On the optimality of gluing over scales

TL;DR

This paper demonstrates the tightness of the scale-gluing lemma by constructing metric spaces that require high distortion to embed into Euclidean or Lp spaces, disproving a previous conjecture and establishing optimal bounds.

Contribution

It provides the first constructions showing the scale-gluing lemma's bounds are tight across all scales, and extends the results to Lp spaces, confirming the optimality of measured descent techniques.

Findings

Constructed metric spaces with high embedding distortion at all scales.

Disproved the conjecture that the scale-gluing lemma bounds are not tight.

Extended tight bounds to Lp spaces for p > 1.

Abstract

We show that for every $\alpha > 0$, there exist $n$-point metric spaces (X,d) where every "scale" admits a Euclidean embedding with distortion at most $\alpha$, 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 $\alpha = \Theta(1)$ and $\alpha = \Theta(\log n)$, but nowhere in between. More specifically, we exhibit $n$-point spaces with doubling constant $\lambda$ 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 obtain a similar tight result for $L_p$ spaces with $p > 1$.