53 research outputs found
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
Degenerate crossing number and signed reversal distance
The degenerate crossing number of a graph is the minimum number of transverse
crossings among all its drawings, where edges are represented as simple arcs
and multiple edges passing through the same point are counted as a single
crossing. Interpreting each crossing as a cross-cap induces an embedding into a
non-orientable surface. In 2007, Mohar showed that the degenerate crossing
number of a graph is at most its non-orientable genus and he conjectured that
these quantities are equal for every graph. He also made the stronger
conjecture that this also holds for any loopless pseudotriangulation with a
fixed embedding scheme.
In this paper, we prove a structure theorem that almost completely classifies
the loopless 2-vertex embedding schemes for which the degenerate crossing
number equals the non-orientable genus. In particular, we provide a
counterexample to Mohar's stronger conjecture, but show that in the vast
majority of the 2-vertex cases, the conjecture does hold.
The reversal distance between two signed permutations is the minimum number
of reversals that transform one permutation to the other one. If we represent
the trajectory of each element of a signed permutation under successive
reversals by a simple arc, we obtain a drawing of a 2-vertex embedding scheme
with degenerate crossings. Our main result is proved by leveraging this
connection and a classical result in genome rearrangement (the
Hannenhali-Pevzner algorithm) and can also be understood as an extension of
this algorithm when the reversals do not necessarily happen in a monotone
order.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023
Genome dedoubling by DCJ and reversal
<p>Abstract</p> <p>Background</p> <p>Segmental duplications in genomes have been studied for many years. Recently, several studies have highlighted a biological phenomenon called <it>breakpoint-duplication</it> that apparently associates a significant proportion of segmental duplications in Mammals, and the Drosophila species group, to breakpoints in rearrangement events.</p> <p>Results</p> <p>In this paper, we introduce and study a combinatorial problem, inspired from the breakpoint-duplication phenomenon, called the <it>Genome Dedoubling Problem.</it> It consists of finding a minimum length rearrangement scenario required to transform a genome with duplicated segments into a non-duplicated genome such that duplications are caused by rearrangement breakpoints. We show that the problem, in the Double-Cut-and-Join (DCJ) and the reversal rearrangement models, can be reduced to an APX-complete problem, and we provide algorithms for the Genome Dedoubling Problem with 2-approximable parts. We apply the methods for the reconstruction of a non-duplicated ancestor of <it>Drosophila yakuba.</it></p> <p>Conclusions</p> <p>We present the <it>Genome Dedoubling Problem</it>, and describe two algorithms solving the problem in the DCJ model, and the reversal model. The usefulness of the problems and the methods are showed through an application to real Drosophila data.</p
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