Prefix Transpositions Research Articles

Improved upper bound for sorting permutations by prefix transpositions

Modelling of chromosomes with permutations has triggered the research of sorting permutations using global rearrangement operations in computational molecular biology. One such rearrangement is transposition which swaps two adjacent substrings. If one of the substrings is restricted to be the prefix of the permutation then that operation is called a prefix transposition. The symmetric prefix transposition distance between two permutations is defined to be the minimum number of prefix transpositions required to transform one into another. Group theory dictates that an upper bound for prefix transposition distance between any pair of permutations in the symmetric group Sn is equivalent to the upper bound for sorting any permutation with prefix transpositions. In this paper, we improve the upper bound for sorting permutations with prefix transpositions to n−log2⁡n.

A new upper bound for sorting permutations with prefix transpositions

Permutations are discrete structures that naturally model a genome where every gene occurs exactly once. In a permutation over the given alphabet [Formula: see text], each symbol of [Formula: see text] appears exactly once. A transposition operation on a given permutation [Formula: see text] exchanges two adjacent sublists of [Formula: see text]. If one of these sublists is restricted to be a prefix then one obtains a prefix transposition. The symmetric group of permutations with [Formula: see text] symbols derived from the alphabet [Formula: see text] is denoted by [Formula: see text]. The symmetric prefix transposition distance between [Formula: see text] and [Formula: see text] is the minimum number of prefix transpositions that are needed to transform [Formula: see text] into [Formula: see text]. It is known that transforming an arbitrary [Formula: see text] into an arbitrary [Formula: see text] is equivalent to sorting some [Formula: see text]. Thus, upper bound for transforming any [Formula: see text] into any [Formula: see text] with prefix transpositions is simply the upper bound to sort any permutation [Formula: see text]. The current upper bound is [Formula: see text] for prefix transposition distance over [Formula: see text]. In this paper, we improve the same to [Formula: see text].

Sorting permutations with transpositions in O(n3) amortized time

The problem of sorting a permutation π over alphabet ∑={1,2,3…n} with the minimum number of transpositions is known to be intractable. That is, the computation of transposition distance i.e. dt(π), is hard. Such sorting is equivalent to determining the distance between any α, β of the symmetric group Sn over ∑ under the transposition operation. It has applications in computer interconnection networks where a node is denoted by a permutation, and genetics where a genome is modeled by a permutation. The maximum distance between any pair of permutations in Sn is the diameter i.e. diam(Γ(Sn)), of the corresponding Cayley graph Γ(Sn). It corresponds to the maximum latency in the computer interconnection network modeled by Γ(Sn) and the maximum genetic dissimilarity between any pair α, β in Sn. We call transpositions, prefix transpositions, suffix transpositions and prefix/suffix transpositions as block-moves. We design an algorithm to compute dt(π) ∀π in Sn and compute diam(Γ(Sn)) with transpositions in O(n!n3) time and O(n!n2) space; at an amortized time of O(n3). The same can be computed for the remaining block moves in O(n2log⁡n) amortized time and O(n!n) space.

Tighter upper bound for sorting permutations with prefix transpositions

Permutations are sequences where each symbol in the given alphabet Σ appears exactly once. A transposition is an operation that exchanges two adjacent sublists in a permutation; if one of these sublists is restricted to be a prefix then one obtains a prefix transposition. Transpositions over permutations have applications in genome rearrangements and computer interconnection networks. The set of permutations with n symbols derived from the alphabet Σ={0,1,…,n−1} form a symmetric group, that we denote by Pn. The prefix transposition distance between two permutations π⁎∈Pn and π∼∈Pn is the minimum number of prefix transpositions, also called moves, needed to transform π⁎ into π∼. A prefix transposition can be modeled by a permutation operation. A permutation is type of sequence and it is also an operation. The generators for prefix transposition operation are a subset of permutation operation. As a result, transforming an arbitrary π⁎∈Pn to an arbitrary π∼∈Pn is equivalent to sorting some πˆ∈Pn. Thus, upper bound for transforming any π⁎∈Pn into any π∼∈Pn with prefix transpositions is simply the upper bound to sort any permutation π∈Pn with moves. In the article that establishes the best known upper bound for prefix transposition distance over Pn as n−log9/2⁡n, explicit proofs were not given for some cases. In this article, we provide the missing proofs, validating the stated upper bound. Furthermore, we establish an upper bound of n−log7/2⁡n for prefix transposition distance over Pn.

Pancake flipping and sorting permutations

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. 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 from practical point of view 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.

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 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.

Prefix transpositions on binary and ternary strings

The problem of sorting by prefix transpositions asks for the minimum number of prefix transpositions required to sort the elements of a given permutation. In this paper, we study a variant of this problem where the prefix transpositions act not on permutations but on strings over an alphabet of fixed size. Here, we determine the minimum number of prefix transpositions required to sort the binary and ternary strings, with polynomial time algorithms for these sorting problems.

Bounding prefix transposition distance for strings and permutations

A transposition is an operation that exchanges two adjacent substrings. Transpositions over permutations, the sequences with no repeated symbols, are related to genome rearrangements. If one of the substrings is restricted to a prefix then it is called a prefix transposition. The prefix transposition distance between a pair of strings (permutations) is the minimum number of prefix transpositions required to transform a given string (permutation) into another given string (permutation). For a permutation of length n, we improve the current prefix transposition distance upper bound of n−log8n to n−log9/2n. For arbitrary strings, we prove new prefix transposition distance upper and lower bounds. For binary strings of length n, we show that n/2 is an upper bound. We show that the prefix transposition distance problem is NP-complete for binary strings.

Pancake Flipping with Two Spatulas

In this paper, we give approximation algorithms for 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 prefix transpositions, prefix transreversals etc. along with the prefix reversals.