On the Hardness of Orthogonal-Order Preserving Graph Drawing

  • Ulrik Brandes
  • Barbara Pampel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5417)


There are several scenarios in which a given drawing of a graph is to be modified subject to preservation constraints. Examples include shape simplification, sketch-based, and dynamic graph layout. While the orthogonal ordering of vertices is a natural and frequently called for preservation constraint, we show that, unfortunately, it results in severe algorithmic difficulties even for the simplest graphs. More precisely, we show that orthogonal-order preserving rectilinear and uniform edge length drawing is \({\mathcal NP}\)-hard even for paths.


  1. 1.
    Agarwal, P., Har-Peled, S., Mustafa, N., Wang, Y.: Near-linear time approximation algorithms for path simplification. Algorithmica 42, 203–219 (2000)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bachmaier, C., Brandes, U., Schlieper, B.: Drawing phylogenetic trees. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 1110–1121. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Boehringer, K.-F., Newbery Paulisch, F.: Using constraints to achieve stability in automatic graph algorithms. In: Proc. of the ACM SIGCHI Conference on Human Factors in Computer Systems, pp. 43–51. WA (1990)Google Scholar
  4. 4.
    Bridgeman, S., Tamassia, R.: A user study in similarity measures for graph drawing. JGAA 6(3), 225–254 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bridgeman, S., Tamassia, R.: Difference metrics for interactive orthogonal graph drawing algorithms. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 57–71. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  6. 6.
    Carlson, J., Eppstein, D.: Trees with convex faces and optimal angles. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 77–88. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Dwyer, T., Koren, Y., Marriott, K.: Stress majorization with orthogonal order constraints. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 141–152. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Douglas, D., Peucker, T.: Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. Canad. Cartog. 10(2), 112–122 (1973)CrossRefGoogle Scholar
  9. 9.
    Eades, R., Lai, W., Misue, K., Sugiyama, K.: Layout adjustment and the mental map. J. Visual Lang. Comput. 6, 183–210 (1995)CrossRefGoogle Scholar
  10. 10.
    Eades, P., Wormald, N.: Fixed edge-length graph drawing is \(\mathcal {NP}\)-hard. Discrete Appl. Math. 28, 111–134 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Garey, M., Johnson, D.: Computers and Intractability. W. H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  12. 12.
    Imai, H., Iri, M.: An optimal algorithm for approximating a piecewise linear function. J. Inform. Process. 9(3), 159–162 (1986)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Lee, Y.-Y., Lin, C.-C., Yen, H.-C.: Mental map preserving graph drawing using simulated annealing. In: Proc. of the 2006 Asia-Pacific Symposium on Inforamtion Visualisation, pp. 179–188. Australian Computer Science (2006)Google Scholar
  14. 14.
    Merrick, D., Gudmundsson, J.: Path simplification for metro map layout. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 258–269. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Neyer, G.: Line simplification with restricted orientations. In: Dehne, F., Gupta, A., Sack, J.-R., Tamassia, R. (eds.) WADS 1999. LNCS, vol. 1663, pp. 13–24. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  16. 16.
    Nöllenburg, M., Wolff, A.: A mixed-integer program for drawing high-quality metro maps. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 321–333. Springer, Heidelberg (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Ulrik Brandes
    • 1
  • Barbara Pampel
    • 1
  1. 1.Department of Computer & Information ScienceUniversity of KonstanzGermany

Personalised recommendations