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(sqrt{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.