Reporting Intersecting Pairs of Polytopes in Two and Three Dimensions

Authors:
Pankaj K. Agarwal;Mark de Berg;Sariel Har-Peled;Mark H. Overmars;Micha Sharir;Jan Vahrenhold
Affiliations:
-;-;-;-;-;-
Venue:
WADS '01 Proceedings of the 7th International Workshop on Algorithms and Data Structures
Year:
2001

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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 log n + k), and an improved randomized algorithm with expected running time O((n log m + k)α(n) log n) (which is faster for small values of k). For d = 3, we present an O(n8/5+Ɛ + k)-time algorithm, for any Ɛ 0. Our algorithms can be modified to count the number of intersecting pairs in O(n4/3 logO(1) n) time for the planar case, and in O(n8/5+Ɛ) time and R3.