Discrete & Computational Geometry

, Volume 26, Issue 2, pp 233-244

First online:

An Optimal Deterministic Algorithm for Computing the Diameter of a Three-Dimensional Point Set

  • E. A. RamosAffiliated withMax-Planck-Institut für Informatik, Stuhlsatzenhausweg 85, 66123 Saarbrücken, Germany

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We describe a deterministic algorithm for computing the diameter of a finite set of points in R 3 , that is, the maximum distance between any pair of points in the set. The algorithm runs in optimal time O(nlog  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(nlog  2 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.