SIAM Journal on Discrete Mathematics
Transforming cabbage into turnip: polynomial algorithm for sorting signed permutations by reversals
Journal of the ACM (JACM)
A Faster and Simpler Algorithm for Sorting Signed Permutations by Reversals
SIAM Journal on Computing
Genome Rearrangements and Sorting by Reversals
SIAM Journal on Computing
A Very Elementary Presentation of the Hannenhalli-Pevzner Theory
CPM '01 Proceedings of the 12th Annual Symposium on Combinatorial Pattern Matching
CPM '96 Proceedings of the 7th Annual Symposium on Combinatorial Pattern Matching
A simpler and faster 1.5-approximation algorithm for sorting by transpositions
Information and Computation
Efficient data structures and a new randomized approach for sorting signed permutations by reversals
CPM'03 Proceedings of the 14th annual conference on Combinatorial pattern matching
A 1.375-approximation algorithm for sorting by transpositions
WABI'05 Proceedings of the 5th International conference on Algorithms in Bioinformatics
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We study the problems of sorting signed permutations by reversals (SBR) and sorting unsigned permutations by transpositions (SBT), which are central problems in computational molecular biology. While a polynomial-time solution for SBR is known, the computational complexity of SBT has been open for more than a decade and is considered a major open problem. In the first efficient solution of SBR, Hannenhalli and Pevzner [HP99] used a graph-theoretic model for representing permutations, called the interleaving graph. This model was crucial to their solution. Here, we define a new model for SBT, which is analogous to the interleaving graph. Our model has some desirable properties that were lacking in earlier models for SBT. These properties make it extremely useful for studying SBT. Using this model, we give a linear-algebraic framework in which SBT can be studied. Specifically, for matrices over any algebraic ring, we define a class of matrices called tight matrices. We show that an efficient algorithm which recognizes tight matrices over a certain ring, $\mathbb{M}$, implies an efficient algorithm that solves SBT on an important class of permutations, called simple permutations. Such an algorithm is likely to lead to an efficient algorithm for SBT that works on all permutations. The problem of recognizing tight matrices is also a generalization of SBR and of a large class of other “sorting by rearrangements” problems, and seems interesting in its own right as. We give an efficient algorithm for recognizing tight symmetric matrices over any field of characteristic 2. We leave as an open problem to find an efficient algorithm for recognizing tight matrices over the ring $\mathbb{M}$.