OPTIMAL PARALLEL RANDOMIZED ALGORITHMS FOR 3-DIMENSIONAL CONVEX HULLS AND RELATED PROBLEMS

Further applications of random sampling techniques which have been used for deriving efficient parallel algorithms are presented by J. H. Reif and S. Sen [Proc. 16th International Conference on Parallel Processing, 1987). This paper presents an optimal parallel randomized algorithm for computing intersection of half spaces in three dimensions. Because of well-known reductions, these methods also yield equally efficient algorithms for fundamental problems like the convex hull in three dimensions, Voronoi diagram of point sites on a plane, and Euclidean minimal spanning tree. The algorithms run in time T = O(log n) for worst-case inputs and use P = O(n) processors in a CREW PRAM model where n is the input size. They are randomized in the sense that they use a total of only polylogarithmic number of random bits and terminate in the claimed time bound with probability 1 - n(-alpha) for any fixed alpha > 0. They are also optimal in P.T product since the sequential time bound for all these problems is OMEGA(n log n). The best known deterministic parallel algorithms for two-dimensional Voronoi-diagram and three-dimensional convex hull run in O(log2 n) and O(log2 n log* n) time, respectively, while using O(n/log n) and O(n) processors, respectively.