Polynomial-time sortable stacks of burnt pancakes

Pancake flipping, a famous open problem in computer science, can be formalised as the problem of sorting a permutation of positive integers using as few prefix reversals as possible. In that context, a prefix reversal of length k reverses the order of the first k elements of the permutation. The burnt variant of pancake flipping involves permutations of signed integers, and reversals in that case not only reverse the order of elements but also invert their signs. Although three decades have now passed since the first works on these problems, neither their computational complexity nor the maximal number of prefix reversals needed to sort a permutation is yet known. In this work, we prove a new lower bound for sorting burnt pancakes, and show that an important class of permutations, known as "simple permutations", can be optimally sorted in polynomial time.Comment: Accepted pending minor revisio

The distribution of cycles in breakpoint graphs of signed permutations

Breakpoint graphs are ubiquitous structures in the field of genome rearrangements. Their cycle decomposition has proved useful in computing and bounding many measures of (dis)similarity between genomes, and studying the distribution of those cycles is therefore critical to gaining insight on the distributions of the genomic distances that rely on it. We extend here the work initiated by Doignon and Labarre, who enumerated unsigned permutations whose breakpoint graph contains $k$ cycles, to signed permutations, and prove explicit formulas for computing the expected value and the variance of the corresponding distributions, both in the unsigned case and in the signed case. We also compare these distributions to those of several well-studied distances, emphasising the cases where approximations obtained in this way stand out. Finally, we show how our results can be used to derive simpler proofs of other previously known results

Lower bounding edit distances between permutations

International audienceA number of fields, including the study of genome rearrangements and the design of interconnection networks, deal with the connected problems of sorting permutations in "as few moves as possible", using a given set of allowed operations, or computing the number of moves the sorting process requires, often referred to as the distance of the permutation. These operations often act on just one or two segments of the permutation, e.g. by reversing one segment or exchanging two segments. The cycle graph of the permutation to sort is a fundamental tool in the theory of genome rearrangements, and has proved useful in settling the complexity of many variants of the above problems. In this paper, we present an algebraic reinterpretation of the cycle graph of a permutation π as an even permutation π, and show how to reformulate our sorting problems in terms of particular factorisations of the latter permutation. Using our framework, we recover known results in a simple and unified way, and obtain a new lower bound on the prefix transposition distance (where a prefix transposition displaces the initial segment of a permutation), which is shown to outperform previous results. Moreover, we use our approach to improve the best known lower bound on the prefix transposition diameter from 2n/3 to ⌊3n/4⌋, and investigate a few relations between some statistics on π and π

Edit Distances and Factorisations of Even Permutations

Abstract. A number of fields, including genome rearrangements and interconnection network design, are concerned with sorting permutations in “as few moves as possible”, using a given set of allowed operations. These often act on just one or two segments of the permutation, e.g. by reversing one segment or exchanging two segments. The cycle graph of the permutation to sort is a fundamental tool in the theory of genome rearrangements. In this paper, we present an algebraic reinterpretation of the cycle graph as an even permutation, and show how to reformulate our sorting problems in terms of particular factorisations of the latter permutation. Using our framework, we recover known results in a simple and unified way, and obtain a new lower bound on the prefix transposition distance (where a prefix transposition displaces the initial segment of a permutation), which is shown to outperform previous results. Moreover, we use our approach to improve the best known lower bound on the prefix transposition diameter from 2n/3 to ⌊ 3n+1

Évolution des génomes par mutations locales et globales : une approche d’alignement

Durant leur évolution, les génomes accumulent des mutations pouvant affecter d’un nucléotide à plusieurs gènes. Les modifications au niveau du nombre et de l’organisation des gènes dans les génomes sont dues à des mutations globales, telles que les duplications, les pertes et les réarrangements. En comparant les ordres de gènes des génomes, il est possible d’inférer les événements évolutifs les plus fréquents, le contenu en gènes des espèces ancestrales ainsi que les histoires évolutives ayant menées aux ordres observés. Dans cette thèse, nous nous intéressons au développement de nouvelles méthodes algorithmiques, par approche d’alignement, afin d’analyser ces différents aspects de l’évolution des génomes. Nous nous intéressons à la comparaison de deux ou d’un ensemble de génomes reliés par une phylogénie, en tenant compte des mutations globales. Pour commencer, nous étudions la complexité théorique de plusieurs variantes du problème de l’alignement de deux ordres de gènes par duplications et pertes, ainsi que de l’approximabilité de ces problèmes. Nous rappelons ensuite les algorithmes exacts, en temps exponentiel, existants, et développons des heuristiques efficaces. Nous proposons, dans un premier temps, DLAlign, une heuristique quadratique pour le problème d’alignement de deux ordres de gènes par duplications et pertes. Ensuite, nous présentons, OrthoAlign, une extension de DLAlign, qui considère, en plus des duplications et pertes, les réarrangements et les substitutions. Nous abordons également le problème de l’alignement phylogénétique de génomes. Pour commencer, l’heuristique OrthoAlign est adaptée afin de permettre l’inférence de génomes ancestraux au noeuds internes d’un arbre phylogénétique. Nous proposons enfin, MultiOrthoAlign, une heuristique plus robuste, basée sur la médiane, pour l’inférence de génomes ancestraux et d’histoires évolutives d’un ensemble de génomes représentés aux feuilles d’un arbre phylogénétique.During the evolution process, genomes accumulate mutations that may affect the genome at different levels, ranging from one base to the overall gene content. Global mutations affecting gene content and organization are mainly duplications, losses and rearrangements. By comparing gene orders, it is possible to infer the most frequent events, the gene content in the ancestral genomes and the evolutionary histories of the observed gene orders. In this thesis, we are interested in developing new algorithmic methods based on an alignment approach for comparing two or a set of genomes represented as gene orders and related through a phylogenetic tree, based on global mutations. We study the theoretical complexity and the approximability of different variants of the two gene orders alignment problem by duplications and losses. Then, we present the existing exact exponential time algorithms, and develop efficient heuristics for these problems. First, we developed DLAlign, a quadratic time heuristic for the two gene orders alignment problem by duplications and losses. Then, we developed OrthoAlign, a generalization of DLAlign, accounting for most genome-wide evolutionary events such as duplications, losses, rearrangements and substitutions. We also study the phylogenetic alignment problem. First, we adapt our heuristic OrthoAlign in order to infer ancestral genomes at the internal nodes of a given phylogenetic tree. Finally, we developed MultiOrthoAlign, a more robust heuristic, based on the median problem, for the inference of ancestral genomes and evolutionary histories of extent genomes labeling leaves of a phylogenetic tree

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

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