Sorting permutations by limited-size operations

Orientadores: Zanoni Dias, Carla Negri LintzmayerDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O resumo poderá ser visualizado no texto completo da tese digitalAbstract: The abstract is available with the full electronic digital documentMestradoCiência da ComputaçãoMestre em Ciência da ComputaçãoCAPE

O problema da ordenação de permutações usando rearranjos de prefixos e sufixos

Orientador: Zanoni DiasTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O Problema das Panquecas tem como objetivo ordenar uma pilha de panquecas que possuem tamanhos distintos realizando o menor número possível de operações. A operação permitida é chamada reversão de prefixo e, quando aplicada, inverte o topo da pilha de panquecas. Tal problema é interessante do ponto de vista combinatório por si só, mas ele também possui algumas aplicações em biologia computacional. Dados dois genomas que compartilham o mesmo número de genes, e assumindo que cada gene aparece apenas uma vez por genoma, podemos representá-los como permutações (pilhas de panquecas também são representadas por permutações). Então, podemos comparar os genomas tentando descobrir como um foi transformado no outro por meio da aplicação de rearranjos de genoma, que são eventos de mutação de grande escala. Reversões e transposições são os tipos mais comumente estudados de rearranjo de genomas e uma reversão de prefixo (ou transposição de prefixo) é um tipo de reversão (ou transposição) que é restrita ao início da permutação. Quando o rearranjo é restrito ao final da permutação, dizemos que ele é um rearranjo de sufixo. Um problema de ordenação de permutações por rearranjos é, portanto, o problema de encontrar uma sequência de rearranjos de custo mínimo que ordene a permutação dada. A abordagem tradicional considera que todos os rearranjos têm o mesmo custo unitário, de forma que o objetivo é tentar encontrar o menor número de rearranjos necessários para ordenar a permutação. Vários esforços foram feitos nos últimos anos considerando essa abordagem. Por outro lado, um rearranjo muito longo (que na verdade é uma mutação) tem mais probabilidade de perturbar o organismo. Portanto, pesos baseados no comprimento do segmento envolvido podem ter um papel importante no processo evolutivo. Dizemos que essa abordagem é ponderada por comprimento e o objetivo nela é tentar encontrar uma sequência de rearranjos cujo custo total (que é a soma do custo de cada rearranjo, que por sua vez depende de seu comprimento) seja mínimo. Nessa tese nós apresentamos os primeiros resultados que envolvem problemas de ordenação de permutações por reversões e transposições de prefixo e sufixo considerando ambas abordagens tradicional e ponderada por comprimento. Na abordagem tradicional, consideramos um total de 10 problemas e desenvolvemos novos resultados para 6 deles. Na abordagem ponderada por comprimento, consideramos um total de 13 problemas e desenvolvemos novos resultados para todos elesAbstract: The goal of the Pancake Flipping problem is to sort a stack of pancakes that have different sizes by performing as few operations as possible. The operation allowed is called prefix reversal and, when applied, flips the top of the stack of pancakes. Such problem is an interesting combinatorial problem by itself, but it has some applications in computational biology. Given two genomes that share the same genes and assuming that each gene appears only once per genome, we can represent them as permutations (stacks of pancakes are also represented by permutations). Then, we can compare the genomes by figuring out how one was transformed into the other through the application of genome rearrangements, which are large scale mutations. Reversals and transpositions are the most commonly studied types of genome rearrangements and a prefix reversal (or prefix transposition) is a type of reversal (or transposition) which is restricted to the beginning of the permutation. When the rearrangement is restricted to the end of the permutation, we say it is a suffix rearrangement. A problem of sorting permutations by rearrangements is, therefore, the problem to find a sequence of rearrangements with minimum cost that sorts a given permutation. The traditional approach considers that all rearrangements have the same unitary cost, in which case the goal is trying to find the minimum number of rearrangements that are needed to sort the permutation. Numerous efforts have been made over the past years regarding this approach. On the other hand, a long rearrangement (which is in fact a mutation) is more likely to disturb the organism. Therefore, weights based on the length of the segment involved may have an important role in the evolutionary process. We say this is the length-weighted approach and the goal is trying to find a sequence of rearrangements whose total cost (the sum of the cost of each rearrangement, which depends on its length) is minimum. In this thesis we present the first results regarding problems of sorting permutations by prefix and suffix reversals and transpositions considering both the traditional and the length-weighted approach. For the traditional approach, we considered a total of 10 problems and developed new results for 6 of them. For the length-weighted approach, we considered a total of 13 problems and developed new results for all of themDoutoradoCiência da ComputaçãoDoutora em Ciência da Computação140017/2013-52013/01172-0FAPESPCNP

Sorting Permutations By Prefix And Suffix Versions Of Reversals And Transpositions

Reversals and transpositions are the most common kinds of genome rearrangements, which allow us to establish the divergence between individuals along evolution. When the rearrangements affect segments from the beginning or from the end of the genome, we say they are prefix or suffix rearrangements, respectively. This paper presents the first approximation algorithms for the problems of Sorting by Prefix Reversals and Suffix Reversals, Sorting by Prefix Transpositions and Suffix Transpositions and Sorting by Prefix Reversals, Prefix Transpositions, Suffix Reversals and Suffix Transpositions, all of them with factor 2. We also present the intermediary algorithms that lead us to the main results. © 2014 Springer-Verlag Berlin Heidelberg.8392 LNCS671682Agencia Nacional de Investigacion e Innovacion (ANII),Centro Latinoamericano de Estudios en Informatica (CLEI),et al.,Google,Universidad de la Republica, CSIC,Yahoo! LabsBafna, V., Pevzner, P.A., Genome rearrangements and sorting by reversals (1993) Proceedings of the 34th Annual Symposium on Foundations of Computer Science FOCS 1993, pp. 148-157Bafna, V., Pevzner, P.A., Sorting by transpositions (1998) SIAM Journal on Discrete Mathematics, 11 (2), pp. 224-240Berman, P., Hannenhalli, S., Karpinski, M., 1.375-approximation algorithm for sorting by reversals (2002) ESA 2002. LNCS, 2461, pp. 200-210. , Möhring, R.H., Raman, R. (eds.) Springer, HeidelbergBulteau, L., Fertin, G., Rusu, I., Pancake flipping is hard (2012) MFCS 2012. LNCS, 7464, pp. 247-258. , Rovan, B., Sassone, V., Widmayer, P. (eds.) Springer, HeidelbergBulteau, L., Fertin, G., Rusu, I., Sorting by transpositions is difficult (2012) SIAM Journal on Computing, 26 (3), pp. 1148-1180Caprara, A., Sorting permutations by reversals and eulerian cycle decompositions (1999) SIAM Journal on Discrete Mathematics, 12 (1), pp. 91-110Dias, Z., Meidanis, J., Sorting by prefix transpositions (2002) SPIRE 2002. LNCS, 2476, pp. 65-76. , Laender, A.H.F., Oliveira, A.L. (eds.) Springer, HeidelbergElias, I., Hartman, T., A 1.375-approximation algorithm for sorting by transpositions (2006) IEEE/ACM Transactions on Computational Biology and Bioinformatics, 3 (4), pp. 369-379Fischer, J., Ginzinger, S.W., A 2-approximation algorithm for sorting by prefix reversals (2005) ESA 2005. LNCS, 3669, pp. 415-425. , Brodal, G.S., Leonardi, S. (eds.) Springer, HeidelbergGalvão, G.R., Dias, Z., On the performance of sorting permutations by prefix operations (2012) Proceedings of the 4th International Conference on Bioinformatics and Computational Biology (BICoB 2012), pp. 102-107. , Las Vegas, Nevada, USASharmin, M., Yeasmin, R., Hasan, M., Rahman, A., Rahman, M.S., Pancake flipping with two spatulas (2010) Electronic Notes in Discrete Mathematics, 36, pp. 231-238. , International Symposium on Combinatorial Optimization ISCO 201

On The Diameter Of Rearrangement Problems

When we consider the Genome Rearrangements area, the problems of finding the distance of a permutation and finding the diameter of all permutations of the same size are the most common studied. In this paper, we considered problems for which no known results were presented regarding their diameters. We present some families of permutations whose distance is identical to the diameter for small sizes. They allowed us to gave bounds for the diameters of the problems we considered, as well as conjectures regarding the exact value. © 2014 Springer International Publishing.8542 LNBI158170Bafna, V., Pevzner, P.A., Genome Rearrangements and Sorting by Reversals (1993) Proceedings of the 34th Annual Symposium on Foundations of Computer Science (FOCS 1993), pp. 148-157Bulteau, L., Fertin, G., Rusu, I., Pancake Flipping is Hard (2012) LNCS, 7464, pp. 247-258. , Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. Springer, HeidelbergBulteau, L., Fertin, G., Rusu, I., Sorting by Transpositions is Difficult (2012) SIAM Journal on Computing, 26 (3), pp. 1148-1180Caprara, A., Sorting Permutations by Reversals and Eulerian Cycle Decompositions (1999) SIAM Journal on Discrete Mathematics, 12 (1), pp. 91-110Chitturi, B., Fahle, W., Meng, Z., Morales, L., Shields, C.O., Sudborough, I.H., Voit, W., An (18/11)n Upper Bound for Sorting by Prefix Reversals (2009) Theoretical Computer Science, 410 (36), pp. 3372-3390Chitturi, B., Sudborough, I.H., Bounding Prefix Transposition Distance for Strings and Permutations (2012) Theoretical Computer Science, 421, pp. 15-24Cibulka, J., On Average and Highest Number of Flips in Pancake Sorting (2011) Theoretical Computer Science, 412 (8-10), pp. 822-834Dias, Z., Meidanis, J., Sorting by Prefix Transpositions (2002) LNCS, 2476, pp. 65-76. , Laender, A.H.F., Oliveira, A.L. (eds.) SPIRE 2002. Springer, HeidelbergElias, I., Hartman, T., A 1.375-Approximation Algorithm for Sorting by Transpositions (2006) 375-Approximation Algorithm for Sorting by Transpositions, 3 (4), pp. 369-379Eriksson, H., Eriksson, K., Karlander, J., Svensson, L., Wastlund, J., Sorting a Bridge Hand (2001) Discrete Mathematics, 241 (1-3), pp. 289-300Fertin, G., Labarre, A., Rusu, I., Tannier, É., Vialette, S., Combinatorics of Genome Rearrangements (2009) Computational Molecular Biology, , MIT PressGalvão, G.R., Dias, Z., Computing Rearrangement Distance of Every Permutation in the Symmetric Group (2011) Proceedings of the 26th ACM Symposium on Applied Computing (SAC 22011), pp. 106-107. , Chu, W.C., Wong, W.E., Palakal, M.J., Hung, C.C. (eds.) ACMGates, W.H., Papadimitriou, C.H., Bounds for Sorting by Prefix Reversal (1979) Discrete Mathematics, 27 (1), pp. 47-57Hannenhalli, S., Pevzner, P.A., Transforming Cabbage into Turnip: Polynomial Algorithm for Sorting Signed Permutations by Reversals (1999) Journal of the ACM, 46 (1), pp. 1-27Heydari, M.H., Sudborough, I.H., On the Diameter of the Pancake Network (1997) Journal of Algorithms, 25 (1), pp. 67-94Labarre, A., Edit Distances and Factorisations of Even Permutations (2008) LNCS, 5193, pp. 635-646. , Halperin, D., Mehlhorn, K. (eds.) ESA 2008. Springer, HeidelbergLintzmayer, C.N., Dias, Z., On Sorting of Signed Permutations by Prefix and Suffix Reversals and Transpositions (2014) Proceedings of the 1st International Conference on Algorithms for Computational Biology (AlCoB 2014), Tarragona, Spain, pp. 1-12. , Dediu, A.H., Martín-Vide, C., Truthe, B. (eds.) SpringerLintzmayer, C.N., Dias, Z., Sorting Permutations by Prefix and Suffix Versions of Reversals and Transpositions (2014) LNCS, 8392, pp. 671-682. , Pardo, A., Viola, A. (eds.) LATIN 2014. Springer, HeidelbergMeidanis, J., Walter, M.M.T., Dias, Z., A Lower Bound on the Reversal and Transposition Diameter (2002) Journal of Computational Biology, 9 (5), pp. 743-745Sharmin, M., Yeasmin, R., Hasan, M., Rahman, A., Rahman, M.S., Pancake Flipping with Two Spatulas (2010) Electronic Notes in Discrete Mathematics, 36, pp. 231-238. , International Symposium on Combinatorial Optimization (ISCO 2010)Walter, M.E.M.T., Dias, Z., Meidanis, J., Reversal and Transposition Distance of Linear Chromosomes (1998) Proceedings of the 5th International Symposium on String Processing and Information Retrieval (SPIRE 1998), pp. 96-102. , IEEE Computer Society, Santa Cru