On the line-separable unit-disk coverage and related problems

Given a set P of n points and a set S of m disks in the plane, the disk coverage problem asks for a smallest subset of disks that together cover all points of P. The problem is NP-hard. In this paper, we consider a line-separable unit-disk version of the problem where all disks have the same radius and their centers are separated from the points of P by a line ℓ. We present an $O\left(\right(n+m)\log (n+m\left)\right)$ time algorithm for the problem. This improves the previously best result of $O(nm+n\log n)$ time. Our techniques also solve the line-constrained version of the problem, where centers of all disks of S are located on a line ℓ while points of P can be anywhere in the plane. Our algorithm runs in $O\left(\right(n+m)\log (m+n)+m\log m\log n)$ time, which improves the previously best result of $O(nm\log (m+n\left)\right)$ time. In addition, our results lead to an algorithm of $O(n^3\log n)$ time for a half-plane coverage problem (given n half-planes and n points, find a smallest subset of half-planes covering all points); this improves the previously best algorithm of $O(n^4\log n)$ time. Further, if all half-planes are lower ones, our algorithm runs in $O(n\log n)$ time while the previously best algorithm takes $O(n^2\log n)$ time.