Noga Alon, Joel H. Spencer

The Probabilistic Method (eBook, PDF)

• Format: PDF

Produktbeschreibung

Praise for the Third Edition "Researchers of any kind of extremal combinatorics or theoretical computer science will welcome the new edition of this book." - MAA Reviews Maintaining a standard of excellence that establishes The Probabilistic Method as the leading reference on probabilistic methods in combinatorics, the Fourth Edition continues to feature a clear writing style, illustrative examples, and illuminating exercises. The new edition includes numerous updates to reflect the most recent developments and advances in discrete mathematics and the connections to other areas in mathematics, theoretical computer science, and statistical physics. Emphasizing the methodology and techniques that enable problem-solving, The Probabilistic Method, Fourth Edition begins with a description of tools applied to probabilistic arguments, including basic techniques that use expectation and variance as well as the more advanced applications of martingales and correlation inequalities. The authors explore where probabilistic techniques have been applied successfully and also examine topical coverage such as discrepancy and random graphs, circuit complexity, computational geometry, and derandomization of randomized algorithms. Written by two well-known authorities in the field, the Fourth Edition features: * Additional exercises throughout with hints and solutions to select problems in an appendix to help readers obtain a deeper understanding of the best methods and techniques * New coverage on topics such as the Local Lemma, Six Standard Deviations result in Discrepancy Theory, Property B, and graph limits * Updated sections to reflect major developments on the newest topics, discussions of the hypergraph container method, and many new references and improved results The Probabilistic Method, Fourth Edition is an ideal textbook for upper-undergraduate and graduate-level students majoring in mathematics, computer science, operations research, and statistics. The Fourth Edition is also an excellent reference for researchers and combinatorists who use probabilistic methods, discrete mathematics, and number theory. Noga Alon, PhD, is Baumritter Professor of Mathematics and Computer Science at Tel Aviv University. He is a member of the Israel National Academy of Sciences and Academia Europaea. A coeditor of the journal Random Structures and Algorithms, Dr. Alon is the recipient of the Polya Prize, The Gödel Prize, The Israel Prize, and the EMET Prize. Joel H. Spencer, PhD, is Professor of Mathematics and Computer Science at the Courant Institute of New York University. He is the cofounder and coeditor of the journal Random Structures and Algorithms and is a Sloane Foundation Fellow. Dr. Spencer has written more than 200 published articles and is the coauthor of Ramsey Theory, Second Edition, also published by Wiley.

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Produktdetails

• Verlag: John Wiley & Sons
• Seitenzahl: 400
• Erscheinungstermin: 28. Oktober 2015
• Englisch
• ISBN-13: 9781119061960
• Artikelnr.: 44058018

Autorenporträt

Noga Alon, PhD, is Baumritter Professor of Mathematics and Computer Science at Tel Aviv University. He is a member of the Israel National Academy of Sciences and Academia Europaea. A coeditor of the journal Random Structures and Algorithms, Dr. Alon is the recipient of the Polya Prize, The Gödel Prize, The Israel Prize, and the EMET Prize. Joel H. Spencer, PhD, is Professor of Mathematics and Computer Science at the Courant Institute of New York University. He is the cofounder and coeditor of the journal Random Structures and Algorithms and is a Sloane Foundation Fellow. Dr. Spencer has written more than 200 published articles and is the coauthor of Ramsey Theory, Second Edition, also published by Wiley.

Inhaltsangabe

PREFACE xiii

ACKNOWLEDGMENTS xv

PART I METHODS 1

1 The Basic Method 3

1.1 The Probabilistic Method, 3

1.2 Graph Theory, 5

1.3 Combinatorics, 9

1.4 Combinatorial Number Theory, 11

1.5 Disjoint Pairs, 12

1.6 Independent Sets and List Coloring, 13

1.7 Exercises, 16

The Erd?os-Ko-Rado Theorem, 18

2 Linearity of Expectation 19

2.1 Basics, 19

2.2 Splitting Graphs, 20

2.3 Two Quickies, 22

2.4 Balancing Vectors, 23

2.5 Unbalancing Lights, 25

2.6 Without Coin Flips, 26

2.7 Exercises, 27

Brégman's Theorem, 29

3 Alterations 31

3.1 Ramsey Numbers, 31

3.2 Independent Sets, 33

3.3 Combinatorial Geometry, 34

3.4 Packing, 35

3.5 Greedy Coloring, 36

3.6 Continuous Time, 38

3.7 Exercises, 41

High Girth and High Chromatic Number, 43

4 The Second Moment 45

4.1 Basics, 45

4.2 Number Theory, 46

4.3 More Basics, 49

4.4 Random Graphs, 51

4.5 Clique Number, 55

4.6 Distinct Sums, 57

4.7 The Rödl nibble, 58

4.8 Exercises, 64

Hamiltonian Paths, 65

5 The Local Lemma 69

5.1 The Lemma, 69

5.2 Property B and Multicolored Sets of Real Numbers, 72

5.3 Lower Bounds for Ramsey Numbers, 73

5.4 A Geometric Result, 75

5.5 The Linear Arboricity of Graphs, 76

5.6 Latin Transversals, 80

5.7 Moser's Fix-It Algorithm, 81

5.8 Exercises, 87

Directed Cycles, 88

6 Correlation Inequalities 89

6.1 The Four Functions Theorem of Ahlswede and Daykin, 90

6.2 The FKG Inequality, 93

6.3 Monotone Properties, 94

6.4 Linear Extensions of Partially Ordered Sets, 97

6.5 Exercises, 99

Turán's Theorem, 100

7 Martingales and Tight Concentration 103

7.1 Definitions, 103

7.2 Large Deviations, 105

7.3 Chromatic Number, 107

7.4 Two General Settings, 109

7.5 Four Illustrations, 113

7.6 Talagrand's Inequality, 116

7.7 Applications of Talagrand's Inequality, 119

7.8 Kim-Vu Polynomial Concentration, 121

7.9 Exercises, 123

Weierstrass Approximation Theorem, 124

8 The Poisson Paradigm 127

8.1 The Janson Inequalities, 127

8.2 The Proofs, 129

8.3 Brun's Sieve, 132

8.4 Large Deviations, 135

8.5 Counting Extensions, 137

8.6 Counting Representations, 139

8.7 Further Inequalities, 142

8.8 Exercises, 143

Local Coloring, 144

9 Quasirandomness 147

9.1 The Quadratic Residue Tournaments, 148

9.2 Eigenvalues and Expanders, 151

9.3 Quasirandom Graphs, 157

9.4 Szemerédi's Regularity Lemma, 165

9.5 Graphons, 170

9.6 Exercises, 172

Random Walks, 174

PART II TOPICS 177

10 Random Graphs 179

10.1 Subgraphs, 180

10.2 Clique Number, 183

10.3 Chromatic Number, 184

10.4 Zero-One Laws, 186

10.5 Exercises, 193

Counting Subgraphs, 195

11 The Erd?os-Rényi Phase Transition 197

11.1 An Overview, 197

11.2 Three Processes, 199

11.3 The Galton-Watson Branching Process, 201

11.4 Analysis of the Poisson Branching Process, 202

11.5 The Graph Branching Model, 204

Rezensionen

"This is an ideal textbook for upper-undergraduate and graduate-level students majoring in mathematics, computer science, operations research, and statistics." (Springer Nature, 2016)