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On the elusiveness of clusters
"... Abstract—Rooted phylogenetic networks are often used to represent conflicting phylogenetic signals. Given a set of clusters, a network is said to represent these clusters in the softwired sense if, for each cluster in the input set, at least one tree embedded in the network contains that cluster. Mo ..."
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Abstract—Rooted phylogenetic networks are often used to represent conflicting phylogenetic signals. Given a set of clusters, a network is said to represent these clusters in the softwired sense if, for each cluster in the input set, at least one tree embedded in the network contains that cluster. Motivated by parsimony we might wish to construct such a network using as few reticulations as possible, or minimizing the level of the network, i.e. the maximum number of reticulations used in any “tangled ” region of the network. Although these are NPhard problems, here we prove that, for every fixedk ≥ 0, it is polynomialtime solvable to construct a phylogenetic network with level equal to k representing a cluster set, or to determine that no such network exists. However, this algorithm does not lend itself to a practical implementation. We also prove that the comparatively efficient CASS algorithm correctly solves this problem (and also minimizes the reticulation number) when input clusters are obtained from two not necessarily binary gene trees on the same set of taxa but does not always minimize level for general cluster sets. Finally, we describe a new algorithm which generates in polynomialtime all binary phylogenetic networks with exactly r reticulations representing a set of input clusters (for every fixed r ≥ 0).
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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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.
Constructing minimal phylogenetic networks from softwired clusters is fixed parameter tractable
, 2011
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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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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.
Constructing Phylogenetic Networks Based on the Isomorphism of Datasets
"... Constructing rooted phylogenetic networks from rooted phylogenetic trees has become an important problem in molecular evolution. So far, many methods have been presented in this area, in which most efficient methods are based on the incompatible graph, such as the CASS, the LNETWORK, and the BIMLR. ..."
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Constructing rooted phylogenetic networks from rooted phylogenetic trees has become an important problem in molecular evolution. So far, many methods have been presented in this area, in which most efficient methods are based on the incompatible graph, such as the CASS, the LNETWORK, and the BIMLR. This paper will research the commonness of the methods based on the incompatible graph, the relationship between incompatible graph and the phylogenetic network, and the topologies of incompatible graphs. We can find out all the simplest datasets for a topology and construct a network for every dataset. For any one dataset C, we can compute a network from the network representing the simplest dataset which is isomorphic to C. This process will save more time for the algorithms when constructing networks.
A New Heuristic Algorithm for MRTC Problem
, 2012
"... A rooted phylogenetic tree is a rooted tree which represents the evolutionary history of currently living species. A rooted binary tree on three leaves is a rooted triplet. The problem of determining whether there exists a rooted phylogenetic tree that contains all of the rooted triplets is polynomi ..."
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A rooted phylogenetic tree is a rooted tree which represents the evolutionary history of currently living species. A rooted binary tree on three leaves is a rooted triplet. The problem of determining whether there exists a rooted phylogenetic tree that contains all of the rooted triplets is polynomial solvable, while the problem of finding a rooted phylogenetic tree that contains the maximum number of rooted triplets is known to be NPhard. This maximization problem is known as the Maximum Rooted Triplets Consistency (MRTC) problem. In this paper we present a new heuristic algorithm for this problem based on the concept of the height function of a tree. We study the performance of our algorithm from both simulation and theoretical viewpoints.
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.