On Small Pair Decompositions for Point Sets

TL;DR

This work studies the minimum-size Well-Separated Pair Decomposition (minWSPD) problem and introduces Approximate Biclique Covers (ABCs) as a relaxation that enables near-linear representations of pairwise distances in general metrics, with linear-sized ABCs in Euclidean spaces and strong SSPD properties. It proves constant-factor approximations for minWSPD in doubling metrics via net-tree WSPDs, analyzes an output-sensitive runtime, and provides hardness results showing NP-hardness for minWSPD in general and in the Euclidean plane. The paper also develops effective 1D results, including a PTAS via SetCover and a simple 3-approximation, and validates the approaches through experiments in one dimension. Overall, the ABC framework broadens the toolkit for compact distance representations, offering practical near-linear constructions while clarifying computational limits and dimensional tradeoffs.

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

Figures (13)

• Figure 2.1: Rightmost example, if the sidelength of the grid is $1$, then $m = \mathcalb{d}_{\min}\mleft(X,Y\mright) = 6$, $M = \mathcalb{d}_{\max}\mleft(X,Y\mright) = 10$, and $\nabla\mleft(X\mright) = \nabla\mleft(Y\mright) =\sqrt{37} \approx 6.08$. Thus, the pair $\{X, Y\}$ is $\tfrac{1}{3}$-stable, as $\frac{M -m}{2m} = \tfrac{1}{3}$.
• Figure 3.1: The set of points in the green cell is paired with the set of points in the red cells (the blue cells are empty).
• Figure 6.1: Left: A few points and their vicinities, for $\varepsilon =0.4$. Right: The left half of the vicinity of a point $p$ can be covered by three squares.
• Figure 6.2: Visualization of the sweep and event status at the beginning of column $p_5$. A search would find $p_4$. When we place an interval over it on the backtrack, we check if the children of a node are covered; if they are, we mark the node as covered. So, covering $p_4$ propagates to the root node, and the next search would terminate at the root.
• Figure 6.3: The five features in different colors and an angle plot with intervals show which direction includes which features. The three arrows highlight directions of the normal vector, which are the only ones with fewer than four features.

Theorems & Definitions (43)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Remark 2.7
• Definition 2.8
• Definition 2.9
• Definition 2.10