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 . We present this result, together with an interesting application due to Linial, London and Rabinovich  for approximating multicommodity flows. We also present a generalization of the negative type condition for Lipschitz ℓ 2-embeddings.
KeywordsPolynomial Time Positive Semidefinite Negative Type Multicommodity Flow Linear Programming Duality
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