Prefix and Suffix Reversals on Strings

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

The problem of sorting permutations by prefix and suffix rearrangements

“…The number in parenthesis in this table is the maximum difference between the prefix and suffix distance and the prefix distance. We note that Fertin et al [29] showed that, for infinitely many values of n, there is a permutation of size n such that the difference between the prefix reversal distance and the prefix and suffix reversal distance is Ω(n).…”

“…In this section we present an algorithm for SbPSR problem, which was recently proved to be NP-hard by Fertin et al [29]. The general idea of our algorithm is, while the permutation is not sorted: try to apply a 1-move or else a 1-move followed by a 0-move; if neither is possible, the permutation has a specific format that can be sorted with at most its number of breakpoint operations.…”

Ordenação de permutações com sinais por reversões e transposições

“…Considerando os eventos de reversão e transposição de prefixo ou sufixo, quase todas as variações possuem complexidade desconhecida, com exceção do problema de Ordenação de Permutações por Reversões de Prefixo ou Sufixo, que pertence à classe de problemas NP-Difíceis [18].…”