Point Location in o(log n) Time, Voronoi Diagrams in o(n log n) Time, and Other Transdichotomous Results in Computational Geometry

Abstract

Given n points in the plane with integer coordinates bounded by U les 2w, we show that the Voronoi diagram can be constructed in O(min {n log n/ log log n, n(radic(log U)}) expected time by a randomized algorithm on the unit-cost RAM with word size w. Similar results are also obtained for many other fundamental problems in computational geometry, such as constructing the convex hull of a 3-dimensional point set, computing the Euclidean minimum spanning tree of a planar point set, triangulating a polygon with holes, and finding intersections among a set of line segments. These are the first results to beat the Omega(n log n) algebraic-decision-tree lower bounds known for these problems. The results are all derived from a new two-dimensional version of fusion trees that can answer point location queries in O(min { log n / log log n, radic(log U)}) time with linear space. Higher-dimensional extensions and applications are also mentioned in the paper