510 research outputs found
Uniqueness, intractability and exact algorithms: reflections on level-k phylogenetic networks
Phylogenetic networks provide a way to describe and visualize evolutionary
histories that have undergone so-called reticulate evolutionary events such as
recombination, hybridization or horizontal gene transfer. The level k of a
network determines how non-treelike the evolution can be, with level-0 networks
being trees. We study the problem of constructing level-k phylogenetic networks
from triplets, i.e. phylogenetic trees for three leaves (taxa). We give, for
each k, a level-k network that is uniquely defined by its triplets. We
demonstrate the applicability of this result by using it to prove that (1) for
all k of at least one it is NP-hard to construct a level-k network consistent
with all input triplets, and (2) for all k it is NP-hard to construct a level-k
network consistent with a maximum number of input triplets, even when the input
is dense. As a response to this intractability we give an exact algorithm for
constructing level-1 networks consistent with a maximum number of input
triplets
When two trees go to war
Rooted phylogenetic networks are often constructed by combining trees,
clusters, triplets or characters into a single network that in some
well-defined sense simultaneously represents them all. We review these four
models and investigate how they are related. In general, the model chosen
influences the minimum number of reticulation events required. However, when
one obtains the input data from two binary trees, we show that the minimum
number of reticulations is independent of the model. The number of
reticulations necessary to represent the trees, triplets, clusters (in the
softwired sense) and characters (with unrestricted multiple crossover
recombination) are all equal. Furthermore, we show that these results also hold
when not the number of reticulations but the level of the constructed network
is minimised. We use these unification results to settle several complexity
questions that have been open in the field for some time. We also give explicit
examples to show that already for data obtained from three binary trees the
models begin to diverge
Constructing level-2 phylogenetic networks from triplets
Jansson and Sung showed that, given a dense set of input triplets T
(representing hypotheses about the local evolutionary relationships of triplets
of species), it is possible to determine in polynomial time whether there
exists a level-1 network consistent with T, and if so to construct such a
network. They also showed that, unlike in the case of trees (i.e. level-0
networks), the problem becomes NP-hard when the input is non-dense. Here we
further extend this work by showing that, when the set of input triplets is
dense, the problem is even polynomial-time solvable for the construction of
level-2 networks. This shows that, assuming density, it is tractable to
construct plausible evolutionary histories from input triplets even when such
histories are heavily non-tree like. This further strengthens the case for the
use of triplet-based methods in the construction of phylogenetic networks. We
also show that, in the non-dense case, the level-2 problem remains NP-hard
Constructing the Simplest Possible Phylogenetic Network from Triplets,
A phylogenetic network is a directed acyclic graph that visualises
an evolutionary history containing so-called reticulations such
as recombinations, hybridisations or lateral gene transfers. Here we consider
the construction of a simplest possible phylogenetic network consistent
with an input set T, where T contains at least one phylogenetic
tree on three leaves (a triplet) for each combination of three taxa. To
quantify the complexity of a network we consider both the total number
of reticulations and the number of reticulations per biconnected component,
called the level of the network. We give polynomial-time algorithms
for constructing a level-1 respectively a level-2 network that contains a
minimum number of reticulations and is consistent with T (if such a
network exists). In addition, we show that if T is precisely equal to the
set of triplets consistent with some network, then we can construct such
a network, which minimises both the level and the total number of reticulations,
in time O(|T|^{k+1}), if k is a fixed upper bound on the level
Phylogenetic Networks Do not Need to Be Complex: Using Fewer Reticulations to Represent Conflicting Clusters
Phylogenetic trees are widely used to display estimates of how groups of
species evolved. Each phylogenetic tree can be seen as a collection of
clusters, subgroups of the species that evolved from a common ancestor. When
phylogenetic trees are obtained for several data sets (e.g. for different
genes), then their clusters are often contradicting. Consequently, the set of
all clusters of such a data set cannot be combined into a single phylogenetic
tree. Phylogenetic networks are a generalization of phylogenetic trees that can
be used to display more complex evolutionary histories, including reticulate
events such as hybridizations, recombinations and horizontal gene transfers.
Here we present the new CASS algorithm that can combine any set of clusters
into a phylogenetic network. We show that the networks constructed by CASS are
usually simpler than networks constructed by other available methods. Moreover,
we show that CASS is guaranteed to produce a network with at most two
reticulations per biconnected component, whenever such a network exists. We
have implemented CASS and integrated it in the freely available Dendroscope
software
Constructing the Simplest Possible Phylogenetic Network from Triplets
A phylogenetic network is a directed acyclic graph that visualises an evolutionary
history containing so-called reticulations such as recombinations, hybridisations or lateral gene
transfers. Here we consider the construction of a simplest possible phylogenetic network consistent
with an input set T, where T contains at least one phylogenetic tree on three leaves (a
triplet) for each combination of three taxa. To quantify the complexity of a network we consider
both the total number of reticulations and the number of reticulations per biconnected component,
called the level of the network. We give polynomial-time algorithms for constructing a
level-1 respectively a level-2 network that contains a minimum number of reticulations and is
consistent with T (if such a network exists). In addition, we show that if T is precisely equal
to the set of triplets consistent with some network, then we can construct such a network with
smallest possible level in time O(|T|^{k+1}), if k is a fixed upper bound on the level of the network
RPNCH: A method for constructing rooted phylogenetic networks from rooted triplets based on height function
Phylogenetic networks are a generalization of phylogenetic trees which permit the representation the non-tree-like events. It is NP-hard to construct an optimal rooted phylogenetic network from a given set of rooted triplets. This paper presents a novel algorithm called RPNCH. For a given set of rooted triplets, RPNCH tries to construct a rooted phylogenetic network with the minimum number of reticulation nodes that contains all the given rooted triplets. The performance of RPNCH algorithm on simulated data is reported here
Level-k Phylogenetic Network can be Constructed from a Dense Triplet Set in Polynomial Time
Given a dense triplet set , there arise two interesting
questions: Does there exists any phylogenetic network consistent with
? And if so, can we find an effective algorithm to construct one?
For cases of networks of levels or 1 or 2, these questions were answered
with effective polynomial algorithms. For higher levels , partial answers
were recently obtained with an time algorithm for
simple networks. In this paper we give a complete answer to the general case.
The main idea is to use a special property of SN-sets in a level-k network. As
a consequence, we can also find the level-k network with the minimum number of
reticulations in polynomial time
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