Sorting Signed Permutations by Reversal (Reversal Sequence)

Years and Authors of Summarized Original Work

• 2004; Tannier, Sagot

Problem Definition

A signed permutation$\uppi$ of size n is a permutation over $\{ - n,\ldots,-1,1\ldots n\}$, where $\uppi _{-i} = -\uppi _{i}$ for all i. We note $\uppi = (\uppi _{1},\ldots,\uppi _{n})$.

The reversal$\uprho =\uprho _{i,j}(1 \leq i \leq j \leq n)$ is an operation that reverses the order and flips the signs of the elements $\uppi _{i},\ldots,\uppi _{j}$ in a permutation $\uppi$:

$$\displaystyle\begin{array}{rcl} & & \uppi \cdot \rho {}\\ & & = (\uppi _{1},\ldots ,\uppi _{i-1},-\uppi _{j},\ldots,-\uppi _{i}\uppi _{j+1},\ldots,\uppi _{n}). {}\\ \end{array}$$

If $\uprho _{1},\ldots,\uprho _{k}$ is a sequence of reversals, it is said to sort a permutation $\uppi$ if $\uppi \cdots \uprho _{1}\cdots \uprho _{k} = Id$, where Id = (1, 2, …, n) is the identity permutation. The length of a shortest sequence of reversals sorting $\uppi$ is called the reversal distance of $\uppi$and is...