An optimal on-line algorithm for metrical task system
Journal of the ACM (JACM)
A polylog(n)-competitive algorithm for metrical task systems
STOC '97 Proceedings of the twenty-ninth 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 threshold of ln n for approximating set cover
Journal of the ACM (JACM)
Better algorithms for unfair metrical task systems and applications
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
A polylogarithmic approximation algorithm for the group Steiner tree problem
Journal of Algorithms
Integrality ratio for group Steiner trees and directed steiner trees
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
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
FOCS '01 Proceedings of the 42nd IEEE 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
Integrality ratio for group Steiner trees and directed steiner trees
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
On metric ramsey-type phenomena
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Polylogarithmic inapproximability
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
A tight bound on approximating arbitrary metrics by tree metrics
Journal of Computer and System Sciences - Special issue: STOC 2003
Ramsey-type theorems for metric spaces with applications to online problems
Journal of Computer and System Sciences - Special issue on FOCS 2001
Embedding metrics into ultrametrics and graphs into spanning trees with constant average distortion
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
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Metric embeddings have become a frequent tool in the design of algorithms. The applicability is often dependent on how high the embedding's distortion is. For example embedding into ultrametrics (or arbitrary trees) requires linear distortion. Using probabilistic metric embeddings, the bound reduces to O(log nlog logn). Yet, the lower bound is still logarithmic.We make a step further in the direction of bypassing this difficulty. We define "multi-embeddings" of metric spaces where a point is mapped onto a set of points, while keeping the target metric being of polynomial size and preserving the distortion of paths. The distortion obtained with such multi-embeddings into ultrametrics is at most O(log Δ log log Δ) where δ is the (normalized) diameter, and probabilistically O(log n log log log n). In particular, for expander graphs, we are able to obtain constant distortions embeddings into trees vs. the Ω(logn) lower bound for all previous notions of embeddings.We demonstrate the algorithmic application of the new embeddings by obtaining improvements for two well-known problems: group Steiner tree and metrical task systems.