## Near-optimal (euclidean) metric compression

##### Author(s)

Indyk, Piotr; Wagner, Tal
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he metric sketching problem is defined as follows. Given a metric on n points, and ϵ > 0, we wish to produce a small size data structure (sketch) that, given any pair of point indices, recovers the distance between the points up to a 1 + ϵ distortion. In this paper we consider metrics induced by l2 and l1 norms whose spread (the ratio of the diameter to the closest pair distance) is bounded by Φ > 0. A well-known dimensionality reduction theorem due to Johnson and Lindenstrauss yields a sketch of size O(ϵ[superscript −2] log(Φn)n log n), i.e., O(ϵ[superscript −2[] log(Φn)n log n) bits per point. We show that this bound is not optimal, and can be substantially improved to O(ϵ[superscript −2] log(1/ϵ) · log n + log log Φ) bits per point. Furthermore, we show that our bound is tight up to a factor of log(1/ϵ).
We also consider sketching of general metrics and provide a sketch of size O(n log(1/ϵ) + log log Φ) bits per point, which we show is optimal.

##### Date issued

2017-01##### Department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science##### Journal

SODA '17 Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

##### Publisher

Association for Computing Machinery

##### Citation

Indyk, Piotr and Tal Wagner. "Near-optimal (euclidean) metric compression." SODA '17 Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, 16-19 September, 2017, Barcelona, Spain, Association for Computing Machinery, 2017.

Version: Original manuscript