On the Optimality of Gluing over Scales

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 Ω( √ α 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 Ω( √ log λ logn), which also shows that the technique of “measured descent” [Krauthgamer, et. al., Geometric and Functional Analysis] is optimal. We extend this to Lp spaces with p > 1, where one requires distortion at least Ω((log n)(log λ)) when q = max{p, 2}, a result which is tight for every p > 1.