Two Optimization Problems for Unit Disks

Abstract

We present an implementation of a recent algorithm to compute shortest-path trees in unit disk graphs in $O(n\log n)$ worst-case time, where $n$ is the number of disks. In the minimum-separation problem, we are given $n$ unit disks and two points $s$ and $t$, not contained in any of the disks, and we want to compute the minimum number of disks one needs to retain so that any curve connecting $s$ to $t$ intersects some of the retained disks. We present a new algorithm solving this problem in $O(n^2{\log }^{3}n)$ worst-case time and its implementation.