Robust algorithms for constructing strongly convex hulls in parallel
Given a set S of n points in the plane, an E-strongly convex delta-hull of S is defined as a convex polygon P with the vertices taken from S such that no point of S lies farther than 6 outside P and such that even if the vertices of P are perturbed by as much as e, P remains convex. This paper presents the first parallel robust method for this generalized convex hull problem (note that the convex hull of S is the 0-strongly convex 0-hull of S). We show that an e-strongly convex O(epsilon + beta)-hull of S can be constructed in O(log(3) n) time using n processors with imprecise computations, where beta is the error unit of primitive operations. This result also implies an improved sequential algorithm. Our algorithm consists of two parts: (1) computing a convex O(epsilon + beta)-hull of n points, in O(log(3) n) time using n processors, and (2) constructing an E-strongly convex O(epsilon + beta)-hull of a convex polygon with n vertices, in O(log(2) n) time with n processors. We also find an approximate bridge of two sets with n points each, in O(log(2) n) time using n processors, which we use as a subroutine. All these algorithms are fundamental and have their own applications. The parallel computational model in this paper is the EREW PRAM. (C) 2002 Elsevier Science B.V. All rights reserved.
robust algorithms convex hull computational geometry pram computational model