Probability and Computing

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Publisher : Cambridge University Press
ISBN 13 : 9780521835404
Total Pages : 372 pages
Book Rating : 4.8/5 (354 download)

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Book Synopsis Probability and Computing by : Michael Mitzenmacher

Download or read book Probability and Computing written by Michael Mitzenmacher and published by Cambridge University Press. This book was released on 2005-01-31 with total page 372 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Probabilistic Analysis of Algorithms

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Publisher : Springer Science & Business Media
ISBN 13 : 1461248000
Total Pages : 254 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Probabilistic Analysis of Algorithms by : Micha Hofri

Download or read book Probabilistic Analysis of Algorithms written by Micha Hofri and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt: Probabilistic Analysis of Algorithms begins with a presentation of the "tools of the trade" currently used in probabilistic analyses, and continues with an applications section in which these tools are used in the analysis ofr selected algorithms. The tools section of the book provides the reader with an arsenal of analytic and numeric computing methods which are then applied to several groups of algorithms to analyze their running time or storage requirements characteristics. Topics covered in the applications section include sorting, communications network protocols and bin packing. While the discussion of the various algorithms is sufficient to motivate their structure, the emphasis throughout is on the probabilistic estimation of their operation under distributional assumptions on their input. Probabilistic Analysis of Algorithms assumes a working knowledge of engineering mathematics, drawing on real and complex analysis, combinatorics and probability theory. While the book is intended primarily as a text for the upper undergraduate and graduate student levels, it contains a wealth of material and should also prove an important reference for researchers. As such it is addressed to computer scientists, mathematicians, operations researchers, and electrical and industrial engineers who are interested in evaluating the probable operation of algorithms, rather than their worst-case behavior.

Randomized Algorithms

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Publisher : Cambridge University Press
ISBN 13 : 1139643134
Total Pages : 496 pages
Book Rating : 4.1/5 (396 download)

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Book Synopsis Randomized Algorithms by : Rajeev Motwani

Download or read book Randomized Algorithms written by Rajeev Motwani and published by Cambridge University Press. This book was released on 1995-08-25 with total page 496 pages. Available in PDF, EPUB and Kindle. Book excerpt: For many applications a randomized algorithm is either the simplest algorithm available, or the fastest, or both. This tutorial presents the basic concepts in the design and analysis of randomized algorithms. The first part of the book presents tools from probability theory and probabilistic analysis that are recurrent in algorithmic applications. Algorithmic examples are given to illustrate the use of each tool in a concrete setting. In the second part of the book, each of the seven chapters focuses on one important area of application of randomized algorithms: data structures; geometric algorithms; graph algorithms; number theory; enumeration; parallel algorithms; and on-line algorithms. A comprehensive and representative selection of the algorithms in these areas is also given. This book should prove invaluable as a reference for researchers and professional programmers, as well as for students.

Concentration of Measure for the Analysis of Randomized Algorithms

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Publisher : Cambridge University Press
ISBN 13 : 1139480995
Total Pages : 213 pages
Book Rating : 4.1/5 (394 download)

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Book Synopsis Concentration of Measure for the Analysis of Randomized Algorithms by : Devdatt P. Dubhashi

Download or read book Concentration of Measure for the Analysis of Randomized Algorithms written by Devdatt P. Dubhashi and published by Cambridge University Press. This book was released on 2009-06-15 with total page 213 pages. Available in PDF, EPUB and Kindle. Book excerpt: Randomized algorithms have become a central part of the algorithms curriculum, based on their increasingly widespread use in modern applications. This book presents a coherent and unified treatment of probabilistic techniques for obtaining high probability estimates on the performance of randomized algorithms. It covers the basic toolkit from the Chernoff–Hoeffding bounds to more sophisticated techniques like martingales and isoperimetric inequalities, as well as some recent developments like Talagrand's inequality, transportation cost inequalities and log-Sobolev inequalities. Along the way, variations on the basic theme are examined, such as Chernoff–Hoeffding bounds in dependent settings. The authors emphasise comparative study of the different methods, highlighting respective strengths and weaknesses in concrete example applications. The exposition is tailored to discrete settings sufficient for the analysis of algorithms, avoiding unnecessary measure-theoretic details, thus making the book accessible to computer scientists as well as probabilists and discrete mathematicians.

The Simplex Method

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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.

Probabilistic Methods for Algorithmic Discrete Mathematics

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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.

Discrete Probability and Algorithms

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Publisher : Springer Science & Business Media
ISBN 13 : 1461208017
Total Pages : 169 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Discrete Probability and Algorithms by : David Aldous

Download or read book Discrete Probability and Algorithms written by David Aldous and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 169 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Practical Analysis of Algorithms

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Publisher : Springer
ISBN 13 : 3319098888
Total Pages : 475 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis Practical Analysis of Algorithms by : Dana Vrajitoru

Download or read book Practical Analysis of Algorithms written by Dana Vrajitoru and published by Springer. This book was released on 2014-09-03 with total page 475 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces the essential concepts of algorithm analysis required by core undergraduate and graduate computer science courses, in addition to providing a review of the fundamental mathematical notions necessary to understand these concepts. Features: includes numerous fully-worked examples and step-by-step proofs, assuming no strong mathematical background; describes the foundation of the analysis of algorithms theory in terms of the big-Oh, Omega, and Theta notations; examines recurrence relations; discusses the concepts of basic operation, traditional loop counting, and best case and worst case complexities; reviews various algorithms of a probabilistic nature, and uses elements of probability theory to compute the average complexity of algorithms such as Quicksort; introduces a variety of classical finite graph algorithms, together with an analysis of their complexity; provides an appendix on probability theory, reviewing the major definitions and theorems used in the book.

Algorithms and Data Structures

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Publisher : Springer Nature
ISBN 13 : 303059758X
Total Pages : 356 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Algorithms and Data Structures by : Helmut Knebl

Download or read book Algorithms and Data Structures written by Helmut Knebl and published by Springer Nature. This book was released on 2020-10-31 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Probabilistic Data Structures and Algorithms for Big Data Applications

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Publisher : BoD – Books on Demand
ISBN 13 : 3748190484
Total Pages : 224 pages
Book Rating : 4.7/5 (481 download)

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Book Synopsis Probabilistic Data Structures and Algorithms for Big Data Applications by : Andrii Gakhov

Download or read book Probabilistic Data Structures and Algorithms for Big Data Applications written by Andrii Gakhov and published by BoD – Books on Demand. This book was released on 2022-08-05 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: A technical book about popular space-efficient data structures and fast algorithms that are extremely useful in modern Big Data applications. The purpose of this book is to introduce technology practitioners, including software architects and developers, as well as technology decision makers to probabilistic data structures and algorithms. Reading this book, you will get a theoretical and practical understanding of probabilistic data structures and learn about their common uses.

Average Case Analysis of Algorithms on Sequences

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Publisher : John Wiley & Sons
ISBN 13 : 1118031024
Total Pages : 580 pages
Book Rating : 4.1/5 (18 download)

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Book Synopsis Average Case Analysis of Algorithms on Sequences by : Wojciech Szpankowski

Download or read book Average Case Analysis of Algorithms on Sequences written by Wojciech Szpankowski and published by John Wiley & Sons. This book was released on 2011-10-14 with total page 580 pages. Available in PDF, EPUB and Kindle. Book excerpt: A timely book on a topic that has witnessed a surge of interest over the last decade, owing in part to several novel applications, most notably in data compression and computational molecular biology. It describes methods employed in average case analysis of algorithms, combining both analytical and probabilistic tools in a single volume. * Tools are illustrated through problems on words with applications to molecular biology, data compression, security, and pattern matching. * Includes chapters on algorithms and data structures on words, probabilistic and analytical models, inclusion-exclusion principles, first and second moment methods, subadditive ergodic theorem and large deviations, elements of information theory, generating functions, complex asymptotic methods, Mellin transform and its applications, and analytic poissonization and depoissonization. * Written by an established researcher with a strong international reputation in the field.

Randomized Algorithms for Analysis and Control of Uncertain Systems

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Publisher : Springer Science & Business Media
ISBN 13 : 1447146107
Total Pages : 363 pages
Book Rating : 4.4/5 (471 download)

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Book Synopsis Randomized Algorithms for Analysis and Control of Uncertain Systems by : Roberto Tempo

Download or read book Randomized Algorithms for Analysis and Control of Uncertain Systems written by Roberto Tempo and published by Springer Science & Business Media. This book was released on 2012-10-21 with total page 363 pages. Available in PDF, EPUB and Kindle. Book excerpt: The presence of uncertainty in a system description has always been a critical issue in control. The main objective of Randomized Algorithms for Analysis and Control of Uncertain Systems, with Applications (Second Edition) is to introduce the reader to the fundamentals of probabilistic methods in the analysis and design of systems subject to deterministic and stochastic uncertainty. The approach propounded by this text guarantees a reduction in the computational complexity of classical control algorithms and in the conservativeness of standard robust control techniques. The second edition has been thoroughly updated to reflect recent research and new applications with chapters on statistical learning theory, sequential methods for control and the scenario approach being completely rewritten. Features: · self-contained treatment explaining Monte Carlo and Las Vegas randomized algorithms from their genesis in the principles of probability theory to their use for system analysis; · development of a novel paradigm for (convex and nonconvex) controller synthesis in the presence of uncertainty and in the context of randomized algorithms; · comprehensive treatment of multivariate sample generation techniques, including consideration of the difficulties involved in obtaining identically and independently distributed samples; · applications of randomized algorithms in various endeavours, such as PageRank computation for the Google Web search engine, unmanned aerial vehicle design (both new in the second edition), congestion control of high-speed communications networks and stability of quantized sampled-data systems. Randomized Algorithms for Analysis and Control of Uncertain Systems (second edition) is certain to interest academic researchers and graduate control students working in probabilistic, robust or optimal control methods and control engineers dealing with system uncertainties. The present book is a very timely contribution to the literature. I have no hesitation in asserting that it will remain a widely cited reference work for many years. M. Vidyasagar

The Probabilistic Method

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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.

An Introduction to the Analysis of Algorithms

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Publisher : Addison-Wesley
ISBN 13 : 0133373487
Total Pages : 735 pages
Book Rating : 4.1/5 (333 download)

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Book Synopsis An Introduction to the Analysis of Algorithms by : Robert Sedgewick

Download or read book An Introduction to the Analysis of Algorithms written by Robert Sedgewick and published by Addison-Wesley. This book was released on 2013-01-18 with total page 735 pages. Available in PDF, EPUB and Kindle. Book excerpt: Despite growing interest, basic information on methods and models for mathematically analyzing algorithms has rarely been directly accessible to practitioners, researchers, or students. An Introduction to the Analysis of Algorithms, Second Edition, organizes and presents that knowledge, fully introducing primary techniques and results in the field. Robert Sedgewick and the late Philippe Flajolet have drawn from both classical mathematics and computer science, integrating discrete mathematics, elementary real analysis, combinatorics, algorithms, and data structures. They emphasize the mathematics needed to support scientific studies that can serve as the basis for predicting algorithm performance and for comparing different algorithms on the basis of performance. Techniques covered in the first half of the book include recurrences, generating functions, asymptotics, and analytic combinatorics. Structures studied in the second half of the book include permutations, trees, strings, tries, and mappings. Numerous examples are included throughout to illustrate applications to the analysis of algorithms that are playing a critical role in the evolution of our modern computational infrastructure. Improvements and additions in this new edition include Upgraded figures and code An all-new chapter introducing analytic combinatorics Simplified derivations via analytic combinatorics throughout The book’s thorough, self-contained coverage will help readers appreciate the field’s challenges, prepare them for advanced results—covered in their monograph Analytic Combinatorics and in Donald Knuth’s The Art of Computer Programming books—and provide the background they need to keep abreast of new research. "[Sedgewick and Flajolet] are not only worldwide leaders of the field, they also are masters of exposition. I am sure that every serious computer scientist will find this book rewarding in many ways." —From the Foreword by Donald E. Knuth

The Probabilistic Method

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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.

Introduction to Algorithms, third edition

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Publisher : MIT Press
ISBN 13 : 0262258102
Total Pages : 1313 pages
Book Rating : 4.2/5 (622 download)

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Book Synopsis Introduction to Algorithms, third edition by : Thomas H. Cormen

Download or read book Introduction to Algorithms, third edition written by Thomas H. Cormen and published by MIT Press. This book was released on 2009-07-31 with total page 1313 pages. Available in PDF, EPUB and Kindle. Book excerpt: The latest edition of the essential text and professional reference, with substantial new material on such topics as vEB trees, multithreaded algorithms, dynamic programming, and edge-based flow. Some books on algorithms are rigorous but incomplete; others cover masses of material but lack rigor. Introduction to Algorithms uniquely combines rigor and comprehensiveness. The book covers a broad range of algorithms in depth, yet makes their design and analysis accessible to all levels of readers. Each chapter is relatively self-contained and can be used as a unit of study. The algorithms are described in English and in a pseudocode designed to be readable by anyone who has done a little programming. The explanations have been kept elementary without sacrificing depth of coverage or mathematical rigor. The first edition became a widely used text in universities worldwide as well as the standard reference for professionals. The second edition featured new chapters on the role of algorithms, probabilistic analysis and randomized algorithms, and linear programming. The third edition has been revised and updated throughout. It includes two completely new chapters, on van Emde Boas trees and multithreaded algorithms, substantial additions to the chapter on recurrence (now called “Divide-and-Conquer”), and an appendix on matrices. It features improved treatment of dynamic programming and greedy algorithms and a new notion of edge-based flow in the material on flow networks. Many exercises and problems have been added for this edition. The international paperback edition is no longer available; the hardcover is available worldwide.

Foundations of Probabilistic Programming

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Publisher : Cambridge University Press
ISBN 13 : 110848851X
Total Pages : 583 pages
Book Rating : 4.1/5 (84 download)

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Book Synopsis Foundations of Probabilistic Programming by : Gilles Barthe

Download or read book Foundations of Probabilistic Programming written by Gilles Barthe and published by Cambridge University Press. This book was released on 2020-12-03 with total page 583 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an overview of the theoretical underpinnings of modern probabilistic programming and presents applications in e.g., machine learning, security, and approximate computing. Comprehensive survey chapters make the material accessible to graduate students and non-experts. This title is also available as Open Access on Cambridge Core.