Optimal Cover of Points by Disks in a Simple Polygon

Abstract

Let $P$ be a simple polygon, and let $Q$ be a set of points in $P$. We present an almost-linear time algorithm for computing a minimum cover of $Q$ by disks that are contained in $P$. We then generalize the algorithm so that it can compute a minimum cover of $Q$ by homothets of any fixed compact convex set ${\cal O}$ of constant description complexity that are contained in $P$. This improves previous results of Katz and Morgenstern [Lecture Notes in Comput. Sci. 5664, 2009, pp. 447-458]. We also consider the minimum disk-cover problem when $Q$ is contained in a (sufficiently narrow) annulus and present a nearly linear algorithm for this case, too.