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tree -decomposition

tree -decomposition. * 5499016 竹内 和樹 * 5499023 藤井 勲. contents. 1,Introduction 2,Main Algorithm 3,yfiles 4,Construction of Tree-Decomposition 5,Analysis of Algorithm 6,From now on Target. Introduction. ~Purpose~. INPUT : G = (V,E). Output : Tree-decomposition ( T , X ). 1. 2.

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tree -decomposition

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  1. tree -decomposition *5499016 竹内 和樹 *5499023 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. 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 Calculate minimal separataing vertex set S for Gi 7 8 3 4 5

  19. Main Algorithm 1 2 6 Calculate minimal separataing vertex set S for Gi 7 8 3 4 5

  20. Main Algorithm 1 2 6 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 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 Calculate minimal separating vertex set S for X . 7 11 8 3 4

  23. Main Algorithm 1 2 6 Calculate minimal separating vertex set S for X . 7 11 8 3 4

  24. Main Algorithm 1 2 6 Calculate minimal separating vertex set S for X . 7 11 8 3 4 Separate Y and Y S =(2,4) 22 21 21

  25. Main Algorithm 1 2 6 Calculate minimal separating vertex set S for X . 7 11 8 3 4 Separate Y and Y S =(2,4) 22 21 21

  26. Main Algorithm 2 6 Calculate minimal separating vertex set S for X . 7 21 8 4

  27. Main Algorithm 2 6 Calculate minimal separating vertex set S for X . 7 21 8 4

  28. Main Algorithm 2 6 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 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 Calculate minimal separating vertex set S for X . 7 31 4

  31. Main Algorithm 2 6 Calculate minimal separating vertex set S for X . 7 31 4

  32. Main Algorithm 2 6 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 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 Calculate minimal separating vertex set S for X . 7 41 4

  35. Main Algorithm 6 Calculate minimal separating vertex set S for X . 7 41 4 ・ ・ ・

  36. Main Algorithm 6 Calculate minimal separating vertex set S for X . 7 41 4 ・ ・ ・ 41 X is clique . r = X = {4,6,7} 1 41

  37. Main Algorithm 2 Calculate minimal separating vertex set S for X . 7 42 4

  38. Main Algorithm 2 Calculate minimal separating vertex set S for Y . 7 42 4 ・ ・ ・

  39. Main Algorithm 2 Calculate minimal separating vertex set S for Y . 7 42 4 ・ ・ ・ 42 X is clique . r = X = {2,4,7} 2 42

  40. Main Algorithm 6 Calculate minimal separating vertex set S for X . 32 8 4 ・ ・ ・ 32 X is clique . r = X = {4,6,8} 3 32

  41. Main Algorithm 1 2 Calculate minimal separating vertex set S for X . 22 3 4

  42. Main Algorithm 1 2 Calculate minimal separating vertex set S for X . 22 3 4

  43. Main Algorithm 1 2 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 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 Calculate minimal separating vertex set S for X . 33 3 4 ・ ・ ・ 33 X is clique . r = X = {2,3,4} 4 33

  46. Main Algorithm 1 2 Calculate minimal separating vertex set S for X . 34 3 ・ ・ ・ 34 X is clique . r = X = {1,2,3} 5 34

  47. Main Algorithm Calculate minimal separating vertex set S for X . 12 4 ・ ・ ・ 5 12 X is clique . 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

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