Exact and approximation algorithms for sorting by reversals, with application to genome rearrangement
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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 wordsComputational biology Approximation algorithms Branch- and-bound algorithms Experimental analysis of algorithms Edit distance Permutations Sorting by reversals Chromosome inversions Genome rearrangements
- [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
- Dobzhansky, T.Genetics of the Evolutionary Process. Columbia University Press, New York, 1970.Google Scholar
- 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
- Golan, H. Personal communication, 1991.Google 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
- Knuth, D. E.The Art of Computer Programming, Vol. 3. Addison-Wesley, Reading, MA, 1973.Google Scholar
- 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
- 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
- Sessions, S. K. Chromosomes: molecular cytogenetics. InMolecular Systematics, D. M. Hillis and C. Moritz, eds., Sinauer, Sunderland, MA, 1990, pp. 156–204.Google Scholar
- 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