A Couple of Simple Algorithms for $k$-Dispersion

TL;DR

This work advances the algorithmic understanding of $k$-dispersion by presenting a unified combinatorial framework that yields a plane algorithm with running time $O(n^{k-1}\log n)$ for all $k$, and a $\mathbb{R}^3$ algorithm with parity-dependent bounds $O(n^{k-1}\log n)$ (even $k$) or $O(n^{k-1}\log^2 n)$ (odd $k)$. It extends these results to $d$-space via a diameter-based refinement, and compares against Fast Matrix Multiplication approaches in graphs, providing explicit exponents for the running times. Additionally, the paper offers a linear-time, high-probability $0.99$-approximation for random planar point sets under suitable $k$ ranges, highlighting practical efficiency in average-case scenarios. Collectively, the results push sub-$n^k$ performance boundaries for several regimes of $k$ and dimensionality, while outlining avenues for further improving constants and exploring approximation thresholds in the plane.

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.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• proof : Proof of Theorem \ref{['thm:plane']}
• proof : Proof of Theorem \ref{['thm:3-space']}
• proof : Proof of Theorem \ref{['thm:fmm']}
• proof : Proof of Theorem \ref{['thm:random']}