Advertisement

On the Expected Diameter, Width, and Complexity of a Stochastic Convex-Hull

  • Jie Xue
  • Yuan Li
  • Ravi Janardan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

Abstract

We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of n points in \(\mathbb {R}^d\) each of which has an existence probability, a SCH refers to the convex hull of a realization of the dataset, i.e., a random sample including each point with its existence probability. We are interested in computing certain expected statistics of a SCH, including diameter, width, and combinatorial complexity. For diameter, we establish the first deterministic 1.633-approximation algorithm with a time complexity polynomial in both n and d. For width, two approximation algorithms are provided: a deterministic O(1)-approximation running in \(O(n^{d+1} \log n)\) time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact \(O(n^d)\)-time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest.

Keywords

Uncertain data Expectation Diameter Width Combinatorial complexity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agarwal, P.K., Aronov, B., Har-Peled, S., Phillips, J.M., Yi, K., Zhang, W.: Nearest neighbor searching under uncertainty II. In: Proc. of the 32nd PODS. ACM (2013)Google Scholar
  2. 2.
    Agarwal, P.K., Cheng, S.W., Yi, K.: Range searching on uncertain data. ACM Transactions on Algorithms (TALG) 8(4), 43 (2012)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Agarwal, P.K., Har-Peled, S., Suri, S., Yıldız, H., Zhang, W.: Convex hulls under uncertainty. In: Schulz, A.S., Wagner, D. (eds.) ESA 2014. LNCS, vol. 8737, pp. 37–48. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-44777-2_4 Google Scholar
  4. 4.
    Fink, M., Hershberger, J., Kumar, N., Suri, S.: Hyperplane separability and convexity of probabilistic point sets. In: Proc. of the 32nd SoCG. ACM (2016)Google Scholar
  5. 5.
    Huang, L., Li, J.: Approximating the expected values for combinatorial optimization problems over stochastic points. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 910–921. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-47672-7_74 CrossRefGoogle Scholar
  6. 6.
    Huang, L., Li, J., Phillips, J.M., Wang, H.: \(\epsilon \)-kernel coresets for stochastic points. arXiv preprint arXiv:1411.0194 (2014)
  7. 7.
    Jørgensen, A., Löffler, M., Phillips, J.M.: Geometric computations on indecisive points. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 536–547. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-22300-6_45 CrossRefGoogle Scholar
  8. 8.
    Kamousi, P., Chan, T., Suri, S.: Stochastic minimum spanning trees in euclidean spaces. In: Proc. of the 27th SoCG, pp. 65–74. ACM (2011)Google Scholar
  9. 9.
    Kamousi, P., Chan, T.M., Suri, S.: Closest pair and the post office problem for stochastic points. Computational Geometry 47(2), 214–223 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Li, C., Fan, C., Luo, J., Zhong, F., Zhu, B.: Expected computations on color spanning sets. Journal of Combinatorial Optimization 29(3), 589–604 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Löffler, M., van Kreveld, M.: Largest and smallest convex hulls for imprecise points. Algorithmica 56(2), 235–269 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Suri, S., Verbeek, K.: On the most likely Voronoi Diagram and nearest neighbor searching. In: Ahn, H.-K., Shin, C.-S. (eds.) ISAAC 2014. LNCS, vol. 8889, pp. 338–350. Springer, Cham (2014). doi: 10.1007/978-3-319-13075-0_27 Google Scholar
  13. 13.
    Suri, S., Verbeek, K., Yıldız, H.: On the most likely convex hull of uncertain points. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 791–802. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-40450-4_67 CrossRefGoogle Scholar
  14. 14.
    Xue, J., Li, Y., Janardan, R.: On the separability of stochastic geometric objects, with applications. In: Proc. of the 32nd SoCG. ACM (2016)Google Scholar
  15. 15.
    Xue, J., Li, Y.: Colored stochastic dominance problems. arXiv preprint arXiv:1612.06954 (2016)
  16. 16.
    Xue, J., Li, Y.: Stochastic closest-pair problem and most-likely nearest-neighbor search in tree spaces. arXiv preprint arXiv:1612.04890 (2016)
  17. 17.
    Xue, J., Li, Y., Janardan, R.: On the expected diameter, width, and complexity of a stochastic convex-hull. arXiv preprint arXiv:1704.07028 (2017)

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of Minnesota — Twin CitiesMinneapolisUSA

Personalised recommendations