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Constructing a Tree from Homeomorphic Subtrees, with Applications to Computational Evolutionary Biology
"... We are given a set T = fT1 ; T2 ; : : : ; Tkg of rooted binary trees, each T i leaflabeled by a subset L(T i ) ae f1; 2; : : : ; ng. If T is a tree on f1; 2; : : : ; ng, we let TjL denote the minimal subtree of T induced by the nodes of L and all their ancestors. The consensus tree problem asks wh ..."
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We are given a set T = fT1 ; T2 ; : : : ; Tkg of rooted binary trees, each T i leaflabeled by a subset L(T i ) ae f1; 2; : : : ; ng. If T is a tree on f1; 2; : : : ; ng, we let TjL denote the minimal subtree of T induced by the nodes of L and all their ancestors. The consensus tree problem asks whether there exists a tree T such that for every i, T jL(T i ) is homeomorphic to T i . We present algorithms which test if a given set of trees has a consensus tree and if so, construct one. The deterministic algorithm takes time minfO(Nn 1=2 ); O(N + n 2 log n)g, where N = P i jT i j, and uses linear space. The randomized algorithm takes time O(N log 3 n) and uses linear space. The previous best for this problem was an 1981 O(Nn) algorithm by Aho et al. Our faster deterministic algorithm uses a new efficient algorithm for the following interesting dynamic graph problem: Given a graph G with n nodes and m edges and a sequence of b batches of one or more edge deletions, then a...
An O(n log n) algorithm for the maximum agreement subtree problem for binary trees
 SIAM Journal on Computing
, 1996
"... Abstract. The maximum agreement subtree problem is the following. Given two rooted trees whose leaves are drawn from the same set of items (e.g., species), find the largest subset of these items so that the portions of the two trees restricted to these items are isomorphic. We consider the case whic ..."
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Abstract. The maximum agreement subtree problem is the following. Given two rooted trees whose leaves are drawn from the same set of items (e.g., species), find the largest subset of these items so that the portions of the two trees restricted to these items are isomorphic. We consider the case which occurs frequently in practice, i.e., the case when the trees are binary, and give an O(n log n) time algorithm for this problem.
An even faster and more unifying algorithm for comparing trees via unbalanced bipartite matchings
 Journal of Algorithms
"... A widely used method for determining the similarity of two labeled trees is to compute a maximum agreement subtree of the two trees. Previous work on this similarity measure is only concerned with the comparison of labeled trees of two special kinds, namely, uniformly labeled trees (i.e., trees with ..."
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Cited by 15 (6 self)
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A widely used method for determining the similarity of two labeled trees is to compute a maximum agreement subtree of the two trees. Previous work on this similarity measure is only concerned with the comparison of labeled trees of two special kinds, namely, uniformly labeled trees (i.e., trees with all their nodes labeled by the same symbol) and evolutionary trees (i.e., leaflabeled trees with distinct symbols for distinct leaves). This paper presents an algorithm for comparing trees that are labeled in an arbitrary manner. In addition to this generality, this algorithm is faster than the previous algorithms. Another contribution of this paper is on maximum weight bipartite matchings. We show how to speed up the best known matching algorithms when the input graphs are nodeunbalanced or weightunbalanced. Based on these enhancements, we obtain an efficient algorithm for a new matching problem called the hierarchical bipartite matching problem, which is at the core of our maximum agreement subtree algorithm. 1
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.
General Techniques for Comparing Unrooted Evolutionary Trees
, 1997
"... This paper presents two sets of techniques for comparing unrooted evolutionary trees, namely, label compression and fourway dynamic programming. The technique of fourway dynamic programming transforms existing algorithms for computing rooted maximum agreement subtrees into new ones for unrooted t ..."
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Cited by 6 (2 self)
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This paper presents two sets of techniques for comparing unrooted evolutionary trees, namely, label compression and fourway dynamic programming. The technique of fourway dynamic programming transforms existing algorithms for computing rooted maximum agreement subtrees into new ones for unrooted trees. Let n be the size of the two input trees. This technique leads to an O(n log n)time algorithm for unrooted trees whose degrees are bounded by a constant, matching the best known complexity for the rooted binary case. The technique of label compression is not based on dynamic programming. With this technique, we obtain an O(n 1:5 log n)time algorithm for unrooted trees with arbitrary degrees, also matching the best algorithm for the rooted unbounded degree case. 1 Introduction An evolutionary tree is a tree whose leaves are labeled with distinct symbols representing species. Evolutionary trees are useful for modeling the evolutionary relationship of species. Many mathematical biol...
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.
From constrained to unconstrained maximum agreement subtree in linear time
 ALGORITHMICA
, 2008
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