Deterministic algorithms for 3-D diameter and some 2-D lower envelopes

Abstract

We present a deterministic algorithm for computing the diameter of a set of n points in R3; its running time O(n log n) is worst-case optimal. This improves previous deterministic algorithms by Ramos (1997) and Bespamyatnikh (1998), both with running time O(n log2 n), and matches the running time of a randomized algorithm by Clarkson and Shor (1989). We also present a deterministic algorithm for computing the lower envelope of n functions of 2 variables, for a class of functions with certain restrictions; if the functions in the class have lower envelope with worst-case complexity O(f2(n)), the running time is O(f2(n) log n), in general, and O(f2(n)) when f2(n) = Ω(n1+ε) for any small fraction ε>0. The algorithms follow a divide-and-conquer approach based on deterministic sampling with the essential feature that planar graph separators are used to group subproblems in order to limit the growth of the total subproblem size.