No abstract

International audienceThe Sorting by Prefix Reversals problem consists insorting the elements of a given permutation π with a minimum numberof prefix reversals, i.e. reversals that always imply the leftmost elementof π. A natural extension of this problem is to consider strings (inwhich any letter may appear several times) rather than permutations. Instrings, three different types of problems arise: grouping (starting from astring S, transform it so that all identical letters are consecutive), sorting(a constrained version of grouping, in which the target string must belexicographically ordered) and rearranging (given two strings S and T,transform S into T). In this paper, we study these three problems, underan algorithmic viewpoint, in the setting where two operations (ratherthan one) are allowed: namely, prefix and suffix reversals - where a suffixreversal must always imply the rightmost element of the string. Wefirst give elements of comparison between the “prefix reversals only” caseand our case. The algorithmic results we obtain on these three problemsdepend on the size k of the alphabet on which the strings are built. Inparticular, we show that the grouping problem is in P for k ∈ [2; 4] andwhen n − k = O(1), where n is the length of the string. We also showthat the grouping problem admits a PTAS for any constant k, and is2-approximable for any k. Concerning sorting, it is in P for k ∈ [2; 3],admits a PTAS for constant k, and is NP-hard for k = n. Finally, concerningthe rearranging problem, we show that it is NP-hard, both fork = O(1) and k = n. We also show that the three problems are FPTwhen the parameter is the maximum number of blocks over the sourceand target strings