Numerical stability of geometric algorithms
SCG '87 Proceedings of the third annual symposium on Computational geometry
Efficient parallel algorithms
Verifiable implementation of geometric algorithms using finite precision arithmetic
Artificial Intelligence - Special issue on geometric reasoning
Recipes for geometry and numerical analysis - Part I: an empirical study
SCG '88 Proceedings of the fourth annual symposium on Computational geometry
Calculating approximate curve arrangements using rounded arithmetic
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Epsilon geometry: building robust algorithms from imprecise computations
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
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Given a set S of n points in the plane, an ε-strongly convex δ-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 δ outside P and such that even if the vertices of P are perturbed by as much as ε, 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 ε-strongly convex O(ε + β)-hull of S can be constructed in O(log3n) time using n processors with imprecise computations, where β 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(ε + β) -hull of n points, in O(log3n) time using n processors, and (2) constructing an ε-strongly convex O(ε + β)-hull of a convex polygon with n vertices, in O(log2n) time with n processors. We also find an approximate bridge of two sets with n points each, in O(log2n) 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.