Distance scales , embeddings , and metrics of negative type ∗

We introduce a new number of new techniques for the construction of low-distortion embeddings of a finite metric space. These include a generic Gluing Lemma which avoids the overhead typically incurred from the näıve concatenation of maps for different scales of a space. We also give a significantly 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 offers a simple hyperplane rounding algorithm for the computation of an O( √ log n)-approximation to the Sparsest Cut problem with uniform demands, and has a number of other applications to embeddings and approximation algorithms.