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Sinclair A. Algorithms for Random Generation and Counting. A Markov Chain Approach
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Birkhäuser, 1993, -158 p.This monograph is a slightly revised version of my Ph.D. thesis (86), completed in the Department of Computer Science at the University of Edinburgh in June 1988, with an additional chapter summarising more recent developments. Some of the material has appeared in the form of papers.
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 problems appear to be intractable for many important structures, so interest has focused on finding effcient randomised algorithms that solve them approximately, 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: simulate 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 optimisation. The efficiency of the technique in any application depends crucially on the rate of convergence of the Markov chain. Since the number of states is typically extremely large, the chain should reach equilibrium after exploring only a tiny fraction of its state space; chains with this property are called rapidly mixing. A major portion of the monograph is devoted to developing new methods for analysing the rate of convergence, which are of independent mathematical interest. The methods are quite general and lay the foundations for the analysis of the time-dependent behaviour of complex Markov chains arising in the above applications.Preliminaries
Markov Chains and Rapid Mixing
Direct Applications
Indirect Applications
Recent developments
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 problems appear to be intractable for many important structures, so interest has focused on finding effcient randomised algorithms that solve them approximately, 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: simulate 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 optimisation. The efficiency of the technique in any application depends crucially on the rate of convergence of the Markov chain. Since the number of states is typically extremely large, the chain should reach equilibrium after exploring only a tiny fraction of its state space; chains with this property are called rapidly mixing. A major portion of the monograph is devoted to developing new methods for analysing the rate of convergence, which are of independent mathematical interest. The methods are quite general and lay the foundations for the analysis of the time-dependent behaviour of complex Markov chains arising in the above applications.Preliminaries
Markov Chains and Rapid Mixing
Direct Applications
Indirect Applications
Recent developments
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