Embedding Metrics into Ultrametrics and Graphs into Spanning Trees with Constant Average Distortion

TL;DR

This paper introduces new embedding techniques that achieve constant average distortion for metric spaces and graphs, with bounds that are tight and applicable to ultrametrics and spanning trees, improving approximation quality.

Contribution

It presents the first scaling distortion embeddings with constant average distortion and tight bounds for ultrametrics and spanning trees, advancing the understanding of metric and graph approximations.

Findings

Embedding into ultrametrics with $O( oot{1/ ext{epsilon}})$ distortion

Existence of spanning trees with $O( oot{1/ ext{epsilon}})$ scaling distortion

Probabilistic embeddings into spanning trees with $ ilde{O}( ext{log}^2(1/ ext{epsilon}))$ distortion

Abstract

This paper addresses the basic question of how well can a tree approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling distortion embeddings where the distortion scales as a function of $\epsilon$, with the guarantee that for each $\epsilon$ the distortion of a fraction $1-\epsilon$ of all pairs is bounded accordingly. Such a bound implies, in particular, that the \emph{average distortion} and $\ell_q$-distortions are small. Specifically, our embeddings have \emph{constant} average distortion and $O(\sqrt{\log n})$ $\ell_2$-distortion. This follows from the following results: we prove that any metric space embeds into an ultrametric with scaling distortion $O(\sqrt{1/\epsilon})$. For the graph setting we prove that any weighted graph contains a spanning tree with scaling distortion $O(\sqrt{1/\epsilon})$. These bounds are tight even for embedding in arbitrary trees. For probabilistic embedding into spanning trees we prove a scaling distortion of $\tilde{O}(\log^2 (1/\epsilon))$, which implies \emph{constant} $\ell_q$-distortion for every fixed $q<\infty$.