20 research outputs found
A quadratic kernel for computing the hybridization number of multiple trees
It has recently been shown that the NP-hard problem of calculating the
minimum number of hybridization events that is needed to explain a set of
rooted binary phylogenetic trees by means of a hybridization network is
fixed-parameter tractable if an instance of the problem consists of precisely
two such trees. In this paper, we show that this problem remains
fixed-parameter tractable for an arbitrarily large set of rooted binary
phylogenetic trees. In particular, we present a quadratic kernel
SPRIT: Identifying horizontal gene transfer in rooted phylogenetic trees
<p>Abstract</p> <p>Background</p> <p>Phylogenetic trees based on sequences from a set of taxa can be incongruent due to horizontal gene transfer (HGT). By identifying the HGT events, we can reconcile the gene trees and derive a taxon tree that adequately represents the species' evolutionary history. One HGT can be represented by a rooted Subtree Prune and Regraft (<smcaps>R</smcaps>SPR) operation and the number of <smcaps>R</smcaps>SPRs separating two trees corresponds to the minimum number of HGT events. Identifying the minimum number of <smcaps>R</smcaps>SPRs separating two trees is NP-hard, but the problem can be reduced to fixed parameter tractable. A number of heuristic and two exact approaches to identifying the minimum number of <smcaps>R</smcaps>SPRs have been proposed. This is the first implementation delivering an exact solution as well as the intermediate trees connecting the input trees.</p> <p>Results</p> <p>We present the SPR Identification Tool (SPRIT), a novel algorithm that solves the fixed parameter tractable minimum <smcaps>R</smcaps>SPR problem and its GPL licensed Java implementation. The algorithm can be used in two ways, exhaustive search that guarantees the minimum <smcaps>R</smcaps>SPR distance and a heuristic approach that guarantees finding a solution, but not necessarily the minimum one. We benchmarked SPRIT against other software in two different settings, small to medium sized trees i.e. five to one hundred taxa and large trees i.e. thousands of taxa. In the small to medium tree size setting with random artificial incongruence, SPRIT's heuristic mode outperforms the other software by always delivering a solution with a low overestimation of the <smcaps>R</smcaps>SPR distance. In the large tree setting SPRIT compares well to the alternatives when benchmarked on finding a minimum solution within a reasonable time. SPRIT presents both the minimum <smcaps>R</smcaps>SPR distance and the intermediate trees.</p> <p>Conclusions</p> <p>When used in exhaustive search mode, SPRIT identifies the minimum number of <smcaps>R</smcaps>SPRs needed to reconcile two incongruent rooted trees. SPRIT also performs quick approximations of the minimum <smcaps>R</smcaps>SPR distance, which are comparable to, and often better than, purely heuristic solutions. Put together, SPRIT is an excellent tool for identification of HGT events and pinpointing which taxa have been involved in HGT.</p
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
A first step towards computing all hybridization networks for two rooted binary phylogenetic trees
Recently, considerable effort has been put into developing fast algorithms to
reconstruct a rooted phylogenetic network that explains two rooted phylogenetic
trees and has a minimum number of hybridization vertices. With the standard
approach to tackle this problem being combinatorial, the reconstructed network
is rarely unique. From a biological point of view, it is therefore of
importance to not only compute one network, but all possible networks. In this
paper, we make a first step towards approaching this goal by presenting the
first algorithm---called allMAAFs---that calculates all
maximum-acyclic-agreement forests for two rooted binary phylogenetic trees on
the same set of taxa.Comment: 21 pages, 5 figure
On unrooted and root-uncertain variants of several well-known phylogenetic network problems
The hybridization number problem requires us to embed a set of binary rooted
phylogenetic trees into a binary rooted phylogenetic network such that the
number of nodes with indegree two is minimized. However, from a biological
point of view accurately inferring the root location in a phylogenetic tree is
notoriously difficult and poor root placement can artificially inflate the
hybridization number. To this end we study a number of relaxed variants of this
problem. We start by showing that the fundamental problem of determining
whether an \emph{unrooted} phylogenetic network displays (i.e. embeds) an
\emph{unrooted} phylogenetic tree, is NP-hard. On the positive side we show
that this problem is FPT in reticulation number. In the rooted case the
corresponding FPT result is trivial, but here we require more subtle
argumentation. Next we show that the hybridization number problem for unrooted
networks (when given two unrooted trees) is equivalent to the problem of
computing the Tree Bisection and Reconnect (TBR) distance of the two unrooted
trees. In the third part of the paper we consider the "root uncertain" variant
of hybridization number. Here we are free to choose the root location in each
of a set of unrooted input trees such that the hybridization number of the
resulting rooted trees is minimized. On the negative side we show that this
problem is APX-hard. On the positive side, we show that the problem is FPT in
the hybridization number, via kernelization, for any number of input trees.Comment: 28 pages, 8 Figure
A tight kernel for computing the tree bisection and reconnection distance between two phylogenetic trees
In 2001 Allen and Steel showed that, if subtree and chain reduction rules
have been applied to two unrooted phylogenetic trees, the reduced trees will
have at most 28k taxa where k is the TBR (Tree Bisection and Reconnection)
distance between the two trees. Here we reanalyse Allen and Steel's
kernelization algorithm and prove that the reduced instances will in fact have
at most 15k-9 taxa. Moreover we show, by describing a family of instances which
have exactly 15k-9 taxa after reduction, that this new bound is tight. These
instances also have no common clusters, showing that a third
commonly-encountered reduction rule, the cluster reduction, cannot further
reduce the size of the kernel in the worst case. To achieve these results we
introduce and use "unrooted generators" which are analogues of rooted
structures that have appeared earlier in the phylogenetic networks literature.
Using similar argumentation we show that, for the minimum hybridization problem
on two rooted trees, 9k-2 is a tight bound (when subtree and chain reduction
rules have been applied) and 9k-4 is a tight bound (when, additionally, the
cluster reduction has been applied) on the number of taxa, where k is the
hybridization number of the two trees.Comment: One figure added, two small typos fixed. This version to appear in
SIDMA (SIAM Journal on Discrete Mathematics