Parallel Computational Geometry: An Approach using Randomization

We describe very general methods for designing efficient parallel algorithms for problems in computational geometry. Although our main focus is the PRAM, we provide strong evidence that these techniques yield equally efficient algorithms in more concrete computing models like Butterfly networks. The algorithms exploit random sampling and randomized techniques for solving a wide class of fundamental problems from computational geometry like convex hulls, Voronoi diagrams, triangulation, point-location and arrangements. In addition, the algorithms on the Butterfly network rely critically on an efficient randomized multisearching algorithm. Our description emphasizes the algorithmic techniques rather than a detailed treatment of the individual problems.