Algorithms for the Line-Constrained Disk Coverage and Related Problems

Abstract

Given a set P of n points and a set S of m weighted disks in the plane, the disk coverage problem asks for a subset of disks of minimum total weight that cover all points of P. The problem is NP-hard. In this paper, we consider a line-constrained version in which all disks are centered on a line L (while points of P can be anywhere in the plane). We present an $O((m+n)\log (m+n)+\kappa \log m)$ time algorithm for the problem, where $\kappa$ is the number of pairs of disks that intersect. For the unit-disk case where all disks have the same radius, the running time can be reduced to $O((n+m)\log (m+n))$. In addition, we solve in $O((m+n)\log (m+n))$ time the $L_{\infty }$ and $L_1$ cases of the problem, in which the disks are squares and diamonds, respectively. Using our techniques, we further solve two other geometric coverage problems. Given in the plane a set P of n points and a set S of n weighted half-planes, we solve in $O(n^4\log n)$ time the problem of finding a subset of half-planes to cover P so that their total weight is minimized. This improves the previous best algorithm of $O(n^5)$ time by almost a linear factor. If all half-planes are lower ones, our algorithm runs in $O(n^2\log n)$ time, which improves the previous best algorithm of $O(n^4)$ time by almost a quadratic factor.

This research was supported in part by NSF under Grant CCF-2005323. A full version of this paper is available at https://arxiv.org/abs/2104.14680.