Neighborhood-Preserving Mapping between Trees

  • Jan Baumbach
  • Jiong Guo
  • Rashid Ibragimov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)


We introduce a variation of the graph isomorphism problem, where, given two graphs G 1 = (V 1,E 1) and G 2 = (V 2,E 2) and three integers l, d, and k, we seek for a set D ⊆ V 1 and a one-to-one mapping f:V 1 → V 2 such that |D| ≤ k and for every vertex v ∈ V 1 ∖ D and every vertex \(u\in N_{G_1}^l(v)\setminus D\) we have \(f(u)\in N_{G_2}^d(f(v))\). Here, for a graph G and a vertex v, we use \(N_{G}^i(v)\) to denote the set of vertices which have distance at most i to v in G. We call this problem Neighborhood-Preserving Mapping (NPM). The main result of this paper is a complete dichotomy of the classical complexity of NPM on trees with respect to different values of l,d,k. Additionally, we present two dynamic programming algorithms for the case that one of the input trees is a path.


tree edit distance graph algorithms complexity graph matching 


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  1. 1.
    Heath, A.P., Kavraki, L.E.: Computational challenges in systems biology. Computer Science Review 3, 1–17 (2009)CrossRefGoogle Scholar
  2. 2.
    Bunke, H., Riesen, K.: Recent advances in graph-based pattern recognition with applications in document analysis. Pattern Recognition 44, 1057–1067 (2011)MATHCrossRefGoogle Scholar
  3. 3.
    Riesen, K., Bunke, H.: Approximate graph edit distance computation by means of bipartite graph matching. Image and Vision Computing 27, 950–959 (2009)CrossRefGoogle Scholar
  4. 4.
    Akutsu, T., Fukagawa, D., Halldórsson, M.M., Takasu, A., Tanaka, K.: Approximation and parameterized algorithms for common subtrees and edit distance between unordered trees. Theor. Comput. Sci. 470, 10–22 (2013)MATHCrossRefGoogle 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)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bille, P.: A survey on tree edit distance and related problems. Theor. Comput. Sci. 337, 217–239 (2005)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Lozano, A., Pinter, R.Y., Rokhlenko, O., Valiente, G., Ziv-Ukelson, M.: Seeded tree alignment. IEEE/ACM Transactions on Computational Biology and Bioinformatics 5, 503–513 (2008)CrossRefGoogle Scholar
  8. 8.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman (1979)Google Scholar
  9. 9.
    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)MATHGoogle 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

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