Partitioning arrangements of lines II: Applications

Abstract

In this paper we present efficient deterministic algorithms for various problems involving lines or segments in the plane, using the partitioning algorithm described in a companion paper [A3]. These applications include: (i) anO(m2/3n2/3 · log2/3n · log?/3 (m/?n)+(m+n) logn) algorithm to compute all incidences betweenm points andn lines, where ? is a constant <3.33; (ii) anO(m2/3n2/3 · log5/3n · log?/3 (m/?n)+(m+n) logn) algorithm to computem faces in an arrangement ofn lines; (iii) anO(n4/3 log(?+2)/3n) algorithm to count the number of intersections in a set ofn segments; (iv) anO(n4/3 log(? + 2)/3n) algorithm to count red-blue intersections between two sets of segments, and (v) anO(n3/2 log?/3n) algorithm to compute spanning trees with low stabbing number for a set ofn points. We also present an algorithm that, given set ofn points in the plane, preprocesses it, in timeO(n?m log?+1/2n), into a data structure of sizeO(m) forn logn≤m≤n2, so that the number of points ofS lying inside a query triangle can be computed inO((n/?m) log3/2n) time.