Finite Volume Spaces and Sparsification

We introduce and study finite d-volumes the high dimensional generalization of finite metric spaces. Having developed a suitable combinatorial machinery, we define l1-volumes and show that they contain Euclidean volumes and hypertree volumes. We show that they can approximate any d-volume with O(n) multiplicative distortion. On the other hand, contrary to Bourgain’s theorem for d = 1, there exists a 2-volume that on n vertices that cannot be approximated by any l1-volume with distortion smaller than Ω̃(n). We further address the problem of l1-dimension reduction in the context of l1 volumes, and show that this phenomenon does occur, although not to the same striking degree as it does for Euclidean metrics and volumes. In particular, we show that any l1 metric on n points can be (1 + ǫ)-approximated by a sum of O(n/ǫ) cut metrics, improving over the best previously known bound of O(n logn) due to Schechtman. In order to deal with dimension reduction, we extend the techniques and ideas introduced by Karger and Benczúr, and Spielman et al. in the context of graph Sparsification, and develop general methods with a wide range of applications. ACM classes: G.2.0.; G.2.1; F.2.2