SSDE: Fast Graph Drawing Using Sampled Spectral Distance Embedding

  • Ali Çivril
  • Malik Magdon-Ismail
  • Eli Bocek-Rivele
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4372)

Abstract

We present a fast spectral graph drawing algorithm for drawing undirected connected graphs. Classical Multi-Dimensional Scaling yields a quadratic-time spectral algorithm, which approximates the real distances of the nodes in the final drawing with their graph theoretical distances. We build from this idea to develop the linear-time spectral graph drawing algorithm SSDE. We reduce the space and time complexity of the spectral decomposition by approximating the distance matrix with the product of three smaller matrices, which are formed by sampling rows and columns of the distance matrix. The main advantages of our algorithm are that it is very fast and it gives aesthetically pleasing results, when compared to other spectral graph drawing algorithms. The runtime for typical 105 node graphs is about one second and for 106 node graphs about ten seconds.

References

  1. 1.
  2. 2.
  3. 3.
  4. 4.
    Borg, I., Groenen, P.: Modern Multidimensional Scaling. Springer, Heidelberg (1997)MATHGoogle Scholar
  5. 5.
    Çivril, A., Magdon-Ismail, M., Bocek-Rivele, E.: SDE: Graph drawing using spectral distance embedding. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 512–513. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Fruchterman, T.M.J., Reingold, E.M.: Graph drawing by force-directed placement. Software - Practice And Experience 21(11), 1129–1164 (1991)CrossRefGoogle Scholar
  7. 7.
    Golub, G.H., Van Loan, C.H.: Matrix Computations. Johns Hopkins U. Press, Baltimore (1996)MATHGoogle Scholar
  8. 8.
    Hall, K.M.: An r-dimensional quadratic placement algorithm. Management Science 17, 219–229 (1970)MATHGoogle Scholar
  9. 9.
    Harel, D., Koren, Y.: A fast multi-scale method for drawing large graphs. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 183–196. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Harel, D., Koren, Y.: Graph drawing by high-dimensional embedding. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, Springer, Heidelberg (2002)CrossRefGoogle Scholar
  11. 11.
    Johnson, W., Lindenstrauss, J.: Extensions of lipschitz maps into a hilbert space. Contemp. Math. 26, 189–206 (1984)MATHMathSciNetGoogle Scholar
  12. 12.
    Kamada, T., Kawai, S.: An algorithm for drawing general undirected graphs. Information Processing Letters 31(1), 7–15 (1989)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Kaufmann, M., Wagner, D. (eds.): Drawing Graphs. LNCS, vol. 2025. Springer, Heidelberg (2001)MATHGoogle Scholar
  14. 14.
    Koren, Y.: On spectral graph drawing. In: Warnow, T.J., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 496–508. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. 15.
    Koren, Y.: One dimensional layout optimization, with applications to graph drawing by axis separation. Computational Geometry: Theory and Applications 32, 115–138 (2005)MATHMathSciNetGoogle Scholar
  16. 16.
    Koren, Y., Harel, D., Carmel, L.: Drawing huge graphs by algebraic multigrid optimization. Multiscale Modeling and Simulation 1(4), 645–673 (2003)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kruskal, J.B., Seery, J.B.: Designing network diagrams. In: Proc. First General Conference on Social Graphics (1980)Google Scholar
  18. 18.
    Maeda, J., Murata, K.: Restoration of band-limited images by an iterative regularized pseudoinverse method. Journal of Optical Society of America 1(1), 28–34 (1984)CrossRefGoogle Scholar
  19. 19.
    Matousek, J.: Open problems on embeddings of finite metric spaces. Discr. Comput. Geom. (to appear)Google Scholar
  20. 20.
    Drineas, P., Kannan, R., Mahoney, M.W.: Fast Monte Carlo algorithms for matrices III: Computing a compressed approximate matrix decomposition. SIAM Journal on Computing 36(1), 184–206 (2006)MATHMathSciNetGoogle Scholar
  21. 21.
    Platt, J.C.: FastMap, MetricMap, and landmarkMDS are all Nystrom algorithms. In: Proc. 10th Int. Workshop on Artificial Intelligence and Statistics, pp. 261–268 (2005)Google Scholar
  22. 22.
    Tollis, I.G., Di Battista, G., Eades, P., Tamassia, R.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, Englewood Cliffs (1999)MATHGoogle Scholar
  23. 23.
    Vazirani, V.: Approximation Algorithms. Springer, Heidelberg (2001)Google Scholar
  24. 24.
    Walshaw, C.: A multilevel algorithm for force-directed graph drawing. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Ali Çivril
    • 1
  • Malik Magdon-Ismail
    • 1
  • Eli Bocek-Rivele
    • 1
  1. 1.Computer Science Department, RPI, 110 8th Street, Troy, NY 12180 

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