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Covering Uncertain Points in a Tree

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Abstract

In this paper, 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 appear 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 for the problem, 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., \(M=\sum _{P_i\in {{\mathcal {P}}}}m_i\). In addition, by using this algorithm, we solve a k-center problem on T for the uncertain points of \({{\mathcal {P}}}\).

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References

  1. Agarwal, P.K., Cheng, S.-W., Tao, Y., Yi, K.: Indexing uncertain data. In: Proceedings of the 28th Symposium on Principles of Database Systems (PODS), pp. 137–146 (2009)

  2. Agarwal, P.K., Efrat, A., Sankararaman, S., Zhang, W.: Nearest-neighbor searching under uncertainty. In: Proceedings of the 31st Symposium on Principles of Database Systems (PODS), pp. 225–236 (2012)

  3. Agarwal, P.K., Har-Peled, S., Suri, S., Yıldız, H., Zhang, W.: Convex hulls under uncertainty. In: Proceedings of the 22nd Annual European Symposium on Algorithms (ESA), pp. 37–48 (2014)

  4. Agarwal, P.K., Sharir, M.: Efficient algorithms for geometric optimization. ACM Comput. Surv. 30(4), 412–458 (1998)

    Article  Google Scholar 

  5. Averbakh, I., Bereg, S.: Facility location problems with uncertainty on the plane. Discrete Optim. 2, 3–34 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Averbakh, I., Berman, O.: Minimax regret \(p\)-center location on a network with demand uncertainty. Locat. Sci. 5, 247–254 (1997)

    Article  MATH  Google Scholar 

  7. Bender, M., Farach-Colton, M.: The LCA problem revisited. In: Proceedings of the 4th Latin American Symposium on Theoretical Informatics, pp. 88–94 (2000)

  8. Bender, M.A., Farach-Colton, M.: The level ancestor problem simplied. Theor. Comput. Sci. 321, 5–12 (2004)

    Article  MATH  Google Scholar 

  9. Bereg, S., Bhattacharya, B., Das, S., Kameda, T., Mahapatra, P.R.S., Song, Z.: Optimizing squares covering a set of points. Theoretical Computer Science (2015) (in press)

  10. Brodal, G., Jacob, R.: Dynamic planar convex hull. In: Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science (FOCS), pp. 617–626 (2002)

  11. Chan, T.M., Hu, N.: Geometric red-blue set cover for unit squares and related problems. Comput. Geom. 48(5), 380–385 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chazelle, B., Guibas, L.: Fractional cascading: I. A data structuring technique. Algorithmica 1(1), 133–162 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cheng, R., Chen, J., Xie, X.: Cleaning uncertain data with quality guarantees. Proc. VLDB Endow. 1(1), 722–735 (2008)

    Article  Google Scholar 

  14. Cheng, R., Xia, Y., Prabhakar, S., Shah, R., Vitter, J.S.: Efficient indexing methods for probabilistic threshold queries over uncertain data. In: Proceedings of the 30th International Conference on Very Large Data Bases (VLDB), pp. 876–887 (2004)

  15. Cole, R.: Slowing down sorting networks to obtain faster sorting algorithms. J. ACM 34(1), 200–208 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  16. de Berg, M., Roeloffzen, M., Speckmann, B.: Kinetic 2-centers in the black-box model. In: Proceedings of the 29th Annual Symposium on Computational Geometry (SoCG), pp. 145–154 (2013)

  17. Dong, X., Halevy, A.Y., Yu, C.: Data integration with uncertainty. In: Proceedings of the 33rd International Conference on Very Large Data Bases, pp. 687–698 (2007)

  18. Frederickson, G.N.: Parametric search and locating supply centers in trees. In: Proceedings of the 2nd International Workshop on Algorithms and Data Structures (WADS), pp. 299–319 (1991)

  19. Frederickson, G.N., Johnson, D.B.: Finding \(k\)th paths and \(p\)-centers by generating and searching good data structures. J. Algorithms 4(1), 61–80 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gonzalez, T.F.: Covering a set of points in multidimensional space. Inf. Process. Lett. 40(4), 181–188 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestors. SIAM J. Comput. 13, 338–355 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32(1), 130–136 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  23. Huang, L., Li, J.: Stochastic \(k\)-center and \(j\)-flat-center problems. In: Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 110–129 (2017)

  24. Jørgensen, A., Löffler, M., Phillips, J.M.: Geometric computations on indecisive points. In: Proceedings of the 12nd Algorithms and Data Structures Symposium (WADS), pp. 536–547 (2011)

  25. Kamousi, P., Chan, T.M., Suri, S.: Closest pair and the post office problem for stochastic points. In: Proceedings of the 12nd Algorithms and Data Structures Symposium (WADS), pp. 548–559 (2011)

  26. Kamousi, P., Chan, T.M., Suri, S.: Stochastic minimum spanning trees in Euclidean spaces. In: Proceedings of the 27th Annual Symposium on Computational Geometry (SoCG), pp. 65–74 (2011)

  27. Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems. I: the \(p\)-centers. SIAM J. Appl. Math. 37(3), 513–538 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems. II: the \(p\)-medians. SIAM J. Appl. Math. 37(3), 539–560 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kim, S.-S., Bae, S.W., Ahn, H.-K.: Covering a point set by two disjoint rectangles. Int. J. Comput. Geom. Appl. 21, 313–330 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Löffler, M., van Kreveld, M.: Largest bounding box, smallest diameter, and related problems on imprecise points. Comput. Geom. Theory Appl. 43(4), 419–433 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Megiddo, N.: Linear-time algorithms for linear programming in \(R^3\) and related problems. SIAM J. Comput. 12(4), 759–776 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  32. Megiddo, N., Tamir, A.: New results on the complexity of \(p\)-centre problems. SIAM J. Comput. 12(4), 751–758 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  33. Megiddo, N., Tamir, A., Zemel, E., Chandrasekaran, R.: An \(O(n \log ^2 n)\) algorithm for the \(k\)-th longest path in a tree with applications to location problems. SIAM J. Comput. 10, 328–337 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  34. Mustafa, N.H., Ray, S.: PTAS for geometric hitting set problems via local search. In: Proceedings of the 25th Annual Symposium on Computational Geometry (SoCG), pp. 17–22 (2009)

  35. Suri, S., Verbeek, K.: On the most likely voronoi diagram and nearest neighbor searching. In: Proceedings of the 25th International Symposium on Algorithms and Computation (ISAAC), pp. 338–350 (2014)

  36. Suri, S., Verbeek, K., Yıldız, H.: On the most likely convex hull of uncertain points. In: Proceedings of the 21st European Symposium on Algorithms (ESA), pp. 791–802 (2013)

  37. Tao, Y., Xiao, X., Cheng, R.: Range search on multidimensional uncertain data. ACM Trans. Database Syst. 32, 15 (2007)

    Article  Google Scholar 

  38. Wang, H.: Minmax regret 1-facility location on uncertain path networks. Eur. J. Oper. Res. 239, 636–643 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang, H., Zhang, J.: One-dimensional \(k\)-center on uncertain data. Theor. Comput. Sci. 602, 114–124 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang, H., Zhang, J.: Computing the center of uncertain points on tree networks. Algorithmica 78(1), 232–254 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang, H., Zhang, J.: An \(O(n \log n)\)-time algorithm for the \(k\)-center problem in trees. In: Proceedings of the 34th International Symposium on Computational Geometry (SoCG), pp. 72:1–72:15 (2018)

  42. Yiu, M.L., Mamoulis, N., Dai, X., Tao, Y., Vaitis, M.: Efficient evaluation of probabilistic advanced spatial queries on existentially uncertain data. IEEE Trans. Knowl. Data Eng. 21, 108–122 (2009)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, Utah State University, Logan, UT, 84322, USA

    Haitao Wang

  2. Department of Computer Science, University of Texas - Rio Grande Valley, Edinburg, TX, 78539, USA

    Jingru Zhang

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  1. Haitao Wang
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  2. Jingru Zhang
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Correspondence to Jingru Zhang.

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A preliminary version of this paper appeared in the Proceedings of the 15th Algorithms and Data Structures Symposium (WADS 2017). This research was supported in part by NSF under Grant CCF-1317143.

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Wang, H., Zhang, J. Covering Uncertain Points in a Tree. Algorithmica 81, 2346–2376 (2019). https://doi.org/10.1007/s00453-018-00537-6

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