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.

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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., leaf-labeled 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 node-unbalanced or weight-unbalanced. 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

This paper presents two sets of techniques for comparing unrooted evolutionary trees, namely, label compression and four-way dynamic programming. The technique of four-way 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...

. 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 time-bound 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...

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.

We study how to label the vertices of a tree in such a way that we can decide the distance of two vertices in the tree given only their labels. For trees, Gavoille et al. [7] proved that for any such distance labelling scheme, the maximum label length is at least 1 8 log2 n − O(log n) bits. They also gave a separatorbased labelling scheme that has the optimal label length Θ(log n · log(Hn(T))), where Hn(T) is the height of the tree. In this paper, we present two new distance labelling schemes that not only achieve the optimal label length Θ(log n · log(Hn(T))), but also have a much smaller expected label length under certain tree distributions. With these new schemes, we also can efficiently find the least common ancestor of any two vertices based on their labels only.

We propose and study the Maximum Constrained Agreement Sub-tree (MCAST) problem, which is a variant of the classical Maximum Agreement Subtree (MAST) problem. Our problem allows users to ap-ply their domain knowledge to control the construction of the agreement subtrees in order to get better results. We show that the MCAST prob-lem can be reduced to the MAST problem in linear time and thus we have algorithms for MCAST with running times matching the fastest known algorithms for MAST.