Representação combinatória e algébrica das permutações na análise do problema de rearranjo de genomas por reversões

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2010.Na genômica comparativa soluções algoríıtmicas eficientes para o problema da dist ância de rearranjo de genomas são uma ferramenta importante para o desenvolvimento de software que permite estabelecer relacionamento evolutivo entre organismos, por exemplo para a constr~ção de árvores filogenéticas de organismos. Existem diversas operações sobre palavras que modelam mutações ocorridas nos genes dos seres vivos (e.g. reversões, transposições, troca de blocos, etc.). Restritos à operação de reversão o problema de rearranjo de genomas ´é NP-difícil. Sendo assim, é plausível considerar algoritmos de aproximação. O algoritmo conhecido que melhor aproxima a solução do problema de rearranjo via reversões tem raio de 1.375. No seu artigo seminal, Bafna e Pevzner apresentam soluções O(n2) de raios de aproximação 1.5 para permutações com sinal e 7 4 para permutações sem sinal. Neste trabalho propõ-se um uso cuidadoso e discriminado entre as representações combinatória (palavras e grafos de pontos de quebra) e algébrica (ciclos de permuta ções) das permutações, que contribuirão para analisar com precisão e de maneira adequada diversas características do problema da distância de reversão e das soluções apresentadas por Bafna e Pevzner. _________________________________________________________________________________ ABSTRACTEfficient algorithmic solutions to the problem of genome rearrangements in comparative genomics are an important tool for the development of software allowing one to estabilish the evolution link between organisms, for instance the construction of phylogenetic trees. There are several string operations modelling mutations occurring inside genes (e.g. reversals, transpositions and block interchange, etc.). When restricted to the reversal operation, the problem of genome rearrangement is NP-hard. Thus, polynomially bounded approximated algorithms are considered to be admissible solutions. The best known approximated algorithm to solve the rearrangement problem through reversals has approximation ratio of 1.375. In their seminal paper, Bafna and Pevzner presented O(n2) solutions of ratio 1.5 for signed permutations and 7 4 for unsigned permutations. This work proposes a careful discrimination between the combinatory (strings and breakpoint graphs) and the algebraic (cycles of permutations) representations of permutations to analyse precisely, and in an adequate way, many of the problem characteristics and solutions presented by Bafna and Pevzner

Complexity of Cayley Distance and other General Metrics on Permutation Groups 1

Abstract—Permutation groups arise as important structures in group theory because many algebraic properties about them are well-known, which makes modeling natural phenomena by permutations of practical interest. Usability of the involved algebraic notions is illustrated by problems such as genome rearrangement by reversals for which it is well-known that for the case of unsigned and signed sorting by reversals the time complexity is, respectively, NP-hard and P. Reversal distance is a particular metric and in this work more general metrics on permutation groups are considered emphasizing on the Cayley distance. In particular, we point out an error in one of the polynomial reductions applied in Pinch’s approach attempting to proof that the subgroup distance problem for Cayley distance is NP-complete and following his approach we present a simplified and correct proof of this fact. Although, recently a shorter and more general proof than Pinch’s one was given by Buchheim, Cameron and Wu, we believe the correction of Pinch’s proof presented in this paper is of great interest because it correctly relates the Cayley distance problem with a maximal routing problem giving an additional perspective in relation to Buchheim et al. recent proof from which only the usual logical satisfiability perspective of distance problems is observable. I

Complexity of Cayley Distance and other General Metrics on Permutation Groups 1

Abstract. Permutation groups arise as important structures in group theory because many algebraic properties about them are well-known, which makes modelling natural phenomena by permutations of practical interest. This paper reviews the complexity of some problems involving permutation groups. Usability of the involved algebraic notions is illustrated by problems such as genome rearragement by reversals for which it is well-known that for the case of unsigned and signed sorting by reversals the time complexity is, respectively, NP-hard and P. Reversal distance is a particular metric and in this work more general metrics on permutation groups are considered emphasizing on the Cayley distance. In particular, we point out an error in one of the polynomial reductions applied in Pinch’s approach atempting to proof that the subgroup distance problem for Cayley distance is NP-complete and following his approach we present a simplified and correct proof of this fact.

Reconfigurable Heterogeneous Parallel Island Models

Heterogeneous Parallel Island Models (HePIMs) run different bio-inspired algorithms (BAs) in their islands. From a variety of communication topologies and migration policies fine-tuned for homogeneous PIMs (HoPIMs), which run the same BA in all their islands, previous work introduced HePIMs that provided competitive quality solutions regarding the best-adapted BA in HoPIMs. This work goes a step forward, maintaining the population diversity provided by HePIMs, and increasing their flexibility, allowing BA reconfiguration on islands during execution: according to their performance, islands may substitute their BAs dynamically during the evolutionary process. Experiments with the introduced architectures (RecHePIMs) were applied to the NP-hard problem of sorting permutations by reversals, using four different BAs, namely, simple Genetic Algorithm, Double-point crossover Genetic Algorithm, Differential Evolution, and self-adjusting Particle Swarm Optimization. The results showed that the new reconfigurable heterogeneous models compute better quality solutions than the HePIMs closing the gap with the HoPIM running the best-adapted BA