Time and Space Efficient Discovery of Maximal Geometric Graphs

  • Hiroki Arimura
  • Takeaki Uno
  • Shinichi Shimozono
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4755)


A geometric graph is a labeled graph whose vertices are points in the 2D plane with an isomorphism invariant under geometric transformations such as translation, rotation, and scaling. While Kuramochi and Karypis (ICDM2002) extensively studied the frequent pattern mining problem for geometric subgraphs, the maximal graph mining has not been considered so far. In this paper, we study the maximal (or closed) graph mining problem for the general class of geometric graphs in the 2D plane by extending the framework of Kuramochi and Karypis. Combining techniques of canonical encoding and a depth-first search tree for the class of maximal patterns, we present a polynomial delay and polynomial space algorithm, MaxGeo, that enumerates all maximal subgraphs in a given input geometric graph without duplicates. This is the first result establishing the output-sensitive complexity of closed graph mining for geometric graphs. We also show that the frequent graph mining problem is also solvable in polynomial delay and polynomial time.


geometric graphs closed graph mining depth-first search rightmost expansion polynomial delay polynomial space enumeration algorithms 


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  1. 1.
    Aho, A.V., Hopcroft, J.E., Ullman, J.D.: Data Structures and Algorithms (1983)Google Scholar
  2. 2.
    Akutsu, T., Tamaki, H., Tokuyama, T.: Distribution of distances and triangles in a point set and algorithms for computing the largest common point sets. Discr. & Comp. Geom. 20(3), 307–331 (1998)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Arimura, H., Uno, T.: An output-polynomial time algorithm for mining frequent closed attribute trees. In: Kramer, S., Pfahringer, B. (eds.) ILP 2005. LNCS (LNAI), vol. 3625, pp. 1–19. Springer, Heidelberg (2005)Google Scholar
  4. 4.
    Arimura, H., Uno, T.: A polynomial space and polynomial delay algorithm for enumeration of maximal motifs in a sequence. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, Springer, Heidelberg (2005)Google Scholar
  5. 5.
    Arimura, H., Uno, T.: Effcient algorithms for mining maximal flexible patterns in texts and sequences, TCS-TR-A-06-20, DCS, Hokkaido Univeristy 2006 (submitting)Google Scholar
  6. 6.
    Asai, T., Abe, K., Kawasoe, S., Arimura, H., Sakamoto, H., Arikawa, S.: Efficient substructure discovery from large semi-structured data. In: Proc. SDM 2002 (2002)Google Scholar
  7. 7.
    Avis, D., Fukuda, K.: Reverse search for enumeration. Discrete App. Math. 65, 21–46 (1996)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chi, Y., Muntz, R.R., Nijssen, S., Kok, J.N.: Frequent subtree mining – An overview. Fundam. Inform. 66(1-2), 161–198 (2005)MATHGoogle Scholar
  9. 9.
    Chi, Y., Yang, Y., Xia, Y., Muntz, R.R.: CMTreeMiner: mining both closed and maximal frequent subtrees. In: Dai, H., Srikant, R., Zhang, C. (eds.) PAKDD 2004. LNCS (LNAI), vol. 3056, Springer, Heidelberg (2004)Google Scholar
  10. 10.
    Garriga, G.C., Khardon, R., De Raedt, L.: On mining closed sets in multi-relational data. In: Proc. IJCAI 2007, pp. 804–809 (2007)Google Scholar
  11. 11.
    Guerra, C.: Vision and image processing algorithms. In: Algorithms and Theory of Computation Handbook, ch. 22, vol. f 22-1–22-23, CRC Press (1999)Google Scholar
  12. 12.
    Inokuchi, A., Washio, T., Motoda, H.: An apriori-based algorithm for mining frequent substructures from graph data. In: Zighed, A.D.A., Komorowski, J., Żytkow, J.M. (eds.) PKDD 2000. LNCS (LNAI), vol. 1910, pp. 13–23. Springer, Heidelberg (2000)Google Scholar
  13. 13.
    Jain, A.: Fundamentals of Digital Image Processing. Prentice-Hall, Englewood Cliffs (1986)Google Scholar
  14. 14.
    Khardon, R.: Learning function-free horn expressions. Machine Learning 37(3), 241–275 (1999)MATHCrossRefGoogle Scholar
  15. 15.
    Kuramochi, M., Karypis, G.: Discovering frequent geometric subgraphs. In: Proc. IEEE ICDM 2002, pp. 258–265 (2002)Google Scholar
  16. 16.
    Nakano, S.: Efficient generation of plane trees. Information Processing Letters 84, 167–172 (2002)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Nijssen, S., Kok, J.N.: Effcient discovery of frequent unordered trees. In: Proc. MGTS 2003 (2003)Google Scholar
  18. 18.
    Pasquier, N., Bastide, Y., Taouil, R., Lakhal, L.: Discovering frequent closed itemsets for association rules. In: Beeri, C., Bruneman, P. (eds.) ICDT 1999. LNCS, vol. 1540, pp. 398–416. Springer, Heidelberg (1999)Google Scholar
  19. 19.
    Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, Heidelberg (1985)Google Scholar
  20. 20.
    Termier, A., Rousset, M.-C., Sebag, M.: Dryade: a new approach for discovering closed frequent trees in heterogeneous tree databases. In: Proc. ICMD 2004 (2004)Google Scholar
  21. 21.
    Tsuda, K., Kudo, T.: Clustering graphs by weighted substructure mining. In: Proc. ICML 2006, pp. 953–960 (2006)Google Scholar
  22. 22.
    Uno, T., Asai, T., Uchida, Y., Arimura, H.: An efficient algorithm for enumerating closed patterns in transaction databases. In: Suzuki, E., Arikawa, S. (eds.) DS 2004. LNCS (LNAI), vol. 3245, pp. 16–30. Springer, Heidelberg (2004)Google Scholar
  23. 23.
    Wang, J., Han, J.: BIDE: Efficient Mining of Frequent Closed Sequences. In: Proc. IEEE ICDE 2004, pp. 79–90 (2004)Google Scholar
  24. 24.
    Washio, T., Motoda, H.: State of the art of graph-based data mining. SIGKDD Explor. 5(1), 59–68 (2003)CrossRefGoogle Scholar
  25. 25.
    Yan, X., Han, J.: CloseGraph: mining closed frequent graph patterns. In: Proc. KDD 2003 (2003)Google Scholar
  26. 26.
    Yang, G.: The complexity of mining maximal frequent itemsets and maximal frequent patterns. In: Proc. KDD 2004, pp. 344–353 (2004)Google Scholar
  27. 27.
    Zaki, M.J.: Efficiently mining frequent trees in a forest. In: Proc. KDD 2002, pp. 71–80 (2002)Google Scholar
  28. 28.
    Zaki, M.J., Aggarwal, C.C.: XRules: an effective structural classifier for XML data. In: Proc. KDD 2003, pp. 316–325 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Hiroki Arimura
    • 1
  • Takeaki Uno
    • 2
  • Shinichi Shimozono
    • 3
  1. 1.Hokkaido University, Kita 14-jo, Nishi 9-chome, Sapporo 060-0814Japan
  2. 2.National Institute of Informatics, Tokyo 101–8430Japan
  3. 3.Kyushu Institute of Technology, Kawazu 680-4, Iizuka 820-8502Japan

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