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A New Template for Solving p-Median Problems for Trees in Sub-quadratic Time

(Extended Abstract)
  • Robert Benkoczi
  • Binay Bhattacharya
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3669)

Abstract

We propose an O(n log p + 2 n) algorithm for solving the well-known p-Median problem for trees. Our analysis relies on the fact that p is considered constant (in practice, very often p << n). This is the first result in almost 25 years that proposes a new algorithm for solving this problem, opening up several new avenues for research.

Keywords

Cost Function Search Tree Facility Location Dynamic Programming Algorithm Facility Location Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Alstrup, S., Holm, J., Thorup, M.: Maintaining center and median in dynamic trees. In: Halldórsson, M.M. (ed.) SWAT 2000. LNCS, vol. 1851, pp. 46–56. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  2. 2.
    Arora, S., Raghavan, P., Rao, S.: Approximation schemes for euclidean k-medians and related problems. In: Proc. 30th Annual ACM Symposium on Theory of Computing (STOC 1998), pp. 106–113 (1998)Google Scholar
  3. 3.
    Arya, V., Garg, N., Khandekar, R., Meyerson, A., Mungala, K., Pandit, V.: Local seach heuristic for k-median and facility location problems. In: Proc. 33rd Annual ACM Symposium on Theory of Computing, pp. 21–29 (2001)Google Scholar
  4. 4.
    Bartal, Y.: On approximating arbitrary metrics by tree metrics. In: Proc. STOC, pp. 183–193 (1998)Google Scholar
  5. 5.
    Benkoczi, R.: Cardinality constrained facility location problems in trees. PhD thesis, School of Computing Science, Simon Fraser University, Burnaby, BC, Canada (May 2004)Google Scholar
  6. 6.
    Benkoczi, R.R., Bhattacharya, B.K., Chrobak, M., Larmore, L., Rytter, W.: Faster algorithms for k-median problems in trees. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 218–227. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Bespamyatnikh, S., Bhattacharya, B., Keil, M., Kirkpatrick, D., Segal, M.: Efficient algorithms for centers and medians in interval and circulararc graphs. NETWORKS 39(3), 144–152 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Boland, R.P.: Polygon visibility decompositions with applications. PhD thesis, University of Ottawa, Ottawa, Canada (2002)Google Scholar
  9. 9.
    Breton, D.: Facility location optimization problems in trees. Master’s thesis, School of Computing Science, Simon Fraser University, Canada (2002)Google Scholar
  10. 10.
    Charikar, M., Guha, S.: Improved combinatorial algorithms for facility location and k-median problems. In: Proc. 40th Symposium on Foundations of Computer Science (FOCS 1999), pp. 378–388 (1999)Google Scholar
  11. 11.
    Cole, R., Vishkin, U.: The accelerated centroid decomposition technique for optimal parallel tree evaluation in logarithmic time. Algorithmica 3, 329–346 (1988)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Drezner, Z., Klamroth, K., Schobel, A., Wesolowsky, G.O.: Facility Location: Applications and Theory. In: The Weber Problem, pp. 1–36. Springer, Heidelberg (2002)Google Scholar
  13. 13.
    Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing (2003)Google Scholar
  14. 14.
    Gavish, R., Sridhar, S.: Computing the 2-median on tree networks in O(nlogn) time. Networks 26, 305–317 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Goldman, A.J.: Optimal center location in simple networks. Trans. Sci. 5, 212–221 (1971)CrossRefGoogle Scholar
  16. 16.
    Goldman, A.J., Witzgall, C.J.: A localization theorem for optimal facility placement. Trans. Sci. 1, 106–109 (1970)MathSciNetGoogle Scholar
  17. 17.
    Granot, D., Skorin-Kapov, D.: On some optimization problems on k-trees and partial k-trees. Discrete Applied Mathematics 48(2), 129–145 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Gurevich, Y., Stockmeyer, L., Vishkin, U.: Solving NP-hard problems on graphs that are almost trees and an application to facility location problems. Journal of the ACM 31(3), 459–473 (1984)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Hale, T.S., Moberg, C.R.: Location science research: A review. Annals of Operations Research 123, 21–35 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Hassin, R., Tamir, A.: Efficient algorithms for optimization and selection on series-parallel graphs. SIAM Journal of Algebraic Discrete Methods 7, 379–389 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Holm, J., de Lichtenberg, K.: Top-trees and dynamic graph algorithms. Technical Report 17, Univ. of Copenhagen, Dept. of Computer Science (1998)Google Scholar
  22. 22.
    Jain, K., Vazirani, V.: Primal-dual approximation algorithms for metric facility location and k-median problems (March 1999) (Manuscript)Google Scholar
  23. 23.
    Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems II: The p-medians. SIAM Journal on Applied Mathematics 37, 539–560 (1979)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Lin, J.-H., Vitter, J.S.: Approximation algorithms for geometric median problems. Information Processing Letters 44, 245–249 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Lin, J.-H., Vitter, J.S.: ε-approximations with minimum packing constraint violation. In: Proceedings of the 24th Annual ACM Symposium on Theory of Computing, pp. 771–782 (1992)Google Scholar
  26. 26.
    Robertson, N., Seymour, P.D.: Graph minors. I. excluding a forest. J. Combin. Theory Ser. B 35, 39–61 (1983)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Shah, R., Farach-Colton, M.: Undiscretized dynamic programming: faster algorithms for facility location and related problems on trees. In: Proc. 13th Annual Symposium on Discrete Algorithms (SODA), pp. 108–115 (2002)Google Scholar
  28. 28.
    Shah, R., Langerman, S., Lodha, S.: Algorithms for efficient filtering in content-based multicast. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 428–439. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  29. 29.
    Tamir, A.: An O(pn 2) algorithm for the p-median and related problems on tree graphs. Operations Research Letters 19, 59–64 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Tamir, A., Pérez-Brito, D., Moreno-Pérez, J.A.: A polynomial algorithm for the p-centdian problem on a tree. Networks 32, 255–262 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Thorup, M.: Quick k-median, k-center, and facility location for sparse graphs. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 249–260. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Robert Benkoczi
    • 1
  • Binay Bhattacharya
    • 2
  1. 1.School of ComputingQueen’s University KingstonCanada
  2. 2.School of Computing ScienceSimon Fraser UniversityBurnabyCanada

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