New Constructions of SSPDs and their Applications

TL;DR

The paper tackles the challenge of constructing semi-separated pair decompositions (SSPDs) with near-linear total weight and limited per-point participation, and extends these constructions to metric spaces of low doubling dimension. It provides two main approaches: a simple, clearer SSPD with O(log^2 n) per-point participation and O(n log^2 n) weight, and an optimal, randomized-partition SSPD achieving O(log n) per-point participation with high probability, both applicable beyond Euclidean space. Leveraging these SSPDs, the authors derive new spanner constructions with excellent sparsity and small separators, including a near-linear-edge t-spanner with O(n) edges and maximum degree O(log^2 n) and a separator of size O(n^{1-1/d}/ε^d). The results advance the state of SSPDs and their practical deployment in spanners and divide-and-conquer strategies, particularly in spaces with low doubling dimension.

Abstract

$\renewcommand{\Re}{\mathbb{R}}$We present a new optimal construction of a semi-separated pair decomposition (i.e., SSPD) for a set of $n$ points in $\Re^d$. In the new construction each point participates in a few pairs, and it extends easily to spaces with low doubling dimension. This is the first optimal construction with these properties. As an application of the new construction, for a fixed $t>1$, we present a new construction of a $t$-spanner with $O(n)$ edges and maximum degree $O(\log^2 n)$ that has a separator of size $O\pth{n^{1-1/d}}$.

Figures (6)

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Theorems & Definitions (31)

• Definition 2.1: Pair decomposition.
• Definition 2.2
• Definition 2.3: WSPD
• Definition 2.4: SSPD
• Lemma 2.5
• Lemma 2.6: hm-fcnld-06
• Lemma 2.7
• Lemma 2.8
• Remark 2.9
• Remark 2.10