Introduction

We shall introduce the problem of sorting signed permutations by reversals. A signed permutation is a permutation $\pi = (\pi_1,\ldots,\pi_n)$ on the integers $\{1,\ldots,n\}$, where each number is also assigned a sign of plus or minus. A reversal, $\rho(i,j)$ on $\pi$ transforms $\pi$ to

$$\begin{eqnarray*}\pi'=\pi\cdot \rho(i,j) = (\pi_1,\ldots,\pi_{i-1},-\pi_j, - \pi_{j-1},\ldots,- \pi_i,\pi_{j+1},\ldots,\pi_n). \end{eqnarray*}$$

This conforms with the usual definition of the product between permutations (i.e., composition), defining $\rho(i,j) = (1,2,\ldots,i-1,-j,-(j-1),\ldots,-i,j+1.\ldots,n)$. As in the case of unsigned permutations, the minimum number of reversals needed to transform one permutation to another is called the reversal distance between them. The problem of sorting signed permutations by reversals is defined as follows:

Problem 11.5   $\;$
INPUT: A signed permutation $\pi$.
QUESTION: What is the reversal distance between $\pi$ and the signed identity permutation $(+1,+2,\ldots,+n)$ ?

Our motivation for studying this problem comes from genome comparison problems. Due to the fast progress in the Human Genome Project, genetic and DNA data is accumulating rapidly, and consequently the ability to compare genomes of different species has grown dramatically. One of the most promising ways of checking large scale similarity between genomes is to compare the order of appearance of identical genes in the two species. Dobzhansky and Sturtevant have shown in 1938 [9] evidence of inversions in chromosomes of Drosophila. In the 1980's, Palmer [18,19,20,21,14] has demonstrated that different species may have essentially the same genes, but the gene order may differ between species. A mathematical description of this problem suggests that genes along a chromosome can be thought of as points along a line. Numbers identify the particular genes, and as genes have directionality, denoted by signs corresponding to their orientation. The difference in order between genomes can be explained by some reversals between them. These reversals correspond to evolutionary changes along the history between the two genomes, so the number of reversals represent the evolutionary distance between the species. Hence, given two such permutations, their reversal distance measures their evolutionary distance. Studies of problem 11.5 resulted in a 1.5 polynomial approximation algorithm [17]. This approximation factor was improved later even more to the value of 1.375  [12]. In 1995, Hannenhalli and Pevzner [13] have shown that the problem of sorting a signed permutation by reversals is polynomial, and can be done in O(n⁴) time. More recently, Berman and Hannenhalli [4] described a faster implementation that finds a minimum sequence of reversals in $O(n^{2}\alpha(n))$ time, where $\alpha$ is the inverse Ackerman's function [1]. In this lecture we present a O(n²) algorithm for sorting a signed permutation of n elements, thereby improving upon the previous bound. In fact, if the reversal distance is r, our algorithm requires $O(n\cdot r+n\alpha(n))$ time [16].