Axis-by-Axis Stress Minimization

  • Yehuda Koren
  • David Harel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2912)


Graph drawing algorithms based on minimizing the so-called stress energy strive to place nodes in accordance with target distances. They were first introduced to the graph drawing field by Kamada and Kawai [11], and they had previously been used to visualize general kinds of data by multidimensional scaling. In this paper we suggest a novel algorithm for the minimization of the Stress energy. Unlike prior stress-minimization algorithms, our algorithm is suitable for a one-dimensional layout, where one axis of the drawing is already given and an additional axis needs to be computed. This 1-D drawing capability of the algorithm is a consequence of replacing the traditional node-by-node optimization with a more global axis-by-axis optimization. Moreover, our algorithm can be used for multidimensional graph drawing, where it has time and space complexity advantages compared with other stress minimization algorithms.


  1. 1.
    Carmel, L., Harel, D., Koren, Y.: Drawing Directed Graphs Using One-Dimensional Optimization. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 193–206. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. 2.
    Carmel, L., Harel, D., Koren, Y.: Combining Hierarchy and Energy for Drawing Directed Graphs. IEEE Transactions on Visualization and Computer Graphics, IEEE (in press)Google Scholar
  3. 3.
    Cohen, J.D.: Drawing Graphs to Convey Proximity: an Incremental Arrangement Method. ACM Transactions on Computer-Human Interaction 4, 197–229 (1997)CrossRefGoogle Scholar
  4. 4.
    Gajer, P., Goodrich, M.T., Kobourov, S.G.: A Multi-dimensional Approach to Force-Directed Layouts of Large Graphs. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 211–221. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. 5.
    Hadany, R., Harel, D.: A Multi-Scale Method for Drawing Graphs Nicely. Discrete Applied Mathematics 113, 3–21 (2001)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Harel, D., Koren, Y.: A Fast Multi-Scale Method for Drawing Large Graphs. Journal of Graph Algorithms and Applications 6, 179–202 (2002)MATHMathSciNetGoogle Scholar
  7. 7.
    Harel, D., Koren, Y.: Graph Drawing by High-Dimensional Embedding. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 207–219. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  8. 8.
    Koren, Y., Carmel, L., Harel, D.: ACE: A Fast Multiscale Eigenvectors Computation for Drawing Huge Graphs. In: Proc. IEEE Information Visualization (InfoVis 2002), pp. 137–144. IEEE, Los Alamitos (2002)Google Scholar
  9. 9.
    Koren, Y., Harel, D.: One-Dimensional Graph Drawing: Part I — Drawing Graphs by Axis Separation. Technical report MCS03-08, Faculty of Math. and Computer Science, The Weizmann Institute of Science (2003)Google Scholar
  10. 10.
    Koren, Y.: Graph Drawing by Subspace Optimization (to be published)Google Scholar
  11. 11.
    Kamada, T., Kawai, S.: An Algorithm for Drawing General Undirected Graphs. Information Processing Letters 31, 7–15 (1989)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Sammon, J.W.: A Nonlinear Mapping for Data Structure Analysis. IEEE Trans. on Computers 18, 401–409 (1969)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Yehuda Koren
    • 1
  • David Harel
    • 1
  1. 1.Dept. of Computer Science and Applied MathematicsThe Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations