Greedy spanners are optimal in 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.

15 pages, 2 figures, submitted to SoCG 2018