On Small Pair Decompositions for Point Sets

TL;DR

This paper investigates the minWSPD problem for point sets, providing approximation algorithms in Euclidean and doubling metrics, and introduces a new pair decomposition with significantly smaller size in general metric spaces.

Contribution

It presents constant approximation algorithms for minWSPD in low-dimensional Euclidean and doubling metrics, and introduces a new, smaller pair decomposition for general metric spaces.

Findings

Constant approximation algorithms for minWSPD in Euclidean space.

A new pair decomposition with size $O( n/ ext{ε} imes ext{log} n)$ in general metric spaces.

Improved bounds for pair decompositions in Euclidean space.

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 $O( d \tfrac{n}{\varepsilon}\log \tfrac{1}{\varepsilon } )$.