150 research outputs found
A barrier for further approximating Sorting By Transpositions
The Transposition Distance Problem (TDP) is a classical problem in genome
rearrangements which seeks to determine the minimum number of transpositions
needed to transform a linear chromosome into another represented by the
permutations and , respectively. This paper focuses on the
equivalent problem of Sorting By Transpositions (SBT), where is the
identity permutation . Specifically, we investigate palisades, a family
of permutations that are "hard" to sort, as they require numerous
transpositions above the celebrated lower bound devised by Bafna and Pevzner.
By determining the transposition distance of palisades, we were able to provide
the exact transposition diameter for -permutations (TD3), a special subset
of the Symmetric Group , essential for the study of approximate solutions
for SBT using the simplification technique. The exact value for TD3 has
remained unknown since Elias and Hartman showed an upper bound for it. Another
consequence of determining the transposition distance of palisades is that,
using as lower bound the one by Bafna and Pevzner, it is impossible to
guarantee approximation ratios lower than when approximating SBT. This
finding has significant implications for the study of SBT, as this problem has
been subject of intense research efforts for the past 25 years
Ordering Metro Lines by Block Crossings
A problem that arises in drawings of transportation networks is to minimize
the number of crossings between different transportation lines. While this can
be done efficiently under specific constraints, not all solutions are visually
equivalent. We suggest merging crossings into block crossings, that is,
crossings of two neighboring groups of consecutive lines. Unfortunately,
minimizing the total number of block crossings is NP-hard even for very simple
graphs. We give approximation algorithms for special classes of graphs and an
asymptotically worst-case optimal algorithm for block crossings on general
graphs. That is, we bound the number of block crossings that our algorithm
needs and construct worst-case instances on which the number of block crossings
that is necessary in any solution is asymptotically the same as our bound
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
Sorting by Block Moves
The research in this thesis is focused on the problem of Block Sorting, which has applications in Computational Biology and in Optical Character Recognition (OCR). A block in a permutation is a maximal sequence of consecutive elements that are also consecutive in the identity permutation. BLOCK SORTING is the process of transforming an arbitrary permutation to the identity permutation through a sequence of block moves. Given an arbitrary permutation π and an integer m, the Block Sorting Problem, or the problem of deciding whether the transformation can be accomplished in at most m block moves has been shown to be NP-hard. After being known to be 3-approximable for over a decade, block sorting has been researched extensively and now there are several 2-approximation algorithms for its solution. This work introduces new structures on a permutation, which are called runs and ordered pairs, and are used to develop two new approximation algorithms. Both the new algorithms are 2-approximation algorithms, yielding the approximation ratio equal to the current best. This work also includes an analysis of both the new algorithms showing they are 2-approximation algorithms
Are There Rearrangement Hotspots in the Human Genome?
In a landmark paper, Nadeau and Taylor [18] formulated the random breakage model (RBM) of chromosome evolution that postulates that there are no rearrangement hotspots in the human genome. In the next two decades, numerous studies with progressively increasing levels of resolution made RBM the de facto theory of chromosome evolution. Despite the fact that RBM had prophetic prediction power, it was recently refuted by Pevzner and Tesler [4], who introduced the fragile breakage model (FBM), postulating that the human genome is a mosaic of solid regions (with low propensity for rearrangements) and fragile regions (rearrangement hotspots). However, the rebuttal of RBM caused a controversy and led to a split among researchers studying genome evolution. In particular, it remains unclear whether some complex rearrangements (e.g., transpositions) can create an appearance of rearrangement hotspots. We contribute to the ongoing debate by analyzing multi-break rearrangements that break a genome into multiple fragments and further glue them together in a new order. In particular, we demonstrate that (1) even if transpositions were a dominant force in mammalian evolution, the arguments in favor of FBM still stand, and (2) the ‘‘gene deletion’’ argument against FBM is flawed
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