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.


  1. 1.
    Biedl, T.C., Kant, G.: A better heuristic for orthogonal graph drawings. Computational Geometry 9(3), 159–180 (1998)CrossRefMATHGoogle 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)CrossRefMATHGoogle 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)CrossRefMATHGoogle 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)CrossRefMATHGoogle 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)MATHGoogle Scholar
  14. 14.
    Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4(4), 373–395 (1984)CrossRefMATHGoogle 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 1994CrossRefMATHGoogle 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)CrossRefMATHGoogle 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)CrossRefMATHGoogle 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)CrossRefMATHGoogle 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)CrossRefMATHGoogle 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)CrossRefMATHGoogle 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)CrossRefMATHGoogle 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