Nearly Tight Low Stretch Spanning Trees

TL;DR

This paper introduces a method to construct low-stretch spanning trees with nearly tight bounds, using a novel approach involving highway construction and a strong diameter decomposition, improving graph spanning tree efficiency.

Contribution

It presents a new probabilistic approach for building low-stretch spanning trees with nearly optimal bounds, utilizing highway networks and a strong diameter decomposition.

Findings

Expected stretch bounded by  O(   log n) for any graph.

New probabilistic decomposition theorem for strong diameter.

Method applicable to all graphs with n points.

Abstract

We prove that any graph $G$ with $n$ points has a distribution $\mathcal{T}$ over spanning trees such that for any edge $(u,v)$ the expected stretch $E_{T \sim \mathcal{T}}[d_T(u,v)/d_G(u,v)]$ is bounded by $\tilde{O}(\log n)$. Our result is obtained via a new approach of building ``highways'' between portals and a new strong diameter probabilistic decomposition theorem.