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15
Constructing level2 phylogenetic networks from triplets
, 2009
"... Jansson and Sung showed that, given a dense set of input triplets T (representing hypotheses about the local evolutionary relationships of triplets of taxa), it is possible to determine in polynomial time whether there exists a level1 network consistent with T, and if so, to construct such a netwo ..."
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Cited by 33 (9 self)
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Jansson and Sung showed that, given a dense set of input triplets T (representing hypotheses about the local evolutionary relationships of triplets of taxa), it is possible to determine in polynomial time whether there exists a level1 network consistent with T, and if so, to construct such a network [24]. Here, we extend this work by showing that this problem is even polynomial time solvable for the construction of level2 networks. This shows that, assuming density, it is tractable to construct plausible evolutionary histories from input triplets even when such histories are heavily nontreelike. This further strengthens the case for the use of tripletbased methods in the construction of phylogenetic networks. We also implemented the algorithm and applied it to yeast data.
Constructing the simplest possible phylogenetic network from triplets,”
 Algorithmica,
, 2011
"... Abstract A phylogenetic network is a directed acyclic graph that visualizes an evolutionary history containing socalled reticulations such as recombinations, hybridizations or lateral gene transfers. Here we consider the construction of a simplest possible phylogenetic network consistent with an i ..."
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Cited by 20 (5 self)
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Abstract A phylogenetic network is a directed acyclic graph that visualizes an evolutionary history containing socalled reticulations such as recombinations, hybridizations 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 polynomialtime algorithms for constructing a level1 respectively a level2 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.
Uniqueness, intractability and exact algorithms: reflections on levelk phylogenetic networks
, 2009
"... Phylogenetic networks provide a way to describe and visualize evolutionary histories that have undergone socalled reticulate evolutionary events such as recombination, hybridization or horizontal gene transfer. The level k of a network determines how nontreelike the evolution can be, with level0 ..."
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Cited by 16 (6 self)
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Phylogenetic networks provide a way to describe and visualize evolutionary histories that have undergone socalled reticulate evolutionary events such as recombination, hybridization or horizontal gene transfer. The level k of a network determines how nontreelike the evolution can be, with level0 networks being trees. We study the problem of constructing levelk phylogenetic networks from triplets, i.e. phylogenetic trees for three leaves (taxa). We give, for each k, a levelk 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 ≥ 1it is NPhard to construct a levelk network consistent with all input triplets, and (2) for all k ≥ 0 it is NPhard to construct a levelk 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 level1 networks consistent with a maximum number of input triplets.
New Results on Optimizing Rooted Triplets Consistency
"... Abstract. A set of phylogenetic trees with overlapping leaf sets is consistent if it can be merged without conflicts into a supertree. In this paper, we study the polynomialtime approximability of two related optimization problems called the maximum rooted triplets consistency problem (MaxRTC) and ..."
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Cited by 13 (2 self)
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Abstract. A set of phylogenetic trees with overlapping leaf sets is consistent if it can be merged without conflicts into a supertree. In this paper, we study the polynomialtime approximability of two related optimization problems called the maximum rooted triplets consistency problem (MaxRTC) and the minimum rooted triplets inconsistency problem (MinRTI) in which the input is a set R of rooted triplets, and where the objectives are to find a largest cardinality subset of R which is consistent and a smallest cardinality subset of R whose removal from R results in a consistent set, respectively. We first show that a simple modification to Wu’s BestPairMergeFirst heuristic [25] results in a bottomupbased 3approximation for MaxRTC. We then demonstrate how any approximation algorithm for MinRTI could be used to approximate MaxRTC, and thus obtain the first polynomialtime approximation algorithm for MaxRTC with approximation ratio smaller than 3. Next, we prove that for a set of rooted triplets generated under a uniform random model, the maximum fraction of triplets which can be consistent with any tree is approximately one third, and then provide a deterministic construction of a triplet set having a similar property which is subsequently used to prove that both MaxRTC and MinRTI are NPhard even if restricted to minimally dense instances. Finally, we prove that MinRTI cannot be approximated within a ratio of Ω(log n) in polynomial time, unless P = NP. 1
LEVELK PHYLOGENETIC NETWORK CAN BE CONSTRUCTED FROM A DENSE TRIPLET SET IN POLYNOMIAL TIME
, 2009
"... Given a dense triplet set T, there arise two interesting questions [7]: Does there exists any phylogenetic network consistent with T? And if so, can we find an effective algorithm to construct one? For cases of networks of levels k = 0 or 1 or 2, these questions were answered in [1, 6, 7, 8, 10] wit ..."
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Cited by 10 (1 self)
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Given a dense triplet set T, there arise two interesting questions [7]: Does there exists any phylogenetic network consistent with T? And if so, can we find an effective algorithm to construct one? For cases of networks of levels k = 0 or 1 or 2, these questions were answered in [1, 6, 7, 8, 10] with effective polynomial algorithms. For higher levels k, partial answers were recently obtained in [11] with an O(T  k+1) time algorithm for simple networks. In this paper we give a complete answer to the general case, solving a problem of [7]. The main idea is to use a special property of SNsets in a levelk network. As a consequence, we can also find the levelk network with the minimum number of reticulations in polynomial time.
A practical algorithm for reconstructing level1 phylogenetic networks
 IEEE/ACM Transactions on Computational Biology and Bioinformatics
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QUARTETS AND UNROOTED PHYLOGENETIC NETWORKS
, 2012
"... Accepted (Day Month Year) Phylogenetic networks were introduced to describe evolution in the presence of exchanges of genetic material between coexisting species or individuals. Split networks in particular were introduced as a special kind of abstract network to visualize conflicts between phylogen ..."
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Cited by 1 (0 self)
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Accepted (Day Month Year) Phylogenetic networks were introduced to describe evolution in the presence of exchanges of genetic material between coexisting species or individuals. Split networks in particular were introduced as a special kind of abstract network to visualize conflicts between phylogenetic trees which may correspond to such exchanges. More recently, methods were designed to reconstruct explicit phylogenetic networks (whose vertices can be interpreted as biological events) from triplet data. In this article, we link abstract and explicit networks through their combinatorial properties, by introducing the unrooted analogue of levelk networks. In particular, we give an equivalence theorem between circular split systems and unrooted level1 networks. We also show how to adapt to quartets some existing results on triplets, in order to reconstruct unrooted levelk phylogenetic networks. These results give an interesting perspective on the combinatorics of phylogenetic networks and also raise algorithmic and combinatorial questions.
A practical algorithm for reconstructing . . .
, 2010
"... 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 level1 phylogenetic networks a type of network slig ..."
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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 level1 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 exponentialtime exact and a greedy algorithm both of which are of independent theoretical interest. Most importantly, LEV1ATHAN runs in polynomial time and always constructs a level1 network. If the data are 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 level1 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.
LEVELK PHYLOGENETIC NETWORK CAN BE CONSTRUCTED FROM A DENSE TRIPLET SET IN POLYNOMIAL TIME THUHIEN TO AND MICHEL HABIB
"... Abstract. Given a dense triplet set T, there arise two interesting questions [7]: Does there exists any phylogenetic network consistent with T? And if so, can we find an effective algorithm to construct one? For cases of networks of levels k = 0 or 1 or 2, these questions were answered in [1, 6, 7, ..."
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Abstract. Given a dense triplet set T, there arise two interesting questions [7]: Does there exists any phylogenetic network consistent with T? And if so, can we find an effective algorithm to construct one? For cases of networks of levels k = 0 or 1 or 2, these questions were answered in [1, 6, 7, 8, 10] with effective polynomial algorithms. For higher levels k, partial answers were recently obtained in [11] with an O(T  k+1) time algorithm for simple networks. In this paper we give a complete answer to the general case, solving a problem of [7]. The main idea is to use a special property of SNsets in a levelk network. As a consequence, we can also find the levelk network with the minimum number of reticulations in polynomial time. hal00352360, version 1 12 Jan 2009 1.