Optimal Randomized Parallel Algorithms for Computational Geometry I

TLDR: 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 [3] 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 [22] Reif and Valiant [21] for flashsort algorithm to perform divide and conquer in a plane very efficiently leading to the improved performance by our approach.