Average-case analysis of perfect sorting by reversals

TL;DR

This paper demonstrates that, despite NP-hardness in the worst case, perfect sorting by reversals can be efficiently performed on average, and provides asymptotic analysis for commuting permutations.

Contribution

It shows average-case polynomial-time sorting for perfect reversals and derives asymptotic formulas for commuting permutations.

Findings

Sorting can be done in polynomial time with probability one.

Asymptotic expressions for average length and number of reversals.

Analysis focuses on commuting permutations as a special case.

Abstract

A sequence of reversals that takes a signed permutation to the identity is perfect if at no step a common interval is broken. Determining a parsimonious perfect sequence of reversals that sorts a signed permutation is NP-hard. Here we show that, despite this worst-case analysis, with probability one, sorting can be done in polynomial time. Further, we find asymptotic expressions for the average length and number of reversals in commuting permutations, an interesting sub-class of signed permutations.