The probabilistic method:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Hoboken ; New Jersey
Wiley
[2016]
|
| Edition: | Fourth edition |
| Series: | Wiley series in discrete mathematics and optimization
|
| Subjects: | |
| Online Access: | Beschreibung für Leser Inhaltsverzeichnis |
| Physical Description: | xiv, 375 Seiten Diagramme |
| ISBN: | 9781119061953 1119061954 |
Staff View
MARC
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Record in the Search Index
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Contents PREFACE xiii ACKNOWLEDGMENTS METHODS 1 The Basic Method 3 PART I 1 XV 1.1 1.2 1.3 1.4 1.5 1.6 1.7 The Probabilistic Method, 3 Graph Theory, 5 Combinatorics, 9 Combinatorial Number Theory, 11 Disjoint Pairs, 12 Independent Sets and List Coloring, 13 Exercises, 16 The Erdős-Ko-Rado Theorem, 18 2 Linearity of Expectation 2.1 2.2 2.3 2.4 Basics, 19 Splitting Graphs, 20 Two Quickies, 22 Balancing Vectors, 23 19
viii CONTENTS 2.5 2.6 2.7 Unbalancing Lights, 25 Without Coin Flips, 26 Exercises, 27 Bregman’s Theorem, 29 3 Alterations 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Ramsey Numbers, 31 Independent Sets, 33 Combinatorial Geometry, 34 Packing, 35 Greedy Coloring, 36 Continuous Time, 38 Exercises, 41 High Girth and High Chromatic Number, 43 4 The Second Moment 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 45 Basics, 45 Number Theory, 46 More Basics, 49 Random Graphs, 51 Clique Number, 55 Distinct Sums, 57 The Rödl nibble, 58 Exercises, 64 Hamiltonian Paths, 65 5 The Local Lemma 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 69 The Lemma, 69 Property В and Multicolored Sets of Real Numbers, 72 Lower Bounds for Ramsey Numbers, 73 A Geometric Result, 75 The Linear Arboricity of Graphs, 76 Latin Transversals, 80 Moser’s Fix-It Algorithm, 81 Exercises, 87 Directed Cycles, 88 6 Correlation Inequalities 6.1 6.2 6.3 The Four Functions Theorem of Ahiswede and Daykin, 90 The FKG Inequality, 93 MonotoneProperties, 94 89
ix CONTENTS 6.4 6.5 Linear Extensions of Partially Ordered Sets, 97 Exercises, 99 Turán’s Theorem, 100 7 Martingales and Tight Concentration 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 103 Definitions, 103 Large Deviations, 105 Chromatic Number, 107 Two General Settings, 109 Four Illustrations, 113 Talagrand’s Inequality, 116 Applications of Talagrand’s Inequality, 119 Kim-Vu Polynomial Concentration, 121 Exercises, 123 Weierstrass Approximation Theorem, 124 8 The Poisson Paradigm 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 127 The Janson Inequalities, 127 The Proofs, 129 Brun’s Sieve, 132 Large Deviations, 135 Counting Extensions, 137 Counting Representations, 139 Further Inequalities, 142 Exercises, 143 Local Coloring, 144 9 Quasirandomness 9.1 9.2 9.3 9.4 9.5 9.6 147 The Quadratic Residue Tournaments, 148 Eigenvalues and Expanders, 151 Quasirandom Graphs, 157 Szemerédi’s Regularity Lemma, 165 Graphons, 170 Exercises, 172 Random Walks, 174 PART П 10 TOPICS Random Graphs 10.1 Subgraphs, 180 177 179
CONTENTS 10.2 10.3 10.4 10.5 Clique Number, 183 Chromatic Number, 184 Zero-One Laws, 186 Exercises, 193 Counting Subgraphs, 195 11 The Erdős-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 11.6 The Graph and Poisson Processes Compared, 205 11.7 The Parametrization Explained, 207 11.8 The Subcriticai Regions, 208 11.9 The Supercritical Regimes, 209 11.10 The Critical Window, 212 11.11 Analogies to Classical Percolation Theory, 214 11.12 Exercises, 219 Long paths in the supercritical regime, 220 12 Circuit Complexity 12.1 12.2 12.3 12.4 12.5 12.6 223 Preliminaries, 223 Random Restrictions and Bounded-Depth Circuits, 225 More on Bounded-Depth Circuits, 229 Monotone Circuits, 232 Formulae, 235 Exercises, 236 Maximal Antichains, 237 13 Discrepancy 13.1 13.2 13.3 13.4 13.5 13.6 239 Basics, 239 Six Standard Deviations Suffice, 241 Linear and Hereditary Discrepancy, 245 Lower Bounds, 248 The Beck-Fiala Theorem, 250 Exercises, 251 Unbalancing Lights, 253 14 Geometry 14.1 The Greatest Angle Among Points in Euclidean Spaces, 256 255
xi CONTENTS 14.2 14.3 14.4 14.5 14.6 Empty Triangles Determined by Points in the Plane, 257 Geometrical Realizations of Sign Matrices, 259 e-Nets and VC-Dimensions of Range Spaces, 261 Dual Shatter Functions and Discrepancy, 266 Exercises, 269 Efficient Packing, 270 15 Codes, Games, and Entropy 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 273 Codes, 273 Liar Game, 276 Tenure Game, 278 Balancing Vector Game, 279 Nonadaptive Algorithms, 281 Half Liar Game, 282 Entropy, 284 Exercises, 289 An Extremal Graph, 291 16 Derandomization 293 16.1 The Method of Conditional Probabilities, 293 16.2 d-Wise Independent Random Variables in Small Sample Spaces, 297 16.3 Exercises, 302 Crossing Numbers, Incidences, Sums and Products, 303 17 Graph Property Testing 17.1 17.2 17.3 17.4 17.5 307 Property Testing, 307 Testing Colorability, 308 Testing Triangle-Freeness, 312 Characterizing the Testable Graph Properties, 314 Exercises, 316 Turan Numbers and Dependent Random Choice, 317 Appendix A.l A.2 A.3 A Bounding of Large Deviations Chernoff Bounds, 321 Lower Bounds, 330 Exercises, 334 Triangle-Free Graphs Have Large Independence Numbers, 336 321
xii CONTENTS Appendix B.l B.2 В.З B.4 В Paul Erdős 339 Papers, 339 Conjectures, 341 On Erdős, 342 Uncle Paul, 343 The Rich Get Richer, 346 Appendix C Hints to Selected Exercises 349 REFERENCES 355 AUTHOR INDEX 367 SUBJECT INDEX 371 |
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| author | Alon, Noga Spencer, Joel H. 1946- |
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| discipline | Mathematik |
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| spelling | Alon, Noga Verfasser (DE-588)170483320 aut The probabilistic method Noga Alon (School of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel) ; Joel H. Spencer (Courant Institute of Mathematical Sciences, New York University, New York, USA) Fourth edition Hoboken ; New Jersey Wiley [2016] © 2016 xiv, 375 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Wiley series in discrete mathematics and optimization Kombinatorische Analysis (DE-588)4164746-4 gnd rswk-swf Wahrscheinlichkeit (DE-588)4137007-7 gnd rswk-swf Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 gnd rswk-swf Discrete Mathematics Wahrscheinlichkeitsrechnung u. mathematische Statistik Kombinatorik Combinatorics Mathematics Mathematik Diskrete Mathematik Wahrscheinlichkeitsrechnung Statistics Statistik Probability & Mathematical Statistics Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 s DE-604 Wahrscheinlichkeit (DE-588)4137007-7 s Kombinatorische Analysis (DE-588)4164746-4 s Spencer, Joel H. 1946- Verfasser (DE-588)133091546 aut John Wiley and Sons (DE-588)4101395-5 pbl Erscheint auch als Online-Ausgabe, PDF 978-1-119-06196-0 Erscheint auch als Online-Ausgabe, EPUB 978-1-119-06207-3 http://www.wiley-vch.de/publish/dt/books/ISBN978-1-119-06195-3/ Verlag Beschreibung für Leser Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028883070&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Alon, Noga Spencer, Joel H. 1946- The probabilistic method Kombinatorische Analysis (DE-588)4164746-4 gnd Wahrscheinlichkeit (DE-588)4137007-7 gnd Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 gnd |
| subject_GND | (DE-588)4164746-4 (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 (School of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel) ; Joel H. Spencer (Courant Institute of Mathematical Sciences, New York University, New York, USA) |
| title_fullStr | The probabilistic method Noga Alon (School of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel) ; Joel H. Spencer (Courant Institute of Mathematical Sciences, New York University, New York, USA) |
| title_full_unstemmed | The probabilistic method Noga Alon (School of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel) ; Joel H. Spencer (Courant Institute of Mathematical Sciences, New York University, New York, USA) |
| title_short | The probabilistic method |
| title_sort | the probabilistic method |
| topic | Kombinatorische Analysis (DE-588)4164746-4 gnd Wahrscheinlichkeit (DE-588)4137007-7 gnd Kombinatorische Wahrscheinlichkeitstheorie (DE-588)4132446-8 gnd |
| topic_facet | Kombinatorische Analysis Wahrscheinlichkeit Kombinatorische Wahrscheinlichkeitstheorie |
| url | http://www.wiley-vch.de/publish/dt/books/ISBN978-1-119-06195-3/ http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028883070&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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