Abstract

... as a subgraph a spanning tree into which the edges of G can be embedded with average stretch exp (O ( √ log n log log n)), and that there exists an n-vertex graph G such that all its spanning trees have average stretch Ω(log n). Closing the exponential gap between these upper and lower bounds is listed as one of the long-standing open questions in the area of low-distortion embeddings of metrics (Matousek 2002). We significantly reduce this gap by constructing a spanning tree in G of average stretch O((log n log log n) 2). Moreover, we show that this tree can be constructed in time O(m 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 diagonally dominant linear systems of Spielman and Teng from to m2 (O( √ log n log log n)) log(1/ɛ) m log O(1) n log(1/ɛ), and to O(n(log n log log n) 2 log(1/ɛ)) when the system is planar. Applying a recent reduction of Boman, Hendrickson and Vavasis, this provides an O(n(log n log log n) 2 log(1/ɛ)) time algorithm for solving the linear systems that arise when applying the finite element method to solve twodimensional elliptic partial differential equations. Our result can also be used to improve several earlier approximation algorithms that use low-stretch spanning trees.

Citations