Average-case analysis of perfect sorting by reversals (Journal Version)

TL;DR

This paper analyzes the average-case complexity of a biologically motivated sorting algorithm by reversals, showing it runs in polynomial time on average and providing detailed statistical insights into its performance.

Contribution

It introduces an average-case analysis of the perfect sorting by reversals algorithm using combinatorial properties of strong interval trees, establishing polynomial runtime with high probability.

Findings

Algorithm runs in polynomial time on average.

Expected reversal length and count are precisely characterized.

For long permutations, the algorithm is polynomial with probability one.

Abstract

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. B\'erard 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.