Abstract
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 \(\varOmega (\sqrt {\alpha\log n})\). This shows that the scale-gluing lemma (Lee in SODA’05: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 92–101, Society for Industrial and Applied Mathematics, Philadelphia, 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(log n), but nowhere in between.
More specifically, we exhibit n-point spaces with doubling constant λ requiring Euclidean distortion \(\varOmega (\sqrt{\log\lambda\log n})\), which also shows that the technique of “measured descent” (Krauthgamer et al. in Geom. Funct. Anal. 15(4):839–858, 2005) is optimal. We extend this to L p spaces with p>1, where one requires distortion at least Ω((log n)1/q(log λ)1−1/q) when q=max {p,2}, a result which is tight for every p>1.
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Jaffe, A., Lee, J.R. & Moharrami, M. On the Optimality of Gluing over Scales. Discrete Comput Geom 46, 270–282 (2011). https://doi.org/10.1007/s00454-011-9359-3
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DOI: https://doi.org/10.1007/s00454-011-9359-3