Covering Tree with Stars

  • Jan Baumbach
  • Jiong Guo
  • Rashid Ibragimov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7936)


We study the tree edit distance problem with edge deletions and edge insertions as edit operations. We reformulate a special case of this problem as Covering Tree with Stars (CTS): given a tree T and a set \(\cal{S}\) of stars, can we connect the stars in \(\cal{S}\) by adding edges between them such that the resulting tree is isomorphic to T? We prove that in the general setting, CST is NP-complete, which implies that the tree edit distance considered here is also NP-hard, even when both input trees having diameters bounded by 10. We also show that, when the number of distinct stars is bounded by a constant k, CTS can be solved in polynomial time by presenting a dynamic programming algorithm running in \(O(|V(T)|^2\cdot k\cdot |V({\cal S})|^{2k})\) time.


graph algorithms tree edit distance NP-completeness dynamic programming 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Conte, D., Foggia, P., Sansone, C., Vento, M.: Thirty Years of Graph Matching in Pattern Recognition. International Journal of Pattern Recognition and Artificial Intelligence (2004)Google Scholar
  2. 2.
    Gao, X., Xiao, B., Tao, D., Li, X.: A survey of graph edit distance. Pattern Analysis & Applications 13(1), 113–129 (2010)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bunke, H., Riesen, K.: Graph Edit Distance – Optimal and Suboptimal Algorithms with Applications, pp. 113–143. Wiley-VCH Verlag GmbH & Co. KGaA (2009)Google Scholar
  4. 4.
    Garey, M.R., Johnson, D.S.: Computers and Intractability, A Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  5. 5.
    Akutsu, T., Fukagawa, D., Takasu, A., Tamura, T.: Exact algorithms for computing the tree edit distance between unordered trees. Theor. Comput. Sci. 412, 352–364 (2011)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bille, P.: A survey on tree edit distance and related problems. Theor. Comput. Sci. 337(1-3), 217–239 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Akutsu, T.: Tree Edit Distance Problems: Algorithms and Applications to Bioinformatics. IEICE Transactions on Information and Systems 93, 208–218 (2010)CrossRefGoogle Scholar
  8. 8.
    Demaine, E.D., Mozes, S., Rossman, B., Weimann, O.: An optimal decomposition algorithm for tree edit distance. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 146–157. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Zhang, K., Statman, R., Shasha, D.: On the editing distance between unordered labeled trees. Information Processing Letters 42(3), 133–139 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Zhang, K., Jiang, T.: Some MAX SNP-hard results concerning unordered labeled trees. Information Processing Letters 49(5), 249–254 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Akutsu, T., Fukagawa, D., Halldórsson, M.M., Takasu, A., Tanaka, K.: Approximation and parameterized algorithms for common subtrees and edit distance between unordered trees. Theoretical Computer Science 470, 10–22 (2013)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Blin, G., Sikora, F., Vialette, S.: Querying graphs in protein-protein interactions networks using feedback vertex set. IEEE/ACM Transactions on Computational Biology and Bioinformatics 7(4), 628–635 (2010)CrossRefGoogle Scholar
  13. 13.
    Natanzon, A., Shamir, R., Sharan, R.: Complexity classification of some edge modification problems. Discrete Applied Mathematics 113(1), 109–128 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Sharan, R.: Graph modification problems and their applications to genomic research. PhD thesis, School of Computer Science (2002)Google Scholar
  15. 15.
    Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Computer Algorithms, 1st edn. Addison-Wesley Longman Publishing Co., Inc., Boston (1974)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Jan Baumbach
    • 1
    • 3
  • Jiong Guo
    • 2
  • Rashid Ibragimov
    • 1
  1. 1.Max Planck Institute für InformatikSaarbrückenGermany
  2. 2.Universität des SaarlandesSaarbrückenGermany
  3. 3.University of Southern DenmarkOdense MDenmark

Personalised recommendations