Abstract
A special class of graphs is introduced in this paper. The graphs belonging to this class are characterised by the existence of unique node labels. A number of matching algorithms for graphs with unique node labels are developed. It is shown that problems such as graph isomorphism, subgraph isomorphism, maximum common subgraph (MCS) and graph edit distance (GED) have a computational complexity that is only quadratic in the number of nodes. Moreover, computing the median of a set of graphs is only linear in the cardinality of the set. In a series of experiments, it is demonstrated that the proposed algorithms run very fast in practice. The considered class makes the matching of large graphs, consisting of thousands of nodes, computationally tractable. We also discuss an application of the considered class of graphs and related matching algorithms to the classification and detection of abnormal events in computer networks.
Similar content being viewed by others
References
(2001) Special section on graph algorithms and computer vision. IEEE Trans PAMI 23(10)
(2003) Special issue on graph-based representations in pattern recognition. Pattern Recognit Lett 24(8)
(2004) Special issue on graph matching in pattern recognition and machine vision. Int J Pattern Recognit Artif Intell 18(3)
McKay B (1981) Practical graph isomorphism. Congressus Numerantium 30:45–87
Ullman JR (1976) An algorithm for subgraph isomorphism. J ACM 23(1):31–42
Levi G (1972) A note on the derivation of maximal common subgraphs of two directed or undirected graphs. Calcolo 9:341–354
McGregor J (1982) Backtrack search algorithms and the maximal common subgraph problem. Software Pract Experience 12(1):23–34
Messmer BT, Bunke H (1998) A new algorithm for error-tolerant subgraph isomorphism detection. IEEE Trans Pattern Anal Machine Intell 20:493–504
Sanfeliu A, Fu KS (1983) A distance measure between attributed relational graphs for pattern recognition. IEEE Trans Syst Man Cybern 13(3):353–362
Cordella LP, Foggia P, Sansone C, Vento M (2001) An improved algorithm for matching large graphs. In: Proceedings of the 3rd IAPR-TC15 workshop on graph based representations in pattern recognition, Naples, Italy, May 2001, pp 149–159
Larrosa J, Valiente G (2002) Constraint satisfaction algorithms for graph pattern matching. Math Struct Comput Sci 12:403–422
Christmas WJ, Kittler J, Petrou M (1995) Structural matching in computer vision using probabilistic relaxation. IEEE Trans PAMI 8:749–764
Wilson RC, Hancock E (1997) Structural matching by discrete relaxation. IEEE Trans PAMI 19:634–648
Cross A, Wilson R, Hancock E (1997) Inexact graph matching with genetic search. Pattern Recognit 30:953–970
Wang I, Fan K-C, Horng J-T (1997) Genetic-based search for error-correcting graph isomorphism. IEEE Trans SMC 27:588–597
Luo B, Hancock E (2001) Structural graph matching using the EM algorithm and singular value decomposition. IEEE Trans PAMI 23:1120–1136
Kosinov S, Caelli T (2002) Inexact multisubgraph matching using graph eigenspace and clustering models. In: Caelli T, Amin A, Duin R, Kamel M, de Ridder D (eds) Structural, syntactic, and statistical pattern recognition, LNCS 2396. Springer, Berlin Heidelberg New York, pp 133–142
Luo B, Wilson R, Hancock E (2002) Spectral feature vectors for graph clustering. In: Caelli T, Amin A, Duin R, Kamel M, de Ridder D (eds) Structural, syntactic, and statistical pattern recognition, LNCS 2396. Springer, Berlin Heidelberg New York, pp 83–93
Pelillo M, Jagota A (1995) Feasible and infeasible maxima in a quadratic program for maximum clique. J Art Neural Netw 2(4):411–420
Hopcroft JE, Wong JK (1974) Linear time algorithm for isomorphism of planar graphs. In: Proceedings of the 6th annual ACM symposium on theory of computing, Seattle, Washington, April/May 1974, pp 172–184
Jiang X, Bunke H (1996) Including geometry in graph representations: a quadratic-time graph isomorphism algorithm and its application. In: Perner P, Wang P, Rosenfeld A (eds) Advances in structural and syntactic pattern recognition, LNCS 1121. Springer, Berlin Heidelberg New York, pp 110–119
Luks EM (1982) Isomorphism of graphs of bounded valence can be tested in polynomial time. J Comput Syst Sci 25:42–65
Pelillo M (2002) Matching free trees, maximal cliques, and monotone game dynamics. IEEE Trans PAMI 24(11):1535–1541
Shokonfandeh A, Dickinson S (2001) A unified framework for indexing and matching hierarchical shape structures. In: Arcelli C, Cordella L, Sanniti di Baja G (eds) Visual form 2001, LNCS 2059. Springer, Berlin Heidelberg New York, pp 67–84
Schenker A, Last M, Bunke H, Kandel A (2003) Clustering of web documents using a graph model. In: A Antonacopoulos, H Jianying (eds) Web document analysis: challenges and opportunities. World Scientific, River Edge, New Jersey
Schenker A, Last M, Bunke H, Kandel A (2003) Classification of web documents using a graph model. In: Proceedings of the 7th international conference on document analysis and recognition, Edinburgh, Scotland, August 2003, pp 472–476
Schenker A, Last M, Bunke H, Kandel A (2004) Classification of web documents using graph matching. Pattern Recognit Artif Intell 18(3):475–496
Dickinson P, Bunke H, Dadej A, Kraetzl M (2003) On graphs with unique node labels. In: Hancock E, Vento M (eds) Proceedings of the 4th IAPR international workshop on graph based representations in pattern recognition (GbRPR 2003), York, UK, June/July 2003. Springer, Berlin Heidelberg New York, pp 13–23
Jiang X, Munger A, Bunke H (2001) On median graphs: properties, algorithms, and applications. PAMI 23(10):1144–1151
Bunke H (1999) Error correcting graph matching: on the influence of the underlying cost function. IEEE Trans PAMI 21:917–922
Huberman BA, Lukose RM (1997) Social dilemmas and internet congestion. Science 277(5325):535–537
Snow AP, Weiss MBH (1997) Empirical evidence of reliability growth in large-scale networks. Netw Syst Manag 5(2):197–213
Dickinson P, Bunke H, Dadej A, Kraetzl M (2001) Application of median graphs in detection of anomalous change in communication networks. In: Proceedings of the 5th world multiconference on systemics, cybernetics and informatics (SCI 2001) vol 5, Orlando, Florida, July 2001
Shoubridge PJ, Kraetzl M, Wallis WD, Bunke H (2002) Detection of abnormal change in a time series of graphs. J Interconnection Netw 3:85–101
Bunke H, Kraetzl M, Shoubridge PJ, Wallis WD (2002) Measuring change in large enterprise data networks. In: Proceedings of the conference on information, decision and control (IDC 2002), Adelaide, South Australia, February 2002, pp 53–58
Chung FRK, Lu L (2002) Connected components in random graphs with given expected degree sequences. Ann Comb (6):125–145
Tangmunarunkit H, Govindan R, Jamin S, Shenker S, Willinger W (2002) Network topology generators: degree-based vs structural. In: Proceedings of the ACM SIGCOMM 2002 conference on applications, technologies, architectures, and protocols for computer communication, Pittsburgh, Pennsylvania, August 2002
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dickinson, P.J., Bunke, H., Dadej, A. et al. Matching graphs with unique node labels. Pattern Anal Applic 7, 243–254 (2004). https://doi.org/10.1007/s10044-004-0222-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10044-004-0222-5