Tight lower bounds for computing shortest paths on proper interval and bipartite permutation graphs

Abstract

Logarithmic time lower bounds for computing the distance between two arbitrary vertices, in a proper interval graph represented by a family of intervals on a real line, and in a bipartite permutation graph represented by a permutation function, on exclusive write PRAM are proved here. The lower bounds are also valid for these classes of graphs represented by adjacency matrices and for their superclasses. Shortest paths on interval and permutation graphs, which, respectively, strictly contain proper interval and bipartite permutation graphs, are known to be computable in logarithmic time on exclusive write PRAM. It follows that the lower bounds derived here are tight.