Average-case analysis of perfect sorting by reversals

Mathilde Bouvel (1) , Cédric Chauve (2) , Marni Mishna (2) , Dominique Rossin (3)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 SFU - Department of Mathematics [Burnaby]
3 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

Perfect sorting by reversals, a problem originating in computational genomics, is the process of sorting a signed permutation to either the identity or to the reversed identity permutation, by a sequence of reversals that do not break any common interval. Bérard et al. (2007) make use of strong interval trees to describe an algorithm for sorting signed permutations by reversals. Combinatorial properties of this family of trees are essential to the algorithm analysis. Here, we use the expected value of certain tree parameters to prove that the average run-time of the algorithm is at worst, polynomial, and additionally, for sufficiently long permutations, the sorting algorithm runs in polynomial time with probability one. Furthermore, our analysis of the subclass of commuting scenarios yields precise results on the average length of a reversal, and the average number of reversals. A preliminary version of this work appeared in the proceedings of Combinatorial Pattern Matching (CPM) 2009, Lectures Notes in Computer Science, vol. 5577, pp. 314--325, Springer.

Mots clés: Algorithm analysis; genome rearrangements; tree parameters; simple permutations

Domaines: Combinatoire [math.CO]; Bio-informatique [q-bio.QM]; Bio-Informatique, Biologie Systémique [q-bio.QM]

Mathilde Bouvel, Cédric Chauve, Marni Mishna, Dominique Rossin. Average-case analysis of perfect sorting by reversals. Discrete Mathematics, Algorithms and Applications, 2011, 3 (3), pp.369-392. ⟨10.1142/S1793830911001280⟩. ⟨hal-00649761⟩