Alignment with Non-overlapping Inversions in O(n3)-Time

  • Augusto F. Vellozo
  • Carlos E. R. Alves
  • Alair Pereira do Lago
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4175)


Alignments of sequences are widely used for biological sequence comparisons. Only biological events like mutations, insertions and deletions are usually modeled and other biological events like inversions are not automatically detected by the usual alignment algorithms.

Alignment with inversions does not have a known polynomial algorithm and a simplification to the problem that considers only non-overlapping inversions were proposed by Schöniger and Waterman [20] in 1992 as well as a corresponding O(n 6) solution. An improvement to an algorithm with O(n 3 logn)-time complexity was announced in an extended abstract [1] and, in this present paper, we give an algorithm that solves this simplified problem in O(n 3)-time and O(n 2)-space in the more general framework of an edit graph.

Inversions have recently [4,7,13,17] been discovered to be very important in Comparative Genomics and Scherer et al. in 2005 [11] experimentally verified inversions that were found to be polymorphic in the human genome. Moreover, 10% of the 1,576 putative inversions reported overlap RefSeq genes in the human genome. We believe our new algorithms may open the possibility to more detailed studies of inversions on DNA sequences using exact optimization algorithms and we hope this may be particularly interesting if applied to regions around known rearrangements boundaries. Scherer report 29 such cases and prioritize them as candidates for biological and evolutionary studies.


Optimal Path Edit Operation Edit Graph Minimal Total Cost Extended Edge 
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.
    Alves, C.E.R., do Lago, A.P., Vellozo, A.F.: Alignment with non-overlapping inversions in \(O(n\sp 3\log n)\)-time. In: Proceedings of GRACO2005. Electron. Notes Discrete Math, vol. 19, pp. 365–371. Elsevier, Amsterdam (2005)Google Scholar
  2. 2.
    Bader, D.A., Moret, B.M.E., Yan, M.: A linear-time algorithm for computing inversion distance between signed permutations with an experimental study. Journal of Computational Biology 8(5), 483–491 (2001)CrossRefGoogle Scholar
  3. 3.
    Caprara, A.: Sorting permutations by reversals and Eulerian cycle decompositions. SIAM J. Discrete Math. 12(1), 91–110 (1999) (electronic)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cerdeño-Tárraga, Patrick, Crossman, Blakely, Abratt, Lennard, Poxton, Duerden, Harris, Quail, Barron, Clark, Corton, Doggett, Holden, Larke, Line, Lord, Norbertczak, Ormond, Price, Rabbinowitsch, Woodward, Barrell, Parkhill: Extensive DNA inversions in the B. fragilis genome control variable gene expression. Science 307(5714), 1463–1465 (2005)Google Scholar
  5. 5.
    Chen, X., Zheng, J., Fu, Z., Nan, P., Zhong, Y., Lonardi, S., Jiang, T.: Assignment of orthologous genes via genome rearrangement. IEEE/ACM Trans. Comput. Biol. Bioinformatics 2(4), 302–315 (2005)CrossRefGoogle Scholar
  6. 6.
    Christie, D.A.: A 3/2-approximation algorithm for sorting by reversals. In: Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (San Francisco, CA, 1998), pp. 244–252. ACM, New York (1998)Google Scholar
  7. 7.
    Cáceres, Ranz, Barbadilla, Long, Ruiz: Generation of a widespread Drosophila inversion by a transposable element. Science 285(5426), 415–418 (1999)Google Scholar
  8. 8.
    do Lago, A.P., Kulikowski, C.A., Linton, E., Messing, J., Muchnik, I.: Comparative genomics: simultaneous identification of conserved regions and their rearrangements through global optimization. In: The Second University of Sao Paulo/Rutgers University Biotechnology Conference, Rutgers University Inn and Conference Center, New Brunswick, NJ (August 2001)Google Scholar
  9. 9.
    do Lago, A.P., Muchnik, I., Kulikowski, C.: An O(n 4) algorithm for alignment with non-overlapping inversions. In: Second Brazilian Workshop on Bioinformatics, WOB 2003, Macaé, RJ, Brazil (2003),
  10. 10.
    do Lago, A.P., Muchnik, I., Kulikowski, C.: A sparse dynamic programming algorithm for alignment with non-overlapping inversions. Theor. Inform. Appl. 39(1), 175–189 (2005)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Feuk, MacDonald, Tang, Carson, Li, Rao, Khaja, Scherer: Discovery of human inversion polymorphisms by comparative analysis of human and chimpanzee DNA sequence assemblies. PLoS Genet. 1(4), 56 (2005)Google Scholar
  12. 12.
    Gao, Y., Wu, J., Niewiadomski, R., Wang, Y., Chen, Z.-Z., Lin, G.: A space efficient algorithm for sequence alignment with inversions. In: Warnow, T.J., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 57–67. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Graham, Olmstead: Evolutionary significance of an unusual chloroplast DNA inversion found in two basal angiosperm lineages. Curr. Genet. 37(3), 183–188 (2000)Google Scholar
  14. 14.
    Hannenhalli, S., Pevzner, P.A.: Transforming cabbage into turnip: polynomial algorithm for sorting signed permutations by reversals. In: ACM Symposium on Theory of Computing, pp. 178–189. Association for Computing Machinery (1995)Google Scholar
  15. 15.
    Hannenhalli, S., Pevzner, P.A.: Transforming cabbage into turnip: polynomial algorithm for sorting signed permutations by reversals. J. ACM 46(1), 1–27 (1999)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kececioglu, J., Sankoff, D.: Exact and approximation algorithms for sorting by reversals, with application to genome rearrangement. Algorithmica 13(1-2), 180–210 (1995)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kuwahara, Yamashita, Hirakawa, Nakayama, Toh, Okada, Kuhara, Hattori, Hayashi, Ohnishi: Genomic analysis of Bacteroides fragilis reveals extensive DNA inversions regulating cell surface adaptation. Proceedings of the National Academy of Sciences U S A 101(41), 14919–14924 (2004)Google Scholar
  18. 18.
    Landau, G.M., Ziv-Ukelson, M.: On the common substring alignment problem. J. Algorithms 41(2), 338–359 (2001)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Schmidt, J.P.: All highest scoring paths in weighted grid graphs and their application to finding all approximate repeats in strings. SIAM J. Comput. 27(4), 972–992 (1998) (electronic)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Schöniger, M., Waterman, M.S.: A local algorithm for DNA sequence alignment with inversions. Bulletin of Mathematical Biology 54(4), 521–536 (1992)MATHGoogle Scholar
  21. 21.
    Tannier, E., Sagot, M.-F.: Sorting by reversals in subquadratic time. In: Sahinalp, S.C., Muthukrishnan, S.M., Dogrusoz, U. (eds.) CPM 2004. LNCS, vol. 3109, pp. 1–13. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  22. 22.
    Wagner, R.: On the complexity of the extended string-to-string correction problem. In: Seventh ACM Symposium on the Theory of Computation, Association for Computing Machinery (1975)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Augusto F. Vellozo
    • 1
  • Carlos E. R. Alves
    • 2
  • Alair Pereira do Lago
    • 1
  1. 1.Instituto de Matemática e Estatística da Universidade de São Paulo (IME-USP)São PauloBrasil
  2. 2.Universidade São Judas Tadeu (FTCE-USJT)São PauloBrasil

Personalised recommendations