Nearly Tight Low Stretch Spanning Trees

We prove that any graph G with n points has a distribution T over spanning trees such that for any edge (u, v) the expected stretch E
_T~T
[d
_T
(u, nu)/d
_G
(u, nu)] is bounded by Otilde(log n). Our result is obtained via a new approach of building "highways" between portals and a new strong diameter probabilistic decomposition theorem.