A divide and conquer approach to shortest paths in planar layered digraphs

The authors give efficient parallel algorithms to compute shortest-paths in planar layered digraphs. They show that these digraphs admit special kinds of separators, called one-way separators, which allow paths in the graph to cross them only once. They use these separators to give divide-and-conquer solutions to the problem of finding the shortest paths. They first give a simple algorithm that works on the CREW (concurrent-read exclusive-write) PRAM (parallel random-across machine) model and computes the shortest path between any two vertices of an n-node planar layered diagraph in time O (log³ n) using n/log n processors. A CRCW (concurrent-read concurrent-write) version of this algorithm runs in O(log² n log log n ) time and uses O(n/log log n) processors. The authors then improve the time bound to O(log ² n) on the CREW model and O(log n log log n) on the CRCW model. The processor bounds still remain n log n for the CREW model and n/log log n for the CRCW model