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)
The solution space of sorting by reversals
ISBRA'07 Proceedings of the 3rd international conference on Bioinformatics research and applications
Conservation of combinatorial structures in evolution scenarios
RCG'04 Proceedings of the 2004 RECOMB international conference on Comparative Genomics
Algorithmic approaches for genome rearrangement: a review
IEEE Transactions on Systems, Man, and Cybernetics, Part C: Applications and Reviews
RECOMB-CG '08 Proceedings of the international workshop on Comparative Genomics
Swarming along the evolutionary branches sheds light on genome rearrangement scenarios
Proceedings of the 11th Annual conference on Genetic and evolutionary computation
Finding All Sorting Tandem Duplication Random Loss Operations
CPM '09 Proceedings of the 20th Annual Symposium on Combinatorial Pattern Matching
Parking Functions, Labeled Trees and DCJ Sorting Scenarios
RECOMB-CG '09 Proceedings of the International Workshop on Comparative Genomics
Counting All DCJ Sorting Scenarios
RECOMB-CG '09 Proceedings of the International Workshop on Comparative Genomics
Chronological order of reversal events on Rickettsia genus
ISB '10 Proceedings of the International Symposium on Biocomputing
An improved algorithm to enumerate all traces that sort a signed permutation by reversals
Proceedings of the 2010 ACM Symposium on Applied Computing
Finding all sorting tandem duplication random loss operations
Journal of Discrete Algorithms
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
Approximating the number of Double Cut-and-Join scenarios
Theoretical Computer Science
Listing Sorting Sequences of Reversals and Translocations
Proceedings of the International Conference on Bioinformatics, Computational Biology and Biomedical Informatics
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In comparative genomics, algorithms that sort permutations by reversals are often used to propose evolutionary scenarios of rearrangements 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. 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. However, no algorithm exists so far to compute this structure except through the enumeration of all solutions, which takes too much time even for small permutations. Bergeron et al. 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 all the classes 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 an example of how to reduce the number of classes obtained, using further constraints. Finally, we apply our algorithm to analyse the possible scenarios of rearrangement between mammalian sex chromosomes.