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Sorting Circular Permutations by Reversal

  • Andrew Solomon
  • Paul Sutcliffe
  • Raymond Lister
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2748)

Abstract

Unsigned circular permutations are used to represent tours in the traveling salesman problem as well as the arrangement of gene loci in circular chromosomes. The minimum number of segment reversals required to transform one circular permutation into another gives some measure of distance between them which is useful when studying the 2-opt local search landscape for the traveling salesman problem, and, when determining the phylogeny of a group of related organisms. Computing this distance is equivalent to sorting by (a minimum number of) reversals. In this paper we show that sorting circular permutations by reversals can be reduced to the same problem for linear reversals, and that it is NP-hard. These results suggest that for most practical purposes any computational tools available for reversal sort of linear permutations will be sufficiently accurate.

These results entail the development of the algebraic machinery for dealing rigorously with circular permutations.

Keywords

Travel Salesman Problem Circular Permutation Circular Arrangement Inapproximability Result Reversal Distance 
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.
    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
  2. 2.
    Bafna, V., Pevzner, P.A.: Genome rearrangements and sorting by reversals. SIAM Journal on Computing 25, 272–289 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Berman, P., Karpinski, M.: On some tighter inapproximability results (extended abstract). In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 200–209. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  4. 4.
    Boese, K.D.: Cost Versus Distance In the Traveling Salesman Problem, Technical Report CSD-950018, UCLA Computer Science Department (May 1995) Google Scholar
  5. 5.
    Caprara, A.: Sorting Permutations by Reversals and Eulerian Cycle Decompositions. SIAM Journal on Discrete Mathematics 12(1), 91–110 (1999)zbMATHCrossRefMathSciNetGoogle 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, California, pp. 244–252, January 25-27 (1998)Google Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability. W. H. Freeman, New York (1979)zbMATHGoogle Scholar
  8. 8.
    Hannenhalli, S., Pevzner, P.A.: Transforming cabbage into turnip: a polynomial algorithm for sorting signed permutations by reversals. Journal of ACM 46, 1–27 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kececioglu, J., Sankoff, D.: Efficient bounds for oriented chromosome inversion distance. In: Kececioglu, J., Sankoff, D. (eds.) Proceedings of the 5th Symposium on Combinatorial Pattern Matching. LNCS, vol. 807, pp. 307–325. Springer, Heidelberg (1994)Google Scholar
  10. 10.
    Kececioglu, J., Sankoff, D.: Exact and approximation algorithms for sorting by reversals, with application to genome rearrangement. Algorithmica 13, 180–210 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lin, S., Kernighan, B.: An efficient heuristic for the traveling salesman problem. Operations Research 21(2), 498–516 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Meidanis, J., Walter, M.E.M.T., Dias, Z.: Reversal distance of signed circular chromosomes, Technical Report IC-00-23 (December 2000), Instituto de Computação, Universidade Estadual de Campinas, http://www.ic.unicamp.br/ic-tr-ftp/2000/Abstracts.html
  13. 13.
    Micali, S., Vazirani, V.: An \(O(\sqrt{|V|}|E|)\) algorithm for finding maximum matchings in general gaphs. In: Proceedings of the 21st Symposium on Foundations of Computer Science, pp. 17–27 (1980) (cited in [10])Google Scholar
  14. 14.
    Watterson, G., Ewens, W., Hall, T., Morgan, A.: The chromosome inversion problem. J. Theor. Biol. 99, 1–7 (1982)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Andrew Solomon
    • 1
  • Paul Sutcliffe
    • 1
  • Raymond Lister
    • 1
  1. 1.University of TechnologySydneyAustralia

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