Reversal Distance for Strings with Duplicates: Linear Time Approximation using Hitting Set

Abstract

In the last decade there has been an ongoing interest in string comparison problems; to a large extend the interest was stimulated by genome rearrangement problems in computational biology but related problems appear in many other areas of computer science. Particular attention has been given to the problem of sorting by reversals (SBR): given two strings, $A$ and $B$, find the minimum number of reversals that transform the string $A$ into the string $B$ (a reversal $\rho(i,j)$, $i < j$, transforms a string $A=a_1\ldots a_n$ into a string $A'=a_1\ldots a_{i-1} a_{j} a_{j-1} \ldots a_{i} a_{j+1} \ldots a_n$). Closely related is the minimum common string partition problem (MCSP): given two strings, $A$ and $B$, find a minimum size partition of $A$ into substrings $P_1,\ldots,P_l$ (i.e., $A=P_1\ldots P_l$) and a partition of $B$ into substrings $Q_1,\ldots,Q_l$ such that $(Q_1,\ldots,Q_l)$ is a permutation of $(P_1,\ldots,P_l)$. Primarily the SBR problem has been studied for strings in which every symbol appears exactly once (that is, for permutations) and only recently attention has been given to the general case where duplicates of the symbols are allowed. In this paper we consider the problem $k$-SBR, a version of SBR in which each symbol is allowed to appear up to $k$ times in each string, for some $k\geq 1$. The main result of the paper is a $\Theta(k)$-approximation algorithm for $k$-SBR running in time $O(n)$; compared to the previously known algorithm for $k$-SBR, this is an improvement by a factor of $\Theta(k)$ in the approximation ratio, and by a factor of $\Theta(k)$ in the running time. We approach the $k$-SBR by finding an approximation for the $k$-MCSP first and then turning it into a solution for $k$-SBR. Crucial ingredients of our algorithm are the suffix tree data structure and a linear time algorithm for a special case of a disjoint set union problem.

Minimum Common String Partition Parameterized by Partition Size Is Fixed-Parameter Tractable

“…We also show, using several properties of frames as a black box, our main result. Then, in Sections 4 and 5, we fill 10 if |S x | = |S y | : abort branch // no bijection of solid pieces exists 11 if |F x | ≥ k or |F y | ≥ k : abort branch // too many fragile pieces in x (or y) 12 while ∃ unmatched solid piece s ∈ S x : 13 for each unmatched solid piece t in S y : The algorithm is a branching algorithm that extends the constraint C in each branch. In order to simplify the pseudo-code somewhat, we describe the algorithm in such a way that the variables C and β are global variables.…”

Computing Genomic Distances: An Algorithmic Viewpoint

“…In this section, we show the main ideas and arguments of an approximation algorithm provided by Kolman and Waleń [10]. Let P and Q be two balanced strings containing signed integers, let n = m P = m Q , and let k be the maximum number of copies of a gene in P (resp.…”

“…It should be first said that the result from Kolman and Waleń [10] considers a slightly different problem than BAL-FMB, called UMCSP, which stands for Unsigned Minimum Common String Partition. This problem is the following: given two balanced unsigned strings P and Q, find a partition P = {P 1 , P 2 , .…”

“…1. An approximation algorithm for BAL-FMB [10] 2. An exact (thus, exponential) algorithm for UNBAL-FMB [1], written in the form of a 0-1 linear program, the goal being to be able to handle large instances.…”

Um Algoritmo Exato em clusters de GPUs para o Hitting Set Aplicado à Inferência de Redes de Regulação Gênica

“…Diversasáreas de estudo possuem problemas teóricos ou práticos os quais podem ser modelados, em parte ou como um todo, como uma instância do problema da Transversal Mínima (do Inglês, Hitting Set Problem ou HSP). Aárea de Biologia de Sistemas nãoé uma exceção, apresentando aplicações tais como distância reversa genômica [Kolman and Walen 2007], experimentos com reação em cadeia de polimerase [Pearson et al 1996] e inferência de redes de regulação gênica (do Inglês Gene Regulatory Networks ou GRNs) [Ideker et al 2000, Ruchkys andSong 2003]. Ideker et al [Ideker et al 2000], por exemplo, utiliza uma abordagem iterativa de perturbação de sinais, na qual se aplicam perturbações sucessivas nos níveis de expressão dos genes e as alterações consequentes são analisadas com o objetivo de inferir interações regulatórias entre os genes.…”