The Longest Common Subsequence Problem with Crossing-Free Arc-Annotated Sequences

  • Guillaume Blin
  • Minghui Jiang
  • Stéphane Vialette
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7608)


An arc-annotated sequence is a sequence, over a given alphabet, with additional structure described by a set of arcs, each arc joining a pair of positions in the sequence. As a natural extension of the longest common subsequence problem, Evans introduced the Longest Arc-Preserving Common Subsequence (LAPCS) problem as a framework for studying the similarity of arc-annotated sequences. This problem has been studied extensively in the literature due to its potential application for RNA structure comparison, but also because it has a compact definition. In this paper, we focus on the nested case where no two arcs are allowed to cross because it is widely considered the most important variant in practice. Our contributions are three folds: (i) we revisit the nice NP-hardness proof of Lin et al. for LAPCS(Nested, Nested), (ii) we improve the running time of the FPT algorithm of Alber et al. from \(O(3.31^{k_1 + k_2} n)\) to \(O(3^{k_1 + k_2} n)\), where resp. k 1 and k 2 deletions from resp. the first and second sequence are needed to obtain an arc-preserving common subsequence, and (iii) we show that LAPCS(Stem, Stem) is NP-complete for constant alphabet size.


Truth Assignment Annotate Sequence Satisfying Truth Assignment Unary Alphabet Longe Common Subsequence Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alber, J., Gramm, J., Guo, J., Niedermeier, R.: Computing the similarity of two sequences with nested arc annotations. Theoretical Computer Science 312(2-3), 337–358 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Blin, G., Crochemore, M., Vialette, S.: Algorithmic Aspects of Arc-Annotated Sequences. In: Algorithms in Computational Molecular Biology: Techniques, Approaches and Applications. Wiley (2010) (to appear)Google Scholar
  3. 3.
    Blin, G., Denise, A., Dulucq, S., Herrbach, C., Touzet, H.: Alignment of RNA structures. IEEE/ACM Transactions on Computational Biology and Bioinformatics (2008) (to appear)Google Scholar
  4. 4.
    Evans, P.A.: Algorithms and Complexity for Annotated Sequences Analysis. PhD thesis, University of Victoria (1999)Google Scholar
  5. 5.
    Evans, P.A.: Finding Common Subsequences with Arcs and Pseudoknots. In: Crochemore, M., Paterson, M. (eds.) CPM 1999. LNCS, vol. 1645, pp. 270–280. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A guide to the theory of NP-completeness. W.H. Freeman, San Francisco (1979)Google Scholar
  7. 7.
    Guignon, V., Chauve, C., Hamel, S.: An Edit Distance Between RNA Stem-Loops. In: Consens, M.P., Navarro, G. (eds.) SPIRE 2005. LNCS, vol. 3772, pp. 335–347. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Blin, G., Hamel, S., Vialette, S.: Comparing RNA Structures with Biologically Relevant Operations Cannot Be Done without Strong Combinatorial Restrictions. In: Rahman, M. S., Fujita, S. (eds.) WALCOM 2010. LNCS, vol. 5942, pp. 149–160. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Jiang, T., Lin, G., Ma, B., Zhang, K.: A general edit distance between RNA structures. Journal of Computational Biology 9(2), 371–388 (2002)CrossRefGoogle Scholar
  10. 10.
    Jiang, T., Lin, G., Ma, B., Zhang, K.: The Longest Common Subsequence Problem for Arc-Annotated Sequences. In: Giancarlo, R., Sankoff, D. (eds.) CPM 2000. LNCS, vol. 1848, pp. 154–165. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  11. 11.
    Lin, G., Chen, Z.-Z., Jiang, T., Wen, J.: The longest common subsequence problem for sequences with nested arc annotations. J. of Computer and System Sc. 65, 465–480 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Shasha, D., Zhang, K.: Simple fast algorithms for the editing distance between trees and related problems. SIAM Journal on Computing 18(6), 1245–1262 (1989)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Guillaume Blin
    • 1
  • Minghui Jiang
    • 2
  • Stéphane Vialette
    • 1
  1. 1.Université Paris-Est, LIGM - UMR CNRS 8049France
  2. 2.Department of Computer ScienceUtah State UniversityUSA

Personalised recommendations