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The Two-Center Problem of Uncertain Points on Trees

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Combinatorial Optimization and Applications (COCOA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14461))

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Abstract

In this paper, we consider the (weighted) two-center problem of uncertain points on a tree. Given are a tree T and a set \(\mathcal {P}\) of n (weighted) uncertain points each of which has m possible locations on T associated with probabilities. The goal is to compute two points on T, i.e., two centers with respect to \(\mathcal {P}\), so that the maximum (weighted) expected distance of n uncertain points to their own expected closest center is minimized. This problem can be solved in \(O(|T|+ n^{2}\log n\log mn + mn\log ^2 mn \log n)\) time by the algorithm for the general k-center problem. In this paper, we give a more efficient and simple algorithm that solves this problem in \(O(|T| + mn\log mn)\) time.

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References

  1. Alipour, S.: Improvements on approximation algorithms for clustering probabilistic data. Knowl. Inf. Syst. 63, 2719–2740 (2021)

    Article  Google Scholar 

  2. Ben-Moshe, B., Bhattacharya, B., Shi, Q., Tamir, A.: Efficient algorithms for center problems in cactus networks. Theoret. Comput. Sci. 378(3), 237–252 (2007)

    Article  MathSciNet  Google Scholar 

  3. Ben-Moshe, B., Bhattacharya, B., Shi, Q.: An optimal algorithm for the continuous/discrete weighted 2-center problem in trees. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 166–177. Springer, Heidelberg (2006). https://doi.org/10.1007/11682462_19

    Chapter  Google Scholar 

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

    Google Scholar 

  5. Bhattacharya, B., Shi, Q.: Improved algorithms to network \(p\)-center location problems. Comput. Geom. 47(2), 307–315 (2014)

    Article  MathSciNet  Google Scholar 

  6. Chan, T.: Dynamic planar convex hull operations in near-logarithmic amortized time. In: Proceedings of the 40th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 92–99 (1999)

    Google Scholar 

  7. Chan, T.: Klee’s measure problem made easy. In: Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pp. 410–419 (2013)

    Google Scholar 

  8. Chen, D., Wang, H.: A note on searching line arrangements and applications. Inf. Process. Lett. 113, 518–521 (2013)

    Article  MathSciNet  Google Scholar 

  9. Eppstein, D.: Faster construction of planar two-centers. In: Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 131–138 (1997)

    Google Scholar 

  10. Hu, R., Kanani, D., Zhang, J.: Computing the center of uncertain points on cactus graphs. In: Proceedings of the 34th International Workshop on Combinatorial Algorithms, pp. 233–245 (2023)

    Google Scholar 

  11. 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)

    Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Nguyen, Q., Zhang, J.: Line-constrained \(l_{\infty }\) one-center problem on uncertain points. In: Proceedings of the \(3\)rd International Conference on Advanced Information Science and System, vol. 71, pp. 1–5 (2021)

    Google Scholar 

  15. Wang, H.: On the planar two-center problem and circular hulls. Discrete Comput. Geom. 68(4), 1175–1226 (2022)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  17. Wang, H., Zhang, J.: Covering uncertain points on a tree. Algorithmica 81, 2346–2376 (2019)

    Article  MathSciNet  Google Scholar 

  18. Xu, H., Zhang, J.: The two-center problem of uncertain points on a real line. J. Comb. Optim. 45(68) (2023)

    Google Scholar 

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Authors and Affiliations

  1. Cleveland State University, Cleveland, OH, 44115, USA

    Haitao Xu & Jingru Zhang

Authors
  1. Haitao Xu
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  2. Jingru Zhang
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Correspondence to Haitao Xu .

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Editors and Affiliations

  1. University of Texas at Dallas, Richardson, TX, USA

    Weili Wu

  2. Beijing Normal University, Zhuhai, China

    Jianxiong Guo

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Xu, H., Zhang, J. (2024). The Two-Center Problem of Uncertain Points on Trees. In: Wu, W., Guo, J. (eds) Combinatorial Optimization and Applications. COCOA 2023. Lecture Notes in Computer Science, vol 14461. Springer, Cham. https://doi.org/10.1007/978-3-031-49611-0_35

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  • DOI: https://doi.org/10.1007/978-3-031-49611-0_35

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-49610-3

  • Online ISBN: 978-3-031-49611-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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