Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time
Timothy M. Chan, Mihai Ptraşcu · SIAM Journal on Computing · 2009
Given a planar subdivision whose coordinates are integers bounded by $U\leq2^w$, we present a linear-space data structure that can answer point-location queries in $O(\min\{\lg n/\lg\lg n,$ $\sqrt{\lg U/\lg\lg U}\})$ time on the unit-cost random access machine (RAM) with word size w. This is the first result to beat the standard $\Theta(\lg n)$ 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).