We give a randomized parallel algorithm for approximate shortest path computation in an undirected weight ed graph. The algorithm is based on a technique used by Unman and Yannakakis in a parallel algorithm for breadth-first search. It has application, e.g., in approximate solution of multi commodit y flow problems with unit capacities. We also show how to adapt the algorithm to perform better for planar graphs.

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We give efficient parallel algorithms to compute shortest-paths in planar layered digraphs. We show that these digraphs admit special kinds of separators, called one-way separators, which allow paths in the graph to cross them only once. We use these separators to give divide-and-conquer solutions to the problem of finding the shortest paths. We first give a simple algorithm that works on the CREW PRAM model and computes the shortest path between any two vertices of an n-node planar layered digraph in time O(10g3 n) using n l l o g n processors. A CRCW version of this algorithm runs in O(log2 'n log log n) time and uses O(n/ log log n) processors. We then improve the time bound to O(log2 n) on the CREW model and O(lognlog1ogn) on the CRCW model. The processor bounds still remain n/ log n for the CREW model and nlloglogn for the CRCW model.