log-Lists and their applications to sorting by transpositions, reversals and block-interchanges

Rusu, Irena

doi:10.1016/j.tcs.2016.11.012

“…In order to avoid recording all these changes one by one, the solutions proposed in literature in order to perform a (not necessarily balanced) reversal use three types of approaches. They are due to Kaplan and Verbin [8] (needs O( √ n log n) time to perform a reversal whose endpoints are known), Han [7] (needs O( √ n) time for the same task) and Rusu [10] (needs O(log n) time for the same task). The latter one, that we choose for efficiency reasons, uses…”

Section: Algorithm 2 Applylemma3mentioning

confidence: 99%

Sorting Permutations with Fixed Pinnacle Set

Rusu¹

2020

Preprint

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We give a positive answer to a question raised by Davis et al. (Discrete Mathematics 341, 2018), concerning permutations with the same pinnacle set. Given π ∈ S n , a pinnacle of π is an element π i (i = 1, n) such that π i−1 < π i > π i+1 . The question is: given π, π ∈ S n with the same pinnacle set S, is there a sequence of operations that transforms π into π such that all the intermediate permutations have pinnacle set S? We introduce balanced reversals, defined as reversals that do not modify the pinnacle set of the permutation to which they are applied. Then we show that π may be sorted by balanced reversals (i.e. transformed into a standard permutation Id S ), implying that π may be transformed into π using at most 4n − 2 min{p, 3} balanced reversals, where p = |S| ≥ 1. In case p = 0, at most 2n − 1 balanced reversals are needed.

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“…In order to avoid recording all these changes one by one, the solutions proposed in literature in order to perform a (not necessarily balanced) reversal use three types of approaches. They are due to Kaplan and Verbin [8] (needs O( √ n log n) time to perform a reversal whose endpoints are known), Han [7] (needs O( √ n) time for the same task) and Rusu [10] (needs O(log n) time for the same task). The latter one, that we choose for efficiency reasons, uses so-called log-lists.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Sorting Permutations with Fixed Pinnacle Set

Rusu

¹

2020

Electron. J. Combin.

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We give a positive answer to a question raised by Davis et al. (Discrete Mathematics 341, 2018), concerning permutations with the same $\pi_{i-1}<\pi_i>\pi_{i+1}$. The question is: given $\pi,\pi'\in S_n$ with the same pinnacle set $S$, is there a sequence of operations that transforms $\pi$ into $\pi'$ such that all the intermediate permutations have pinnacle set $S$? We introduce {\em balanced reversals}, defined as reversals that do not modify the pinnacle set of the permutation to which they are applied. Then we show that $\pi$ may be sorted by balanced reversals (i.e. transformed into a standard permutation $Id_S$), implying that $\pi$ may be transformed into $\pi'$ using at most $4n-2\min\{p,3\}$ balanced reversals, where $p=|S|\geq 1$. In case $p=0$, at most $2n-1$ balanced reversals are needed.

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“…Galvão and Dias [ 13 ] studied solutions for SBT using three different structures: permutation codes, a concept previously introduced by Benoît-Gagné and Hamel [ 14 ]; breakpoint diagram, 1 introduced by Walter et al [ 15 ]; and longest increasing subsequence, introduced by Guyer et al [ 16 ]. Rusu [ 17 ], on the other hand, used a structure called log-list, formerly devised with the name link-cut trees by Sleator and Tarjan [ 18 ], to derive another 1.375-approximation algorithm for SBT. In addition to these, recently, other studies have been proposed involving variations of the transposition event.…”

Section: Introductionmentioning

confidence: 99%

A new 1.375-approximation algorithm for sorting by transpositions

Silva

¹

,

Kowada

²

,

Rocco

³

et al. 2022

Algorithms Mol Biol

9

0

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Background sorting by transpositions (SBT) is a classical problem in genome rearrangements. In 2012, SBT was proven to be $$\mathcal {NP}$$ NP -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 $$\pi$$ π into a simple permutation$${\hat{\pi }}$$ π ^ , presumably easier to handle with. The permutation $${\hat{\pi }}$$ π ^ is obtained by inserting new symbols into $$\pi$$ π in a way that the lower bound of the transposition distance of $$\pi$$ π is kept on $${\hat{\pi }}$$ π ^ . The simplification is guaranteed to keep the lower bound, not the transposition distance. A sequence of operations sorting $${\hat{\pi }}$$ π ^ can be mimicked to sort $$\pi$$ π . Results and conclusions First, using an algebraic approach, we propose a new upper bound for the transposition distance, which holds for all $$S_n$$ S n . 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 $$S_n$$ S n . 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, $$2\le n \le 12$$ 2 ≤ n ≤ 12 . 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.

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log-Lists and their applications to sorting by transpositions, reversals and block-interchanges

Cited by 7 publications

References 11 publications

Sorting Permutations with Fixed Pinnacle Set

Sorting Permutations with Fixed Pinnacle Set

Sorting Permutations with Fixed Pinnacle Set

A new 1.375-approximation algorithm for sorting by transpositions

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