Transpositions Problem Research Articles

An improved approximation algorithm for the reversal and transposition distance considering gene order and intergenic sizes

BackgroundIn the comparative genomics field, one of the goals is to estimate a sequence of genetic changes capable of transforming a genome into another. Genome rearrangement events are mutations that can alter the genetic content or the arrangement of elements from the genome. Reversal and transposition are two of the most studied genome rearrangement events. A reversal inverts a segment of a genome while a transposition swaps two consecutive segments. Initial studies in the area considered only the order of the genes. Recent works have incorporated other genetic information in the model. In particular, the information regarding the size of intergenic regions, which are structures between each pair of genes and in the extremities of a linear genome.Results and conclusionsIn this work, we investigate the sorting by intergenic reversals and transpositions problem on genomes sharing the same set of genes, considering the cases where the orientation of genes is known and unknown. Besides, we explored a variant of the problem, which generalizes the transposition event. As a result, we present an approximation algorithm that guarantees an approximation factor of 4 for both cases considering the reversal and transposition (classic definition) events, an improvement from the 4.5-approximation previously known for the scenario where the orientation of the genes is unknown. We also present a 3-approximation algorithm by incorporating the generalized transposition event, and we propose a greedy strategy to improve the performance of the algorithms. We performed practical tests adopting simulated data which indicated that the algorithms, in both cases, tend to perform better when compared with the best-known algorithms for the problem. Lastly, we conducted experiments using real genomes to demonstrate the applicability of the algorithms.

Heuristics for the Reversal and Transposition Distance Problem.

We present three heuristics-Sliding Window, Look Ahead, and Iterative Sliding Window-to improve solutions for the Sorting Signed Permutations by Reversals and Transpositions Problem. We investigate the classical version of the problem as well as versions restricted to prefix and prefix or suffix operations. To assess the heuristics based on its improvement, we implemented algorithms described in the literature to provide initial solutions. Although we have a limited number of problems, these heuristics can be applied to many others within the area of genome rearrangement. When time is a crucial factor, Sliding Window is a better choice because it runs in linear time. If the quality of the solution is a priority, Look Ahead should be preferred. Iterative Sliding Window is the most flexible heuristic and allows us to find a trade-off for specific scenarios where running time and solution quality are relevant.

On the Complexity of Sorting by Reversals and Transpositions Problems.

In comparative genomics, rearrangements are mutations that affect a stretch of DNA sequences. Reversals and transpositions are well-known rearrangements, and each has a vast literature. The reversal and transposition distance, that is, the minimum number of reversals and transpositions needed to transform one genome into another is a relevant evolutionary distance. The problem of computing this distance when genomes are represented by permutations was proposed >20 years ago and received the name of sorting by reversals and transpositions problem. It has been the focus of a number of studies, but the computational complexity has remained open until now. We hereby solve this question and prove that it is NP-hard no matter whether genomes are represented by signed or unsigned permutations. In addition, we prove that a usual generalization of this problem, which assigns weights wρ for reversals and wτ for transpositions, is also NP-hard as long as wτ/wρ ≤ 1.5 for both signed and unsigned permutations.

Sorting by Weighted Reversals and Transpositions.

Genome rearrangements are global mutations that change large stretches of DNA sequence throughout genomes. They are rare but accumulate during the evolutionary process leading to organisms with similar genetic material in different places and orientations within the genome. Sorting by Genome Rearrangements problems seek for minimum-length sequences of rearrangements that transform one genome into the other. These problems accept alternative versions that assign weights for each event, and the goal is to find a minimum-weight sequence. We study the Sorting by Weighted Reversals and Transpositions problem on signed permutations. In this study, we use weight 2 for reversals and 3 for transpositions and consider theoretical and practical aspects in our analysis. We present two algorithms with approximation factors of 5/3 and 3/2. We also developed a generic approximation algorithm to deal with different weights for reversals and transpositions, and we show the approximation factor reached in each scenario.

On the sorting by reversals and transpositions problem

On the sorting by reversals and transpositions problem

Sorting by Prefix Reversals and Prefix Transpositions

In this paper, we present a new algorithm for the Sorting by Prefix Reversals and Prefix Transpositions Problem. The previous approximation algorithm was bounded by factor 3, and here we present an asymptotic 2-approximation algorithm. We consider theoretical and practical aspects in our analysis, and we show that our method is better than other approaches in both cases.

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.

Sorting by Transpositions Is Difficult

In comparative genomics, a transposition is an operation that exchanges two consecutive sequences of genes in a genome. The transposition distance between two genomes, that is, the minimum number of transpositions needed to transform a genome into another, is, according to numerous studies, a relevant evolutionary distance. The problem of computing this distance when genomes are represented by permutations is called the Sorting by Transpositions problem, and has been introduced by Bafna and Pevzner in [Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, 1995, pp. 614--623]. It has naturally been the focus of a number of studies (see, for instance, [G. Fertin, A. Labarre, I. Rusu, É. Tannier, and S. Vialette, Combinatorics of Genome Rearrangements, The MIT Press, Cambridge, MA, 2009]), but the computational complexity of this problem has remained undetermined for 15 years. In this paper, we answer this long-standing open question by proving that the Sorting by Transpositions problem is \sf NP-hard. As a corollary of our result, we also prove that the following problem, first described in [D. A. Christie, Genome Rearrangement Problems, Ph.D. thesis, University of Glasgow, Glasgow, Scotland, 1998], is \sf NP-hard: given a permutation $\pi$, is it possible to sort $\pi$ using exactly $d_b(\pi)/3$ transpositions, where $d_b(\pi)$ is the number of breakpoints of $\pi$?