The solution space of sorting by reversals

In comparative genomics, algorithms that sort permutations by reversals are often used to propose evolutionary scenarios of large scale genomic mutations between species. One of the main problems of such methods is that they give one solution while the number of optimal solutions is huge, with no criteria to discriminate among them. Bergeron et al. [4] started to give some structure to the set of optimal solutions, in order to be able to deliver more presentable results than only one solution or a complete list of all solutions. The structure is a way to group solutions into equivalence classes, and to identify in each class one particular representative. However, no algorithm exists so far to compute this set of representatives except through the enumeration of all solutions, which takes too much time even for small permutations. Bergeron et al. [4] state as an open problem the design of such an algorithm. We propose in this paper an answer to this problem, that is, an algorithm which gives one representative for each class of solutions and counts the number of solutions in each class, with a better theoretical and practical complexity than the complete enumeration method. We give several biological examples where the result is more relevant than a unique optimal solution or the list of all solutions.