A Note on Minimum-Sum Coverage by Aligned Disks

TL;DR

This paper improves the algorithmic efficiency for a facility location problem involving covering points with aligned disks under various metrics, reducing the complexity from O(n^4 log n) to O(n^2 log n).

Contribution

It introduces a faster algorithm for minimum-sum coverage of points by aligned disks, applicable for any a > 1 and any Lp metric, enhancing previous methods.

Findings

Achieved an O(n^2 log n) time complexity algorithm.

Applicable for any a > 1 and any Lp metric.

Improved upon previous O(n^4 log n) algorithm.

Abstract

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^a for any a >= 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 n points, and the sum of the cost of the disks is minimized. Alt et al. [1] presented an algorithm in O(n^4 log n) time for any a > 1 under any Lp metric. We present a faster algorithm for this problem in O(n^2 log n) time for any a > 1 and any Lp metric.