## On the parallel complexity of Hamiltonian Cycle and Matching Problem on Dense Graphs (1993)

Venue: | Journal of Algorithms |

Citations: | 12 - 1 self |

### BibTeX

@ARTICLE{Dahlhaus93onthe,

author = {Elias Dahlhaus and Péter Hajnal and Marek Karpinski},

title = {On the parallel complexity of Hamiltonian Cycle and Matching Problem on Dense Graphs},

journal = {Journal of Algorithms},

year = {1993},

volume = {15},

pages = {15--367}

}

### OpenURL

### Abstract

Dirac's classical theorem asserts that, if every vertex of a graph G on n vertices has degree at least n 2 then G has a Hamiltonian cycle. We give a fast parallel algorithm on a CREW \Gamma PRAM to find a Hamiltonian cycle in such graphs. Our algorithm uses a linear number of processors and is optimal up to a polylogarithmic factor. The algorithm works in O(log 4 n) parallel time and uses linear number of processors on a CREW \Gamma PRAM . Our method bears some resemblance to Anderson's RNC algorithm [An] for maximal paths: we, too, start from a system of disjoint paths and try to glue them together. We are, however, able to perform the base step (perfect matching) deterministically. We also prove that a perfect matching in dense graphs can be found in NC 2 . The cost of improved time is a quadratic number of processors. On the negative side, we prove that finding an NC algorithm for perfect matching in slightly less dense graphs (minimum degree is at least ( 1 2 \Gamma ff...