Approximation Algorithms for Sorting λ-Permutations by λ-Operations

Understanding how different two organisms are is one question addressed by the comparative genomics field. A well-accepted way to estimate the evolutionary distance between genomes of two organisms is finding the rearrangement distance, which is the smallest number of rearrangements needed to transform one genome into another. By representing genomes as permutations, one of them can be represented as the identity permutation, and, so, we reduce the problem of transforming one permutation into another to the problem of sorting a permutation using the minimum number of rearrangements. This work investigates the problems of sorting permutations using reversals and/or transpositions, with some additional restrictions of biological relevance. Given a value λ, the problem now is how to sort a λ-permutation, which is a permutation whose elements are less than λ positions away from their correct places (regarding the identity), by applying the minimum number of rearrangements. Each λ-rearrangement must have size, at most, λ, and, when applied to a λ-permutation, the result should also be a λ-permutation. We present algorithms with approximation factors of O(λ2), O(λ), and O(1) for the problems of Sorting λ-Permutations by λ-Reversals, by λ-Transpositions, and by both operations.

Exact upper bound for sorting Rn with LE

A permutation over alphabet [Formula: see text] is a sequence over [Formula: see text], where every element occurs exactly once. [Formula: see text] denotes symmetric group defined over [Formula: see text]. [Formula: see text] denotes the Identity permutation. [Formula: see text] is the reverse permutation i.e., [Formula: see text]. An operation, that we call as an LE operation, has been defined which consists of exactly two generators: set-rotate that we call Rotate and pair-exchange that we call Exchange (OEIS). At least two generators are the required to generate [Formula: see text]. Rotate rotates all elements to the left (moves leftmost element to the right end) and Exchange is the pair-wise exchange of the two leftmost elements. The optimum number of moves for transforming [Formula: see text] into [Formula: see text] with LE operation are known for [Formula: see text]; as listed in OEIS with identity A048200. However, no general upper bound is known. The contributions of this article are: (a) a novel upper bound for the number of moves required to sort [Formula: see text] with LE has been derived; (b) the optimum number of moves to sort the next larger [Formula: see text] i.e., [Formula: see text] has been computed; (c) an algorithm conjectured to compute the optimum number of moves to sort a given [Formula: see text] has been designed.

Sorting permutations by prefix and suffix rearrangements

Some interesting combinatorial problems have been motivated by genome rearrangements, which are mutations that affect large portions of a genome. When we represent genomes as permutations, the goal is to transform a given permutation into the identity permutation with the minimum number of rearrangements. When they affect segments from the beginning (respectively end) of the permutation, they are called prefix (respectively suffix) rearrangements. This paper presents results for rearrangement problems that involve prefix and suffix versions of reversals and transpositions considering unsigned and signed permutations. We give 2-approximation and ([Formula: see text])-approximation algorithms for these problems, where [Formula: see text] is a constant divided by the number of breakpoints (pairs of consecutive elements that should not be consecutive in the identity permutation) in the input permutation. We also give bounds for the diameters concerning these problems and provide ways of improving the practical results of our algorithms.

Bounding the Distinguishing Number of Infinite Graphs and Permutation Groups

A group of permutations $G$ of a set $V$ is $k$-distinguishable if there exists a partition of $V$ into $k$ cells such that only the identity permutation in $G$ fixes setwise all of the cells of the partition. The least cardinal number $k$ such that $(G,V)$ is $k$-distinguishable is its distinguishing number $D(G,V)$. In particular, a graph $\Gamma$ is $k$-distinguishable if its automorphism group $\rm{Aut}(\Gamma)$ satisfies $D(\rm{Aut}(\Gamma),V\Gamma)\leq k$.Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph $\Gamma$ is not $k$-distinguishable (for a given cardinal $k$), then it contains a ball of finite radius whose distinguishing number is at least $k$. Moreover, this lower bound cannot be sharpened, since for any integer $k \geq 3$ there exists an infinite, locally finite, connected graph $\Gamma$ that is not $k$-distinguishable but in which every ball of finite radius is $k$-distinguishable.In the second half of this paper we show that a large distinguishing number for an imprimitive group $G$ is traceable to a high distinguishing number either of a block of imprimitivity or of the induced action by $G$ on the corresponding system of imprimitivity. An immediate application is to automorphism groups of infinite imprimitive graphs. These results are companion to the study of the distinguishing number of infinite primitive groups and graphs in a previous paper by the authors together with T. W. Tucker.

Refined Inversion Statistics on Permutations

We introduce and study new refinements of inversion statistics for permutations, such as $k$-step inversions, (the number of inversions with fixed position differences) and non-inversion sums (the sum of the differences of positions of the non-inversions of a permutation). We also provide a distribution function for non-inversion sums, a distribution function for $k$-step inversions that relates to the Eulerian polynomials, and special cases of distribution functions for other statistics we introduce, such as $(\le\!\!k)$-step inversions and $(k_1,k_2)$-step inversions (that fix the value separation as well as the position). We connect our refinements to other work, such as inversion tops that are $0$ modulo a fixed integer $d$, left boundary sums of paths, and marked meshed patterns. Finally, we use non-inversion sums to show that for every number $n>34$, there is a permutation such that the dot product of that permutation and the identity permutation (of the same length) is $n$.

AVERAGE-CASE ANALYSIS OF PERFECT SORTING BY REVERSALS

Perfect sorting by reversals, a problem originating in computational genomics, is the process of sorting a signed permutation to either the identity or to the reversed identity permutation, by a sequence of reversals that do not break any common interval. Bérard et al. (2007) make use of strong interval trees to describe an algorithm for sorting signed permutations by reversals. Combinatorial properties of this family of trees are essential to the algorithm analysis. Here, we use the expected value of certain tree parameters to prove that the average run-time of the algorithm is at worst, polynomial, and additionally, for sufficiently long permutations, the sorting algorithm runs in polynomial time with probability one. Furthermore, our analysis of the subclass of commuting scenarios yields precise results on the average length of a reversal, and the average number of reversals.

Equivalence Relations of Permutations Generated by Constrained Transpositions

International audience We consider a large family of equivalence relations on permutations in $S_n$ that generalise those discovered by Knuth in his study of the Robinson-Schensted correspondence. In our most general setting, two permutations are equivalent if one can be obtained from the other by a sequence of pattern-replacing moves of prescribed form; however, we limit our focus to patterns where two elements are transposed, conditional upon the presence of a third element of suitable value and location. For some relations of this type, we compute the number of equivalence classes, determine how many $n$-permutations are equivalent to the identity permutation, or characterise this equivalence class. Although our results include familiar integer sequences (e.g., Catalan, Fibonacci, and Tribonacci numbers) and special classes of permutations (layered, connected, and $123$-avoiding), some of the sequences that arise appear to be new. Nous considérons une famille de relations d’équivalence sur l'ensemble $S_n$ des permutations, qui généralisent les relations de Knuth liées à la correspondance Robinson-Schensted. Dans notre contexte général, deux permutations sont considérées comme équivalentes si l'une peut être obtenue de l'autre auprès d'une séquence de remplacements d'un motif par un autre selon des règles précisées. Désormais, nous ne considérons dans l’œuvre actuelle que les motifs qui correspondent à la transposition de deux éléments, conditionné sur la présence d'un élément de valeur et de position approprié. Pour plusieurs exemples de ce problème, nous énumérons les classes d'équivalence, nous déterminons combien de permutations sur $n$ éléments sont équivalentes à l'identité, ou nous précisons la forme des éléments dans cette dernière classe. Bien que nos résultats retrouvent des séquences des entiers très bien connues (nombres de Catalan, de Fibonacci, de Tribonacci...) ainsi que des classes de permutations déjà étudiées (en couches, connexes, sans motif $123$), nous trouvons également des séquences qui paraissent être nouvelles.

PERMUTATION AND ITS PARTIAL TRANSPOSE

Permutation and its partial transpose play important roles in quantum information theory. The Werner state is recognized as a rational solution of the Yang–Baxter equation, and the isotropic state with an adjustable parameter is found to form a braid representation. The set of permutation's partial transposes is an algebra called the PPT algebra, which guides the construction of multipartite symmetric states. The virtual knot theory, having permutation as a virtual crossing, provides a topological language describing quantum computation as having permutation as a swap gate. In this paper, permutation's partial transpose is identified with an idempotent of the Temperley–Lieb algebra. The algebra generated by permutation and its partial transpose is found to be the Brauer algebra. The linear combinations of identity, permutation and its partial transpose can form various projectors describing tangles; braid representations; virtual braid representations underlying common solutions of the braid relation and Yang–Baxter equations; and virtual Temperley–Lieb algebra which is articulated from the graphical viewpoint. They lead to our drawing a picture called the ABPK diagram describing knot theory in terms of its corresponding algebra, braid group and polynomial invariant. The paper also identifies non-trivial unitary braid representations with universal quantum gates, derives a Hamiltonian to determine the evolution of a universal quantum gate, and further computes the Markov trace in terms of a universal quantum gate for a link invariant to detect linking numbers.

APPROXIMATE BLOCK SORTING

We consider the problem BLOCK-SORTING: Given a permutation, sort it by using a minimum number of block moves, where a block is a maximal substring of the permutation which is also a substring of the identity permutation, and a block move repositions the chosen block so that it merges with another block. Although this problem has recently been shown to be NP-hard [3], nothing better than a trivial 3-approximation was known. We present here the first non-trivial approximation algorithm to this problem. For this purpose, we introduce the following optimization problem: Given a set of increasing sequences of distinct elements, merge them into one increasing sequence with a minimum number of block moves. We show that the merging problem has a polynomial time algorithm. Using this, we obtain an O(n3) time 2-approximation algorithm for BLOCK-SORTING. We also observe that BLOCK-SORTING, as well as sorting by transpositions, are fixed-parameter-tractable in the framework of [6].

A permutability problem in infinite groups and Ramsey's theorem

We use Ramsey's theorem to generalise a result of L. Babai and T.S. Sós on Sidon subsets and then use this to prove that for an integer n > 1 the class of groups in which every infinite subset contains a rewritable n-subset coincides with the class of groups in which ever infinite subset contains n mutually disjoint non-empty subsets X1,…,Xn such that X1…Xn∩Xσ(1) …xσ(n) ≠ θ for some non-identity permutation σ on the set {1,…,n}.