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Covering Uncertain Points in a Tree

  • Haitao Wang
  • Jingru Zhang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

Abstract

We consider a coverage problem for uncertain points in a tree. Let T be a tree containing a set \(\mathcal {P}\) of n (weighted) demand points, and the location of each demand point \(P_i\in \mathcal {P}\) is uncertain but is known to be in one of \(m_i\) points on T each associated with a probability. Given a covering range \(\lambda \), the problem is to find a minimum number of points (called centers) on T to build facilities for serving (or covering) these demand points in the sense that for each uncertain point \(P_i\in \mathcal {P}\), the expected distance from \(P_i\) to at least one center is no more than \(\lambda \). The problem has not been studied before. We present an \(O(|T|+M\log ^2 M)\) time algorithm, where |T| is the number of vertices of T and M is the total number of locations of all uncertain points of \(\mathcal {P}\), i.e., \(M=\sum _{P_i\in \mathcal {P}}m_i\).

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References

  1. 1.
    Agarwal, P., Cheng, S.W., Tao, Y., Yi, K.: Indexing uncertain data. In: Proc. of the 28th Symposium on Principles of Database Systems (PODS), pp. 137–146 (2009)Google Scholar
  2. 2.
    Agarwal, P., Efrat, A., Sankararaman, S., Zhang, W.: Nearest-neighbor searching under uncertainty. In: Proc. of the 31st Symposium on Principles of Database Systems (PODS), pp. 225–236 (2012)Google Scholar
  3. 3.
    Agarwal, P., Har-Peled, S., Suri, S., Yıldız, H., Zhang, W.: Convex hulls under uncertainty. In: Proc. of the 22nd Annual European Symposium on Algorithms (ESA), pp. 37–48 (2014)Google Scholar
  4. 4.
    de Berg, M., Roeloffzen, M., Speckmann, B.: Kinetic 2-centers in the black-box model. In: Proc. of the 29th Annual Symposium on Computational Geometry (SoCG), pp. 145–154 (2013)Google Scholar
  5. 5.
    Brodal, G., Jacob, R.: Dynamic planar convex hull. In: Proc. of the 43rd IEEE Symposium on Foundations of Computer Science (FOCS), pp. 617–626 (2002)Google Scholar
  6. 6.
    Cole, R.: Slowing down sorting networks to obtain faster sorting algorithms. Journal of the ACM 34(1), 200–208 (1987)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Huang, L., Li, J.: Stochasitc \(k\)-center and \(j\)-flat-center problems. In: Proc. of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 110–129 (2017)Google Scholar
  8. 8.
    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
  9. 9.
    Kamousi, P., Chan, T.M., Suri, S.: Closest pair and the post office problem for stochastic points. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 548–559. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-22300-6_46 CrossRefGoogle Scholar
  10. 10.
    Kamousi, P., Chan, T., Suri, S.: Stochastic minimum spanning trees in Euclidean spaces. In: Proc. of the 27th Annual Symposium on Computational Geometry (SoCG), pp. 65–74 (2011)Google Scholar
  11. 11.
    Kariv, O., Hakimi, S.: An algorithmic approach to network location problems. I: The \(p\)-centers. SIAM J. on Applied Mathematics 37(3), 513–538 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Löffler, M., van Kreveld, M.: Largest bounding box, smallest diameter, and related problems on imprecise points. Computational Geometry: Theory and Applications 43(4), 419–433 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Megiddo, N.: Linear-time algorithms for linear programming in \(R^3\) and related problems. SIAM Journal on Computing 12(4), 759–776 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Megiddo, N., Tamir, A.: New results on the complexity of \(p\)-centre problems. SIAM Journal on Computing 12(4), 751–758 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Tao, Y., Xiao, X., Cheng, R.: Range search on multidimensional uncertain data. ACM Transactions on Database Systems 32 (2007)Google Scholar
  16. 16.
    Wang, H., Zhang, J.: Computing the center of uncertain points on tree networks. Algorithmica 78(1), 232–254 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wang, H., Zhang, J.: One-dimensional \(k\)-center on uncertain data. Theoretical Computer Science 602, 114–124 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wang, H., Zhang, J.: Covering uncertain points in a tree. arXiv:1704.07497 (2017)

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceUtah State UniversityLoganUSA

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