RESUMO
Genome rearrangements are events where large blocks of DNA exchange pieces during evolution. The analysis of such events is a tool for understanding evolutionary genomics, in whose context many rearrangement distances have been proposed, based on finding the minimum number of rearrangements to transform one genome into another, using some predefined operation. However, when more than two genomes are considered, we have new challenging problems. Studying such problems from a combinatorial point of view has been shown to be a useful tool to approach such problems, for example, the reconstruction of phylogenetic trees. We focus on genome rearrangement problems related to graph convexity. Such an approach is in connection with some other well-known studies on multigenomic models, for example, those based on the median and on the closest string. We propose an association between graph convexities and genome rearrangements in such a way that graph convexity problems deal with input sets of vertices and try to answer questions concerning the closure of such inputs. The concept of closure is useful for studies on genome rearrangement by suggesting mechanisms to reduce the genomic search space. Regarding the computational complexity, and considering the Hamming distance on strings, we solve the following problems: decide if a given set is convex; compute the interval and the convex hull of a given set; and determine the convexity number, interval number, and hull number of a Hamming graph. All such problems are solved for three types of convexities: geodetic, monophonic, and P3. Considering the Cayley distance on permutations, we solve the convexity number and interval determination problems for the geodetic convexity.
Assuntos
Biologia Computacional/métodos , Evolução Molecular , Rearranjo Gênico/genética , Filogenia , Genômica/métodosRESUMO
Sorting by Transpositions is an NP-hard problem for which several polynomial-time approximation algorithms have been developed. Hartman and Shamir (2006) developed a 1.5-approximation [Formula: see text] algorithm, whose running time was improved to O(nlogn) by Feng and Zhu (2007) with a data structure they defined, the permutation tree. Elias and Hartman (2006) developed a 1.375-approximation O(n(2)) algorithm, and Firoz et al. (2011) claimed an improvement to the running time, from O(n(2)) to O(nlogn), by using the permutation tree. We provide counter-examples to the correctness of Firoz et al.'s strategy, showing that it is not possible to reach a component by sufficient extensions using the method proposed by them. In addition, we propose a 1.375-approximation algorithm, modifying Elias and Hartman's approach with the use of permutation trees and achieving O(nlogn) time.