, Volume 79, Issue 2, pp 340–367 | Cite as

Convex Hulls Under Uncertainty

  • Pankaj K. Agarwal
  • Sariel Har-Peled
  • Subhash Suri
  • Hakan Yıldız
  • Wuzhou ZhangEmail author


We study the convex-hull problem in a probabilistic setting, motivated by the need to handle data uncertainty inherent in many applications, including sensor databases, location-based services and computer vision. In our framework, the uncertainty of each input point is described by a probability distribution over a finite number of possible locations including a null location to account for non-existence of the point. Our results include both exact and approximation algorithms for computing the probability of a query point lying inside the convex hull of the input, time–space tradeoffs for the membership queries, a connection between Tukey depth and membership queries, as well as a new notion of \(\beta \)-hull that may be a useful representation of uncertain hulls.


Convex hull Membership probability Tukey depth Uncertainty 


  1. 1.
    Afshani, P., Agarwal, P.K., Arge, L., Larsen, K.G., Phillips, J.M.: (Approximate) uncertain skylines. Theory Comput. Syst. 52(3), 342–366 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agarwal, P.K., Aronov, B., Har-Peled, S., Phillips, J.M., Yi, K., Zhang, W.: Nearest neighbor searching under uncertainty II. In: Proceedings of 32nd ACM Symposium Principles on Database System, pp. 115–126 (2013)Google Scholar
  3. 3.
    Agarwal, P.K., Cheng, S.-W., Yi, K.: Range searching on uncertain data. ACM Trans. Algorithms 8(4), 43:1–43:17 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Agarwal, P.K. , Efrat, A., Sankararaman, S., Zhang, W.: Nearest-neighbor searching under uncertainty. In: Proceedings of 31st ACM Symposium Principles Database Systems, pp. 225–236 (2012)Google Scholar
  5. 5.
    Agarwal, P.K., Sharir, M.: Efficient algorithms for geometric optimization. ACM Comput. Surv. 30, 412–458 (1998)CrossRefGoogle Scholar
  6. 6.
    Agarwal, P.K., Sharir, M.: Arrangements and their applications. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 49–119. Elsevier Science Publishers B.V. North-Holland, Amsterdam (2000)CrossRefGoogle Scholar
  7. 7.
    Agarwal, P.K., Sharir, M., Welzl, E.: Algorithms for center and Tverberg points. ACM Trans. Algorithms 5(1), 5:1–5:20 (2008)MathSciNetGoogle Scholar
  8. 8.
    Cabello, S.: Approximation algorithms for spreading points. J. Algorithms 62(2), 49–73 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chazelle, B.: Cutting hyperplanes for divide-and-conquer. Discrete Comput. Geom. 9(1), 145–158 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chazelle, B., Guibas, L.J., Lee, D.T.: The power of geometric duality. BIT 25(1), 76–90 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chazelle, B.: Cutting hyperplanes for divide-and-conquer. Discrete Comput. Geom. 9(2), 145–158 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dalvi, N.N., Ré, C., Suciu, D.: Probabilistic databases: diamonds in the dirt. Comm. ACM 52(7), 86–94 (2009)CrossRefGoogle Scholar
  13. 13.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  14. 14.
    Eckhoff, J.: Helly, radon, and Carathéodory type theorems. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, pp. 389–448. North-Holland, Amsterdam (1993)CrossRefGoogle Scholar
  15. 15.
    Evans, W.S., Gansner, E.R., Kaufmann, M., Liotta, G., Meijer, H., Spillner, A.: Approximate proximity drawings. In : Proceedings of 19th International Symposium on Graph Drawing, pp. 166–178 (2011)Google Scholar
  16. 16.
    Fink, M., Hershberger, J., Kumar, N., Suri, S.: Hyperplane separability and convexity of probabilistic point sets. In: Proceedings of 32nd Annual Symposium on Computational Geometry, pp. 38:1–38:16 (2016)Google Scholar
  17. 17.
    Franciosa, P.G., Gaibisso, C., Gambosi, G., Talamo, M.: A convex hull algorithm for points with approximately known positions. Int. J. Comput. Geometry Appl. 4(2), 153–163 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Guibas, L.J., Salesin, D., Stolfi, J.: Constructing strongly convex approximate hulls with inaccurate primitives. Algorithmica 9(6), 534–560 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Haussler, D., Welzl, E.: \(\varepsilon \)-nets and simplex range queries. Discrete Comput. Geom. 2, 127–151 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jørgensen, A., Löffler, M., Phillips, J.: Geometric computations on indecisive points. In: Proceedings of 12th Workshop on Algorithms and Data Structures, pp. 536–547 (2011)Google Scholar
  21. 21.
    Kamousi, P., Chan, T.M., Suri, S.: Closest pair and the post office problem for stochastic points. Comput. Geom. Theory Appl. 47(2), 214–223 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kamousi, P., Chan, T.M., Suri, S.: Stochastic minimum spanning trees in euclidean spaces. In: Proceedings of 27th Annual Symposium on Computational Geometry, pp. 65–74 (2011)Google Scholar
  23. 23.
    Löffler, M.: Data imprecision in computational geometry. PhD Thesis, Department of Computer Science, Utrecht University (2009)Google Scholar
  24. 24.
    Löffler, M., Snoeyink, J.: Delaunay triangulations of imprecise pointsin linear time after preprocessing. In: Proceedings of 24th Annual Symposium on Computational Geometry, pp. 298–304 (2008)Google Scholar
  25. 25.
    Löffler, M., van Kreveld, M.J.: Largest bounding box, smallest diameter, and related problems on imprecise points. Comput. Geom. Theory Appl. 43(4), 419–433 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Matoušek, J.: Computing the center of planar point sets. In: Goodman, J.E., Pollack, R., Steiger, W. (eds.) Computational Geometry: Papers from the DIMACS Special Year, pp. 221–230. Amer. Math. Soc., Providence (1991)CrossRefGoogle Scholar
  27. 27.
    Matoušek, J.: Linear optimization queries. J. Algorithms 14(3), 432–448 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  29. 29.
    Nagai, T., Tokura, N.: Tight error bounds of geometric problems on convex objects with imprecise coordinates. In: Proceedings of Japanese Conference on Discrete Computational Geometry, pp. 252–263 (2000)Google Scholar
  30. 30.
    Nagai, T., Yasutome, S., Tokura, N.: Convex hull problem with imprecise input. In: Proceedings of Japanese Conference on Discrete Computational Geometry, pp. 207–219 (1998)Google Scholar
  31. 31.
    Pérez-Lantero, P.: Area and perimeter of the convex hull of stochastic points. CoRR arXiv:1412.5153 (2014)
  32. 32.
    Phillips, J.M.: Small and stable descriptors of distributions for geometric statistical problems. Ph.D. Thesis, Department of Computer Science, Duke University (2009)Google Scholar
  33. 33.
    Seidel, R.: Convex hull computations. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, pp. 495–512. CRC Press, Boca Raton (2004)Google Scholar
  34. 34.
    Sember, J.: Guarantees concerning geometric objects with uncertain imputs. Ph.D. Thesis, Department of Computer Science, University of British Columbia (2011)Google Scholar
  35. 35.
    Suri, S., Verbeek, K., Yildiz, H.: On the most likely convex hull of uncertain points. In: Proceedings of 21st Annual European Symposium on Algorithms, pp. 791–802 (2013)Google Scholar
  36. 36.
    van Kreveld, M.J., Löffler, M.: Approximating largest convex hulls for imprecise points. J. Discrete Algorithms 6(4), 583–594 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wang, H., Zhang, W.: The \(\tau \)-skyline for uncertain data. In: Proceedings of 26th Canadian Conference on Computational Geometry, pp. 326–331 (2014)Google Scholar
  38. 38.
    Zhang, W.: Geometric computing over uncertain data. Ph.D. Thesis, Department of Computer Science, Duke University (2015)Google Scholar
  39. 39.
    Zhao, Z., Yan, D., Ng, W.: A probabilistic convex hull query tool. In: Proceedings of 15th International Conferences on Extending Database Technology, pp. 570–573 (2012)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Pankaj K. Agarwal
    • 1
  • Sariel Har-Peled
    • 2
  • Subhash Suri
    • 3
  • Hakan Yıldız
    • 4
  • Wuzhou Zhang
    • 5
    Email author
  1. 1.Duke UniversityDurhamUSA
  2. 2.University of Illinois at Urbana-ChampaignChampaignUSA
  3. 3.University of California, Santa BarbaraSanta BarbaraUSA
  4. 4.Microsoft CorporationRedmondUSA
  5. 5.Apple Inc.CupertinoUSA

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