Using petal-decompositions to build a low stretch spanning tree

We prove that any weighted graph G = (V, E, w) with n points and m edges has a spanning tree T such that ^\sum _\{ u,v\} \in E ^dT (u,v⁾ = O(m log n log log n). Moreover, such a tree can w(u,v) be found in time O(m log n log log n). Our result is obtained using our new petal-decomposition approach which guarantees that the radius of each cluster in the tree is at most four times the radius of the induced subgraph of the cluster in the original graph.