Gustavo Rodrigues Galvão

Sorting signed permutations by short operations

BackgroundDuring 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.ResultsIn 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.

A general heuristic for genome rearrangement problems.

In this paper, we present a general heuristic for several problems in the genome rearrangement field. Our heuristic does not solve any problem directly, it is rather used to improve the solutions provided by any non-optimal algorithm that solve them. Therefore, we have implemented several algorithms described in the literature and several algorithms developed by ourselves. As a whole, we implemented 23 algorithms for 9 well known problems in the genome rearrangement field. A total of 13 algorithms were implemented for problems that use the notions of prefix and suffix operations. In addition, we worked on 5 algorithms for the classic problem of sorting by transposition and we conclude the experiments by presenting results for 3 approximation algorithms for the sorting by reversals and transpositions problem and 2 approximation algorithms for the sorting by reversals problem. Another algorithm with better approximation ratio can be found for the last genome rearrangement problem, but it is purely theoretical with no practical implementation. The algorithms we implemented in addition to our heuristic lead to the best practical results in each case. In particular, we were able to improve results on the sorting by transpositions problem, which is a very special case because many efforts have been made to generate algorithms with good results in practice and some of these algorithms provide results that equal the optimum solutions in many cases. Our source codes and benchmarks are freely available upon request from the authors so that it will be easier to compare new approaches against our results.

Computing rearrangement distance of every permutation in the symmetric group

We consider the problem of computing rearrangement distance of every permutation in the symmetric group Sn and present a simple algorithm for doing it. By analysing the rearrangement distance distribution computed for different scenarios, we were able to correct the reversal distance distribution given by Kececioglu and Sankoff; disprove a conjecture of Walter, Dias and Meidanis on signed reversal and transposition diameter; and reinforce the conjecture of Dias and Meidanis on prefix transposition diameter. As an attempt to better characterize how rearrangement distances are distributed, two new measures are introduced: the traversal diameter and the longevity. We conjecture results on them and on the diameter of some rearrangement distances.

Sorting Circular Permutations by Super Short Reversals

We consider the problem of sorting a circular permutation by super short reversals (i.e., reversals of length at most 2), a problem that finds application in comparative genomics. Polynomial-time solutions to the unsigned version of this problem are known, but the signed version remained open. In this paper, we present the first polynomial-time solution to the signed version of this problem. Moreover, we perform experiments for inferring phylogenies of two different groups of bacterial species and compare our results with the phylogenies presented in previous works. Finally, to facilitate phylogenetic studies based on the methods studied in this paper, we present a web tool for rearrangement-based phylogenetic inference using short operations, such as super short reversals.

An Audit Tool for Genome Rearrangement Algorithms

We consider the combinatorial problem of sorting a permutation using a minimum number of rearrangement events, which finds application in the estimation of evolutionary distance between species. Many variants of this problem, which we generically refer to as the rearrangement sorting problem, have been tackled in the literature, and for most of them, the best known algorithms are approximations or heuristics. In this article, we present a tool, called GRAAu, to aid in the evaluation of the results produced by these algorithms. To illustrate its application, we use GRAAu to evaluate the results of four approximation algorithms regarding two variants of the rearrangement sorting problem: the problem of sorting by prefix reversals and the problem of sorting by prefix transpositions. As a result, we show that the approximation ratios of three algorithms are tight and conjecture that the approximation ratio of the remaining one is also tight.

Sorting Signed Circular Permutations by Super Short Reversals

We consider the problem of sorting a circular permutation by reversals of length at most 2, a problem that finds application in comparative genomics. Polynomial-time solutions for the unsigned version of this problem are known, but the signed version remained open. In this paper, we present the first polynomial-time solution for the signed version of this problem. Moreover, we perform an experiment for inferring distances and phylogenies for published Yersinia genomes and compare the results with the phylogenies presented in previous works.

Approximation algorithms for sorting by signed short reversals

During evolution, global mutations may modify the gene order in a genome. Such mutations are commonly referred to as rearrangement events. One of the most frequent rearrangement events observed in genomes are reversals, which are responsible for reversing the order and orientation of a sequence of genes. The problem of sorting by reversals, that is, the problem of computing the most parsimonious reversal scenario to transform one genome into another, is a well-studied problem that finds application in comparative genomics. There is a number of works concerning this problem in the literature, but these works generally do not take into account the length of the reversals. Since it has been observed that short reversals are prevalent in the evolution of some species, recent efforts have been made to address this issue algorithmically. In this paper, we add to these efforts by introducing the problem of sorting by signed short reversals and by presenting three approximation algorithms for solving it. Although the worst-case approximation ratios of these algorithms are high, we show that their expected approximation ratios for sorting a random equiprobable signed permutation are much lower. Moreover, we present experimental results on small signed permutations which indicate that the worst-case approximation ratios of these algorithms may be better than those we have been able to prove.

On the Approximation Ratio of Algorithms for Sorting by Transpositions without Using Cycle Graphs

We study the problem of sorting by transpositions, which is related to comparative genomics. Our goal is to determine how good approximation algorithms which do not rely on the cycle graph are when it comes to approximation ratios by implementing three such algorithms. We compare their theoretical approximation ratio to the experimental results obtained by running them for all permutations of up to 13 elements. Our results suggest that the approaches adopted by these algorithms are not promising alternatives in the design of approximation algorithms with low approximation ratios. Furthermore, we prove an approximation bound of 3 for a constrained version of one algorithm, and close a missing gap on the proof for the approximation ratio of another algorithm.