Optimal Randomized Parallel Algorithms for Computational Geometry I
TL;DR: In this paper, the authors present parallel algorithms for 3-D maxima and two-set dominance counting by an application of integer sorting, which have running time of O(logn)$ using $n$ processors, with very high probability.
Abstract: We present parallel algorithms for some fundamental problems in computational geometry which have running time of $O(logn)$ using $n$ processors, with very high probability (approaching 1 as $n~ \rightarrow~ \infty$). These include planar point location, triangulation and trapezoidal decomposition. We also present optimal algorithms for 3-D maxima and two-set dominance counting by an application of integer sorting. Most of these algorithms run on CREW PRAM model and have optimal processor-time product which improve on the previously best known algorithms of Atallah and Goodrich  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  Reif and Valiant  for flashsort algorithm to perform divide and conquer in a plane very efficiently leading to the improved performance by our approach.
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