A new upper bound for sorting permutations with prefix transpositions

Abstract

Permutations are discrete structures that naturally model a genome where every gene occurs exactly once. In a permutation over the given alphabet [Formula: see text], each symbol of [Formula: see text] appears exactly once. A transposition operation on a given permutation [Formula: see text] exchanges two adjacent sublists of [Formula: see text]. If one of these sublists is restricted to be a prefix then one obtains a prefix transposition. The symmetric group of permutations with [Formula: see text] symbols derived from the alphabet [Formula: see text] is denoted by [Formula: see text]. The symmetric prefix transposition distance between [Formula: see text] and [Formula: see text] is the minimum number of prefix transpositions that are needed to transform [Formula: see text] into [Formula: see text]. It is known that transforming an arbitrary [Formula: see text] into an arbitrary [Formula: see text] is equivalent to sorting some [Formula: see text]. Thus, upper bound for transforming any [Formula: see text] into any [Formula: see text] with prefix transpositions is simply the upper bound to sort any permutation [Formula: see text]. The current upper bound is [Formula: see text] for prefix transposition distance over [Formula: see text]. In this paper, we improve the same to [Formula: see text].