Metric decomposition , smooth measures , and clustering

In recent years, randomized decompositions have become a fundamental tool in the design of algorithms on metric spaces. A randomized decomposition is a distribution over partitions of a metric space satisfying certain properties. Two main types of decompositions have emerged: The “padded” decomposition, see e.g. [20, 15, 9, 19], and the “separating” decomposition, e.g. [2, 3, 5, 14]. Here, we show that if a metric space admits the former type of decomposition, then it also admits the latter. This result froms a nontrivial transformation of distributions. Using this technique, we give new approximation algorithms for the 0-extension problem when the input metric (T, d) on terminals admits such a decomposition. For instance, we achieve an O(1) approximation when the metric on terminals is planar or a subset of R under any norm. We then introduce two additional techniques. The first is based on a powerful theorem about the existence of “well-behaved” probability measures on metric spaces. The second yields improved randomized decompositions for n-point subsets of Lp, 1 < p < 2, by utilizing probabilistic techniques of Marcus and Pisier based on p-stable random variables. Both yield improved approximation algorithms for important classes of metric spaces.