Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time ∗ (2008)
BibTeX
@MISC{08transdichotomousresults,
author = {},
title = {Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time ∗},
year = {2008}
}
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Abstract
Given a planar subdivision whose coordinates are integers bounded by U ≤ 2 w, we present a linear-space data structure that can answer point location queries in O(min{lg n / lg lg n, √ lg U/lg lg U}) time on the unit-cost RAM with word size w. Thisisthe first result to beat the standard Θ(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 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). Key words. Computational geometry, word-RAM algorithms, data structures, sorting, searching, convex hulls, Voronoi diagrams, segment intersection AMS subject classifications. 68Q25, 68P05, 68U05 Abbreviated title. Point location in sublogarithmic time
