Max-Min 3-dispersion Problems

Abstract

Given a set P of n points on which facilities can be placed and an integer k, we want to place k facilities on some points so that the minimum distance between facilities is maximized. The problem is called the k-dispersion problem. In this paper, we consider the 3-dispersion problem when P is a set of points on a plane (2-dimensional space). Note that the 2-dispersion problem corresponds to the diameter problem. We give an O(n) time algorithm to solve the 3-dispersion problem in the L_∞ metric, and an O(n) time algorithm to solve the 3-dispersion problem in the L₁ metric. Also, we give an O(n² log n) time algorithm to solve the 3-dispersion problem in the L₂ metric.