A Global k-Level Crossing Reduction Algorithm

  • Christian Bachmaier
  • Franz J. Brandenburg
  • Wolfgang Brunner
  • Ferdinand Hübner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5942)


Directed graphs are commonly drawn by the Sugiyama algorithm, where crossing reduction is a crucial phase. It is done by repeated one-sided 2-level crossing minimizations, which are still \({\mathcal{NP}}\)-hard.

We introduce a global crossing reduction, which at any particular time captures all crossings, especially for long edges. Our approach is based on the sifting technique and improves the level-by-level heuristics in the hierarchic framework by a further reduction of the number of crossings by 5 – 10%. In addition it avoids type 2 conflicts which help to straighten the edges, and has a running time which is quadratic in the size of the input graph independently of dummy vertices. Finally, the approach can directly be extended to cyclic, radial, and clustered level graphs where it achieves similar improvements over the previous algorithms.


Outer Segment Input Graph Binary Decision Diagram Local View Global Median 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bachmaier, C.: A radial adaption of the sugiyama framework for visualizing hierarchical information. IEEE Trans. Vis. Comput. Graphics 13(3), 583–594 (2007)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bachmaier, C., Brunner, W.: Linear time planarity testing and embedding of strongly connected cyclic level graphs. In: Halperin, D., Mehlhorn, K. (eds.) ESA 2008. LNCS, vol. 5193, pp. 136–147. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Baur, M., Brandes, U.: Crossing reduction in circular layouts. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 332–343. Springer, Heidelberg (2004)Google Scholar
  4. 4.
    Brandes, U., Köpf, B.: Fast and simple horizontal coordinate assignment. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 31–44. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Eades, P., Kelly, D.: Heuristics for reducing crossings in 2-layered networks. Ars Combinatorica 21(A), 89–98 (1986)MathSciNetGoogle Scholar
  6. 6.
    Eades, P., Wormald, N.C.: Edge crossings in drawings of bipartite graphs. Algorithmica 11(1), 379–403 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Eiglsperger, M., Siebenhaller, M., Kaufmann, M.: An efficient implementation of sugiyama’s algorithm for layered graph drawing. J. Graph Alg. App. 9(3), 305–325 (2005)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Jünger, M., Lee, E.K., Mutzel, P., Odenthal, T.: A polyhedral approach to the multi-layer crossing minimization problem. In: Di Battista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 13–24. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  9. 9.
    Kaufmann, M., Wagner, D. (eds.): Drawing Graphs. LNCS, vol. 2025. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  10. 10.
    Matuszewski, C., Schönfeld, R., Molitor, P.: Using sifting for k-layer straightline crossing minimization. In: Kratochvíl, J. (ed.) GD 1999. LNCS, vol. 1731, pp. 217–224. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  11. 11.
    Rudell, R.: Dynamic variable ordering for ordered binary decision diagrams. In: Proc. IEEE/ACM International Conference on Computer Aided Design, ICCAD 1993, pp. 42–47. IEEE Computer Society Press, Los Alamitos (1993)Google Scholar
  12. 12.
    Sugiyama, K., Tagawa, S., Toda, M.: Methods for visual understanding of hierarchical system structures. IEEE Trans. Syst., Man, Cybern. 11(2), 109–125 (1981)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Christian Bachmaier
    • 1
  • Franz J. Brandenburg
    • 1
  • Wolfgang Brunner
    • 1
  • Ferdinand Hübner
    • 1
  1. 1.University of PassauGermany

Personalised recommendations