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)+κlog⁡m) time algorithm for the problem, where κ is the number of pairs of disks whose boundaries intersect. Alternatively, we can also solve the problem in O(nmlog⁡(m+n)) time. 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∞ and L1 cases of the problem, in which the disks are squares and diamonds, respectively. We further demonstrate that our techniques can also be used to solve other geometric coverage problems. For example, given in the plane a set P of n points and a set S of n weighted half-planes, we solve in O(n4log⁡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(n5) time by almost a linear factor. If all half-planes are lower ones, then our algorithm runs in O(n2log⁡n) time, which improves the previous best algorithm of O(n4) time by almost a quadratic factor.