# Two Optimization Problems for Unit Disks

## TL;DR

This paper introduces efficient algorithms for shortest-path trees in unit disk graphs and a novel solution for the minimum-separation problem, with implementations demonstrating their practical performance.

## Contribution

It provides an implementation of a recent shortest-path algorithm in unit disk graphs and introduces a new algorithm for the minimum-separation problem with complexity analysis.

## Key findings

- Shortest-path trees computed in $O(n ext{log} n)$ time.
- Minimum-separation problem solved in $O(n^2 ext{log}^3 n)$ time.
- Implementation validates the theoretical algorithms.

## 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.