Sorting Signed Permutations by Reversals in Nearly-Linear Time

TL;DR

This paper presents a nearly-linear time algorithm for sorting signed permutations by reversals, significantly improving the efficiency over previous methods and leveraging dynamic graph connectivity techniques.

Contribution

It introduces a novel nearly-linear time algorithm for sorting signed permutations by reversals, advancing computational genomics methods.

Findings

Achieves $ ext{O}(n ext{ log}^2 n / ext{log log} n)$ time complexity

Leverages dynamic graph connectivity results for algorithm design

Improves upon previous polynomial-time algorithms

Abstract

Given a signed permutation on $n$ elements, we need to sort it with the fewest reversals. This is a fundamental algorithmic problem motivated by applications in comparative genomics, as it allows to accurately model rearrangements in small genomes. The first polynomial-time algorithm was given in the foundational work of Hannenhalli and Pevzner [J. ACM'99]. Their approach was later streamlined and simplified by Kaplan, Shamir, and Tarjan [SIAM J. Comput.'99] and their framework has eventually led to an algorithm that works in $\mathcal{O}(n^{3/2}\sqrt{\log n})$ time given by Tannier, Bergeron, and Sagot [Discr. Appl. Math.'07]. However, the challenge of finding a nearly-linear time algorithm remained unresolved. In this paper, we show how to leverage the results on dynamic graph connectivity to obtain a surprisingly simple $\mathcal{O}(n \log^2 n / \log \log n)$ time algorithm for this problem.