- Research
- Open Access
On the optimality of the neighbor-joining algorithm
- Kord Eickmeyer^{1},
- Peter Huggins^{2},
- Lior Pachter^{2}Email author and
- Ruriko Yoshida^{3}
https://doi.org/10.1186/1748-7188-3-5
© Eickmeyer et al; licensee BioMed Central Ltd. 2008
- Received: 13 November 2007
- Accepted: 30 April 2008
- Published: 30 April 2008
Abstract
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 ${\mathcal{R}}_{+}^{\left(\begin{array}{c}n\\ 2\end{array}\right)}$ 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 neighbor-joining 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.
Keywords
- Lineality Space
- Shift Vector
- Affine Hull
- Consistency Cone
- Polyhedral Subdivision
1 Introduction
The popular neighbor-joining algorithm used for phylogenetic tree reconstruction [1] has recently been "revealed" to be a greedy algorithm for finding the balanced minimum evolution tree associated to a dissimilarity map [2]. This means the following:
where ${\lambda}_{ij}^{T}={\displaystyle {\prod}_{v\in {p}_{ij}^{T}}{(deg(v)-1)}^{-1}}$ if i ≠ j and ${\lambda}_{ij}^{T}=0$. In [3], Day shows that choosing a minimizing tree for (2) from among the (2n-5)!! unrooted binary trees is an NP-hard problem. Yet it is desirable to find algorithms for minimizing (2) because of the following statistical interpretation:
Definition 1.1
Theorem 1.2
([4])Let T be a binary tree with edge lengths given by l: E(T) → $\mathcal{R}$_{+} and $D={\left\{{d}_{ij}\right\}}_{i,j=1}^{n}$a dissimilarity map. If the variance of d_{ ij } is proportional to ${2}^{\left|{p}_{ij}^{T}\right|}$(i. e., var(d_{ ij }) = $c{2}^{\left|{p}_{ij}^{T}\right|}$for some constant c) then (2) is the minimum variance tree length estimator of T. Moreover, the weighted least squares tree length estimate is equal to (2).
This result provides a weighted least squares rationale for the minimization of (2), and highlights the importance of understanding the balanced minimum evolution polytope:
Definition 1.3
The balanced minimum evolution polytope in this case is a triangle in $\mathcal{R}$^{6}. Note that the star-shaped tree is in the interior of the triangle.
For any dissimilarity map, the trees which minimize (2) will be vertices of the balanced minimum evolution polytope; these are always the binary trees. In fact, for such trees ${\lambda}_{ij}^{T}={2}^{1-\left|{p}_{ij}^{T}\right|}$; this is Pauplin's formula [5].
The BME polytope lies in ${\mathcal{R}}^{\left(\begin{array}{c}n\\ 2\end{array}\right)}$ and has dimension $\left(\begin{array}{c}n\\ 2\end{array}\right)$ - n. The normal fan [6] of the BME polytope gives rise to BME cones which form a polyhedral subdivision of the space of dissimilarity maps ${\mathcal{R}}_{+}^{\left(\begin{array}{c}n\\ 2\end{array}\right)}$. They describe, for each tree T, those dissimilarity maps for which T minimizes (2). We provide an introduction to the necessary polyhedral combinatorics in Section 2, and discuss the polytope in more detail in Section 3.
The neighbor-joining algorithm is a greedy algorithm for finding an approximate solution to (2). We omit a detailed description of the algorithm here – readers can consult [2] – but we do mention the crucial fact that the selection criterion is linear in the dissimilarity map [7]. Thus, the NJ algorithm will pick pairs of leaves to merge in a particular order and output a particular tree T if and only if the pairwise distances satisfy a system of linear inequalities, whose solution set forms a polyhedral cone in ${\mathcal{R}}^{\left(\begin{array}{c}n\\ 2\end{array}\right)}$. We call such a cone a neighbor-joining cone. or NJ cone. The NJ algorithm will output a particular tree T if and only if the distance data lies in a union of NJ cones. In Section 4 we show that the NJ cones partition ${\mathcal{R}}^{\left(\begin{array}{c}n\\ 2\end{array}\right)}$, but do not form a fan. This has important implications for the behavior of the NJ algorithm.
Our main result is a comparison of the neighbor-joining cones with the normal fan of the balanced minimum evolution polytope. This means that we characterize those dissimilarity maps for which neighbor-joining, despite being a greedy algorithm, is able to identify the balanced minimum evolution tree. These results are discussed in Section 5.
2 Polyhedral preliminaries
In this section we will introduce some of the elementary polyhedral combinatorics necessary for this paper. For more details see [8].
A convex linear combination is an affine linear combination with nonnegative linear coefficients, i.e. α_{ i }≥ 0 for i = 1, ..., m. The affine hull of a set C ⊆ $\mathcal{R}$^{ d }is the set of all affine linear combinations of vectors from C. The convex hull of C is the set of all convex linear combinations on vectors from C. A set is called affinely closed or an affine space if it equals its affine hull, and it is called convex if it equals its convex hull. Every affine space A ⊂ $\mathcal{R}$^{ d }can be written as
a + V = {a + v : v ⊆ V}
where V ⊆ $\mathcal{R}$^{ d }is a subspace and a ∈ A. V is uniquely determined by A and the affine dimension of A is defined to be the dimension of V.
Given two distinct points x, y ∈ $\mathcal{R}$^{ d }, the set [x, y] = {αx + (1 - α)y : 0 ≤ α ≤ 1} of all convex combinations of x and y is called the interval with endpoints x and y. Then C ⊂ $\mathcal{R}$^{ d }is convex iff [x, y] ⊂ C for any two x, y ∈ C.
A (d - 1) dimensional affine set in $\mathcal{R}$^{ d }is called a hyperplane and every hyperplane can be represented as {x ∈ $\mathcal{R}$^{ d }: n·x = b} for some n ≠ 0 ∈ $\mathcal{R}$^{ d }and b ∈ $\mathcal{R}$, where n·x is the dot-product of n and x. We call n a normal vector of this hyperplane.
Let H := {x ∈ $\mathcal{R}$^{ d }: h·x ≤ b}, where h ≠ 0 ∈ $\mathcal{R}$^{ d }and b ∈ $\mathcal{R}$, be an affine half space. Then if P ⊂ H and P ⋂ {x ∈ $\mathcal{R}$^{ d }: h·x = b} ≠ ∅, then H is called a supporting hyperplane of P. A subset F of P is called a face if F = P or F = P ⋂ H, where H is a supporting hyperplane. Faces of polyhedra are polyhedra and faces of polytopes are polytopes.
Faces of dimension 0 are called vertices, faces of dimension 1 are called edges, and faces of dimension d - 1 are called facets. The f-vector of P is the vector (f_{0}, f_{1}, f_{2}, ...), where f_{ i }is the number of faces of dimension i of P'. For example, consider the 3-dimensional polytope labeled 'C' in Figure 1. This polytope has 6 vertices, 9 edges, and 5 facets (3 quadrilaterals and 2 triangles), and so its f-vector is (6, 9, 5).
for some y_{1}, ..., y_{ N }∈ $\mathcal{R}$^{ d }. This is equivalent to the existence of a matrix A ∈ $\mathcal{R}$^{m × n}such that C = {x : A_{ x }≥ 0}. A cone is pointed if its lineality space is {0}.
Given a face F of a polytope P, the normal cone N(F) is the set of all vectors c for which c·v = max_{x∈P}c·x for all v ∈ F. The collection of relative interiors of normal cones of faces of P partition $\mathcal{R}$^{ d }, and for each face we have dim(F) + dim(N(F)) = d. The collection of normal cones of faces of P is called the normal fan of P.
Given a polyhedron P, the lineality space of P is the set of vectors v for which y + c·v ∈ P for all y ∈ P and c ∈ R. The largest such subspace is called lineality space of P. If a polyhedron P has lineality space V, we can let V' be the orthogonal complement V' (i.e. V ⊕ V' = $\mathcal{R}$^{ d }) and consider the polyhedron P' := P ⋂ V', which has lineality space {0}.
3 The balanced minimum evolution polytope
where its rows are indexed by pairs of leaves (1, 2), (1, 3), (2, 3), (1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5), (4, 5) and its columns are indexed by edges (1, a), (2, a), (3, b), (4, c), (5, c), (a, b), (b, c) with a is an internal node adjacent to leaves 1 and 2, c is an internal node adjacent to leaves 4, 5, and b is an internal node adjacent to nodes 3, a and c. Given edge lengths l : E(T) → $\mathcal{R}$_{+} we let b be the vector with components l(e) as e ranges over E(T). Any dissimilarity map d (encoded as a row vector) can now be written as
d = B_{ T }b + e
where e is a vector of "error" terms that are zero when d is a tree metric.
where ·^{ t }denotes matrix transpose. The length of T with respect to the least squares edge lengths is then
l(T) = v_{ T }·d,
The BME method is equivalent to minimizing the linear functional v_{ T }·d over all BME vectors for all tree topologies T. The BME polytope is the convex hull of all BME vectors in ${\mathcal{R}}^{\left(\begin{array}{c}n\\ 2\end{array}\right)}$. The following facts follow from the definition of the balanced minimum evolution tree:
Lemma 3.1
The vertices of the BME polytope are the BME vectors of binary trees. The BME vector of the star phylogeny lies in the interior of the BME polytope, and all other BME vectors lie on the boundary of the BME polytope.
The normal fan of a BME polytope partitions the space ${\mathcal{R}}^{\left(\begin{array}{c}n\\ 2\end{array}\right)}$ of dissimilarity maps into cones, one for each tree. We call these BME cones. They completely characterize the BME method: T is the BME tree topology if and only if the dissimilarity map D lies in the BME cone of T.
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. Thus the dot-product of a BME vector with a shift vector must necessarily equal 1, and in fact the lineality space of BME cones is spanned by shift vectors. So when we describe a BME cone we will always describe just the pointed component, i.e. modulo the lineality space of shift vectors.
The f-vector for small BME polytopes.
#leaves | dim(BME polytope) | f-vector |
---|---|---|
4 | 2 | (3,3) |
5 | 5 | (15, 105, 250, 210, 52) |
6 | 9 | (105, 5460, ?, ?, ?, 90262) |
7 | 14 | (945, 445410, ?, ?, ?, ?, ?) |
ℬ | ℬ | ℬ |
n | $\left(\begin{array}{c}n\\ 2\end{array}\right)$ - n | ((2n - 5)!!,?, ...) |
4 Neighbor-joining cones
The nodes a, b are replaced by a single node z, and new distances d_{ zk }are obtained by a straightforward linear combination of the original pairwise distances:${d}_{zk}:=\frac{1}{2}({d}_{ak}+{d}_{bk}-{d}_{ab})$. Then the NJ method is applied recursively.
We note that since new distances d_{ zk }are always linear combinations of the previous distances, all Q-criteria computed throughout the NJ algorithm are linear combinations of the original pairwise distances. Thus, for a fixed n, for every possible ordering σ of picked cherries that results in one of the trees T with n leaves there is a polyhedral cone C_{ σ }⊂ ${\mathcal{R}}^{\left(\begin{array}{c}n\\ 2\end{array}\right)}$ of dissimilarity maps. The set of all neighbor-joining cones is denoted by ${\mathcal{C}}_{n}$. Their union ${\cup}_{C\in {\mathcal{C}}_{n}}C$ is all of of ${\mathcal{R}}^{\left(\begin{array}{c}n\\ 2\end{array}\right)}$, and the intersection of any two cones is a subset – but not necessarily a face – of the boundary of each of the cones. Given an input from the interior of C_{ σ }, the NJ algorithm will pick the cherries in the order σ and output the corresponding tree. For inputs d on the boundary of one (and therefore at least two) of the cones, the order in which NJ picks cherries is undefined, because at some point there will be two cherries both of which have minimal Q-criterion. 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.
Example. There is only one unlabeled binary tree with 5 leaves and there are 15 distinct labeled trees. For each labeled tree, there are two ways in which a cherry might be picked by the NJ algorithm in the first step. For instance, neighbor-joining applied to any dissimilarity map in C_{12,45} or C_{45,12} will produce the tree in Figure 2. There are a total of 30 NJ cones for n = 5.
We note that all Q-criteria for shift vectors equal -2, so adding any linear combination of shift vectors to a dissimilarity map does not change the relative values of the Q-criteria. Also, after picking a cherry, the reduced distance matrix of a shift vector is again a shift vector. 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. modulo the lineality space.
Theorem 4.1
The cones in ${\mathcal{C}}_{n}$do not form a fan. In particular, they are not the normal fan of any polytope for n ≥ 5.
The theorem follows from that fact that the NJ cones have rays which are on the boundary of other cones but not rays of them. Thus there are pairs of cones whose intersection is not a face of both cones. We describe the case n = 5 in detail; it also suffices to prove the theorem.
We begin by noting that all of the NJ cones are equivalent under the action of the symmetric group on five elements (S_{5}), where an element of S_{5} permutes the five taxa or, equivalently, the rows and columns of the input distance matrix. Each NJ cone is defined by $\left(\left(\begin{array}{c}5\\ 2\end{array}\right)-1\right)+\left(\left(\begin{array}{c}4\\ 2\end{array}\right)-1\right)=14$ inequalities that are implied by the Q-criteria as the NJ algorithm picks the two cherries. The cones are 5-dimensional, and their intersection with a suitable hyperplane leaves a four dimensional polytope P. The f-vector of P is (14, 32, 27, 9).
The 14 rays of the cone C_{23,45}. Each ray is determined by a vector shown in the second column. The third column shows, for each ray, which cones it belongs to. If a cone is starred then the ray is on the boundary of that cone, but not a ray of it.
Type | rays | Cones |
---|---|---|
I | (-3, 5, -3, -1, 5, -3, -1, 1, 1, -1) (-3, 5, -3, -1, 1, 1, -1, 5, -3, -1) (5, -3, -3, -1, -3, 5, -1, 1, 1, -1) (1, 1, -3, -1, -3, 5, -1, 5, -3, -1) (5, -3, -3, -1, 1, 1, -1, -3, 5, -1) (1, 1, -3, -1, 5, -3, -1, -3, 5, -1) | C_{23,45}, C_{23,15}, C_{23,14}, ${C}_{12,34}^{\ast}$, ${C}_{34,12}^{\ast}$ C_{23,45}, C_{23,15}, C_{23,14}, ${C}_{12,35}^{\ast}$, ${C}_{35,12}^{\ast}$ C_{23,45}, C_{23,15}, C_{23,14}, ${C}_{24,13}^{\ast}$, ${C}_{13,24}^{\ast}$ C_{23,45}, C_{23,15}, C_{23,14}, ${C}_{25,13}^{\ast}$, ${C}_{25,13}^{\ast}$ C_{23,45}, C_{23,15}, C_{23,14}, ${C}_{24,35}^{\ast}$, ${C}_{35,24}^{\ast}$ C_{23,45}, C_{23,15}, C_{23,14}, ${C}_{25,34}^{\ast}$, ${C}_{25,34}^{\ast}$ |
II | (-1, 1, -1, 1, 1, -1, -1, 1, 1, -1) (-1, 1, -1, -1, 1, 1, 1, 1, -1, -1) (1, 1, -1, -1, -1, 1, -1, 1, -1, 1) (1, -1, -1, 1, -1, 1, -1, 1, 1, -1) | C_{12,45}, C_{12,34}, C_{23,45}, C_{23,15}, C_{34,15}, C_{34,12}, C_{45,23}, C_{45,12}, C_{15,34}, C_{15,23} C_{12,45}, C_{12,35}, C_{23,45}, C_{23,14}, C_{35,14}, C_{35,12}, C_{45,23}, C_{45,12}, C_{14,35}, C_{14,23} C_{25,14}, C_{25,13}, C_{23,14}, C_{23,45}, C_{13,45}, C_{13,25}, C_{14,23}, C_{14,25}, C_{45,13}, C_{45,23} C_{24,15}, C_{24,13}, C_{23,15}, C_{23,45}, C_{13,45}, C_{13,24}, C_{15,23}, C_{15,24}, C_{45,13}, C_{45,23} |
III | (1, -1, -1, 1, 1, -1, -1, -1, 3, -1) (1, -1, -1, -1, -1, 3, 1, 1, -1, -1) (1, -1, -1, 1, 1, -1, -1, -1, 3, -1) (1, -1, -1, -1, -1, 3, 1, 1, -1, -1) | C_{23,45}, C_{23,15}, C_{12,45}, C_{12,35}, C_{24,15}, C_{24,35}, C_{35,24}, C_{35,12}, C_{15,24}, C_{15,23}, C_{45,12}, C_{45,23} C_{23,45}, C_{23,14}, C_{12,45}, C_{12,34}, C_{25,14}, C_{25,34}, C_{34,25}, C_{34,12}, C_{14,25}, C_{14,23}, C_{45,12}, C_{45,23} C_{23,45}, C_{23,15}, C_{13,45}, C_{13,25}, C_{34,15}, C_{34,25}, C_{25,34}, C_{25,13}, C_{15,34}, C_{15,23}, C_{45,13}, C_{45,23} C_{23,45}, C_{23,14}, C_{13,45}, C_{13,24}, C_{35,14}, C_{35,24}, C_{24,35}, C_{24,13}, C_{14,35}, C_{14,23}, C_{45,13}, C_{45,23} |
We note that the rays of NJ cones are minimal intersections of NJ cones, and thus give dissimilarity maps for which the NJ algorithm is least stable.
where g_{ ij }is the fraction of different nucleotides between sequences i and j in the second set.
Notice that the vector representation of D_{1} lies in the cone C_{12,45} and the vector representation of D_{2} lies in the cone C_{45,12}. Thus NJ returns the same tree topology for both D_{1} and D_{2}.
However, the vector representation of D_{3} lies in the cone C_{24,15}, which means that neighbor-joining returns a different tree topology for D_{3}. This example provides a distance-based recon-struction analog to the recent mixture model results of [14].
An analysis of the rays of ${\mathcal{C}}_{n}$ suffices to prove Theorem 4.1. but the facet structure of each cone is also informative, and we were able to obtain complete information for n = 5. The types of facets constituting each cone are shown in Figure 1. Each cone consists of one Type A facet, two Type B facets, two Type C facets and four Type D facets. These facets intersect as follows: Type A facets are shared by pairs of cones of the form C_{ab,cd}, C_{cd,ab}. Type B facets are shared by pairs of cones of the form C_{ab,de}, C_{ab,ce}; there are two such pairs for each cone. Two of the square facets of a Type A facet belong to Type B facets, and a pair of Type B facets share a hexagon consisting of six Type I rays. The remaining two square facets of a Type A facet form Type C facets with two Type I rays. The four triangular facets of a Type A facet form Type D facets (Egyptian pyramids) with two Type I rays.
The l_{2} radius is obtained by dividing (4) by min(α, β), so the minimum is attained at α = β = $\frac{1}{2}$
Theorem 4.2
The l_{2} radius of neighbor-joining for 5 taxa is $\frac{1}{\sqrt{3}}$ ≈ 0.5773.
This is slightly larger than the l_{∞} radius of $\frac{1}{2}$ given by Atteson's theorem [15]. It is an interesting problem to compute the l_{2} radius for neighbor-joining with more taxa.
The description of the NJ cones we have provided can also be used in practice to evaluate the robustness of the algorithm when used with a specific dataset. For n = 5, we examined data simulated from subtrees of the two tree models T_{1} and T_{2} in [16] with the Jukes-Cantor model and the Kimura 2-parameter models [6]. For each of 40, 000 simulations, we calculated the ℓ_{2}-distance between the NJ cone of the given tree and the maximum likelihood estimates for the pairwise distances (see supplementary material). These show that in many cases the maximum likelihood estimates lie very close to the boundary. In such cases, one must conclude that the NJ tree is possibly incorrect due to the variance in the distance estimates.
5 Optimality of the neighbor-joining algorithm
In order to study the optimality of the neighbor-joining algorithm, we compared the BME cones with the NJ cones. Such a comparison involves intersecting the cones with the ($\left(\begin{array}{c}n\\ 2\end{array}\right)$ - 1)-sphere (in the first orthant) and then studying the volumes of their intersection by computing the standard Euclidean volume of the resulting surfaces. These surfaces are an intersection of closed hemispheres, i.e. spherical polytopes. Computing Euclidean volumes of (non-spherical) polytopes is a standard problem that is usually solved by triangulating and summing the volumes of the simplices. However there has been no publicly available software developed for computing or approximating volumes of spherical polytopes of dimension > 3 using this method. One possible reason for this is that in higher dimensions the volumes of spherical simplices are given by complicated analytical formulas [17] whose computational complexities are unknown.
We implemented two approaches in MATLAB (using polymake as a preprocessing step) for approximating the volume of a spherical polytope P. One approach is trivial: it simply samples uniformly from the sphere, and counts how many points are inside P. This approach is particularly suitable if P has large volume, or if many spherical polytopes are being simultaneously measured which partition the sphere, as is the case for NJ and BME cones. The second approach is suitable for spherical polytopes having small volume. We used this approach for computing the volumes of consistency cones [18] which we discuss briefly in the Discussion section.
The second approach begins by computing a triangulation of the vertices of P with some additional interior points of P added. This triangulation defines a simplicial mesh M which is obtained by replacing each spherical simplex with the corresponding Euclidean simplex having the same vertices. The volume of M (i.e. the sum of the volumes of the simplices in the mesh) is already an approximation to the volume of P. We refine this estimate by Monte Carlo estimation of the average value of the Jacobian from M to P. This requires sampling uniformly from M, which can be done very quickly in O(m + kd log d + k log k) time, where m is the number of simplices in the mesh, k is the number of samples, and d is the dimension. Briefly, the method partitions the unit interval into m subintervals, where the length of the i th subinterval is proportional to the volume of the i th simplex S_{ i }in the mesh. Then to sample k points from the mesh, first we decide how many of the k samples to draw from each S_{ i }, by sampling uniformly from unit interval k times. For each S_{ i }, we sample ℓ_{ i }points uniformly from S_{ i }where ℓ_{ i }is the number of samples x ∈ [0, 1] which land in the i th subinterval. Sampling uniformly from a single simplex is a classical problem solved in O(d log d) time.
Comparison of NJ and BME cones. The volume estimates for n = 8 do not all add up to exactly 100% due to round-off errors
#taxa | tree shape | #trees | NJ vol | BME vol | NJ accuracy |
---|---|---|---|---|---|
4 | unique | 3 | 100% | 100% | 100% |
5 | unique | 15 | 100% | 100% | 98.06% |
6 | 3-cherry | 15 | 18.49% | 18.57% | 90.39% |
6 | caterpillar | 90 | 81.51% | 81.43% | 91.33% |
7 | 3-cherry | 315 | 45.32% | 44.58% | 82.42% |
7 | caterpillar | 630 | 54.68% | 55.42% | 78.85% |
8 | 4-cherry | 315 | 6.48% | 6.36% | 70.12% |
8 | 3-cherry (two are neighbors) | 2520 | 27.12% | 25.84% | 69.93% |
8 | 3-cherry (none are neighbors) | 2520 | 35.67% | 34.55% | 71.63% |
8 | caterpillar | 5040 | 30.73% | 33.24% | 61.75% |
Figure 4 can be interpreted as follows: For a random dissimilarity map, if the NJ algorithm does not produce the BME tree, then with probability 0.67 it produces the tree on the right, and if not then it almost always produces the tree in the middle. This tree differs from the BME tree significantly. A surprising result is that the tree on the left is almost never the NJ tree. We believe that a deeper understanding of the "mistakes" NJ makes when it does not optimize the balanced minimum evolution criterion may be important in interpreting the results, especially for large trees.
We also computed analogous results for n = 6, 7, 8, 9, 10. They are available, together with the software for computing volumes at [19].
6 Discussion
Theoretical studies of the neighbor-joining algorithm have focused on statistical consistency and the robustness of the algorithm to small perturbations of tree metrics. The paper by [20] established the consistency of NJ, that is, if D_{ T }is a tree metric then NJ outputs the tree T. This result was then extended in [15] and more recently by [18] who show that if D is "close" to a tree metric D_{ T }for some T, then NJ outputs T on input D.
Our results provide a different perspective on the NJ algorithm. Namely, we address the question of the accuracy of the greedy approach for the underlying linear programming problem of BME optimization. This led us to the study of BME polytopes, and the combinatorics of these polytopes is interesting in its own right:
Question 6.1
Is there a combinatorial criterion for two tree topologies forming an edge in the BME polytope, similar to pruning/re-grafting or some other operation on trees? If so, this could be used to define a combinatorial pivoting rule on tree space that could be used in hill-climbing algorithms for phylogenetic reconstruction. Such a pivoting rule would have the advantage that it would be equivalent to performing an edge-walk on the BME polytope. Edge-walking methods are known to perform well in practice for solving linear programs. See [21]for an example of a local search approach to finding minimum evolution trees.
Similarly, a better understanding of the combinatorics of the NJ cones will lead to a clearer view of the strengths and weaknesses of the neighbor-joining algorithm. A basic problem is the following:
Question 6.2
Find a combinatorial description of the NJ cones for general n. How many facets/rays are there?
Our computational results lend new insights into the performance of the NJ and BME algorithms for small trees. We have measured the relative sizes of cones for different shapes of trees, and measured the frequencies of all combinatorial types of discrepancies between BME and NJ trees. In particular, we have observed that the NJ algorithm is least likely to reproduce the BME tree when the BME tree is the caterpillar tree.
Conjecture 6.3
For n > 6, it is the caterpillar tree that yields the smallest ratio of spherical cone volumes vol(NJ ⋂ BME)/vol(BME) where NJ is the spherical cone volume of a union of the NJ cones and BME is the spherical cone volume of the BME cone for a fixed tree. In other words, the caterpillar tree is the most difficult BME tree topology for the NJ algorithm to reproduce.
Another problem we believe is very important is to extend the results shown in Figure 4 to large trees. In other words, to understand how neighbor-joining can fail when it does not succeed in finding the balanced minimum evolution tree.
Question 6.4
What tree topologies is neighbor-joining likely to pick when it fails to construct the balanced minimum evolution tree?
There are many other interesting cones related to distance-based methods that can be considered in this context. For example, in [18], it is shown that the quartet consistency condition is sufficient for neighbor-joining to reconstruct a tree from a dissimilarity map for n ≤ 7 leaves. The quartet consistency conditions define polyhedral cones (consistency cones) in ${\mathcal{R}}^{\left(\begin{array}{c}n\\ 2\end{array}\right)}$; see [18] for details. For n = 4 taxa the consistency cones cover all of ${\mathcal{R}}^{\left(\begin{array}{c}4\\ 2\end{array}\right)}$ showing that quartet consistency explains the behavior of neighbor-joining for all dissimilarity maps. Using the second method outlined in Section 4 we succeeded in computing the volumes of the consistency cones intersected with the first orthant of the sphere for n = 5 taxa. There are 15 cones, all equivalent under orthogonal transformation, and their union covers 27.93% of ${\mathcal{R}}_{+}^{\left(\begin{array}{c}5\\ 2\end{array}\right)}$, measured with respect to spherical volume. In other words, quartet consistency explains the behavior of neighbor-joining on almost $\frac{1}{3}$ of dissimilarity maps.
Such computations are pushing the boundary of computational polyhedral geometry. For n ≥ 6 taxa, triangulating a consistency cone is too unwieldy, although we are confident that spherical volumes could still be computed using polynomial time hit-and-run sampling methods for volume approximation [22]. Such methods are complicated and not yet implemented.
Finally, we comment on the example in Section 3 that shows how different alignments may lead to the same neighbor-joining tree, whereas the neighbor-joining tree constructed from a concatenation of the alignments is different. This result has significant implications for studies where species trees are constructed from multiple gene families by combining the data.
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