Discrete & Computational Geometry

, Volume 5, Issue 6, pp 533–573 | Cite as

Partitioning arrangements of lines II: Applications

  • Pankaj K. Agarwal
Article

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 · logω/3 (m/√n)+(m+n) logn) algorithm to compute all incidences betweenm points andn lines, where ω is a constant <3.33; (ii) anO(m2/3n2/3 · log5/3n · logω/3 (m/√n)+(m+n) logn) algorithm to computem faces in an arrangement ofn lines; (iii) anO(n4/3 log(ω+2)/3n) algorithm to count the number of intersections in a set ofn segments; (iv) anO(n4/3 log(ω + 2)/3n) algorithm to count “red-blue” intersections between two sets of segments, and (v) anO(n3/2 logω/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(nm logω+1/2n), into a data structure of sizeO(m) forn lognmn2, so that the number of points ofS lying inside a query triangle can be computed inO((n/√m) log3/2n) time.

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Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Pankaj K. Agarwal
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityUSA
  2. 2.Department of Computer ScienceDuke UniversityDurhamUSA

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