TY - GEN

T1 - Metric embeddings with relaxed guarantees

AU - Abraham, Ittai

AU - Bartal, Yair

AU - Chan, T. H.Hubert

AU - Dhamdhere, Kedar

AU - Gupta, Anupam

AU - Kleinberg, Jon

AU - Neiman, Ofer

AU - Slivkins, Aleksandrs

PY - 2005/12/1

Y1 - 2005/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33748589199&partnerID=8YFLogxK

U2 - 10.1109/SFCS.2005.51

DO - 10.1109/SFCS.2005.51

M3 - Conference contribution

AN - SCOPUS:33748589199

SN - 0769524680

SN - 9780769524689

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 83

EP - 100

BT - Proceedings - 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005

T2 - 46th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2005

Y2 - 23 October 2005 through 25 October 2005

ER -