Fast probabilistic algorithms for hamiltonian circuits and matchings

The main purpose of this paper is to give techniques for analysing the probabilistic performance of certain kinds of algorithms, and hence to suggest some fast algorithms with provably desirable probabilistic behaviour. The particular problems we consider are: finding Hamiltonian circuits in directed graphs (DHC), finding Hamiltonian circuits in undirected graphs (UHC), and finding perfect matchings in undirected graphs (PM). We show that for each problem there is an algorithm that is extremely fast (0(n(log n)2) for DHC and UHC, and 0(nlog n) for PM), and which with probability tending to one finds a solution in randomly chosen graphs of sufficient density. These results contrast with the known NP-completeness of the first two problems [2,12] and the best worst-case upper bound known of 0(n2.5) for the last [9].

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