Abstract. 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(nlog 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(nlog 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(nlogn) time. 1

Abstract. As data about genomic architecture accumulates, genomic rearrangements have attracted increasing attention. One of the main rearrangement mechanisms, inversions (also called reversals), was characterized by Hannenhalli and Pevzner and this characterization in turn extended by various authors. The characterization relies on the concepts of breakpoints, cycles, and obstructions colorfully named hurdles and fortresses. In this paper, we study the probability of generating a hurdle in the process of sorting a permutation if one does not take special precautions to avoid them (as in a randomized algorithm, for instance). To do this we revisit and extend the work of Caprara and of Bergeron by providing simple and exact characterizations of the probability of encountering a hurdle in a random permutation. Using similar methods we provide the first asymptotically tight analysis of the probability that a fortress exists in a random permutation. Finally, we study other aspects of hurdles, both analytically and through experiments: when are they created in a sequence of sorting inversions, how much later are they detected, and how much work may need to be undone to return to a sorting sequence. 1

The problem of sorting a signed permutation by reversals is inspired and motivated by comparative genomics. Following the first polynomial time solution of this problem, several improvements have been published on the subject. The currently fastest algorithms is defined by the sequence augmentation sorting algorithm using balanced binary tree with running time O(n 3/2 √log n). We give a parallel implementation of the sequence augmentation sorting algorithm on the Mesh of trees architecture.

all sorting reversals in quadratic time