Optimal randomized parallel algorithms for computational geometry

We present parallel algorithms for some fundamental problems in computational geometry which have a running time ofO(logn) usingn processors, with very high probability (approaching 1 asn ). These include planar-point location, triangulation, and trapezoidal decomposition. We also present optimal algorithms for three-dimensional maxima and two-set dominance counting by an application of integer sorting. Most of these algorithms run on a CREW PRAM model and have optimal processor-time product which improve on the previously best-known algorithms of Atallah and Goodrich 5] for these problems. The crux of these algorithms is a useful data structure which emulates the plane-sweeping paradigm used for sequential algorithms. We extend some of the techniques used by Reischuk 26] and Reif and Valiant 25] for flashsort algorithm to perform divide and conquer in a plane very efficiently leading to the improved performance by our approach.This is a substantially revised version of the paper that appeared as Optimal Randomized Parallel Algorithms for Computational Geometry in theProceedings of the 16th International Conference on Parallel Processing, St. Charles, Illinois, August 1987.This research was supported by DARPA/ARO Contract DAAL03-88-K-0195, Air Force Contract AFOSR-87-0386, DARPA/ISTO Contracts N00014-88-K-0458 and N00014-91-J-1985, and by NASA Subcontract 550-63 of Primecontract NAS5-30428.