A Couple of Simple Algorithms for $k$-Dispersion

2025-11-01

AI Summary

The k-dispersion problem seeks to select k points from a set of n points in ℝᵈ to maximize the minimum pairwise Euclidean distance. This paper presents efficient exact and approximation algorithms for d = 2 and d = 3. Methodologically, it integrates geometric divide-and-conquer, ordered pruning, and probabilistic analysis to overcome the combinatorial explosion inherent in brute-force enumeration. For d = 3 and k ≥ 4, it introduces the first sub-O(nᵏ) combinatorial algorithm—achieving O(nᵏ⁻¹ log n) time for even k and O(nᵏ⁻¹ log²n) for odd k. In d = 2, the exact algorithm is optimized to O(nᵏ⁻¹ log n). Furthermore, for n random points uniformly distributed in [0,1]², it provides a linear-time 0.99-approximation algorithm. These results represent significant advances both theoretically—by breaking long-standing enumeration barriers—and practically—through improved runtime efficiency and scalability.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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 $Oleft(n^{k-1} log{n} ight)$ 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 $Oleft(n^{k-1} log{n} ight)$ time, if $k$ is even; and $Oleft(n^{k-1} log^2{n} ight)$ 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.

Problem

• Maximizing minimum distance among k points in Euclidean space
• Developing efficient algorithms for k-dispersion in 2D and 3D
• Providing approximation methods for random point distributions

Innovation

• Algorithm with O(n^(k-1) log n) time for 2D k-dispersion
• Algorithm with O(n^(k-1) log^2 n) time for 3D k-dispersion
• Linear-time 0.99-approximation for random 2D points