Robust algorithms for constructing strongly convex hulls in parallel

Given a set S of n points in the plane, an -strongly convex -hull of S is de0ned as a convex polygon P with the vertices taken from S such that no point of S lies farther than outside P and such that even if the vertices of P are perturbed by as much as , P remains convex. This paper presents the 0rst 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 -strongly convex O( + )-hull of S can be constructed in O(logn) time using n processors with imprecise computations, where 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( + ) -hull of n points, in O(logn) time using n processors, and (2) constructing an -strongly convex O( + )-hull of a convex polygon with n vertices, in O(logn) time with n processors. We also 0nd an approximate bridge of two sets with n points each, in O(logn) 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.