Abstract
Understanding how different two organisms are is one question addressed by the comparative genomics field. A well-accepted way to estimate the evolutionary distance between genomes of two organisms is finding the rearrangement distance, which is the smallest number of rearrangements needed to transform one genome into another. By representing genomes as permutations, one of them can be represented 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 rearrangements. This work investigates the problems of sorting permutations using reversals and/or transpositions, with some additional restrictions of biological relevance. Given a value λ, the problem now is how to sort a λ-permutation, which is a permutation whose elements are less than λ positions away from their correct places (regarding the identity), by applying the minimum number of rearrangements. Each λ-rearrangement must have size, at most, λ, and, when applied to a λ-permutation, the result should also be a λ-permutation. We present algorithms with approximation factors of O(λ2), O(λ), and O(1) for the problems of Sorting λ-Permutations by λ-Reversals, by λ-Transpositions, and by both operations.
Highlights
One challenge in biology is to understand how species evolve, considering that new organisms arise from mutations that occurred in others
Using the principle of parsimony, the minimum number of rearrangements that transform one genome into another, called rearrangement distance, is a widely adopted way to estimate the evolutionary distance between two genomes
We can achieve an approximation factor of O(λ2) with algorithms that always apply an operation of size 2 according to Lemma 3, for unsigned permutations, or Lemma 5, for signed permutations
Summary
One challenge in biology is to understand how species evolve, considering that new organisms arise from mutations that occurred in others. Considering a size limit of 2, the problems of Sorting Unsigned Permutations by Reversals and/or Transpositions are solvable in polynomial time [13]. Considering a size limit of 3, the best known approximation factors for Sorting Unsigned Permutations by Reversals, by Transpositions, and by Reversals and Transpositions are 2 [13], 5/4 [14], and 2 [15], respectively.
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