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Algebraic Theory on Shortest Paths for All Flows

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Combinatorial Optimization and Applications

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9486))

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Abstract

As a mathematical model for the passenger routing problem for ticketing in a railway network, we consider a shortest path problem for a directed graph with edges labeled with a cost and a capacity. The problem is to push flow f from a specified source to all other vertices with the minimum cost for all f values. If there are t different capacity values, we can solve the single source shortest path problem for all f t times in \(O(tm+tn\log n)=O(m^2)\) time when \(t=m\). We improve this time to O(cmn) if edge costs are non-negative integers bounded by c.

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References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Englewood Cliffs (1993)

    MATH  Google Scholar 

  2. Alon, N., Galil, Z., Margalit, O.: On the exponent of the all pairs shortest path problem. In: Proceedings of 32th IEEE FOCS, pp. 569–575 (1991). Also JCSS 54, 255–262 (1997)

    Google Scholar 

  3. Anderegg, L., Eidenbenz, S., Gentenbein, M., Stamm, C., Taylor, D.S., Weber, B., Widmeyer, P.: Routing algorithms: concepts. design choices, and practical considerations. In: ALENEX 2003, pp. 106–118 (2003)

    Google Scholar 

  4. Chakrabarti, A., Chekuri, C., Gupta, A., Kumar, A.: Approximation algorithms for the unsplittable flow problem. Algorithmica 47(1), 53–78 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cherkassky, B.V., Goldberg, A.V., Radzik, T.: Shortest paths algorithms: theory and experimental evaluation. Math. Prog. 73, 129–174 (1996)

    MathSciNet  MATH  Google Scholar 

  6. Denardo, E.V., Fox, B.L.: Shortest-route methods: I. Reaching, pruning, and buckets. Oper. Res. 27, 161–186 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dial, R.B.: Algorithm 360: shortest path forest with topological ordering. Commun. ACM 12, 632–633 (1969)

    Article  Google Scholar 

  8. Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1(269–271), 1343–1345 (1959)

    MathSciNet  MATH  Google Scholar 

  9. Dinits, Y., Garg, N., Goemans, N.: On the single-source unsplittable flow problem. Combinatorica 19(1), 17–41 (1999). Springer

    Article  MathSciNet  MATH  Google Scholar 

  10. Duan, R., Pettie, S.: Fast algorithms for (max, min)-matrix multiplication and bottleneck shortest paths. In: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2009), pp. 384–391 (2009)

    Google Scholar 

  11. Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)

    MATH  Google Scholar 

  12. Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34, 596–615 (1987)

    Article  MathSciNet  Google Scholar 

  13. Gabow, H.N., Tarjan, R.E.: Algorithms for two bottleneck optimization problems. J. Algorithms 9(3), 411–417 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  14. Takaoka, T.: Sharing information for the all pairs shortest path problem. Theor. Comput. Sci. 520, 43–50 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Takaoka, T.: Subcubic cost algorithms for the all pairs shortest path problem. Algorithmica 20(3), 309–318 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Shinn, T.-W., Takaoka, T.: Combining the shortest paths and the bottleneck paths problems. In: ACSC, pp. 13–18 (2014)

    Google Scholar 

  17. Shinn, T.-W., Takaoka, T.: Some extensions of the bottleneck paths problem. In: Pal, S.P., Sadakane, K. (eds.) WALCOM 2014. LNCS, vol. 8344, pp. 176–187. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  18. Shinn, T.-W., Takaoka, T.: Combining all pairs shortest paths and all pairs bottleneck paths problems. In: Pardo, A., Viola, A. (eds.) LATIN 2014. LNCS, vol. 8392, pp. 226–237. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  19. Thorup, M.: Integer priority queues with decrease key in constant time and the single source shortest paths problem. In: STOC 2003, pp. 149–158 (2003)

    Google Scholar 

  20. Zwanevelt, P.J., Kroon, L.G., Romeijn, H.E., Salomon, M., Dauzere-Peres, S., Van Hoesel, S.P.M., Ambergen, H.W.: Routing trains through railway stations: model formulation and algorithms. Transp. Sci. 30(3), 181–194 (1996)

    Article  MATH  Google Scholar 

  21. Zwick, U.: All pairs shortest paths using bridging sets and rectangular matrix multiplication. J. ACM 49(3), 289–317 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Canterbury, Christchurch, New Zealand

    Tadao Takaoka

Authors
  1. Tadao Takaoka
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Correspondence to Tadao Takaoka .

Editor information

Editors and Affiliations

  1. Dept. of Mathematics and Computer Science, Marywood University, Scranton, Pennsylvania, USA

    Zaixin Lu

  2. North Carolina Central University, Durham, North Carolina, USA

    Donghyun Kim

  3. University of Texas at Dallas, Richardson, Texas, USA

    Weili Wu

  4. Texas Southern University, Houston, Texas, USA

    Wei Li

  5. University of Texas at Dallas, Richardson, Texas, USA

    Ding-Zhu Du

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Takaoka, T. (2015). Algebraic Theory on Shortest Paths for All Flows. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, DZ. (eds) Combinatorial Optimization and Applications. Lecture Notes in Computer Science(), vol 9486. Springer, Cham. https://doi.org/10.1007/978-3-319-26626-8_55

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  • DOI: https://doi.org/10.1007/978-3-319-26626-8_55

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-26625-1

  • Online ISBN: 978-3-319-26626-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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