2. Metric spaces
Metric space (X, d) is a set X with a map
d, defined on it, such that:
✗
d(x,y) ≥ 0, d(x,y) = 0 <=> x = y
✗
d(x,y) = d(y,x)
✗
d(x,z) + d(z,y) ≥ d(x,y)
3. Problem formulation
Let (X, d) be a finite metric space.
Realization (G, φ, w) of d on X - graph (G, φ) and a w: E
→ R + , s.t. d(x i , x j ) = Σw(x p , x p+1 ), p = i,... j-1 and x i , x i+1 ,
…, x j is the shortest path between x i and x j .
Let ||G|| = Σw(e),e∊E is the total edge weight of G.
Problem: to find optimal realization, that is the
realization (G, φ, w) with the minimal ||G||.
6. 4-point condition
aa'+bb' ≤ max{ab+a'b', ab' + ba'}
If for a, a', b, b' ∊ X there exists such a
labeling that they satisfy 4-point condition,
then the metric is tree-realizable.
7. Metric cone
The metric cone is the following:
<M(X)> = {d: XxX→R, xy=yx, xx=0}.
Among its extreme rays are the split
metrics.
8. Split metrics
0, if x, y ∊
A, or x, y ∊ B
δA,B(x,y) =
1, otherwise
For 3-points metric it can be easily seen that:
d = αaδ{a},{b,c}+ αbδ{b},{a,c}+ αcδ{c},{a,b}
9. Isolation index
β{a,a'},{b,b'}= ½ max{ab+a'b', a'b+ab', aa'+bb'} - aa' – bb'
αA,B= min {β{a,a'},{b,b'}}, a, a'∊ A, b, b' ∊ B.
If αA,B > 0, for A, B ⊂ X, A∪ B = X, then A, B is called d-split.
10. Totally decomposable metrics
Metrics, that can be represented as
d = Σ α A,B δ A,B
are called totally split-decomposable.
A metric d is split-prime, if it can not be
represented as above.
11. Example of split-prime metric
For a d-split A, B both sets are d-convex (that is if
a, c ∊ X and ab + bc = ac). For this metric it is
not possible to find a bipartition into two dconvex sets.
12. Main theorem
Every symmetric function d: X x X → R on a finite
set X can be expressed in the form
d = d 0 + Σ α A,B δ A,B,
where d 0 is a split-prime metric, A,B – d-splits of
the set X
13. Splits extension
A split A, B of a set X is said to extend the split A',
B', if A'
⊂ A and B' ⊂ B.
It is clear that if A, B extends A', B' then
α A,B ≤ α A',B'
14. Weakly compatible splits
α {t,u},{v,w}, α {t,v},{u,w}, α {t,w},{u,v} can not be all positive.
The system of splits is weakly compatible, if there
are no 4 points t, u, v, w from X and three splits
S 1 , S 2 , S 3 , such that S 1 extends {t,u},{v,w}, S 2
extends {t,v},{u,w}, S 3 extends {t,w},{u,v}.
15. Weakly compatible splits
Theorem: Let S be a collection of weekly
compatible splits of X. Then the split metrics δ A,B
are linearly independent. Thus, S has at most C 2 n
members.
16. 3 types of 5-point metrics
Let |X| = 5, S 2 = {{A, B} ∊ S| min{|A|,|B|}=2 }
Let G is a graph with VG = VX, EG = S 2 .
S 2 are weakly compatible => G is isomorphic to a
subgraph of 5-cycle or 4-cycle plus one isolated
vertex.
18. Type I 5-point metric
For X = {x0, x1, x2, x3, x4} we have:
d=Σ α xi δ xi + Σ α xi,xi+1 δ xi,xi+1
where indices are taken modulo 5.
19. 5-point condition
A metric d is totally decomposable if and only if
α{t, u},{v, w} = α{t, u, x},{v, w} + α{t, u},{v, w, x},
Or
α{t, u},{v, w} ≤ α{t, x},{v, w} + α{t, u},{v, x}
20. 6-point metrics
Each generic metric on six points is either a threedimensional cell complex with
26 vertices, 42
edges, 18 polygons and one 3-cell, or it is a twodimensional cell complex with 25 vertices, 39 edges
and 15 polygons. There are 327 three-dimensional
metrics and 12 two-dimensional metrics.
23. 5-point hypothesis for n-point metrics
Is it correct, that in case of finite metric spaces (X,d)
with |X| > 5 if for every 5 points x0, x1, x2, x3, x4 the
condition
d=Σ α xi δ xi + Σ α xi,xi+1 δ xi,xi+1 (*)
holds, then the optimal realization is represented as a
cactus-tree?
24. 5-point hypothesis for n-point metrics
At least, for 6 points it was checked for each of the 12
2-dimensional metric types that the statement of the
hypothesis is not true, moreover it is not possible to
have the situation that for every five points x0, x1,
x2, x3, x4 the condition (*) holds.
26. References
[1] Bandelt, H.J., Dress, A., A canonical decomposition
theory for Metrics on a Finite set
[2] Sturmfels, B., Yu, J., Classification of Six-Point
Metrics
[3] Koolen, J., Lesser, A., Moulton, V., Optimal
realizations of generic 5-point metrics
