Steiner points in tree metrics don't (really) help
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Bounded Geometries, Fractals, and Low-Distortion Embeddings
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Expander flows, geometric embeddings and graph partitioning
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Approximating the cut-norm via Grothendieck's inequality
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Euclidean distortion and the sparsest cut
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Improved approximation algorithms for minimum-weight vertex separators
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
O(√log n) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Vertex cuts, random walks, and dimension reduction in series-parallel graphs
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Approximation algorithms for embedding general metrics into trees
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Circular partitions with applications to visualization and embeddings
Proceedings of the twenty-fourth annual symposium on Computational geometry
Markov convexity and local rigidity of distorted metrics
Proceedings of the twenty-fourth annual symposium on Computational geometry
How to complete a doubling metric
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Inapproximability for planar embedding problems
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
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We give combinatorial, geometric, and probabilistic characterizations of the distortion of tree metrics into Lp spaces. This requires the development of new embedding techniques, as well as a method for proving distortion lower bounds which is based on the wandering of Markov chains in Banach spaces, and a new metric invariant we call Markov convexity. Trees are thus the first non-trivial class of metric spaces for which one can give a simple and complete characterization of their distortion into a Hilbert space, up to universal constants. Our results also yield an efficient algorithm for constructing such embeddings.