An algorithm to enumerate all sorting reversals

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Abstract
The problem of estimating evolutionary distance from differences in gene order has been distilled to the problem of finding the reversal distance between two signed permutations. During the last decade, much progress was made both in computing reversal distance and in finding a minimum sequence of sorting reversals. For most problem instances, however, many minimum sequences of sorting reversals exist, and obtaining the complete set can be useful in exploring the space of genome rearrangements (e.g., in pursuit of solutions to higher-level problems). The problem of finding all minimum sequences of sorting reversals reduces easily to the problem of finding all sorting reversals of one permutation with respect to another. We derive an efficient algorithm to solve this latter problem, and present experimental results indicating that our algorithm offers a dramatic improvement over the best known alternative. It should be noted that in asymptotic terms the new algorithm does not represent a significant improvement: it requires O(n3) time (where n is the permutation size), while the problem can now be solved trivially in &THgr;(n3) time.

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