Mapping Kernels Between Rooted Labeled Trees Beyond Ordered Trees

  • Kouichi Hirata
  • Tetsuji Kuboyama
  • Takuya Yoshino
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9067)


In this paper, we investigate several mapping kernels to count all of the mappings between two rooted labeled trees beyond ordered trees, that is, cyclically ordered trees such as biordered trees, cyclic-ordered trees and cyclic-biordered trees, and degree-bounded unordered trees. Then, we design the algorithms to compute a top-down mapping kernel, an LCA-preserving segmental mapping kernel, an LCA-preserving mapping kernel, an accordant mapping kernel and an isolated-subtree mapping kernel for biordered trees in O(nm) time and ones for cyclic-ordered and cyclic-biordered trees in O(nmdD) time, where n is the number of nodes in a tree, m is the number of nodes in another tree, D is the maximum value of the degrees in two trees and d is the minimum value of the degrees in two trees. Also we design the algorithms to compute the above kernels for degree-bounded unordered trees in O(nm) time. On the other hand, we show that the problem of computing label-preserving leaf-extended top-down mapping kernel and label-preserving bottom-up mapping kernel is #P-complete.


  1. 1.
    Chawathe, S.S.: Comparing hierarchical data in external memory. In: Proceedings of the VLDB 1999, pp. 90–101 (1999)Google Scholar
  2. 2.
    Gärtner, T.: Kernels for Structured Data. World Scientific Publishing, Singapore (2008)zbMATHGoogle Scholar
  3. 3.
    Hamada, I., Shimada, T., Nakata, D., Hirata, K., Kuboyama, T.: Agreement subtree mapping kernel for phylogenetic trees. In: Nakano, Y., Satoh, K., Bekki, D. (eds.) JSAI-isAI 2013. LNCS, vol. 8417, pp. 321–336. Springer, Heidelberg (2014) Google Scholar
  4. 4.
    Jiang, T., Wang, L., Zhang, K.: Alignment of trees - an alternative to tree edit. Theoret. Comput. Sci. 143, 137–148 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Kan, T., Higuchi, S., Hirata, K.: Segmental mapping and distance for rooted ordered labeled trees. Fundamenta Informaticae 132, 1–23 (2014)MathSciNetGoogle Scholar
  6. 6.
    Kashima, H., Sakamoto, H., Koyanagi, T.: Tree kernels. J. JSAI 21, 1–9 (2006). (in Japanese)Google Scholar
  7. 7.
    Kimura, D., Kuboyama, T., Shibuya, T., Kashima, H.: A subpath kernel for rooted unordered trees. J. JSAI 26, 473–482 (2011). (in Japanese)Google Scholar
  8. 8.
    Kuboyama, T.: Matching and learning in trees. Ph.D. thesis, University of Tokyo (2007)Google Scholar
  9. 9.
    Kuboyama, T., Hirata, K., Aoki-Kinoshita, K.F.: An efficient unordered tree kernel and its application to glycan classification. In: Washio, T., Suzuki, E., Ting, K.M., Inokuchi, A. (eds.) PAKDD 2008. LNCS (LNAI), vol. 5012, pp. 184–195. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  10. 10.
    Kuboyama, T., Shin, K., Kashima, H.: Flexible tree kernels based on counting the number of tree mappings. In: Proceedings of the MLG 2006, pp. 61–72 (2006)Google Scholar
  11. 11.
    Lu, C.L., Su, Z.-Y., Tang, C.Y.: A new measure of edit distance between labeled trees. In: Wang, J. (ed.) COCOON 2001. LNCS, vol. 2108, pp. 338–348. Springer, Heidelberg (2001) CrossRefGoogle Scholar
  12. 12.
    Lu, S.-Y.: A tree-to-tree distance and its application to cluster analysis. IEEE Trans. Pattern Anal. Mach. Intell. 1, 219–224 (1979)CrossRefzbMATHGoogle Scholar
  13. 13.
    Selkow, S.M.: The tree-to-tree editing problem. Inform. Process. Lett. 6, 184–186 (1977)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Shawe-Taylor, J., Cristianini, N.: Kernel methods for pattern analysis. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  15. 15.
    Shin, K.: Engineering positive semedefinite kernels for trees - a framework and a survey. J. JSAI 24, 459–468 (2009). (in Japanese)Google Scholar
  16. 16.
    Shin, K., Cuturi, M., Kuboyama, T.: Mapping kernels for trees. In: Proceedings of ICML 2011 (2011)Google Scholar
  17. 17.
    Shin, K., Kuboyama, T.: A generalization of Haussler’s convolutioin kernel - Mapping kernel and its application to tree kernels. J. Comput. Sci. Tech. 25, 1040–1054 (2010)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Tai, K.-C.: The tree-to-tree correction problem. J. ACM 26, 422–433 (1979)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Valiant, L.G.: The complexity of enumeration and reliablity problems. SIAM J. Comput. 8, 410–421 (1979)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Valiente, G.: An efficient bottom-up distance between trees. In: Proceedings of SPIRE 2001, pp. 212–219 (2001)Google Scholar
  21. 21.
    Wang, J.T.L., Zhang, K.: Finding similar consensus between trees: an algorithm and a distance hierarchy. Pattern Recogn. 34, 127–137 (2001)CrossRefzbMATHGoogle Scholar
  22. 22.
    Yamamoto, Y., Hirata, K., Kuboyama, T.: Tractable and intractable variations of unordered tree edit distance. Int. J. Found. Comput. Sci. 25, 307–329 (2014)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Yoshino, T., Hirata, K.: Hierarchy of segmental and alignable mapping for rooted labeled trees. In: Procedings of DDS 2013, pp. 62–69 (2013)Google Scholar
  24. 24.
    Yoshino, T., Hirata, K.: Alignment of cyclically ordered trees. In: Proceedings of ICPRAM 2015 (2015, to appear)Google Scholar
  25. 25.
    Zhang, K.: Algorithms for the constrained editing distance between ordered labeled trees and related problems. Pattern Recogn. 28, 463–474 (1995)CrossRefGoogle Scholar
  26. 26.
    Zhang, K.: A constrained edit distance between unordered labeled trees. Algorithmica 15, 205–222 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Zhang, K., Wang, J., Shasha, D.: On the editing distance between undirected acyclic graphs. Int. J. Found. Comput. Sci. 7, 43–58 (1996)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Kouichi Hirata
    • 1
  • Tetsuji Kuboyama
    • 2
  • Takuya Yoshino
    • 1
  1. 1.Kyushu Institute of TechnologyIizukaJapan
  2. 2.Gakushuin UniversityToshimaJapan

Personalised recommendations