Computing the Quartet Distance between Evolutionary Trees in Time O(n log n)

Abstract

Evolutionary trees describing the relationship for a set of species are central in evolutionary biology, and quantifying differences between evolutionary trees is therefore an important task. The quartet distance is a distance measure between trees previously proposed by Estabrook, McMorris, and Meacham. The quartet distance between two unrooted evolutionary trees is the number of quartet topology differences between the two trees, where a quartet topology is the topological subtree induced by four species. In this paper we present an algorithm for computing the quartet distance between two unrooted evolutionary trees of n species, where all internal nodes have degree three, in time O(n log n). The previous best algorithm for the problem uses time O(n 2 ). Introduction.The evolutionary relationship for a set of species can be described by an evolutionary tree, which is a rooted tree where the leaves correspond to the species, and the internal nodes correspond to speciation events, i.e. the points in time where the evolution has diverged in different directions. The direction of the evolution is described by the location of the root, which corresponds to the most recent common ancestor for all the species, and the duration of evolutionary periods is described by assigning lengths to the edges. The true evolutionary tree for a set of species is most often unknown; estimating it from obtainable information about the species, e.g. genomic data, is of great interest in evolutionary biology. The problem of computationally estimating aspects of the true evolutionary tree requires a model describing how to use the available information about the species in question. Given a model, the problem of estimating aspects of the true evolutionary tree is referred to as constructing the evolutionary tree in that model. Many models and construction methods are available, see Chapter 17 of [11] for an overview.An important aspect of the true evolutionary tree for a set of species is its undirected tree topology induced by ignoring the location of the root and the length of the edges. Many models and methods are concerned with estimating just this tree topology, usually under the further assumption that all internal nodes have degree three. We say that such models and methods are concerned with constructing the unrooted evolutionary tree of

Polyhedral Geometry of Phylogenetic Rogue Taxa

“…0 The quartet distance is analogous to the Robinson-Foulds distance but with the role of splits replaced by that of quartets (induced subtrees of size four) contained in a tree. The naive algorithm for computation is O(n 4 ), although it be computed in O(n 2 ) via a simple algorithm (Bryant et al 2000) and in O(n log n) via a more complex algorithm (Brodal et al 2004). In this paper, s will have one more taxon than t; we accommodate this difference for the Robinson-Foulds and quartet distances by simply taking the induced tree on s given by the set of lower taxa.…”

On the Scalability of Computing Triplet and Quartet Distances

“…Brodal et al [2] improved the quartet calculation runtime to O(n lg 2 n) for binary trees, and subsequently impro>ved this to O(n lg n) [3].…”

“…The algorithm for calculating the quartet distance for binary trees in time O(n lg 2 n) from [2] has been implemented and documented to be useful in practice [10]. 1 The algorithm calculating the quartet distance for trees of arbitrary degree in time O(n 2.688 ) has also been implemented [11].…”

Towards a Practical O(n logn) Phylogeny Algorithm

“…Given a ground-truth topology for a set of taxa and a second topology on the http://www.almob.org/content/7/1/32 same taxa, the Robinson-Foulds quality is the fraction of the non-trivial splits (internal edges) of one topology found in the other. The quartet quality (see, e.g., [15,16]) is the fraction of the n 4 quartets that have the same result in both tree topologies. The Robinson-Foulds measure is fragile: a single misplaced taxon can ruin all of the splits of the tree.…”