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.
Categories: cs.CG