A light metric spanner

Abstract

It has long been known that $d$-dimensional Euclidean point sets admit $(1+\epsilon)$-stretch spanners with lightness $W_E = \epsilon^{-O(d)}$, that is total edge weight at most $W_E$ times the weight of the minimum spaning tree of the set [DHN93]. Whether or not a similar result holds for metric spaces with low doubling dimension has remained an important open problem, and has resisted numerous attempts at resolution. In this paper, we resolve the question in the affirmative, and show that doubling spaces admit $(1+\epsilon)$-stretch spanners with lightness $W_D = (ddim/\epsilon)^{O(ddim)}$. Important in its own right, our result also implies a much faster polynomial-time approximation scheme for the traveling salesman problemin doubling metric spaces, improving upon the bound presented in [BGK-12].