Probabilistic Methods In Combinatorics

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The Probabilistic Method

Author : Noga Alon
Publisher : John Wiley & Sons
ISBN 13 : 1118210441
Total Pages : 257 pages
Book Rating : 4.1/5 (182 download)

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Book Synopsis The Probabilistic Method by : Noga Alon

Download or read book The Probabilistic Method written by Noga Alon and published by John Wiley & Sons. This book was released on 2011-09-20 with total page 257 pages. Available in PDF, EPUB and Kindle. Book excerpt: Praise for the Second Edition: "Serious researchers in combinatorics or algorithm design will wish to read the book in its entirety...the book may also be enjoyed on a lighter level since the different chapters are largely independent and so it is possible to pick out gems in one's own area..." —Formal Aspects of Computing This Third Edition of The Probabilistic Method reflects the most recent developments in the field while maintaining the standard of excellence that established this book as the leading reference on probabilistic methods in combinatorics. Maintaining its clear writing style, illustrative examples, and practical exercises, this new edition emphasizes methodology, enabling readers to use probabilistic techniques for solving problems in such fields as theoretical computer science, mathematics, and statistical physics. The book begins with a description of tools applied in probabilistic arguments, including basic techniques that use expectation and variance as well as the more recent applications of martingales and correlation inequalities. Next, the authors examine where probabilistic techniques have been applied successfully, exploring such topics as discrepancy and random graphs, circuit complexity, computational geometry, and derandomization of randomized algorithms. Sections labeled "The Probabilistic Lens" offer additional insights into the application of the probabilistic approach, and the appendix has been updated to include methodologies for finding lower bounds for Large Deviations. The Third Edition also features: A new chapter on graph property testing, which is a current topic that incorporates combinatorial, probabilistic, and algorithmic techniques An elementary approach using probabilistic techniques to the powerful Szemerédi Regularity Lemma and its applications New sections devoted to percolation and liar games A new chapter that provides a modern treatment of the Erdös-Rényi phase transition in the Random Graph Process Written by two leading authorities in the field, The Probabilistic Method, Third Edition is an ideal reference for researchers in combinatorics and algorithm design who would like to better understand the use of probabilistic methods. The book's numerous exercises and examples also make it an excellent textbook for graduate-level courses in mathematics and computer science.

The Probabilistic Method

Author : Noga Alon
Publisher : John Wiley & Sons
ISBN 13 : 1119062071
Total Pages : 396 pages
Book Rating : 4.1/5 (19 download)

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Book Synopsis The Probabilistic Method by : Noga Alon

Download or read book The Probabilistic Method written by Noga Alon and published by John Wiley & Sons. This book was released on 2015-11-02 with total page 396 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Probabilistic Methods for Algorithmic Discrete Mathematics

Author : Michel Habib
Publisher : Springer Science & Business Media
ISBN 13 : 3662127881
Total Pages : 342 pages
Book Rating : 4.6/5 (621 download)

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Book Synopsis Probabilistic Methods for Algorithmic Discrete Mathematics by : Michel Habib

Download or read book Probabilistic Methods for Algorithmic Discrete Mathematics written by Michel Habib and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: Leave nothing to chance. This cliche embodies the common belief that ran domness has no place in carefully planned methodologies, every step should be spelled out, each i dotted and each t crossed. In discrete mathematics at least, nothing could be further from the truth. Introducing random choices into algorithms can improve their performance. The application of proba bilistic tools has led to the resolution of combinatorial problems which had resisted attack for decades. The chapters in this volume explore and celebrate this fact. Our intention was to bring together, for the first time, accessible discus sions of the disparate ways in which probabilistic ideas are enriching discrete mathematics. These discussions are aimed at mathematicians with a good combinatorial background but require only a passing acquaintance with the basic definitions in probability (e.g. expected value, conditional probability). A reader who already has a firm grasp on the area will be interested in the original research, novel syntheses, and discussions of ongoing developments scattered throughout the book. Some of the most convincing demonstrations of the power of these tech niques are randomized algorithms for estimating quantities which are hard to compute exactly. One example is the randomized algorithm of Dyer, Frieze and Kannan for estimating the volume of a polyhedron. To illustrate these techniques, we consider a simple related problem. Suppose S is some region of the unit square defined by a system of polynomial inequalities: Pi (x. y) ~ o.

Graph Colouring and the Probabilistic Method

Author : Michael Molloy
Publisher : Springer Science & Business Media
ISBN 13 : 3642040160
Total Pages : 320 pages
Book Rating : 4.6/5 (42 download)

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Book Synopsis Graph Colouring and the Probabilistic Method by : Michael Molloy

Download or read book Graph Colouring and the Probabilistic Method written by Michael Molloy and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: Over the past decade, many major advances have been made in the field of graph coloring via the probabilistic method. This monograph, by two of the best on the topic, provides an accessible and unified treatment of these results, using tools such as the Lovasz Local Lemma and Talagrand's concentration inequality.

The Probabilistic Method

Author : Noga Alon
Publisher : John Wiley & Sons
ISBN 13 : 0471653985
Total Pages : 322 pages
Book Rating : 4.4/5 (716 download)

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Book Synopsis The Probabilistic Method by : Noga Alon

Download or read book The Probabilistic Method written by Noga Alon and published by John Wiley & Sons. This book was released on 2004-04-05 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: The leading reference on probabilistic methods in combinatorics-now expanded and updated When it was first published in 1991, The Probabilistic Method became instantly the standard reference on one of the most powerful and widely used tools in combinatorics. Still without competition nearly a decade later, this new edition brings you up to speed on recent developments, while adding useful exercises and over 30% new material. It continues to emphasize the basic elements of the methodology, discussing in a remarkably clear and informal style both algorithmic and classical methods as well as modern applications. The Probabilistic Method, Second Edition begins with basic techniques that use expectation and variance, as well as the more recent martingales and correlation inequalities, then explores areas where probabilistic techniques proved successful, including discrepancy and random graphs as well as cutting-edge topics in theoretical computer science. A series of proofs, or "probabilistic lenses," are interspersed throughout the book, offering added insight into the application of the probabilistic approach. New and revised coverage includes: * Several improved as well as new results * A continuous approach to discrete probabilistic problems * Talagrand's Inequality and other novel concentration results * A discussion of the connection between discrepancy and VC-dimension * Several combinatorial applications of the entropy function and its properties * A new section on the life and work of Paul Erdös-the developer of the probabilistic method

Proceedings of the International Congress of Mathematicians

Author : S.D. Chatterji
Publisher : Birkhäuser
ISBN 13 : 3034890788
Total Pages : 1669 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Proceedings of the International Congress of Mathematicians by : S.D. Chatterji

Download or read book Proceedings of the International Congress of Mathematicians written by S.D. Chatterji and published by Birkhäuser. This book was released on 2012-12-06 with total page 1669 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the first ICM was held in Zürich in 1897, it has become the pinnacle of mathematical gatherings. It aims at giving an overview of the current state of different branches of mathematics and its applications as well as an insight into the treatment of special problems of exceptional importance. The proceedings of the ICMs have provided a rich chronology of mathematical development in all its branches and a unique documentation of contemporary research. They form an indispensable part of every mathematical library. The Proceedings of the International Congress of Mathematicians 1994, held in Zürich from August 3rd to 11th, 1994, are published in two volumes. Volume I contains an account of the organization of the Congress, the list of ordinary members, the reports on the work of the Fields Medalists and the Nevanlinna Prize Winner, the plenary one-hour addresses, and the invited addresses presented at Section Meetings 1 - 6. Volume II contains the invited address for Section Meetings 7 - 19. A complete author index is included in both volumes. '...the content of these impressive two volumes sheds a certain light on the present state of mathematical sciences and anybody doing research in mathematics should look carefully at these Proceedings. For young people beginning research, this is even more important, so these are a must for any serious mathematics library. The graphical presentation is, as always with Birkhäuser, excellent....' (Revue Roumaine de Mathematiques pures et Appliquées)

Ten Lectures on the Probabilistic Method

Author : Joel Spencer
Publisher : SIAM
ISBN 13 : 9781611970074
Total Pages : 98 pages
Book Rating : 4.9/5 (7 download)

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Book Synopsis Ten Lectures on the Probabilistic Method by : Joel Spencer

Download or read book Ten Lectures on the Probabilistic Method written by Joel Spencer and published by SIAM. This book was released on 1994-01-01 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: This update of the 1987 title of the same name is an examination of what is currently known about the probabilistic method, written by one of its principal developers. Based on the notes from Spencer's 1986 series of ten lectures, this new edition contains an additional lecture: The Janson inequalities. These inequalities allow accurate approximation of extremely small probabilities. A new algorithmic approach to the Lovasz Local Lemma, attributed to Jozsef Beck, has been added to Lecture 8, as well. Throughout the monograph, Spencer retains the informal style of his original lecture notes and emphasizes the methodology, shunning the more technical "best possible" results in favor of clearer exposition. The book is not encyclopedic--it contains only those examples that clearly display the methodology. The probabilistic method is a powerful tool in graph theory, combinatorics, and theoretical computer science. It allows one to prove the existence of objects with certain properties (e.g., colorings) by showing that an appropriately defined random object has positive probability of having those properties.

Polynomial Methods in Combinatorics

Author : Larry Guth
Publisher : American Mathematical Soc.
ISBN 13 : 1470428903
Total Pages : 287 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Polynomial Methods in Combinatorics by : Larry Guth

Download or read book Polynomial Methods in Combinatorics written by Larry Guth and published by American Mathematical Soc.. This book was released on 2016-06-10 with total page 287 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book.

Probabilistic Methods in combinatorics

Author : Paul Erdös
Publisher :
ISBN 13 :
Total Pages : 106 pages
Book Rating : 4.:/5 (246 download)

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Book Synopsis Probabilistic Methods in combinatorics by : Paul Erdös

Download or read book Probabilistic Methods in combinatorics written by Paul Erdös and published by . This book was released on 1974 with total page 106 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Random Trees

Author : Michael Drmota
Publisher : Springer Science & Business Media
ISBN 13 : 3211753575
Total Pages : 466 pages
Book Rating : 4.2/5 (117 download)

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Book Synopsis Random Trees by : Michael Drmota

Download or read book Random Trees written by Michael Drmota and published by Springer Science & Business Media. This book was released on 2009-04-16 with total page 466 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book is to provide a thorough introduction to various aspects of trees in random settings and a systematic treatment of the mathematical analysis techniques involved. It should serve as a reference book as well as a basis for future research.

A Walk Through Combinatorics

Author : Mikl¢s B¢na
Publisher : World Scientific
ISBN 13 : 9812568859
Total Pages : 492 pages
Book Rating : 4.8/5 (125 download)

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Book Synopsis A Walk Through Combinatorics by : Mikl¢s B¢na

Download or read book A Walk Through Combinatorics written by Mikl¢s B¢na and published by World Scientific. This book was released on 2006 with total page 492 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a textbook for an introductory combinatorics course that can take up one or two semesters. An extensive list of problems, ranging from routine exercises to research questions, is included. In each section, there are also exercises that contain material not explicitly discussed in the preceding text, so as to provide instructors with extra choices if they want to shift the emphasis of their course. Just as with the first edition, the new edition walks the reader through the classic parts of combinatorial enumeration and graph theory, while also discussing some recent progress in the area: on the one hand, providing material that will help students learn the basic techniques, and on the other hand, showing that some questions at the forefront of research are comprehensible and accessible for the talented and hard-working undergraduate. The basic topics discussed are: the twelvefold way, cycles in permutations, the formula of inclusion and exclusion, the notion of graphs and trees, matchings and Eulerian and Hamiltonian cycles. The selected advanced topics are: Ramsey theory, pattern avoidance, the probabilistic method, partially ordered sets, and algorithms and complexity. As the goal of the book is to encourage students to learn more combinatorics, every effort has been made to provide them with a not only useful, but also enjoyable and engaging reading.

Analytic Combinatorics

Author : Philippe Flajolet
Publisher : Cambridge University Press
ISBN 13 : 1139477161
Total Pages : 825 pages
Book Rating : 4.1/5 (394 download)

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Book Synopsis Analytic Combinatorics by : Philippe Flajolet

Download or read book Analytic Combinatorics written by Philippe Flajolet and published by Cambridge University Press. This book was released on 2009-01-15 with total page 825 pages. Available in PDF, EPUB and Kindle. Book excerpt: Analytic combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures. The theory has emerged over recent decades as essential both for the analysis of algorithms and for the study of scientific models in many disciplines, including probability theory, statistical physics, computational biology, and information theory. With a careful combination of symbolic enumeration methods and complex analysis, drawing heavily on generating functions, results of sweeping generality emerge that can be applied in particular to fundamental structures such as permutations, sequences, strings, walks, paths, trees, graphs and maps. This account is the definitive treatment of the topic. The authors give full coverage of the underlying mathematics and a thorough treatment of both classical and modern applications of the theory. The text is complemented with exercises, examples, appendices and notes to aid understanding. The book can be used for an advanced undergraduate or a graduate course, or for self-study.

Graph Searching Games and Probabilistic Methods

Author : Anthony Bonato
Publisher : CRC Press
ISBN 13 : 135181477X
Total Pages : 346 pages
Book Rating : 4.3/5 (518 download)

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Book Synopsis Graph Searching Games and Probabilistic Methods by : Anthony Bonato

Download or read book Graph Searching Games and Probabilistic Methods written by Anthony Bonato and published by CRC Press. This book was released on 2017-11-28 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: Graph Searching Games and Probabilistic Methods is the first book that focuses on the intersection of graph searching games and probabilistic methods. The book explores various applications of these powerful mathematical tools to games and processes such as Cops and Robbers, Zombie and Survivors, and Firefighting. Written in an engaging style, the book is accessible to a wide audience including mathematicians and computer scientists. Readers will find that the book provides state-of-the-art results, techniques, and directions in graph searching games, especially from the point of view of probabilistic methods. The authors describe three directions while providing numerous examples, which include: • Playing a deterministic game on a random board. • Players making random moves. • Probabilistic methods used to analyze a deterministic game.

Extremal Combinatorics

Author : Stasys Jukna
Publisher : Springer Science & Business Media
ISBN 13 : 3662046504
Total Pages : 389 pages
Book Rating : 4.6/5 (62 download)

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Book Synopsis Extremal Combinatorics by : Stasys Jukna

Download or read book Extremal Combinatorics written by Stasys Jukna and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 389 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a concise, up-to-date introduction to extremal combinatorics for non-specialists. Strong emphasis is made on theorems with particularly elegant and informative proofs which may be called the gems of the theory. A wide spectrum of the most powerful combinatorial tools is presented, including methods of extremal set theory, the linear algebra method, the probabilistic method and fragments of Ramsey theory. A thorough discussion of recent applications to computer science illustrates the inherent usefulness of these methods.

Extremal Graph Theory

Author : Bela Bollobas
Publisher : Courier Corporation
ISBN 13 : 0486317587
Total Pages : 512 pages
Book Rating : 4.4/5 (863 download)

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Book Synopsis Extremal Graph Theory by : Bela Bollobas

Download or read book Extremal Graph Theory written by Bela Bollobas and published by Courier Corporation. This book was released on 2013-07-02 with total page 512 pages. Available in PDF, EPUB and Kindle. Book excerpt: The ever-expanding field of extremal graph theory encompasses a diverse array of problem-solving methods, including applications to economics, computer science, and optimization theory. This volume, based on a series of lectures delivered to graduate students at the University of Cambridge, presents a concise yet comprehensive treatment of extremal graph theory. Unlike most graph theory treatises, this text features complete proofs for almost all of its results. Further insights into theory are provided by the numerous exercises of varying degrees of difficulty that accompany each chapter. Although geared toward mathematicians and research students, much of Extremal Graph Theory is accessible even to undergraduate students of mathematics. Pure mathematicians will find this text a valuable resource in terms of its unusually large collection of results and proofs, and professionals in other fields with an interest in the applications of graph theory will also appreciate its precision and scope.

Combinatorics of Finite Sets

Author : Ian Anderson
Publisher : Courier Corporation
ISBN 13 : 9780486422572
Total Pages : 276 pages
Book Rating : 4.4/5 (225 download)

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Book Synopsis Combinatorics of Finite Sets by : Ian Anderson

Download or read book Combinatorics of Finite Sets written by Ian Anderson and published by Courier Corporation. This book was released on 2002-01-01 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: Among other subjects explored are the Clements-Lindström extension of the Kruskal-Katona theorem to multisets and the Greene-Kleitmen result concerning k-saturated chain partitions of general partially ordered sets. Includes exercises and solutions.

The Simplex Method

Author : Karl Heinz Borgwardt
Publisher : Springer Science & Business Media
ISBN 13 : 3642615783
Total Pages : 279 pages
Book Rating : 4.6/5 (426 download)

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Book Synopsis The Simplex Method by : Karl Heinz Borgwardt

Download or read book The Simplex Method written by Karl Heinz Borgwardt and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 279 pages. Available in PDF, EPUB and Kindle. Book excerpt: For more than 35 years now, George B. Dantzig's Simplex-Method has been the most efficient mathematical tool for solving linear programming problems. It is proba bly that mathematical algorithm for which the most computation time on computers is spent. This fact explains the great interest of experts and of the public to understand the method and its efficiency. But there are linear programming problems which will not be solved by a given variant of the Simplex-Method in an acceptable time. The discrepancy between this (negative) theoretical result and the good practical behaviour of the method has caused a great fascination for many years. While the "worst-case analysis" of some variants of the method shows that this is not a "good" algorithm in the usual sense of complexity theory, it seems to be useful to apply other criteria for a judgement concerning the quality of the algorithm. One of these criteria is the average computation time, which amounts to an anal ysis of the average number of elementary arithmetic computations and of the number of pivot steps. A rigid analysis of the average behaviour may be very helpful for the decision which algorithm and which variant shall be used in practical applications. The subject and purpose of this book is to explain the great efficiency in prac tice by assuming certain distributions on the "real-world" -problems. Other stochastic models are realistic as well and so this analysis should be considered as one of many possibilities.