THE ALIGNED K-CENTER PROBLEM
In this paper we study several instances of the aligned k-center 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 constrained 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, or a polygon. We first consider the simplest version of the problem where the line I: is given in advance; we can sole this problem in time O(n log(2) n). In the case where only the direction of is fixed, we give an O(n(2) log(2) n)-time algorithm. When l is an arbitrary line, we give a randomized algorithm with expected running time O(n(4) log(2) n). Then we present (1+epsilon)-approximation algorithms for these three problems. When we denote T (k, epsilon) = (k/epsilon(2)+(k/epsilon) log k) log(1/epsilon), these algorithms run in O(n log k+T (k, epsilon)) time, O(n log k vertical bar T(k. epsilon)/epsilon) time, and O(n log k vertical bar T(k. epsilon)/epsilon(2)) time, respectively. For k = O(n(1/ 3) / log n), we also give randomized algorithms with expected running times O(n + (k/epsilon(2)) log(1/epsilon)), O(n + (k/epsilon(3)) log(1/epsilon)), and O(n + (k/epsilon(4)) log(1/epsilon)), respectively.
the k-center problem approximate algorithm