Lower bounding edit distances between permutations

TL;DR

This paper introduces an algebraic approach to permutation sorting problems, providing new lower bounds on edit distances and diameters relevant to genome rearrangements and network design.

Contribution

It offers a novel algebraic reinterpretation of the cycle graph as an even permutation and uses this to unify existing results and improve lower bounds on prefix transposition metrics.

Findings

New lower bound on prefix transposition distance outperforms previous results.

Improved lower bound on prefix transposition diameter from 2n/3 to 3n/4.

Unified algebraic framework simplifies analysis of permutation sorting problems.

Abstract

A number of fields, including the study of genome rearrangements and the design of interconnection networks, deal with the connected problems of sorting permutations in "as few moves as possible", using a given set of allowed operations, or computing the number of moves the sorting process requires, often referred to as the \emph{distance} of the permutation. These operations often act on just one or two segments of the permutation, e.g. by reversing one segment or exchanging two segments. The \emph{cycle graph} of the permutation to sort is a fundamental tool in the theory of genome rearrangements, and has proved useful in settling the complexity of many variants of the above problems. In this paper, we present an algebraic reinterpretation of the cycle graph of a permutation $\pi$ as an even permutation $\bar{\pi}$, and show how to reformulate our sorting problems in terms of particular factorisations of the latter permutation. Using our framework, we recover known results in a simple and unified way, and obtain a new lower bound on the \emph{prefix transposition distance} (where a \emph{prefix transposition} displaces the initial segment of a permutation), which is shown to outperform previous results. Moreover, we use our approach to improve the best known lower bound on the \emph{prefix transposition diameter} from $2n/3$ to $\lfloor3n/4\rfloor$, and investigate a few relations between some statistics on $\pi$ and $\bar{\pi}$.