NDLI: Metric Embeddings with Relaxed Guarantees

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Gupta, Anupam Slivkins, Aleksandrs Kleinberg, Jon Dhamdhere, Kedar Chan, T.-H. Hubert
Copyright Year	2009
Abstract	We consider the problem of embedding finite metrics with slack: We seek to produce embeddings with small dimension and distortion while allowing a (small) constant fraction of all distances to be arbitrarily distorted. This definition is motivated by recent research in the networking community, which achieved striking empirical success at embedding Internet latencies with low distortion into low-dimensional Euclidean space, provided that some small slack is allowed. Answering an open question of Kleinberg, Slivkins, and Wexler [in Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, 2004], we show that provable guarantees of this type can in fact be achieved in general: Any finite metric space can be embedded, with constant slack and constant distortion, into constant-dimensional Euclidean space. We then show that there exist stronger embeddings into $\ell_1$ which exhibit gracefully degrading distortion: There is a single embedding into $\ell_1$ that achieves distortion at most $O(\log\frac{1}{\epsilon})$ on all but at most-1.5pt an $\epsilon$ fraction of distances simultaneously for all $\epsilon>0$. We extend this with distortion1pt $O(\log\frac{1}{\epsilon})^{1/p}$ to maps into general $\ell_p$, $p\geq1$, for several classes of metrics, including those with bounded doubling dimension and those arising from the shortest-path metric of a graph with an excluded minor. Finally, we show that many of our constructions are tight and give a general technique to obtain lower bounds for $\epsilon$-slack embeddings from lower bounds for low-distortion embeddings.
Starting Page	2303
Ending Page	2329
Page Count	27
File Format	PDF
ISSN	00975397
DOI	10.1137/060670511
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	6
Volume Number	38
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2009-03-04
Access Restriction	Subscribed
Subject Keyword	Analysis of algorithms and problem complexity randomized algorithms metric decompositions Randomized algorithms Embedding metric spaces low-distortion embeddings Analysis of algorithms metric embeddings
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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