Skip to main content

Bounds on the Transposition Distance for Lonely Permutations

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNBI,volume 6268)

Abstract

The problem of determining the transposition distance of permutations is a notoriously challenging one; to this date, neither there exists a polynomial algorithm for solving it, nor a proof that it is NP-hard. Moreover, there are no tight bounds on the transposition distance of permutations in general. Our proposed approach merges two successful strategies: the classical reality and desire diagram proposed by Bafna and Pevzner and the more recent toric equivalence relation proposed by Eriksson et al. We focus on unitary toric equivalence classes and the corresponding lonely permutations. In a previous paper, we considered the case n + 1 prime, proved that the reality and desire diagram of such lonely permutations has just one odd cycle and succeed in identifying in this subset of lonely permutations, new permutations for which the transposition distance is computed. The present paper extends regularity properties of the cycle structure for general n, yielding tight bounds for the transposition distance of lonely permutations. The subset of lonely permutations that are 3-permutations is characterized and consequently an upper bound is obtained for their transposition distances.

Keywords

  • Cycle Length
  • Tight Bound
  • Cyclic Shift
  • Cycle Structure
  • Classical Reality

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-642-15060-9_4
  • Chapter length: 12 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   69.99
Price excludes VAT (USA)
  • ISBN: 978-3-642-15060-9
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   89.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bafna, V., Pevzner, P.A.: Sorting by transpositions. SIAM J. Disc. Math. 11(2), 224–240 (1998)

    MATH  CrossRef  MathSciNet  Google Scholar 

  2. Sankoff, D., Leduc, G., Antoine, N., Paquin, B., Lang, B.F., Cedergren, R.: Gene order comparisons for phylogenetic inference: evolution of the mitochondrial genome. Proc. Natl. Acad. Sci. 89(14), 6575–6579 (1992)

    CrossRef  Google Scholar 

  3. Boore, J.L.: The duplication/random loss model for gene rearrangement exemplified by mitochondrial genomes of deuterostome animals. In: Sankoff, D., Nadeau, J.H. (eds.) Comparative Genomics, pp. 133–148. Kluwer Academic Publishers, Dordrecht (2000)

    Google Scholar 

  4. Elias, I., Hartman, T.: A 1.375-approximation algorithm for sorting by transpositions. IEEE/ACM Trans. Comput. Biol. and Bioinformatics 3(4), 369–379 (2006)

    CrossRef  Google Scholar 

  5. Labarre, A.: New bounds and tractable instances for the transposition distance. IEEE/ACM Trans. Comput. Biol. and Bioinformatics 3(4), 380–394 (2006)

    CrossRef  Google Scholar 

  6. Eriksson, H., Eriksson, K., Karlander, J., Svensson, L., Wästlund, J.: Sorting a bridge hand. Discrete Mathematics 241(1), 289–300 (2001)

    MATH  CrossRef  MathSciNet  Google Scholar 

  7. Hausen, R., Faria, L., Figueiredo, C.M., Kowada, L.A.: On the toric graph as a tool to handle the problem of sorting by transpositions. In: Bazzan, A.L.C., Craven, M., Martins, N.F. (eds.) BSB 2008. LNCS (LNBI), vol. 5167, pp. 79–91. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  8. Hultman, A.: Toric Permutations. Master’s thesis, Department of Mathematics, KTH, Stockholm, Sweden (1999)

    Google Scholar 

  9. Hausen, R., Faria, L., Figueiredo, C.M., Kowada, L.A.: Unitary toric classes, the reality and desire diagram, and sorting by transpositions. SIAM J. Disc. Math. (to appear, 2010)

    Google Scholar 

  10. Meidanis, J., Walter, M.E.M.T., Dias, Z.: Transposition distance between a permutation and its reverse. In: Baeza-Yates, R. (ed.) Proc. 4th South American Workshop on String Processing, pp. 70–79. Carleton University Press (1997)

    Google Scholar 

  11. Christie, D.A.: Genome Rearrangement Problems. PhD thesis, University of Glasgow (1999)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kowada, L.A.B., de A. Hausen, R., de Figueiredo, C.M.H. (2010). Bounds on the Transposition Distance for Lonely Permutations. In: Ferreira, C.E., Miyano, S., Stadler, P.F. (eds) Advances in Bioinformatics and Computational Biology. BSB 2010. Lecture Notes in Computer Science(), vol 6268. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15060-9_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-15060-9_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15059-3

  • Online ISBN: 978-3-642-15060-9

  • eBook Packages: Computer ScienceComputer Science (R0)