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Cambridge, Massachusettes

Oct. 11, 2003 to Oct. 14, 2003

ISBN: 0-7695-2040-5

pp: 534

Anupam Gupta , Carnegie Mellon University

Robert Krauthgamer , University of California at Berkeley

James R. Lee , University of California at Berkeley

ABSTRACT

<p>The doubling constant of a metric space (X, d) is the smallest value \lambda such that every ball in X can be covered by \lambda balls of half the radius. The doubling dimension of X is then defined as \dim (X) = \log _2 \lambda. A metric (or sequence of metrics) is called doubling precisely when its doubling dimension is bounded. This is a robust class of metric spaces which contains many families of metrics that occur in applied settings.</p> <p>We give tight bounds for embedding doubling metrics into (low-dimensional) normed spaces. We consider both general doubling metrics, as well as more restricted families such as those arising from trees, from graphs excluding a fixed minor, and from snowflaked metrics. Our techniques include decomposition theorems for doubling metrics, and an analysis of a fractal in the plane due to Laakso [21]. Finally, we discuss some applications and point out a central open question regarding dimensionality reduction in L<sub>2</sub>.</p>

INDEX TERMS

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CITATION

Anupam Gupta,
Robert Krauthgamer,
James R. Lee,
"Bounded Geometries, Fractals, and Low-Distortion Embeddings",

*FOCS*, 2003, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science 2003, pp. 534, doi:10.1109/SFCS.2003.1238226