A note on minimum-sum coverage by aligned disks

In this paper, we consider a facility location problem to find a minimum-sum coverage of n points by disks centered at a fixed line. The cost of a disk with radius r has a form of a non-decreasing function f(r) = r for any α ≥ 1. The goal is to find a set of disks under Lp metric such that the disks are centered on the x-axis, their union covers the points, and the sum of the cost of the disks is minimized. Alt et al. [1] presented an algorithm running in O(n log n) time for any α > 1 under any Lp metric. We present a faster algorithm for this problem running in O(n log n) time for any α > 1 and any Lp metric.