, Volume 21, Issue 4, pp 797–827 | Cite as

Uncertain Voronoi cell computation based on space decomposition

  • Klaus Arthur Schmid
  • Andreas Zufle
  • Tobias Emrich
  • Matthias Renz
  • Reynold Cheng


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.


Uncertain data kNN query Nearest neighbor query Voronoi cells Voronoi decomposition 



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.


  1. 1.
    Chow CY, Mokbel MF, Aref WG (2009) Casper*: query processing for location services without compromising privacy. ACM TODS 34(4):24CrossRefGoogle Scholar
  2. 2.
    Beskales G, Soliman MA, IIyas IF (2008) Efficient search for the top-k probable nearest neighbors in uncertain databases. Proc VLDB Endowment 1(1):326–339CrossRefGoogle Scholar
  3. 3.
    Cheng R, Xie X, Yiu ML, Chen J, Sun L (2010) Uv-diagram: a voronoi diagram for uncertain data. In: ICDE, IEEE, pp 796–807Google Scholar
  4. 4.
    Ali ME, Tanin E, Zhang R, Kotagiri R (2012) Probabilistic voronoi diagrams for probabilistic moving nearest neighbor queries. DKE 75:1–33CrossRefGoogle Scholar
  5. 5.
    Bernecker T, Emrich T, Kriegel HP, Mamoulis N, Renz M, Züfle A (2011) A novel probabilistic pruning approach to speed up similarity queries in uncertain databases. In: Proceedings of the 27th international conference on data engineering (ICDE). Hannover, pp 339–350Google Scholar
  6. 6.
    Zhang P, Cheng R, Mamoulis N, Renz M, Züfle A, Tang Y, Emrich T (2013) Voronoi-based nearest neighbor search for multi-dimensional uncertain databases. In: Proceedings of the 29th International conference on data engineering (ICDE). Brisbane, pp 158–169Google Scholar
  7. 7.
    Yuan J, Zheng Y, Zhang C, Xie W, Xie X, Sun G, Huang Y (2010) T-drive: Driving directions based on taxi trajectories. In: SIGSPATIAL, pp 99–108Google Scholar
  8. 8.
    Yuan J, Zheng Y, Xie X, Sun G (2011) Driving with knowledge from the physical world. In: SIGKDD, pp 316–324Google Scholar
  9. 9.
    Niedermayer J, Züfle A, Emrich T, Renz M, Mamoulis N, Chen L, Kriegel HP (2013) Probabilistic nearest neighbor queries on uncertain moving object trajectories. Proc VLDB Endowment 7(3):205–216CrossRefGoogle Scholar
  10. 10.
    Emrich T, Kriegel HP, Kröger P, Renz M, Züfle A (2009) Incremental reverse nearest neighbor ranking in vector spaces. In: Proceedings of the 11th International symposium on spatial and temporal databases (SSTD). Aalborg, pp 265–282Google Scholar
  11. 11.
    Beckmann N, Kriegel HP, Schneider R, Seeger B (1990) The R*-Tree: An efficient and robust access method for points and rectangles. In: Proceedings of the ACM International conference on management of data (SIGMOD). Atlantic City, pp 322–331Google Scholar
  12. 12.
    Orenstein JA, Merrett TH (1984) A class of data structures for associative searching. In: ACM SIGACT-SIGMOD. ACM, pp 181–190Google Scholar
  13. 13.
    Schmid KA, Emrich T, Züfle A, Renz M, Cheng R (2015) Approximate uv computation based on space decomposition. In: Proceedings of the 14th International symposium on spatial and temporal databases (SSTD). Hong KongGoogle Scholar
  14. 14.
    Cheng R, Kalashnikov DV, Prabhakar S (2004) Querying imprecise data in moving object environments. In: IEEE Transactions on knowledge and data engineeringGoogle Scholar
  15. 15.
    Li J, Saha B, Deshpande A (2009) A unified approach to ranking in probabilistic databases. Proc VLDB Endowment 2(1):502–513CrossRefGoogle Scholar
  16. 16.
    Bernecker T, Kriegel HP, Mamoulis N, Renz M, Züfle A (2010) Scalable probabilistic similarity ranking in uncertain databases. IEEE Trans Knowl Data Eng 22(9):1234–1246CrossRefGoogle Scholar
  17. 17.
    Cheema MA, Lin X, Wang W, Zhang W, Pei J (2010) Probabilistic reverse nearest neighbor queries on uncertain data. IEEE Trans Knowl Data Eng 22 (4):550–564CrossRefGoogle Scholar
  18. 18.
    Aurenhammer F (1991) Voronoi diagrams-a survey of a fundamental geometric data structure. ACM CSUR 23(3):345–405CrossRefGoogle Scholar
  19. 19.
    Sharifzadeh M, Shahabi C (2010) Vor-tree: R-trees with voronoi diagrams for efficient processing of spatial nearest neighbor queries. VLDB Endowment 3 (1–2):1231–1242CrossRefGoogle Scholar
  20. 20.
    Zheng B, Xu J, Lee WC, Lee L (2006) Grid-partition index: a hybrid method for nearest-neighbor queries in wireless location-based services. VLDB J 15 (1):21–39CrossRefGoogle Scholar
  21. 21.
    Nutanong S, Zhang R, Tanin E, Kulik L (2008) The v*-diagram: a query-dependent approach to moving knn queries. VLDB Endowment 1(1):1095–1106CrossRefGoogle Scholar
  22. 22.
    Sharifzadeh M, Shahabi C (2009) Approximate voronoi cell computation on spatial data streams. VLDB J 18(1):57–75CrossRefGoogle Scholar
  23. 23.
    Akdogan A, Demiryurek U, Banaei-Kashani F, Shahabi C (2010) Voronoi-based geospatial query processing with mapreduce. In: IEEE CloudCom. IEEE, pp 9–16Google Scholar
  24. 24.
    Kolahdouzan M, Shahabi C (2004) Voronoi-based k nearest neighbor search for spatial network databases. In: VLDB Endowment, VLDB Endowment, pp 840–851Google Scholar
  25. 25.
    Roussopoulos N, Kelley S, Vincent F (1995) Nearest neighbor queries. In: ACM SIGMOD, vol 24. ACM, pp 71–79Google Scholar
  26. 26.
    Emrich T, Kriegel HP, Kröger P, Renz M, Züfle A (2010) Boosting spatial pruning: on optimal pruning of MBRs. In: Proceedings of the ACM international conference on management of data (SIGMOD). Indianapolis, pp 39–50Google Scholar
  27. 27.
    Hjaltason GR, Samet H (1995) Ranking in spatial databases. In: Proceedings of the 4th international symposium on large spatial databases (SSD). Portland, pp 83–95Google Scholar
  28. 28.
    Ṡaltenis S, Jensen CS, Leutenegger ST, Lopez MA (2000) Indexing the positions of continuously moving objects, vol 29. ACMGoogle Scholar
  29. 29.
    Zhang M, Chen S, Jensen CS, Ooi BC, Zhang Z (2009) Effectively indexing uncertain moving objects for predictive queries. Proc VLDB Endowment 2 (1):1198–1209CrossRefGoogle Scholar
  30. 30.
    Emrich T, Kriegel HP, Mamoulis N, Renz M, Züfle A (2012) Indexing uncertain spatio-temporal data. In: Proceedings of the 21st ACM conference on information and knowledge management (CIKM). Maui, pp 395–404Google Scholar
  31. 31.
    Lian X, Chen L (2009) Efficient processing of probabilistic reverse nearest neighbor queries over uncertain data. VLDB J Int J Very Large Data Bases 18(3):787–808CrossRefGoogle Scholar
  32. 32.
    Emrich T, Graf F, Kriegel HP, Schubert M, Thoma M (2010) Optimizing all-nearest-neighbor queries with trigonometric pruning. In: Proceedings of the 22nd international conference on scientific and statistical database management (SSDBM). Heidelberg, pp 501–518Google Scholar
  33. 33.
    Achtert E, Kriegel HP, Schubert E, Zimek A (2013) Interactive data mining with 3D-Parallel-Coordinate-Trees. In: Proceedings of the ACM international conference on management of data (SIGMOD). New York City, pp 1009–1012Google Scholar
  34. 34.
    Dalvi N, Suciu D (2007) Efficient query evaluation on probabilistic databases. VLDB J 16(4):523–544CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institute for InformaticsLudwig-Maximilians-Universität MünchenMünchenGermany
  2. 2.George Mason UniversityFairfaxUSA
  3. 3.Department of Computer ScienceUniversity of Hong KongPok Fu LamHong Kong

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