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On the Complexity of Constructing Evolutionary Trees
, 1999
"... In this paper we study a few important tree optimization problems with applications to computational biology. These problems ask for trees that are consistent with an as large part of the given data as possible. We show that the maximum homeomorphic agreement subtree problem cannot be approximated w ..."
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Cited by 17 (8 self)
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In this paper we study a few important tree optimization problems with applications to computational biology. These problems ask for trees that are consistent with an as large part of the given data as possible. We show that the maximum homeomorphic agreement subtree problem cannot be approximated within a factor of N ffl , where N is the input size, for any 0 ffl ! 1 9 in polynomial time unless P=NP, even if all the given trees are of height 2. On the other hand, we present an O(N log N)time heuristic for the restriction of this problem to instances with O(1) trees of height O(1) yielding solutions within a constant factor of the optimum. We prove that the maximum inferred consensus tree problem is NPcomplete, and provide a simple, fast heuristic for it yielding solutions within one third of the optimum. We also present a more specialized polynomialtime heuristic for the maximum inferred local consensus tree problem.
Rooted Maximum Agreement Supertrees
, 2005
"... Given a set T of rooted, unordered trees, where each Ti ∈ T is distinctly leaflabeled by a set �(Ti) and where the sets �(Ti) may overlap, the maximum agreement supertree problem (MASP) is to construct a distinctly leaflabeled tree Q with leaf set �(Q) ⊆ ∪Ti ∈T �(Ti) such that �(Q)  is maximiz ..."
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Cited by 11 (2 self)
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Given a set T of rooted, unordered trees, where each Ti ∈ T is distinctly leaflabeled by a set �(Ti) and where the sets �(Ti) may overlap, the maximum agreement supertree problem (MASP) is to construct a distinctly leaflabeled tree Q with leaf set �(Q) ⊆ ∪Ti ∈T �(Ti) such that �(Q)  is maximized and for each Ti ∈ T, the topological restriction of Ti to �(Q) is isomorphic to the topological restriction of Q to �(Ti). Let n = � �∪Ti ∈T �(Ti) � � , k =T , and D = maxTi ∈T {deg(Ti)}. We first show that MASP with k = 2 can be solved in O ( √ Dn log(2n/D)) time, which is O(n log n) when D = O(1) and O(n1.5) when D is unrestricted. We then present an algorithm for MASP with D = 2 whose running time is polynomial if k = O(1). On the other hand, we prove that MASP is NPhard for any fixed k ≥ 3 when D is unrestricted, and also NPhard for any fixed D ≥ 2 when k is unrestricted even if each input tree is required to contain at most three leaves. Finally, we describe a polynomialtime (n/log n)approximation algorithm for MASP.
A structurebased search engine for phylogenetic databases
 In SSDBM (Scientific Database Management
, 2002
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Balanced Randomized Tree Splitting with Applications to Evolutionary Tree Constructions
 In Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science
, 1999
"... . We present a new technique called balanced randomized tree splitting. It is useful in constructing unknown trees recursively. By applying it we obtain two new results on e#cient construction of evolutionary trees: a new upper timebound on the problem of constructing an evolutionary tree from ..."
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Cited by 5 (2 self)
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. We present a new technique called balanced randomized tree splitting. It is useful in constructing unknown trees recursively. By applying it we obtain two new results on e#cient construction of evolutionary trees: a new upper timebound on the problem of constructing an evolutionary tree from experiments, and a relatively fast approximation algorithm for the maximum agreement subtree problem for binary trees for which the maximum number of leaves in an optimal solution is large. We also present new lower bounds for the problem of constructing an evolutionary tree from experiments and for the problem of constructing a tree from an ultrametric distance matrix. 1 Introduction Several of the known e#cient algorithms for trees rely on their excellent separator properties. It is well known that each tree contains a vertex whose removal splits it into components of balanced size. Unfortunately, finding such a vertex usually requires the knowledge of the tree. In this paper, we co...
Approximating the Maximum Isomorphic Agreement Subtree is Hard
"... The Maximum Isomorphic Agreement Subtree (MIT) problem is one of the simplest versions of the Maximum Interval Weight Agreement Subtree method (MIWT) which is used to compare phylogenies. More precisely MIT allows to provide a subset of the species such that the exact distances between species in ..."
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Cited by 3 (0 self)
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The Maximum Isomorphic Agreement Subtree (MIT) problem is one of the simplest versions of the Maximum Interval Weight Agreement Subtree method (MIWT) which is used to compare phylogenies. More precisely MIT allows to provide a subset of the species such that the exact distances between species in such subset are preserved among all evolutionary trees considered. In this paper, the approximation complexity of the MIT problem is investigated, showing that it cannot be approximated in polynomial time within factor log n for any > 0 unless NPDTIME(2 ) for instances containing three trees. Moreover, we show that such result can be strengthened whenever instances of the MIT problem can contain an arbitrary number of trees, since MIT shares the same approximation lower bound of MAX CLIQUE.
Lecturer: Prof. Sung Wing Kin Scribe: Vu Quang Hai 10.1 Phylogenetic Network 10.1.1 Limitation of phylogenetic tree
"... Ford Doolittle [14] said that “Molecular phylogeneticists will have failed to find the ‘true tree’, not because their moethods are inadequate or because they have chosen the wrong gens, but because the history of life cannot properly be represetnted as a tree.” Therefore, scientists have to come up ..."
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Ford Doolittle [14] said that “Molecular phylogeneticists will have failed to find the ‘true tree’, not because their moethods are inadequate or because they have chosen the wrong gens, but because the history of life cannot properly be represetnted as a tree.” Therefore, scientists have to come up with more realistic assumption. • Evolution is infact more than mutaiton. We have other types of evolutions. – Hybridization: This process happens when two diffent species produce an offspring that has the genes from both parents. E.g. tiger + lion → tiglion. – Horizontal gene transfer or lateral gene transfer. This is the process of transfer a portion of genome from one species (donor) to another (recipient). E.g. Evolution of influenza. • Phylogenetic tree cannot model those types of evolutions. Definition 1 Phylogenetic network is a generalization of phylogenetic tree in which nodes may have more than one parent. A network N is a directed acyclic graph (i.e. no cycle) such that • Each node has indegree 1 or 2 (except the root) and outdegree at most 2. • No node has both indegree 1 and out degree 1. • All nodes with out degree 0 are distinctly labeled (”leave”). From Figure 10.1, we can see that X1,.., X4 are the leaves. At the top of the tree is the root whose in degree is zero. Some of the internal nodes have in degree of 2 from two different parents, which are called hybrid nodes. Definition 2 A network is called galled network when all cylces in the phylogenetic network is nodedisjoint. 101