Efficient Parallel Algorithms for Planar st-Graphs1

Planar st-graphs find applications in a number of areas. In this paper we present efficient parallel algorithms for solving several fundamental problems on planar st-graphs. The problems we consider include all-pairs shortest paths in weighted planar st-graphs, single-source shortest paths in weighted planar layered digraphs (which can be reduced to single-source shortest paths in certain special planar st-graphs), and depthfirst search in planar st-graphs. Our parallel shortest path techniques exploit the specific geometric and graphic structures of planar st-graphs, and involve schemes for partitioning planar st-graphs into subgraphs in a way that ensures that the resulting path length matrices have a monotonicity property [1], [2]. The parallel algorithms we obtain are a considerable improvement over the previously best known solutions (when they are applied to these st-graph problems), and are in fact relatively simple. The parallel computational models we use are the CREW PRAM and EREW PRAM.