## Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time (2008)

Citations: | 10 - 3 self |

### BibTeX

@MISC{Chan08transdichotomousresults,

author = {Timothy M. Chan and et al.},

title = {Transdichotomous Results in Computational Geometry, I: Point Location in Sublogarithmic Time },

year = {2008}

}

### OpenURL

### 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).