The Probabilistic Method (Wiley Series in Discrete Mathematics and Optimization)
Language: English
Pages: 400
ISBN: 1119061954
Format: PDF / Kindle (mobi) / ePub
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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dangerous or, possibly, many of its points were saved because neighboring (in G) sets were dangerous. The probability of a particular Ai becoming dangerous is at most 21""™1 since for this to occur the first ri\ coin flips determining colors of j £ Ai must come up the same. (We only have inequality THE ALGORITHMIC ASPECT 81 since in addition n\ points of Ai must be reached before being saved.) Let V be an independent set in G; that is, the Ai eV are mutually disjoint. Then the probability
a,i+2i-i) differ in at least ( | — e)l coordinates. 3. Let G = (V, E) be a simple graph and suppose each v £ V is associated with a set S(v) of colors of size at least lOd, where d > 1. Suppose, in addition, that for each v e V and c G S(v) there are at most d neighbors u of v such that c lies in S(u). Prove that there is a proper coloring of G assigning to each vértex v a color from its class S(v). 4. Let G = (V, E) be a eyele of length An and let V = Vi U V2 U • • • U Vn be a partition of its
follows. For each x £ L, a(x) = n(x)f(x), 7(x) = n(x)f(x)g(x), /3(x) = p,(x)g(x), 6{x) = p,{x). We claim that these functions satisfy the hypothesis of the Ahlswede-Daykin Theorem, stated in Corollary 6.1.2. Indeed, if x, y G L then, by the supermodularity of p, and since / and g are increasing, a(x)(3(y) = p{x)f{x)p(y)g{y) < p(xVy)f(x)g{y)p(xAy) < l¿(x V y)f(x V y)g(x V y)p{x A y) = ^{x V y)5{x A y). Therefore by Corollary 6.1.2 (with X = Y = L), a(L)(3(L) < 7 (L)5(L), which is the
of G. Let Cx be the event A 2 SXJ/2 and X x the corresponding indicator random variable. We use Janson's Inequality to bound E [Xx] — Pr [Cx\. Here p = o(l) so e = o(l). £) Pr [Bxyz] = M a s defined above. Dependency xj/z ~ xuv occurs if and only if the sets overlap (other than in x). Henee A = Y, P r [Bxyz A Bxv*'] = 0 ( » V ) = o(l) y,z.2' 2 3+ sincep = n ~ / ° ^ ' . Thus c n E[*x] Now define X the number of vértices x not lying in a triangle. Then from linearity of expectation, E [X] =
to the formulation of Section 8.1. Our object is to derive large deviation results on X similar to those in Appendix A. Given a point in the probability space (i.e., a selection of R) we cali an Índex set J C I a disjoint family (abbreviated disfam) if • Bj for every j e J. • For no j , f e J is j ~ j ' . If, in addition, • If j ' ^ J and Bj' then j ~ j ' for some j e J, then we cali J a maximal disjoint family (maxdisfam). We give some general results on the possible sizes of maxdisfams. The
