A new minimum cost flow algorithm with applications to graph drawing

  • Ashim Garg
  • Roberto Tamassia
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1190)

Abstract

Let N be a single-source single-sink flow network with n nodes, m arcs, and positive arc costs. We present a pseudo-polynomial algorithm that computes a maximum flow of minimum cost for N in time O(χ3/4m√log n), where χ is the cost of the flow. This improves upon previously known methods for networks where the minimum cost of the flow is small. We also show an application of our flow algorithm to a well-known graph drawing problem. Namely, we show how to compute a planar orthogonal drawing with the minimum number of bends for an n- vertex embedded planar graph in time O(n7/4√log n). This is the first subquadratic algorithm for bend minimization. The previous best bound for this problem was O(n2 log n) [19].

References

  1. 1.
    R.K. Ahuja, T.L. Magnanti, and J.B. Orlin. Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs, NJ, 1993.Google Scholar
  2. 2.
    T. Biedl and G. Kant. A better heuristic for orthogonal graph drawings. In Proc. 2nd Annu. European Sympos. Algorithms (ESA '94), volume 855 of Lecture Notes in Computer Science, pages 24–35. Springer-Verlag, 1994.Google Scholar
  3. 3.
    G. Di Battista, P. Eades, R. Tamassia, and I. G. Tollis. Algorithms for drawing graphs: an annotated bibliography. Comput. Geom. Theory Appl., 4:235–282, 1994.Google Scholar
  4. 4.
    G. Di Battista, A. Garg, G. Liotta, R. Tamassia, E. Tassinari, and F. Vargiu. An experimental comparison of three graph drawing algorithms. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 306–315, 1995.Google Scholar
  5. 5.
    G. Di Battista, A. Giammarco, G. Santucci, and R. Tamassia. The architecture of Diagram Server. In Proc. IEEE Workshop on Visual Languages (VL '90), pages 60–65, 1990.Google Scholar
  6. 6.
    G. Di Battista, G. Liotta, and F. Vargiu. Spirality of orthogonal representations and optimal drawings of series-parallel graphs and 3-planar graphs. In Proc. Workshop Algorithms Data Struct., volume 709 of Lecture Notes in Computer Science, pages 151–162. Springer-Verlag, 1993.Google Scholar
  7. 7.
    G. Di Battista, G. Liotta, and F. Vargiu. Diagram Server. J. Visual Lang. Comput., 6(3):275–298, 1995. (special issue on Graph Visualization, edited by I. F. Cruz and P. Eades).CrossRefGoogle Scholar
  8. 8.
    S. Even and G. Granot. Grid layouts of block diagrams — bounding the number of bends in each connection. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 64–75. Springer-Verlag, 1995.Google Scholar
  9. 9.
    L.R. Ford and D.R. Fulkerson. A primal-dual algorithm for the capacitated hitchcock problem. Naval Research Logistics Quarterly, 4:47–54, 1957.Google Scholar
  10. 10.
    L.R. Ford and D.R. Fulkerson. Flows in Networks. Princeton University Press, Princeton, NJ, 1962.Google Scholar
  11. 11.
    U. Fößmeier and M. Kaufmann. On bend-minimum orthogonal upward drawing of directed planar graphs. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 52–63. Springer-Verlag, 1995.Google Scholar
  12. 12.
    A. Garg and R. Tamassia. On the computational complexity of upward and rectilinear planarity testing. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 286–297. Springer-Verlag, 1995.Google Scholar
  13. 13.
    G. Kant. Drawing planar graphs using the canonical ordering. Algorithmica, 16:4–32, 1996. (special issue on Graph Drawing, edited by G. Di Battista and R. Tamassia).CrossRefGoogle Scholar
  14. 14.
    Y. Liu, P. Marchioro, R. Petreschi, and B. Simeone. Theoretical results on at most 1-bend embeddability of graphs. Technical report, Dipartimento di Statistica, Univ. di Roma “La Sapienza”, 1990.Google Scholar
  15. 15.
    K. Mehlhorn. Graph Algorithms and NP-Completeness, volume 2 of Data Structures and Algorithms. Springer-Verlag, Heidelberg, West Germany, 1984.Google Scholar
  16. 16.
    A. Papakostas and I. G. Tollis. Improved algorithms and bounds for orthogonal drawings. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 40–51. Springer-Verlag, 1995.Google Scholar
  17. 17.
    D.D. Sleator. An O(nm log n) Algorithm for Maximum Network Flow. PhD thesis, Dept. Comput. Sci., Stanford Univ., Palo Alto, California, 1980. Google Scholar
  18. 18.
    J. A. Storer. On minimal node-cost planar embeddings. Networks, 14:181–212, 1984.Google Scholar
  19. 19.
    R. Tamassia. On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput, 16(3):421–444, 1987.CrossRefGoogle Scholar
  20. 20.
    R. Tamassia and I. G. Tollis. Planar grid embedding in linear time. IEEE Trans. Circuits Syst., CAS-36(9):1230–1234, 1989.CrossRefGoogle Scholar
  21. 21.
    R. E. Tarjan. Data Structures and Network Algorithms, volume 44 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial Applied Mathematics, 1983.Google Scholar
  22. 22.
    L. Valiant. Universality considerations in VLSI circuits. IEEE Trans. Comput., C-30(2):135–140, 1981.Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Ashim Garg
    • 1
  • Roberto Tamassia
    • 1
  1. 1.Department of Computer ScienceBrown UniversityProvidenceUSA

Personalised recommendations