A 1.375-Approximation Algorithm for Sorting by Transpositions

  • Isaac Elias
  • Tzvika Hartman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3692)


Sorting permutations by transpositions is an important problem in genome rearrangements. A transposition is a rearrangement operation in which a segment is cut out of the permutation and pasted in a different location. The complexity of this problem is still open and it has been a ten-year-old open problem to improve the best known 1.5-approximation algorithm. In this paper we provide a 1.375-approximation algorithm for sorting by transpositions. The algorithm is based on a new upper bound on the diameter of 3-permutations. In addition, we present some new results regarding the transposition diameter: We improve the lower bound for the transposition diameter of the symmetric group, and determine the exact transposition diameter of 2-permutations and simple permutations.


Symmetric Group Genome Rearrangement Algorithm Sort Open Gate Circular Permutation 
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.


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  1. 1.
  2. 2.
    Appel, K., Haken, W.: Every planar map is four colorable part I: Discharging. Illinois Journal of Mathematics 21, 429–490 (1977)zbMATHMathSciNetGoogle Scholar
  3. 3.
    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
  4. 4.
    Bafna, V., Pevzner, P.A.: Genome rearragements and sorting by reversals. SIAM Journal on Computing 25(2), 272–289 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bafna, V., Pevzner, P.A.: Sorting by transpositions. SIAM Journal on Discrete Mathematics 11(2), 224–240 (1998); Preliminary version in the Proceedings of SODA (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bergeron, A.: A very elementary presentation of the Hannenhalli-Pevzner theory. In: Amir, A., Landau, G.M. (eds.) CPM 2001. LNCS, vol. 2089, pp. 106–117. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Berman, P., Hannanhalli, S., Karpinski, M.: 1.375-approximation algorithm for sorting by reversals. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 200–210. Springer, Heidelberg (2002)Google Scholar
  8. 8.
    Caprara, A.: Sorting permutations by reversals and Eulerian cycle decompositions. SIAM Journal on Discrete Mathematics 12(1), 91–110 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Christie, D.A.: Genome Rearrangement Problems. PhD thesis, University of Glasgow (1999)Google Scholar
  10. 10.
    Eriksson, H., Eriksson, K., Karlander, J., Svensson, L., Wastlund, J.: Sorting a bridge hand. Discrete Mathematics 241(1-3), 289–300 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hannenhalli, S., Pevzner, P.: Transforming cabbage into turnip: Polynomial algorithm for sorting signed permutations by reversals. Journal of the ACM 46, 1–27 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hartman, T.: A simpler 1.5-approximation algorithm for sorting by transpositions. In: Baeza-Yates, R., Chávez, E., Crochemore, M. (eds.) CPM 2003. LNCS, vol. 2676, pp. 156–169. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Hoot, S.B., Palmer, J.D.: Structural rearrangements, including parallel inversions, within the chloroplast genome of Anemone and related genera. J. Molecular Evooution 38, 274–281 (1994)Google Scholar
  14. 14.
    Kaplan, H., Shamir, R., Tarjan, R.E.: Faster and simpler algorithm for sorting signed permutations by reversals. SIAM Journal of Computing 29(3), 880–892 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lin, G.H., Xue, G.: Signed genome rearrangements by reversals and transpositions: Models and approximations. Theoretical Computer Science 259, 513–531 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Meidanis, J., Walter, M.E., Dias, Z.: Transposition distance between a permutation and its reverse. In: Proceedings of the 4th South American Workshop on String Processing, pp. 70–79 (1997)Google Scholar
  17. 17.
    Palmer, J.D., Herbon, L.A.: Tricircular mitochondrial genomes of Brassica and Raphanus: reversal of repeat configurations by inversion. Nucleic Acids Research 14, 9755–9764 (1986)CrossRefGoogle Scholar
  18. 18.
    Pevzner, P.A.: Computational Molecular Biology: An Algorithmic Approach. MIT Press, Cambridge (2000)zbMATHGoogle Scholar
  19. 19.
    Sankoff, D., El-Mabrouk, N.: Genome rearrangement. In: Current Topics in Computational Molecular Biology. MIT Press, Cambridge (2002)Google Scholar
  20. 20.
    Shamir, R.: Algorithms in molecular biology: Lecture notes (2002), Available at
  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.
    Zwick, U.: Computer assisted proof of optimal approximability results. In: Symposium On Discrete Mathematics (SODA 2002), pp. 496–505 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Isaac Elias
    • 1
  • Tzvika Hartman
    • 2
  1. 1.Dept. of Numerical Analysis and Computer ScienceRoyal Institute of TechnologyStockholmSweden
  2. 2.Dept. of Molecular GeneticsWeizmann Institute of ScienceRehovotIsrael

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