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The transposition distance problem is a classical problem in genome rearrangements, which seeks to determine the minimum number of transpositions needed to transform a linear chromosome into another represented by the permutations π and σ, respectively. This article focuses on the equivalent problem of sorting by transpositions (SBT), where σ is the identity permutation ι. Specifically, we investigate palisades, a family of permutations that are "hard" to sort, as they require numerous transpositions above the celebrated lower bound devised by Bafna and Pevzner. By determining the transposition distance of palisades, we were able to provide the exact transposition diameter for 3-permutations (TD3), a special subset of the symmetric group Sn, essential for the study of approximate solutions for SBT using the simplification technique. The exact value for TD3 has remained unknown since Elias and Hartman showed an upper bound for it. Another consequence of determining the transposition distance of palisades is that, using as lower bound the one by Bafna and Pevzner, it is impossible to guarantee approximation ratios lower than 1.375 when approximating SBT. This finding has significant implications for the study of SBT, as this problem has been the subject of intense research efforts for the past 25 years.

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RESUMO

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.

RESUMO

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.

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RESUMO

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.

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