Parallelizability of Some P-Complete Geometric Problems in the EREW-PRAM

Abstract

P-complete problems seem to have no parallel algorithm which runs in polylogarithmic time using a polynomial number of processors. A P-complete problem is in class EP (Efficient and Polynomially fast) if and only if there exists a cost optimal algorithm to solve it in T(n) = O(t(n))Ɛ) (Ɛ>1) using P(n) processors such that T(n) × P(n) = O(t(n)), where t(n) is the time complexity of the fastest sequential algorithm which solves the problem. The goal of our research is to find EPparallel algorithms for P-complete problems. In this paper we consider two P-complete geometric problems in the plane. First we consider the convex layers problem of a set S of n points. Let k be the number of the convex layers of S. When 1 ≤ k ≤ nƐ/2 (0 < Ɛ < 1) we can find the convex layers of S in O( nlogn/p) time using p processors, where 1 ≤ p ≤ n1-Ɛ/2. Next, we consider the envelope layers problem of a set S of n line segments. Let k be the number of the envelope layers of S. When 1 ≤ k ≤ nƐ/2 (0 < Ɛ < 1), we propose an algorithm for computing the envelope layers of S in O( nα(n) log3 n/p) time using pprocessors, where 1 ≤p ≤ n1-Ɛ/2, and α(n) is the functional inverse of Ackermann's function which grows extremely slowly. The computational model we use in this paper is the EREW-PRAM. Our first algorithm, for the convex layers problem, belongs to EP, and the second one, for the envelope layers problem, belongs to the class EP if a small factor of log n is ignored.