Bipartite Graph Matching for Computing the Edit Distance of Graphs

  • Kaspar Riesen
  • Michel Neuhaus
  • Horst Bunke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4538)

Abstract

In the field of structural pattern recognition graphs constitute a very common and powerful way of representing patterns. In contrast to string representations, graphs allow us to describe relational information in the patterns under consideration. One of the main drawbacks of graph representations is that the computation of standard graph similarity measures is exponential in the number of involved nodes. Hence, such computations are feasible for rather small graphs only. One of the most flexible error-tolerant graph similarity measures is based on graph edit distance. In this paper we propose an approach for the efficient compuation of edit distance based on bipartite graph matching by means of Munkres’ algorithm, sometimes referred to as the Hungarian algorithm. Our proposed algorithm runs in polynomial time, but provides only suboptimal edit distance results. The reason for its suboptimality is that implied edge operations are not considered during the process of finding the optimal node assignment. In experiments on semi-artificial and real data we demonstrate the speedup of our proposed method over a traditional tree search based algorithm for graph edit distance computation. Also we show that classification accuracy remains nearly unaffected.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Conte, D., Foggia, P., Sansone, C., Vento, M.: Thirty years of graph matching in pattern recognition. Int. Journal of Pattern Recognition and Artificial Intelligence 18(3), 265–298 (2004)CrossRefGoogle Scholar
  2. 2.
    Umeyama, S.: An eigendecomposition approach to weighted graph matching problems. IEEE Transactions on Pattern Analysis and Machine Intelligence 10(5), 695–703 (1988)MATHCrossRefGoogle Scholar
  3. 3.
    Luo, B., Wilson, R., Hancock, E.R.: Spectral embedding of graphs. Pattern Recognition 36(10), 2213–2223 (2003)MATHCrossRefGoogle Scholar
  4. 4.
    Christmas, W.J., Kittler, J., Petrou, M.: Structural matching in computer vision using probabilistic relaxation. IEEE Transactions on Pattern Analysis and Machine Intelligence 17(8), 749–764 (1995)CrossRefGoogle Scholar
  5. 5.
    Suganthan, P.N., Teoh, E.K., Mital, D.P.: Pattern recognition by graph matching using the potts MFT neural networks. Pattern Recognition 28(7), 997–1009 (1995)CrossRefGoogle Scholar
  6. 6.
    Cross, A., Wilson, R., Hancock, E.: Inexact graph matching using genetic search. Pattern Recognition 30(6), 953–970 (1997)CrossRefGoogle Scholar
  7. 7.
    Gori, M., Maggini, M., Sarti, L.: Exact and approximate graph matching using random walks. IEEE Transactions on Pattern Analysis and Machine Intelligence 27(7), 1100–1111 (2005)CrossRefGoogle Scholar
  8. 8.
    Bunke, H., Allermann, G.: Inexact graph matching for structural pattern recognition. Pattern Recognition Letters 1, 245–253 (1983)MATHCrossRefGoogle Scholar
  9. 9.
    Sanfeliu, A., Fu, K.S.: A distance measure between attributed relational graphs for pattern recognition. IEEE Transactions on Systems, Man, and Cybernetics (Part B) 13(3), 353–363 (1983)MATHGoogle Scholar
  10. 10.
    Neuhaus, M., Bunke, H.: Edit distance based kernel functions for structural pattern classification. Pattern Recognition 39(10), 1852–1863 (2006)MATHCrossRefGoogle Scholar
  11. 11.
    Neuhaus, M., Bunke, H.: An error-tolerant approximate matching algorithm for attributed planar graphs and its application to fingerprint classification. In: Fred, A., Caelli, T.M., Duin, R., Campilho, A., de Ridder, D. (eds.) Structural, Syntactic, and Statistical Pattern Recognition. LNCS, vol. 3138, pp. 180–189. Springer, Heidelberg (2004)Google Scholar
  12. 12.
    Ambauen, R., Fischer, S., Bunke, H.: Graph edit distance with node splitting and merging and its application to diatom identification. In: Hancock, E., Vento, M. (eds.) GbRPR 2003. LNCS, vol. 2726, pp. 95–106. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Robles-Kelly, A., Hancock, E.R.: Graph edit distance from spectral seriation. IEEE Transactions on Pattern Analysis and Machine Intelligence 27(3), 365–378 (2005)CrossRefGoogle Scholar
  14. 14.
    Boeres, M.C., Ribeiro, C.C., Bloch, I.: A randomized heuristic for scene recognition by graph matching. In: Ribeiro, C.C., Martins, S.L. (eds.) WEA 2004. LNCS, vol. 3059, pp. 100–113. Springer, Heidelberg (2004)Google Scholar
  15. 15.
    Sorlin, S., Solnon, C.: Reactive tabu search for measuring graph similarity. In: Brun, L., Vento, M. (eds.) GbRPR 2005. LNCS, vol. 3434, pp. 172–182. Springer, Heidelberg (2005)Google Scholar
  16. 16.
    Justice, D., Hero, A.: A binary linear programming formulation of the graph edit distance. IEEE Trans. on Pattern Analysis ans Machine Intelligence 28(8), 1200–1214 (2006)CrossRefGoogle Scholar
  17. 17.
    Munkres, J.: Algorithms for the assignment and transportation problems. Journal of the Society for Industrial and Applied Mathematics 5, 32–38 (1957)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Hart, P.E., Nilsson, N.J., Raphael, B.: A formal basis for the heuristic determination of minimum cost paths. IEEE Transactions of Systems, Science, and Cybernetics 4(2), 100–107 (1968)CrossRefGoogle Scholar
  19. 19.
    Neuhaus, M., Riesen, K., Bunke, H.: Fast suboptimal algorithms for the computation of graph edit distance. In: Ribeiro, C.C., Martins, S.L. (eds.) WEA 2004. LNCS, vol. 3059, pp. 163–172. Springer, Heidelberg (2005)Google Scholar
  20. 20.
    Le Saux, B., Bunke, H.: Feature selection for graph-based image classifiers. In: Marques, J.S., Pérez de la Blanca, N., Pina, P. (eds.) IbPRIA 2005. LNCS, vol. 3523, pp. 147–154. Springer, Heidelberg (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Kaspar Riesen
    • 1
  • Michel Neuhaus
    • 1
  • Horst Bunke
    • 1
  1. 1.Department of Computer Science, University of Bern, Neubrückstrasse 10, CH-3012 BernSwitzerland

Personalised recommendations