A time-optimal solution for the path cover problem on cographs

  • Authors:

  • Koji Nakano;Stephan Olariu;Albert Y. Zomaya

  • Affiliations:

  • School of Information Science, Japan Advanced Institute of Science and Technology, Tatsunokuchi, Ishikawa 923-1292, Japan;Department of Computer Science, Old Dominion University, Norfolk, VA;Department of Electrical and Electronic Engineering, The University of Western Australia, Perth, WA 6970, Australia

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2003

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Abstract

We show that the notoriously difficult problem of finding and reporting the smallest number of vertex-disjoint paths that cover the vertices of a graph can be solved time- and work-optimally for cographs. Our result implies that for this class of graphs the task of finding a Hamiltonian path can be solved time- and work-optimally in parallel.It was open for more than 10 years to find a time- and work-optimal parallel solution for this important problem. Our contribution is to offer an optimal solution to this important problem. We begin by showing that any algorithm that solves an instance of size n of the problem must take Ω(logn) time on the CREW, even if an infinite number of processors are available. We then go on to show that this time lower bound is tight by devising an EREW algorithm that, given an n-vertex cograph G represented by its cotree, finds and reports all the paths in a minimum path cover in O(logn) time using n/logn processors.