A tight bound on approximating arbitrary metrics by tree metrics

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 (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. Categories and Subject Descriptors G.2.2 [Discrete Mathematics]: Graph Theory--Graph Algorithms General Terms Algorithms, Theory Keywords Metrics, Embeddings, Tree metrics

• Type: Conference
• Year: 2003
• Where: STOC
• Authors: Jittat Fakcharoenphol, Satish Rao, Kunal Talwar
• Keywords: Algorithms, Log Log N, Point Metric Space, STOC 2003, Tree Metrics