New Constructions of SSPDs and their Applications

TL;DR

This paper introduces an optimal semi-separated pair decomposition construction for point sets in Euclidean space, which is efficient and extendable to low doubling dimension spaces, and applies it to create sparse, low-degree spanners with small separators.

Contribution

It presents the first optimal construction of SSPDs with limited participation per point and extendability to low doubling dimension spaces.

Findings

Constructed an optimal SSPD with each point in few pairs.

Extended the SSPD construction to low doubling dimension spaces.

Developed a new t-spanner with O(n) edges, O(log^2 n) degree, and small separators.

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 $O\pth{n^{1-1/d}}$.