A tight bound on approximating arbitrary metrics by tree metrics

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.