A tight bound on approximating arbitrary metrics by tree metrics

  • Authors:
  • Jittat Fakcharoenphol;Satish Rao;Kunal Talwar

  • Affiliations:
  • Kasetsart University, Bangkok, Thailand and University of California, Berkeley;Computer Science Division, University of California, Berkeley, CA;Computer Science Division, University of California, Berkeley, CA

  • Venue:
  • Journal of Computer and System Sciences - Special issue: STOC 2003
  • Year:
  • 2004

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Abstract

In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion Ω(log n)-distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buy-at-bulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.