In Section 8.5, we introduced the idea of approximating one kind of metric with another one; namely, we looked at the idea of approximating a general metric with a tree metric. Here we will consider approximating a general metric with another metric more general than a tree metric, namely, an ℓ1-embeddable metric. We show that we can approximate any metric (V, d) with an ℓ1-embeddable metric with distortion O(log n), where n = |V|. The ℓ1-embeddable metrics have a particularly close connection to cuts; we show that any such metric is a convex combination of the cut semimetrics we discussed at the beginning of Chapter 8. We show that the low-distortion embeddings into ℓ1-embeddable metrics have applications to cut problems by giving an approximation algorithm for the sparsest cut problem.

In Section 15.2, we give an algorithm that finds a packing of trees called cut-trees into a graph; this packing allows us to solve a particular routing problem. In the subsequent section, we show that the cut-tree packing can be used in a way analogous to the probabilistic approximation of metrics by tree metrics in Section 8.5. In that section, we showed that given an algorithm to solve a problem on a tree metric, we could provide an approximate solution for the problem in a general metric with only an additional factor of O(log n) in the performance guarantee.