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Continuous nearest-neighbor queries with location uncertainty

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Abstract

In this paper, we consider the problem of evaluating the continuous query of finding the \(k\) nearest objects with respect to a given point object \(O_{q}\) among a set of \(n\) moving point-objects. The query returns a sequence of answer-pairs, namely pairs of the form \((I,\, S)\) such that \(I\) is a time interval and \(S\) is the set of objects that are closest to \(O_{q}\) during \(I\). When there is uncertainty associated with the locations of the moving objects, \(S\) is the set of all the objects that are possibly the \(k\) nearest neighbors. We analyze the lower bound and the upper bound on the maximum number of answer-pairs, for the certain case and the uncertain case, respectively. Then, we consider two different types of algorithms. The first is off-line algorithms that compute a priori all the answer-pairs. The second type is on-line algorithms that at any time return the current answer-pair. We present algorithms for the certain case and the uncertain case, respectively, and analyze their complexity. We experimentally compare different algorithms using a database of 1 million objects derived from real-world GPS traces.

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Notes

  1. Better complexity is known for kinetic heap when the distance functions are pseudo-lines (i.e., any pair of distance functions intersect each other at most once) (see [6]). In our problem, the square-distance functions are parabolas. Observe that each parabola can be cut into two pseudo-lines. However, this will make the square-distance functions partially defined, whereas the analysis in [6] assumes totally defined functions.

  2. In the computational geometry literature, the level of a point \(p\) is defined to be the number of curves lying strictly below \(p\). In this paper, we define the level to be the number of curves not above \(p\), so that the \(k\)-level corresponds to the \(k\)-th nearest neighbor. But this definition is purely terminological without affecting any results.

  3. See [7] for the definition of Ackermann’s function.

  4. Strictly speaking, the minimum possible distance is zero when \(d_{i}(t)-r_{i}\le 0\), which occurs when the location of \(\hbox {O}_{\mathrm{q}}\) at t falls into the uncertainty region \(\hbox {U}_{\mathrm{i}}\hbox {(t)}\). In this case, the min-curve is a parabola truncated by the horizontal line y=0. However, for simplicity of presentation, we assume that \(d_{i}(t)-r_{i}\) is greater than zero for all values of t. But all the results in this paper hold for the case where \(d_{i}(t)-r_{i}\) may be smaller than or equal to zero.

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Acknowledgments

This research was supported in part by the US Department of Transportation National University Rail Center (NURAIL); Illinois Department of Transportation (METSI); and National Science Foundation grants IIS-1213013, CCF-1216096, DGE-0549489, IIP-1315169, CCF-0916438, CNS-1035914, CCF-1319754, CNS-1314485.

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Authors and Affiliations

  1. Department of Computer Science, University of Illinois at Chicago, Chicago, IL, USA

    A. Prasad Sistla & Ouri Wolfson

  2. HERE, Chicago, IL, USA

    Bo Xu

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  1. A. Prasad Sistla
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Correspondence to Bo Xu.

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Sistla, A.P., Wolfson, O. & Xu, B. Continuous nearest-neighbor queries with location uncertainty. The VLDB Journal 24, 25–50 (2015). https://doi.org/10.1007/s00778-014-0361-2

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