169 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
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
A Note on Encodings of Phylogenetic Networks of Bounded Level
Driven by the need for better models that allow one to shed light into the
question how life's diversity has evolved, phylogenetic networks have now
joined phylogenetic trees in the center of phylogenetics research. Like
phylogenetic trees, such networks canonically induce collections of
phylogenetic trees, clusters, and triplets, respectively. Thus it is not
surprising that many network approaches aim to reconstruct a phylogenetic
network from such collections. Related to the well-studied perfect phylogeny
problem, the following question is of fundamental importance in this context:
When does one of the above collections encode (i.e. uniquely describe) the
network that induces it? In this note, we present a complete answer to this
question for the special case of a level-1 (phylogenetic) network by
characterizing those level-1 networks for which an encoding in terms of one (or
equivalently all) of the above collections exists. Given that this type of
network forms the first layer of the rich hierarchy of level-k networks, k a
non-negative integer, it is natural to wonder whether our arguments could be
extended to members of that hierarchy for higher values for k. By giving
examples, we show that this is not the case
A Practical Algorithm for Reconstructing Level-1 Phylogenetic Networks
Recently much attention has been devoted to the construction of phylogenetic
networks which generalize phylogenetic trees in order to accommodate complex
evolutionary processes. Here we present an efficient, practical algorithm for
reconstructing level-1 phylogenetic networks - a type of network slightly more
general than a phylogenetic tree - from triplets. Our algorithm has been made
publicly available as the program LEV1ATHAN. It combines ideas from several
known theoretical algorithms for phylogenetic tree and network reconstruction
with two novel subroutines. Namely, an exponential-time exact and a greedy
algorithm both of which are of independent theoretical interest. Most
importantly, LEV1ATHAN runs in polynomial time and always constructs a level-1
network. If the data is consistent with a phylogenetic tree, then the algorithm
constructs such a tree. Moreover, if the input triplet set is dense and, in
addition, is fully consistent with some level-1 network, it will find such a
network. The potential of LEV1ATHAN is explored by means of an extensive
simulation study and a biological data set. One of our conclusions is that
LEV1ATHAN is able to construct networks consistent with a high percentage of
input triplets, even when these input triplets are affected by a low to
moderate level of noise
Reconstructing phylogenetic level-1 networks from nondense binet and trinet sets
Binets and trinets are phylogenetic networks with two and three leaves, respectively. Here we consider the problem of deciding if there exists a binary level-1 phylogenetic network displaying a given set T of binary binets or trinets over a taxon set X, and constructing such a network whenever it exists. We show that this is NP-hard for trinets but polynomial-time solvable for binets. Moreover, we show that the problem is still polynomial-time solvable for inputs consisting of binets and trinets as long as the cycles in the trinets have size three. Finally, we present an O(3^{|X|} poly(|X|)) time algorithm for general sets of binets and trinets. The latter two algorithms generalise to instances containing level-1 networks with arbitrarily many leaves, and thus provide some of the first supernetwork algorithms for computing networks from a set of rooted 1 phylogenetic networks
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
Reconstructing a phylogenetic level-1 network from quartets
We describe a method that will reconstruct an unrooted binary phylogenetic
level-1 network on n taxa from the set of all quartets containing a certain
fixed taxon, in O(n^3) time. We also present a more general method which can
handle more diverse quartet data, but which takes O(n^6) time. Both methods
proceed by solving a certain system of linear equations over GF(2).
For a general dense quartet set (containing at least one quartet on every
four taxa) our O(n^6) algorithm constructs a phylogenetic level-1 network
consistent with the quartet set if such a network exists and returns an (O(n^2)
sized) certificate of inconsistency otherwise. This answers a question raised
by Gambette, Berry and Paul regarding the complexity of reconstructing a
level-1 network from a dense quartet set
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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