Advertisement

Theory of Computing Systems

, Volume 52, Issue 3, pp 342–366 | Cite as

(Approximate) Uncertain Skylines

  • Peyman Afshani
  • Pankaj K. Agarwal
  • Lars Arge
  • Kasper Green Larsen
  • Jeff M. PhillipsEmail author
Article

Abstract

Given a set of points with uncertain locations, we consider the problem of computing the probability of each point lying on the skyline, that is, the probability that it is not dominated by any other input point. If each point’s uncertainty is described as a probability distribution over a discrete set of locations, we improve the best known exact solution. We also suggest why we believe our solution might be optimal. Next, we describe simple, near-linear time approximation algorithms for computing the probability of each point lying on the skyline. In addition, some of our methods can be adapted to construct data structures that can efficiently determine the probability of a query point lying on the skyline.

Keywords

Data structures Approximation algorithms Computational geometry 

References

  1. 1.
    Afshani, P., Agarwal, P.K., Arge, L., Larsen, K.G., Phillips, J.M.: (Appoximate) uncertain skylines. In: 14th International Conference on Database Theory (2011) Google Scholar
  2. 2.
    Agarwal, P.K., Sharir, M.: Arrangements of surfaces in higher dimensions. In: Sack, J., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 49–119. North-Holland, Amsterdam (2000) CrossRefGoogle Scholar
  3. 3.
    Agrawal, P., Benjelloun, O., Sarma, A.D., Hayworth, C., Nabar, S., Sugihara, T., Widom, J.: Trio: a system for data, uncertainty, and lineage. In: ACM Symposium on Principles of Database Systems (2006) Google Scholar
  4. 4.
    Atallah, M.J., Qi, Y.: Computing all skyline probabilities for uncertain data. In: ACM Symposium on Principles of Database Systems, pp. 279–287 (2009) Google Scholar
  5. 5.
    Bentley, J.L.: Multidimensional binary search trees used for associative searching. Commun. ACM 18(9), 509–517 (1975) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Boissonnat, J.-D., Sharir, M., Tagansky, B., Yvinec, M.: Voronoi diagrams in higher dimensions under certain polyhedral distance functions. Discrete Comput. Geom. 19, 485–519 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Börzsönyi, S., Kossman, D., Stocker, K.: The skyline operator. In: IEEE International Conference on Data Engineering (2001) Google Scholar
  8. 8.
    Chan, C.-Y., Jagadish, H.V., Tan, K.-L., Tung, A.K.H., Zhang, Z.: Finding k-dominant skylines in high dimensional space. In: ACM-SIGMOD International Conference on Management of Data (2006) Google Scholar
  9. 9.
    Cheng, R., Kalashnikov, D.V., Prabhakar, S.: Evaluating probabilitic queries over imprecise data. In: ACM-SIGMOD International Conference on Management of Data (2003) Google Scholar
  10. 10.
    Cormode, G., Garafalakis, M.: Histograms and wavelets of probabilitic data. In: IEEE International Conference on Data Engineering (2009) Google Scholar
  11. 11.
    Cormode, G., Deligiannakis, A., Garafalakis, M., McGregor, A.: Probabilistic histograms for probabilistic data. In: International Conference on Very Large Data Bases (2009) Google Scholar
  12. 12.
    Cormode, G., Li, F., Yi, K.: Semantics of ranking queries for probabilistic data and expected ranks. In: IEEE International Conference on Data Engineering (2009) Google Scholar
  13. 13.
    Dalvi, N., Suciu, D.: Efficient query evaluation on probabilitic databases. VLDB J. 16, 523–544 (2007) CrossRefGoogle Scholar
  14. 14.
    Das Sarma, A., Lall, A., Nanongkai, D., Xu, J.: Randomized multi-pass streaming skyline algorithms. In: International Conference on Very Large Data Bases (2009) Google Scholar
  15. 15.
    Das Sarma, A., Lall, A., Nanongkai, D., Lipton, R.J., Xu, J.: Representative skylines using threshold-based preference distributions. In: IEEE International Conference on Data Engineering (2011) Google Scholar
  16. 16.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry Algorithms and Applications. Springer, Berlin (2008) zbMATHGoogle Scholar
  17. 17.
    Edelsbunner, H., Guibas, L.J., Stolfi, J.: Optimal point location on a monotone subdivision. SIAM J. Comput. 15, 317–340 (1986) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Koltun, V., Papadimitriou, C.H.: Approximately dominating representatives. Theor. Comput. Sci. 371, 148–154 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Kossman, D., Ramsak, F., Rost, S.: Shooting stars in the sky: an optimal algorithm for skyline queries. In: International Conference on Very Large Data Bases (2002) Google Scholar
  20. 20.
    Kung, H.T., Luccio, F., Preparata, F.P.: On finding the maxima of a set of vectors. J. ACM 22(4), 469–476 (1975) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Li, J., Saha, B., Deshpande, A.: A unified approach to ranking in probabilistic databases. In: International Conference on Very Large Data Bases (2009) Google Scholar
  22. 22.
    Lian, X., Chen, L.: Monochromatic and bichromatic reverse skyline search over uncertain databases. In: ACM-SIGMOD International Conference on Management of Data (2008) Google Scholar
  23. 23.
    Löffler, M., Phillips, J.M.: Shape fitting of point sets with probability distributions. In: European Symposium on Algorithms (2009) Google Scholar
  24. 24.
    Löffler, M., Snoeyink, J.: Delaunay triangulations of imprecise points in linear time after preprocessing. In: Symposium on Computational Geometry, pp. 298–304 (2008) Google Scholar
  25. 25.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995) zbMATHCrossRefGoogle Scholar
  26. 26.
    Nanongkai, D., Das Sarma, A., Lall, A., Lipton, R.J., Xu, J.: Regret-minimizing representative databases. In: International Conference on Very Large Data Bases (2010) Google Scholar
  27. 27.
    Papadias, D., Tao, Y., Fu, G., Seeger, B.: An optimal and progressive algorithm for skyline queries. In: ACM-SIGMOD International Conference on Management of Data (2003) Google Scholar
  28. 28.
    Pei, J., Jiang, B., Lin, X., Yuan, Y.: Probabilistic skylines on uncertain data. In: International Conference on Very Large Data Bases (2007) Google Scholar
  29. 29.
    Preparata, F.P., Shamos, M.I.: Computational Geometry an Introduction. Springer, Berlin (1985) Google Scholar
  30. 30.
    Tan, K.-L., Eng, P.-K., Ooi, B.C.: Efficient progressive skyline computation. In: International Conference on Very Large Data Bases (2001) Google Scholar
  31. 31.
    Tao, Y., Cheng, R., Xiao, X., Ngai, W.K., Kao, B., Prabhakar, S.: Indexing multi-dimensional uncertain data with arbitrary probability density functions. In: International Conference on Very Large Data Bases (2005) Google Scholar
  32. 32.
    Willard, D.E.: New data structures for orthogonal range queries. SIAM J. Comput. 14(1), 232–253 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Zhang, W., Lin, X., Zhang, Y., Wang, W., Yu, J.X.: Probabilistic skyline operator over sliding windows. In: IEEE International Conference on Data Engineering (2009) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Peyman Afshani
    • 1
  • Pankaj K. Agarwal
    • 2
  • Lars Arge
    • 3
  • Kasper Green Larsen
    • 3
  • Jeff M. Phillips
    • 4
    Email author
  1. 1.Faculty of Computer ScienceDalhousie UniversityHalifaxCanada
  2. 2.Department of Computer ScienceDuke UniversityDurhamUSA
  3. 3.MADALGO & Department of Computer ScienceUniversity of AarhusAarhus NDenmark
  4. 4.School of ComputingUniversity of UtahSalt Lake CityUSA

Personalised recommendations