Abstract
Let (X, d) be a finite semimetric space on n := |X| points. In general, (X, d) cannot be isometrically embedded into some ℓ 1-space. However, (X, d) admits always an embedding into some ℓ 1-space, where the distances are preserved up to a multiplicative factor whose size is of the order log n. This result is due to Bourgain [1985]. We present this result, together with an interesting application due to Linial, London and Rabinovich [1994] for approximating multicommodity flows. We also present a generalization of the negative type condition for Lipschitz ℓ 2-embeddings.
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© 1997 Springer-Verlag Berlin Heidelberg
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Deza, M.M., Laurent, M. (1997). Lipschitz Embeddings. In: Geometry of Cuts and Metrics. Algorithms and Combinatorics, vol 15. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04295-9_10
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DOI: https://doi.org/10.1007/978-3-642-04295-9_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04294-2
Online ISBN: 978-3-642-04295-9
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