Lower-Stretch Spanning Trees

Michael Elkin, Yuval Emek, Daniel A. Spielman, Shang Hua Teng

Research output: Contribution to journalArticlepeer-review

91 Scopus citations


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 n log log n). Moreover, we show that this tree can be constructed in time O (m log n + n log2 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 m2(√ to m logO(1) n, and to O(n log2 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.

Original languageEnglish
Pages (from-to)608-628
Number of pages21
JournalSIAM Journal on Computing
Issue number2
StatePublished - 7 Nov 2008


  • Low-distortion embeddings
  • Low-stretch spanning trees
  • Probabilistic tree metrics

ASJC Scopus subject areas

  • General Computer Science
  • General Mathematics


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