Perfect Sorting by Reversals Is Not Always Difficult

Abstract

Abstract. This paper investigates the problem of conservation of combinatorial structures in genome rearrangement scenarios. We characterize a class of signed permutations for which one can compute in polynomial time a reversal scenario that conserves all common intervals, and that is parsimonious among such scenarios. The general problem is believed to be NP-hard. We show that there exists a class of permutations for which this computation can be done in linear time with a very simple algorithm, and, for a larger class of signed permutations, the computation can be achieved in subquadratic time. We apply these methods permutations obtained from the X chromosomes of the human, mouse and rat.

A more efficient algorithm for perfect sorting by reversals

“…Our main result in this note is an algorithm that computes a parsimonious perfect scenario for a signed permutation more efficiently that the algorithm described in [1]. A very similar use of the notion of positive, negative and neutral permutations was used in [8], but in a framework that did not take advantage of the structure provided by the strong interval tree, in particular to propagate signs.…”

“…It was shown in [1] that, for a given signed permutation P , if T S (P ) is unambiguous, then Algorithm 1 below computes a parsimonious perfect scenario for P in worst-case time O(n n log(n)), while Algorithm 2 can handle the case where T S (P ) is ambiguous in O(2 p n n log(n)) worst-case time, where p is the number of unsigned prime vertices in T S (P ).…”

“…We first describe the algorithms given in [1] to compute a perfect scenario for a given signed permutation P (Algorithms 1 and 2). The basis of the algorithm is to give signs, + or −, to the vertices of T S (P ) using the following set of rules:…”

“…The correctness of Algorithm 2 (see [1,Theorem 4]) induces the following obvious lemma, that we state for the sake of completeness, as it will serve as a basis for the invariant of the algorithm we describe in Section 3.2.…”

“…It was latter improved in [1] by showing that the problem is fixed parameterized tractable [6] (FPT): i.e. the complexity function is f (p).n O(1) for some parameter p. The parameter p proposed in [1] is the number p of prime edges of the strong interval tree of the signed permutation to be sorted (see definition in Section 2). The algorithm proposed in this note is still FPT, but the parameter we use is always smaller than or equal to the number of prime edges.…”