, 13:180 | Cite as

Exact and approximation algorithms for sorting by reversals, with application to genome rearrangement

  • J. Kececioglu
  • D. Sankoff


Motivated by the problem in computational biology of reconstructing the series of chromosome inversions by which one organism evolved from another, we consider the problem of computing the shortest series of reversals that transform one permutation to another. The permutations describe the order of genes on corresponding chromosomes, and areversal takes an arbitrary substring of elements, and reverses their order.

For this problem, we develop two algorithms: a greedy approximation algorithm, that finds a solution provably close to optimal inO(n 2) time and0(n) space forn-element permutations, and a branch- and-bound exact algorithm, that finds an optimal solution in0(mL(n, n)) time and0(n 2) space, wherem is the size of the branch- and-bound search tree, andL(n, n) is the time to solve a linear program ofn variables andn constraints. The greedy algorithm is the first to come within a constant factor of the optimum; it guarantees a solution that uses no more than twice the minimum number of reversals. The lower and upper bounds of the branch- and-bound algorithm are a novel application of maximum-weight matchings, shortest paths, and linear programming.

In a series of experiments, we study the performance of an implementation on random permutations, and permutations generated by random reversals. For permutations differing byk random reversals, we find that the average upper bound on reversal distance estimatesk to within one reversal fork<1/2n andn<100. For the difficult case of random permutations, we find that the average difference between the upper and lower bounds is less than three reversals forn<50. Due to the tightness of these bounds, we can solve, to optimality, problems on 30 elements in a few minutes of computer time. This approaches the scale of mitochondrial genomes.

Key words

Computational biology Approximation algorithms Branch- and-bound algorithms Experimental analysis of algorithms Edit distance Permutations Sorting by reversals Chromosome inversions Genome rearrangements 


  1. [1]
    Aigner, M., and D. B. West. Sorting by insertion of leading elements.Journal of Combinatorial Theory, Series A,45, 306–309, 1987.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Amato, N., M. Blum, S. Irani, and R. Rubinfeld. Reversing trains: a turn of the century sorting problem.Journal of Algorithms,10, 413–428, 1989.MATHCrossRefMathSciNetGoogle Scholar
  3. [2a]
    Bafna, V., and P. A. Pevzner. Genome rearrangements and sorting by reversals.Proceedings of the 34th Symposium on Foundations of Computer Science, November 1993, pp. 148–157.Google Scholar
  4. [3]
    Bibb, M. J., R. A. van Etten, C. T. Wright, M. W. Walberg, and D. A. Clayton. Sequence and gene organization of mouse mitochondrial DNA.Cell,26, 167–180, 1981.CrossRefGoogle Scholar
  5. [4]
    Dobzhansky, T.Genetics of the Evolutionary Process. Columbia University Press, New York, 1970.Google Scholar
  6. [5]
    Driscoll, J. R., and M. L. Furst. Computing short generator sequences.Information and Computation,72, 117–132, 1987.MATHCrossRefMathSciNetGoogle Scholar
  7. [6]
    Even, S., and O. Goldreich. The minimum-length generator sequence problem is NP-hard.Journal of Algorithms,2, 311–313, 1981.MATHCrossRefMathSciNetGoogle Scholar
  8. [7]
    Furst, M., J. Hopcroft, and E. Luks. Polynomial-time algorithms for permutation groups.Proceedings of the 21st Symposium on Foundations of Computer Science, 1980, pp. 36–41.Google Scholar
  9. [8]
    Garey, M. R., and D. S. Johnson.Computers and Intractability: A Guide to The Theory of NP-Completeness. Freeman, New York, 1979.MATHGoogle Scholar
  10. [9]
    Gates, W. H., and C. H. Papadimitriou. Bounds for sorting by prefix reversal.Discrete Mathematics,27, 47–57, 1979.CrossRefMathSciNetGoogle Scholar
  11. [10]
    Golan, H. Personal communication, 1991.Google Scholar
  12. [11]
    Jerrum, M. R. The complexity of finding minimum-length generator sequences.Theoretical Computer Science,36, 265–289, 1985.MATHCrossRefMathSciNetGoogle Scholar
  13. [12]
    Johnson, D. B. Finding all the elementary circuits of a directed graph.SIAM Journal on Computing,4(1), 77–84, 1975.MATHCrossRefMathSciNetGoogle Scholar
  14. [12a]
    Kececioglu, J., and D. Sankoff. Exact and approximation algorithms for the inversion distance between two chromosomes.Proceedings of the 4th Symposium on Combinatorial Pattern Matching, Lecture Notes in Computer Science, vol. 684, Springer-Verlag, Berlin, June 1993, pp. 87–105. (An earlier version appeared as “Exact and approximation algorithms for sorting by reversals,” Technical Report 1824, Centre de recherches mathématiques, Université de Montréal, July 1992).Google Scholar
  15. [12b]
    Kececioglu, J., and D. Sankoff. Efficient bounds for oriented chromosome-inversion distance.Proceedings of the 5th Symposium on Combinatorial Pattern Matching, Lecture Notes in Computer Science, vol. 807, Springer-Verlag, Berlin, June 1994, pp. 307–325.Google Scholar
  16. [13]
    Knuth, D. E.The Art of Computer Programming, Vol. 3. Addison-Wesley, Reading, MA, 1973.Google Scholar
  17. [14]
    Mannila, H. Measures of presortedness and optimal sorting algorithms,IEEE Transactions on Computers, 34, 318–325, 1985.MATHMathSciNetCrossRefGoogle Scholar
  18. [15]
    Micali, S. and V. Vazirani. Ano(√¦V¦·¦E¦) algorithm for finding maximum matchings in general graphs.Proceedings of the 21st Symposium on Foundations of Computer Science, 1980, pp. 17–27.Google Scholar
  19. [16]
    Nadeau, J. H., and B. A. Taylor. Lengths of chromosomal segments conserved since divergence of man and mouse.Proceedings of the National Academy of Sciences of the USA,81, 814, 1984.CrossRefGoogle Scholar
  20. [17]
    O'Brien, S. J., ed.Genetic Maps: Locus Maps of Complex Genomes. 6th edition. Cold Spring Harbor Laboratory Press, Cold Spring Harbor, NY, 1993.Google Scholar
  21. [18]
    Palmer, J. D., B. Osorio, and W. F. Thompson. Evolutionary significance of inversions in legume chloroplast DNAs.Current Genetics,14, 65–74, 1988.CrossRefGoogle Scholar
  22. [19]
    Sankoff, D., G. Leduc, N. Antoine, B. Paquin, B. F. Lang, and R. Cedergren. Gene order comparisons for phylogenetic inference: evolution of the mitochondrial genome.Proceedings of the National Academy of Sciences of the USA,89, 6575–6579, 1992.CrossRefGoogle Scholar
  23. [20]
    Schöniger, M., and M. S. Waterman. A local algorithm for DNA sequence alignment with inversions.Bulletin of Mathematical Biology,54, 521–536, 1992.MATHGoogle Scholar
  24. [21]
    Sessions, S. K. Chromosomes: molecular cytogenetics. InMolecular Systematics, D. M. Hillis and C. Moritz, eds., Sinauer, Sunderland, MA, 1990, pp. 156–204.Google Scholar
  25. [22]
    Tichy, W. F. The string-to-string correction problem with block moves.ACM Transactions on Computer Systems,2(4), 309–321, 1984.CrossRefMathSciNetGoogle Scholar
  26. [23]
    Wagner, R. A. On the complexity of the extended string-to-string correction problem. InTime Warps, String Edits, and Macromolecules: The Theory and Practice of Sequence Comparison, D. Sankoft and J. B. Kruskal, eds., Addison-Wesley, Reading, MA, 1983, pp. 215–235.Google Scholar
  27. [24]
    Watterson, G. A., W. J. Ewens, T. E. Hall, and A. Morgan. The chromosome inversion problem.Journal of Theoretical Biology,99, 1–7, 1982.CrossRefGoogle Scholar
  28. [25]
    Wolstenholme, D. R., J. L. MacFarlane, R. Okimoto, D. O. Clary, and J. A. Wahieithner. Bizarre tRNAs inferred from DNA sequences of mitochondrial genomes of nematode worms.Proceedings of the National Academy of Sciences of the USA,84, 1324–1328, 1987.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1995

Authors and Affiliations

  • J. Kececioglu
    • 1
  • D. Sankoff
    • 2
  1. 1.Department of Computer ScienceThe University of GeorgiaAthensUSA
  2. 2.Centre de recherches mathématiquesUniversité de MontréalMontréalCanada

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