Local embeddings of metric spaces
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Note: On compact representations of All-Pairs-Shortest-Path-Distance matrices
Theoretical Computer Science
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On approximate distance labels and routing schemes with affine stretch
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Spectral sparsification via random spanners
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Approximate distance oracles with improved preprocessing time
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Fast, precise and dynamic distance queries
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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This paper addresses the non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce and construct optimal Ramsey partitions, and use them to show that for every \varepsilon\in (0, 1), any n-point metric space has a subset of size n^{1 - \varepsilon }which embeds into Hilbert space with distortion O(1/\varepsilon). This result is best possible and improves part of the metric Ramsey theorem of Bartal, Linial, Mendel and Naor [5], in addition to considerably simplifying its proof. We use our new Ramsey partitions to design approximate distance oracles with a universal constant query time, closing a gap left open by Thorup and Zwick in [26]. Namely, we show that for any n point metric space X, and k \geqslant1, there exists an O(k)-approximate distance oracle whose storage requirement is O(n^{1 + 1/k} ), and whose query time is a universal constant. We also discuss applications to various other geometric data structures, and the relation to well separated pair decompositions.