Fast Parallel Algorithms for Finding Hamiltonian Paths and Cycles in a Tournament

Abstract

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Finding Hamiltonian paths in tournaments on clusters

“…A trivial, but useful, fact is that any induced subgraph of a tournament is also a tournament. Related work on tournaments can be found in [5,15,17]. We start by stating the theorem for the Hamiltonian path [17].…”

Cycles and paths in semicomplete multipartite digraphs, theorems, and algorithms: a survey

“…We refer the reader to [50] for a discussion of N C-algorithms. D. Soroker [68] studies the parallel complexity of the above mentioned problems. He proved the following: Theorem 6.1 There are N C-algorithms for the Hamiltonian path and Hamiltonian cycle problems in tournaments.…”

“…The parallel running time of the procedure is O(log n) using O(n 2 / log n) processors in the CRCW model. J. Bang-Jensen, Y. Manoussakis and C. Thomassen [16] obtained a polynomial algorithm solving the problem (which appears in [60,68]) of deciding the existence of a Hamiltonian path with prescribed initial and terminal vertices in a tournament. Obviously, the last problem is equivalent to the problem of existence of a Hamiltonian cycle containing a given arc in a tournament.…”

The Hamilton circuit problem on grids

“…Their many applications, though, have caused a large body of research to be directed both towards finding efficient heuristics and towards trying to single out classes of graphs on which the Hamilton cycle problem is polynomially solvable: There are classes of graphs that all the members are Hamiltonian (le., they have a Hamilton cycle). One such class is the tournaments (complete directed graphs) [9], another class is the dense graphs [6], [7]. Moreover, the Hamilton cycle problem is studied on the context of parallel computation; in [10] they present an optimal polylogarithmic parallel algorithm that computes a Hamilton cycle on dense graphs.…”