The probabilistic method:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Hoboken, NJ
Wiley
2008
|
| Edition: | 3. ed. |
| Series: | Wiley-interscience series in discrete mathematics and optimization
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 352 S. |
| ISBN: | 9780470170205 |
Staff View
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Record in the Search Index
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|---|---|
| adam_text | Titel: The probabilistic method
Autor: Alon, Noga
Jahr: 2008
Contents
PREFACE xiii
ACKNOWLEDGMENTS xv
PART I METHODS
1 The Basic Method 1
1.1 The Probabilistic Method 1
1.2 Graph Theory 1
1.3 Combinatorics 3
1.4 Combinatorial Number Theory 9
1.5 Disjoint Pairs 10
1.6 Exercises 11
The Probabilistic Lens: The Erdos-Ko-Rado Theorem 13
2 Linearity of Expectation 15
2.1 Basics 15
2.2 Splitting Graphs 16
2.3 Two Quickies 18
2.4 Balancing Vectors 19
2.5 Unbalancing Lights 21
2.6 Without Coin Flips 22
2.7 Exercises 23
The Probabilistic Lens: Bregman s Theorem 24
vii
viH CONTENTS
3 Alterations 27
3.1 Ramsey Numbers 27
3.2 Independent Sets 29
3.3 Combinatorial Geometry 30
3.4 Packing 31
3.5 Recoloring 32
3.6 Continuous Time 35
3.7 Exercises 39
The Probabilistic Lens: High Girth and High Chromatic Number 41
4 The Second Moment 43
4.1 Basics 43
4.2 Number Theory 44
4.3 More Basics 47
4.4 Random Graphs 49
4.5 Clique Number 53
4.6 Distinct Sums 54
4.7 The Rodl Nibble 56
4.8 Exercises 61
The Probabilistic Lens: Hamiltonian Paths 63
5 The Local Lemma 67
5.1 The Lemma 67
5.2 Property B and Multicolored Sets of Real Numbers 70
5.3 Lower Bounds for Ramsey Numbers 71
5.4 A Geometric Result 73
5.5 The Linear Arboricity of Graphs 74
5.6 Latin Transversals 78
5.7 The Algorithmic Aspect 79
5.8 Exercises 82
The Probabilistic Lens: Directed Cycles 83
6 Correlation Inequalities 85
6.1 The Four Functions Theorem of Ahlswede and Daykin 86
6.2 The FKG Inequality 89
6.3 Monotone Properties 90
6.4 Linear Extensions of Partially Ordered Sets 92
6.5 Exercises 94
The Probabilistic Lens: Turan s Theorem 95
CONTENTS IX
7 Martingales and Tight Concentration 97
7.1 Definitions 97
7.2 Large Deviations 99
7.3 Chromatic Number 101
7.4 Two General Settings 103
7.5 Four Illustrations 107
7.6 Talagrand s Inequality 109
7.7 Applications of Talagrand s Inequality 113
7.8 Kim-Vu Polynomial Concentration 115
7.9 Exercises 116
The Probabilistic Lens: Weierstrass Approximation Theorem 117
8 The Poisson Paradigm 119
8.1 The Janson Inequalities 119
8.2 The Proofs 121
8.3 Bran s Sieve 124
8.4 Large Deviations 127
8.5 Counting Extensions 129
8.6 Counting Representations 130
8.7 Further Inequalities 133
8.8 Exercises 135
The Probabilistic Lens: Local Coloring 136
9 Pseudorandomness 139
9.1 The Quadratic Residue Tournaments 140
9.2 Eigenvalues and Expanders 143
9.3 Quasirandom Graphs 149
9.4 Exercises 156
The Probabilistic Lens: Random Walks 157
PART II TOPICS
10 Random Graphs 161
10.1 Subgraphs 162
10.2 Clique Number 164
10.3 Chromatic Number 166
10.4 Zero-One Laws 167
10.5 Exercises 175
The Probabilistic Lens: Counting Subgraphs 177
X CONTENTS
11 The Erdos-Renyi Phase Transition 179
11.1 An Overview 180
11.2 Three Processes 182
11.3 The Galton-Watson Branching Process 183
11.4 Analysis of the Poisson Branching Process 184
11.5 The Graph Branching Model 186
11.6 The Graph and Poisson Processes Compared 187
11.7 The Parametrization Explained 190
11.8 The Subcritical Regimes 190
11.9 The Supercritical Regimes 191
11.10 The Critical Window 194
11.11 Analogies to Classical Percolation Theory 197
11.12 Exercises 201
The Probabilistic Lens: The Rich Get Richer 203
12 Circuit Complexity 205
12.1 Preliminaries 205
12.2 Random Restrictions and Bounded-Depth Circuits 207
12.3 More on Bounded-Depth Circuits 211
12.4 Monotone Circuits 214
12.5 Formulae 217
12.6 Exercises 218
The Probabilistic Lens: Maximal Antichains 219
13 Discrepancy 221
13.1 Basics 221
13.2 Six Standard Deviations Suffice 223
13.3 Linear and Hereditary Discrepancy 226
13.4 Lower Bounds 229
13.5 The Beck-Fiala Theorem 231
13.6 Exercises 232
The Probabilistic Lens: Unbalancing Lights 234
14 Geometry 237
14.1 The Greatest Angle Among Points in Euclidean Spaces 238
14.2 Empty Triangles Determined by Points in the Plane 239
14.3 Geometrical Realizations of Sign Matrices 241
14.4 e-Nets and VC-Dimensions of Range Spaces 243
CONTENTS Xi
14.5 Dual Shatter Functions and Discrepancy 248
14.6 Exercises 251
The Probabilistic Lens: Efficient Packing 252
15 Codes, Games and Entropy 255
15.1 Codes 255
15.2 Liar Game 258
15.3 Tenure Game 260
15.4 Balancing Vector Game 261
15.5 Nonadaptive Algorithms 264
15.6 Half Liar Game 264
15.7 Entropy 266
15.8 Exercises 271
The Probabilistic Lens: An Extremal Graph 273
16 Derandomization 275
16.1 The Method of Conditional Probabilities 275
16.2 d-Wise Independent Random Variables in Small 280
Sample Spaces
16.3 Exercises 284
The Probabilistic Lens: Crossing Numbers, Incidences, Sums and Products 285
17 Graph Property Testing 289
17.1 Property Testing 289
17.2 Testing Colorability 290
17.3 Szemeredi s Regularity Lemma 294
17.4 Testing Triangle-Freeness 298
17.5 Characterizing the Testable Graph Properties 300
17.6 Exercises 302
The Probabilistic Lens: Turdn Numbers and Dependent Random Choice 303
Appendix A: Bounding of Large Deviations 307
A.I Chernoff Bounds 307
A.2 Lower Bounds 315
A.3 Exercises 320
The Probabilistic Lens: Triangle-Free Graphs Have Large 321
Independence Numbers
Xii CONTENTS
Appendix B: Paul Erdos 323
B.I Papers 323
B.2 Conjectures 325
B.3 On Erdos 326
B.4 Uncle Paul 327
References 331
Author Index 345
Subject Index 349
|
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| dewey-search | 511/.6 |
| dewey-sort | 3511 16 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 3. ed. |
| format | Book |
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| spelling | Alon, Noga Verfasser aut The probabilistic method Noga Alon ; Joel H. Spencer 3. ed. Hoboken, NJ Wiley 2008 XV, 352 S. txt rdacontent n rdamedia nc rdacarrier Wiley-interscience series in discrete mathematics and optimization Combinatorial analysis Probabilities Kombinatorische Analysis (DE-588)4164746-4 gnd rswk-swf Kombinatorik (DE-588)4031824-2 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Wahrscheinlichkeit (DE-588)4137007-7 gnd rswk-swf Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 gnd rswk-swf Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 s DE-604 Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s Wahrscheinlichkeit (DE-588)4137007-7 s Kombinatorische Analysis (DE-588)4164746-4 s Kombinatorik (DE-588)4031824-2 s 1\p DE-604 Spencer, Joel H. 1946- Verfasser (DE-588)133091546 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017105401&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Alon, Noga Spencer, Joel H. 1946- The probabilistic method Combinatorial analysis Probabilities Kombinatorische Analysis (DE-588)4164746-4 gnd Kombinatorik (DE-588)4031824-2 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Wahrscheinlichkeit (DE-588)4137007-7 gnd Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 gnd |
| subject_GND | (DE-588)4164746-4 (DE-588)4031824-2 (DE-588)4064324-4 (DE-588)4137007-7 (DE-588)4132446-8 |
| title | The probabilistic method |
| title_auth | The probabilistic method |
| title_exact_search | The probabilistic method |
| title_full | The probabilistic method Noga Alon ; Joel H. Spencer |
| title_fullStr | The probabilistic method Noga Alon ; Joel H. Spencer |
| title_full_unstemmed | The probabilistic method Noga Alon ; Joel H. Spencer |
| title_short | The probabilistic method |
| title_sort | the probabilistic method |
| topic | Combinatorial analysis Probabilities Kombinatorische Analysis (DE-588)4164746-4 gnd Kombinatorik (DE-588)4031824-2 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Wahrscheinlichkeit (DE-588)4137007-7 gnd Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 gnd |
| topic_facet | Combinatorial analysis Probabilities Kombinatorische Analysis Kombinatorik Wahrscheinlichkeitsrechnung Wahrscheinlichkeit Kombinatorische Wahrscheinlichkeitstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017105401&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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