Optimal, output-sensitive algorithms for constructing planar hulls in parallel

Abstract

In this paper we focus on the problem of designing very fast parallel algorithms for the planar convex hull problem that achieve the optimal O( nlog H) work-bound for input size n and output size H. Our algorithms are designed for the arbitrary CRCW PRAM model. We first describe a very simple O(log nlog H) time optimal deterministic algorithm for the planar hulls which is an improvement over the previously known Ω( log 2n) time algorithm for small outputs. For larger values of H, we can achieve a running time of O(log nloglog n) steps with optimal work. We also present a fast randomized algorithm that runs in expected time O(log H·loglog n) and does optimal O( nlog H) work. For logH = Ω( loglogn) , we can achieve the optimal running time of O(log H) while simultaneously keeping the work optimal. When log H is o(log n), our results improve upon the previously best known Θ(log n) expected time randomized algorithm of Ghouse and Goodrich. The randomized algorithms do not assume any input distribution and the running times hold with high probability.