## Parallel Computational Geometry by Selim G. Akl, Kelly A. Lyons

By Selim G. Akl, Kelly A. Lyons

It is a unified, educational description of the main usual versions of parallel computation and their software to difficulties in computational geometry. each one bankruptcy deals an in-depth research of an issue in computational geometry and provides parallel algorithms to resolve them. Comparative tables summarize many of the algorithms built to unravel every one challenge. a variety of versions of parallel computation to increase the algorithms - parallel random entry desktop (PRAM) - are thought of, in addition to a number of networks for interconnecting processors on a parallel desktop.

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For example, an algorithm appearing in [Nath8O] runs in 0(K logn) time with 0(n±+I/K) processors, 1 < K < logn, thus duplicating the performance of the CREW PRAM algorithm in [Chow8l]. Also described in [Nath8O] is an O(N)-processor algorithm, I < N < n, which runs in 0(n/N log n + log n log N) time. An algorithm in [Akl84] uses 0(n 1 -) processors, 0 < E < 1, and runs in 0(nL logh) time, where h is the number of edges on the convex hull. It is shown in [Mill88] how the CREW PRAM multiway divide-and-conquer algorithm of [Atal86a] and [Agga88] can be modified to achieve the same performance on the weaker EREW PRAM.

3 n). There are also parallel algorithms for computing the convex hull of a set of points in three dimensions on network models. Two algorithms are given in [Chow8O] that run on a CCC network. The first runs in 0 (log4 n) time using O(n) processors, and the second runs in 0(K log 3 n) time using 0(n1+1/K) processors. A three-dimensional convex hull algorithm that runs in O(n 1/2 logn) time on an n 1/2 x n1/2 mesh is presented in [Dehn88b]. If each processor in the mesh is allowed 0(logn) space, the algorithm runs in 0 (n' /2) time.

The vertices of CH(P) are points of P such that every point of P is either a vertex of CH(P) or lies inside CH(P). 1 shows a set P of points and the convex hull of P. Without doubt the most popular problem among designers of sequential computational geometric algorithms, constructing convex hulls enjoyed a similar attention in parallel computing. In fact, it appears to be the first problem in computational geometry for which parallel algorithms were designed [Nath8O, Chow8l, Akl82]. To simplify our subsequent discussion we make two assumptions: 1.