Bounded Geometries, Fractals, and Low-Distortion Embeddings

The doubling constant of a metric space (X, d) is the smallest value λ such that every ball in X can be covered by λ balls of half the radius. The doubling dimension of X is then defined as dim(X) = log2 λ. 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. 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 [20]. Finally, we discuss some applications and point out a central open question regarding dimensionality reduction in L2.

• Type: Conference
• Year: 2003
• Where: FOCS
• Authors: Anupam Gupta, Robert Krauthgamer, James R. Lee
• Keywords: Doubling Metrics, FOCS 2003, General Doubling Metrics, Metric Spaces, Theoretical Computer Science