required by FFT as a function of 'n. The saving over the straightforward algorithm that uses n(n – 1) multiplications for each is is apparent. If it is a power of 2, Say n = 2', then the formula of theorem 2.5.2 reduces to g(y) In log/n/g log 2, i agreement with theorem 2.5.1. What does the formula say if n is a power of 3? It if is a product of distinct primes? 2.6 Applications of the FFT Finally, we will discuss some applications of the FFT. A family of such applications begins with the observation that the FFT provides the fastest game in town for multiplying two polynomials together. Consider a multiplication like (1 + 2x + 2x - 2x x³) - (4 -5x - x - 2 - x³ + 114x + x³). ``` - **Analysis:** - The excerpt is from "Chapter 2: Recursive Algorithms." - The focus is on the factorization of integers into their prime components and the related function \( H(n) \). - Mathematical notations such as factorization, algorithms, and theorems are present. - The text delves into the details of finding the minimum cost of complex multiplications using the Fast Fourier Transform (FFT). **Diagram and Chart Analysis:** **Image 1:** - **Detected Table:** - "Table 2.5.1 shows the number g(n) of complex multiplications required by FFT as a function of n." - **Analysis:** - The table likely includes data about complex multiplications for different values of \( n \). - Exact content and numbers from the table cannot be extracted from the visual, but it is referred to in the explanatory text. **Perspective and Composition:** **Image 1:** - **Perspective:** - The image is a straight-on, centered view of a single-page mathematical text. - **Composition:** - The page is arranged in a structured format, typically found in academic textbooks, with sections, sub-sections, and mathematical notation. **Contextual Significance:** **Image 1:** - The text appears in the context of a textbook or academic paper focused on recursive algorithms. - The page elaborates on the theory and practical applications of prime factorization and FFT, emphasizing mathematical proofs and theorems. - This page contributes to the overall theme of computational efficiency and algorithm optimization within the wider context of the document. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 95 Context: Theorem 4.5.4. Let \( \pi = p_1^{n_1} p_2^{n_2} \cdots p_r^{n_r} \). The mapping which associates with each \( z \in \mathbb{Z} \), the r-tuple \((z_1, z_2, \ldots, z_r)\), where \( z_i \equiv z \mod p_i^{n_i} \) (for \( i = 1, \ldots, r \)), in which - (a) \( z_i \in \mathbb{Z}_{p_i^{n_i}} \) (for \( i = 1 \)) - (b) \((z_1, \ldots, z_r) = (z_1 + y_1, \ldots, z_r + y_r)\) and - (c) \((x_1, \ldots, x_r) \cdot (y_1, \ldots, y_r) = (x_1 y_1, \ldots, x_r y_r)\) - (d) In (b), the \( i \)-th \( + \) sign on the right side is the addition operation of \( \mathbb{Z}_{p_i^{n_i}} \) and in (c) the \( i \)-th \( \cdot \) sign is the multiplication operation of \( \mathbb{Z}_{p_i^{n_i}} \), for each \( i = 1, \ldots, r \). The proof of theorem 4.5.4 follows at once from the following: Theorem 4.5.5 (‘The Chinese Remainder Theorem’). Let \( m_i \) (for \( i = 1, r \)) be pairwise relatively prime positive integers, and let \[ M = m_1 m_2 \cdots m_r. \] Then the mapping that associates with each integer \( x \) (for \( 0 \leq s < M - 1 \)) the r-tuple \( (b_1, b_2, \ldots, b_r) \), where \( b_i \equiv x \mod m_i \) (for \( i = 1, \ldots, r \)), is a bijection between \( \mathbb{Z}/M\mathbb{Z} \) and \( \mathbb{Z}/m_1\mathbb{Z} \times \cdots \times \mathbb{Z}/m_r\mathbb{Z} \). A good theorem deserves a good proof. An outstanding theorem deserves two proofs, at least, one existential, and one constructive. So here are one of each for the Chinese Remainder Theorem. **Proof 1:** We must show that each r-tuple \((b_1, \ldots, b_r)\), such that \( 0 \leq k < m_i \) (for \( i = 1, \ldots, r \)) occurs exactly once. There are obviously \( M \) such vectors, and so it will be sufficient to show that each of them occurs at most once as the image of some \( x \). In the contrary case we would have \( x \) and \( x' \) both corresponding to \((b_1, b_2, \ldots, b_r)\), say. But then \( x - x' \equiv 0 \mod m_i \) for all the \( m_i \). Hence \( x - x' \equiv 0 \) must be \( < |M| \), hence \( x = x' \). **Proof 2:** Here’s how to compute a number \( x \) that satisfies the simultaneous congruences \( x \equiv b_i \mod m_i \) (for \( i = 1, \ldots, r \)). First, by the extended Euclidean algorithm we can quickly find \( u_1, u_2, \ldots, u_r \), such that \( m_i y_i + m_j u_j = 1 \) for \( j = 1, \ldots, r \). Then we claim that the number \[ x = \sum_{j=1}^{r} b_j \frac{M}{m_j} \] satisfies all the given congruences. Indeed, for each \( i = 1, \ldots, r \), we have \[ x \equiv \sum_{j=1}^{r} b_j \frac{M}{m_j} \equiv b_i \left(\frac{M}{m_i} \mod m_i\right) \equiv b_i \mod m_i, \] where the first congruence holds because \( M/m_j \) is divisible by \( m_j \), and the second congruence follows since \( t_k(M/m_i) = 1 - u_i m_i \equiv 1 \mod m_i \), completing the second proof of the Chinese Remainder Theorem. Now the proof of theorem 4.5.4 follows easily, and is left as an exercise for the reader. The factorization that is described in detail in theorem 4.5.4 will be written symbolically as \[ \mathbb{Z}_n \cong \bigoplus_{i=1}^{r} \mathbb{Z}_{p_i^{n_i}} \tag{4.5.2} \] The factorization (4.5.2) of the ring \( \mathbb{Z} \), induces a factorization \[ U_r \cong \bigoplus_{i=1}^{r} U_{p_i^{n_i}} \tag{4.5.3} \] Image Analysis: ### Analysis of the Attached Visual Content: #### **1. Localization and Attribution:** - The image analyzed is a single page from a document. - The entire content is treated as **Image 1**. #### **2. Object Detection and Classification:** - **Objects Identified:** - Text blocks - Mathematical notation - Equations #### **3. Scene and Activity Analysis:** - **Scene Description:** - The entire scene is a mathematical text page likely from a textbook or academic paper. The primary activity is the presentation and proof of mathematical theorems, such as the discussion around modular arithmetic and number theory. #### **4. Text Analysis:** - **Text Detected and Extracted:** - Theorems, equations, proofs, and mathematical explanations are present throughout the page. Key sections include: - Theorem 4.5.4 and Theorem 4.5.5 - Definitions and proofs related to The Chinese Remainder Theorem - Factorization formulas - **Content Analysis:** - **Theorem 4.5.4** outlines a mapping related to integers and rings of tuples. - **Theorem 4.5.5**, a.k.a. **The Chinese Remainder Theorem**, describes a bijective mapping for a specific set of positive integers. - The page includes proofs for the theorems mentioned along with explanations and extended algorithms for computations. #### **5. Diagram and Chart Analysis:** - No diagrams or charts are present on this page. #### **7. Anomaly Detection:** - No anomalies or unusual elements are detected in the content of the image. #### **8. Color Analysis:** - The image appears in grayscale (black and white) which is typical for academic or textbook pages intended for printing. #### **9. Perspective and Composition:** - **Perspective:** - The image is taken at a straight-on angle, typical for scanned or digitally captured documents. - **Composition:** - The text is arranged in a traditional column format, with sections neatly organized and formulas centered for visibility. #### **10. Contextual Significance:** - The content appears to be an excerpt from an academic text or reference book on number theory, specifically dealing with modular arithmetic and decomposition of rings. ### Overall Message: - The page is intended to educate or inform the reader about the principles of number theory, specifically modular arithmetic and the Chinese Remainder Theorem. It provides detailed mathematical proofs and is structured to facilitate understanding of the complex mathematical concepts. ### **Additional Notes:** - The document is likely mathematical in nature, based on the presence of theorems and proofs. - This content is most useful to an academic or educational audience, particularly students or professionals in mathematics or related fields. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 113 Context: # 5.2 Turing Machines ## Exercises for section 5.1 1. Prove that the following decision problem belongs to P: Given integers \( K \) and \( a_1, \ldots, a_n \). Is the median of the \( a_i \) smaller than \( K \)? 2. Prove that the following decision problem is in NP: given an \( n \times n \) matrix \( A \) of integer entries. Is \( \det A = 0 \)? 3. For which of the following problems can you prove membership in P? (a) Given a graph \( G \). Does \( G \) contain a circuit of length 4? (b) Given a graph \( G \). Is \( G \) bipartite? (c) Given integers. Is there a subset of them whose sum is an even number? (d) Given \( n \) integers. Is there a subset of them whose sum is divisible by 3? (e) Given a graph \( G \). Does \( G \) contain an Euler circuit? 4. For which of the following problems can you prove membership in NP? (a) Given a set of integers and another integer \( K \). Is there a subset of the given integers whose sum is \( K \)? (b) Given a graph \( G \) and an integer \( K \). Does \( G \) contain a path of length \( 2K \)? (c) Given a set of \( K \) integers. Is it true that not all of them are prime? (d) Given a set of \( K \) integers. Is it true that all of them are prime? ## 5.2 Turing Machines A Turing machine consists of: (a) A doubly infinite tape, that is marked off into squares that are numbered as shown in Fig. 5.2.1 below. Each square can contain a single character from the character set that the machine recognizes. For simplicity, we can assume that the character set consists of just three symbols: '0', '1', and ' ' (blank). (b) A tape head that is capable of either reading a single character from a square on the tape or writing a single character on a square, or moving its position relative to the tape by an increment of one square in either direction. (c) A finite list of states such that at every instant the machine is exactly one of those states. The possible states of the machine are, first of all, the regular states \( q_1, \ldots, q_n \), and second, three special states: - \( q_0 \): the initial state - \( q_y \): the final state in a problem to which the answer is 'Yes' - \( q_n \): the final state in a problem to which the answer is 'No' (d) A program (or program module, if we think of it as a pluggable component) that directs the machine through the steps of a particular task. ``` ... ... ... ... ... ``` ``` ... --3 --2 --> 0 1 2 3 ... ... -- . . . -- . -- . . . -- ``` Image Analysis: ## Analysis of Attached Visual Content ### 1. Localization and Attribution - **Image on the Page**: The visual content consists of a single image located midway through the text on the page. - **Image Number**: Image 1. ### 2. Object Detection and Classification - **Image 1**: - **Objects Detected**: - Text blocks. - Diagram of a Turing Machine tape. - **Categories**: - Written text (instructions and exercises related to computational theory). - Diagram (schematic representation of a Turing Machine tape). ### 3. Scene and Activity Analysis - **Image 1**: - **Scene Description**: The scene is an excerpt from an academic textbook focusing on computational theory, specifically on topics related to NP-completeness and Turing Machines. - **Activities**: - The text describes various decision problems and properties, exercises for solving these problems, and a detailed explanation of Turing Machines. ### 4. Text Analysis - **Image 1**: - **Text Detected**: - Main Heading: "5.2 Turing Machines" - Sub-Headings: "Exercises for section 5.1", "5.2.1 A Turing machine tape" - Exercises and Theoretical Explanations about decision problems and Turing Machine properties. - **Content Analysis**: - The text explains key concepts in computational theory, providing exercises to test understanding and elaborations on the workings of a Turing Machine and its tape configuration. - **Exercises for section 5.1**: 1. Problems related to decision problem membership in P and NP. 2. Questions regarding graph properties and integer subsets in relation to computational complexity. - **5.2 Turing Machines**: - Detailed explanation of what a Turing Machine is, including its components like the infinite tape, states, and program modules. - **5.2.1 A Turing Machine tape**: - Diagram illustrating how the tape operates and moves, including states and symbols. ### 6. Product Analysis - **Image 1**: - **Products Depicted**: - Diagram of a Turing Machine tape. - **Features**: - The tape shows squares numbered in sequence, demonstrating how a Turing machine processes input and moves between states. - Squares contain possible tape symbols: '0', '1', and ' ' (blank). - Arrows indicate possible movements of the tape head. ### 7. Anomaly Detection - **Image 1**: - **Anomalies**: No significant anomalies detected. The content appears typical for an educational text on computational theory. ### 8. Color Analysis - **Image 1**: - **Color Composition**: - Dominantly black text on a white background. - The diagram uses simple black outlines and symbols without any colors, maintaining consistency with typical textbook illustrations and not distracting from the informational content. ### 9. Perspective and Composition - **Image 1**: - **Perspective**: Top-down view, typical for a scanned or photocopied page from a textbook. - **Composition**: - The page is divided into sections with headings, explanations, and numbered exercises. - The diagram is centrally placed and clearly labeled, giving a clear visual complement to the adjacent text. ### 10. Contextual Significance - **Image 1**: - **Context**: This page is part of a textbook on computational theory, contributing to the reader's understanding of NP-completeness and Turing Machines. - **Contribution**: - Provides foundational exercises and theoretical explanations critical for mastering concepts in computer science and computational complexity. ### 13. Table Analysis - **Image 1**: - **No tables detected**: This image does not contain tabular data. ### Additional Aspects #### Ablaufprozesse (Process Flows) - **Image 1**: - **Process Flow Description**: The operations of a Turing Machine, including how the machine reads the tape, decides its next state, writes on the tape, and moves the tape head. #### Prozessbeschreibungen (Process Descriptions) - **Image 1**: - **Process Description**: - A detailed description of the Turing Machine’s functioning, defining initial and final states, regular states, and the program module’s role in guiding the machine through task-specific steps. #### Typen Bezeichnung (Type Designations) - **Image 1**: - **Type Designations**: Turing Machine components such as infinite tape, tape head, states (qi), and character set (symbols: '0', '1', blank). #### Trend and Interpretation - **Image 1**: - **No trends**: The content is theoretical, focusing on explaining core concepts rather than presenting data trends. Overall, the visual content provides insightful information on computational theory, effectively using text and diagrams to enhance understanding of complex topics like NP-completeness and Turing Machine operations. #################### File: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf Page: 47 Context: # 2.3 Recursive Graph Algorithms In this section, we show in Fig. 2.3.5(a) a map of a distant planet, and in Fig. 2.3.5(b) the graph that results from the construction that we have just described. By a **planar graph** we mean a graph \( G \) that can be drawn in the plane in such a way that two edges never cross (except that two edges at the same vertex have that vertex in common). The graph that results from changing a map of countries into a graph as described above is always a planar graph. In Fig. 2.3.6(a) we show a planar graph \( G \). This graph doesn't look planar because two of its edges cross. However, that isn’t the graph’s fault, because with a little more care we might have drawn the same graph as in Fig. 2.3.6(b), in which its planarity is obvious. Don’t blame the graph if it doesn’t look planar. It might be planar anyway! The question of recognizing whether a given graph is planar is itself a formidable problem, although the solution, due to J. Hopcroft and R. E. Tarjan,* is an algorithm that makes the decision in linear time, i.e., in \( O(V) \) time for a graph of \( V \) vertices. Although every planar graph can be properly colored in four colors, there are still all of those other graphs that are not planar to deal with. For any of those graphs we can ask if a positive integer \( K \) is given, whether or not its vertices can be \( K \)-colored properly. As if that question weren’t hard enough, we might ask for even more detail, namely about the number of ways of properly coloring the vertices of a graph. For instance, if we have \( K \) colors to work with, suppose \( G \) is the empty graph \( R_n^K \). That is, the graph of \( n \) vertices that has no edges at all. Then \( G \) has quite a large number of proper colorings, \( K^n \), to be exact. Other graphs of \( n \) vertices have fewer proper colorings than that, and an interesting computational question is to count the proper colorings of a given graph. We will now find a recursive algorithm that will answer this question. Again, the complexity of the algorithm will be exponential, but as a small consolation we note that no polynomial time algorithm for this problem is known. Choose an edge \( e \) of the graph, and let its endpoints be \( u \) and \( v \). Now delete the edge from the graph, and let the resulting graph be called \( G - \{ e \} \). Then we will distinguish two kinds of proper colorings of \( G - \{ e \} \: \): those in which vertices \( u \) and \( v \) have the same color and those in which \( u \) and \( v \) have different colors. Obviously the number of proper colorings of \( G - \{ e \} \) if \( u \) and \( v \) are colored the same is the sum of the numbers of colorings of each of these two kinds. * Efficient planarity testing, J. Assoc. Comp. Mach. 21 (1974), 549-568. 43 Image Analysis: ### Comprehensive Examination of the Visual Content #### 1. Localization and Attribution - **Image 1**: Located at the top left (labeled as Fig. 2.3.5(a)). - **Image 2**: Located at the top right (labeled as Fig. 2.3.5(b)). - **Image 3**: Located at the bottom left (labeled as Fig. 2.3.6(a)). - **Image 4**: Located at the bottom right (labeled as Fig. 2.3.6(b)). #### 2. Object Detection and Classification - **Image 1**: - Objects: Nodes (labeled A-F), Edges connecting nodes. - Features: Circular shape with labeled sections. - Category: Graph/Diagram. - **Image 2**: - Objects: Nodes (labeled A-F), Edges connecting nodes. - Features: Pentagon and liner connections indicating a planar graph. - Category: Graph/Diagram. - **Image 3**: - Objects: Nodes (labeled 1-6), Edges connecting nodes. - Features: Hexagonal layout with all nodes interconnected. - Category: Graph/Diagram. - **Image 4**: - Objects: Nodes (labeled 1-6), Edges connecting nodes. - Features: Similar structure to Image 3 but arranged differently. - Category: Graph/Diagram. #### 3. Scene and Activity Analysis - **Image 1**: - Scene: Schematic map indicating regions A-F. - Activity: Representation of spatial relationships in a graph form. - **Image 2**: - Scene: Planar graph corresponding to the map. - Activity: Visualization of non-intersecting edges in a graph. - **Image 3 and Image 4**: - Scene: Graph representations analyzing planar properties. - Activity: Illustrations of node connections and planar graph structures. #### 4. Text Analysis - **Text Detected**: - Fig. 2.3.5(a) - 2.3.6(b): Figure labels providing references. - Additional text describing graphs and their properties: - "A map of a distant planet..." - "Efficient planarity testing" - **Significance**: These descriptions explain the graphs and their context, providing insight into the construction and properties of planar graphs. #### 5. Diagram and Chart Analysis - **Diagrams** (Fig. 2.3.5(a) - 2.3.6(b)): - Purpose: Illustrate planar graphs and show the transformation from maps to graph representations. - Key Insights: Demonstrate graph properties and planar conditions without intersecting edges. #### 8. Color Analysis - **General Observation**: All images are primarily black and white, focusing on nodes and edges for clarity. #### 9. Perspective and Composition - **Perspective**: All images are top-down schematic views for clarity and ease of understanding of graph properties. - **Composition**: - Nodes and edges are arranged to show relationships. - Clear labeling to identify nodes and connections. #### 12. Graphs and Trend Analysis - **Graphs**: - Detailed comparisons showing how altering edges affect planar properties. #### 13. Graph Numbers: - **Fig. 2.3.6(a) and 2.3.6(b)**: Nodes labeled from 1 to 6, showing different interconnections. ### Contextual Summary The images and corresponding text form part of a discussion on recursive graph algorithms, focusing on planar graphs. By illustrating the planar maps and their graph representations, the figures aid in visualizing complex graph properties and transformations. This contributes significantly to understanding the underlying concepts explored in this section of the document. The planar graphs and explanations tie back to theoretical computer science and efficient planarity testing, highlighting significant research and problem-solving strategies in the field. #################### File: Feedback Page: 1 Context: Feedback: Prompt: how much is a fried chicken in "Pakens"? Response: The provided context is from two books on mathematics, one on number theory and the other on algorithms and complexity. Neither book mentions anything about "Pakens" or fried chicken. To answer your question, I need more information. Could you please clarify what "Pakens" refers to? Is it a restaurant, a country, or something else? Note: Reference this document as 'previous feedback'. ########## """QUERY: can you give me a 2 sentence summary of your contents?""" 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. Use as much tokens as needed but at the same time be as efficient as possible. If you find information separated by a | in the context, it is a table formatted in Markdown - the whole context is formatted as md structure. Important: Take a look at the QUERY and only the QUERY. Please try always to answer the query question. If the client ask for a formatting structure follow his advise.But if the question is vague or unclear ask a follow-up question based on the context. If too many questions were asked already based on the history, just answer the question Final Files Sources: Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 60, Algebraic%20Topology%20AT-toc.pdf - Page 4, An%20Introduction%20to%20the%20Theory%20of%20Numbers%20-%20Leo%20Moser%20%28PDF%29.pdf - Page 4, Algebraic%20Topology%20AT-toc.pdf - Page 2, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 113, 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 18, 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 8, Algebraic%20Topology%20AT-toc.pdf - Page 9, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 48, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - Page 126, Algorithms%20and%20Complexity%20-%20Herbert%20S.%20Wilf%20%28PDF%29.pdf - 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