Greedy spanners are optimal in doubling metrics

TL;DR

This paper proves that the greedy spanner algorithm produces near-optimal sparse graphs in doubling metrics, extending previous Euclidean space results and simplifying the proof using the packing property.

Contribution

It demonstrates that greedy spanners are optimal in doubling metrics, resolving an open problem and providing a simpler proof framework.

Findings

Greedy spanner constructs a $(1+psilon)$-spanner with weight close to the MST.

The result generalizes Euclidean space findings to doubling metrics.

The proof relies solely on the packing property of doubling metrics.

Abstract

We show that the greedy spanner algorithm constructs a $(1+\epsilon)$-spanner of weight $\epsilon^{-O(d)}w(\mathrm{MST})$ for a point set in metrics of doubling dimension $d$, resolving an open problem posed by Gottlieb. Our result generalizes the result by Narasimhan and Smid who showed that a point set in $d$-dimension Euclidean space has a $(1+\epsilon)$-spanner of weight at most $\epsilon^{-O(d)}w(\mathrm{MST})$. Our proof only uses the packing property of doubling metrics and thus implies a much simpler proof for the same result in Euclidean space.