*In which we prove Bourgain’s theorem.*

Today we prove the following theorem.

Theorem 1 (Bourgain)Let be a semimetric defined over a finite set . Then there exists a mapping such that, for every two elements ,

where is an absolute constant. Given , the mapping can be found with high probability in randomized polynomial time in .

Together with the results that we proved in the last lecture, this implies that an optimal solution to the Leighton-Rao relaxation can be rounded to an -approximate solution to the sparsest cut problem. This was the best known approximation algorithm for sparsest cut for 15 years, until the Arora-Rao-Vazirani algorithm, which will be our next topic.