Parallel algorithms for the hamiltonian cycle and hamiltonian path problems in semicomplete bipartite digraphs
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We give anO(log4 n)-timeO(n 2)-processor CRCW PRAM algorithm to find a hamiltonian cycle in a strong semicomplete bipartite digraph,B, provided that a factor ofB (i.e., a collection of vertex disjoint cycles covering the vertex set ofB) is computed in a preprocessing step. The factor is found (if it exists) using a bipartite matching algorithm, hence placing the whole algorithm in the class Random-NC.
We show that any parallel algorithm which can check the existence of a hamiltonian cycle in a strong semicomplete bipartite digraph in timeO(r(n)) usingp(n) processors can be used to check the existence of a perfect matching in a bipartite graph in timeO(r(n)+n 2 /p(n)) usingp(n) processors. Hence, our problem belongs to the class NC if and only if perfect matching in bipartite graphs belongs to NC.
We also consider the problem of finding a hamiltonian path in a semicomplete bipartite digraph.
Key WordsGraph algorithms Hamilton cycle Parallel algorithms Semicomplete bipartite graphs Randomized algorithms
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