Lower-Stretch Spanning Trees

We show that every weighted connected graph $G$ contains as a subgraph a spanning tree into which the edges of $G$ can be embedded with average stretch $O (\log^{2} n \log \log n)$. Moreover, we show that this tree can be constructed in time $O (m \log n + n \log^2 n)$ in general, and in time $O (m \log n)$ if the input graph is unweighted. The main ingredient in our construction is a novel graph decomposition technique. Our new algorithm can be immediately used to improve the running time of the recent solver for symmetric diagonally dominant linear systems of Spielman and Teng from $ m 2^{(O (\sqrt{\log n\log\log n})) }$ to $m \log^{O (1)}n$, and to $O ( n \log^{2} n \log \log n)$ when the system is planar. Our result can also be used to improve several earlier approximation algorithms that use low-stretch spanning trees.