A new algorithm for subgraph optimal isomorphism

Abstract In this paper a new algorithm for subgraph isomorphism is proposed. The main idea of the new algorithm is to decompose the graphs to be matched into smaller subgraphs. The matching process is then done at the level of the decomposed subgraphs based on the concept of error-correcting transformations. The cost of matching two graphs is defined as the minimum of a weighted bipartite graph constructed from the decomposed subgraphs. The average computational complexity of the proposed algorithm is found to be O ( N 4 ). The results of the application of the new algorithm show that the new technique is quite efficient and, in many respects, superior to similar existing techniques in the literature. Besides, it overcomes most of the problems encountered in using tree search algorithms. The new algorithm is also suitable for parallel processing implementation.

[1]  Hans L. Boblaender.  Polynomial algorithms for graph isomorphism and chromatic index on partial k -trees , 1990 .

[2]  Larry S. Davis,et al.  Shape Matching Using Relaxation Techniques , 1979, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[3]  Josiane Zerubia,et al.  DPA: a deterministic approach to the MAP problem , 1995, IEEE Trans. Image Process..

[4]  Mohan M. Trivedi,et al.  Graph matching using a direct classification of node attendance , 1996, Pattern Recognit..

[5]  Eiichi Tanaka,et al.  Theoretical aspects of syntactic pattern recognition , 1995, Pattern Recognit..

[6]  Steven Gold,et al.  A Graduated Assignment Algorithm for Graph Matching , 1996, IEEE Trans. Pattern Anal. Mach. Intell..

[7]  Eam Khwang Teoh,et al.  Pattern recognition by graph matching using the Potts MFT neural networks , 1995, Pattern Recognit..

[8]  Ai-Jia Hsieh,et al.  Bipartite weighted matching for on-line handwritten Chinese character recognition , 1995, Pattern Recognit..

[9]  H. B. Mittal,et al.  A Fast Backtrack Algorithm for Graph Isomorphism , 1988, Inf. Process. Lett..

[10]  Frank Harary,et al.  Graph Theory , 2016 .

[11]  Nasser M. Nasrabadi,et al.  Object recognition using multilayer Hopfield neural network , 1997, IEEE Trans. Image Process..

[12]  H. Kuhn.  The Hungarian method for the assignment problem , 1955 .

[13]  Ming S. Hung,et al.  Solving the Assignment Problem by Relaxation , 1980, Oper. Res..

[14]  Theodosios Pavlidis,et al.  Linear and Context-Free Graph Grammars , 1972, JACM.

[15]  King-Sun Fu,et al.  A distance measure between attributed relational graphs for pattern recognition , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[16]  Curtis R. Cook,et al.  First Order Graph Grammars , 1974, SIAM J. Comput..

[17]  Tamotsu Kasai,et al.  An efficient algorithm for the linear assignment problem , 1990 .

[18]  Andrew K. C. Wong,et al.  An algorithm for graph optimal monomorphism , 1990, IEEE Trans. Syst. Man Cybern..

[19]  Shinji Umeyama,et al.  An Eigendecomposition Approach to Weighted Graph Matching Problems , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

[20]  David Avis,et al.  A survey of heuristics for the weighted matching problem , 1983, Networks.

[21]  Alan C. Shaw,et al.  A Formal Picture Description Scheme as a Basis for Picture Processing Systems , 1969, Inf. Control..

[22]  King-Sun Fu,et al.  An Image Understanding System Using Attributed Symbolic Representation and Inexact Graph-Matching , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[23]  Uwe Schöning.  Graph Isomorphism is in the Low Hierarchy , 1988, J. Comput. Syst. Sci..

[24]  Derek G. Corneil,et al.  The graph isomorphism disease , 1977, J. Graph Theory.

[25]  Alan C. Shaw,et al.  Parsing of Graph-Representable Pictures , 1970, JACM.

[26]  King-Sun Fu,et al.  Subgraph error-correcting isomorphisms for syntactic pattern recognition , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[27]  Robert M. Haralick,et al.  Structural Descriptions and Inexact Matching , 1981, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[28]  V. G. Timkovskii.  Some recognition problems related to graph isomorphism , 1988 .

[29]  Theodosios Pavlidis,et al.  Representation of figures by labeled graphs , 1972, Pattern Recognit..

[30]  H. Almohamad.  A polynomial transform for matching pairs of weighted graphs , 1991 .

[31]  Salih O. Duffuaa,et al.  A Linear Programming Approach for the Weighted Graph Matching Problem , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[32]  Les Kitchen,et al.  Relaxation Applied to Matching Quantitative Relational Structures , 1978 .

[33]  John Stroud.  Airports of the world , 1980 .

[34]  King-Sun Fu,et al.  A graph distance measure for image analysis , 1984, IEEE Transactions on Systems, Man, and Cybernetics.

[35]  Andrew K. C. Wong,et al.  Graph Optimal Monomorphism Algorithms , 1980, IEEE Transactions on Systems, Man, and Cybernetics.

[36]  Richard M. Karp,et al.  An algorithm to solve the m × n assignment problem in expected time O(mn log n) , 1980, Networks.

[37]  Narsingh Deo,et al.  Graph Theory with Applications to Engineering and Computer Science , 1975, Networks.

[38]  Stan Z. Li,et al.  Matching: Invariant to translations, rotations and scale changes , 1992, Pattern Recognit..

[39]  N.R. Malik,et al.  Graph theory with applications to engineering and computer science , 1975, Proceedings of the IEEE.

[40]  Robert J. Schalkoff,et al.  Pattern recognition - statistical, structural and neural approaches , 1991 .

[41]  Jianer Chen,et al.  A Linear Time Algorithm for Isomorphism of Graphs of Bounded Average Genus , 1992, WG.

[42]  William J. Christmas,et al.  Structural Matching in Computer Vision Using Probabilistic Relaxation , 1995, IEEE Trans. Pattern Anal. Mach. Intell..