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:
Let D={dij}i,j=1n
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraqKaeyypa0Jaei4EaSNaemizaq2aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGG9bqFdaqhaaWcbaGaemyAaKMaeiilaWIaemOAaOMaeyypa0JaeGymaedabaGaemOBa4gaaaaa@3C4B@ be a dissimilarity map (this is an n × n symmetric matrix with zeroes on the diagonals and non-negative real entries). The balanced minimum evolution problem is to find the unrooted binary tree T with n leaves that minimizes
1
|
o
(
T
)
|
∑
(
x
1
,
...
,
x
n
)
∈
o
(
T
)
[
1
2
∑
i
=
1
n
d
x
i
x
i
+
1
]
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5EB5@
Here o(T) is the set of all cyclic permutations of the leaves that arise from planar embeddings of T and xi are leaves of T. Denote by pijT
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiCaa3aa0baaSqaaiabdMgaPjabdQgaQbqaaiabdsfaubaaaaa@3154@ the set of internal vertices in a tree T on the path between i and j. Then (1) is equivalent to minimizing
∑
i
j
λ
i
j
T
d
i
j
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaabuaeaacqaH7oaBdaqhaaWcbaGaemyAaKMaemOAaOgabaGaemivaqfaaOGaemizaq2aaSbaaSqaaiabdMgaPjabdQgaQbqabaaabaGaemyAaKMaemOAaOgabeqdcqGHris5aaaa@3AFC@
where λijT=∏v∈pijT(deg(v)−1)−1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4UdW2aa0baaSqaaiabdMgaPjabdQgaQbqaaiabdsfaubaakiabg2da9maarababaGaeiikaGIaemizaqMaemyzauMaem4zaCMaeiikaGIaemODayNaeiykaKIaeyOeI0IaeGymaeJaeiykaKYaaWbaaSqabeaacqGHsislcqaIXaqmaaaabaGaemODayNaeyicI4SaemiCaa3aa0baaWqaaiabdMgaPjabdQgaQbqaaiabdsfaubaaaSqab0Gaey4dIunaaaa@49B5@ if i ≠ j and λijT=0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4UdW2aa0baaSqaaiabdMgaPjabdQgaQbqaaiabdsfaubaakiabg2da9iabicdaWaaa@339D@. 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
Let T be a tree with n leaves and l: E(T) → R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@an assignment of lengths to the edges. Then the length l(T) of T is defined to be
l
(
T
)
=
∑
e
∈
E
(
T
)
l
(
e
)
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiBaWMaeiikaGIaemivaqLaeiykaKIaeyypa0ZaaabuaeaacqWGSbaBcqGGOaakcqWGLbqzcqGGPaqkaSqaaiabdwgaLjabgIGiolabdweafjabcIcaOiabdsfaujabcMcaPaqab0GaeyyeIuoakiabc6caUaaa@3FB1@
Theorem 1.2
(4)Let T be a binary tree with edge lengths given by l: E(T) → R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@+ and D={dij}i,j=1n
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiraqKaeyypa0Jaei4EaSNaemizaq2aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGG9bqFdaqhaaWcbaGaemyAaKMaeiilaWIaemOAaOMaeyypa0JaeGymaedabaGaemOBa4gaaaaa@3C4B@a dissimilarity map. If the variance of dij is proportional to 2|pijT|
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeGOmaiZaaWbaaSqabeaadaabdaqaaiabdchaWnaaDaaameaacqWGPbqAcqWGQbGAaeaacqWGubavaaaaliaawEa7caGLiWoaaaaaaa@35A1@(i. e., var(dij) = c2|pijT|
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4yamMaeGOmaiZaaWbaaSqabeaadaabdaqaaiabdchaWnaaDaaameaacqWGPbqAcqWGQbGAaeaacqWGubavaaaaliaawEa7caGLiWoaaaaaaa@36F0@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 is the convex hull of the vectors
{
[
λ
12
T
,
λ
13
T
,
...
,
λ
i
j
T
,
...
,
λ
n
−
1
,
n
T
]
:
T
i
s
a
t
r
e
e
w
i
t
h
n
l
e
a
v
e
s
}
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaiWaaeaadaWadaqaaiabeU7aSnaaDaaaleaacqaIXaqmcqaIYaGmaeaacqWGubavaaGccqGGSaalcqaH7oaBdaqhaaWcbaGaeGymaeJaeG4mamdabaGaemivaqfaaOGaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaeq4UdW2aa0baaSqaaiabdMgaPjabdQgaQbqaaiabdsfaubaakiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiabeU7aSnaaDaaaleaacqWGUbGBcqGHsislcqaIXaqmcqGGSaalcqWGUbGBaeaacqWGubavaaaakiaawUfacaGLDbaacqGG6aGocqWGubavcqqGGaaicqWGPbqAcqWGZbWCcqqGGaaicqWGHbqycqqGGaaicqWG0baDcqWGYbGCcqWGLbqzcqWGLbqzcqqGGaaicqWG3bWDcqWGPbqAcqWG0baDcqWGObaAcqqGGaaicqWGUbGBcqqGGaaicqWGSbaBcqWGLbqzcqWGHbqycqWG2bGDcqWGLbqzcqWGZbWCaiaawUhacaGL9baaaaa@721A@
Example. There are four trees with n = 4 leaves. They are the 3 binary trees and the star-shaped tree. In this case the balanced minimum evolution polytope is the convex hull of the vectors:
[
1
2
,
1
4
,
1
4
,
1
4
,
1
4
,
1
2
]
T is the tree with leaves
1
,
2
seperated from
3
,
4
,
[
1
4
,
1
2
,
1
4
,
1
4
,
1
2
,
1
4
]
T is the tree with leaves
1
,
3
seperated from
2
,
4
,
[
1
4
,
1
4
,
1
2
,
1
2
,
1
4
,
1
4
]
T is the tree with leaves
1
,
4
seperated from
2
,
3
,
[
1
3
,
1
3
,
1
3
,
1
3
,
1
3
,
1
3
]
T is the star-shaped tree
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5203@
The balanced minimum evolution polytope in this case is a triangle in R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@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 λijT=21−|pijT|
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4UdW2aa0baaSqaaiabdMgaPjabdQgaQbqaaiabdsfaubaakiabg2da9iabikdaYmaaCaaaleqabaGaeGymaeJaeyOeI0YaaqWaaeaacqWGWbaCdaqhaaadbaGaemyAaKMaemOAaOgabaGaemivaqfaaaWccaGLhWUaayjcSdaaaaaa@3E58@; this is Pauplin's formula 5.
The BME polytope lies in R(n2)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHi1aaWbaaSqabeaadaqadaqaauaabeqaceaaaeaacqWGUbGBaeaacqaIYaGmaaaacaGLOaGaayzkaaaaaaaa@3ABF@ and has dimension (n2)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaeWaaeaafaqabeGabaaabaGaemOBa4gabaGaeGOmaidaaaGaayjkaiaawMcaaaaa@2FC2@ - 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 R+(n2)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHi1aa0baaSqaaiabgUcaRaqaamaabmaabaqbaeqabiqaaaqaaiabd6gaUbqaaiabikdaYaaaaiaawIcacaGLPaaaaaaaaa@3BA1@. 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 R(n2)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHi1aaWbaaSqabeaadaqadaqaauaabeqaceaaaeaacqWGUbGBaeaacqaIYaGmaaaacaGLOaGaayzkaaaaaaaa@3ABF@. 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 R(n2)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHi1aaWbaaSqabeaadaqadaqaauaabeqaceaaaeaacqWGUbGBaeaacqaIYaGmaaaacaGLOaGaayzkaaaaaaaa@3ABF@, 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.
Let {y1, y2, ..., ym} be a finite set of points in R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@d. An affine linear combination is a linear combination of the form
y
=
∑
i
=
1
m
α
i
y
i
,
where
∑
i
=
1
m
α
i
=
1.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabeWaaaqaaiabdMha5jabg2da9maaqahabaGaeqySde2aaSbaaSqaaiabdMgaPbqabaGccqWG5bqEdaWgaaWcbaGaemyAaKgabeaaaeaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGTbqBa0GaeyyeIuoakiabcYcaSaqaaiabbEha3jabbIgaOjabbwgaLjabbkhaYjabbwgaLbqaamaaqahabaGaeqySde2aaSbaaSqaaiabdMgaPbqabaGccqGH9aqpcqaIXaqmcqGGUaGlaSqaaiabdMgaPjabg2da9iabigdaXaqaaiabd2gaTbqdcqGHris5aaaaaaa@5093@
A convex linear combination is an affine linear combination with nonnegative linear coefficients, i.e. αi ≥ 0 for i = 1, ..., m. The affine hull of a set C ⊆ R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@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 ⊂ R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@d can be written as
a + V = {a + v : v ⊆ V}
where V ⊆ R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@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 ∈ R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@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 ⊂ R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@d is convex iff [x, y] ⊂ C for any two x, y ∈ C.
Let A1, A2, ..., AN ∈ R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@d and let b1, b2, ..., bN ∈ R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@. Then the set
P
:
=
{
x
∈
R
d
:
A
i
⋅
x
≤
b
i
for
i
=
1
,
2
,
...
,
N
}
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiuaaLaeiOoaOJaeyypa0Jaei4EaSNaemiEaGNaeyicI48enfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHi1aaWbaaSqabeaacqWGKbazaaGccqGG6aGocqWGbbqqdaWgaaWcbaGaemyAaKgabeaakiabgwSixlabdIha4jabgsMiJkabdkgaInaaBaaaleaacqWGPbqAaeqaaOGaeeiiaaIaeeOzayMaee4Ba8MaeeOCaiNaeeiiaaIaemyAaKMaeyypa0JaeGymaeJaeiilaWIaeGOmaiJaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaemOta4KaeiyFa0haaa@5DF6@
is called a polyhedron. The convex hull of a finite set of points in R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@d is called a polytope and the Weyl-Minkowski Theorem says that a polytope is a bounded polyhedron 9. Polytopes are familiar objects in geometry. In the plane, polytopes are precisely the convex polygons. In R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@3, examples of polytopes are shown in Figure 1. The dimension dim P of a polytope or polyhedron P is defined to be the dimension of the affine hull of P.
<p>Figure 1</p>
The four types of facets of P
The four types of facets of P.
![]()
A (d - 1) dimensional affine set in R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@d is called a hyperplane and every hyperplane can be represented as {x ∈ R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@d: n·x = b} for some n ≠ 0 ∈ R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@d and b ∈ R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@, where n·x is the dot-product of n and x. We call n a normal vector of this hyperplane.
Let H := {x ∈ R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@d : h·x ≤ b}, where h ≠ 0 ∈ R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@d and b ∈ R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@, be an affine half space. Then if P ⊂ H and P ⋂ {x ∈ R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@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 (f0, f1, f2, ...), where fi 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).
A polyhedron C is a cone if it can be written as
C
=
{
∑
i
=
1
N
α
i
y
i
:
α
i
≥
0
for
i
=
1
,
...
,
N
}
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4qamKaeyypa0ZaaiWaaeaadaaeWbqaaiabeg7aHnaaBaaaleaacqWGPbqAaeqaaOGaemyEaK3aaSbaaSqaaiabdMgaPbqabaGccqGG6aGocqaHXoqydaWgaaWcbaGaemyAaKgabeaakiabgwMiZkabicdaWiabbccaGiabbAgaMjabb+gaVjabbkhaYjabbccaGiabdMgaPjabg2da9iabigdaXiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiabd6eaobWcbaGaemyAaKMaeyypa0JaeGymaedabaGaemOta4eaniabggHiLdaakiaawUhacaGL9baaaaa@52DA@
for some y1, ..., yN ∈ R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@d. This is equivalent to the existence of a matrix A ∈ R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@m × n such that C = {x : Ax ≥ 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 = maxx∈P c·x for all v ∈ F. The collection of relative interiors of normal cones of faces of P partition R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@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' = R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@d) and consider the polyhedron P' := P ⋂ V', which has lineality space {0}.
3 The balanced minimum evolution polytope
Throughout this paper we work with binary unrooted trees on n leaves labeled {1, ..., n}. Such trees are also known as phylogenetic X-trees. We refer the reader to 10 for more detail about such trees, and for related definitions. Recall there are 2n - 3 edges in an unrooted tree with n leaves. For a fixed tree topology T, let BT be the (n2)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaeWaaeaafaqabeGabaaabaGaemOBa4gabaGaeGOmaidaaaGaayjkaiaawMcaaaaa@2FC2@ × (2n - 3) matrix with rows indexed by pairs of leaves and columns indexed by edges in T defined as follows:
B
T
(
{
a
,
b
}
,
e
)
=
{
1
if edge
e
is in the path from leaf
a
to leaf
b
,
0
otherwise
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@826B@
For example, for the tree in Figure 2,
<p>Figure 2</p>
A tree with five leaves
A tree with five leaves.
![]()
B
T
=
(
1
1
0
0
0
0
0
1
0
1
0
0
1
0
0
1
1
0
0
1
0
1
0
0
1
0
1
1
0
1
0
1
0
1
1
0
0
1
1
0
0
1
1
0
0
0
1
1
1
0
1
0
0
1
1
1
0
0
1
0
1
0
1
0
0
0
1
1
0
0
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOqai0aaSbaaSqaaiabdsfaubqabaGccqGH9aqpdaqadaqaauaabeqakCaaaaaaaeaacqaIXaqmaeaacqaIXaqmaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIXaqmaeaacqaIWaamaeaacqaIXaqmaeaacqaIWaamaeaacqaIWaamaeaacqaIXaqmaeaacqaIWaamaeaacqaIWaamaeaacqaIXaqmaeaacqaIXaqmaeaacqaIWaamaeaacqaIWaamaeaacqaIXaqmaeaacqaIWaamaeaacqaIXaqmaeaacqaIWaamaeaacqaIWaamaeaacqaIXaqmaeaacqaIWaamaeaacqaIXaqmaeaacqaIXaqmaeaacqaIWaamaeaacqaIXaqmaeaacqaIWaamaeaacqaIXaqmaeaacqaIWaamaeaacqaIXaqmaeaacqaIXaqmaeaacqaIWaamaeaacqaIWaamaeaacqaIXaqmaeaacqaIXaqmaeaacqaIWaamaeaacqaIWaamaeaacqaIXaqmaeaacqaIXaqmaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIXaqmaeaacqaIXaqmaeaacqaIXaqmaeaacqaIWaamaeaacqaIXaqmaeaacqaIWaamaeaacqaIWaamaeaacqaIXaqmaeaacqaIXaqmaeaacqaIXaqmaeaacqaIWaamaeaacqaIWaamaeaacqaIXaqmaeaacqaIWaamaeaacqaIXaqmaeaacqaIWaamaeaacqaIXaqmaeaacqaIWaamaeaacqaIWaamaeaacqaIWaamaeaacqaIXaqmaeaacqaIXaqmaeaacqaIWaamaeaacqaIWaamaaaacaGLOaGaayzkaaaaaa@72D9@
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) → R
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHifaaa@36A5@+ 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 = BT b + e
where e is a vector of "error" terms that are zero when d is a tree metric.
The weighted least squares solution for the edge lengths b assuming a variance matrix V with off-diagonal entries vij=λijT
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemODay3aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGH9aqpcqaH7oaBdaqhaaWcbaGaemyAaKMaemOAaOgabaGaemivaqfaaaaa@3708@ (as defined in the introduction) and dissimilarity map d is given by
b
^
=
(
B
T
t
V
−
1
B
T
)
−
1
B
T
t
V
−
1
d
,
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaacbeGaf8NyaiMbaKaacqGH9aqpcqGGOaakcqWGcbGqdaqhaaWcbaGaemivaqfabaGaemiDaqhaaOGaemOvay1aaWbaaSqabeaacqGHsislcqaIXaqmaaGccqWGcbGqdaWgaaWcbaGaemivaqfabeaakiabcMcaPmaaCaaaleqabaGaeyOeI0IaeGymaedaaOGaemOqai0aa0baaSqaaiabdsfaubqaaiabdsha0baakiabdAfawnaaCaaaleqabaGaeyOeI0IaeGymaedaaOGae8hzaqMaeiilaWcaaa@4551@
where ·t denotes matrix transpose. The length of T with respect to the least squares edge lengths is then
l(T) = vT·d,
where vT=V−1BT(BTtV−1BT)−1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaacbeGae8NDay3aaSbaaSqaaiabdsfaubqabaGccqGH9aqpcqWGwbGvdaahaaWcbeqaaiabgkHiTiabigdaXaaakiabdkeacnaaBaaaleaacqWGubavaeqaaOGaeiikaGIaemOqai0aa0baaSqaaiabdsfaubqaaiabdsha0baakiabdAfawnaaCaaaleqabaGaeyOeI0IaeGymaedaaOGaemOqai0aaSbaaSqaaiabdsfaubqabaGccqGGPaqkdaahaaWcbeqaaiabgkHiTiabigdaXaaaaaa@4327@1 and 1 is the vector of all 1's. We call the vectors vT the balanced minimum evolution vectors (or BME vectors). In the case of Figure 2, the BME vector is
v
T
=
[
1
2
,
1
4
,
1
4
,
1
4
,
1
4
,
1
4
,
1
4
,
1
4
,
1
4
,
1
2
]
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@4F85@
The BME method is equivalent to minimizing the linear functional vT·d over all BME vectors for all tree topologies T. The BME polytope is the convex hull of all BME vectors in R(n2)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHi1aaWbaaSqabeaadaqadaqaauaabeqaceaaaeaacqWGUbGBaeaacqaIYaGmaaaacaGLOaGaayzkaaaaaaaa@3ABF@. 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 R(n2)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHi1aaWbaaSqabeaadaqadaqaauaabeqaceaaaeaacqWGUbGBaeaacqaIYaGmaaaacaGLOaGaayzkaaaaaaaa@3ABF@ 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 sa 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, (vT)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.
As part of our computational study, we computed the BME polytope and BME cones for trees with n = 4, 5, 6, 7, 8 leaves using the software polymake 12. In Table 1 we display some of the components of f-vectors we were able to compute. This provides information about the polytopes: Recall that the ith component of the f-vector of a polytope is the number of faces of dimension i - 1. For example, the first component in each vector in Table 1 is the number of 0-dimensional faces (vertices) of the corresponding BME polytope, i.e., the number of binary trees.
<p>Table 1</p>
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
(n2)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaeWaaeaafaqabeGabaaabaGaemOBa4gabaGaeGOmaidaaaGaayjkaiaawMcaaaaa@2FC2@ - n
((2n - 5)!!,?, ...)
We found that the edge graph of the BME polytope is the complete graph for n = 4, 5, 6 which means that for every pair of trees T1 and T2 with the same number (≤ 6) of leaves, there is a dissimilarity map for which T1 and T2 are (the only) co-optimal BME trees. However, for n = 7, the BME polytope does in fact have one combinatorial type of non-edge. Namely, two bifurcating trees with seven leaves and three cherries (two leaves adjacent to the same node in the tree) will form a non-edge if and only if they are related by two leaf exchanges as depicted in Figure 3. This completely characterizes the non-edges for n = 7. It is an interesting open problem to characterize the non-edges of the BME polytope in general.
<p>Figure 3</p>
The non-edges on the BME polytope for n = 7
The non-edges on the BME polytope for n = 7. Two trees will form a non-edge if and only if they are trees that have three cherries, and differ by the pair of leaf exchanges shown in the figure. There are two ways to perform each leaf-exchange, so each binary tree with three cherries is not adjacent to 4 trees.
![]()
4 Neighbor-joining cones
The neighbor-joining algorithm takes as input a dissimilarity map and outputs a tree. The tree is constructed "one cherry at a time". In each step the algorithm chooses a pair of leaves a and b that minimize the Q-criterion, which is defined by the formula
q
a
b
:
=
(
n
−
2
)
d
a
b
−
∑
k
=
1
N
d
a
k
−
∑
k
=
1
n
d
k
b
.
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyCae3aaSbaaSqaaiabdggaHjabdkgaIbqabaGccqGG6aGocqGH9aqpcqGGOaakcqWGUbGBcqGHsislcqaIYaGmcqGGPaqkcqWGKbazdaWgaaWcbaGaemyyaeMaemOyaigabeaakiabgkHiTmaaqahabaGaemizaq2aaSbaaSqaaiabdggaHjabdUgaRbqabaaabaGaem4AaSMaeyypa0JaeGymaedabaGaemOta4eaniabggHiLdGccqGHsisldaaeWbqaaiabdsgaKnaaBaaaleaacqWGRbWAcqWGIbGyaeqaaaqaaiabdUgaRjabg2da9iabigdaXaqaaiabd6gaUbqdcqGHris5aOGaeiOla4caaa@5437@
The nodes a, b are replaced by a single node z, and new distances dzk are obtained by a straightforward linear combination of the original pairwise distances:dzk:=12(dak+dbk−dab)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemizaq2aaSbaaSqaaiabdQha6jabdUgaRbqabaGccqGG6aGocqGH9aqpjuaGdaWcaaqaaiabigdaXaqaaiabikdaYaaakiabcIcaOiabdsgaKnaaBaaaleaacqWGHbqycqWGRbWAaeqaaOGaey4kaSIaemizaq2aaSbaaSqaaiabdkgaIjabdUgaRbqabaGccqGHsislcqWGKbazdaWgaaWcbaGaemyyaeMaemOyaigabeaakiabcMcaPaaa@44C8@. Then the NJ method is applied recursively.
We note that since new distances dzk 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 Cσ ⊂ R(n2)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae83gHi1aaWbaaSqabeaadaqadaqaauaabeqaceaaaeaacqWGUbGBaeaacqaIYaGmaaaacaGLOaGaayzkaaaaaaaa@3ABF@ of dissimilarity maps. The set of all neighbor-joining cones is denoted by Cn
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8NaXp0aaSbaaSqaaiabd6gaUbqabaaaaa@38DE@. Their union ∪C∈CnC
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaambeaeaacqWGdbWqaSqaaiabdoeadjabgIGioprtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=jq8dnaaBaaameaacqWGUbGBaeqaaaWcbeqdcqWIQisvaaaa@3E00@ is all of of R(n2<