Abstract
In comparative genomics, a transposition is an operation that exchanges two consecutive sequences of genes in a genome. The transposition distance, that is, the minimum number of transpositions needed to transform a genome into another, can be considered as a relevant evolutionary distance. The problem of computing this distance when genomes are represented by permutations, called the Sorting by Transpositions problem (SBT), has been introduced by Bafna and Pevzner [3] in 1995. It has naturally been the focus of a number of studies, but the computational complexity of this problem has remained undetermined for 15 years.
In this paper, we answer this long-standing open question by proving that the Sorting by Transpositions problem is NP-hard. As a corollary of our result, we also prove that the following problem from [10] is NP-hard: given a permutation π, is it possible to sort π using d b (π)/3 permutations, where d b (π) is the number of breakpoints of π?
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References
Amir, A., Aumann, Y., Benson, G., Levy, A., Lipsky, O., Porat, E., Skiena, S., Vishne, U.: Pattern matching with address errors: Rearrangement distances. J. Comput. Syst. Sci. 75(6), 359–370 (2009)
Amir, A., Aumann, Y., Indyk, P., Levy, A., Porat, E.: Efficient computations of ℓ1 and ℓ?8? rearrangement distances. In: Ziviani, N., Baeza-Yates, R.A. (eds.) SPIRE 2007. LNCS, vol. 4726, pp. 39–49. Springer, Heidelberg (2007)
Bafna, V., Pevzner, P.A.: Sorting permutations by transpositions. In: SODA, pp. 614–623 (1995)
Bafna, V., Pevzner, P.A.: Sorting by transpositions. SIAM J. Discrete Math. 11(2), 224–240 (1998)
Benoît-Gagné, M., Hamel, S.: A new and faster method of sorting by transpositions. In: Ma, B., Zhang, K. (eds.) CPM 2007. LNCS, vol. 4580, pp. 131–141. Springer, Heidelberg (2007)
Bongartz, D.: Algorithmic Aspects of Some Combinatorial Problems in Bioinformatics. PhD thesis, RWTH Aachen University, Germany (2006)
Bulteau, L., Fertin, G., Rusu, I.: Sorting by Transpositions is Difficult. CoRR abs/1011.1157 (2010)
Chitturi, B., Sudborough, I.H.: Bounding prefix transposition distance for strings and permutations. In: HICSS, p. 468. IEEE Computer Society, Los Alamitos (2008)
Christie, D.A.: Sorting permutations by block-interchanges. Inf. Process. Lett. 60(4), 165–169 (1996)
Christie, D.A.: Genome Rearrangement Problems. PhD thesis, University of Glasgow, Scotland (1998)
Christie, D.A., Irving, R.W.: Sorting strings by reversals and by transpositions. SIAM J. Discrete Math. 14(2), 193–206 (2001)
Cormode, G., Muthukrishnan, S.: The string edit distance matching problem with moves. In: SODA, pp. 667–676 (2002)
Dias, Z., Meidanis, J.: Sorting by prefix transpositions. In: Laender, A.H.F., Oliveira, A.L. (eds.) SPIRE 2002. LNCS, vol. 2476, pp. 65–76. Springer, Heidelberg (2002)
Elias, I., Hartman, T.: A 1.375-approximation algorithm for sorting by transpositions. IEEE/ACM Trans. Comput. Biology Bioinform. 3(4), 369–379 (2006)
Eriksson, H., Eriksson, K., Karlander, J., Svensson, L.J., Wästlund, J.: Sorting a bridge hand. Discrete Mathematics 241(1-3), 289–300 (2001)
Feng, J., Zhu, D.: Faster algorithms for sorting by transpositions and sorting by block interchanges. ACM Transactions on Algorithms 3(3) (2007)
Fertin, G., Labarre, A., Rusu, I., Tannier, É., Vialette, S.: Combinatorics of genome rearrangements. The MIT Press, Cambridge (2009)
Gu, Q.-P., Peng, S., Chen, Q.M.: Sorting permutations and its applications in genome analysis. Lectures on Mathematics in the Life Science, vol. 26, pp. 191–201 (1999)
Guyer, S.A., Heath, L.S., Vergara, J.P.: Subsequence and run heuristics for sorting by transpositions. Technical report, Virginia State University (1997)
Hartman, T., Shamir, R.: A simpler and faster 1.5-approximation algorithm for sorting by transpositions. Inf. Comput. 204(2), 275–290 (2006)
Kolman, P., Waleń, T.: Reversal distance for strings with duplicates: Linear time approximation using hitting set. In: Erlebach, T., Kaklamanis, C. (eds.) WAOA 2006. LNCS, vol. 4368, pp. 279–289. Springer, Heidelberg (2007)
Labarre, A.: New bounds and tractable instances for the transposition distance. IEEE/ACM Trans. Comput. Biology Bioinform. 3(4), 380–394 (2006)
Labarre, A.: Edit distances and factorisations of even permutations. In: Halperin, D., Mehlhorn, K. (eds.) ESA 2008. LNCS, vol. 5193, pp. 635–646. Springer, Heidelberg (2008)
Qi, X.-Q.: Combinatorial Algorithms of Genome Rearrangements in Bioinformatics. PhD thesis, University of Shandong, China (2006)
Radcliffe, A.J., Scott, A.D., Wilmer, A.L.: Reversals and transpositions over finite alphabets. SIAM J. Discret. Math. 19, 224–244 (2005)
Shapira, D., Storer, J.A.: Edit distance with move operations. In: Apostolico, A., Takeda, M. (eds.) CPM 2002. LNCS, vol. 2373, pp. 85–98. Springer, Heidelberg (2002)
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Bulteau, L., Fertin, G., Rusu, I. (2011). Sorting by Transpositions Is Difficult. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_55
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DOI: https://doi.org/10.1007/978-3-642-22006-7_55
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