An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm
SIAM Journal on Computing
Small distortion and volume preserving embeddings for planar and Euclidean metrics
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Lectures on Discrete Geometry
Bounded Geometries, Fractals, and Low-Distortion Embeddings
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Euclidean distortion and the sparsest cut
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
On distance scales, embeddings, and efficient relaxations of the cut cone
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Embeddings of negative-type metrics and an improved approximation to generalized sparsest cut
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Volume Distortion for Subsets of Euclidean Spaces
Discrete & Computational Geometry
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We show that for every *** 0, there exist n -point metric spaces (X ,d ) where every "scale" admits a Euclidean embedding with distortion at most *** , 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 *** = ***(1) and *** = ***(logn ), but nowhere in between. More specifically, we exhibit n -point spaces with doubling constant *** 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 ***((logn )1/q (log*** )1 *** 1/q ) when q = max {p ,2}, a result which is tight for every p 1.