# Optimal randomized parallel algorithms for computational geometry

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## Abstract

We present parallel algorithms for some fundamental problems in computational geometry which have a running time of*O*(log*n*) using*n* processors, with very high probability (approaching 1 as*n* → ∞). 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.

## Key words

Randomized Parallel algorithm Computational geometry Point location Triangulation Trapezoidal decomposition## Preview

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