Sorting Signed Permutations by Reversal (Reversal Sequence)

Keywords and Synonyms

Sorting by inversions          

Problem Definition

A signed permutation π of size n is a permutation over ${ \{-n,\dots,-1,1\dots n\} }$, where ${ \pi_{-i}=-\pi_i }$ for all i.

The reversal ${ \rho=\rho_{i,j} }$ (${ 1\le i\le j\le n }$) is an operation that reverses the order and flips the signs of the elements ${ \pi_i,\dots,\pi_j }$ in a permutation π:

$$ \pi\cdot\rho = (\pi_{1},\dots,\pi_{i-1},-\pi_{j},\dots,-\pi_{i},\pi_{j+1},\dots,\pi_{n})\;. $$

If ${ \rho_1,\dots,\rho_k }$ is a sequence of reversals, it is said to sort a permutation π if ${ \pi \cdot \rho_1 \cdots \rho_k = Id }$, where ${ Id=(1,\dots,n) }$ is the identity permutation. The length of a shortest sequence of reversals sorting π is called the reversal distance of π, and is denoted by d(π).

If the computation of d(π) is solved in linear time [2] (see the entry “reversal distance”), the computation of a sequence of size d(π) that sorts π is more complicated and no linear...