Parallelizability of Some P-Complete Problems
IPDPS '00 Proceedings of the 15 IPDPS 2000 Workshops on Parallel and Distributed Processing
On a class of O(n2) problems in computational geometry
Computational Geometry: Theory and Applications
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We show that in the worst case, /spl Omega/(n/sup d/) sidedness queries are required to determine whether a set of n points in R/sup d/ is affinely degenerate, i.e., whether it contains d+1 points on a common hyperplane. This matches known upper bounds. We give a straightforward adversary argument, based on the explicit construction of a point set containing /spl Omega/(n/sup d/) "collapsible" simplices, any one of which can be made degenerate without changing the orientation of any other simplex. As an immediate corollary, we have an /spl Omega/(n/sup d/) lower bound on the number of sidedness queries required to determine the order type of a set of n points in R/sup d/. Using similar techniques, we also show that /spl Omega/(n/sup d+1/) in-sphere queries are required to decide the existence of spherical degeneracies in a set of n points in R/sup d/.