An optimal deterministic algorithm for computing the diameter of a three-dimensional point set

Abstract

We describe a deterministic algorithm for computing the diameter of a finite set of points in ℝ3, that is, the maximum distance between any pair of points in the set. The algorithm runs in optimal time O(n log n) for a set of n points. The first optimal, but randomized, algorithm for this problem was proposed more than 10 years ago by Clarkson and Shor [11] in their ground-breaking paper on geometric applications of random sampling. Our algorithm is relatively simple except for a procedure by Matoušek [25] for the efficient deterministic construction of epsilon-nets. This work improves previous deterministic algorithms by Ramos [31] and Bespamyatnikh [7], both with running time O(n log2 n). The diameter algorithm appears to be the last one in Clarkson and Shor's paper that up to now had no deterministic counterpart with a matching running time.