A simple framework on sorting permutations

TL;DR

This paper introduces a unified framework for analyzing permutation distance problems relevant to genome evolution, providing new lower bounds and connecting various permutation metrics.

Contribution

It offers a general formulation for lower bounds on permutation distances and links reversal distance of signed permutations to block-interchange distance, advancing the theoretical understanding.

Findings

Derived a general lower bound framework for transposition and block-interchange distances.

Translated reversal distance of signed permutations into a block-interchange problem, obtaining a new lower bound.

Identified new combinatorial problems related to permutation products inspired by the framework.

Abstract

In this paper we present a simple framework to study various distance problems of permutations, including the transposition and block-interchange distance of permutations as well as the reversal distance of signed permutations. These problems are very important in the study of the evolution of genomes. We give a general formulation for lower bounds of the transposition and block-interchange distance from which the existing lower bounds obtained by Bafna and Pevzner, and Christie can be easily derived. As to the reversal distance of signed permutations, we translate it into a block-interchange distance problem of permutations so that we obtain a new lower bound. Furthermore, studying distance problems via our framework motivates several interesting combinatorial problems related to product of permutations, some of which are studied in this paper as well.