A Graph-Theoretic Game and its Application to the $k$-Server Problem
SIAM Journal on Computing
Rounding via trees: deterministic approximation algorithms for group Steiner trees and k-median
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
On approximating arbitrary metrices by tree metrics
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
A polylogarithmic approximation algorithm for the group Steiner tree problem
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms for the 0-extension problem
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Embedding k-outerplanar graphs into ℓ1
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
A tight bound on approximating arbitrary metrics by tree metrics
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Approximating a Finite Metric by a Small Number of Tree Metrics
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Probabilistic approximation of metric spaces and its algorithmic applications
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Algorithmic Applications of Low-Distortion Geometric Embeddings
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Cuts, Trees and ℓ1-Embeddings of Graphs*
Combinatorica
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Approximation algorithms for conflict-free vehicle routing
ESA'11 Proceedings of the 19th European conference on Algorithms
Low distortion delaunay embedding of trees in hyperbolic plane
GD'11 Proceedings of the 19th international conference on Graph Drawing
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Over the past decade, numerous algorithms have been developed using the fact that the distances in any n-point metric (V, d) can be approximated to within O(log n) by distributions D over trees on the point set V [3, 10]. However, when the metric (V, d) is the shortest-path metric of an edge weighted graph G = (V, E), a natural requirement is to obtain such a result where the support of the distribution D is only over subtrees of G. For a long time, the best result satisfying this stronger requirement was a exp {√log n log log n} distortion result of Alon et al. [1]. In a recent breakthrough, Elkin et al. [9] improved the distortion to O(log2 n log log n). (The best lower bound on the distortion is Ω(log n), say, for the n-vertex grid [1].)In this paper, we give a construction that improves the distortion to O(log2 n), improving slightly on the EEST construction. The main contribution of this paper is in the analysis: we use an algorithm which is similar to one used by EEST to give a distortion of O(log3 n), but using a new probabilistic analysis, we eliminate one of the logarithmic factors. The ideas and techniques we use to obtain this logarithmic improvement seem orthogonal to those used earlier in such situations---e.g., Seymour's decomposition scheme [4, 9] or the cutting procedures of CKR/FRT [5, 10], both which do not seem to give a guarantee of better than O(log2 n log log n) for this problem. We hope that our ideas (perhaps in conjunction with some of these others) will ultimately lead to an O(log n) distortion embedding of graph metrics into distributions over their spanning trees.