An Optimal Deterministic Algorithm for Computing the Diameter of a Three-Dimensional Point Set
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- Ramos, E. Discrete Comput Geom (2001) 26: 233. doi:10.1007/s00454-001-0029-8
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  in their ground-breaking paper on geometric applications of random sampling. Our algorithm is relatively simple except for a procedure by Matoušek  for the efficient deterministic construction of epsilon-nets. This work improves previous deterministic algorithms by Ramos  and Bespamyatnikh , 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.