Accelerated Bend Minimization

  • Sabine Cornelsen
  • Andreas Karrenbauer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7034)


We present an \(\mathcal O( n^{3/2})\) algorithm for minimizing the number of bends in an orthogonal drawing of a plane graph. It has been posed as a long standing open problem at Graph Drawing 2003, whether the bound of \(\mathcal O(n^{7/4}\sqrt{\log n})\) shown by Garg and Tamassia in 1996 could be improved. To answer this question, we show how to solve the uncapacitated min-cost flow problem on a planar bidirected graph with bounded costs and face sizes in \(\mathcal O(n^{3/2})\) time.


Planar Graph Graph Draw Minimum Cost Flow Orthogonal Representation Residual Network 
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.


  1. 1.
    Biedl, T.C., Kant, G.: A better heuristic for orthogonal graph drawings. Computational Geometry 9(3), 159–180 (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM Journal on Computing 31(2), 601–625 (2001)CrossRefzbMATHGoogle Scholar
  3. 3.
    Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM Journal on Computing 16, 421–444 (1987)CrossRefzbMATHGoogle Scholar
  4. 4.
    Fößmeier, U., Kaufmann, M.: Drawing High Degree Graphs with Low Bend Numbers. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 254–266. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  5. 5.
    Klau, G.W., Mutzel, P.: Quasi orthogonal drawing of planar graphs. Technical Report MPI-I-98-1-013, Max-Planck-Institut für Informatik, Saarbrücken, Germany (1998),
  6. 6.
    Tamassia, R., Di Battista, G., Batini, C.: Automatic graph drawing and readability of diagrams. IEEE Transactions on Systems, Man and Cybernetics 18(1), 61–79 (1988)CrossRefGoogle Scholar
  7. 7.
    Bertolazzi, P., Di Battista, G., Didimo, W.: Computing orthogonal drawings with the minimum number of bends. IEEE Transactions on Computers 49(8), 826–840 (2000)CrossRefGoogle Scholar
  8. 8.
    Brandes, U., Cornelsen, S., Fieß, C., Wagner, D.: How to draw the minimum cuts of a planar graph. Computational Geometry: Theory and Applications 29(2), 117–133 (2004)CrossRefzbMATHGoogle Scholar
  9. 9.
    Lütke-Hüttmann, D.: Knickminimales Zeichnen 4-planarer Clustergraphen. Master’s thesis, Universität des Saarlandes (1999) (Diplomarbeit)Google Scholar
  10. 10.
    Brandes, U., Wagner, D.: Dynamic Grid Embedding with Few Bends and Changes. In: Chwa, K.-Y., Ibarra, O.H. (eds.) ISAAC 1998. LNCS, vol. 1533, pp. 89–98. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  11. 11.
    Brandes, U., Eiglsperger, M., Kaufmann, M., Wagner, D.: Sketch-Driven Orthogonal Graph Drawing. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 1–11. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  12. 12.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows. Prentice-Hall (1993)Google Scholar
  13. 13.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg (2003)zbMATHGoogle Scholar
  14. 14.
    Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4(4), 373–395 (1984)CrossRefzbMATHGoogle Scholar
  15. 15.
    Imai, H., Iwano, K.: Efficient Sequential and Parallel Algorithms for Planar Minimum Cost Flow. In: Asano, T., Imai, H., Ibaraki, T., Nishizeki, T. (eds.) SIGAL 1990. LNCS, vol. 450, pp. 21–30. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  16. 16.
    Henzinger, M.R., Klein, P., Rao, S., Subramanian, S.: Faster shortest-path algorithms for planar graphs. Journal of Computer and System Sciences 55, 3–23 (1997); Special Issue on Selected Papers from STOC 1994CrossRefzbMATHGoogle Scholar
  17. 17.
    Fakcharoenphol, J., Rao, S.: Planar graphs, negative weight edges, shortest paths, and near linear time. J. Comput. Syst. Sci. 72, 868–889 (2006)CrossRefzbMATHGoogle Scholar
  18. 18.
    Klein, P., Mozes, S., Weimann, O.: Shortest paths in directed planar graphs with negative lengths: a linear-space O(n log2 n)-time algorithm. In: Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, pp. 236–245. SIAM, Philadelphia (2009)CrossRefGoogle Scholar
  19. 19.
    Mozes, S., Wulff-Nilsen, C.: Shortest Paths in Planar Graphs with Real Lengths in O(n log2 n / loglogn) Time. In: de Berg, M., Meyer, U. (eds.) ESA 2010. LNCS, vol. 6347, pp. 206–217. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  20. 20.
    Weihe, K.: Maximum (s,t)-flows in planar networks in O(V log V) time. J. Comput. Syst. Sci. 55, 454–475 (1997)CrossRefzbMATHGoogle Scholar
  21. 21.
    Borradaile, G., Klein, P.: An O(n log n) algorithm for maximum st-flow in a directed planar graph. J. ACM 56, 9:1–9:30 (2009)Google Scholar
  22. 22.
    Hassin, R.: Maximum flow in (s,t) planar networks. Information Processing Letters 13(3), 107 (1981)CrossRefGoogle Scholar
  23. 23.
    Miller, G.L., Naor, J.: Flow in planar graphs with multiple sources and sinks. SIAM J. Comput. 24, 1002–1017 (1995)CrossRefzbMATHGoogle Scholar
  24. 24.
    Garg, A., Tamassia, R.: A New Minimum Cost Flow Algorithm with Applications to Graph Drawing. In: North, S.C. (ed.) GD 1996. LNCS, vol. 1190, pp. 201–213. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  25. 25.
    Brandenburg, F.J., Eppstein, D., Goodrich, M.T., Kobourov, S.G., Liotta, G., Mutzel, P.: Selected Open Problems in Graph Drawing. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 515–539. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  26. 26.
    Miller, G.L.: Finding small simple cycle separators for 2-connected planar graphs. Journal of Computer and System Sciences 32(4), 265–279 (1986)CrossRefzbMATHGoogle Scholar
  27. 27.
    Borradaile, G., Klein, P., Mozes, S., Nussbaum, Y., Wulff-Nilsen, C.: Multiple-source multiple-sink maximum flow in directed planar graphs in near-linear time. In: Proceedings of the 52nd Annual Symposium on Foundations of Computer Science, FOCS  2011 (to appear, 2011)Google Scholar
  28. 28.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall (1999)Google Scholar
  29. 29.
    Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press (1962)Google Scholar
  30. 30.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271 (1959)CrossRefzbMATHGoogle Scholar
  31. 31.
    Tazari, S., Müller-Hannemann, M.: Shortest paths in linear time on minor-closed graph classes, with an application to steiner tree approximation. Discrete Applied Mathematics 157(4), 673–684 (2009)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sabine Cornelsen
    • 1
  • Andreas Karrenbauer
    • 1
  1. 1.Department of Computer & Information ScienceUniversity of KonstanzGermany

Personalised recommendations