Two Algorithms for the Sum of Diameters Problem and a Related Problem
Given a set S of points in the plane, we consider the problem of partitioning S into two subsets such that the sum of their diameters is minimized. We present two algorithms with time complexities O(n log 2 n / log log n) and O(n log n / (1 - ∊)), where ∊, 0 < ∊ < 1, is a real number that is dependent on the density of the point set. In almost all practical instances, the second algorithm runs in optimal O(n log n) time, improving all previous results in the case of nonsparse point sets. These bounds follow immediately from two corresponding algorithms with the same time complexities for the following problem: given a set of points S = {p1, p2, …,pn} in the plane sorted in increasing distance from p1, compute the sequence of diameters d1, d2, …, dn, where di= Diam {p1, …, pi} is the diameter of the first i points, 1 ≤ i ≤ n.