Sorting λ-Permutations by λ-Operations

The understanding of how different two organisms are is one of the challenging tasks of modern science. A well accepted way to estimate the evolutionary distance between two organisms is estimating the rearrangement distance, which is the smallest number of rearrangements needed to transform one genome into another. If we represent genomes as permutations, we can represent one as the identity permutation and so we reduce the problem of transforming one permutation into another to the problem of sorting a permutation using the minimum number of operations. In this work, we study the problems of sorting permutations using reversals and/or transpositions, with some additional restrictions of biological relevance. Given a value lambda, the problem now is how to sort a lambda-permutation, which is a permutation where all elements are less than lambda positions away from their correct places (regarding the identity), by applying the minimum number of operations. Each lambda-operation must have size at most. and, when applied over a lambda-permutation, the result should also be a lambda-permutation. We present algorithms with approximation factors of O(lambda(2)), O(lambda), and O(1) for the problems of Sorting lambda-Permutations by lambda-Reversals, by lambda-Transpositions and by both operations.