An efficient parallel algorithm for shortest paths in planar layered digraphs

Abstract

Computing shortest paths in a directed graph has received considerable attention in the sequential RAM model of computation. However, developing a polylog-time parallel algorithm that is close to the sequential optimal in terms of the total work done remains an elusive goal. We present a first step in this direction by giving efficient parallel algorithms for shortest paths in planar layered digraphs. We show that these graphs admit special kinds of separators called {\em one-way} separators which allow the paths in the graph to cross it only once. We use these separators to give divide and conquer solutions to the problem of finding the shortest paths between any two vertices. We first give a simple algorithm that works in the CREW model and computes the shortest path between any two vertices in an $n$-node planar layered digraph in time $O(\log^3 n)$ using $n/ \log n$ processors. A CRCW version of this algorithm runs in time $O(\log^2 n \log \log n)$ and uses $n/ \log \log n$ processors. We then use results of Aggarwal and Park [1] and Atallah [4] to improve the time bound to $O(\log^2 n)$ in the CREW model and $O(\log n \log \log n)$ in the CRCW model. The processor bounds still remain as $n/ \log n$ for the CREW model and $n/ \log \log n$ for the CRCW model.