Transposition Distance Based On The Algebraic Formalism

In computational biology, genome rearrangements is a field in which we study mutational events affecting large portions of a genome. One such event is the transposition, that changes the position of contiguous blocks of genes inside a chromosome. This event generates the problem of transposition distance, that is to find the minimal number of transpositions transforming one chromosome into another. It is not known whether this problem is -hard or has a polynomial time algorithm. Some approximation algorithms have been proposed in the literature, whose proofs are based on exhaustive analysis of graphical properties of suitable cycle graphs. In this paper, we follow a different, more formal approach to the problem, and present a 1.5-approximation algorithm using an algebraic formalism. Besides showing the feasibility of the approach, the presented algorithm exhibits good results, as our experiments show. © 2008 Springer-Verlag Berlin Heidelberg.5167 LNBI115126Bader, D.A., Moret, B.M.E., Yan, M., A linear-time algorithm for computing inversion distance between signed permutations with an experimental study (2001) Journal of Computational Biology, 8 (5), pp. 483-491Bafna, V., Pevzner, P.A., Sorting by transpositions (1995) Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 614-623. , San Francisco, USA, JanuaryBafna, V., Pevzner, P.A., Sorting by transpositions (1998) SIAM Journal on Discrete Mathematics, 11 (2), pp. 224-240Benoît-Gagné, M., Hamel, S.: A new and faster method of sorting by transpositions. In: Ma, B., Zhang, K. (eds.) CPM 2007. LNCS, 4580, pp. 131-141. Springer, Heidelberg (2007)Christie, D.A., Sorting permutations by block-interchanges (1996) Information Processing Letters, 60 (4), pp. 165-169Christie, D.A., (1998) Genome Rearrangement Problems, , PhD thesis, Glasgow UniversityElias, I., Hartman, T.: A 1.375-approximation algorithm for sorting by transpositions. In: Casadio, R., Myers, G. (eds.) WABI 2005. LNCS (LNBI), 3692, pp. 204-215. Springer, Heidelberg (2005)Hannenhalli, S., Pevzner, P.A., Transforming men into mice (polynomial algorithm for genomic distance problem) (1995) Proceedings of the 36th Annual Symposium on Foundations of Computer Science (FOCS, pp. 581-592. , October, IEEE Computer Society Press, Los Alamitos () 1995Hartman, T.: A simpler 1.5-approximation algorithm for sorting by transpositions. In: Baeza-Yates, R., Chávez, E., Crochemore, M. (eds.) CPM 2003. LNCS, 2676, pp. 156-169. Springer, Heidelberg (2003)Hartman, T., Shamir, R.: A simpler and faster 1.5-approximation algorithm for sorting by transpositions. In: Proceedings of CPM 2003, pp. 156-169 (2003) (extended version)Honda, M.I., (2004) Implementation of the algorithm of Hartman for the problem of sorting by transpositions, , Master's thesis, Department of Computer Science, University of Brasilia in portugueseMeidanis, J., Dias, Z., An alternative algebraic formalism for genome rearrangements (2000) Comparative Genomics: Empirical and Analyitical Approaches to Gene Order Dynamics, Map Alignment and Evolution of Gene Families, pp. 213-223. , Sankoff, D, Nadeau, J.H, eds, Kluwer Academic Publishers, Dordrecht NovemberMeidanis, J., Walter, M.E.M.T., Dias, Z., Transposition distance between a permutation and its reverse (1997) Proceedings of the 4th South American Workshop on String Processing (WSP 1997), pp. 70-79. , Baeza-Yates, R, ed, Valparaiso, Chile, pp, Carleton University PressMira, C., Meidanis, J., Algebraic formalism for genome rearrangements (part 1) (2005), Technical Report IC-05-10, Institute of Computing, University of Campinas JuneMira, C.V.G., Meidanis, J., Analysis of sorting by transpositions based on algebraic formalism (2004) The Eighth Annual International Conference on Research in Computational Molecular Biology (RECOMB, , MarchWalter, M.E.M.T., Curado, L.R.A.F., Oliveira, A.G.: Working on the problem of sorting by transpositions on genome rearrangements. In: Baeza-Yates, R., Chávez, E., Crochemore, M. (eds.) CPM 2003. LNCS, 2676, pp. 372-383. Springer, Heidelberg (2003)Walter, M.E.M.T., Dias, Z., Meidanis, J., A new approach for approximating the transposition distance (2000) String Processing and Information Retrieval, pp. 199-208. , SPIREWalter, M.E.M.T., Oliveira, E.T.G., Extending the theory of Bafna and Pevzner for the problem of sorting by transpositions (2002) Tendências em Matemática Aplicada e Computacional - TEMA - SBMAC, 3 (1), pp. 213-222. , in portugueseWalter, M.E.M.T., Soares, L.S.N., Dias, Z., Branch-and-bound algorithms for the problem of sorting by transpositions on genome rearrangements (2006) Proceedings of the 26th Congress of the Brazilian Computer Society, XXXIII Seminário integrado de hardware e software - SEMISH, pp. 69-8

Constraint Programming Models For Transposition Distance Problem

Genome Rearrangements addresses the problem of finding the minimum number of global operations, such as transpositions, reversals, fusions and fissions that transform a given genome into another. In this paper we deal with transposition events, which are events that change the position of two contiguous block of genes in the same chromosome. The transposition event generates the transposition distance problem, that is to find the minimum number of transposition that transform one genome (or chromosome) into another. Although some tractables instances were found [20,14], it is not known if an exact polynomial time algorithm exists. Recently, Dias and Souza [9] proposed polynomial-sized Integer Linear Programming (ILP) models for rearrangement distance problems where events are restricted to reversals, transpositions or a combination of both. In this work we devise a slight different approach. We present some Constraint Logic Programming (CLP) models for transposition distance based on known bounds to the problem. © 2009 Springer Berlin Heidelberg.5676 LNBI1323Apt, K., Wallace, M., (2007) Constraints Logic Programming using Eclipse, , CambridgeBafna, V., Pevzner, P.A., Sorting by reversals: Genome rearrangements in plant organelles and evolutionary history of X chromosome (1995) Molecular Biology and Evolution, 12 (2), pp. 239-246Bafna, V., Pevzner, P.A., Sorting by Transpositions (1998) SIAM Journal on Discrete Mathematics, 11 (2), pp. 224-240Benoǐt-Gagńe, M., Hamel, S.: A New and Faster Method of Sorting by Transpositions. In: Ma, B., Zhang, K. (eds.) CPM 2007. LNCS, 4580, pp. 131-141. Springer, Heidelberg (2007)Caprara, A., Lancia, G., Ng, S.-K., A Column-Generation Based Branch-and-Bound Algorithm for Sorting by Reversals (1999) DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 47, pp. 213-226. , The American Mathematical SocietyCaprara, A., Lancia, G., Ng, S.-K., Sorting Permutations by Reversals through Branch-and-Price (1999), Technical Report OR-99-1, DEIS, Operations Research Group, University of BolognaCaprara, A., Lancia, G., Ng, S.-K., Fast Practical Solution of Sorting by Reversals (2000) Proceedings of the 11th ACM-SIAM Annual Symposium on Discrete Algorithms (SODA, pp. 12-21. , San Francisco, USA, pp, ACM Press, New YorkChristie, D.A., (1998) Genome Rearrangement Problems, , PhD thesis, Glasgow UniversityDias, Z., Souza, C., Polynomial-sized ILP Models for Rearrangement Distance Problems (2007) BSB 2007 Poster ProceedingsDobzhansky, T., Sturtevant, A.H., Inversions in the third chromosome of wild races of Drosophila pseudoobscura, and their use in the study of the history of the species (1936) Proceedings of the National Academy of Science, 22, pp. 448-450The Eclipse Constraint Programming System, , http://www.eclipse-clp.org, March 2009Elias, I., Hartmn, T., A 1.375-Approximation Algorithm for Sorting by Transpositions (2006) Comput, pp. 369-379. , Trans, Biol. 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In: Bazzan, A.L.C., Craven, M., Martins, N.F. (eds.) BSB 2008. LNCS (LNBI), 5167, pp. 79-91. Springer, Heidelberg (2008)Kececioglu, J.D., Ravi, R.: Of Mice and Men: Algorithms for Evolutionary Distances Between Genomes with Translocation. In: Proceedings of the 6th Annual Symposium on Discrete Algorithms, January 1995, pp. 604-613. ACM Press, New York (1995)Labarre, A., New Bounds and Tractable Instances for the Transposition Distance (2006) IEEE/ACM Trans. Comput. Biol. Bioinformatics, 3 (4), pp. 380-394Marriott, K., Stuckey, P.J., (1998) Programming with Constraints: An Introduction, , MIT Press, CambridgeMira, C.V.G., Dias, Z., Santos, H.P., Pinto, G.A., Walter, M.E.: Transposition Distance Based on the Algebraic Formalism. In: Bazzan, A.L.C., Craven, M., Martins, N.F. (eds.) BSB 2008. LNCS (LNBI), 5167, pp. 115-126. Springer, Heidelberg (2008)Palmer, J.D., Herbon, L.A., Plant mitochondrial DNA evolves rapidly in structure, but slowly in sequence (1988) Journal of Molecular Evolution, 27, pp. 87-97Walter, M.E.M.T., Dias, Z., Meidanis, J., A New Approach for Approximating the Transposition Distance (2000) Proceedings of the String Processing and Information Retrieval (SPIRE, , Septembe