On a lower bound for sorting signed permutations by reversals

TL;DR

This paper proves the equivalence of two lower bounds for sorting signed permutations by reversals, confirms a related conjecture on skew-symmetric plane permutations, and interprets the lower bound in terms of topological genera.

Contribution

It establishes the equality of two different lower bounds for reversal distance and confirms a conjecture on skew-symmetric plane permutations, linking combinatorics and topology.

Findings

The two lower bounds for reversal distance are proven to be equal.

A conjecture on skew-symmetric plane permutations is confirmed.

The lower bound is interpreted as the topological genus of associated surfaces.

Abstract

Computing the reversal distances of signed permutations is an important topic in Bioinformatics. Recently, a new lower bound for the reversal distance was obtained via the plane permutation framework. This lower bound appears different from the existing lower bound obtained by Bafna and Pevzner through breakpoint graphs. In this paper, we prove that the two lower bounds are equal. Moreover, we confirm a related conjecture on skew-symmetric plane permutations, which can be restated as follows: let $p=(0,-1,-2,\ldots -n,n,n-1,\ldots 1)$ and let $$ \tilde{s}=(0,a_1,a_2,\ldots a_n,-a_n,-a_{n-1},\ldots -a_1) $$ be any long cycle on the set $\{-n,-n+1,\ldots 0,1,\ldots n\}$. Then, $n$ and $a_n$ are always in the same cycle of the product $p\tilde{s}$. Furthermore, we show the new lower bound via plane permutations can be interpreted as the topological genera of orientable surfaces associated to signed permutations.