# 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) an*O*(*m*^{2/3}*n*^{2/3} · log^{2/3}*n* · log^{ω/3} (*m*/√*n*)+(*m*+*n*) log*n*) algorithm to compute all incidences between*m* points and*n* lines, where ω is a constant <3.33; (ii) an*O*(*m*^{2/3}*n*^{2/3} · log^{5/3}*n* · log^{ω/3} (*m*/√*n*)+(*m*+*n*) log*n*) algorithm to compute*m* faces in an arrangement of*n* lines; (iii) an*O*(*n*^{4/3} log^{(ω+2)/3}*n*) algorithm to count the number of intersections in a set of*n* segments; (iv) an*O*(*n*^{4/3} log^{(ω + 2)/3}*n*) algorithm to count “red-blue” intersections between two sets of segments, and (v) an*O*(*n*^{3/2} log^{ω/3}*n*) algorithm to compute spanning trees with low stabbing number for a set of*n* points. We also present an algorithm that, given set of*n* points in the plane, preprocesses it, in time*O*(*n*√*m* log^{ω+1/2}*n*), into a data structure of size*O*(*m*) for*n* log*n*≤*m*≤*n*^{2}, so that the number of points of*S* lying inside a query triangle can be computed in*O*((*n*/√*m*) log^{3/2}*n*) time.

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