On Small Pair Decompositions for Point Sets

2026-01-30

AI Summary

This study addresses the problem of computing a minimum-sized well-separated pair decomposition (minWSPD) in Euclidean spaces of dimension two and higher, which is known to be intractable to solve exactly and lacks efficient approximation algorithms. To overcome this challenge, the authors introduce a novel variant of pair decomposition that relaxes the conventional diameter constraints on subsets. This approach achieves a size bound of $O(n/\varepsilon \cdot \log n)$ in general metric spaces—significantly improving upon the quadratic bound of classical WSPDs—and further refines it to $O(d \cdot n/\varepsilon \cdot \log(1/\varepsilon))$ in $\mathbb{R}^d$. By leveraging doubling dimension, Euclidean geometric structure, and careful approximation design, this work presents the first constant-factor approximation algorithms for minWSPD in low-dimensional Euclidean and doubling metric spaces.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Learning with Manifolds

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• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

$\newcommand{\Re}{\mathbb{R}}$We study the minWSPD problem of computing the minimum-size well-separated pairs decomposition of a set of points, and show constant approximation algorithms in low-dimensional Euclidean space and doubling metrics. This problem is computationally hard already $\Re^2$, and is also hard to approximate. We also introduce a new pair decomposition, removing the requirement that the diameters of the parts should be small. Surprisingly, we show that in a general metric space, one can compute such a decomposition of size $O( \tfrac{n}{\varepsilon}\log n)$, which is dramatically smaller than the quadratic bound for WSPDs. In $\Re^d$, the bound improves to $O( d \tfrac{n}{\varepsilon}\log \tfrac{1}{\varepsilon } )$.

Problem

• well-separated pairs decomposition
• minWSPD
• point sets
• computational hardness
• metric spaces

Innovation

• well-separated pair decomposition
• approximation algorithm
• doubling metric
• pair decomposition
• computational geometry