Improved Algorithms for Computing k-Sink on Dynamic Flow Path Networks

  • Binay Bhattacharya
  • Mordecai J. Golin
  • Yuya Higashikawa
  • Tsunehiko Kameda
  • Naoki Katoh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)


We address the problem of locating k sinks on dynamic flow path networks with n vertices in such a way that the evacuation completion time to them is minimized. Our two algorithms run in \(O(n\log n + k^2\log ^4 n)\) and \(O(n\log ^3 n)\) time, respectively. When all edges have the same capacity, we also present two algorithms which run in \(O(n + k^2\log ^2n)\) time and \(O(n\log n)\) time, respectively. These algorithms together improve upon the previously most efficient algorithms, which have time complexities \(O(kn\log ^2n)\) [1] and O(kn) [11], in the general and uniform edge capacity cases, respectively. The above results are achieved by organizing relevant data for subpaths in a strategic way during preprocessing, and the final results are obtained by extracting/merging them in an efficient manner.


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  1. 1.
    Arumugam, G.P., Augustine, J., Golin, M.J., Srikanthan, P.: A polynomial time algorithm for minimax-regret evacuation on a dynamic path (2014). arXiv:1404,5448v1
  2. 2.
    Benkoczi, R., Bhattacharya, B., Chrobak, M., Larmore, L.L., Rytter, W.: Faster algorithms for k-medians in trees. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 218–227. Springer, Heidelberg (2003). doi: 10.1007/978-3-540-45138-9_16 CrossRefGoogle Scholar
  3. 3.
    Bhattacharya, B., Kameda, T.: Improved algorithms for computing minmax regret sinks on path and tree networks. Theoretical Computer Science 607, 411–425 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cheng, S.-W., Higashikawa, Y., Katoh, N., Ni, G., Su, B., Xu, Y.: Minimax regret 1-sink location problems in dynamic path networks. In: Chan, T.-H.H., Lau, L.C., Trevisan, L. (eds.) TAMC 2013. LNCS, vol. 7876, pp. 121–132. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-38236-9_12 CrossRefGoogle Scholar
  5. 5.
    Ford, L.R., Fulkerson, D.R.: Constructing maximal dynamic flows from static flows. Operations research 6(3), 419–433 (1958)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Frederickson, G.N.: Optimal algorithms for tree partitioning. In: Proc. 2nd ACM-SIAM Symp. Discrete Algorithms, pp. 168–177 (1991)Google Scholar
  7. 7.
    Frederickson, G.N., Johnson, D.B.: Finding \(k\)th paths and \(p\)-centers by generating and searching good data structures. J. Algorithms 4, 61–80 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hamacher, H.W., Tjandra, S.A.: Mathematical modeling of evacuation problems: a state of the art. In: Pedestrian and Evacuation Dynamics, pp. 227–266. Springer Verlag (2002)Google Scholar
  9. 9.
    Higashikawa, Y.: Studies on the space exploration and the sink location under incomplete information towards applications to evacuation planning. PhD thesis, Kyoto University, Japan (2014)Google Scholar
  10. 10.
    Higashikawa, Y., Golin, M.J., Katoh, N.: Minimax regret sink location problem in dynamic tree networks with uniform capacity. J. of Graph Algorithms and Applications 18(4), 539–555 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Higashikawa, Y., Golin, M. J., Katoh, N.: Multiple sink location problems in dynamic path networks. Theoretical Computer Science 607, 2–15 (2015)Google Scholar
  12. 12.
    Hoppe, B., Tardos, É.: The quickest transshipment problem. Mathematics of Operations Research 25(1), 36–62 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kamiyama, N., Katoh, N., Takizawa, A.: An efficient algorithm for evacuation problem in dynamic network flows with uniform arc capacity. IEICE Transactions 89-D(8), 2372–2379 (2006)Google Scholar
  14. 14.
    Mamada, S., Makino, K., Fujishige, S.: Optimal sink location problem for dynamic flows in a tree network. IEICE Trans. Fundamentals E85-A, 1020–1025 (2002)Google Scholar
  15. 15.
    Mamada, S., Uno, T., Makino, K., Fujishige, S.: An \({O}(n\log ^2 n)\) algorithm for a sink location problem in dynamic tree networks. Discrete Applied Mathematics 154, 2387–2401 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Megiddo, N.: Combinatorial optimization with rational objective functions. Math. Oper. Res. 4, 414–424 (1979)Google Scholar
  17. 17.
    Megiddo, N., Tamir, A.: New results on the complexity of \(p\)-center problems. SIAM J. Comput. 12, 751–758 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Skutella, M.: An introduction to network flows over time. In: Research Trends in Combinatorial Optimization, pp. 451–482. Springer (2009)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Binay Bhattacharya
    • 1
  • Mordecai J. Golin
    • 2
  • Yuya Higashikawa
    • 3
  • Tsunehiko Kameda
    • 1
  • Naoki Katoh
    • 4
  1. 1.School of Computing ScienceSimon Fraser UniversityBurnabyCanada
  2. 2.Dept. of Computer ScienceHong Kong Univ. of Science and TechnologyHong KongChina
  3. 3.Dept. of Information and System EngineeringChuo UniversityTokyoJapan
  4. 4.School of Science and TechnologyKwansei Gakuin UniversityHyogoJapan

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