Metric Embeddings with Relaxed Guarantees

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 1 which exhibit gracefully degrading distortion: these is a single embedding into 1 that achieves distortion at most O(log 1 ) on all but at most an fraction of distances, simultaneously for all &g...

• Type: Conference
• Year: 2005
• Where: FOCS
• Authors: Ittai Abraham, Yair Bartal, Hubert T.-H. Chan, Kedar Dhamdhere, Anupam Gupta, Jon M. Kleinberg, Ofer Neiman, Aleksandrs Slivkins
• Keywords: FOCS 2005, Theoretical Computer Science