Dissimilarity Vectors of Trees and Their Tropical
Linear Spaces
Benjamin Iriarte Giraldo
Massachusetts Institute of Technology
FPSAC 2011, Reykjavik, Iceland
Basic Notions
• A weighted n-tree T : A trivalent tree
with set of leaves [n] = {1, . . . , n} and
s.t. every edge e is assigned a weight
w (e), where
w (e) ∈ R>0 if e is internal.
(F. Cools) Let Cm ⊆ Sm be the set of
cyclic permutations. Take
{i1 , i2 , . . . , im } ∈
[n]
m
. Assume
WLOG that 1 = i1 , 2 = i2 , ...,m = im .
Then, d12...m
=
1
2
· min {d1σ(1) + dσ(1)σ2 (1) +
σ∈Cm
dσ2 (1)σ3 (1) +dσ3 (1)σ4 (1) +· · ·+dσm−1 (1)σm (1) }.
•
d ∈
[n]
R m
and dσ is the total weight of the subtree
of T spanned by the set of leaves σ.
In particular, we have the nice equation
dijk = 12 (dij + dik + djk ) for all
1 ≤ i < j < k ≤ n, from which we
obtain (I.G.) thatforall
{i, j, k, l, p} ∈
[n]
R m
•d ∈
is called the
m-dissimilarity vector of T .
[n]
5
,
1
(2dijk + 2dijl + 2dijp
3
− dipl − dikl − dikp
dij =
− djpl − djkl − djkp + 2dplk ).
This is shown for a 6-tree, for m = 2 and m = 3.
6
5
100
−5
d12
d36
d45
d24
2e
1
3
5
2
1
2
−π
2
−1
e
4
3
[6]
We obtain a vector d ∈ R 2 , the 2-dissimilarity vector.
= 53 − π
= −1 + 12 + 2e + 100
= 2 + 2e − 5
= −π + 2 + 12 + e
...
6
5
100
−5
d123 =
2e
1
3
5
−π+2−1
d136 = 35 + 2 − 1 + 12 + 2e + 100
d456 = 2 + 2e − 5 + 100
d245 = −π + 2 + 12 + e + 2e − 5
..
.
3
5
2
1
2
−π
2
−1
e
4
3
[6]
We obtain a vector d ∈ R 3 , the 3-dissimilarity vector.
d123...(10)(11)(12) = d12 + d23 + . . . + d(10)(11) + d(11)(12) + d(12)1
11
12
6
5
7
10
8
1
9
4
2
3
Tropical Algebraic Geometry
• Let K be the a.c. field of elements
ω=
k=p
X
ck t
• Let R = R ∪ {−∞}.
• Tropical semiring (R, ⊕, ).
a ⊕ b = max{a, b}
ab =a+b
• Tropicalization of a Polynomial
f ∈ K[X1 , . . . , XN ].
• Let A ⊆ ZN
≥0 be finite.
k/q
−∞
p ∈ Z, cp 6= 0, q ∈ Z+ and ck ∈ C
Consider the valuation
val: K 7→ Q ∪ {−∞} by which
f =
val(ω) = p/q and val(0) = −∞
trop(f ) =
Example
2
5
1/2
α
α
c α X 1 1 . . . XN N
M
α
α
val(cα )x1 1 · · ·xN N
α∈A
Example
6
ω = t + t + t and val(ω) = 6
ω = (1+i)t
X
α∈A
−(i)t
2/3
+(2)t
5/3
√
+( 2
f ∈ K[X , Y , Z ]
1
f = iXY + (t 5 + t
√
5/2
− 3i)t
and val(ω) = 5/2
ω=t
1/2
+t
+1+t
−5/2
−1/2
+t
−1
+t
−3/2
+t
. . . and val(ω) = 1/2
−2
−1
trop (f ) = max{x+y ,
2
+ 2i)XYZ − Z
1
5
4
+x+y +2z, 4z}
• Tropical Hypersurface defined by
trop(f ).
T ( trop(f )) := {x ∈
N
R
• The Tropical Variety of an ideal I of
K[X1 , . . . , XN ]:
| the maximum of trop(f ) is attained at least twice}
T (I ) =
Example (Cont.)
T ( trop(f ))
= {x, y , z | x + y =
1
5
+ x + y + 2z ≥ 4z}
∪ {x, y , z | x + y = 4z ≥
∪ {x, y , z |
1
5
1
5
\
T ( trop(f ))
f ∈I
+ x + y + 2z}
• Fundamental Theorem of Tropical
Algebraic Geometry:
T (I ) =
closure of {(val(C1 ), . . . , val(CN )) | (C1 , . . . , CN ) ∈
V (I ) ⊆ KN }
• T ( trop(f )) and T (I ) often
considered in
+ x + y + 2z = 4z ≥ x + y }
N−1
TP
N
= R /(1, 1, . . . , 1)R
Tropical Linear Spaces
• Plücker ideal Im,n in K[Pi1 i2 ...im | {i1 , i2 , . . . , im } ∈
[n]
m
], consisting of the algebraic relations-syzygies
m×n
among the maximal minors of a generic matrix in K
. 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]
for all S ∈
m−2
and {i, j, k, l} ∈
[n]\S
4
• The tropical Grassmannian-(m, n):
Gm,n = T (Im,n )
• A vector p ∈
[n]
R m
satisfies the tropical Plücker relations if and only if:
\
p∈
T
trop (PSij PSkl − PSik PSjl + PSil PSjk )
[n]
S∈
m−2
[n]\S
{i,j,k,l}∈
4
[n]
• Assume p ∈ R m
[n]
(R m
satisfies the tropical Plücker relations.
Notation: Canonical basis of Rn : {e1 , . . . , en }. For I ∈
P
[n]
, eI =
i∈I ei .
m
• (D. Speyer) Every face of the regular subdivision of the m-hypersimplex Hm := chull{eI | I ∈
[n]
} ⊆ Rn
m
induced by p is matroidal.
• Three objects:
{i1 <i2 <···<im <im+1 }∈


T 
(pi i ...ib ...i i
xir )
r
m m+1
12
m+1
M
\
I. L(p) :=
[n]
m+1
r =1
II. Pp := {x ∈ Rn |xi1 + xi2 + · · · + xim ≥ pi1 i2 ...im for all {i1 , i2 , . . . , im } ∈ [n]m }
III. Pp0 := {x ∈ Pp | If y ≤ x and y ∈ Pp , then y = x}
• Facts (D. 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).
Main Problems
[n]
• Given a vector d ∈ R m , is d the m-dissimilarity vector of an n-tree?
[n]
Observation: For p ∈ R 2 , p ∈ G2,n if and only if p satisfies the tropical Plücker relations. Quadrics form a
Tropical Basis for I2,n .
[n]
• (P. Buneman) Four point condition theorem, modern statement: A vector d ∈ R 2
if and only if d ∈ G2,n .
is a 2-dissimilarity vector
[n]
R 2
• (A. Dress) A vector d ∈
is a 2-dissimilarity vector if and only if Td is a tree.
(B. Sturmfels) In general, quadratics DO NOT form a tropical basis of Im,n for m > 2, only a Gröbner Basis.
• (L. Pachter, D. Speyer) Problem 1: If m > 2, decide whether the set of m-dissimilarity vectors of n-trees is
contained in Gm,n .
[n]
• (B. Sturmfels) Problem 2: Given an m-dissimilarity vector d ∈ R m , study the tropical linear space L(d), or
equivalently Pd0 . In particular, Td .
Ultrametrics
• Ultrametric n-tree:
i. rooted
ii. binary
iii. leaves [n]
iv. nonnegative real weight w (e) for each edge e
v. λ-equidistant, for some λ > 0
induces a metric on [n].
• Ultrametric: A metric space S with distance d : S × S 7→ R≥0 for which, dxz ≤ max{dxy , dyz } for all
x, y , z ∈ S.
• Both definitions are “equivalent”.
Theorem (I.G.)
[n]
Let m > 2. If d ∈ R m
is an m-dissimilarity vector, then d ∈ Gm,n .
Corollary
[n]
If d ∈ R m is an m-dissimilarity vector, d satisfies the tropical Plücker relations, L(d) can be defined and has
“nice” properties, in particular, it satisfies the f -vector conjecture of Speyer.
• Our approach produces a complete flag of tropical linear spaces.
r
1
2
3
6
29
6
7
3
2
1
4
2
1
3
2
3
1
3
1
6
4
5
1
6
6
7
The Tropical Linear Space
• Let 1 < m < n. Fix a weighted n-tree
T with m-dissimilarity vector d. We will
describe the tropical linear space L(d).
Let in(T ) be the internal subtree of T .
Example
We consider the combinatorial tree T of
the figure, where in(T ) is shown in blue.
Notation: ⊆st denotes the relation
being a subtree of.
Example
In the figure, we have S ⊆st T , shown
in red.
Definition
For S ⊆st T , the extended tree S. The set of leaves of S is called the set of extended leaves of S.
Example
In the figure, we present S (green and red) corresponding to S (red).
• Any S ⊆st T determines two collections H and R of subsets of the set of leaves of T ,
L(T ) = [n] = {1, 2, . . . , n}.
i. Members of H are indexed by the leaves of S. We have i ∈ H` ∈ 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 ∈ R` ∈ 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.
Definition
Suppose S ⊆st T . If the collection H associated to S is a partition of [n], we say that S is a full subtree of T .
Example
5
5
4
4
3
3
6
2
6
2
12
12
7
7
1
1
11
11
8
8
10
For the subtree S of the figure, we have
H=
{{1}, {3}, {6}, {7, 8, 9, 10, 11, 12}}.
Note the case of leaves 2, 4, 5.
10
9
9
For
the subtree S of the figure, we have R =
{{1}, {2}, {3}, {4}, {5}, {6}, {7, 8, 9, 10, 11}, {12}}.
Definition
Let i ∈ [m − 1] ∪ {0} and S ⊆st T .
We say that (S, A) is an (m, i)-good
pair of T if:
5
4
i. i ≤ |L(S)| ≤
m − 1,
3
ii. m − 1 + i ≤
|V(S)|,
6
2
iii. A ⊆ L(S) with
|A| = i, and
12
1
iv. L(S) ∩ L(T ) ⊆
A.
7
8
10
Definition
For all i ∈ [m − 1] ∪ {0}, define
11
9
Ti := {(S, A) | (S, A) is an (m, i)-good pair of T },
in
Ti
:= {(S, A) ∈ Ti | S ⊆st in(T )},
T∗ := {S | S is a full subtree of T with m leaves},
In this case, the subtree in red is a
full subtree of our tree, and we have H =
{{1, 2, 3, 4, 5}, {6}, {7, 8}, {9, 10, 11}, {12}}.
in
T∗ := {S ∈ T∗ | S ⊆st in(T )}.
Theorem (I.G.)
• Let i ∈ [m − 2] ∪ {0}. Then, the set of i-dimensional faces of Pd0 is in bijection with the set Ti , and the set of
i-dimensional faces of Td is in bijection with the set Tin
i .
• The set of (m − 1)-dimensional faces of Pd0 is in bijection with the set Tm−1 t T∗ , and the set of
in
(m − 1)-dimensional faces of Td is in bijection with the set Tin
m−1 t T∗ .
In particular, the vertices of Pd0 correspond to trees S ⊆st in(T ) with at most m − 1 leaves and at least m − 1
vertices.
Theorem (I.G.)
• Let i ∈ [m − 1] ∪ {0}. The matroid MF of the face F of Pd0 corresponding to an (m, i)-good pair (S, A) of T ,
has a collection of bases which consists of all A ∈ [n]
such that:
m
|A ∩ H| = 1 for all H ∈ H s.t. H = H` for some ` ∈ A,
|A ∩ H| ≥ 1 for all H ∈ H s.t. H = H` for some ` ∈ L(S)\A,
|A ∩ R| ≤ 1 for all R ∈ R.
If F is bounded, then its face lattice is isomorphic to the face lattice of the i-dimensional cubein Rn .
• Now, consider the (m − 1)-dimensional pyrope Pm−1 := conv [−1, 0]m−1 ∪ [1, 0]m−1 , i.e. the convex
hull of the 0-1 and 0-(−1) cubes in Rm−1 .
The matroid MF of an (m − 1)-dimensional face F of Pd0 corresponding to a full subtree S of T with m leaves has
a collection of bases which consists of all A ∈ [n]
such that:
m
|A ∩ H| = 1 for all H ∈ H, so it is a transversal matroid.
If F is bounded, then its face lattice is isomorphic to the face lattice of the (m − 1)-dimensional pyrope.
Example (Matroid
Inequalities)
4 ≤1
5
≤1
4 ≤1
5
≤1
3
3 ≥1
=1
2 ≤1
=1
6
≥1
2 ≤1
≥1
6
≥1
12 ≤ 1
12 ≤ 1
≤1
≥1
≤1
1
≥1
1
7
7
11
11
8
10
8
10
Inequalities describing a vertex of the
tropical linear space.
9
Inequalities describing a 2-dimensional
face of the tropical linear space.
9
Example (Tight
Spans)
8
7
6
9
5
1
4
We present its tight span for m = 2.
2
Consider the tree in the figure.
3
We present its tight span for m = 4.
We present its tight span for m = 3.