Group Theory Viewpoint

From a group theory point of view, the sorting of signed permutations can be viewed as follows: Consider S_n, the symmetric group (group of all permutations) on n elements. The set $\{\rho(i,j)\}$ of all possible reversals is a set of generators of S_n, Therefore, from the group theory point of view, problem 11.5 is a special case of the following general problem:

Problem 11.6   $\;$
INPUT: Two permutations $\pi_1,\pi_2 \in S_n$, and a set $\{g_1,\ldots,g_k\}$ of generators.
QUESTION: What is their distance, i.e.,what is the shortest product of generators that transforms $\pi_1$ into $\pi_2$ ?

Even and Goldreich have shown that this problem is NP-Hard [10]. Jerrum Has showed that this problem is also PSPACE-complete [15].

Problem 11.7   $\;$
INPUT: A set $\{g_1,\ldots,g_k\}$ of generators.
QUESTION: What is the diameter of S_n, where the diameter is the longest distance between two permutations.

Gates and Papadimitriou have shown [11] that by using only prefix reversals as generators, the diameter can be bounded by $\frac{17}{16} n \leq diameter \leq \frac{5}{3}n+\frac{5}{3}$.