Progress in Theoretical Computer Science

A Markov Chain Approach

By: A. Sinclair

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Hardcover | 1 February 1993 | Edition Number 2

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160 Pages

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This monograph is a slightly revised version of my PhD thesis [86], com pleted in the Department of Computer Science at the University of Edin burgh in June 1988, with an additional chapter summarising more recent developments. Some of the material has appeared in the form of papers [50,88]. The underlying theme of the monograph is the study of two classical problems: counting the elements of a finite set of combinatorial structures, and generating them uniformly at random. In their exact form, these prob lems appear to be intractable for many important structures, so interest has focused on finding efficient randomised algorithms that solve them ap proxim~ly, with a small probability of error. For most natural structures the two problems are intimately connected at this level of approximation, so it is natural to study them together. At the heart of the monograph is a single algorithmic paradigm: sim ulate a Markov chain whose states are combinatorial structures and which converges to a known probability distribution over them. This technique has applications not only in combinatorial counting and generation, but also in several other areas such as statistical physics and combinatorial optimi sation. The efficiency of the technique in any application depends crucially on the rate of convergence of the Markov chain.

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You Can Find This Book In

Non-Fiction Mathematics Applied Mathematics Stochastics Algebra Probability & Statistics Science Physics Computing & I.T.Databases Data Capture & Analysis Computer Programming & Software Development

Algorithms & Data Structures Computer Science Mathematical Theory of Computation Maths for Computer Scientists Calculus & Mathematical Analysis Numerical Analysis

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Paperback

Published: 1st November 2012

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Synopsis	p. 1
Preliminaries	p. 7
Some basic definitions	p. 7
Notions of tractability	p. 12
An extended model	p. 18
Counting, generation and self-reducibility	p. 28
An interesting class of relations	p. 36
Markov chains and rapid mixing	p. 42
The Markov chain approach to generation problems	p. 43
Conductance and the rate of convergence	p. 46
A characterisation of rapid mixing	p. 56
Direct Applications	p. 63
Some simple examples	p. 63
Approximating the permanent	p. 70
Monomer-dimer systems	p. 85
Concluding remarks	p. 96
Indirect Applications	p. 101
A robust notion of approximate counting	p. 102
Self-embeddable relations	p. 112
Graphs with specified degrees	p. 114
Appendix: Recent developments	p. 126
Bibliography	p. 136
Index	p. 144
Table of Contents provided by Blackwell. All Rights Reserved.

Progress in Theoretical Computer Science

A Markov Chain Approach

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Hardcover

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How to return your order

Defective items

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Other Editions and Formats

Paperback

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