In genome rearrangements, the mutational event transposition swaps two adjacent blocks of genes in one chromosome. The Transposition Distance Problem (TDP) aims to find the minimum number of transpositions required to transform one chromosome into another, both represented as permutations. The TDP can be reduced to the problem of Sorting by Transpositions (SBT).
SBT is $\mathcal{NP}$-hard and the best approximation algorithm with a $1.375$ ratio was proposed by Elias and Hartman. Their algorithm employs simplification, a technique used to transform an input permutation $π$ into a simple permutation $\hatπ$, presumably easier to handle with. The permutation $\hatπ$ is obtained by inserting new symbols into $π$ in a way that the lower bound of the transposition distance of $π$ is kept on $\hatπ$. The simplification is guaranteed to keep the lower bound, not the transposition distance.
In this paper, we first show that the algorithm of Elias and Hartman (EH algorithm) may require one extra transposition above the approximation ratio of $1.375$, depending on how the input permutation is simplified. Next, using an algebraic approach, we propose a new upper bound for the transposition distance and a new $1.375$-approximation algorithm to solve SBT skipping simplification and ensuring the approximation ratio of $1.375$ for all $S_n$.
We implemented our algorithm and EH's. Regarding the implementation of the EH algorithm, two issues needed to be fixed. We tested both algorithms against all permutations of size $n$, $2\leq n \leq 12$. The results show that the EH algorithm exceeds the approximation ratio of $1.375$ for permutations with a size greater than $7$. Finally, we investigate the performance of both implementations on longer permutations of maximum length $500$.
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, is, according to numerous studies, a relevant evolutionary distance. The problem of computing this distance when genomes are represented by permutations, called the Sorting by Transpositions problem, has been introduced by Bafna and Pevzner in 1995. It has naturally been the focus of a number of studies, but the computational complexity of this problem has rema…
Sorting a Permutation by Transpositions (SPbT) is an important problem in Bioinformtics. In this paper, we improve the running time of the best known approximation algorithm for SPbT. We use the permutation tree data structure of Feng and Zhu and improve the running time of the 1.375 Approximation Algorithm for SPbT of Elias and Hartman to $O(n\log n)$. The previous running time of EH algorithm was $O(n^2)$.
The set of all permutations with $n$ symbols is a symmetric group denoted by $S_n$. A transposition tree, $T$, is a spanning tree over its $n$ vertices $V_T=${$1, 2, 3, \ldots n$} where the vertices are the positions of a permutation $π$ and $π$ is in $S_n$. $T$ is the operation and the edge set $E_T$ denotes the corresponding generator set. The goal is to sort a given permutation $π$ with $T$. The number of generators of $E_T$ that suffices to sort any $π\in S_n$ constitutes an upper bound. It is an upper bound, on the diameter of the corresponding Cayley graph $Γ$ i.e. $diam(Γ)$. A precise u…
In this paper we study several variations of the \emph{pancake flipping problem}, which is also well known as the problem of \emph{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…
Link-cut trees have been introduced by D.D. Sleator and R.E. Tarjan (Journal of Computer and System Sciences, 1983) with the aim of efficiently maintaining a forest of vertex-disjoint dynamic rooted trees under cut and link operations. These operations respectively disconnect a subtree from a tree, and join two trees by an edge. Additionally, link-cut trees allow to change the root of a tree and to perform a number of updates and queries on cost values defined on the arcs of the trees. All these operations are performed in $O(\log\, n)$ amortized or worst-case time, depending on the implementa…
A number of fields, including the study of genome rearrangements and the design of interconnection networks, deal with the connected problems of sorting permutations in "as few moves as possible", using a given set of allowed operations, or computing the number of moves the sorting process requires, often referred to as the \emph{distance} of the permutation. These operations often act on just one or two segments of the permutation, e.g. by reversing one segment or exchanging two segments. The \emph{cycle graph} of the permutation to sort is a fundamental tool in the theory of genome rearrange…
Block Sorting is a well studied problem, motivated by its applications in Optical Character Recognition (OCR), and Computational Biology. Block Sorting has been shown to be NP-Hard, and two separate polynomial time 2-approximation algorithms have been designed for the problem. But questions like whether a better approximation algorithm can be designed, and whether the problem is APX-Hard have been open for quite a while now. In this work we answer the latter question by proving Block Sorting to be Max-SNP-Hard (APX-Hard). The APX-Hardness result is based on a linear reduction of Max-3SAT to Bl…
We initiate the study of sorting permutations using prefix block-interchanges, which exchange any prefix of a permutation with another non-intersecting interval. The goal is to transform a given permutation into the identity permutation using as few such operations as possible. We give a 2-approximation algorithm for this problem, show how to obtain improved lower and upper bounds on the corresponding distance, and determine the largest possible value for that distance.
We empirically study sorting in the evolving data model. In this model, a sorting algorithm maintains an approximation to the sorted order of a list of data items while simultaneously, with each comparison made by the algorithm, an adversary randomly swaps the order of adjacent items in the true sorted order. Previous work studies only two versions of quicksort, and has a gap between the lower bound of Omega(n) and the best upper bound of O(n log log n). The experiments we perform in this paper provide empirical evidence that some quadratic-time algorithms such as insertion sort and bubble sor…
We consider the offline sorting buffer problem. The input is a sequence of items of different types. All items must be processed one by one by a server. The server is equipped with a random-access buffer of limited capacity which can be used to rearrange items. The problem is to design a scheduling strategy that decides upon the order in which items from the buffer are sent to the server. Each type change incurs unit cost, and thus, the cost minimizing objective is to minimize the total number of type changes for serving the entire sequence. This problem is motivated by various applications in…
We study maximum selection and sorting of $n$ numbers using pairwise comparators that output the larger of their two inputs if the inputs are more than a given threshold apart, and output an adversarially-chosen input otherwise. We consider two adversarial models. A non-adaptive adversary that decides on the outcomes in advance based solely on the inputs, and an adaptive adversary that can decide on the outcome of each query depending on previous queries and outcomes. Against the non-adaptive adversary, we derive a maximum-selection algorithm that uses at most $2n$ comparisons in expectation, …
Let $A$ be a sequence of $n \geq 0$ real numbers. A subsequence of $A$ is a sequence of contiguous elements of $A$. A \emph{maximum scoring subsequence} of $A$ is a subsequence with largest sum of its elements, which can be found in O(n) time by Kadane's dynamic programming algorithm. We consider in this paper two problems involving maximal scoring subsequences of a sequence. Both of these problems arise in the context of buffer memory minimization in computer networks. The first one, which is called {\sc Insertion in a Sequence with Scores (ISS)}, consists in inserting a given real number $x$…
Given a graph where every vertex has exactly one labeled token, how can we most quickly execute a given permutation on the tokens? In (sequential) token swapping, the goal is to use the shortest possible sequence of swaps, each of which exchanges the tokens at the two endpoints of an edge of the graph. In parallel token swapping, the goal is to use the fewest rounds, each of which consists of one or more swaps on the edges of a matching. We prove that both of these problems remain NP-hard when the graph is restricted to be a tree. These token swapping problems have been studied by disparate gr…
The rank aggregation problem has received significant recent attention within the computer science community. Its applications today range far beyond the original aim of building metasearch engines to problems in machine learning, recommendation systems and more. Several algorithms have been proposed for these problems, and in many cases approximation guarantees have been proven for them. However, it is also known that some Markov chain based algorithms (MC1, MC2, MC3, MC4) perform extremely well in practice, yet had no known performance guarantees. We prove supra-constant lower bounds on appr…
We study the Ordered k-Median problem, in which the solution is evaluated by first sorting the client connection costs and then multiplying them with a predefined non-increasing weight vector (higher connection costs are taken with larger weights). Since the 1990s, this problem has been studied extensively in the discrete optimization and operations research communities and has emerged as a framework unifying many fundamental clustering and location problems such as k-Median and k-Center. This generality, however, renders the problem intriguing from the algorithmic perspective and obtaining no…
A top-list is a possibly incomplete ranking of elements: only a subset of the elements are ranked, with all unranked elements tied for last. Top-list aggregation, a generalization of the well-known rank aggregation problem, takes as input a collection of top-lists and aggregates them into a single complete ranking, aiming to minimize the number of upsets (pairs ranked in opposite order in the input and in the output). In this paper, we give simple approximation algorithms for top-list aggregation. * We generalize the footrule algorithm for rank aggregation. * Using inspiration from approval vo…
We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In contrast to the classical technique of DeMillo, Lipton, Schwartz, and Zippel, this methodology can decrease the error probability by increasing the precision of the approximations instead of using more random bits. Consequently, randomized algorithms that use the classical technique can generally be improved using the new methodology. To demonstrate the methodolo…
In data-parallel computing frameworks, intermediate parallel data is often produced at various stages which needs to be transferred among servers in the datacenter network (e.g. the shuffle phase in MapReduce). A stage often cannot start or be completed unless all the required data pieces from the preceding stage are received. \emph{Coflow} is a recently proposed networking abstraction to capture such communication patterns. We consider the problem of efficiently scheduling coflows with release dates in a shared datacenter network so as to minimize the total weighted completion time of coflows…
The complexity of an algorithm is an important parameter to determine its effi-ciency. They are of different types viz. Time complexity, Space complexity, etc. However, none of them consider the execution path as a complexity measure. Ashok et al, firstly proposed the notion of the Path Complexity of a pro-gram/algorithm, which defined based on the number of execution paths as a function of the input size. However, the notion of path complexity of the pro-gram, cannot apply to the object-oriented environment. Therefore, Anupam et al, has extended the notion of path complexity to the class as f…
SCAN (Structural Clustering Algorithm for Networks) is a well-studied, widely used graph clustering algorithm. For large graphs, however, sequential SCAN variants are prohibitively slow, and parallel SCAN variants do not effectively share work among queries with different SCAN parameter settings. Since users of SCAN often explore many parameter settings to find good clusterings, it is worthwhile to precompute an index that speeds up queries. This paper presents a practical and provably efficient parallel index-based SCAN algorithm based on GS*-Index, a recent sequential algorithm. Our parallel…
