Verfügbarkeit: Discrete mathematics

Discrete mathematics:

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Bibliographische Detailangaben
1. Verfasser:	Aigner, Martin 1942-2023 (VerfasserIn)
Format:	Buch
Sprache:	English German
Veröffentlicht:	Providence, R.I. American Mathematical Society 2007
Schlagworte:	Informatique - Mathématiques Informatik Mathematik Computer science > Mathematics Diskrete Mathematik Aufgabensammlung
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XII, 388 S. graph. Darst. 26 cm
ISBN:	9780821841518

Internformat

MARC


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Datensatz im Suchindex

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adam_text	Contents Prefaces ix Part 1. Counting Chapter 1. Fundamentals 7 §1.1 . Elementary Counting Principles §1.2 . 10 §1.3 . The Fundamental Counting Coefficients Permutations §1.4 . Recurrence Equations 17 §1.5 . Discrete Probability 23 Existence Theorems Exercises for Chapter 1 29 §1.6 . Chapter 2. Summation 7 14 33 41 §2.1 . Direct Methods 41 §2.2 . The Calculus of Finite Differences 46 §2.3 §2.4 . . Inversion Inclusion-Exclusion 52 55 Exercises for Chapter 2 Chapter 3. Generating Functions 60 65 §3.1 . Definitions and Examples 65 §3.2 . Solving Recurrences 67 §3.3 . Generating Functions of Exponential Type 74 Exercises for Chapter 3 76 v Contents vi Chapter 4. Counting Patterns 81 §4.1 . Symmetries 81 §4.2 . Statement of the Problem 84 §4.3 . Patterns and the Cycle Indicator 86 §4.4 . Polya’s Theorem 88 Exercises for Chapter 4 Chapter 5. 94 Asymptotic Analysis 99 99 §5.1 . The Growth of Functions §5.2 . Order of Magnitude of Recurrence Relations 103 §5.3 . Running Times of Algorithms 106 Exercises for Chapter 5 Bibliography for Part 1 109 113 Part 2. Graphs and Algorithms Chapter 6. Graphs 119 §6.1 . Definitions and Examples 119 §6.2 . Representation of Graphs 124 §6.3 . Paths and Circuits 126 §6.4 . Directed Graphs 129 Exercises for Chapter 6 Chapter 7. 132 Trees 137 §7.1 . What Is a Tree? 137 §7.2 . Breadth-First and Depth-First Search 141 §7.3 . Minimal Spanning Trees 143 §7.4 . The Shortest Path in a Graph 146 Exercises for Chapter 7 Chapter 8. 148 Matchings and Networks 153 §8.1 . Matchings in Bipartite Graphs 153 §8.2 . Construction of Optimal Matchings 157 §8.3 . Flows in Networks 164 §8.4 . Eulerian Graphs and the Traveling Salesman Problem 170 §8.5 . The Complexity Classes P and NP 178 Exercises for Chapter 8 181 Contents vii Chapter 9. Searching and Sorting 187 §9.1 . Search Problems and Decision Trees 187 §9.2 . The Fundamental Theoremof Search Theory 191 §9.3 . Sorting Lists 197 §9.4 . Binary Search Trees 203 Exercises for Chapter 9 Chapter 10. 208 General Optimization Methods 215 §10.1 . Backtracking 215 §10.2 . Dynamic Programming 219 §10.3 . The Greedy Algorithm 226 Exercises for Chapter 10 Bibliography for Part 2 229 233 Part 3. Algebraic Systems Chapter 11. Boolean Algebras 239 §11.1 . Definition and Properties 239 §11.2 . Propositional Logic and Boolean Functions 241 §11.3 . Logical Nets 246 §11.4 . Boolean Lattices, Orders, and Hypergraphs 249 Exercises for Chapter 11 Chapter 12. 255 Modular Arithmetic 259 §12.1 . Calculating with Congruences 259 §12.2 . Finite Fields 262 §12.3 . Latin Squares 265 §12.4 . Combinatorial Designs 268 Exercises for Chapter 12 Chapter 13. 276 Coding 281 §13.1 . Statement of the Problem 281 §13.2 . Source Encoding 282 §13.3 . Error Detection and Correction 284 §13.4 . Linear Codes 289 §13.5 . Cyclic Codes 294 Exercises for Chapter 13 297 Contents viii Chapter 14. Cryptography 303 303 306 312 §14.1 . Cryptosystems §14.2 . Linear Shift Registers §14.3 . Public-Key Cryptosystems §14.4 . Zero-Knowledge Protocols Exercises for Chapter 14 Chapter 15. Linear Optimization §15.1 . Examples and Definitions §15.2 §15.3 . . Duality The Fundamental Theorem of Linear Optimization §15.4 §15.5 . . Admissible Solutions and Optimal Solutions The Simplex Algorithm §15.6 . Integer Linear Optimization Exercises for Chapter 15 316 319 323 323 325 331 336 340 347 349 Bibliography for Part 3 353 Solutions to Selected Exercises 355 Index 383
adam_txt	Contents Prefaces ix Part 1. Counting Chapter 1. Fundamentals 7 §1.1 . Elementary Counting Principles §1.2 . 10 §1.3 . The Fundamental Counting Coefficients Permutations §1.4 . Recurrence Equations 17 §1.5 . Discrete Probability 23 Existence Theorems Exercises for Chapter 1 29 §1.6 . Chapter 2. Summation 7 14 33 41 §2.1 . Direct Methods 41 §2.2 . The Calculus of Finite Differences 46 §2.3 §2.4 . . Inversion Inclusion-Exclusion 52 55 Exercises for Chapter 2 Chapter 3. Generating Functions 60 65 §3.1 . Definitions and Examples 65 §3.2 . Solving Recurrences 67 §3.3 . Generating Functions of Exponential Type 74 Exercises for Chapter 3 76 v Contents vi Chapter 4. Counting Patterns 81 §4.1 . Symmetries 81 §4.2 . Statement of the Problem 84 §4.3 . Patterns and the Cycle Indicator 86 §4.4 . Polya’s Theorem 88 Exercises for Chapter 4 Chapter 5. 94 Asymptotic Analysis 99 99 §5.1 . The Growth of Functions §5.2 . Order of Magnitude of Recurrence Relations 103 §5.3 . Running Times of Algorithms 106 Exercises for Chapter 5 Bibliography for Part 1 109 113 Part 2. Graphs and Algorithms Chapter 6. Graphs 119 §6.1 . Definitions and Examples 119 §6.2 . Representation of Graphs 124 §6.3 . Paths and Circuits 126 §6.4 . Directed Graphs 129 Exercises for Chapter 6 Chapter 7. 132 Trees 137 §7.1 . What Is a Tree? 137 §7.2 . Breadth-First and Depth-First Search 141 §7.3 . Minimal Spanning Trees 143 §7.4 . The Shortest Path in a Graph 146 Exercises for Chapter 7 Chapter 8. 148 Matchings and Networks 153 §8.1 . Matchings in Bipartite Graphs 153 §8.2 . Construction of Optimal Matchings 157 §8.3 . Flows in Networks 164 §8.4 . Eulerian Graphs and the Traveling Salesman Problem 170 §8.5 . The Complexity Classes P and NP 178 Exercises for Chapter 8 181 Contents vii Chapter 9. Searching and Sorting 187 §9.1 . Search Problems and Decision Trees 187 §9.2 . The Fundamental Theoremof Search Theory 191 §9.3 . Sorting Lists 197 §9.4 . Binary Search Trees 203 Exercises for Chapter 9 Chapter 10. 208 General Optimization Methods 215 §10.1 . Backtracking 215 §10.2 . Dynamic Programming 219 §10.3 . The Greedy Algorithm 226 Exercises for Chapter 10 Bibliography for Part 2 229 233 Part 3. Algebraic Systems Chapter 11. Boolean Algebras 239 §11.1 . Definition and Properties 239 §11.2 . Propositional Logic and Boolean Functions 241 §11.3 . Logical Nets 246 §11.4 . Boolean Lattices, Orders, and Hypergraphs 249 Exercises for Chapter 11 Chapter 12. 255 Modular Arithmetic 259 §12.1 . Calculating with Congruences 259 §12.2 . Finite Fields 262 §12.3 . Latin Squares 265 §12.4 . Combinatorial Designs 268 Exercises for Chapter 12 Chapter 13. 276 Coding 281 §13.1 . Statement of the Problem 281 §13.2 . Source Encoding 282 §13.3 . Error Detection and Correction 284 §13.4 . Linear Codes 289 §13.5 . Cyclic Codes 294 Exercises for Chapter 13 297 Contents viii Chapter 14. Cryptography 303 303 306 312 §14.1 . Cryptosystems §14.2 . Linear Shift Registers §14.3 . Public-Key Cryptosystems §14.4 . Zero-Knowledge Protocols Exercises for Chapter 14 Chapter 15. Linear Optimization §15.1 . Examples and Definitions §15.2 §15.3 . . Duality The Fundamental Theorem of Linear Optimization §15.4 §15.5 . . Admissible Solutions and Optimal Solutions The Simplex Algorithm §15.6 . Integer Linear Optimization Exercises for Chapter 15 316 319 323 323 325 331 336 340 347 349 Bibliography for Part 3 353 Solutions to Selected Exercises 355 Index 383
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