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This monograph is a barely revised mannequin of my PhD thesis , com pleted in the Division of Computer Science on the School of Edin burgh in June 1988, with an extra chapter summarising newer developments. A few of the supplies has appeared in the kind of papers [50,88]. The underlying theme of the monograph is the analysis of two classical points: counting the climate of a finite set of combinatorial buildings, and producing them uniformly at random. Of their precise sort, these prob lems seem like intractable for many important buildings, so curiosity has focused on discovering surroundings pleasant randomised algorithms that clear up them ap proxim~ly, with a small probability of error. For a lot of pure buildings the two points are intimately associated at this diploma of approximation, so it is pure to assessment them collectively. On the coronary coronary heart of the monograph is a single algorithmic paradigm: sim ulate a Markov chain whose states are combinatorial buildings and which converges to a acknowledged probability distribution over them. This technique has functions not solely in combinatorial counting and period, however in addition in a lot of totally different areas harking back to statistical physics and combinatorial optimi sation. The effectivity of the tactic in any software depends upon crucially on the speed of convergence of the Markov chain.
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