Prefix and suffix reversals on strings

The Sorting by Prefix Reversals problem consists in sorting the elements of a given permutation π using a minimum number of prefix reversals, i.e. reversals that always involve the leftmost element of π. A natural extension of this problem is to consider strings, in which any letter may appear several times, rather than permutations. In strings, three different types of problems arise: grouping (given a string S, transform it so that all identical letters are consecutive), sorting (a constrained version of grouping, in which the target string must be lexicographically ordered) and rearranging (given two strings S and T, transform S into T). In this paper, we study these three problems, under an algorithmic viewpoint, in the setting where two operations, rather than one, are allowed: namely, prefix and suffix reversals - where a suffix reversal must always involve the rightmost element of the string. We first compare the “prefix reversals only” case to our case, before presenting a series of algorithmic results on these three problems concerning polynomiality, constant ratio approximation algorithms, NP-hardness and fixed-parameterized tractability. These results depend on the size k of the alphabet on which the strings are built, with a particular focus on small-sized alphabet instances (i.e., k=O(1)) and big-sized alphabet instances (i.e. n−k=O(1), where n is the length of the input string(s)).