Four-point Conditions Research Articles

The k-weak hierarchical representations: an extension of the indexed closed weak hierarchies

Several approaches have been proposed for the purpose of proving that different classes of dissimilarities (e.g. ultrametrics) can be represented by certain types of stratified clusterings which are easily visualized (e.g. indexed hierarchies). These approaches differ in the choice of the clusters that are used to represent a dissimilarity coefficient. More precisely, the clusters may be defined as the maximal linked subsets, also called M L-sets; equally they may be defined as a particular type of 2-ball. In this paper, we first introduce the notion of a k-ball, thereby extending the notion of a 2-ball. For an arbitrary dissimilarity coefficient, we establish some properties of the k-balls that pinpoint the connection between them and the M L-sets. We also introduce the (2, k)-point condition ( k⩾1) which is an extension of the Bandelt four-point condition. For k⩾2, we prove that the dissimilarities satisfying the (2, k)-point condition are in one–one correspondence with a class of stratified clusterings, called k-weak hierarchical representations, whose main characteristic is that the intersection of ( k+1) arbitrary clusters may be reduced to the intersection of some k of these clusters.

How to Account for Reticulation Events in Phylogenetic Analysis: A Comparison of Distance-Based Methods

This paper presents a review of mathematical techniques capable of representing reticulate events in phylogenetics. Two families of methods are identified; they relax either the ultrametric inequality defining dendrograms or the four-point condition defining additive trees. Pyramids and weak hierarchies are techniques developed to fit dendrograms with overlapping clusters. Splitsgraphs and reticulograms are extensions of the additive tree model; they allow one to fit a dissimilarity matrix using a graph containing reticulations. The four methods are applied to a data set; the results are compared and discussed in a phylogenetic setting.

On some relations between 2-trees and tree metrics

A tree function (TF) t on a finite set X is a real function on the set of the pairs of elements of X satisfying the four-point condition: for all distinct x, y, z, w ∈ X, t( xy)+ t( zw)⩽ max{ t( xz) + t( yw), t( xw) + t( yz)}. Equivalently, t is representable by the lengths of the paths between the leaves of a valued tree T ℓ. TFs are a straightforward generalization of the tree dissimilarities and tree metrics of the literature. A graph Θ is a 2- tree if it belongs to the following class Q : an edge-graph belongs to Q : if Θ′ ∈ Q and yz is an edge of Θ′, then the graph obtained by the addition to Θ l of a new vertex x adjacent to y and z belongs to Q . These graphs, and the more general k-trees, have been studied in the literature as generalizations of trees. It is first explicited here how to make a TF t Θ, d correspond to any positively valued 2-tree Θ d on X. Then, given a tree dissimilarity t, the set Q( t) of the 2-trees Θ such that t = t Θ, t is studied. Any element of Q( t) gives a way of summarizing t by its restriction to a minimal subset of entries. Several characterizations and properties of the elements of Q( t) are given. We describe five classes of such elements, including two new ones. Associated with a dissimilarity of the general type, these classes of 2-trees lead to methods for the recognition and fitting of tree dissimilarities.

R-Trees and Symmetric Differences of Sets

The supremum of the symmetric differencex▵y:= (x\\y)∪(y\\x) of subsetsx,yof R satisfies the so-called four-point condition; that is, for allx,x′,y,y′ ⊆ R, one has sup (x▵x′) + sup(y▵y′)≤ max { sup(x▵y) + sup(x′ ▵y′), sup (x▵y′) + sup(x′ ▵y)}. It follows that the setEof all subsets of R which are bounded from above forms a valuated matroid relative to the mapv:E×E→{-∞}∪R : (x,y)→sup (x▵y). Hence, according to T-theory, there exists an R-treeT(E,v)uniquely determined by (E,v) up to isometry, the ends of which correspond in a one-to-one fashion to the elements of the completion of (E,v). In addition, the points ofT(E,v)can be identified with those bounded subsets of R which contain their infimum.Here, we show that these observations hold true in a much more general setting: given an arbitrary non-empty setBand an arbitrary mapr:B→{-∞}∪R, the map P(B) × P(B)→R∪{±∞}: (x,y)→supr(x▵y) again satisfies the four-point condition; so any non-empty setZof subsets ofBwith supr(x▵y)<∞ for allx,y∈Zforms a valuated matroid of rank 2 relative to this map and, therefore, gives rise to an R-tree.It is shown here that every valuated matroid of rank 2 can be realized in this way by choosing an appropriate system (B,r:B→{-∞}∪R ,Z⊆ P(B)); consequently, since every R-tree can be embedded isometrically intoT(E,v)for some valuated matroid (E,v), every R-tree can, in principle, be described in terms of such a system.

A six-point condition for ordinal matrices.

Ordinal assertions in an evolutionary context are of the form "species s is more similar to species x than the species y" and can be deduced from a distance matrix M of interspecies dissimilarities (M[s, x] < M[s, y]). Given species x and y, the ordinal binary character cxy of M is defined by cxy(s) = 1 if and only if M[s,x] < M[s, y], for all species s. In this paper we present several results concerning the inference of evolutionary trees or phylogenies from ordinal assertions. In particular, we present. A six-point condition that characterizes those distance matrices whose ordinal binary characters are pairwise compatible. This characterization is analogous to the four-point condition for additive matrices. An optimal O(n2) algorithm, where n is the number of species, for recovering a phylogeny that realizes the ordinal binary characters of a distance matrix that satisfies the six-point condition. An NP-completeness result on determining if there is a phylogeny that realizes k or more of the ordinary binary characters of a given distance matrix.

Estimating Phylogenies from Lacunose Distance Matrices: Additive is Superior to Ultrametric Estimation

Lapointe and Kirsch (1995) have recently explored the possibility of reconstructing phylogenetic trees from lacunose distance matrices. They have shown that missing cells can be estimated using the ultrametric property of distances, and that reliable trees can be derived from such filled matrices. Here, we extend their work by introducing a novel way to estimate distances based on the four-point condition of additive matrices. A simulation study was designed to assess whether the additive procedure is superior to the ultrametric one in recovering distances randomly deleted from complete distance matrices. Our results clearly indicate that the topologies and branch lengths of the trees derived from matrices which were estimated additively are better recovered than those of trees derived from matrices estimated ultrametrically; the original distances are also better recovered with the additive procedure. Except in the case of small matrices with many missing cells for which both methods perform equally well, the additive is generally superior to the ultrametric method for estimating missing cells in distance matrices prior to phylogenetic reconstruction.

On the Existence of Two Solutions of the Four-Point Problem

We have proved the existence of at least two different solutions to the equation u″ = ƒ( t, u, u′, s), where s ∈ R . The solutions fulfill four-point conditions u( a) = u( c), u( d) = u( b), where a < c ≤ d < b, a, b, c, d ∈ R .