Light Spanners for High Dimensional Norms via Stochastic Decompositions

TL;DR

This paper develops new light and sparse spanners for high-dimensional normed spaces, especially for subsets of lp spaces, using stochastic decompositions to improve upon previous bounds and handle various metric families.

Contribution

It introduces a method to construct light, sparse spanners in high-dimensional normed spaces based on stochastic decompositions, extending results to broader metric families.

Findings

Constructed lp space spanners with near-optimal size and lightness.

Established a tradeoff between decomposability parameters and spanner lightness.

Reduced exponential dependency on dimension for certain metric families.

Abstract

Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har-Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any $n$-point Euclidean metric has an $O(t)$-spanner with $\tilde{O}(n^{1+1/t^2})$ edges, little is known. In this paper we study several aspects of spanners in high dimensional normed spaces. First, we build spanners for finite subsets of $\ell_p$ with $1<p\le 2$. Second, our construction yields a spanner which is both sparse and also {\em light}, i.e., its total weight is not much larger than that of the minimum spanning tree. In particular, we show that any $n$-point subset of $\ell_p$ for $1<p\le 2$ has an $O(t)$-spanner with $n^{1+\tilde{O}(1/t^p)}$ edges and lightness $n^{\tilde{O}(1/t^p)}$. In fact, our results are more general, and they apply to any metric space admitting a certain low diameter stochastic decomposition. It is known that arbitrary metric spaces have an $O(t)$-spanner with lightness $O(n^{1/t})$. We exhibit the following tradeoff: metrics with decomposability parameter $\nu=\nu(t)$ admit an $O(t)$-spanner with lightness $\tilde{O}(\nu^{1/t})$. For example, $n$-point Euclidean metrics have $\nu\le n^{1/t}$, metrics with doubling constant $\lambda$ have $\nu\le\lambda$, and graphs of genus $g$ have $\nu\le g$. While these families do admit a ($1+\epsilon$)-spanner, its lightness depend exponentially on the dimension (resp. $\log g$). Our construction alleviates this exponential dependency, at the cost of incurring larger stretch.