Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time

Abstract

Given a planar subdivision whose coordinates are integers bounded by U ≤ 2w [U less than or equal to 2 superscript w], we present a linear-space data structure that can answer point-location queries in O(min{lg n/ lg lg n, sq rt lg U/ lg lgU}) [O(min {lg n / lg lg n, square root of lg U/lg lg U})] time on the unit-cost random access machine (RAM) with word size w. This is the first result to beat the standard Θ [theta](lg n) bound for infinite precision models. As a consequence, we obtain the first o(n lg n) (randomized) algorithms for many fundamental problems in computational geometry for arbitrary integer input on the word RAM, including: constructing the convex hull of a three-dimensional (3D) point set, computing the Voronoi diagram or the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. Higher-dimensional extensions and applications are also discussed. Though computational geometry with bounded precision input has been investigated for a long time, improvements have been limited largely to problems of an orthogonal flavor. Our results surpass this long-standing limitation, answering, for example, a question of Willard (SODA'92).