9,372 research outputs found
A lower bound on the reversal and transposition diameter
One possible model to study genome evolution is to represent genomes as permutations of genes and compute distances based on the minimum number of certain operations (re-arrangements) needed to transform one permutation into another. Under this model, the shorter the distance, the closer the genomes are. Two operations that have been extensively studied are the reversal and the transposition. A reversal is an operation that reverses the order of the genes on a certain portion of the permutation. A transposition is an operation that 'cuts' a certain portion of the permutation and 'pastes' it elsewhere in the same permutation. In this note, we show that the reversal and transposition distance of the signed permutation pi(n) = (-1 -2... -(n - 1) -n) with respect to the identity is [n/2] +2 for all n greater than or equal to 3. We conjecture that this value is the diameter of the permutation group under these operations.9574374
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
Lower bounding edit distances between permutations
International audienceA number of fields, including the study of genome rearrangements and the design of interconnection networks, deal with the connected problems of sorting permutations in "as few moves as possible", using a given set of allowed operations, or computing the number of moves the sorting process requires, often referred to as the distance of the permutation. These operations often act on just one or two segments of the permutation, e.g. by reversing one segment or exchanging two segments. The cycle graph of the permutation to sort is a fundamental tool in the theory of genome rearrangements, and has proved useful in settling the complexity of many variants of the above problems. In this paper, we present an algebraic reinterpretation of the cycle graph of a permutation π as an even permutation π, and show how to reformulate our sorting problems in terms of particular factorisations of the latter permutation. Using our framework, we recover known results in a simple and unified way, and obtain a new lower bound on the prefix transposition distance (where a prefix transposition displaces the initial segment of a permutation), which is shown to outperform previous results. Moreover, we use our approach to improve the best known lower bound on the prefix transposition diameter from 2n/3 to ⌊3n/4⌋, and investigate a few relations between some statistics on π and π
Diameter of Cayley graphs of permutation groups generated by transposition trees
Let be a Cayley graph of the permutation group generated by a
transposition tree on vertices. In an oft-cited paper
\cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is
shown that the diameter of the Cayley graph is bounded as
\diam(\Gamma) \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n
\dist_T(i,\pi(i))}, where the maximization is over all permutations ,
denotes the number of cycles in , and \dist_T is the distance
function in . In this work, we first assess the performance (the sharpness
and strictness) of this upper bound. We show that the upper bound is sharp for
all trees of maximum diameter and also for all trees of minimum diameter, and
we exhibit some families of trees for which the bound is strict. We then show
that for every , there exists a tree on vertices, such that the
difference between the upper bound and the true diameter value is at least
.
Observe that evaluating this upper bound requires on the order of (times
a polynomial) computations. We provide an algorithm that obtains an estimate of
the diameter, but which requires only on the order of (polynomial in)
computations; furthermore, the value obtained by our algorithm is less than or
equal to the previously known diameter upper bound. This result is possible
because our algorithm works directly with the transposition tree on
vertices and does not require examining any of the permutations (only the proof
requires examining the permutations). For all families of trees examined so
far, the value computed by our algorithm happens to also be an upper
bound on the diameter, i.e.
\diam(\Gamma) \le \beta \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n
\dist_T(i,\pi(i))}.Comment: This is an extension of arXiv:1106.535
Linear Transformations Between Colorings in Chordal Graphs
Let k and d be such that k >= d+2. Consider two k-colorings of a d-degenerate graph G. Can we transform one into the other by recoloring one vertex at each step while maintaining a proper coloring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length.
If k=d+2, we know that there exists graphs for which a quadratic number of recolorings is needed. And when k=2d+2, there always exists a linear transformation. In this paper, we prove that, as long as k >= d+4, there exists a transformation of length at most f(Delta) * n between any pair of k-colorings of chordal graphs (where Delta denotes the maximum degree of the graph). The proof is constructive and provides a linear time algorithm that, given two k-colorings c_1,c_2 computes a linear transformation between c_1 and c_2
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