Efficient Parallel Algorithms for Shortest Paths in Planar Graphs

Efficient parallel algorithms are presented, on the CREW PRAM model, for generating a succinct encoding of all pairs shortest path information in a directed planar graph G with real-valued edge costs but no negative cycles. We assume that a planar embedding of G is given, together with a set of q faces that cover all the vertices. Then our algorithm runs in O(log2n+log3q) time and employs O(nq) processors. O(log2n) time, n-processor algorithms are presented for various subproblems, including that of generating all pairs shortest path information in a directed outerplanar graph. Our work is based on the fundamental hammock-decomposition technique of G. Frederickson. We achieve this decomposition in O(log2n) parallel time by using O(n) processors. The hammock-decomposition seems to be a fundamental operation that may help in improving efficiency of many parallel (and sequential) graph algorithms. Our algorithms avoid the matrix powering (sometimes called the transitive closure bottleneck) thus lead to a considerably smaller number of processors, and tighter processor-time products.

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