Distance scales, embeddings, and metrics of negative type ⁄ (preliminary draft)

We introduce a new number of new techniques for the construction of low-distortion embeddings of a flnite metric space. These include a generic Gluing Lemma which avoids the overhead typically incurred from the na˜‡ve concatenation of maps for difierent scales of a space. We also give a signiflcantly improved and quantitatively optimal version of the main structural theorem of Arora, Rao, and Vazirani on separated sets in metrics of negative type. The latter result ofiers a simple hyperplane rounding algorithm for the computation of an O( p logn)-approximation to the Sparsest Cut problem with uniform demands, and has a number of other applications to embeddings and approximation algorithms.

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