How to Achieve an Equivalent Simple Permutation in Linear Time

The problem of Sorting signed permutations by reversals is a well studied problem in computational biology. The first polynomial time algorithm was presented by Hannenhalli and Pevzner in 1995 [5]. The algorithm was improved several times, and nowadays the most efficient algorithm has a subquadratic running time [9, 8]. Simple permutations played an important role in the development of these algorithms. Although the latest result of Tannier et al. [8] does not require simple permutations the preliminary version of their algorithm [9] as well as the first polynomial time algorithm of Hannenhalli and Pevzner [5] use the structure of simple permutations. However, the latter algorithms require a precomputation that transforms a permutation into an equivalent simple permutation. To the best of our knowledge, all published algorithms for this transformation have at least a quadratic running time. For further investigations on genome rearrangement problems, the existence of a fast algorithm for the transformation could be crucial. In this paper, we present a linear time algorithm for the transformation.