This Article 
 Bibliographic References 
 Add to: 
Matching Free Trees, Maximal Cliques, and Monotone Game Dynamics
November 2002 (vol. 24 no. 11)
pp. 1535-1541

Abstract—Motivated by our recent work on rooted tree matching, in this paper we provide a solution to the problem of matching two free (i.e., unrooted) trees by constructing an association graph whose maximal cliques are in one-to-one correspondence with maximal common subtrees. We then solve the problem using simple payoff-monotonic dynamics from evolutionary game theory. We illustrate the power of the approach by matching articulated and deformed shapes described by shape-axis trees. Experiments on hundreds of larger, uniformly random trees are also presented. The results are impressive: despite the inherent inability of these simple dynamics to escape from local optima, they always returned a globally optimal solution.

[1] J.E. Ash, P.A. Chubb, S.E. Ward, S.M. Welford, and P. Willett, Communication, Storage and Retrieval of Chemical Information. Chichester, UK: Ellis Horwood, 1985.
[2] D.H. Ballard and C.M. Brown, Computer Vision, Prentice Hall, Upper Saddle River, N.J., 1982.
[3] H. Blum and R.N. Nagel, “Shape Description Using Weighted Symmetric Axis Features,” Pattern Recognition, vol. 10, pp. 167-180, 1978.
[4] I.M. Bomze, “Evolution Towards the Maximum Clique,” J. Global Optimization, vol. 10, pp. 143-164, 1997.
[5] I.M. Bomze, M. Budinich, P.M. Pardalos, and M. Pelillo, “The Maximum Clique Problem,” Handbook of Combinatorial Optimization (supplement vol. A), D.-Z. Du and P.M. Pardalos, eds., pp. 1-74, Boston: Kluwer, 1999.
[6] H. Bunke and K. Shearer, “A Graph-Distance Metric Based on the Maximal Common Subgraph,” Pattern Recognition Letters, vol. 19, nos. 3-4, pp. 255-259, 1998.
[7] H. Bunke, “Error Correcting Graph Matching: On the Influence of the Underlying Cost Function,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 9, pp. 917-922, Sept. 1999.
[8] T.H. Cormen,C.E. Leiserson, and R.L. Rivest,Introduction to Algorithms.Cambridge, Mass.: MIT Press/McGraw-Hill, 1990.
[9] R. Duda, P. Hart, and D. Stork, Pattern Classification. New York: John Wiley&Sons, 2001.
[10] D. Fudenberg and D.K. Levine, The Theory of Learning in Games. Cambridge, Mass.: MIT Press, 1998.
[11] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness.New York: W.H. Freeman, 1979.
[12] S. Gold and A. Rangarajan, “A Graduated Assignment Algorithm for Graph Matching,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18, no. 4, pp. 377-388, Apr. 1996.
[13] J. Hofbauer, “Imitation Dynamics for Games,” Collegium Budapest, preprint, 1995.
[14] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics. Cambridge, UK: Cambridge Univ. Press, 1998.
[15] P. Klein, “Computing the Edit-Distance between Unrooted Ordered Trees,” Proc. of Sixth European Symp. Algorithms, pp. 91-102, 1998.
[16] P. Klein, S. Tirthapura, D. Sharvit, and B. Kimia, “A Tree-Edit-Distance Algorithm for Comparing Simple, Closed Shapes,” Proc. 10th Ann. ACM-SIAM Symp. Discrete Algorithms (SODA), pp. 696-704, 2000.
[17] T. Liu and D. Geiger, Approximate Tree Matching and Shape Similarity Proc. Int'l Conf. Computer Vision, pp. 456-462, 1999.
[18] T. Liu, D. Geiger, and R. Kohn, “Representation and Self-Similarity of Shapes,” Proc. Int'l Conf. Computer Vision, Bombay, 1998.
[19] T.S. Motzkin and E.G. Straus, “Maxima for Graphs and a New Proof of a Theorem of Turán,” Canadian J. Math., vol. 17, pp. 533-540, 1965.
[20] M. Pelillo, “Replicator Equations, Maximal Cliques, and Graph Isomorphism,” Neural Computation, vol. 11, no. 8, pp. 2023-2045, 1999.
[21] M. Pelillo, K. Siddiqi, and S.W. Zucker, “Matching Hierarchical Structures Using Association Graphs,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 11, pp. 1105-1120, 1999.
[22] M. Pelillo, K. Siddiqi, and S.W. Zucker, “Many-to-Many Matching of Attributed Trees Using Association Graphs and Game Dynamics,” Visual Form 2001, C. Arcelli, L.P. Cordella, and G. Sanniti di Baja, eds., pp. 583-593, Berlin: Springer, 2001.
[23] J.W. Weibull, Evolutionary Game Theory. Cambridge, Mass.: MIT Press, 1995.
[24] H. Wilf, “The Uniform Selection of Free Trees,” J. Algorithms, vol. 2, pp. 204-207, 1981.
[25] K. Zhang, J.T.L. Wang, and D. Shasha, “On the Editing Distance between Undirected Acyclic Graphs,” Int'l J. Foundations of Computer Science, vol. 7, no. 1, pp. 43-57, 1996.
[26] S.C. Zhu and A.L. Yuille, “FORMS: A Flexible Object Recognition and Modeling System,” Int'l J. Computer Vision, vol. 20, no. 3, Dec. 1996

Index Terms:
Graph matching, combinatorial optimization, quadratic programming, dynamical systems, evolutionary game theory, shape recognition.
Marcello Pelillo, "Matching Free Trees, Maximal Cliques, and Monotone Game Dynamics," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 24, no. 11, pp. 1535-1541, Nov. 2002, doi:10.1109/TPAMI.2002.1046176
Usage of this product signifies your acceptance of the Terms of Use.