A Couple of Simple Algorithms for $k$-Dispersion

TL;DR

This paper introduces simple algorithms for the $k$-dispersion problem, providing exact solutions in specific dimensions and approximate solutions efficiently for random points, advancing the computational understanding of this geometric problem.

Contribution

The paper extends known algorithms for $k$-dispersion to arbitrary $k$ in the plane and three dimensions, and offers a fast approximation method for random points in the unit square.

Findings

Exact algorithms with $O(n^{k-1} ext{log} n)$ time for planar $k$-dispersion.

Exact algorithms with $O(n^{k-1} ext{log} n)$ or $O(n^{k-1} ext{log}^2 n)$ time in 3D depending on $k$.

A high-probability $0.99$-approximation algorithm running in $O(n)$ time for random points in $[0,1]^2$.

Abstract

Given a set $P$ of $n$ points in $\mathbf{R}^d$, and a positive integer $k \leq n$, the $k$-dispersion problem is that of selecting $k$ of the given points so that the minimum inter-point distance among them is maximized (under Euclidean distances). Among others, we show the following: (I) Given a set $P$ of $n$ points in the plane, and a positive integer $k \geq 2$, the $k$-dispersion problem can be solved by an algorithm running in $O\left(n^{k-1} \log{n}\right)$ time. This extends an earlier result for $k=3$, due to Horiyama, Nakano, Saitoh, Suetsugu, Suzuki, Uehara, Uno, and Wasa (2021) to arbitrary $k$. In particular, it improves on previous running times for small $k$. (II) Given a set $P$ of $n$ points in $\mathbf{R}^3$, and a positive integer $k \geq 2$, the $k$-dispersion problem can be solved by an algorithm running in $O\left(n^{k-1} \log{n}\right)$ time, if $k$ is even; and $O\left(n^{k-1} \log^2{n}\right)$ time, if $k$ is odd. For $k \geq 4$, no combinatorial algorithm running in $o(n^k)$ time was known for this problem. (III) Let $P$ be a set of $n$ random points uniformly distributed in $[0,1]^2$. Then under suitable conditions, a $0.99$-approximation for $k$-dispersion can be computed in $O(n)$ time with high probability.