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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 log s n), and matches the running time of a randomized algorithm by Clarkson and Shot (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)logn), in general, and O(f2(n)) when f2(n) = ~(n x+~) for any small fraction e > 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.

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