A new 1.375-approximation algorithm for sorting by transpositions.

BACKGROUND: SORTING BY TRANSPOSITIONS (SBT) is a classical problem in genome rearrangements. In 2012, SBT was proven to be [Formula: see text]-hard and the best approximation algorithm with a 1.375 ratio was proposed in 2006 by Elias and Hartman (EH algorithm). Their algorithm employs simplification, a technique used to transform an input permutation [Formula: see text] into a simple permutation [Formula: see text], presumably easier to handle with. The permutation [Formula: see text] is obtained by inserting new symbols into [Formula: see text] in a way that the lower bound of the transposition distance of [Formula: see text] is kept on [Formula: see text]. The simplification is guaranteed to keep the lower bound, not the transposition distance. A sequence of operations sorting [Formula: see text] can be mimicked to sort [Formula: see text]. RESULTS AND CONCLUSIONS: First, using an algebraic approach, we propose a new upper bound for the transposition distance, which holds for all [Formula: see text]. Next, motivated by a problem identified in the EH algorithm, which causes it, in scenarios involving how the input permutation is simplified, to require one extra transposition above the 1.375-approximation ratio, we propose a new approximation algorithm to solve SBT ensuring the 1.375-approximation ratio for all [Formula: see text]. We implemented our algorithm and EH's. Regarding the implementation of the EH algorithm, two other issues were identified and needed to be fixed. We tested both algorithms against all permutations of size n, [Formula: see text]. The results show that the EH algorithm exceeds the approximation ratio of 1.375 for permutations with a size greater than 7. The percentage of computed distances that are equal to transposition distance, computed by the implemented algorithms are also compared with others available in the literature. Finally, we investigate the performance of both implementations on longer permutations of maximum length 500. From the experiments, we conclude that maximum and the average distances computed by our algorithm are a little better than the ones computed by the EH algorithm and the running times of both algorithms are similar, despite the time complexity of our algorithm being higher.

New Genome Similarity Measures based on Conserved Gene Adjacencies.

Many important questions in molecular biology, evolution, and biomedicine can be addressed by comparative genomic approaches. One of the basic tasks when comparing genomes is the definition of measures of similarity (or dissimilarity) between two genomes, for example, to elucidate the phylogenetic relationships between species. The power of different genome comparison methods varies with the underlying formal model of a genome. The simplest models impose the strong restriction that each genome under study must contain the same genes, each in exactly one copy. More realistic models allow several copies of a gene in a genome. One speaks of gene families, and comparative genomic methods that allow this kind of input are called gene family-based. The most powerful-but also most complex-models avoid this preprocessing of the input data and instead integrate the family assignment within the comparative analysis. Such methods are called gene family-free. In this article, we study an intermediate approach between family-based and family-free genomic similarity measures. Introducing this simpler model, called gene connections, we focus on the combinatorial aspects of gene family-free genome comparison. While in most cases, the computational costs to the general family-free case are the same, we also find an instance where the gene connections model has lower complexity. Within the gene connections model, we define three variants of genomic similarity measures that have different expression powers. We give polynomial-time algorithms for two of them, while we show NP-hardness for the third, most powerful one. We also generalize the measures and algorithms to make them more robust against recent local disruptions in gene order. Our theoretical findings are supported by experimental results, proving the applicability and performance of our newly defined similarity measures.

A faster 1.375-approximation algorithm for sorting by transpositions.

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.