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20
Inferring a level1 phylogenetic network from a dense set of rooted triplets
, 2006
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Algorithms for combining rooted triplets into a galled phylogenetic network
 SIAM Journal on Computing
, 2005
"... Abstract. This paper considers the problem of determining whether a given set T of rooted triplets can be merged without conflicts into a galled phylogenetic network and, if so, constructing such a network. When the input T is dense, we solve the problem in O(T ) time, which is optimal since the s ..."
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Cited by 26 (7 self)
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Abstract. This paper considers the problem of determining whether a given set T of rooted triplets can be merged without conflicts into a galled phylogenetic network and, if so, constructing such a network. When the input T is dense, we solve the problem in O(T ) time, which is optimal since the size of the input is Θ(T ). In comparison, the previously fastest algorithm for this problem runs in O(T  2) time. We also develop an optimal O(T )time algorithm for enumerating all simple phylogenetic networks leaflabeled by L that are consistent with T, where L is the set of leaf labels in T, which is used by our main algorithm. Next, we prove that the problem becomes NPhard if extended to nondense inputs, even for the special case of simple phylogenetic networks. We also show that for every positive integer n, there exists some set T of rooted triplets on n leaves such that any galled network can be consistent with at most 0.4883 ·T  of the rooted triplets in T. On the other hand, we provide a polynomialtime approximation algorithm that always outputs a galled network consistent with at least a factor of 5 (> 0.4166) of the rooted triplets in T.
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.
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.
INFERRING PHYLOGENETIC RELATIONSHIPS AVOIDING FORBIDDEN ROOTED TRIPLETS
, 2006
"... To construct a phylogenetic tree or phylogenetic network for describing the evolutionary history of a set of species is a wellstudied problem in computational biology. One previously proposed method to infer a phylogenetic tree/network for a large set of species is by merging a collection of known ..."
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Cited by 6 (2 self)
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To construct a phylogenetic tree or phylogenetic network for describing the evolutionary history of a set of species is a wellstudied problem in computational biology. One previously proposed method to infer a phylogenetic tree/network for a large set of species is by merging a collection of known smaller phylogenetic trees on overlapping sets of species so that no (or as little as possible) branching information is lost. However, little work has been done so far on inferring a phylogenetic tree/network from a specified set of trees when in addition, certain evolutionary relationships among the species are known to be highly unlikely. In this paper, we consider the problem of constructing a phylogenetic tree/network which is consistent with all of the rooted triplets in a given set C and none of the rooted triplets in another given set F. Although NPhard in the general case, we provide some efficient exact and approximation algorithms for a number of biologically meaningful variants of the problem.
The maximum agreement of two nested phylogenetic networks
 New Topics in Theoretical Computer Science, chap. 4, Nova Publishers, 2008
"... Given a set N of phylogenetic networks, the maximum agreement phylogenetic subnetwork problem (MASN) asks for a subnetwork embedded in every Ni ∈ N with as many leaves as possible. MASN can be used to identify shared branching structure among phylogenetic networks or to measure their similarity. In ..."
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Cited by 6 (4 self)
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Given a set N of phylogenetic networks, the maximum agreement phylogenetic subnetwork problem (MASN) asks for a subnetwork embedded in every Ni ∈ N with as many leaves as possible. MASN can be used to identify shared branching structure among phylogenetic networks or to measure their similarity. In this chapter, we prove that the general case of MASN is NPhard already for two phylogenetic networks (in fact, even if one of the two input networks is a binary tree), but that the problem can be solved efficiently if each of the two input phylogenetic networks exhibits a nested structure. For this purpose, we introduce the concept of a nested phylogenetic network and study some of its underlying fundamental combinatorial properties. We first show that the total number of nodes V (N)  in any nested phylogenetic network N with n leaves and nesting depth d is O(n(d + 1)). We then describe a simple algorithm for testing if a given phylogenetic network is nested, and if so, determining its nesting depth in O(V (N)  · (d + 1)) time. Next, we present a polynomialtime algorithm for MASN for two nested phylogenetic networks N1,N2. Its running time is O(V (N1)  ·
Reconstruction of certain phylogenetic networks from the genomes at their leaves
 Journal of Theoretical Biology
, 2008
"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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Cited by 6 (3 self)
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit:
Reconstructing recombination network from sequence data: The small parsimony problem
 IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB
, 2006
"... The small parsimony problem is studied for reconstructing recombination networks from sequence data. The small parsimony problem is polynomialtime solvable for phylogenetic trees. However, the problem is proved NPhard even for galled recombination networks. A dynamic programming algorithm is also ..."
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Cited by 4 (0 self)
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The small parsimony problem is studied for reconstructing recombination networks from sequence data. The small parsimony problem is polynomialtime solvable for phylogenetic trees. However, the problem is proved NPhard even for galled recombination networks. A dynamic programming algorithm is also developed to solve the small parsimony problem. It takes O(dn2 3h) time on an input recombination network over lengthd sequences in which there are h recombination and n − h tree nodes.