An efficient parallel algorithm for the longest path problem in meshes

The longest path problem is the problem of finding a simple path with the maximum number of vertices in a given graph, and so far it has been solved polynomially only for a few classes of graphs. This problem generalizes the well-known Hamiltonian path problem, hence it is NP-hard in general graphs. In this paper, first we give a sequential linear-time algorithm for the longest path problem in meshes. Then based on this algorithm, we present a constant-time parallel algorithm for the problem, which can be run on every parallel machine.