Reporting intersecting pairs of convex polytopes in two and three dimensions

Authors:
Pankaj K. Agarwal;Mark de Berg;Sariel Har-Peled;Mark H. Overmars;Micha Sharir;Jan Vahrenhold
Affiliations:
Department of Computer Science, Duke University, Durham, NC;Institute of Information and Computing Sciences, Utrecht University, P.O. Box 80.089, 3508 TB Utrecht, The Netherlands;Department of Computer Science, DCL 2111, University of Illinois, 1304 West Springfield Ave., Urbana, IL;Institute of Information and Computing Sciences, Utrecht University, P.O. Box 80.089, 3508 TB Utrecht, The Netherlands;School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel and Courant Institute of Mathematical Sciences, New York University, New York, NY;Westfälische Wilhelms-Universität Münster, Institut für Informatik, 48149 Münster, Germany
Venue:
Computational Geometry: Theory and Applications
Year:
2002

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Abstract

Let P = {P1....,Pm) be a set of m convex polytopes in Rd, for d = 2, 3, with a total of n vertices. We present output-sensitive algorithms for reporting all k pairs of indices (i, j) such that Pi intersects Pj. For the planar case we describe a simple algorithm with running time O(n4/3 log2+ε n + k), for any constant ε 0, and an improved randomized algorithm with expected running time O((n logm + k)α(n)logn) (which is faster for small values of k). For d = 3, we present an O(n8/5+ε + k)-time algorithm, for any ε 0. Our algorithms can be modified to count the number of intersecting pairs in O(n4/3 log2+ε n) time for the planar case, and in O(n8/5+ε) time for the three-dimensional case.