Chapter 18 - Parallel Computational Geometry: An Approach using Randomization

Abstract

This chapter discusses the general methods used for designing efficient parallel algorithms for problems in computational geometry. The chapter focuses on parallel random access machine (PRAM) model. Algorithms exploit random sampling and randomized techniques for solving a wide class of fundamental problems from computational geometry, such as convex hulls, Voronoi diagrams, triangulation, point-location, and arrangements. In addition, the algorithms on the Butterfly network rely critically on an efficient randomized multi-searching algorithm. Randomization is a powerful tool in the design of both sequential and parallel algorithms. An objective in designing randomized algorithms is to ensure that the number of “bad” algorithms is a relatively small fraction of the total number of algorithms. Divide-and-conquer is the most commonly used technique for designing parallel algorithms. Its basic idea is analogous to sequential algorithm design where the original problem is sub-divided into smaller subproblems, and then the solutions of the subproblems are combined to obtain a solution to the original problem. The smaller subproblems are solved recursively until a sub-problem size becomes smaller than a predetermined threshold.