Skip to main content
SpringerLink
Log in

Use of dynamic trees in a network simplex algorithm for the maximum flow problem

  • Published:
Mathematical Programming Submit manuscript

Abstract

Goldfarb and Hao (1990) have proposed a pivot rule for the primal network simplex algorithm that will solve a maximum flow problem on ann-vertex,m-arc network in at mostnm pivots and O(n 2 m) time. In this paper we describe how to extend the dynamic tree data structure of Sleator and Tarjan (1983, 1985) to reduce the running time of this algorithm to O(nm logn). This bound is less than a logarithmic factor larger than those of the fastest known algorithms for the problem. Our extension of dynamic trees is interesting in its own right and may well have additional applications.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R.K. Ahuja and J.B. Orlin, “A fast and simple algorithm for the maximum flow problem,”Operations Research 37 (1989) 748–759.

    Google Scholar 

  2. R.K. Ahuja, J.B. Orlin and R.E. Tarjan, “Improved time bounds for the maximum flow problem,”SIAM Journal on Computing 18 (1989) 939–954.

    Google Scholar 

  3. V. Chvatal,Linear Programming (Freeman, New York, 1983).

    Google Scholar 

  4. W.H. Cunningham, “A network simplex method,”Mathematical Programming 1 (1976) 105–116.

    Google Scholar 

  5. G.L. Dantzig,Linear Programming and Extensions (Princeton University Press, Princeton, NJ, 1963).

    Google Scholar 

  6. E.A. Dinic, “Algorithm for solution of a problem of maximum flow in networks with power estimation,”Soviet Mathematics Doklady 11 (1970) 1277–1280.

    Google Scholar 

  7. L.R. Ford, Jr. and D.R. Fulkerson,Flows in Networks (Princeton University Press, Princeton, NJ, 1962).

    Google Scholar 

  8. D.R. Fulkerson and G.B. Dantzig, “Computations of maximal flows in networks,”Naval Research Logistics Quarterly 2 (1955) 277–283.

    Google Scholar 

  9. A.V. Goldberg, “A new max-flow algorithm,” Technical Report MIT/LCS/TM-291, Laboratory for Computer Science, Massachusetts Institute of Technology (Cambridge, MA, 1985).

    Google Scholar 

  10. A.V. Goldberg and R.E. Tarjan, “A new approach to the maximum flow problem,”Journal of the Association of Computing Machinery 35 (1988) 921–940.

    Google Scholar 

  11. D. Goldfarb and M.D. Grigoriadis, “A computational comparison of the Dinic and network simplex methods for maximum flow,”Annals of Operations Research 13 (1988) 83–123.

    Google Scholar 

  12. D. Goldfarb and J. Hao, “A primal simplex algorithm that solves the maximum flow problem in at mostnm pivots and O(n 2 m) time,”,Mathematical Programming 47 (1990) 353–365.

    Google Scholar 

  13. M.D. Grigoriadis, “An efficient implementation of the primal simplex method,”Mathematical Programming Study 26 (1986) 83–111.

    Google Scholar 

  14. J.L. Kennington and R.V. Helgason,Algorithms for Network Programming (Wiley, New York, 1980).

    Google Scholar 

  15. E.L. Lawler,Combinatorial Optimization: Networks and Matroids (Holt, Reinhart, and Winston, New York, 1976).

    Google Scholar 

  16. C.H. Papadimitriou and K. Steiglitz,Combinatorial Optimization: Algorithms and Complexity (Prentice-Hall, Englewood Cliffs, NJ, 1982).

    Google Scholar 

  17. D.D. Sleator and R.E. Tarjan, “A data structure for dynamic trees,”Journal of Computer and System Sciences 26 (1983) 362–391.

    Google Scholar 

  18. D.D. Sleator and R.E. Tarjan, “Self-adjusting binary search trees,”Journal of the Association of Computing Machinery 32 (1985) 652–686.

    Google Scholar 

  19. R.E. Tarjan,Data Structures and Network Algorithms (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1983).

    Google Scholar 

  20. R.E. Tarjan, “Amortized computational complexity,”SIAM Journal on Algebraic and Discrete Methods 6 (1985) 306–318.

    Google Scholar 

  21. R.E. Tarjan, “Efficiency of the primal network simplex algorithm for the minimum-cost circulation problem,” to appear in:Mathematics of Operations Research (1991).

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Stanford University, 94305, Stanford, CA, USA

    Andrew V. Goldberg

  2. Department of Computer Science, Rutgers University, 08903, New Brunswick, NJ, USA

    Michael D. Grigoriadis

  3. Department of Computer Science, Princeton University, 08544, Princeton, NJ, USA

    Robert E. Tarjan

  4. AT&T Bell Laboratories, 07974, Murray Hill, NY, USA

    Robert E. Tarjan

Authors
  1. Andrew V. Goldberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael D. Grigoriadis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Robert E. Tarjan
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research partially supported by a Presidential Young Investigator Award from the National Science Foundation, Grant No. CCR-8858097, an IBM Faculty Development Award, and AT&T Bell Laboratories.

Research partially supported by the Office of Naval Research, Contract No. N00014-87-K-0467.

Research partially supported by the National Science Foundation, Grant No. DCR-8605961, and the Office of Naval Research, Contract No. N00014-87-K-0467.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goldberg, A.V., Grigoriadis, M.D. & Tarjan, R.E. Use of dynamic trees in a network simplex algorithm for the maximum flow problem. Mathematical Programming 50, 277–290 (1991). https://doi.org/10.1007/BF01594940

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01594940

Key words

Search

Navigation