Abstract: During the last years, the genomes of more and more
species have been sequenced, providing data for phylogenetic recon-
struction based on genome rearrangement measures. A main task in
all phylogenetic reconstruction algorithms is to solve the median of
three problem. Although this problem is NP-hard even for the sim-
plest distance measures, there are exact algorithms for the breakpoint
median and the reversal median that are fast enough for practical use.
In this paper, this approach is extended to the transposition median as
well as to the weighted reversal and transposition median. Although
there is no exact polynomial algorithm known even for the pairwise
distances, we will show that it is in most cases possible to solve
these problems exactly within reasonable time by using a branch and
bound algorithm.
Abstract: Star graphs are Cayley graphs of symmetric groups of permutations, with transpositions as the generating sets. A star graph is a preferred interconnection network topology to a hypercube for its ability to connect a greater number of nodes with lower degree. However, an attractive property of the hypercube is that it has a Hamiltonian decomposition, i.e. its edges can be partitioned into disjoint Hamiltonian cycles, and therefore a simple routing can be found in the case of an edge failure. The existence of Hamiltonian cycles in Cayley graphs has been known for some time. So far, there are no published results on the much stronger condition of the existence of Hamiltonian decompositions. In this paper, we give a construction of a Hamiltonian decomposition of the star graph 5-star of degree 4, by defining an automorphism for 5-star and a Hamiltonian cycle which is edge-disjoint with its image under the automorphism.
Abstract: Bioinformatics and computational biology involve
the use of techniques including applied mathematics,
informatics, statistics, computer science, artificial intelligence,
chemistry, and biochemistry to solve biological problems
usually on the molecular level. Research in computational
biology often overlaps with systems biology. Major research
efforts in the field include sequence alignment, gene finding,
genome assembly, protein structure alignment, protein structure
prediction, prediction of gene expression and proteinprotein
interactions, and the modeling of evolution. Various
global rearrangements of permutations, such as reversals and
transpositions,have recently become of interest because of their
applications in computational molecular biology. A reversal is
an operation that reverses the order of a substring of a permutation.
A transposition is an operation that swaps two adjacent
substrings of a permutation. The problem of determining the
smallest number of reversals required to transform a given
permutation into the identity permutation is called sorting by
reversals. Similar problems can be defined for transpositions
and other global rearrangements. In this work we perform a
study about some genome rearrangement primitives. We show
how a genome is modelled by a permutation, introduce some
of the existing primitives and the lower and upper bounds
on them. We then provide a comparison of the introduced
primitives.