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International Conference on Similarity Search and Applications

SISAP 2020: Similarity Search and Applications pp 111–125Cite as

Optimal Metric Search Is Equivalent to the Minimum Dominating Set Problem

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Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 12440))

Abstract

In metric search, worst-case analysis is of little value, as the search invariably degenerates to a linear scan for ill-behaved data. Consequently, much effort has been expended on more nuanced descriptions of what performance might in fact be attainable, including heuristic baselines like the AESA family, as well as statistical proxies such as intrinsic dimensionality. This paper gets to the heart of the matter with an exact characterization of the best performance actually achievable for any given data set and query. Specifically, linear-time objective-preserving reductions are established in both directions between optimal metric search and the minimum dominating set problem, whose greedy approximation becomes the equivalent of an oracle-based AESA, repeatedly selecting the pivot that eliminates the most of the remaining points. As an illustration, the AESA heuristic is adapted to downplay the role of previously eliminated points, yielding some modest performance improvements over the original, as well as its younger relative iAESA2.

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Notes

  1. 1.

    Other measures include the distance exponent  [24] and the ball-overlap factor  [21].

  2. 2.

    Personal communication, July 2012.

  3. 3.

    Note that the reductions are to and from two different versions of the dominating set problem (the directed and undirected version, respectively). At the price of slightly looser bounds, one could stick with just one of these.

  4. 4.

    This is the worst case given that the optimal number of distance computations is some value \(\gamma \), not the more general, non-informative worst-case of \(\Omega (n)\).

  5. 5.

    Optimal kNN with upper bounds does not map as cleanly to dominating sets.

  6. 6.

    The undirected version is most commonly discussed, with a reduction, e.g., from set covering  [13, Th. A.1]. A similar reduction to the directed version is straightforward.

  7. 7.

    In terms of vertices, not edges.

  8. 8.

    Note that only the lower bound is relevant, as the upper bound is always greater than the search radius.

  9. 9.

    If the new distance is allowed to use the original graph as part of its definition, the reduction can be performed in constant time—it is merely a reinterpretation.

  10. 10.

    The upper bound is easily shown by reinterpreting the minimum dominating set problem for a directed graph G = (V, E) as the problem of covering V with the closed out-neighborhoods of G, translating the standard set covering approximation  [26].

  11. 11.

    That is, for any range search instance, there is a directed graph with the objects as its nodes for which the equivalence holds. Reducing in the other direction preserves the objective value, but not necessarily the number of nodes/objects.

  12. 12.

    Chávez et al. say that such independence is a “reasonable approximation”  [5].

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Acknowledgements

The author would like to thank Ole Edsberg, both for discussions providing the initial idea for this paper, and for substantial later input. He would also like to thank Jon Marius Venstad and Bilegsaikhan Naidan for reading early drafts of the paper and providing feedback.

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Authors and Affiliations

  1. Norwegian University of Science and Technology, Trondheim, Norway

    Magnus Lie Hetland

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  1. Magnus Lie Hetland
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Correspondence to Magnus Lie Hetland .

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Editors and Affiliations

  1. National Institute of Informatics, Tokyo, Japan

    Shin'ichi Satoh

  2. ISTI-CNR, Pisa, Italy

    Lucia Vadicamo

  3. University of Southern Denmark, Odense M, Denmark

    Arthur Zimek

  4. ISTI-CNR, Pisa, Italy

    Fabio Carrara

  5. University of Bologna, Bologna, Italy

    Ilaria Bartolini

  6. IT University of Copenhagen, Copenhagen, Denmark

    Martin Aumüller

  7. IT University of Copenhagen, Copenhagen, Denmark

    Björn Þór Jónsson

  8. IT University of Copenhagen, Copenhagen, Denmark

    Rasmus Pagh

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Hetland, M.L. (2020). Optimal Metric Search Is Equivalent to the Minimum Dominating Set Problem. In: Satoh, S., et al. Similarity Search and Applications. SISAP 2020. Lecture Notes in Computer Science(), vol 12440. Springer, Cham. https://doi.org/10.1007/978-3-030-60936-8_9

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  • DOI: https://doi.org/10.1007/978-3-030-60936-8_9

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