Fast Approximation and Randomized Algorithms for Diameter

We consider approximation of diameter of a set S of n points in dimension m. Eg̃eciog̃lu and Kalantari [8] have shown that given any p ∈ S, by computing its farthest in S, say q, and in turn the farthest point of q, say q′, we have diam(S) ≤ √ 3 d(q, q′). Furthermore, iteratively replacing p with an appropriately selected point on the line segment pq, in at most t ≤ n additional iterations, the constant bound factor is improved to c∗ = √ 5− 2 √ 3 ≈ 1.24. Here we prove when m = 2, t = 1. This suggests in practice a few iterations may produce good solutions in any dimension. Here we also propose a randomized version and present large scale computational results with these algorithm for arbitrary m. The algorithms outperform many existing algorithms. On sets of data as large as 1, 000, 000 points, the proposed algorithms compute solutions to within an absolute error of 10−4.