Counting linear extensions is #P-complete
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Transforming cabbage into turnip: polynomial algorithm for sorting signed permutations by reversals
Journal of the ACM (JACM)
The Book of Traces
WABI '02 Proceedings of the Second International Workshop on Algorithms in Bioinformatics
Advances on sorting by reversals
Discrete Applied Mathematics
Perfect Sorting by Reversals Is Not Always Difficult
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
Evolution under Reversals: Parsimony and Conservation of Common Intervals
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
Algorithmic approaches for genome rearrangement: a review
IEEE Transactions on Systems, Man, and Cybernetics, Part C: Applications and Reviews
Minimum Common String Partition Parameterized
WABI '08 Proceedings of the 8th international workshop on Algorithms in Bioinformatics
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
Listing all sorting reversals in quadratic time
WABI'10 Proceedings of the 10th international conference on Algorithms in bioinformatics
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
Listing all parsimonious reversal sequences: new algorithms and perspectives
RECOMB-CG'10 Proceedings of the 2010 international conference on Comparative genomics
Partial enumeration of solutions traces for the problem of sorting by signed reversals
Proceedings of the 2nd ACM Conference on Bioinformatics, Computational Biology and Biomedicine
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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.