 
Summary: On the optimality of gluing over scales
Alex Ja#e # James R. Lee # Mohammad Moharrami #
Abstract
We show that for every # > 0, there exist npoint metric spaces (X, d) where every ``scale''
admits a Euclidean embedding with distortion at most #, but the whole space requires distortion
at
least# # # log n). This shows that the scalegluing 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 # = #(1) and # = #(log n), but nowhere in between.
More specifically, we exhibit npoint spaces with doubling constant # requiring Euclidean dis
tortion# # log # 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 #ast n) 1/q (log #) 11/q ) when q = max{p, 2}, a result
which is tight for every p > 1.
1 Introduction
Suppose one is given a collection of mappings from some finite metric space (X, d) into a Euclidean
space, each of which reflects the geometry at some ``scale'' of X. Is there a nontrivial way of
gluing these mappings together to form a global mapping which reflects the entire geometry of X?
The answers to such questions have played a fundamental role in the bestknown approximation
algorithms for Sparsest Cut [6, 9, 4, 1] and Graph Bandwidth [16, 6, 10], and have found applications
