Reporting intersecting pairs of convex polytopes in two and three dimensions

Pankaj K. Agarwal, Mark de Berg, Sariel Har-Peled, Mark H. Overmars, Micha Sharir, Jan Vahrenhold
2002 Computational geometry  
Let P = {P 1 , . . . , P m } be a set of m convex polytopes in R d , 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 P i intersects P j . For the planar case we describe a simple algorithm with running time O(n 4/3 log 2+ε n + k), for any constant ε > 0, 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
more » ... O(n 8/5+ε + k)-time algorithm, for any ε > 0. Our algorithms can be modified to count the number of intersecting pairs in O(n 4/3 log 2+ε n) time for the planar case, and in O(n 8/5+ε ) time for the three-dimensional case. 
doi:10.1016/s0925-7721(02)00049-4 fatcat:wl2swhgeubdfrnh3oyg7gm6kr4