Short Proofs for Cut-and-Paste Sorting of Permutations

We consider the problem of determining the maximum number of moves required to sort a permutation of $[n]$ using cut-and-paste operations, in which a segment is cut out and then pasted into the remaining string, possibly reversed. We give short proofs that every permutation of $[n]$ can be transformed to the identity in at most \flr{2n/3} such moves and that some permutations require at least \flr{n/2} moves.Comment: 7 pages, 2 figure

Are There Rearrangement Hotspots in the Human Genome?

In a landmark paper, Nadeau and Taylor [18] formulated the random breakage model (RBM) of chromosome evolution that postulates that there are no rearrangement hotspots in the human genome. In the next two decades, numerous studies with progressively increasing levels of resolution made RBM the de facto theory of chromosome evolution. Despite the fact that RBM had prophetic prediction power, it was recently refuted by Pevzner and Tesler [4], who introduced the fragile breakage model (FBM), postulating that the human genome is a mosaic of solid regions (with low propensity for rearrangements) and fragile regions (rearrangement hotspots). However, the rebuttal of RBM caused a controversy and led to a split among researchers studying genome evolution. In particular, it remains unclear whether some complex rearrangements (e.g., transpositions) can create an appearance of rearrangement hotspots. We contribute to the ongoing debate by analyzing multi-break rearrangements that break a genome into multiple fragments and further glue them together in a new order. In particular, we demonstrate that (1) even if transpositions were a dominant force in mammalian evolution, the arguments in favor of FBM still stand, and (2) the ‘‘gene deletion’’ argument against FBM is flawed

Rearranjo de genomas : uma coletanea de artigos

Orientador : João MeidanisTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Hoje em dia, estão disponíveis, publicamente, uma imensa quantidade de informações genéticas. O desafio atual da Genômica é processar estes dados de forma a obter conclusões biológicas relevantes. Uma das maneiras de estruturar estas informações é através de comparação de genomas, que busca semelhanças e diferenças entre os genomas de dois ou mais organismos. Neste contexto, a área de Rearranjo de Genomas vem recebendo bastante atenção ultimamente. Uma forma de comparar genomas é através da distância de rearranjo, determinada pelo número mínimo de eventos de rearranjo que podem explicar as diferenças entre dois genomas. Os principais estudos em distância de rearranjo envolvem eventos de reversões e transposições. A presente coletânea é composta de oito artigos, contendo vários resultados importantes sobre Rearranjo de Genomas. Estes trabalhos foram apresentados em seis conferências, sendo uma nacional e cinco internacionais. Dois destes trabalhos serão publicados em importantes revistas internacionais e outro foi incluído como um capítulo de um livro. Nossas principais contribuições podem ser divididas em dois grupos: um novo formalismo algébrico e uma série de resultados envolvendo o evento de transposição. A nova teoria algébrica relaciona a teoria de Rearranjo de Genomas com a de grupos de permutações. Nossa intenção foi estabelecer um formalismo algébrico que simplificasse a obtenção de novos resultados, até hoje, muito baseados na construção de diagramas. Estudamos o evento de transposição de várias formas. Além de apresentarmos resultados sobre a distância de transposição entre uma permutação e sua inversa, também estudamos o problema de rearranjo envolvendo transposições e reversões simultaneamente, construindo algoritmos de aproximação e estabelecendo uma conjectura sobre o diâmetro. Usamos o formalismo algébrico para mostrar que é possível determinar a distância de fusão, fissão e transposição em tempo polinomial. Este é o primeiro resultado polinomial conhecido para um problema de rearranjo envolvendo o evento de transposição. Por último, introduzimos dois novos problemas de rearranjo: o problema de distância sintênica envolvendo fusões e fissões, e o problema de transposição de prefixos. Para ambos apresentamos resultados significativos, que avançam o conhecimento na áreaAbstract: Nowadays, a huge amount of genetic information is public1y available. Genomic's current challenge is to process this information in order to obtain relevant biological conc1usions. One possible way of structuring this information is through genome comparison, where we seek similarities and differences among the genomes of two or more organisms. In this context, the area of Genome Rearrangements has received considerable attention lately. One way of comparing genomes is given by the rearrangement distance, which is determined by the minimum number of rearrangement events that explain the differences between two genomes. The main studies in rearrangement distance involve reversal and transposition events. The present collection is composed of eight artic1es, containing several important results on Genome Rearrangements. These papers were presented in six conferences, one with Brazilian scope and five with international scope. Two of these works will be published in important international journals, and one other work appeared as a book chapter. Our main contributions can be divided into two groups: a new algebraic formalism and a series of results involving the transposition event. The new algebraic theory relates the genome rearrangement theory to the theory of permutation groups. Our intention was to establish an algebraic formalism that simplifies the creation of new results, up to now excessively based on the construction of diagrams. We studied the transposition event in several ways. Besides presenting results on the transpositions distance between a permutation and its inverse, we also studied the rearrangement problem involving transpositions and reversals simultaneously, constructing approximation algorithms and proposing a conjecture on the diameter. We used the algebraic formalism to show that it is possible to determine the distance of fusion, fission, and transposition in polynomial time. This is the first polynomial time result for a rearrangement problem involving the transposition event. Finally, we introduced two now rearrangement problems: the syntenic distance problem involving fission and fusion, and the prefix transposition problem. For each one of these problems we present significant results, widening the knowledge in this areaDoutoradoDoutor em Ciência da Computaçã

Sorting signed permutations by short operations

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Background: During evolution, global mutations may alter the order and the orientation of the genes in a genome. Such mutations are referred to as rearrangement events, or simply operations. In unichromosomal genomes, the most common operations are reversals, which are responsible for reversing the order and orientation of a sequence of genes, and transpositions, which are responsible for switching the location of two contiguous portions of a genome. The problem of computing the minimum sequence of operations that transforms one genome into another - which is equivalent to the problem of sorting a permutation into the identity permutation - is a well-studied problem that finds application in comparative genomics. There are a number of works concerning this problem in the literature, but they generally do not take into account the length of the operations (i.e. the number of genes affected by the operations). Since it has been observed that short operations are prevalent in the evolution of some species, algorithms that efficiently solve this problem in the special case of short operations are of interest. Results: In this paper, we investigate the problem of sorting a signed permutation by short operations. More precisely, we study four flavors of this problem: (i) the problem of sorting a signed permutation by reversals of length at most 2; (ii) the problem of sorting a signed permutation by reversals of length at most 3; (iii) the problem of sorting a signed permutation by reversals and transpositions of length at most 2; and (iv) the problem of sorting a signed permutation by reversals and transpositions of length at most 3. We present polynomial-time solutions for problems (i) and (iii), a 5-approximation for problem (ii), and a 3-approximation for problem (iv). Moreover, we show that the expected approximation ratio of the 5-approximation algorithm is not greater than 3 for random signed permutations with more than 12 elements. Finally, we present experimental results that show that the approximation ratios of the approximation algorithms cannot be smaller than 3. In particular, this means that the approximation ratio of the 3-approximation algorithm is tight.During evolution, global mutations may alter the order and the orientation of the genes in a genome. Such mutations are referred to as rearrangement events, or simply operations. In unichromosomal genomes, the most common operations are reversals, which a10117CAPES - COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIORFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOCNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOCoordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)FAPESP [2014/04718-6]CNPq [303947/2008-0, 477692/2012-5]CNPq [477692/2012-5, 306730/2012-0, 483370/2013-4]FAPESP [2013/08293-7]SEM INFORMAÇÃO2014/04718-6; 2013/08293-7303947/2008-0; 477692/2012-5; 306730/2012-0; 477692/2012-5; 483370/2013-

Transposition distances between genomes

Orientador: João MeidanisDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Uma das principais formas de se medir a distância evolutiva entre espécies é avaliando-se quão transformado um genoma foi em relação a outro. Tais transformações são conhecidas como rearranjos de genoma. Neste trabalho estaremos analisando o rearranjo chamado de transposição, evento que troca de posição dois blocos consecutivos de genes de um mesmo cromossomo. Mais especificamente, buscamos encontrar o número mínimo de transposições que transforma um cromos somo em outro, valor conhecido como distância de transposição. Matematicamente, consideramos os cromossomos como permutações e o problema de se transformar uma permutação em outra pode ser visto como uma ordenação. Em nosso estudo, introduzimos uma operação de remoção de elementos, ferramenta ainda pouco explorada no estudo da distância de transposição, mas que nos possibilitou obter um limite superior para a distância de transposição. Também sugerimos novas formas de se utilizar a remoção de elementos e a análise de subseqüências de permutações. Como forma de se tentar obter novos conhecimentos sobre o problema da distância de transposição, consideramos a variação do problema em que somente transposições de prefixo são permitidas. Zanoni Dias propôs um algoritmo polinomial que transforma qualquer permutação de comprimento n em sua reversa em f3n/41 passos, porém sem uma prova completa. Modificamos esse algoritmo, mantendo o número de passos, e apresentamos uma prova completa da correção do algoritmo modificado. Ainda no problema de distância de transposição de prefixo, analisamos as permutações cujas distâncias se igualam ao limite inferior de distância de pontos de quebra. Tais permutações são fáceis de serem ordenadas por transposições de prefixo em tempo polinomial, pois possuem ordenação ótima única e bem definida. Ao final chegamos à conclusão que as permutações que são fáceis de se ordenar no problema de transposições de prefixo também são fáceis no problema de transposições, o que prova que a variação do problema auxilia no estudo do problema originalAbstract: One of the main ways of measuring the evolution distance among species is to evaluate how large chunks have moved when comparing two genomes. Such changes are know as genome rearrangements. In this work we analyze a rearrangement event called transposition that changes the position of two consecutive blocks of genes in a chromosome. More specifically, we look for the minimum number of transpositions needed to transform a chromosome into another. This value is called the transposition distance. Mathematically, chromosomes are regarded as permutations and changing one genome into another can be seen as a sorting problem. In our study, we introduce an operation of element removal from permutations, which has not been fully explored, but allowed us to find an upper bound for the transposition distance. We also suggest new ways of making use of element removal and the analysis of permutation subsequences. In the hope of obtaining new knowledge about the problem of transposition distance, we considered the variation of the problem where we allow prefix transpositions only. Zanoni Dias developed a polynomial algorithm that changes any permutation into its reverse using f3n/41 steps, but without a proof of its correctness. We have modified this algorithm, keeping the number of steps, and presented a complete proof of the correction of the modified algorithm. Still about the prefix transposition distance, we have analyzed those permutations whose distance equals the breakpoint lower bound. Such permutations are easily sorted by prefix transpositions in polynomial time, since they have a unique and well-defined optimum sorting. Finally, we concluded that the permutations that are easily sorted in the prefix transposition problem are also easily sorted in the transposition problem, which proves that the variation of the problem helps the study of the original problemMestradoMestre em Ciência da Computaçã

Applications of heuristic search on phylogeny reconstruction problems

Phylogenies or evolutionary trees for a given family of species show the evolutionary relationships between these species. The leaves denote the given species, the internal nodes denote their common ancestors and the edges denote the genetic relationships. Species can be identified by their whole genomes and the evolutionary relations between species can be measured by the number of rearrangement events (i.e. mutations) that transform one genome into another. One approach to infer phylogeny from genomic data is by solving median genome problems for three genomes, or the genome rearrangement problem for pairs of genomes, while trying to minimize the total evolutionary distance among the given species. In this thesis, we have developed and implemented two search based algorithms for phylogeny reconstruction problem based on solving median genome problems for circular genomes of the same length without gene duplication. In order to show applicability and effectiveness of our algorithms, we have tested them with randomly generated instances and two real data sets: mitochondrial genomes of Metazoa and chloroplast genomes of Campanulaceae