Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time
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Chan-2009-TRANSDICHOTOMOUS RES.pdf
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Author(s)
Chan, Timothy M.
Patrascu, Mihai
Date Issued
July 2009
Journal
SIAM Journal on Computing
Publisher
Society for Industrial and Applied Mathematics (SIAM)..
Citation
Chan, Timothy M., and Mihai Pǎtrascu. “Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time.” SIAM Journal on Computing 39.2 (2009) : 703. 2009 Society for Industrial and Applied Mathematics
Version
Final published version
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).
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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DOI of Published Version
https://doi.org/10.1137/07068669x
