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Let (X,d X ) be an n-point metric space. We show that there exists a distribution over non-contractive embeddings into trees f:Xi¾?Tsuch that for every xi¾? X, where Cis a universal constant. Conversely we show that the above quadratic dependence on logncannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location.

Maximum Gradient Embeddings and Monotone Clustering