Probabilistic Methods For Algorithmic Discrete Mathematics

Author : Michel Habib,Colin McDiarmid,Jorge Ramirez-Alfonsin,Bruce Reed
Publisher : Springer Science & Business Media
Page : 342 pages
File Size : 49,5 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9783662127889

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Probabilistic Methods for Algorithmic Discrete Mathematics by Michel Habib,Colin McDiarmid,Jorge Ramirez-Alfonsin,Bruce Reed Pdf

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.

Author : Noga Alon,Joel H. Spencer
Publisher : John Wiley & Sons
Page : 400 pages
File Size : 42,5 Mb
Release : 2015-10-28
Category : Mathematics
ISBN : 9781119061960

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The Probabilistic Method by Noga Alon,Joel H. Spencer Pdf

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.

Author : Helmut Knebl
Publisher : Springer Nature
Page : 349 pages
File Size : 46,7 Mb
Release : 2020-10-31
Category : Computers
ISBN : 9783030597580

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Algorithms and Data Structures by Helmut Knebl Pdf

This is a central topic in any computer science curriculum. To distinguish this textbook from others, the author considers probabilistic methods as being fundamental for the construction of simple and efficient algorithms, and in each chapter at least one problem is solved using a randomized algorithm. Data structures are discussed to the extent needed for the implementation of the algorithms. The specific algorithms examined were chosen because of their wide field of application. This book originates from lectures for undergraduate and graduate students. The text assumes experience in programming algorithms, especially with elementary data structures such as chained lists, queues, and stacks. It also assumes familiarity with mathematical methods, although the author summarizes some basic notations and results from probability theory and related mathematical terminology in the appendices. He includes many examples to explain the individual steps of the algorithms, and he concludes each chapter with numerous exercises.

Author : Joel Spencer
Publisher : SIAM
Page : 98 pages
File Size : 44,6 Mb
Release : 1994-01-01
Category : Mathematics
ISBN : 1611970075

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Ten Lectures on the Probabilistic Method by Joel Spencer Pdf

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.

Author : Michael Mitzenmacher
Publisher : Unknown
Page : 368 pages
File Size : 48,6 Mb
Release : 2005
Category : Electronic
ISBN : OCLC:1137344100

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Probability and Computing by Michael Mitzenmacher Pdf

Randomization and probabilistic techniques play an important role in modern computer science, with applications ranging from combinatorial optimization and machine learning to communication networks and secure protocols. This textbook is designed to accompany a one- or two-semester course for advanced undergraduates or beginning graduate students in computer science and applied mathematics. It gives an excellent introduction to the probabilistic techniques and paradigms used in the development of probabilistic algorithms and analyses. It assumes only an elementary background in discrete mathematics and gives a rigorous yet accessible treatment of the material, with numerous examples and applications. The first half of the book covers core material, including random sampling, expectations, Markov's inequality, Chevyshev's inequality, Chernoff bounds, balls and bins models, the probabilistic method, and Markov chains. In the second half, the authors delve into more advanced topics such as continuous probability, applications of limited independence, entropy, Markov chain Monte Carlo methods, coupling, martingales, and balanced allocations. With its comprehensive selection of topics, along with many examples and exercises, this book is an indispensable teaching tool.

Author : Michael Mitzenmacher,Eli Upfal
Publisher : Cambridge University Press
Page : 372 pages
File Size : 41,6 Mb
Release : 2005-01-31
Category : Computers
ISBN : 9781139643788

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Probability and Computing by Michael Mitzenmacher,Eli Upfal Pdf

Randomization and probabilistic techniques play an important role in modern computer science, with applications ranging from combinatorial optimization and machine learning to communication networks and secure protocols. This 2005 textbook is designed to accompany a one- or two-semester course for advanced undergraduates or beginning graduate students in computer science and applied mathematics. It gives an excellent introduction to the probabilistic techniques and paradigms used in the development of probabilistic algorithms and analyses. It assumes only an elementary background in discrete mathematics and gives a rigorous yet accessible treatment of the material, with numerous examples and applications. The first half of the book covers core material, including random sampling, expectations, Markov's inequality, Chevyshev's inequality, Chernoff bounds, the probabilistic method and Markov chains. The second half covers more advanced topics such as continuous probability, applications of limited independence, entropy, Markov chain Monte Carlo methods and balanced allocations. With its comprehensive selection of topics, along with many examples and exercises, this book is an indispensable teaching tool.

Author : V. F. Kolchin,V. Ya Kozlov,V. V. Mazalov,Yu. L. Pavlov,Yu. V. Prokhorov
Publisher : Walter de Gruyter GmbH & Co KG
Page : 400 pages
File Size : 50,5 Mb
Release : 2020-05-18
Category : Mathematics
ISBN : 9783112314104

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Probabilistic Methods in Discrete Mathematics by V. F. Kolchin,V. Ya Kozlov,V. V. Mazalov,Yu. L. Pavlov,Yu. V. Prokhorov Pdf

No detailed description available for "Probabilistic Methods in Discrete Mathematics".

Author : Stephen B. Maurer,Anthony Ralston
Publisher : CRC Press
Page : 793 pages
File Size : 54,5 Mb
Release : 2005-01-21
Category : Computers
ISBN : 9781439863756

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Discrete Algorithmic Mathematics by Stephen B. Maurer,Anthony Ralston Pdf

Thoroughly revised for a one-semester course, this well-known and highly regarded book is an outstanding text for undergraduate discrete mathematics. It has been updated with new or extended discussions of order notation, generating functions, chaos, aspects of statistics, and computational biology. Written in a lively, clear style that talks to th

Author : Michael Molloy,Bruce Reed
Publisher : Springer Science & Business Media
Page : 320 pages
File Size : 47,8 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9783642040160

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Graph Colouring and the Probabilistic Method by Michael Molloy,Bruce Reed Pdf

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.

Author : V.F. Kolchin,V. Ya. Kozlov,Yu. L Pavlov,Yu. V. Prokhorov
Publisher : Walter de Gruyter GmbH & Co KG
Page : 480 pages
File Size : 43,5 Mb
Release : 2020-05-18
Category : Mathematics
ISBN : 9783112318980

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Progress in Pure and Applied Discrete Mathematics, Vol. 1: Probabilistic Methods in Discrete Mathematics by V.F. Kolchin,V. Ya. Kozlov,Yu. L Pavlov,Yu. V. Prokhorov Pdf

No detailed description available for "Progress in Pure and Applied Discrete Mathematics, Vol. 1: Probabilistic Methods in Discrete Mathematics".

Author : Daniele Gardy,Abdelkader Mokkadem
Publisher : Birkhäuser
Page : 337 pages
File Size : 41,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034884051

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Mathematics and Computer Science by Daniele Gardy,Abdelkader Mokkadem Pdf

This is the first book where mathematics and computer science are directly confronted and joined to tackle intricate problems in computer science with deep mathematical approaches. It contains a collection of refereed papers presented at the Colloquium on Mathematics and Computer Science held at the University of Versailles-St-Quentin on September 18-20, 2000. The colloquium was a meeting place for researchers in mathematics and computer science and thus an important opportunity to exchange ideas and points of view, and to present new approaches and new results in the common areas such as algorithms analysis, trees, combinatorics, optimization, performance evaluation and probabilities. The book is intended for a large public in applied mathematics, discrete mathematics and computer science, including researchers, teachers, graduate students and engineers. It provides an overview of the current questions in computer science and related modern mathematical methods. The range of applications is very wide and reaches beyond computer science.

Author : Béla Bollobás,Fan R. K. Chung
Publisher : American Mathematical Soc.
Page : 196 pages
File Size : 48,7 Mb
Release : 1991
Category : Mathematics
ISBN : 9780821855003

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Probabilistic Combinatorics and Its Applications by Béla Bollobás,Fan R. K. Chung Pdf

Probabilistic methods have become a vital tool in the arsenal of every combinatorialist. The theory of random graphs is still a prime area for the use of probabilistic methods, and, over the years, these methods have also proved of paramount importance in many associated areas such as the design and analysis of computer algorithms. In recent years, probabilistic combinatorics has undergone revolutionary changes as the result of the appearance of some exciting new techniques such as martingale inequalities, discrete isoperimetric inequalities, Fourier analysis on groups, eigenvalue techniques, branching processes, and rapidly mixing Markov chains. The aim of this volume is to review briefly the classical results in the theory of random graphs and to present several of the important recent developments in probabilistic combinatorics, together with some applications. The first paper contains a brief introduction to the theory of random graphs.The second paper reviews explicit constructions of random-like graphs and discusses graphs having a variety of useful properties. Isoperimetric inequalities, of paramount importance in probabilistic combinatorics, are covered in the third paper. The chromatic number of random graphs is presented in the fourth paper, together with a beautiful inequality due to Janson and the important and powerful Stein-Chen method for Poisson approximation. The aim of the fifth paper is to present a number of powerful new methods for proving that a Markov chain is 'rapidly mixing' and to survey various related questions, while the sixth paper looks at the same topic in a very different context. For the random walk on the cube, the convergence to the stable distribution is best analyzed through Fourier analysis; the final paper examines this topic and proceeds to several more sophisticated applications. Open problems can be found throughout each paper.

Author : Vladimir A. Dobrushkin
Publisher : CRC Press
Page : 824 pages
File Size : 49,7 Mb
Release : 2016-03-09
Category : Computers
ISBN : 9781420068306

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Methods in Algorithmic Analysis by Vladimir A. Dobrushkin Pdf

Explores the Impact of the Analysis of Algorithms on Many Areas within and beyond Computer Science A flexible, interactive teaching format enhanced by a large selection of examples and exercises Developed from the author’s own graduate-level course, Methods in Algorithmic Analysis presents numerous theories, techniques, and methods used for analyzing algorithms. It exposes students to mathematical techniques and methods that are practical and relevant to theoretical aspects of computer science. After introducing basic mathematical and combinatorial methods, the text focuses on various aspects of probability, including finite sets, random variables, distributions, Bayes’ theorem, and Chebyshev inequality. It explores the role of recurrences in computer science, numerical analysis, engineering, and discrete mathematics applications. The author then describes the powerful tool of generating functions, which is demonstrated in enumeration problems, such as probabilistic algorithms, compositions and partitions of integers, and shuffling. He also discusses the symbolic method, the principle of inclusion and exclusion, and its applications. The book goes on to show how strings can be manipulated and counted, how the finite state machine and Markov chains can help solve probabilistic and combinatorial problems, how to derive asymptotic results, and how convergence and singularities play leading roles in deducing asymptotic information from generating functions. The final chapter presents the definitions and properties of the mathematical infrastructure needed to accommodate generating functions. Accompanied by more than 1,000 examples and exercises, this comprehensive, classroom-tested text develops students’ understanding of the mathematical methodology behind the analysis of algorithms. It emphasizes the important relation between continuous (classical) mathematics and discrete mathematics, which is the basis of computer science.

Author : David Aldous,Persi Diaconis,Joel Spencer,J. Michael Steele
Publisher : Springer Science & Business Media
Page : 169 pages
File Size : 53,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461208013

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Discrete Probability and Algorithms by David Aldous,Persi Diaconis,Joel Spencer,J. Michael Steele Pdf

Discrete probability theory and the theory of algorithms have become close partners over the last ten years, though the roots of this partnership go back much longer. The papers in this volume address the latest developments in this active field. They are from the IMA Workshops "Probability and Algorithms" and "The Finite Markov Chain Renaissance." They represent the current thinking of many of the world's leading experts in the field. Researchers and graduate students in probability, computer science, combinatorics, and optimization theory will all be interested in this collection of articles. The techniques developed and surveyed in this volume are still undergoing rapid development, and many of the articles of the collection offer an expositionally pleasant entree into a research area of growing importance.

Author : Joel H. Spencer
Publisher : Unknown
Page : 92 pages
File Size : 48,6 Mb
Release : 1987
Category : Combinatorial analysis
ISBN : UCAL:B4405691

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Ten Lectures on the Probabilistic Method by Joel H. Spencer Pdf