1,665 research outputs found
Exact and Approximation Algorithms for Computing Reversal Distances in Genome Rearrangement
Genome rearrangement is a research area capturing wide attention in molecular biology. The reversal distance problem is one of the most widely studied models of genome rearrangements in inferring the evolutionary relationship between two genomes at chromosome level. The problem of estimating reversal distance between two genomes is modeled as sorting by reversals. While the problem of sorting signed permutations can have polynomial time solutions, the problem of sorting unsigned permutations has been proven to be NP-hard [4]. This work introduces an exact greedy algorithm for sorting by reversals focusing on unsigned permutations. An improved method of producing cycle decompositions for a 3/2-approximation algorithm and the consideration of 3-cycles for reversal sequences are also presented in this paper
Polynomial-time sortable stacks of burnt pancakes
Pancake flipping, a famous open problem in computer science, can be
formalised as the problem of sorting a permutation of positive integers using
as few prefix reversals as possible. In that context, a prefix reversal of
length k reverses the order of the first k elements of the permutation. The
burnt variant of pancake flipping involves permutations of signed integers, and
reversals in that case not only reverse the order of elements but also invert
their signs. Although three decades have now passed since the first works on
these problems, neither their computational complexity nor the maximal number
of prefix reversals needed to sort a permutation is yet known. In this work, we
prove a new lower bound for sorting burnt pancakes, and show that an important
class of permutations, known as "simple permutations", can be optimally sorted
in polynomial time.Comment: Accepted pending minor revisio
The distribution of cycles in breakpoint graphs of signed permutations
Breakpoint graphs are ubiquitous structures in the field of genome
rearrangements. Their cycle decomposition has proved useful in computing and
bounding many measures of (dis)similarity between genomes, and studying the
distribution of those cycles is therefore critical to gaining insight on the
distributions of the genomic distances that rely on it. We extend here the work
initiated by Doignon and Labarre, who enumerated unsigned permutations whose
breakpoint graph contains cycles, to signed permutations, and prove
explicit formulas for computing the expected value and the variance of the
corresponding distributions, both in the unsigned case and in the signed case.
We also compare these distributions to those of several well-studied distances,
emphasising the cases where approximations obtained in this way stand out.
Finally, we show how our results can be used to derive simpler proofs of other
previously known results
Average-case analysis of perfect sorting by reversals (Journal Version)
Perfect sorting by reversals, a problem originating in computational
genomics, is the process of sorting a signed permutation to either the identity
or to the reversed identity permutation, by a sequence of reversals that do not
break any common interval. B\'erard et al. (2007) make use of strong interval
trees to describe an algorithm for sorting signed permutations by reversals.
Combinatorial properties of this family of trees are essential to the algorithm
analysis. Here, we use the expected value of certain tree parameters to prove
that the average run-time of the algorithm is at worst, polynomial, and
additionally, for sufficiently long permutations, the sorting algorithm runs in
polynomial time with probability one. Furthermore, our analysis of the subclass
of commuting scenarios yields precise results on the average length of a
reversal, and the average number of reversals.Comment: A preliminary version of this work appeared in the proceedings of
Combinatorial Pattern Matching (CPM) 2009. See arXiv:0901.2847; Discrete
Mathematics, Algorithms and Applications, vol. 3(3), 201
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