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Uncertain Voronoi cell computation based on space decomposition

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Abstract

To facilitate (k)-Nearest Neighbor queries, the concept of Voronoi decomposition is widely used. In this work, we propose solutions to extend the concept of Voronoi-cells to uncertain data. Due to data uncertainty, the location, the shape and the extent of a Voronoi cell are random variables. To facilitate reliable query processing despite the presence of uncertainty, we employ the concept of possible-Voronoi cells and introduce the novel concept of guaranteed-Voronoi cells: The possible-Voronoi cell of an object U consists of all points in space that have a non-zero probability of having U as their nearest-neighbor; and the guaranteed-Voronoi cell, which consists of all points in space which must have U as their nearest-neighbor. Since exact computation of both types of Voronoi cells is computationally hard, we propose approximate solutions. Therefore, we employ hierarchical access methods for both data and object space. Our proposed algorithm descends both index structures simultaneously, constantly trying to prune branches in both trees by employing the concept of spatial domination. To support (k)-Nearest Neighbor queries having k > 1, this work further pioneers solutions towards the computation of higher-order possible and higher-order guaranteed Voronoi cells, which consist of all points in space which may (respectively must) have U as one of their k-nearest neighbors. For this purpose, we develop three algorithms to explore our index structures and show that the approach that descends both index structures in parallel yields the fastest query processing times. Our experiments show that we are able to approximate uncertain Voronoi cells of any order much more effectively than the state-of-the-art while improving run-time performance. Since our approach is the first to compute guaranteed-Voronoi cells and higher order (possible and guaranteed) Voronoi cells, we extend the existing state-of-the-art solutions to these concepts, in order to allow a fair experimental evaluation.

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Notes

  1. 1 The later case can not be guaranteed by the approach of [6] due to the numeric nature of their approach.

  2. 2 We use Euclidean distance in all examples and illustrations, but any L p norm can be applied.

  3. recall that if \(e_{max}^{{\mathcal {D}}}\) corresponds to case 3, then there exists no R ∗-entry such that case 4 holds

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Acknowledgments

Part of the research leading to these results has received funding from the Deutsche Forschungsgemeinschaft (DFG) under grant number RE 266/5-1 and from the DAAD supported by BMBF under grant number 57055388. Reynold Cheng was supported by the Research Grants Council of Hong Kong (RGC Project (HKU 711110)).

Andreas Zufle has been supported by National Science Foundation AitF grant CCF-1637541.

Reynold Cheng was supported by the Research Grants Council of HK (Project HKU 17205115) and HKU (Projects 102009508 and 104004129). We would like to thank the reviewers for their insightful comments.

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Authors and Affiliations

  1. Institute for Informatics, Ludwig-Maximilians-Universität München, München, Germany

    Klaus Arthur Schmid & Tobias Emrich

  2. George Mason University, Fairfax, VA, USA

    Andreas Zufle & Matthias Renz

  3. Department of Computer Science, University of Hong Kong, Pok Fu Lam, Hong Kong

    Reynold Cheng

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  1. Klaus Arthur Schmid
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  2. Andreas Zufle
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  3. Tobias Emrich
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  4. Matthias Renz
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  5. Reynold Cheng
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Correspondence to Klaus Arthur Schmid.

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Schmid, K.A., Zufle, A., Emrich, T. et al. Uncertain Voronoi cell computation based on space decomposition. Geoinformatica 21, 797–827 (2017). https://doi.org/10.1007/s10707-017-0293-2

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  • DOI: https://doi.org/10.1007/s10707-017-0293-2

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