Sequence comparison in molecular biology is in the beginning of a major paradigm shift—a shift from gene comparison based on local mutations (i.e., insertions, deletions, and substitutions of nucleotides) to chromosome comparison based on global rearrangements (i.e., inversions and transpositions of fragments). The classical methods of sequence comparison do not work for global rearrangements, and little is known in computer science about the edit distance between sequences if global rearrangements are allowed. In the simplest form, the problem of gene rearrangements corresponds to sorting by reversals, i.e., sorting of an array using reversals of arbitrary fragments. Recently, Kececioglu and Sankoff gave the first approximation algorithm for sorting by reversals with guaranteed error bound 2 and identified open problems related to chromosome rearrangements. One of these problems is Gollan’s conjecture on the reversal diameter of the symmetric group. This paper proves the conjecture. Further, the problem of expected reversal distance between two random permutations is investigated. The reversal distance between two random permutations is shown to be very close to the reversal diameter, thereby indicating that reversal distance provides a good separation between related and nonrelated sequences in molecular evolution studies. The gene rearrangement problem forces us to consider reversals of signed permutations, as the genes in DNA could be positively or negatively oriented. An approximation algorithm for signed permutation is presented, which provides a performance guarantee of $\tfrac{3}{2}$ . Finally, using the signed permutations approach, an approximation algorithm for sorting by reversals is described which achieves a performance guarantee of $\tfrac{7}{4}$.

MSC codes

  1. 68Q25
  2. 68Q05


  1. computational molecular biology
  2. sorting by reversals
  3. genome rearrangements

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Martin Aigner, Douglas B. West, Sorting by insertion of leading elements, J. Combin. Theory Ser. A, 45 (1987), 306–309
Nancy Amato, Manuel Blum, Sandra Irani, Ronitt Rubinfeld, Reversing trains: a turn of the century sorting problem, J. Algorithms, 10 (1989), 413–428
V. Bafna, P. Pevzner, Sorting by reversals: Genome rearrangements in plant organelles and evolutionary history of X chromosome, Molec. Biol. Evolution, 12 (1995), 239–246
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Introduction to algorithms, The MIT Electrical Engineering and Computer Science Series, MIT Press, Cambridge, MA, 1990xx+1028
S. Even, O. Goldreich, The minimum-length generator sequence problem is NP-hard, J. Algorithms, 2 (1981), 311–313
C. Fauron, M. Havlik, The maize mitochondrial genome of the normal type and the cytoplasmic male sterile type T have very different organization, Current Genetics, 15 (1989), 149–154
William H. Gates, Christos H. Papadimitriou, Bounds for sorting by prefix reversal, Discrete Math., 27 (1979), 47–57
E. Gyori, G. Turan, Stack of pancakes, Studia Sci. Math. Hungar., 13 (1978), 133–137 (1981)
S. Hannenhalli, C. Chappey, E. Koonin, P. Pevzner, Genome sequence comparison and scenarios for genome rearrangement: A test case, Genomics, (1995),
R. J. Hoffmann, J. L. Boore, W. M. Brown, A novel mitochondrial genome organization for the blue mussel, Mytilus edulis, Genetics, 131 (1992), 397–412
John E. Hopcroft, R. Karp, An $n\sp{5/2}$ algorithm for maximum matchings in bipartite graphs, SIAM J. Comput., 2 (1973), 225–231
S. B. Hoot, J. D. Palmer, Structural rearrangements including parallel inversions within the chloroplast genome of anemone and related genera, J. Molec. Evolution, 38 (1994), 274–281
Mark R. Jerrum, The complexity of finding minimum-length generator sequences, Theoret. Comput. Sci., 36 (1985), 265–289
S. Karlin, E. S. Mocarski, G. A. Schachtel, Molecular evolution of herpesviruses: Genomic and protein sequence comparisons, J. Virology, 68 (1994), 1886–1902
John Kececioglu, David Sankoff, A. Apostolico, M. Crochemore, Z. Galil, U. Manber, Exact and approximation algorithms for the inversion distance between two chromosomesCombinatorial pattern matching (Padova, 1993), Lecture Notes in Comput. Sci., Vol. 684, Springer, Berlin, 1993, 87–105, Proc. 4th Annual Symposium, Italy
J. Kececioglu, D. Sankoff, Exact and approximation algorithms for sorting by reversals, with application to genome rearrangement, Algorithmica, 13 (1995), 180–210
John Kececioglu, David Sankoff, M. Crochemore, D. Gusfield, Efficient bounds for oriented chromosome inversion distanceCombinatorial pattern matching (Asilomar, CA, 1994), Lecture Notes in Comput. Sci., Vol. 807, Springer, Berlin, 1994, 307–325, Proc. 5th Annual Symposium
E. B. Knox, S. R. Downie, J. D. Palmer, Chloroplast genome rearrangements and evolution of giant lobelias from herbaceous ancestors, Molec. Biol. Evolution, 10 (1993), 414–430
E. V. Koonin, V. V. Dolja, Evolution and taxonomy of positive-strand RNA viruses: Implications of comparative analysis: of amino acid sequences, Crit. Rev. Biochem. Molec. Biol., 28 (1993), 375–430
Anton Kotzig, Moves without forbidden transitions in a graph, Mat. Časopis Sloven. Akad. Vied, 18 (1968), 76–80
M. F. Lyon, X-chromosome inactivation and the location and expression of X-linked genes, Amer. J. Human Genetics, 42 (1988), 8–16
J. D. Palmer, L. Herbon, Unicircular structure of the Brassica hirta mitochondrial genome, Current Genetics, 11 (1987), 565–570
J. D. Palmer, L. Herbon, Plant mitochondrial DNA evolves rapidly in structure, but slowly in sequence, J. Molec. Evolution, 27 (1988), 65–74
P. Pevzner, DNA physical mapping and alternating Eulerian cycles in colored graphs, Algorithmica, 13 (1995), 77–105
L. A. Raubeson, R. K. Jansen, Chloroplast DNA evidence on the ancient evolutionary split in vascular land plants, Science, 255 (1992), 1697–1699
D. Sankoff, G. Leduc, N. Antoine, B. Paquin, B. F. Lang, R. Cedergren, Gene order comparisons for phylogenetic inference: Evolution of the mitochondrial genome, Proc. Natl. Acad. Sci. U.S.A., 89 (1992), 6575–6579
D. Sankoff, Analytical approaches to genomic evolution, Biochimie, 7 (1993), 409–413
J. Whiting, M. Pliley, J. Farmer, D. Jeffery, In situ hybridization analysis of chromosomal homologies in Drosophila melanogaster and Drosophila virilis, Genetics, 122 (1989), 99–109
G. A. Watterson, W. J. Ewens, T. E. Hall, A. Morgan, The chromosome inversion problem, J. Theoret. Biol., 99 (1982), 1–7

Information & Authors


Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 272 - 289
ISSN (online): 1095-7111


Submitted: 17 June 1993
Accepted: 22 August 1994
Published online: 13 July 2006

MSC codes

  1. 68Q25
  2. 68Q05


  1. computational molecular biology
  2. sorting by reversals
  3. genome rearrangements



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