Computing k-Centers On a Line

In this paper we consider several instances of the k-center on a line problem where the goal is, given a set of points S in the plane and a parameter k ≥ 1, to find k disks with centers on a line l such that their union covers S and the maximum radius of the disks is minimized. This problem is a constraint version of the well-known k-center problem in which the centers are constrained to lie in a particular region such as a segment, a line, and a polygon. We first consider the simplest version of the problem where the line l is given in advance; we can solve this problem in O(n log n) time. We then investigate the cases where only the orientation of the line l is fixed and where the line l can be arbitrary. We can solve these problems in O(n log n) time and in O(n log n) expected time, respectively. For the last two problems, we present (1 + ε)-approximation algorithms, which run in O( 1 ε n log n) time and O( 1 ε 2 n log 2 n) time, respectively.