Computing the Local Consensus of Trees

Abstract

Abstract. The inference of consensus from a set of evolutionary trees is a fundamental problem in a number of fields such as biology and historical linguistics, and many models for inferring this consensus have been proposed. In this paper we present a model for deriving what we call a local consensus tree T from a set of trees T . The model we propose presumes a function f , called a total local consensus function, which determines for every triple A of species, the form that the local consensus tree should take on A. We show that all local consensus trees, when they exist, can be constructed in polynomial time and that many fundamental problems can be solved in linear time. We also consider partial local consensus functions and study optimization problems under this model. We present linear time algorithms for several variations. Finally we point out that the local consensus approach ties together many previous approaches to constructing consensus trees.Key words. algorithms, graphs, evolutionary trees AMS subject classifications. 05C05, 68Q25, 92-08, 92B05PII. S0097539795287642 1.Introduction. An evolutionary tree (also called a phylogeny or phylogenetic tree) for a species set S is a rooted tree with |S| = n leaves labeled by distinct elements in S. Because evolutionary history is difficult to determine (it is both computationally difficult as most optimization problems in this area are NP-hard and scientifically difficult as well since a range of approaches appropriate to different types of data exist), a common approach to solving this problem is to apply many different algorithms to a given data set, or to different data sets representing the same species set, and then look for common elements from the set of trees which are returned.There is extensive literature about inferring consensus from ordered sets of trees, with much attention paid to the properties of the rules for inferring the consensus. In this paper, we will make an explicit assumption that the consensus rule be independent of the ordering of the trees in the input; i.e., we will presume that the input to the consensus problem is an unordered multiset of evolutionary trees, each leaf-labelled by the elements in S. We call this input a profile, noting that in this paper the terminology is restricted in meaning as we have indicated.Several consensus methods are described in the literature for deriving one tree from a profile of evolutionary trees. These methods include maximum agreement subtrees [16,19,13,24,14], strict consensus trees [4,9], median trees (also known as majority trees) [5], compatibility trees [10,11,12], the Nelson tree [22], and the Adams consensus [1].The algorithms for some of these are implemented in standard packages and are in use; most common, perhaps, are strict and majority consensus tree approaches.

Constructing the R* Consensus Tree of Two Trees in Subcubic Time

“…We remark that for the restricted case where both input trees are binary, the problem becomes much easier because then the R* consensus tree is equivalent to the so-called RV-I tree in [14]. The RV-I tree differs slightly from the R* consensus tree in general in that if ab|c belongs to one of t (T 1 ) and t (T 2 ), and a|b|c belongs to the other, then ab|c may never be consistent with the output tree; however, for the special case of binary trees, neither t (T 1 ) nor t (T 2 ) contain any fan triplets so R maj equals the set of all rooted triplets that are consistent with both T 1 and T 2 , and this is precisely the "set of triples which are resolved identically in T 1 and T 2 " in Definition 5.1 in [14].…”

“…One advantage of the R* consensus tree is that it provides a statistically consistent estimator of the species tree topology when combining a set of gene trees, as recently demonstrated by Degnan et al [8]; moreover, R* consensus outperformed other methods such as majority-rule consensus in the study conducted by [8]. For the case of two input trees, it is known [3] that the R* consensus tree is equivalent to the RV-III tree introduced in [14].…”

“…27(4): 705-732, 2001; Kannan et al in SIAM J. Comput. 27(6):1695-1724, 1998. In this manuscript, we describe a new O(n 2 √ log n)-time algorithm to solve the problem.…”

“…A consensus tree is a single phylogenetic tree which summarizes the branching information contained in an input collection of phylogenetic trees with identical leaf label sets. Consensus trees are useful when different data sets or different tree inference methods have produced a set of trees with the same leaf labels and slightly conflicting structures, yet a single tree is required to represent all of them [1,3,7,9,14,18] (indeed, phylogenetic analyses often output alternative trees for the same set of species [1]). Also, by exclusion, a consensus tree indicates areas of conflict in the input trees [3].…”

“…The goal of this paper is to develop a fast algorithm for constructing the R* consensus tree for two input trees T 1 and T 2 with a leaf label set L of cardinality n. The previous algorithms for this problem require Θ(n 3 ) time [4,14], which was believed to be optimal because when T 1 and T 2 have similar branching structures, |R maj | = Ω(n 3 ). Our main result is that the time complexity of the problem is in fact subcubic.…”