Approximation algorithms for sorting by signed short reversals

Abstract

During evolution, global mutations may modify the gene order in a genome. Such mutations are commonly referred to as rearrangement events. One of the most frequent rearrangement events observed in genomes are reversals, which are responsible for reversing the order and orientation of a sequence of genes. The problem of sorting by reversals, that is, the problem of computing the most parsimonious reversal scenario to transform one genome into another, is a well-studied problem that finds application in comparative genomics. There is a number of works concerning this problem in the literature, but these works generally do not take into account the length of the reversals. Since it has been observed that short reversals are prevalent in the evolution of some species, recent efforts have been made to address this issue algorithmically. In this paper, we add to these efforts by introducing the problem of sorting by signed short reversals and by presenting three approximation algorithms for solving it. Although the worst-case approximation ratios of these algorithms are high, we show that their expected approximation ratios for sorting a random equiprobable signed permutation are much lower. Moreover, we present experimental results on small signed permutations which indicate that the worst-case approximation ratios of these algorithms may be better than those we have been able to prove.

Sorting signed permutations by short operations

“…In this section, we present basic definitions that are used throughout this paper, generally following [ 28 ]. Let n be a positive integer.…”

“…In preliminary work, Galvão and Dias [ 28 ] investigated the problem of sorting a signed permutation by reversals of length at most 3 and presented three approximation algorithms, the best one having an approximation factor of 9. In this paper, we not only present an approximation algorithm with a better approximation factor, but also consider other bounded variants.…”

Algorithms for Sorting by Reversals or Transpositions, with Application to Genome Rearrangement

“…In preliminary work, we [Galvão and Dias 2014a] investigated the problem of sorting a signed permutation by reversals of length at most 3 and presented three approximation algorithms, the best one having an approximation factor of 9. In Chapter 5, we not only present an approximation algorithm with a better approximation factor, but also consider other variants of the problem.…”

Sorting Permutations by Prefix and Suffix Versions of Reversals and Transpositions

“…Dias and Meidanis [7] presented a 2-approximation algorithm for Sbpt, here called 2-PT, which always removes one breakpoint with one prefix transposition. They also demonstrated that there is at most one prefix transposition that removes two breakpoints at once, which leads to a greedy 2-approximation algorithm for the problem, called 2-PTg [10]: first it tries to remove two breakpoints and if this is not possible, it removes only one, as 2-PT does. Therefore, both guarantee that d Θp (Θ) ⊕ b Θp (Θ)− 1, since the last transposition always removes two breakpoints [7], and both are 2-approximation algorithms.…”