Layout with Circular and Other Non-linear Constraints Using Procrustes Projection

  • Tim Dwyer
  • George Robertson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5849)

Abstract

Recent work on constrained graph layout has involved projection of simple two-variable linear equality and inequality constraints in the context of majorization or gradient-projection based optimization. While useful classes of containment, alignment and rectangular non-overlap constraints could be built using this framework, a severe limitation was that the layout used an axis-separation approach such that all constraints had to be axis aligned. In this paper we use techniques from Procrustes Analysis to extend the gradient-projection approach to useful types of non-linear constraints. The constraints require subgraphs to be locally fixed into various geometries—such as circular cycles or local layout obtained by a combinatorial algorithm (e.g. orthogonal or layered-directed)—but then allow these sub-graph geometries to be integrated into a larger layout through translation, rotation and scaling.

References

  1. 1.
    Baur, M., Brandes, U.: Multi-circular layout of micro/macro graphs. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 255–267. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    Becker, M.Y., Rojas, I.: A graph layout algorithm for drawing metabolic pathways. Bioinformatics 17(5), 461–467 (2001)CrossRefGoogle Scholar
  3. 3.
    Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1999)MATHGoogle Scholar
  4. 4.
    Borg, I., Groenen, P.J.F.: Modern Multidimensional Scaling: Theory and Applications, 2nd edn. Springer, Heidelberg (2005)MATHGoogle Scholar
  5. 5.
    Dwyer, T.: Scalable, versatile and simple constrained graph layout. In: Proc. Eurographics/IEEE-VGTC Symp. on Visualization (Eurovis 2009). IEEE, Los Alamitos (2009) (to appear)Google Scholar
  6. 6.
    Dwyer, T., Koren, Y., Marriott, K.: Drawing directed graphs using quadratic programming. IEEE Transactions on Visualization and Computer Graphics 12(4), 536–548 (2006)CrossRefGoogle Scholar
  7. 7.
    Dwyer, T., Koren, Y., Marriott, K.: IPSep-CoLa: an incremental procedure for separation constraint layout of graphs. IEEE Transactions on Visualization and Computer Graphics 12(5), 821–828 (2006)CrossRefGoogle Scholar
  8. 8.
    Dwyer, T., Marriott, K., Schreiber, F., Stuckey, P.J., Woodward, M., Wybrow, M.: Exploration of networks using overview+detail with constraint-based cooperative layout. IEEE Transactions on Visualization and Computer Graphics 14(6), 1293–1300 (2008)CrossRefGoogle Scholar
  9. 9.
    Dwyer, T., Marriott, K., Stuckey, P.: Fast node overlap removal. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 153–164. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Dwyer, T., Marriott, K., Wybrow, M.: Integrating edge routing into force-directed layout. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 8–19. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Dwyer, T., Marriott, K., Wybrow, M.: Dunnart: A constraint-based network diagram authoring tool. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 420–431. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Dwyer, T., Marriott, K., Wybrow, M.: Topology preserving constrained graph layout. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 230–241. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  13. 13.
    Everson, R.: Orthogonal, but not orthonormal, procrustes problems. Advances in Computational Mathematics (submitted) (1998), http://secamlocal.ex.ac.uk/people/staff/reverson/uploads/Site/procrustes.pdf
  14. 14.
    Friedrich, C., Eades, P.: Graph drawing in motion. Graph Algorithms and Applications 6(3), 353–370 (2002)MATHMathSciNetGoogle Scholar
  15. 15.
    Hu, Y.: Efficient and high quality force-directed graph drawing. The Mathematica Journal 10(1), 37–71 (2005)Google Scholar
  16. 16.
    Jakobsen, T.: Advanced character physics. In: San Jose Games Developers’ Conference (2001), http://www.gamasutra.com/resource_guide/20030121/jacobson_01.shtml
  17. 17.
    Lauther, U.: Multipole-based force approximation revisited - a simple but fast implementation using a dynamized enclosing-circle-enhanced k-d-tree. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 20–29. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  18. 18.
    Müller, M., Heidelberger, B., Hennix, M., Ratcliff, J.: Position based dynamics. In: Proc. of Virtual Reality Interactions and Physical Simulations (VRIPhys), pp. 71–80 (2006)Google Scholar
  19. 19.
    Six, J.M., Tollis, I.G.: A framework for user-grouped circular drawings. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 135–146. Springer, Heidelberg (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Tim Dwyer
    • 1
  • George Robertson
    • 1
  1. 1.Microsoft ResearchRedmondUSA

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