Group Theory Viewpoint

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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 10.2 is a special case of the following general problem:

$\begin{problem} $\;$\\ {\bf INPUT:} Two permutations $\pi_1,\pi_2 \in S_n$ , an... ...product of generators that transforms $\pi_1$\space into $\pi_2$ ? \end{problem}$

Even and Goldreich showed that this problem is NP-Hard [9]. Jerrum generalized this result by proving it PSPACE-complete [14].

$\begin{problem} $\;$\\ {\bf INPUT:} A set $\{g_1,\ldots,g_k\}$\space of generat... ...e {\em diameter} is the longest distance between two permutations? \end{problem}$

Gates and Papadimitriou have shown [10] that by using only prefix reversals as generators, the diameter can be bounded by $\frac{17}{16} n \leq diameter \leq \frac{5}{3}n+\frac{5}{3}$ .

Peer Itsik
2001-01-17