Abstract
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.
Supported by the Deutsche Forschungsgemeinschaft (DFG), grant BR835/15-1.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bachmaier, C.: A radial adaption of the sugiyama framework for visualizing hierarchical information. IEEE Trans. Vis. Comput. Graphics 13(3), 583–594 (2007)
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)
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)
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)
Eades, P., Kelly, D.: Heuristics for reducing crossings in 2-layered networks. Ars Combinatorica 21(A), 89–98 (1986)
Eades, P., Wormald, N.C.: Edge crossings in drawings of bipartite graphs. Algorithmica 11(1), 379–403 (1994)
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)
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)
Kaufmann, M., Wagner, D. (eds.): Drawing Graphs. LNCS, vol. 2025. Springer, Heidelberg (2001)
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)
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)
Sugiyama, K., Tagawa, S., Toda, M.: Methods for visual understanding of hierarchical system structures. IEEE Trans. Syst., Man, Cybern. 11(2), 109–125 (1981)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bachmaier, C., Brandenburg, F.J., Brunner, W., Hübner, F. (2010). A Global k-Level Crossing Reduction Algorithm. In: Rahman, M.S., Fujita, S. (eds) WALCOM: Algorithms and Computation. WALCOM 2010. Lecture Notes in Computer Science, vol 5942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11440-3_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-11440-3_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11439-7
Online ISBN: 978-3-642-11440-3
eBook Packages: Computer ScienceComputer Science (R0)