Parallel Convex Hull Computation by Generalised Regular Sampling

摘要

The model of bulk-synchronous parallel (BSP) computation is an emerging paradigm of general-purpose parallel computing. We propose the first optimal deterministic BSP algorithm for computing the convex hull of a set of points in three-dimensional Euclidean space. Our algorithm is based on known fundamental results from combinatorial geometry, concerning small-sized, efficiently constructible ε-nets and ε-approximations of a given point set. The algorithm generalises the technique of regular sampling, used previously for sorting and two-dimensional convex hull computation. The cost of the simple algorithm is optimal only for extremely large inputs; we show how to reduce the required input size by applying regular sampling in a multi-level fashion.