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46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05) (2005)
Pittsburgh, Pennsylvania, USA
Oct. 23, 2005 to Oct. 25, 2005
ISBN: 0-7695-2468-0
pp: 83-100
Ittai Abraham , Ittai Abraham
Yair Bartal , Yair Bartal
T-H. Hubert Chan , T-H. Hubert Chan
Kedar Dhamdhere Dhamdhere , Kedar Dhamdhere
Anupam Gupta , Anupam Gupta
Jon Kleinberg , Jon Kleinberg
Ofer Neiman , Ofer Neiman
Aleksandrs Slivkins , Aleksandrs Slivkins
<p>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.</p> <p>Answering an open question of Kleinberg, Slivkins, and Wexler [29], we show that provable guarantees of this type can in fact be achieved in general: any finite metric 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: these is a single embedding into \ell 1 that achieves distortion at most O(\log \frac{1}{\varepsilon }) on all but at most an \varepsilon fraction of distances, simultaneously for all \varepsilon > 0. We extend this with distortion O(\log \frac{1}{\varepsilon }) to maps into general \ell p, p \ge 1 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 \varepsilon-slack embeddings from lower bounds for low-distortion embeddings.</p>

K. D. Dhamdhere et al., "Metric Embeddings with Relaxed Guarantees," 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)(FOCS), Pittsburgh, Pennsylvania, USA, 2005, pp. 83-100.
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