Download presentation
Presentation is loading. Please wait.
1
tree -decomposition * 竹内 和樹 * 藤井 勲
2
contents 1,Introduction 2,Main Algorithm 3,yfiles
4,Construction of Tree-Decomposition 5,Analysis of Algorithm 6,From now on Target
3
Introduction ~Purpose~ INPUT : G = (V,E)
Output : Tree-decomposition ( T , X ) 1 2 6 1,2,3 2,3,4 4,6,8 7 8 3 4 4,5 2,4,7 4,6,7 5
4
Introduction ( T , X ) : T = ( I , F ) , X = { Xi : i ∈ I }
(1), ∪ Xi = V i∈I (2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi and w ∈ Xi, (3), for all i,j,k ∈ I if j is on the path from i to k in T , then Xi ∩Xk ⊆ Xj
5
Introduction (1), ∪ Xi = V i∈I 1 2 6 1,2,3 2,3,4 4,6,8 7 8 3 4 4,5 2,4,7 4,6,7 5
6
Introduction ( T , X ) : T = ( I , F ) , X = { Xi : i ∈ I }
(1), ∪ Xi = V i∈I (2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi and w ∈ Xi, (3), for all i,j,k ∈ I if j is on the path from i to k in T , then Xi ∩Xk ⊆ Xj
7
Introduction (2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi and w ∈ Xi, 1 2 6 1,2,3 2,3,4 4,6,8 7 8 3 4 4,5 2,4,7 4,6,7 5
8
Introduction (2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi and w ∈ Xi, 1 2 6 1,2,3 2,3,4 4,6,8 7 8 3 4 4,5 2,4,7 4,6,7 5
9
Introduction (2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi and w ∈ Xi, 1 2 6 1,2,3 2,3,4 4,6,8 7 8 3 4 4,5 2,4,7 4,6,7 5
10
Introduction ( T , X ) : T = ( I , F ) , X = { Xi : i ∈ I }
(1), ∪ Xi = V i∈I (2), for every edge {v,w} ∈E, there is an i ∈ I with v ∈Xi and w ∈ Xi, (3), for all i,j,k ∈ I if j is on the path from i to k in T , then Xi ∩Xk ⊆ Xj
11
Introduction (3), for all i,j,k ∈ I if j is on the path from i to k in T , then Xi ∩Xk ⊆ Xj 1 2 6 1,2,3 2,3,4 4,6,8 7 8 3 4 4,5 2,4,7 4,6,7 5
12
Introduction (3), for all i,j,k ∈ I if j is on the path from i to k in T , then Xi ∩Xk ⊆ Xj 1 2 6 1,2,3 2,3,4 4,6,8 7 8 3 4 4,5 2,4,7 4,6,7 5
13
Introduction (3), for all i,j,k ∈ I if j is on the path from i to k in T , then Xi ∩Xk ⊆ Xj 1 2 6 1,2,3 2,3,4 4,6,8 7 8 3 4 4,5 2, 4,7 4,6,7 5
14
Main Algorithm ~ An st-separating set ~
a set S ⊆ V {s,t} with the property that any path from s to t 1 2
15
Main Algorithm ~ Minimum st-separating set ~
The minimum number of vertices in an st-separating set
16
Calculate minimal separating
Main Algorithm ~Outline~ Input Graph G = ( V , E ) Construct( T , X ) With | I | = 1 and X = V 1 no ∃i∈ I with Gi not a clique Output :current Tree-decomposition yes Calculate minimal separating Vertex set S for Gi: m components Yi1,.....,Yim Define Xi0 : = S Xij : Yij ∪ S, for j = 1,....,m Construct new ( T , X )
17
Main Algorithm 1 2 6 7 8 3 4 5 Input graph G = (V,I)
18
Main Algorithm 1 2 6 7 8 3 4 5 Calculate minimal separataing
vertex set S for Gi 7 8 3 4 5
19
Main Algorithm 1 2 6 7 8 3 4 5 Calculate minimal separataing
vertex set S for Gi 7 8 3 4 5
20
Main Algorithm 1 2 6 7 8 3 4 5 Calculate minimal separating
vertex set S for Gi . 7 8 3 4 5 Separate Y and Y S =(4) 11 12 11
21
Main Algorithm 1 2 6 7 8 3 4 5 Calculate minimal separating
vertex set S for Gi . 7 8 3 4 5 Separate Y and Y S =(4) 11 12 11
22
Main Algorithm 1 2 6 7 8 3 4 Calculate minimal separating
vertex set S for X . 7 11 8 3 4
23
Main Algorithm 1 2 6 7 8 3 4 Calculate minimal separating
vertex set S for X . 7 11 8 3 4
24
Main Algorithm 1 2 6 7 8 3 4 Calculate minimal separating
vertex set S for X . 7 11 8 3 4 Separate Y and Y S =(2,4) 21 22 21
25
Main Algorithm 1 2 6 7 8 3 4 Calculate minimal separating
vertex set S for X . 7 11 8 3 4 Separate Y and Y S =(2,4) 21 22 21
26
Main Algorithm 2 6 7 8 4 Calculate minimal separating
vertex set S for X . 7 21 8 4
27
Main Algorithm 2 6 7 8 4 Calculate minimal separating
vertex set S for X . 7 21 8 4
28
Main Algorithm 2 6 7 8 4 Calculate minimal separating
vertex set S for X . 7 21 8 4 Separate Y and Y S =(4,6) 31 32 31
29
Main Algorithm 2 6 7 8 4 Calculate minimal separating
vertex set S for X . 7 21 8 4 Separate Y and Y S =(4,6) 31 32 31
30
Main Algorithm 2 6 7 4 Calculate minimal separating
vertex set S for X . 7 31 4
31
Main Algorithm 2 6 7 4 Calculate minimal separating
vertex set S for X . 7 31 4
32
Main Algorithm 2 6 7 4 Calculate minimal separating
vertex set S for X . 7 31 4 Separate Y and Y S =(4,6) 41 42 41
33
Main Algorithm 2 6 7 4 Calculate minimal separating
vertex set S for X . 7 31 4 Separate Y and Y S =(4,6) 41 42 41
34
Main Algorithm 6 7 4 Calculate minimal separating vertex set S for X .
41 4
35
Main Algorithm 6 7 4 Calculate minimal separating vertex set S for X .
41 4 ・ ・ ・
36
Main Algorithm 6 7 4 Calculate minimal separating vertex set S for X .
41 4 ・ ・ ・ X is clique . 41 r = X = {4,6,7} 1 41
37
Main Algorithm 2 7 4 Calculate minimal separating vertex set S for X .
42 4
38
Main Algorithm 2 7 4 Calculate minimal separating vertex set S for Y .
42 4 ・ ・ ・
39
Main Algorithm 2 7 4 Calculate minimal separating vertex set S for Y .
42 4 ・ ・ ・ X is clique . 42 r = X = {2,4,7} 2 42
40
Main Algorithm 6 8 4 Calculate minimal separating vertex set S for X .
32 8 4 ・ ・ ・ X is clique . 32 r = X = {4,6,8} 3 32
41
Main Algorithm 1 2 3 4 Calculate minimal separating
vertex set S for X . 22 3 4
42
Main Algorithm 1 2 3 4 Calculate minimal separating
vertex set S for X . 22 3 4
43
Main Algorithm 1 2 3 4 Calculate minimal separating
vertex set S for X . 22 3 4 Separate Y and Y S =(2,3) 33 34 32
44
Main Algorithm 1 2 3 4 Calculate minimal separating
vertex set S for X . 22 3 4 Separate Y and Y S =(2,3) 33 34 32
45
Main Algorithm 2 3 4 Calculate minimal separating vertex set S for X .
33 3 4 ・ ・ ・ X is clique . 33 r = X = {2,3,4} 4 33
46
Main Algorithm 1 2 3 Calculate minimal separating vertex set S for X .
34 3 ・ ・ ・ X is clique . 34 r = X = {1,2,3} 5 34
47
Main Algorithm 4 5 Calculate minimal separating vertex set S for X . ・
12 4 ・ ・ ・ 5 X is clique . 12 r = X = {1,2,3} 6 12
48
Main Algorithm All vertex sets are clique . Connect r ,…, r . 1 6
49
Main Algorithm All vertex sets are clique . S = (2,3) S = (2,4)
32 21 Connect r ,…, r . 1 6 1,2,3 2,3,4 4,6,8 S = (4,6) 31 4,5 2,4,7 4,6,7 S = (4) S = (4,7) 1 41
50
Analysis of algorithm
51
From now on Target ・自分達の algorithm の検討
・yfiles を用いての tree-decomposition の ダイナミックな実装 ・Heuristic の検討 ??動的な同型性判定??
52
Ex) Given a graph G =(V,E), construct tree- decomposition (T,X).
1,2,4 7,8 1 2 4 7 3 1,3,4 4,6,7 5 6 8 3,4,5 4,5,6
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.