Journal of Combinatorial Optimization

, Volume 29, Issue 1, pp 141–152 | Cite as

Covering tree with stars



We study the tree edit distance (TED) 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 \(\mathcal {S}\) of stars, can we connect the stars in \(\mathcal {S}\) 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 TED considered here is also NP-hard, even when both input trees have 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 \(O(|V(T)|^2\cdot k\cdot |V(\mathcal{S})|^{2k})\) time.


Graph algorithms Tree edit distance NP-completeness  Dynamic programming 



Partially supported by the DFG Excellence Cluster MMCI and International Max Planck Research School.


  1. Aho AV, Hopcroft JE, Ullman JD (1974) The design and analysis of computer algorithms. Addison-Wesley, ReadingMATHGoogle Scholar
  2. Akutsu T (2010) Tree edit distance problems: algorithms and applications to bioinformatics. IEICE Trans Inf Syst 93:208–218CrossRefGoogle Scholar
  3. Akutsu T, Fukagawa D, Takasu A, Tamura T (2011) Exact algorithms for computing the tree edit distance between unordered trees. Theor Comput Sci 412(4–5):352–364CrossRefMATHMathSciNetGoogle Scholar
  4. Bille P (2005) A survey on tree edit distance and related problems. Theor Comput Sci 337(1–3):217–239CrossRefMATHMathSciNetGoogle Scholar
  5. Blin G, Sikora F, Vialette S (2010) Querying graphs in protein–protein interactions networks using feedback vertex set. IEEE/ACM Trans Comput Biol Bioinform 7(4):628–635CrossRefGoogle Scholar
  6. Bunke H, Riesen K (2009) Graph edit distance—optimal and suboptimal algorithms with applications. Wiley-VCH Verlag GmbH and Co. KGaA, WeinheimGoogle Scholar
  7. Conte D, Foggia P, Sansone C, Vento M (2004) Thirty years of graph matching in pattern recognition. IJPRAI 18(3):265–298Google Scholar
  8. Demaine E, Mozes S, Rossman B, Weimann O (2007) An optimal decomposition algorithm for tree edit distance. In: Automata, , languages and programming. Lecture notes in computer science, vol 4596. Springer, Berlin, pp 146–157Google Scholar
  9. Gao X, Xiao B, Tao D, Li X (2010) A survey of graph edit distance. Pattern Anal Appl 13(1):113–129CrossRefMathSciNetGoogle Scholar
  10. Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman and Company, San FranciscoMATHGoogle Scholar
  11. Kirkpatrick DG, Hell P (1978) On the completeness of a generalized matching problem. In: Proceedings of the tenth annual ACM symposium on theory of computing, STOC’78. ACM, New York, pp 240–245Google Scholar
  12. Natanzon A, Shamir R, Sharan R (2001) Complexity classification of some edge modification problems. Discret Appl Math 113(1):109–128CrossRefMATHMathSciNetGoogle Scholar
  13. Pawlik M, Augsten N (2011) RTED: a robust algorithm for the tree edit distance. PVLDB 5(4):334–345Google Scholar
  14. Sharan R (2002) Graph modification problems and their applications to genomic research. PhD Thesis, Tel-Aviv UniversityGoogle Scholar
  15. Zhang K, Jiang T (1994) Some MAX SNP-hard results concerning unordered labeled trees. Inf Process Lett 49(5):249–254CrossRefMATHGoogle Scholar
  16. Zhang K, Statman R, Shasha D (1992) On the editing distance between unordered labeled trees. Inf Process Lett 42(3):133–139CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.University of Southern DenmarkOdense MDenmark
  2. 2.Universität des SaarlandesSaarbrückenGermany
  3. 3.Max Planck Institute für InformatikSaarbrückenGermany

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