The Geometry of the Neighbor-Joining Algorithm for Small Trees

Eickmeyer, Kord; Yoshida, Ruriko

doi:10.1007/978-3-540-85101-1_7

Cited by 9 publications

(10 citation statements)

References 16 publications

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“…As in the work of Eickmeyer & Yoshida, 40 we partition the four-dimensional space of possible values of ( λ , q 1 , q 2 , q 3 ) according to the tree topologies produced by neighbor-joining.…”

Section: The Neighbor-joining Algorithm In An Admixture Scenariomentioning

confidence: 99%

The Behavior of Admixed Populations in Neighbor-Joining Inference of Population Trees

et al. 2012

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Neighbor-joining is one of the most widely used methods for constructing evolutionary trees. This approach from phylogenetics is often employed in population genetics, where distance matrices obtained from allele frequencies are used to produce a representation of population relationships in the form of a tree. In phylogenetics, the utility of neighbor-joining derives partly from a result that for a class of distance matrices including those that are additive or tree-like-generated by summing weights over the edges connecting pairs of taxa in a tree to obtain pairwise distancesapplication of neighbor-joining recovers exactly the underlying tree. For populations within a species, however, migration and admixture can produce distance matrices that reflect more complex processes than those obtained from the bifurcating trees typical in the multispecies context. Admixed populations-populations descended from recent mixture of groups that have long been separated-have been observed to be located centrally in inferred neighbor-joining trees, with short external branches incident to the path connecting their source populations. Here, using a simple model, we explore mathematically the behavior of an admixed population under neighbor-joining. We show that with an additive distance matrix, a population admixed among two source populations necessarily lies on the path between the sources. Relaxing the additivity requirement, we examine the smallest nontrivial case-four populations, one of which is admixed between two of the other three-showing that the two source populations never merge with each other before one of them merges with the admixed population. Furthermore, the distance on the constructed tree between the admixed population and either source population is always smaller than the distance between the source populations, and the external branch for the admixed population is always incident to the path connecting the sources. We define three properties that hold for four taxa and that we hypothesize are satisfied under more general conditions: antecedence of clustering, intermediacy of distances, and intermediacy of path lengths. Our findings can inform interpretations of neighbor-joining trees with admixed groups, and they provide an explanation for patterns observed in trees of human populations.

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Section: The Neighbor-joining Algorithm In An Admixture Scenariomentioning

confidence: 99%

The Behavior of Admixed Populations in Neighbor-Joining Inference of Population Trees

et al. 2012

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“…For a leaf node a in a binary unrooted tree, the shift vector s a is the dissimilarity map in which a is at distance 1 from all other leaves, and all other distances are 0 (see [ 11 ] for the description of shift vectors). According to [ 5 ], for a tree T , ( v T ) ab gives the probability that a will immediately precede b in a random circular ordering of T .…”

Section: The Balanced Minimum Evolution Polytopementioning

confidence: 99%

“…We call the cones C σ neighbor-joining cones , or NJ cones . See [ 11 ] for the hyperplane representation of NJ cones and descriptions how to construct each cone.…”

Section: Neighbor-joining Conesmentioning

confidence: 99%

“…Thus, for any input vector d , the behavior of the NJ algorithm on d will be the same as on d + s if s is any linear combination of shift vectors. In fact it can be shown that the lineality space of NJ cones is spanned by shift vectors, just as for BME cones [ 11 ]. So from now on, when we refer to NJ cones, we will mean the pointed portion of the cone, i.e.…”

Section: Neighbor-joining Conesmentioning

confidence: 99%

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On the optimality of the neighbor-joining algorithm

Eickmeyer

Huggins

Pachter

et al. 2008

Algorithms Mol Biol

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The popular neighbor-joining (NJ) algorithm used in phylogenetics is a greedy algorithm for finding the balanced minimum evolution (BME) tree associated to a dissimilarity map. From this point of view, NJ is "optimal" when the algorithm outputs the tree which minimizes the balanced minimum evolution criterion. We use the fact that the NJ tree topology and the BME tree topology are determined by polyhedral subdivisions of the spaces of dissimilarity maps to study the optimality of the neighbor-joining algorithm. In particular, we investigate and compare the polyhedral subdivisions for n ≤ 8. This requires the measurement of volumes of spherical polytopes in high dimension, which we obtain using a combination of Monte Carlo methods and polyhedral algorithms. Our results include a demonstration that highly unrelated trees can be co-optimal in BME reconstruction, and that NJ regions are not convex. We obtain the l 2 radius for neighborjoining for n = 5 and we conjecture that the ability of the neighbor-joining algorithm to recover the BME tree depends on the diameter of the BME tree.

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“…It is agglomerative, so it constructs ancestral relationships between taxa by clustering the most closely related taxa at each step until a complete phylogeny is formed. The performance of the NJ algorithm has been studied from multiple mathematical and biological perspectives [4,12,13,15]. Moreover, some statistical conditions have been given to guarantee a good performance of the algorithm for different types of biological data [16,21].…”

Section: Introductionmentioning

confidence: 99%

Combinatorial and computational investigations of Neighbor-Joining bias

Davidson¹,

Campo²

2020

Preprint

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The Neighbor-Joining algorithm is a popular distance-based phylogenetic method that computes a tree metric from a dissimilarity map arising from biological data. Realizing dissimilarity maps as points in Euclidean space, the algorithm partitions the input space into polyhedral regions indexed by the combinatorial type of the trees returned. A full combinatorial description of these regions has not been found yet; different sequences of Neighbor-Joining agglomeration events can produce the same combinatorial tree, therefore associating multiple geometric regions to the same algorithmic output. We resolve this confusion by defining agglomeration orders on trees, leading to a bijection between distinct regions of the output space and weighted Motzkin paths. As a result, we give a formula for the number of polyhedral regions depending only on the number of taxa. We conclude with a computational comparison between these polyhedral regions, to unveil biases introduced in any implementation of the algorithm.

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The Geometry of the Neighbor-Joining Algorithm for Small Trees

Cited by 9 publications

References 16 publications

The Behavior of Admixed Populations in Neighbor-Joining Inference of Population Trees

The Behavior of Admixed Populations in Neighbor-Joining Inference of Population Trees

On the optimality of the neighbor-joining algorithm

Combinatorial and computational investigations of Neighbor-Joining bias

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