Background
Problems
Results
Dissimilarity Vectors of Trees and Their
Tropical Linear Spaces
Benjamin Iriarte
San Francisco State University
San Francisco CA USA
M.A. Mathematics, Thesis Defense
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Basic Notions
• A weighted n-tree T : A tree with set of leaves [n] = {1, . . . , n} and
s.t. all edges e are given a weight w(e):
w(e) 2 R>0 if e is internal.
•
[n]
d 2 R( m )
d = (d )
2([n]
m)
and d is the total weight of the smallest subtree of T which contains
the leaves .
[n]
• d 2 R( m ) is the m-dissimilarity vector of T .
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Suppose we start with a tree, each of whose edges has a
corresponding associated positive weight. This is shown for an
11-tree.
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This introduces the notion of dissimilarity vector. The notion
can be generalized.
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Relation between 2-dissimilarity and m-dissimilarity
vectors of T for m > 2
(Cools)
Let Cm ✓ Sm be the set of cyclic permutations.
Take {i1 , i2 , . . . , im } 2
[n]
m
.
Assume WLOG that 1 = i1 , 2 = i2 , ...,m = im .
Then
d12...m
=
1
· min {d
2 2Cm 1
(1) +d (1)
2 (1)
+d
2 (1) 3 (1)
+d
3 (1) 4 (1)
+· · ·+d
m 1 (1) m (1)
}
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In particular, we have the nice equation
1
dijk = (dij + dik + djk )
2
for all 1  i < j < k  n.
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• (I.G.) Corollary
For all {i, j, k , l, p} 2
[n]
5
dij =
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we have:
1
(2dijk + 2dijl + 2dijp
3
dipl dikl dikp
djpl
djkl
djkp
+ 2dplk )
(It is possible to characterize 3-dissimilarity vectors from this
corollary, using Buneman)
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Tropical Algebraic Geometry
• (Ardila, Pachter, Speyer, Sturmfels) Tropical Geometry is a
powerful tool to study weighted trees.
• Let K be the a.c. field of elements
!=
kX
=p
ck t k /q
1
p 2 Z, cp 6= 0, q 2 Z+ and ck 2 C
Consider the valuation val: K 7! Q [ { 1} by which
val(!) = p/q and val(0) =
Examples:
! = (1 + i)t 1/2
! = t 1/2 + 1 + t
1
! = t 2 + t 5 + t 6 and val(!) = 6
p
p
(i)t 2/3 + (2)t 5/3 + ( 2
3i)t 5/2 and val(!) = 5/2
1/2
+t
1
+t
3/2
+t
2
+t
5/2 . . .
and val(!) = 1/2
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• Let R = R [ { 1}.
• Tropical semiring (R, , ).
a
b = max{a, b}
a
b =a+b
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• Tropicalization of a Polynomial f 2 K[X1 , . . . , XN ].
• Let A ✓ ZN 0 be finite.
f =
X
c↵ X1↵1 . . . XN↵N
↵2A
trop(f ) =
M
val(c↵ )
x1↵1
↵2A
···
xN↵N
Example:
f 2 K[X , Y , Z ]
1
+ 2i)XYZ 2 Z 4
1
trop(f ) = max{x + y , + x + y + 2z, 4z}
5
f = iXY + (t 5 + t
1
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• Tropical Hypersurface defined by trop(f ).
T ( trop(f )) := {x 2
N
R | the maximum of trop(f ) is attained at least twice}
Example (Cont.):
T ( trop(f ))
1
+ x + y + 2z 4z}
5
1
[ {x, y , z | x + y = 4z
+ x + y + 2z}
5
1
[ {x, y , z | + x + y + 2z = 4z x + y }
5
= {x, y , z | x + y =
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• The Tropical Variety of an ideal I of K[X1 , . . . , XN ]:
\
T (I) =
T ( trop(f ))
f 2I
• Fundamental Theorem of Tropical Algebraic Geometry:
T (I) = closure of {( val(C1 ), . . . , val(CN )) | (C1 , . . . , CN ) 2
V (I) ✓ KN }
• T ( trop(f )) and T (I) often considered in
TPN
1
N
= R /(1, 1, . . . , 1)R
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Tropical Linear Spaces
• Plücker ideal Im,n in K[Pi1 i2 ...im | {i1 , i2 , . . . , im } 2 [n]
m ],
consisting of the algebraic relations-syzygies among the
maximal minors of a generic matrix in Km⇥n . Elements of Im,n
are called Plücker relations.
• Quadrics form a Gröbner basis for Im,n , among which we find
the three-term Plücker relations:
PSij PSkl PSik PSjl + PSil PSjk
✓
◆
✓
◆
[n]
[n]\S
for all S 2
and {i, j, k , l} 2
m 2
4
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• The tropical Grassmannian-(m, n):
Gm,n = T (Im,n )
• A vector p 2 R
only if:
\
p2
([n]
m)
S2 m[n]2
{i,j,k ,l}2 [n]\S
4
(
T
)
(
[n]
satisfies the tropical Plücker relations if and
trop(PSij PSkl
PSik PSjl + PSil PSjk )
)
• Assume p 2 R( m ) ( R
relations.
([n]
m)
satisfies the tropical Plücker
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[n]
n
notation:
P Canonical basis of R : {e1 , . . . , en }. For I 2 m ,
eI = i2I ei .
• (Speyer) Every face of the regular subdivision of the
n
m-hypersimplex Hm := chull{eI | I 2 [n]
m } ✓ R induced by p
is matroidal.
• Three objects:
I. L(p) :=
\
[n]
{i1 <i2 <···<im <im+1 }2(m+1
)
T
m+1
M
r =1
(pi
b
1 i2 ...ir ...im im+1
II. Pp := {x 2 Rn |xi1 + xi2 + · · · + xim
pi1 i2 ...im for all {i1 , i2 , . . . , im } 2 [n]m }
!
xir )
III. Pp0 := {x 2 Pp | If y  x and y 2 Pp , then y = x}
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• Facts (Speyer):
i. L(p)“=”Pp0 in TPn 1 .
ii. Pp0 is an (m 1)-dimensional pure and contractible
polyhedral complex.
iii. Tropical linear space associated to p.
• The tight span Tp is the complex of bounded faces of Pp (or
Pp0 , or L(p) projectively). To define Tp , we can relax the
requirement that p satisfies the tropical Plücker relations. Face
enumeration of Tp and the f -vector conjecture (Speyer).
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Main Question
[n]
• Given a vector d 2 R( m ) , is d the m-dissimilarity vector of
an n-tree?
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The case m = 2
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[n]
observation: For p 2 R( 2 ) , p 2 G2,n if and only if p satisfies the
tropical Plücker relations. Quadrics form a Tropical Basis for
I2,n .
[n]
• (Buneman) Four point condition theorem: A vector d 2 R( 2 )
is a 2-dissimilarity vector if and only if d 2 G2,n .
[n]
• (Dress) A vector d 2 R( 2 ) is a 2-dissimilarity vector if and
only if Td is a tree.
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m > 2 and problems
(Sturmfels) In general, quadratics DO NOT form a tropical basis
of Im,n for m > 2, only a Gröbner Basis.
• (Pachter, Speyer) Problem 1: If m > 2, decide whether the
set of m-dissimilarity vectors of n-trees is contained in Gm,n .
• (Ardila, Sturmfels) Problem 2: Given an m-dissimilarity vector
[n]
d 2 R( m ) , study the tight span Td . If Problem 1 has a positive
answer (or at least, if d satisfies the tropical Plücker relations),
this amounts to studying the bounded faces of Pd0 .
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Ultrametrics
• Ultrametric n-tree:
i. rooted
ii. binary
iii. leaves [n]
iv. nonnegative real weight w(e) for each edge e
v.
-equidistant, for some
>0
vi. induces a metric on [n].
• Ultrametric: A metric space S with distance d : S ⇥ S 7! R
for which, dxz  max{dxy , dyz } for all x, y , z 2 S.
• Both definitions are “equivalent”.
0
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The Tropical Grassmannian
notation: For a matrix M 2 Km⇥n and ⇢ 2 [n]
m , M⇢ denotes the
maximal minor of M coming from columns ⇢.
[n]
• (I.G.) Theorem: Let m > 2. If d 2 R( m ) is an m-dissimilarity
vector, then d 2 Gm,n .
Sketch of proof:
[n]
I. Fix a weighted n-tree T . Define d 0 2 R( 2 ) by:
dij0 = 2t ⇤ + dij
djn
din for all {i, j} 2
[n]
2
For t ⇤ 2 R>0 sufficiently large, d 0 induces an ultrametric on [n],
so d 0 is the m-dissimilarity vector of an ultrametric tree.
II. There exists a full rank matrix M 0 2 Km⇥n s.t. val(M⇢0 ) = d⇢0
for all ⇢ 2 [n]
m .
III. There exists a diagonal matrix D 2 Kn⇥n (given by
Dii = din t ⇤ for all i 2 [n 1] and Dnn = t ⇤ ) such that
M = M 0 D satisfies val(M⇢ ) = d⇢ for all ⇢ 2 [n]
m .
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[n]
• (I.G.) Corollary: If d 2 R( m ) is an m-dissimilarity vector, d
satisfies the tropical Plücker relations, L(d) can be defined and
studied through Td . The f -vector conjecture of Speyer applies.
• Our approach produces a complete flag of tropical linear
spaces.
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The tight span
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• Let 1 < m < n. Fix a weighted n-tree T with m-dissimilarity
vector d. We will describe the tight span Td .
Let in(T ) be the internal subtree of T .
Example:
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notation: ✓st denotes the relation being a subtree of.
definition: For S ✓st in(T ), the extended tree S. The set of
leaves of S is called the set of extended leaves of S.
Example:
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• S ✓st in(T ) determines two collections H and R of subsets of
leaves of T .
i. members of H are indexed by the leaves of S. We
have i 2 H` 2 H if and only if the minimal path
from i to S meets S at `.
ii. members of R are indexed by the extended leaves
of S. We have i 2 R` 2 R if and only if the minimal
path from i to S meets S at `.
observations:
I. R is a partition of the set [n] of leaves of T .
II. H is a collection of disjoint subsets of [n], but
generally not a partition of [n].
III. Every member of H is the disjoint union of at least
two elements of R.
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definition: Let 0  i  m 1. A good pair (S, A) with |A| = i
is given by:
i. S ✓st in(T ) with at most m 1 leaves.
ii. A a subset of the leaves of S.
iii. the number of extended leaves of S not adjacent
to an element of A is at least m + 1 i.
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• (I.G.) Theorem: A good pair (S, A) with |A| = i uniquely
determines an i-dimensional face of Td . The face lattice of this
face is isomorphic to the face lattice of the i-dimensional cube.
The set of bases of the matroid associated to our face is the
collection of all m-sets I 2 [n]
m satisfying:
|I \ H` | = 1 for all leaves ` 2 A.
|I \ H` |
1 for all leaves ` of S s.t. ` 62 A.
|I \ R` |  1 for all leaves ` of S.
Good pairs give rise to all i-dimensional faces of Td whenever
0  i < m 1. In general, good pairs give rise only to a subset
of the (m 1)-dimensional faces of Td .
If T is trivalent, S ✓st in(T ), and A is an i-subset of the leaves
of S, then (S, A) is a good pair if and only if S has at most
m 1 leaves and at least m 1 + i vertices.
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• Corollary: The vertices of Td correspond to trees S ✓st in(T )
with at most m 1 leaves and at least m + 1 extended leaves.
For any such S, the coordinates of the associated vertex x are
given by:
!(S)
xi = !(i, S) +
for all i 2 [n]
m
If T is trivalent, the vertices of Td correspond to trees
S ✓st in(T ) with at most m 1 leaves and least m 1 vertices.
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definition: Suppose S ✓st in(T ). If the collection H associated
to S is a partition of [n], we say that S is a full subtree of in(T ).
Example:
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• (I.G.) Theorem: The (m 1)-dimensional faces of Td which
do not arise from good pairs, are in one to one correspondence
with the full m-subtrees S ✓st in(T ).
This faces are line segments for m = 2, hexagons for m = 3,
rhombic dodecahedrons for m = 4, and they are simple if and
only if m < 4.
The matroid associated to such face, coming from S, is the
one whose set of bases are the full transversals of the collection
H associated to S (Notice |H| = m). It is a transversal matroid.
The f -vector of the face is given by:
f
fm
=1
⇣
fi = 2m
1
1
=1
i
⌘ ✓m ◆
2
for all i with 0  i  m
i
2
If T is trivalent, full m-subtrees of in(T ) correspond to trees
S ✓st in(T ) with m leaves and 2(m 1) vertices.
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Let S ✓st in(T ) and A a subset of the leaves of S.
definition: Let SA be the subtree of S spanned by the vertices
in V(S)\A.
• (I.G.) Theorem: Let F and F 0 be faces of Td . Suppose that F
comes from a good pair (S, A) and that F 0 comes from a good
pair (S 0 , A0 ). Then, F \ F 0 6= ; if and only if:
0
SA ✓st S 0 and SA
0 ✓st S
Moreover, in the case when our faces meet, F \ F 0 comes from
the good pair (S \ S 0 , A \ A0 ). In particular, F \ F 0 is of
dimension |A \ A0 |.
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Tight spans of trees
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Tree-matroids
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Thanks!
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