On the Optimality of Gluing over Scales

Abstract

We show that for every α > 0, there exist n-point metric spaces (X,d) where every “scale” admits a Euclidean embedding with distortion at most α, but the whole space requires distortion at least $\Omega(\sqrt{\alpha \log n})$. This shows that the scale-gluing lemma [Lee, SODA 2005] is tight, and disproves a conjecture stated there. This matching upper bound was known to be tight at both endpoints, i.e. when α = Θ(1) and α = Θ(logn), but nowhere in between.

More specifically, we exhibit n-point spaces with doubling constant λ requiring Euclidean distortion $\Omega(\sqrt{\log \lambda \log n})$, which also shows that the technique of “measured descent” [Krauthgamer, et. al., Geometric and Functional Analysis] is optimal. We extend this to L _p spaces with p > 1, where one requires distortion at least Ω((logn)^1/q (logλ)^{1 − 1/q}) when q =  max {p,2}, a result which is tight for every p > 1.

Research partially supported by NSF CCF-0644037.