3 research outputs found
The Geometry of the Neighbor-Joining Algorithm for Small Trees
In 2007, Eickmeyer et al. showed that the tree topologies outputted by the
Neighbor-Joining (NJ) algorithm and the balanced minimum evolution (BME) method
for phylogenetic reconstruction are each determined by a polyhedral subdivision
of the space of dissimilarity maps , where is the
number of taxa. In this paper, we will analyze the behavior of the
Neighbor-Joining algorithm on five and six taxa and study the geometry and
combinatorics of the polyhedral subdivision of the space of dissimilarity maps
for six taxa as well as hyperplane representations of each polyhedral
subdivision. We also study simulations for one of the questions stated by
Eickmeyer et al., that is, the robustness of the NJ algorithm to small
perturbations of tree metrics, with tree models which are known to be hard to
be reconstructed via the NJ algorithm.Comment: 15 page
Optimality of the Neighbor Joining Algorithm and Faces of the Balanced Minimum Evolution Polytope
Balanced minimum evolution (BME) is a statistically consistent distance-based
method to reconstruct a phylogenetic tree from an alignment of molecular data.
In 2000, Pauplin showed that the BME method is equivalent to optimizing a
linear functional over the BME polytope, the convex hull of the BME vectors
obtained from Pauplin's formula applied to all binary trees. The BME method is
related to the Neighbor Joining (NJ) algorithm, now known to be a greedy
optimization of the BME principle. Further, the NJ and BME algorithms have been
studied previously to understand when the NJ Algorithm returns a BME tree for
small numbers of taxa. In this paper we aim to elucidate the structure of the
BME polytope and strengthen knowledge of the connection between the BME method
and NJ Algorithm. We first prove that any subtree-prune-regraft move from a
binary tree to another binary tree corresponds to an edge of the BME polytope.
Moreover, we describe an entire family of faces parametrized by disjoint
clades. We show that these {\em clade-faces} are smaller dimensional BME
polytopes themselves. Finally, we show that for any order of joining nodes to
form a tree, there exists an associated distance matrix (i.e., dissimilarity
map) for which the NJ Algorithm returns the BME tree. More strongly, we show
that the BME cone and every NJ cone associated to a tree have an
intersection of positive measure.Comment: 24 pages,4 figur
Combinatorial and computational investigations of Neighbor-Joining bias
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.Comment: 18 pages, 11 figure