Résumé
The study of genomic inversions (or reversals) has been a mainstay of computational genomics for nearly 20 years. After the initial breakthrough of Hannenhalli and Pevzner, who gave the first polynomial-time algorithm for sorting signed permutations by inversions, improved algorithms have been designed, culminating with an optimal linear-time algorithm for computing the inversion distance and a subquadratic algorithm for providing a shortest sequence of inversions-also known as sorting by inversions. Remaining open was the question of whether sorting by inversions Could be done in O(n log n) time.
In this paper, we present a qualified answer to this question, by providing two new sorting algorithms, a simple and fast randomized algorithm and a deterministic refinement. The deterministic algorithm runs in time O(n log n + kn), where k is a data-dependent parameter. We provide the results of extensive experiments showing that both the average and the standard deviation for k are small constants, independent of the size of the permutation. We conclude (but do not prove) that almost all signed permutations can be sorted by inversions in O(n log n) time.