On Small Pair Decompositions for Point Sets

$\newcommand{\Re}{\mathbb{R}}$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 $O( d \tfrac{n}{\varepsilon}\log \tfrac{1}{\varepsilon } )$.

💡 Research Summary

The paper investigates two intertwined problems concerning pair decompositions of point sets. The first problem, termed minWSPD, asks for a minimum‑size well‑separated pair decomposition (WSPD) for a given set P of n points and a separation parameter ε ∈ (0,1). The second problem relaxes the classic WSPD definition by dropping the requirement that the diameters of the two sides of each pair be small; this leads to the notion of an Approximate Biclique Cover (ABC), where a pair {X,Y} is ε‑stable if every distance d(x,y) with x∈X, y∈Y lies within a (1+ε) factor of the minimum distance between the two sides.

Hardness Results.
The authors prove that minWSPD is NP‑hard already in the Euclidean plane. Specifically, for ε≈1/(4n) it is NP‑complete to decide whether a 1/ε‑WSPD of a prescribed size exists. Moreover, they show that in general metric spaces computing the minimum 3‑WSPD is NP‑hard and, via a padding argument, that the problem cannot be approximated within a factor better than (1‑o(1))·ln n unless P=NP. These results stem from a reduction to Set Cover: the family of all maximal well‑separated pairs forms the ground set, and each pair covers a biclique of point pairs. Consequently, the greedy Set‑Cover algorithm yields only an O(log n) approximation, matching known lower bounds for Set Cover.

Approximate Biclique Covers.
The paper introduces ABCs, which require only that distances inside a pair be approximately equal, without any bound on the diameters of X or Y. The authors present a simple greedy construction that, for any finite metric space, produces an ε‑stable cover of size O(n·ε⁻¹·log n). In Euclidean space ℝᵈ the bound improves to O(d·n·ε⁻¹·log (1/ε)). Remarkably, the same construction also yields a 1/ε‑semi‑separated pair decomposition (SSPD), improving the previously known O(n·ε⁻ᵈ) bound to a linear‑in‑n bound with only polynomial dependence on d. The trade‑off is that the cover is not a partition (pairs may overlap) and the total weight (sum of |X|+|Y| over all pairs) can be quadratic in the worst case.

Near‑Optimal WSPD in Doubling Metrics.
For metric spaces with constant doubling dimension, the classic net‑tree algorithm of Har‑Peled and Mendel builds a WSPD of size O(ε⁻ᵈ·n). The authors prove that this algorithm is essentially optimal: the produced WSPD has size within a constant factor of the optimum (opt) and runs in O(opt + n log n) time, which is output‑sensitive and improves upon the previous O(ε⁻ᵈ·n + n log n) bound.

One‑Dimensional Case.
When points lie on the real line, the authors exploit the fact that the family of maximal well‑separated pairs is polynomial in size. They formulate minWSPD as a Set‑Cover instance and apply known PTAS techniques to obtain a (1+δ)‑approximation in time n·O(1/δ²). They also give a deterministic O(n log n) algorithm achieving a 3‑approximation via a sweep‑line method, and show how to convert a covering WSPD into a partition WSPD with a constant‑factor loss.

Experimental Evaluation.
Six algorithms for constructing WSPDs in one dimension are implemented: greedy Set‑Cover, the Callahan‑Kosaraju algorithm, two variants of the 3‑approximation, and two integer‑programming based methods (one producing disjoint covers). Experiments on synthetic point sets demonstrate that the ABC‑based approaches produce substantially smaller decompositions than traditional methods while maintaining comparable running times.

Contributions and Outlook.
The work establishes the intrinsic computational difficulty of minimizing WSPD size, introduces a more flexible decomposition (ABC) that achieves near‑linear size in general metrics, and shows that existing net‑tree constructions are essentially optimal for doubling spaces. Open directions include reducing the weight of ABCs, extending the techniques to higher‑dimensional geometric settings, and applying the new decompositions to problems such as approximate nearest‑neighbor search, clustering, and graph spanner construction.