On the Line-Separable Unit-Disk Coverage and Related Problems
Gang Liu, Haitao Wang

TL;DR
This paper presents faster algorithms for line-separable unit-disk coverage problems and related geometric covering problems, improving previous computational complexity bounds significantly.
Contribution
The authors develop new algorithms with improved time complexities for line-separable unit-disk coverage, line-constrained coverage, and half-plane coverage problems.
Findings
Achieved an $O((n+m)\log(n+m))$ time algorithm for line-separable unit-disk coverage.
Improved the line-constrained coverage algorithm to $O((n+m)\log (m+ n)+m \log m\log n)$ time.
Reduced the half-plane coverage problem to $O(n^3\log n)$ and $O(n\log n)$ time for special cases.
Abstract
Given a set of points and a set of disks in the plane, the disk coverage problem asks for a smallest subset of disks that together cover all points of . 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 by a line . We present an time algorithm for the problem. This improves the previously best result of time. Our techniques also solve the line-constrained version of the problem, where centers of all disks of are located on a line while points of can be anywhere in the plane. Our algorithm runs in time, which improves the previously best result of time. In addition, our results lead to an algorithm of …
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