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

- 320 Downloads
- 91 Citations

## Abstract

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 a*reversal* 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 in*O*(*n* ^{2}) time and*0(n)* space for*n*-element permutations, and a branch- and-bound exact algorithm, that finds an optimal solution in*0(mL(n, n))* time and*0*(*n* ^{2}) space, where*m* is the size of the branch- and-bound search tree, and*L(n, n)* is the time to solve a linear program of*n* variables and*n* 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 by*k* random reversals, we find that the average upper bound on reversal distance estimates*k* to within one reversal for*k*<1/2n and*n*<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 for*n*<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## References

- [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]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 - [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 - [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 - [4]Dobzhansky, T.
*Genetics of the Evolutionary Process.*Columbia University Press, New York, 1970.Google Scholar - [5]Driscoll, J. R., and M. L. Furst. Computing short generator sequences.
*Information and Computation*,**72**, 117–132, 1987.MATHCrossRefMathSciNetGoogle Scholar - [6]Even, S., and O. Goldreich. The minimum-length generator sequence problem is NP-hard.
*Journal of Algorithms*,**2**, 311–313, 1981.MATHCrossRefMathSciNetGoogle Scholar - [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 - [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 - [9]Gates, W. H., and C. H. Papadimitriou. Bounds for sorting by prefix reversal.
*Discrete Mathematics*,**27**, 47–57, 1979.CrossRefMathSciNetGoogle Scholar - [10]Golan, H. Personal communication, 1991.Google Scholar
- [11]Jerrum, M. R. The complexity of finding minimum-length generator sequences.
*Theoretical Computer Science*,**36**, 265–289, 1985.MATHCrossRefMathSciNetGoogle Scholar - [12]Johnson, D. B. Finding all the elementary circuits of a directed graph.
*SIAM Journal on Computing*,**4**(1), 77–84, 1975.MATHCrossRefMathSciNetGoogle Scholar - [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 - [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 - [13]Knuth, D. E.
*The Art of Computer Programming*, Vol. 3. Addison-Wesley, Reading, MA, 1973.Google Scholar - [14]Mannila, H. Measures of presortedness and optimal sorting algorithms,
*IEEE Transactions on Computers*, 34, 318–325, 1985.MATHMathSciNetCrossRefGoogle Scholar - [15]Micali, S. and V. Vazirani. An
*o*(√¦*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 - [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 - [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 - [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 - [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 - [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 - [21]Sessions, S. K. Chromosomes: molecular cytogenetics. In
*Molecular Systematics*, D. M. Hillis and C. Moritz, eds., Sinauer, Sunderland, MA, 1990, pp. 156–204.Google Scholar - [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 - [23]Wagner, R. A. On the complexity of the extended string-to-string correction problem. In
*Time 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 - [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 - [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