Details

The Probabilistic Method


The Probabilistic Method


Wiley Series in Discrete Mathematics and Optimization 4. Aufl.

von: Noga Alon, Joel H. Spencer

91,99 €

Verlag: Wiley
Format: PDF
Veröffentl.: 28.10.2015
ISBN/EAN: 9781119061960
Sprache: englisch
Anzahl Seiten: 400

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Beschreibungen

<p><b>Praise for the <i>Third Edition</i></b></p> <p><b>“Researchers of any kind of extremal combinatorics or theoretical computer science will welcome the new edition of this book.” </b><b>- <i>MAA Reviews</i></b></p> <p>Maintaining a standard of excellence that establishes <i>The Probabilistic Method </i>as the leading reference on probabilistic methods in combinatorics, the <i>Fourth Edition </i>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.</p> <p>Emphasizing the methodology and techniques that enable problem-solving, <i>The Probabilistic Method, Fourth Edition </i>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 <i>Fourth Edition </i>features:</p> <ul> <li>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</li> <li>New coverage on topics such as the Local Lemma, Six Standard Deviations result in Discrepancy Theory, Property B, and graph limits</li> <li>Updated sections to reflect major developments on the newest topics, discussions of the hypergraph container method, and many new references and improved results</li> </ul> <p><i>The Probabilistic Method, Fourth Edition </i>is an ideal textbook for upper-undergraduate and graduate-level students majoring in mathematics, computer science, operations research, and statistics. The <i>Fourth Edition </i>is also an excellent reference for researchers and combinatorists who use probabilistic methods, discrete mathematics, and number theory.</p> <p><b>Noga Alon, PhD,</b> 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 <i>Random Structures and Algorithms</i>, Dr. Alon is the recipient of the Polya Prize, The Gödel Prize, The Israel Prize, and the EMET Prize.</p> <p><b>Joel H. Spencer, PhD,</b> is Professor of Mathematics and Computer Science at the Courant Institute of New York University. He is the cofounder and coeditor of the journal <i>Random Structures</i> <i>and Algorithms </i>and is a Sloane Foundation Fellow. Dr. Spencer has written more than 200 published articles and is the coauthor of <i>Ramsey Theory, Second Edition</i>, also published by Wiley.</p>
<p>PREFACE xiii</p> <p>ACKNOWLEDGMENTS xv</p> <p><b>PART I METHODS 1</b></p> <p><b>1 The Basic Method 3</b></p> <p>1.1 The Probabilistic Method, 3</p> <p>1.2 Graph Theory, 5</p> <p>1.3 Combinatorics, 9</p> <p>1.4 Combinatorial Number Theory, 11</p> <p>1.5 Disjoint Pairs, 12</p> <p>1.6 Independent Sets and List Coloring, 13</p> <p>1.7 Exercises, 16</p> <p><b><i>The Erd˝os–Ko–Rado Theorem, 18</i></b></p> <p><b>2 Linearity of Expectation 19</b></p> <p>2.1 Basics, 19</p> <p>2.2 Splitting Graphs, 20</p> <p>2.3 Two Quickies, 22</p> <p>2.4 Balancing Vectors, 23</p> <p>2.5 Unbalancing Lights, 25</p> <p>2.6 Without Coin Flips, 26</p> <p>2.7 Exercises, 27</p> <p><b><i>Brégman’s Theorem, 29</i></b></p> <p><b>3 Alterations 31</b></p> <p>3.1 Ramsey Numbers, 31</p> <p>3.2 Independent Sets, 33</p> <p>3.3 Combinatorial Geometry, 34</p> <p>3.4 Packing, 35</p> <p>3.5 Greedy Coloring, 36</p> <p>3.6 Continuous Time, 38</p> <p>3.7 Exercises, 41</p> <p><b><i>High Girth and High Chromatic Number, 43</i></b></p> <p><b>4 The Second Moment 45</b></p> <p>4.1 Basics, 45</p> <p>4.2 Number Theory, 46</p> <p>4.3 More Basics, 49</p> <p>4.4 Random Graphs, 51</p> <p>4.5 Clique Number, 55</p> <p>4.6 Distinct Sums, 57</p> <p>4.7 The Rödl nibble, 58</p> <p>4.8 Exercises, 64</p> <p><b><i>Hamiltonian Paths, 65</i></b></p> <p><b>5 The Local Lemma 69</b></p> <p>5.1 The Lemma, 69</p> <p>5.2 Property B and Multicolored Sets of Real Numbers, 72</p> <p>5.3 Lower Bounds for Ramsey Numbers, 73</p> <p>5.4 A Geometric Result, 75</p> <p>5.5 The Linear Arboricity of Graphs, 76</p> <p>5.6 Latin Transversals, 80</p> <p>5.7 Moser’s Fix-It Algorithm, 81</p> <p>5.8 Exercises, 87</p> <p><b><i>Directed Cycles, 88</i></b></p> <p><b>6 Correlation Inequalities 89</b></p> <p>6.1 The Four Functions Theorem of Ahlswede and Daykin, 90</p> <p>6.2 The FKG Inequality, 93</p> <p>6.3 Monotone Properties, 94</p> <p>6.4 Linear Extensions of Partially Ordered Sets, 97</p> <p>6.5 Exercises, 99</p> <p><b><i>Turán’s Theorem, 100</i></b></p> <p><b>7 Martingales and Tight Concentration 103</b></p> <p>7.1 Definitions, 103</p> <p>7.2 Large Deviations, 105</p> <p>7.3 Chromatic Number, 107</p> <p>7.4 Two General Settings, 109</p> <p>7.5 Four Illustrations, 113</p> <p>7.6 Talagrand’s Inequality, 116</p> <p>7.7 Applications of Talagrand’s Inequality, 119</p> <p>7.8 Kim–Vu Polynomial Concentration, 121</p> <p>7.9 Exercises, 123</p> <p><b><i>Weierstrass Approximation Theorem, 124</i></b></p> <p><b>8 The Poisson Paradigm 127</b></p> <p>8.1 The Janson Inequalities, 127</p> <p>8.2 The Proofs, 129</p> <p>8.3 Brun’s Sieve, 132</p> <p>8.4 Large Deviations, 135</p> <p>8.5 Counting Extensions, 137</p> <p>8.6 Counting Representations, 139</p> <p>8.7 Further Inequalities, 142</p> <p>8.8 Exercises, 143</p> <p><b><i>Local Coloring, 144</i></b></p> <p><b>9 Quasirandomness 147</b></p> <p>9.1 The Quadratic Residue Tournaments, 148</p> <p>9.2 Eigenvalues and Expanders, 151</p> <p>9.3 Quasirandom Graphs, 157</p> <p>9.4 Szemerédi’s Regularity Lemma, 165</p> <p>9.5 Graphons, 170</p> <p>9.6 Exercises, 172</p> <p><b><i>Random Walks, 174</i></b></p> <p><b>PART II TOPICS 177</b></p> <p><b>10 Random Graphs 179</b></p> <p>10.1 Subgraphs, 180</p> <p>10.2 Clique Number, 183</p> <p>10.3 Chromatic Number, 184</p> <p>10.4 Zero–One Laws, 186</p> <p>10.5 Exercises, 193</p> <p><b><i>Counting Subgraphs, 195</i></b></p> <p><b>11 The Erd˝os–Rényi Phase Transition 197</b></p> <p>11.1 An Overview, 197</p> <p>11.2 Three Processes, 199</p> <p>11.3 The Galton–Watson Branching Process, 201</p> <p>11.4 Analysis of the Poisson Branching Process, 202</p> <p>11.5 The Graph Branching Model, 204</p> <p>11.6 The Graph and Poisson Processes Compared, 205</p> <p>11.7 The Parametrization Explained, 207</p> <p>11.8 The Subcritical Regions, 208</p> <p>11.9 The Supercritical Regimes, 209</p> <p>11.10 The Critical Window, 212</p> <p>11.11 Analogies to Classical Percolation Theory, 214</p> <p>11.12 Exercises, 219</p> <p><b><i>Long paths in the supercritical regime, 220</i></b></p> <p><b>12 Circuit Complexity 223</b></p> <p>12.1 Preliminaries, 223</p> <p>12.2 Random Restrictions and Bounded-Depth Circuits, 225</p> <p>12.3 More on Bounded-Depth Circuits, 229</p> <p>12.4 Monotone Circuits, 232</p> <p>12.5 Formulae, 235</p> <p>12.6 Exercises, 236</p> <p><b><i>Maximal Antichains, 237</i></b></p> <p><b>13 Discrepancy 239</b></p> <p>13.1 Basics, 239</p> <p>13.2 Six Standard Deviations Suffice, 241</p> <p>13.3 Linear and Hereditary Discrepancy, 245</p> <p>13.4 Lower Bounds, 248</p> <p>13.5 The Beck–Fiala Theorem, 250</p> <p>13.6 Exercises, 251</p> <p><b><i>Unbalancing Lights, 253</i></b></p> <p><b>14 Geometry 255</b></p> <p>14.1 The Greatest Angle Among Points in Euclidean Spaces, 256</p> <p>14.2 Empty Triangles Determined by Points in the Plane, 257</p> <p>14.3 Geometrical Realizations of Sign Matrices, 259</p> <p>14.4 𝜖-Nets and VC-Dimensions of Range Spaces, 261</p> <p>14.5 Dual Shatter Functions and Discrepancy, 266</p> <p>14.6 Exercises, 269</p> <p><b><i>Efficient Packing, 270</i></b></p> <p><b>15 Codes, Games, and Entropy 273</b></p> <p>15.1 Codes, 273</p> <p>15.2 Liar Game, 276</p> <p>15.3 Tenure Game, 278</p> <p>15.4 Balancing Vector Game, 279</p> <p>15.5 Nonadaptive Algorithms, 281</p> <p>15.6 Half Liar Game, 282</p> <p>15.7 Entropy, 284</p> <p>15.8 Exercises, 289</p> <p><b><i>An Extremal Graph, 291</i></b></p> <p><b>16 Derandomization 293</b></p> <p>16.1 The Method of Conditional Probabilities, 293</p> <p>16.2 d-Wise Independent Random Variables in Small Sample Spaces, 297</p> <p>16.3 Exercises, 302</p> <p><b><i>Crossing Numbers, Incidences, Sums and Products, 303</i></b></p> <p><b>17 Graph Property Testing 307</b></p> <p>17.1 Property Testing, 307</p> <p>17.2 Testing Colorability, 308</p> <p>17.3 Testing Triangle-Freeness, 312</p> <p>17.4 Characterizing the Testable Graph Properties, 314</p> <p>17.5 Exercises, 316</p> <p><b><i>Turán Numbers and Dependent Random Choice, 317</i></b></p> <p>Appendix A Bounding of Large Deviations 321</p> <p>A.1 Chernoff Bounds, 321</p> <p>A.2 Lower Bounds, 330</p> <p>A.3 Exercises, 334</p> <p><b><i>Triangle-Free Graphs Have Large Independence Numbers, 336</i></b></p> <p>Appendix B Paul Erd˝os 339</p> <p>B.1 Papers, 339</p> <p>B.2 Conjectures, 341</p> <p>B.3 On Erd˝os, 342</p> <p>B.4 Uncle Paul, 343</p> <p><b><i>The Rich Get Richer, 346</i></b></p> <p>Appendix C Hints to Selected Exercises 349</p> <p>REFERENCES 355</p> <p>AUTHOR INDEX 367</p> <p>SUBJECT INDEX 371</p>
<p>"This is an ideal textbook for upper-undergraduate and graduate-level students majoring in mathematics, computer science, operations research, and statistics." (<i>Springer Nature</i>, 2016)</p>
<p><b>Noga Alon, PhD,</b> 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 <i>Random Structures and Algorithms</i>, Dr. Alon is the recipient of the Polya Prize, The Gödel Prize, The Israel Prize, and the EMET Prize.</p> <p><b>Joel H. Spencer, PhD,</b> is Professor of Mathematics and Computer Science at the Courant Institute of New York University. He is the cofounder and coeditor of the journal <i>Random Structures</i> <i>and Algorithms </i>and is a Sloane Foundation Fellow. Dr. Spencer has written more than 200 published articles and is the coauthor of <i>Ramsey Theory, Second Edition</i>, also published by Wiley.</p>
<p><b>Praise for the <i>Third Edition</i></b></p> <p><b>“Researchers of any kind of extremal combinatorics or theoretical computer science will welcome the new edition of this book.” </b><b>- <i>MAA Reviews</i></b></p> <p>Maintaining a standard of excellence that establishes <i>The Probabilistic Method </i>as the leading reference on probabilistic methods in combinatorics, the <i>Fourth Edition </i>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.</p> <p>Emphasizing the methodology and techniques that enable problem-solving, <i>The Probabilistic Method, Fourth Edition </i>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 <i>Fourth Edition </i>features:</p> <ul> <li>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</li> <li>New coverage on topics such as the Local Lemma, Six Standard Deviations result in Discrepancy Theory, Property B, and graph limits</li> <li>Updated sections to reflect major developments on the newest topics, discussions of the hypergraph container method, and many new references and improved results</li> </ul> <p><i>The Probabilistic Method, Fourth Edition </i>is an ideal textbook for upper-undergraduate and graduate-level students majoring in mathematics, computer science, operations research, and statistics. The <i>Fourth Edition </i>is also an excellent reference for researchers and combinatorists who use probabilistic methods, discrete mathematics, and number theory.</p>

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