Metric embeddings with relaxed guarantees

Ittai Abraham, Yair Bartal, T. H.Hubert Chan, Kedar Dhamdhere, Anupam Gupta, Jon Kleinberg, Ofer Neiman, Aleksandrs Slivkins

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

52 Scopus citations

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 [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 l1 which exhibit gracefully degrading distortion: these is a single embedding into l1 that achieves distortion at most O(log 1/ε) on all but at most an e fraction of distances, simultaneously for all ε > 0. We extend this with distortion O(log 1/ε) 1/p to maps into general lp, p ≥ 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 ε-slack embeddings from lower bounds for low-distortion embeddings.

Original languageEnglish
Title of host publicationProceedings - 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005
Pages83-100
Number of pages18
DOIs
StatePublished - 1 Dec 2005
Externally publishedYes
Event46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005 - Pittsburgh, PA, United States
Duration: 23 Oct 200525 Oct 2005

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2005
ISSN (Print)0272-5428

Conference

Conference46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005
Country/TerritoryUnited States
CityPittsburgh, PA
Period23/10/0525/10/05

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