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A Simpler 1.5-Approximation Algorithm for Sorting by Transpositions

  • Tzvika Hartman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2676)

Abstract

An important problem in genome rearrangements is sorting permutations by transpositions. Its complexity is still open, and two rather complicated 1.5-approximation algorithms for sorting linear permutations are known (Bafna and Pevzner, 96 and Christie, 98). In this paper, we observe that the problem of sorting circular permutations by transpositions is equivalent to the problem of sorting linear permutations by transpositions. Hence, all algorithms for sorting linear permutations by transpositions can be used to sort circular permutations. Our main result is a new 1.5-approximation algorithm, which is considerably simpler than the previous ones, and achieves running time which is equal to the best known. Moreover, the analysis of the algorithm is significantly less involved, and provides a good starting point for studying related open problems.

Keywords

Genome Rearrangement Circular Permutation Identity Permutation Oriented Cycle Black 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.

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References

  1. 1.
    D.A. Bader, B. M.E. Moret, and M. Yan. 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
  2. 2.
    V. Bafna and P. A. Pevzner. Genome rearragements and sorting by reversals. SIAM Journal on Computing, 25(2):272–289, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    V. Bafna and P. A. Pevzner. Sorting by transpositions. SIAM Journal on Discrete Mathematics, 11(2):224–240, May 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    A. Bergeron. A very elementary presentation of the Hannenhalli-Pevzner theory. In Proc. 12th Annual Symposium on Combinaotrial Pattern Matching (CPM’ 01), 2001.Google Scholar
  5. 5.
    P. Berman, S. Hannanhalli, and M. Karpinski. 1.375-approximation algorithm for sorting by reversals. In Proc. of 10th Eurpean Symposium on Algoriths, (ESA’02), pages 200–210. Springer, 2002. LNCS 2461.Google Scholar
  6. 6.
    A. Caprara. Sorting permutations by reversals and Eulerian cycle decompositions. SIAM Journal on Discrete Mathematics, 12(1):91–110, February 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    D.A. Christie. A 3/2-approximation algorithm for sorting by reversals. In Proc. ninth annual ACM-SIAM Symp. on Discrete Algorithms (SODA 98), pages 244–252. ACM Press, 1998.Google Scholar
  8. 8.
    D.A. Christie. Genome Rearrangement Problems. PhD thesis, University of Glasgow, 1999.Google Scholar
  9. 9.
    H. Eriksson, K. Eriksson, J. Karlander, L. Svensson, and J. Wastlund. Sorting a bridge hand. Discrete Mathematics, 241(1–3):289–300, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Q.P. Gu, S. Peng, and H. Sudborough. A 2-approximation algorithm for genome rearrangements by reversals and transpositions. Theoretical Computer Science, 210(2):327–339, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    S. Hannenhalli and P. Pevzner. Transforming cabbage into turnip: Polynomial algorithm for sorting signed permutations by reversals. Journal of the ACM, 46:1–27, 1999. (Preliminary version in Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing 1995 (STOC 95), pages 178-189).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    S.B. Hoot and J. D. Palmer. Structural rearrangements, including parallel inversions, within the chloroplast genome of Anemone and related genera. J. Molecular Evooution, 38:274–281, 1994.Google Scholar
  13. 13.
    H. Kaplan, R. Shamir, and R. E. Tarjan. Faster and simpler algorithm for sorting signed permutations by reversals. SIAM Journal of Computing, 29(3):880–892, 2000. (Preliminary version in Proceedings of the eighth annual ACM-SIAM Symposium on Discrete Algorithms 1997 (SODA 97), ACM Press, pages 344–351).zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    G.H. Lin and G. Xue. Signed genome rearrangements by reversals and transpositions: Models and approximations. Theoretical Computer Science, 259:513–531, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    J. Meidanis, M. E. Walter, and Z. Dias. Reversal distance of signed circular chromosomes. manuscript, 2000.Google Scholar
  16. 16.
    J.D. Palmer and L. A. Herbon. Tricircular mitochondrial genomes of Brassica and Raphanus: reversal of repeat configurations by inversion. Nucleic Acids Research, 14:9755–9764, 1986.CrossRefGoogle Scholar
  17. 17.
    P.A. Pevzner. Computational Molecular Biology: An Algorithmic Approach. MIT Press, 2000.Google Scholar
  18. 18.
    D. Sankoff and N. El-Mabrouk. Genome rearrangement. In T. Jiang, T. Smith, Y. Xu, and M. Q. Zhang, editors, Current Topics in Computational Molecular Biology. MIT Press, 2002.Google Scholar
  19. 19.
    J. Setubal and J. Meidanis. Introduction to Computational Biology. PWS Publishing Co., 1997.Google Scholar
  20. 20.
    R. Shamir. Algorithms in molecular biology: Lecture notes, 2002. Available at http://www.math.tau.ac.il/~rshamir/algmb/01/algmb01.html.
  21. 21.
    M.E. Walter, Z. Dias, and J. Meidanis. Reversal and transposition distance of linear chromosomes. In String Processing and Information Retrieval: A South American Symposium (SPIRE 98), 1998.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Tzvika Hartman
    • 1
  1. 1.Dept. of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael

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