111 Lemmas 18.9 and 18.10 give:
Theorem 18.11 There exists a -distortion `1-embedding for metric (V d) if and only if there
exists a -approximate cut packing for it. Moreover, the number of non-zero cuts and the dimension of the `1 -embedding are polynomially related.
Corollary 18.12 Metric (V d) is isometrically `1-embeddable i there exists an exact cut packing for it.
We have already shown that metric obtained for the instance in Example 18.3 does not have an exact cut packing, so, it is not isometrically `1 -embeddable. However, we will show that any metric has an O(log n)-distortion `1-embedding this fact lies at the heart of the approximation algorithm for the sparsest cut problem.
Low distortion `1-embeddings for metrics First consider the following one dimensional embedding for metric (V d): pick a set S V , and de ne the coordinate of vertex v to be (v ) = mins S d(s v ), i.e., the length of the shortest edge from v to S . This mapping does not stretch any edge: 2
Lemma 18.13 For the one dimensional embedding given above, 8u v 2 V j (u) ; (v)j d(u v):
Proof : Let s1 and s2 be the closest vertices of S to u and v respectively. W.l.o.g. assume that d(s1 u) d(s2 v ). Then, j (u) ; (v)j = d(s2 v ) ; d(s1 u) d(s2 v ) ; d(s2 u) d(u v ). The last inequality follows by triangle inequality. 2 More generally, consider the following m-dimensional embedding: Pick m subsets of V , S1 : : : Sm, and de ne the ith coordinate of vertex v to be i (v ) = (mins Si d(s v ))=m notice the scaling factor of m used. The additivity of `1 metric, together with Lemma 18.13, imply that this mapping also 2
does not stretch any edge.
Ensuring that a single edge is not over-shrunk The remaining task is to choose the sets in such a way that no edge shrinks by a factor of more than O(log n). It is natural to use randomization for picking the sets. Let us rst ensure that a single edge (u v ) is not over-shrunk. For this purpose, de ne the expected contribution of set Si to the `1 length of edge (u v ) to be E (j i(u) ; i (v )j). For simplicity, assume that n is a power of 2 let n = 2l. For 2 i l + 1, set Si is formed by picking each vertex of V with probability 1=2i. The embedding w.r.t. these sets works for the single edge (u v ), with high probability. The proof of this fact involves cleverly taking into consideration the expected contribution of each set. For di erent metrics, di erent sets have large contribution. In order to develop intuition for the proof, we rst illustrate this through a series of examples. Example 18.14 In the following three metrics, d(u v) = 1. Relative to u and v, the remaining (n ; 2) vertices are placed as shown in the gure below.