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Approximate Nearest Neighbor Search for \(\ell _p\)-Spaces \((2<p<\infty )\) via Embeddings

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LATIN 2018: Theoretical Informatics (LATIN 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10807))

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Abstract

While the problem of approximate nearest neighbor search has been well-studied for Euclidean space and \(\ell _1\), few non-trivial algorithms are known for \(\ell _p\) when \(2<p<\infty \). In this paper, we revisit this fundamental problem and present approximate nearest-neighbor search algorithms which give the best known approximation factor guarantees in this setting.

Y. Bartal is supported in part by an Israel Science Foundation grant #1817/17.

L.-A. Gottlieb is supported in part by an Israel Science Foundation grant #755/15.

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Notes

  1. 1.

    This is a space equipped with a Minkowski norm, which defines the distance between two d-dimensional vectors x, y as \(\Vert x-y\Vert _p = (\sum _{i=1}^d |x_i-y_i|^p)^{1/p}\).

  2. 2.

    If \(d = O(\log n)\) then AVDs may be used, and if \(d = n^{\varOmega (1)}\) then comparing the query point q to each point in V in a brute-force manner can be done in \(O(dn) = d^{O(1)}\) time. (We recall also that there exists an oblivious mapping for all \(\ell _p\) that embeds \(\ell _p^m\) into \(\ell _p^d\) for \(d = {n \atopwithdelims ()2}\) dimensions [5, 15].) We also assume that \(d=2^{o({{\mathrm{ddim}}})}\), as otherwise a constant-factor approximation can be computed in polynomial time [13, 19].

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Acknowledgements

We thank Sariel Har-Peled, Piotr Indyk, Robi Krauthgamer, Assaf Naor and Gideon Schechtman for helpful conversations.

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Authors and Affiliations

  1. Hebrew University, Jerusalem, Israel

    Yair Bartal

  2. Ariel University, Ariel, Israel

    Lee-Ad Gottlieb

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  1. Yair Bartal
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  2. Lee-Ad Gottlieb
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Correspondence to Lee-Ad Gottlieb .

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Editors and Affiliations

  1. Stony Brook University, Stony Brook, New York, USA

    Michael A. Bender

  2. Rutgers University, New Brunswick, New Jersey, USA

    Martín Farach-Colton

  3. Pace University, New York, New York, USA

    Miguel A. Mosteiro

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Bartal, Y., Gottlieb, LA. (2018). Approximate Nearest Neighbor Search for \(\ell _p\)-Spaces \((2<p<\infty )\) via Embeddings. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_10

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  • DOI: https://doi.org/10.1007/978-3-319-77404-6_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-77403-9

  • Online ISBN: 978-3-319-77404-6

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