Sorting by Reversals in Subquadratic Time

The problem of sorting a signed permutation by reversals is inspired by genome rearrangements in computational molecular biology. Given two genomes represented as two signed permutations of the same elements (e.g. orthologous genes), the problem consists in finding a most parsimonious scenario of reversals that transforms one genome into the other. We propose a method for sorting a signed permutation by reversals in time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(n\sqrt{n\log n})$\end{document}. The best known algorithms run in time O(n2), the main obstacle to an improvement being a costly operation of detection of so-called “safe” reversals. We bypass this detection and, using the same data structure as a previous random approximation algorithm, we achieve the same subquadratic complexity for finding an exact optimal solution. This answers an open question by Ozery-Flato and Shamir whether a subquadratic complexity could ever be achieved for solving the problem.