Optimal, output-sensitive algorithms for constructing upper envelope of line segments in parallel

Abstract

In this paper we focus on the problem of designing very fast parallel algorithms for constructing the upper envelope of straight-line segments that achieve the O(n logH) work-bound for input size n and output size H. Our algorithms are designed for the arbitrary CRCW PRAM model. We first describe an O(log n.(logH + log log n)) time deterministic algorithm for the problem, that achieves O(n logH) work bound for H = Ω(log n). We present a fast randomized algorithm that runs in expected time O(logH.log n) with high probability and does O(n logH) work. For log H = Ω(log log n), we can achieve the running time of O(log H) while simultaneously keeping the work optimal. We also present a fast randomized algorithm that runs in Õ(log n/ log k) time with nk processors, k > log^Ω(1) n. The algorithms do not assume any input distribution and the running times hold with high probability.