The Probabilistic Method (4版)

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【簡介】 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. 【目錄】 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 11.6 The Graph and Poisson Processes Compared, 205 11.7 The Parametrization Explained, 207 11.8 The Subcritical 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 223 12.1 Preliminaries, 223 12.2 Random Restrictions and Bounded-Depth Circuits, 225 12.3 More on Bounded-Depth Circuits, 229 12.4 Monotone Circuits, 232 12.5 Formulae, 235 12.6 Exercises, 236 Maximal Antichains, 237 13 Discrepancy 239 13.1 Basics, 239 13.2 Six Standard Deviations Suffice, 241 13.3 Linear and Hereditary Discrepancy, 245 13.4 Lower Bounds, 248 13.5 The Beck–Fiala Theorem, 250 13.6 Exercises, 251 Unbalancing Lights, 253 14 Geometry 255 14.1 The Greatest Angle Among Points in Euclidean Spaces, 256 14.2 Empty Triangles Determined by Points in the Plane, 257 14.3 Geometrical Realizations of Sign Matrices, 259 14.4 𝜖-Nets and VC-Dimensions of Range Spaces, 261 14.5 Dual Shatter Functions and Discrepancy, 266 14.6 Exercises, 269 Efficient Packing, 270 15 Codes, Games, and Entropy 273 15.1 Codes, 273 15.2 Liar Game, 276 15.3 Tenure Game, 278 15.4 Balancing Vector Game, 279 15.5 Nonadaptive Algorithms, 281 15.6 Half Liar Game, 282 15.7 Entropy, 284 15.8 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 307 17.1 Property Testing, 307 17.2 Testing Colorability, 308 17.3 Testing Triangle-Freeness, 312 17.4 Characterizing the Testable Graph Properties, 314 17.5 Exercises, 316 Turán Numbers and Dependent Random Choice, 317 Appendix A Bounding of Large Deviations 321 A.1 Chernoff Bounds, 321 A.2 Lower Bounds, 330 A.3 Exercises, 334 Triangle-Free Graphs Have Large Independence Numbers, 336 Appendix B Paul Erd˝os 339 B.1 Papers, 339 B.2 Conjectures, 341 B.3 On Erd˝os, 342 B.4 Uncle Paul, 343 The Rich Get Richer, 346 Appendix C Hints to Selected Exercises 349 REFERENCES 355 AUTHOR INDEX 367 SUBJECT INDEX 371

PROBABILISTIC MACHINE LEARNING: AN INTRODUCTION (1版)

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Probabilistic Machine Learning 系列名：Adaptive Computation and Machine Learning series ISBN13：9780262046824 出版社：Mit Pr 作者：Kevin P. Murphy 裝訂／頁數：精裝／864頁 出版日：2022/02/01