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Nearest-Neighbor Searching Under Uncertainty I

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Abstract

Nearest-neighbor queries, which ask for returning the nearest neighbor of a query point in a set of points, are important and widely studied in many fields because of a wide range of applications. In many of these applications, such as sensor databases, location based services, face recognition, and mobile data, the location of data is imprecise. We therefore study nearest-neighbor queries in a probabilistic framework in which the location of each input point and/or query point is specified as a probability density function and the goal is to return the point that minimizes the expected distance, which we refer to as the expected nearest neighbor (\(\mathop {\mathrm {ENN}}\)). We present methods for computing an exact \(\mathop {\mathrm {ENN}}\) or an \(\varepsilon \)-approximate \(\mathop {\mathrm {ENN}}\), for a given error parameter \(0<\varepsilon < 1\), under different distance functions. These methods build a data structure of near-linear size and answer \(\mathop {\mathrm {ENN}}\) queries in polylogarithmic or sublinear time, depending on the underlying function. As far as we know, these are the first nontrivial methods for answering exact or \(\varepsilon \)-approximate \(\mathop {\mathrm {ENN}}\) queries with provable performance guarantees. Moreover, we extend our results to answer exact or \(\varepsilon \)-approximate k-\(\mathop {\mathrm {ENN}}\) queries.

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Notes

  1. If the location of data is precise, we refer to it as certain.

  2. The squared Euclidean distance between two points \(p, q \in {\mathbb {R}}^d\) is \(\Vert p-q \Vert ^2\) where \(\Vert \cdot \Vert \) is the \(L_2\) metric.

  3. We note that we are assuming a model of computation in which some basic primitive operations on functions of constant-description complexity (e.g. Gaussian distribution) can be performed in O(1) time.

  4. If there are multiple points of \(\mathcal {P}\) at the ith smallest expected distance from Q, then we break ties arbitrarily and choose \(\varphi _i(\mathcal {P}, Q)\) to be one of these points. We do the same for \(\varphi (\mathcal {P},Q)\).

  5. For simplicity, we focus on P having a continuous pdf; the argument holds for a discrete pdf as well.

  6. If a square \(\square \) appears in multiple \(\mathcal {B}_i\)’s, we keep only one copy of \(\square \) in \(\mathcal {B}_{\mathrm {in}}\) and \(\delta _{\square }\) is the minimum of the values associated with the different copies of \(\square \).

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Acknowledgements

The work of P. Agarwal and W. Zhang was supported by NSF under Grants CCF-09-40671, CCF-10-12254, and CCF-11-61359, by ARO Grants W911NF-07-1-0376 and W911NF-08-1-0452, and by an ERDC contract W9132V-11-C-0003. The work of A. Efrat and S. Sankararaman was supported by NSF CAREER Grant 0348000.

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Authors and Affiliations

  1. Department of Computer Science, Duke University, P.O. Box 90129, Durham, NC, 27708-0129, USA

    Pankaj K. Agarwal

  2. Department of Computer Science, University of Arizona, P.O. Box 210077, Tucson, AZ, 85721-0077, USA

    Alon Efrat

  3. Akamai Technologies, 150 Broadway, Cambridge, MA, 02142, USA

    Swaminathan Sankararaman

  4. Apple Inc., 1 Infinite Loop, Cupertino, CA, 95014, USA

    Wuzhou Zhang

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  1. Pankaj K. Agarwal
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  2. Alon Efrat
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  3. Swaminathan Sankararaman
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  4. Wuzhou Zhang
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Correspondence to Swaminathan Sankararaman.

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Editor in Charge: Kenneth Clarkson

A preliminary version appeared as “Nearest-neighbor searching under uncertainty, Proc. 31st ACM Sympos. Principles of Database Systems, 2012, 225–236”.

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Agarwal, P.K., Efrat, A., Sankararaman, S. et al. Nearest-Neighbor Searching Under Uncertainty I. Discrete Comput Geom 58, 705–745 (2017). https://doi.org/10.1007/s00454-017-9903-x

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