Coarse Grained Parallel Graph Planarity Testing

We present a coarse grained parallel algorithm for planarity testing and planar embedding. The algorithm requires O(log p) communication rounds and linear sequential work per round. It assumes that the local memory per processor, N=p, is larger than p for some xed > 0. This assumption is true for all commercially available multiprocessors. Our result implies a BSP algorithm with O(log p) supersteps, O(g log(p) p ) communication, and O(log(p) p ) local computation. Our algorithm is based on the general structure of a previous PRAM method presented by Klein, using a parallel implementation of PQ-trees. The main contribution of this paper lies in the study of many of the individual PRAM steps that are very inefcient on existing commercial parallel machines. We present non-trivial, eÆcient, CGM implementations of the various parts of the overall strategy proposed by Klein. Our main result is a parallel planarity testing algorithm which is much more practical and eÆcient on commercially available multi-