A 1.375-Approximation Algorithm for Sorting by Transpositions

Elias, Isaac; Hartman, Tzvika

doi:10.1109/tcbb.2006.44

Cited by 154 publications

(102 citation statements)

References 21 publications

(14 reference statements)

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“…In order to explain the existence of essentially the same set of genes but differences in their order in different species, several rearrangement operations have been suggested, including reversals [6][7][8], block interchange [9], transpositions [10][11][12], transreversals [13], fission and fusion [14], prefix transposition [15], etc.…”

Section: Introductionmentioning

confidence: 99%

Pancake Flipping with Two Spatulas

Sharmin

Yeasmin

Hasan

et al. 2010

Electronic Notes in Discrete Mathematics

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Abstract. In this paper we study several variations of the pancake flipping problem, which is also well known as the problem of sorting by prefix reversals. We consider the variations in the sorting process by adding with prefix reversals other similar operations such as prefix transpositions and prefix transreversals. These type of sorting problems have applications in interconnection networks and computational biology. We first study the problem of sorting unsigned permutations by prefix reversals and prefix transpositions and present a 3-approximation algorithm for this problem. Then we give a 2-approximation algorithm for sorting by prefix reversals and prefix transreversals. We also provide a 3-approximation algorithm for sorting by prefix reversals and prefix transpositions where the operations are always applied at the unsorted suffix of the permutation. We further analyze the problem in more practical way and show quantitatively how approximation ratios of our algorithms improve with the increase of number of prefix reversals applied by optimal algorithms. Finally, we present experimental results to support our analysis.

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Section: Introductionmentioning

confidence: 99%

Pancake Flipping with Two Spatulas

Sharmin

Yeasmin

Hasan

et al. 2010

Electronic Notes in Discrete Mathematics

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“…the operations chosen. Common operations studied are transpositions [3,6,13,16,30], block-interchanges [5,13,14,23,31] and reversals [2,4,7,9,25,26], etc.…”

Section: Transposition Distance Of Permutationsmentioning

confidence: 99%

Plane Permutations and Applications to a Result of Zagier--Stanley and Distances of Permutations

Chen¹,

Reidys²

2016

SIAM J. Discrete Math.

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In this paper, we introduce plane permutations, i.e. pairs p = (s, π) where s is an n-cycle and π is an arbitrary permutation, represented as a two-row array. Accordingly a plane permutation gives rise to three distinct permutations: the permutation induced by the upper horizontal (s), the vertical (π) and the diagonal (D p ) of the array. The latter can also be viewed as the three permutations of a hypermap. In particular, a map corresponds to a plane permutation, in which the diagonal is a fixed point-free involution. We study the transposition action on plane permutations obtained by permuting their diagonal-blocks. We establish basic properties of plane permutations and study transpositions and exceedances and derive various enumerative results. In particular, we prove a recurrence for the number of plane permutations having a fixed diagonal and k cycles in the vertical, generalizing Chapuy's recursion for maps filtered by the genus. As applications of this framework, we present a combinatorial proof of a result of Zagier and Stanley, on the number of n-cycles ω, for which the product ω(1 2 · · · n) has exactly k cycles. Furthermore, we integrate studies on the transposition and block-interchange distance of permutations as well as the reversal distance of signed permutations. Plane permutations allow us to generalize and recover various lower bounds for transposition and block-interchange distances and to connect reversals with block-interchanges.

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“…Since its introduction by Bafna and Pevzner [3,4], the complexity class of this problem has never been established. Hence a number of studies [4,9,17,19,13,5,15] aim at designing approximation algorithms or heuristics, the best known fixed-ratio algorithm being a 1.375-approximation [13]. Other works [18,9,14,21,13,5] aim at computing bounds on the transposition distance of a permutation.…”

Section: Introductionmentioning

confidence: 99%

“…Hence a number of studies [4,9,17,19,13,5,15] aim at designing approximation algorithms or heuristics, the best known fixed-ratio algorithm being a 1.375-approximation [13]. Other works [18,9,14,21,13,5] aim at computing bounds on the transposition distance of a permutation. Studies have also been devoted to variants of this problem, by considering, for example, prefix transpositions [12,22,7] (in which one of the blocks has to be a prefix of the sequence), or distance between strings [10,11,25,24,20] (where multiple occurrences of each element are allowed in the sequences), possibly with weighted or prefix transpositions [23,6,1,2,7].…”

Section: Introductionmentioning

confidence: 99%

Sorting by Transpositions Is Difficult

Bulteau

Fertin

Rusu

2011

Automata, Languages and Programming

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Abstract.In comparative genomics, a transposition is an operation that exchanges two consecutive sequences of genes in a genome. The transposition distance, that is, the minimum number of transpositions needed to transform a genome into another, can be considered as a relevant evolutionary distance. The problem of computing this distance when genomes are represented by permutations, called the SORTING BY TRANSPOSITIONS problem (SBT), has been introduced by Bafna and Pevzner [3] in 1995. It has naturally been the focus of a number of studies, 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 NP-hard. As a corollary of our result, we also prove that the following problem from [9] is NP-hard: given a permutation π, is it possible to sort π using d b (π)/3 permutations, where d b (π) is the number of breakpoints of π?

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A 1.375-Approximation Algorithm for Sorting by Transpositions

Cited by 154 publications

References 21 publications

Pancake Flipping with Two Spatulas

Pancake Flipping with Two Spatulas

Plane Permutations and Applications to a Result of Zagier--Stanley and Distances of Permutations

Sorting by Transpositions Is Difficult

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