Abstract
In the field of structural pattern recognition, graphs provide us with a common and powerful way to represent objects. Yet, one of the main drawbacks of graph representation is that the computation of standard graph similarity measures is exponential in the number of involved nodes. Hence, such computations are feasible for small graphs only. The present paper considers the problem of graph isomorphism, i.e. checking two graphs for identity. A novel approach for the efficient computation of graph isomorphism is presented. The proposed algorithm is based on bipartite graph matching by means of Munkres’ algorithm. The algorithmic framework is suboptimal in the sense of possibly rejecting pairs of graphs without making a decision. As an advantage, however, it offers polynomial runtime. In experiments on two TC-15 graph sets we demonstrate substantial speedups of our proposed method over several standard procedures for graph isomorphism, such as Ullmann’s method, the VF2 algorithm, and Nauty. Furthermore, although the computational framework for isomorphism is suboptimal, we show that the proposed algorithm rejects only very few pairs of graphs.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Ullmann, J.: An algorithm for subgraph isomorphism. Journal of the Association for Computing Machinery 23(1), 31–42 (1976)
Cordella, L.P., Foggia, P., Sansone, C., Vento, M.: An improved algorithm for matching large graphs. In: Proc. 3rd Int. Workshop on Graph Based Representations in Pattern Recognition (2001)
Larrosa, J., Valiente, G.: Constraint satisfaction algorithms for graph pattern matching. Mathematical Structures in Computer Science 12(4), 403–422 (2002)
McKay, B.: Practical graph isomorphism. Congressus Numerantium 30, 45–87 (1981)
Emms, D., Hancock, E., Wilson, R.: A correspondence measure for graph matching using the discrete quantum walk. In: Escolano, F., Vento, M. (eds.) GbRPR 2007. LNCS, vol. 4538, pp. 81–91. Springer, Heidelberg (2007)
Messmer, B., Bunke, H.: A decision tree approach to graph and subgraph isomorphism detection. Pattern Recognition 32, 1979–1998 (2008)
Umeyama, S.: An eigendecomposition approach to weighted graph matching problems. IEEE Transactions on Pattern Analysis and Machine Intelligence 10(5), 695–703 (1988)
Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman and Co., New York (1979)
Aho, A., Hopcroft, J., Ullman, J.: The Design and Analysis of Computer Algorithms. Addison Wesley, Reading (1974)
Hopcroft, J., Wong, J.: Linear time algorithm for isomorphism of planar graphs. In: Proc. 6th Annual ACM Symposium on Theory of Computing, pp. 172–184 (1974)
Luks, E.: Isomorphism of graphs of bounded valence can be tested in polynomial time. Journal of Computer and Systems Sciences 25, 42–65 (1982)
Jiang, X., Bunke, H.: Optimal quadratic-time isomorphism of ordered graphs. Pattern Recognition 32(17), 1273–1283 (1999)
Dickinson, P., Bunke, H., Dadej, A., Kraetzl, M.: Matching graphs with unique node labels. Pattern Analysis and Applications 7(3), 243–254 (2004)
Ebeling, C.: Gemini ii: A second generation layout validation tool. In: IEEE International Conference on Computer Aided Design, pp. 322–325 (1988)
Bunke, H.: Error correcting graph matching: On the influence of the underlying cost function. IEEE Transactions on Pattern Analysis and Machine Intelligence 21(9), 917–911 (1999)
Riesen, K., Bunke, H.: Approximate graph edit distance computation by means of bipartite graph matching. In: Image and Vision Computing (2008) (accepted for publication)
Eshera, M., Fu, K.: A graph distance measure for image analysis. IEEE Transactions on Systems, Man, and Cybernetics (Part B) 14(3), 398–408 (1984)
Munkres, J.: Algorithms for the assignment and transportation problems. Journal of the Society for Industrial and Applied Mathematics 5, 32–38 (1957)
Foggia, P., Sansone, C., Vento, M.: A database of graphs for isomorphism and subgraph isomorphism benchmarking. In: Proc. 3rd Int. Workshop on Graph Based Representations in Pattern Recognition, pp. 176–187 (2001)
Cordella, L., Foggia, P., Sansone, C., Vento, M.: A (sub)graph isomorphism algorithm for matching large graphs. IEEE Trans. on Pattern Analysis and Machine Intelligence 26(20), 1367–1372 (2004)
Morgan, H.: The generation of a unique machine description for chemical structures-a technique developed at chemical abstracts service. Journal of Chemical Documentation 5(2), 107–113 (1965)
Mahé, P., Ueda, N., Akutsu, T.: Graph kernels for molecular structures – activity relationship analysis with support vector machines. Journal of Chemical Information and Modeling 45(4), 939–951 (2005)
Foggia, P., Sansone, C., Vento, M.: A performance comparison of five algorithms for graph isomorphism. In: Jolion, J., Kropatsch, W., Vento, M. (eds.) Proc. 3rd Int. Workshop on Graph Based Representations in Pattern Recognition, pp. 188–199 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Riesen, K., Fankhauser, S., Bunke, H., Dickinson, P. (2009). Efficient Suboptimal Graph Isomorphism. In: Torsello, A., Escolano, F., Brun, L. (eds) Graph-Based Representations in Pattern Recognition. GbRPR 2009. Lecture Notes in Computer Science, vol 5534. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02124-4_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-02124-4_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02123-7
Online ISBN: 978-3-642-02124-4
eBook Packages: Computer ScienceComputer Science (R0)