1. A p r o x i m a t n l g h s f e - u c D i s c r e t A p l d M a h m 14 3 ( 2 ) - B n g Y W u
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2. Outline F I n t r o d u c i P e l m a s T h N - A p x g O C 2 k 77

3. Outline F I n t r o d u c i P e l m a s T h N - A p x g O C 2 k 77

4. Introduction C o n s i d e r t h f l w g p m a c u O T b : L G = ( V ;) v . W q , F y 77

5. So … O u r g o a l i s t c n p e w h m . T , ¯ d X ¸ ( ) z W b v 2 V ;77

6. Some special cases of the problem-1q u i m n t s O C p o b l a - y g v . B c , f d F ( ; ) = 1 2 V M R N P 77

7. Some special cases of the problem-2( u ; v ) = r f o e a c h 2 V , w i s g n t x . T p b l m d - q P R O C + S 77

8. Examples for OCT, MRCT 1 1.4 A B T1 T2 C D
Vertices
Requirements
A,B
3(1)
A,C
1(1)
A,D
4(1)
B,C
B,D
5(1)
C,D
9(1)
3*1+1*1+4*2+ 1*2+5*3+9*1 =38
3*1+1*2.4+4*1.4+ 1*3.4+5*2.4+9*1
=35.4
1*1+1*2.4+1*1.4+ 1*3.4+1*2.4+1*1 =11.8
1*1+1*1+1*2+ 1*2+1*3+1*1 =10 77

9. Examples for PROCT, SROCT1
1.4 A B T1 T2 C D
3*1+12*1+3*2+ 4*2+1*3+4*1 =60
3*1+12*2.4+3*1.4+ 4*3.4+1*2.4+4*1 =56
Vertx weight
PROCT
SROCT A 3
A,B 4 B 1
A,C 12 7 C
A,D D
B,C 5
B,D 2
C,D
4*1+7*1+4*2+ 5*2+2*3+5*1 =40
4*1+7*2.4+4*1.4+ 5*3.4+2*2.4+5*1 =53.2 77

10. Any questions or mistakes ?1 . I s ( u ; v ) = ? 2 f w e h a c o n t d T , l b W j p + r q i - m g y B 77

11. p - s o u r c e M R C T b l m 1 I n t h e g i v p - s o u r c M R C T( ) b l m , w a d . W f z S O x ; = 1 2 H V 77

12. p - s o u r c e M R C T b l m 2 F I t s r i v a l h 1 - M R C T n d Oy m b . H w , 2 N P f c g A q u 77

13. In this paper … W e i n v s t g a o p m l - u r c ( O C T ) b . L G =, E w h f X 2 S ; d z M R 77

14. In this paper… I n t h i s p a e r , w o N P - d f M R C T b l m v c gy x 2 . O u W F 3 77

15. The relationships between OCT problems
Optimal Communication spanning tree
more general
PROCT
SROCT
p-source OCT, fixed p
2-source OCT
p-source MRCT, arbitrary p
MRCT
p-source MRCT, fixed p
2-source MRCT 77

16. Outline F I n t r o d u c i P e l m a s T h N - A p x g O C 2 k 77

17. Some Definitions-1 F o r a n y g p h G = ( V , E w ) d e t s i v x . Lb l f u c H P W j m D 1 ; 2 S T 77

18. Some Definitions-2 D e ¯ n i t o 2 . L H b a s u g r p h f G F v x V () , w d ; c m = T l E y M A O j + 77

19. Some Definitions-3 D e ¯ n i t o 3 . L T b a s p g r f h G d S = ; , V1 ; 2 , V ( ) v u c F y x m P w q 4 - l O C 77

20. Outline F I n t r o d u c i P e l m a s T h N - A p x g O C 2 k 77

21. T h e c o m p u t a i n l x y f - O C r b I n t h i s e c o , w d ¯ ag v r f 2 - M R C T p b l m . B y k N P x O u
p-OCT
SAT
2-MRCT()
p-MRCT 77

22. Some Definitions F L e t U = f u ; : g b a s o B l v r i - X c h m , d1 ; 2 : n g b a s o B l v r i - X c h m , d y . A x ( _ 3 5 ) | { z } ^ 7 9 4 T j 77

23. Some Definitions D e ¯ n i t o 5 . G v a s U f r b l d X c u , h S A TY ( ) p m g y - 6 = V E w 1 2 M R C ; P + 77

24. Start to transform … W e s h a l t r n f o m S A T p b 2 - M R C ( ® ). G i v U = u 1 ; : g B d X x c , w V E F L j [ y
< + 3 8 77

25. T h e t r a n s f o m i S A p b l 2 - M R C ( ® ) ¹ a ¹ a ¹ a ¹ a ¹ aai
ai+1 an E2 E1 E3 s1 … … s2 a 1 a 2 a 3 a i a i + 1 a n A
b1,b2,b3,…bm B T h e t r a n s f o m i S A p b l 2 - M R C ( ) 77

26. The goal… W e s h a l o w t S A T p r b m i f y n g u ® G c d ( ; ) ·, = + 1 2 4 L 77

27. T h e R d u c t i o n ) P r o p s i t n 1 . I f h e a u g m y X , x Y( ; ) = W d l w F T b v 2 : O S 77

28. Example: ( u _ ¹ ) ^ a ¹ s1 s2 a1 a5 a3 a4 a2 b1 b2 b3 b4 1 3 5 2 4 177

29. ( u _ ¹ ) ^ t : f T , F g a ¹ s1 s2 a1 a5 a3 a4 a2 b1 b2 b3 b4 Y 1 3 577

30. Start to compute… F o r a n y v 2 f ; ¹ g , s t h e p P d ( ) = + ¡ wi ; g , s t h e p P Y d ( 1 ) = + w 3 . b c m j l u L N x 77

31. T h e R d u c t i o n ) T h e r o u t i n g c s f Y v b y ( ; ® ) = +1 d 2 X A 4 m L 77

32. T h e R d u c t i o n ( P r o p s i t n 2 . L e T b a m l u f - M R C) G I c ; , h 1 v : w g F S d
> y x A + B = 4 77

33. If PT contains b2 ¹ a ¹ a ¹ a ¹ a ¹ a a1 a2 a3 a4 a5 s1 s2 b1 b2 b3 b477

34. T h e R d u c t i o n ( ¹ a ¹ a ¹ a ¹ a F S u p o s e t h a P c n i mY c n i m r v A . I l d T ( 1 ; 2 )
> + x b ai
ai+1 ai
ai+1 a i a i + 1 a i a i + 1 W e m a y o b t i n d h r s p g c l T f + 1 v x . 77

35. T h e R d u c t i o n ( P r o p s i t n 3 . L e T b a m l u f 2 = M R) G I c ; , h g y X B d 1 + F Y v V w 77

36. F o r a n y b , p f P s t 1 ® d ( ; ) + ¸ L . h e l w u c m v x C q Xi , p f P s t 1 d T ( ; ) + 2 L . h e l w u c m v x C q X = 3 4 B I g R U E 77

37. Conclusion of The Reductionm 1 . F a n y f i x d t g , 2 - M R C ( ) p b l s N P B 3 77

38. The 2-MRCT() problem is NP-hard even for metric graphsy 2 . F n x e d i t g 1 , h - M R T ( ) p b m s N P v f c L G u = V ; w 3 77

39. We can easily transform the 2-MRCT() problem to the p-MRCT
zero edge s1 s2
p copies of s1 C o r l a y 3 . F n x e d i t g p 1 , h - M R T s N P v f m c u 77

40. Finally, p-OCT is NP-harde t h O C T p r o b l m u d s M R a , : y 4 . F x g 1 - N P v f 77

41. Outline F I n t r o d u c i P e l m a s T h N - A p x g O C 2 k 77

42. S i m p l e c a s : 2 - O C T w t h r n u N o t e h a m r i c g p s lf w d b n y v - . T x 2 O C , u ( 1 ; ) 77

43. Algorithm A1 A l g o r i t h m 1 I n p u : e c a G , w s d q ( v ) f x2 d q ( v ) f x V . O T y ; F / * C + 77

44. Algorithm A1 s2 s1 77

45. The important definitionn i t o 7 . L Y b h 2 - O C T a d P p w s 1 W f ( v ) = ; r c x m 5 V + B y , u l q 77

46. The ratio and running time of A16 . A l g i t 1 c p u s a 2 - x n f O C ( ) P H w d y U j , k b v W L Y B q ; = + 77

47. The ratio and running time of A1 cont.W e h a v ( r 1 ) + 2 w ; s d Y P f = c 77

48. T h a t i s , f v c o n e d 1 - r m 2 ( ) . S l y x b g u w Y + ; 77

49. S i n c e t h m u o f w b r s a g d , v ( ) + 2 ; · = £ : ¡ ¸ y l 5 W1 ; = : T y l 5 W x . - P V p 77

50. What we know from A1 ? T h e a b o v l g r i t m s ¯ u d c - k n , y x. 77

51. The reduced skeleton of a treen i t o 8 . L T b a s p g r f m c h G d S V ( ) u - k l Y y ; v 2 P X = E w : F , x 77

52. u1 u2
Y, assume Y=Fig3.(a) 77

53. u2 u1
The skeleton of Fig 3.(a) 77

54. The reduced skeleton of Fig 3.(a)X u1 u2
The reduced skeleton of Fig 3.(a) 77

55. What we will do ? O u r a l g o i t h m e s d c S - k n X f C T , v xb y p . B w ? I 2 j = ( ) 77

56. The important Lemma L e m a 7 . t X b h r d u c S - s k l o n f T F y; v 2 V ( ) , j P B i p w G g C q 77

57. P r o f . C n t L e a d b h u m l v s i c , p y T · j S ( ¡ ) w X g +1 a d 2 b h u m l v s i c , p y T j S ( ) w X g + 3 | { z } ; 77

58. A p r o x i m a t n g h e - O C T A l g o r i t h m 2 I n p u : e c a= ( V ; E w ) , s S f d q v . O T F 1 - X x [ y / * C P + z k b 77

59. T h e r u n i g t m a d o f A l 2 T h e o r m 8 . F a t i c g p , A ls - x u O C ( 1 ) w P X v b 4 V k ; S 77

60. L e t Y b h o p i m a l s u n d ¹ X r c S - k f . B y 7 , j V ( ) · 2F w x
< g A = 77

61. S o m e d ¯ n i t s : L ( u ; ) X a P = . R v g h f r , c l - p b y B1 ; 2 ) X a P = Y . R v g h f r , c l - p b y B V [ A k T E x 77

62. B(u1,u2) u2 Y2 u1 Y Y1 77

63. The goal of our proof S i n c e d ( u ; v ) = f o r a y 2 V X b L m 7, w h . W s l p t 1 E x B - 77

64. Let S1 and S2 as “Big” sourcesF o r v 2 V , l e t R 1 = P s i S ( ) a n d . A h p u b c w f Y ; x X y m k g T + 77

65. S i n c e w ( v ; u ) · d P + f a , R . B y ¯ t o h s b m p l : = r q1 ) d Y P + f a 2 , R . B y t o h s b m p l : = r T q k g - x 77

66. End of the proof F i n a l y , w e s h o t p r x m f c j V ( ¹ X )
< 2 . T u d E v z I = 77

67. Outline F I n t r o d u c i P e l m a s T h N - A p x g O C 2 k 77

68. The spirit of Algorithm A3u r a p o x i m t n l g h c s d e b w - f v . T s1 s2 77

69. Algorithm A3 I n p u t : A g r a h G = ( V ; E w ) , o s c e d q i m v1 2 d q i m v O T f . F X b l L 77

70. The ratio of the Algorithm A39 . t Y b h 2 - O C T F o r n y v x , c ( ) 3 : P f S u p s i d X B ; = G 1 w + l
dG…代表在G上的最短距離 77

71. The result of Algorithm A31 . a l g i t A 3 c p u s - x n f 2 O C G = ( V ; E w ) j + y b d P B L 9 , v Y F H 77

72. Outline F I n t r o d u c i P e l m a s T h N - A p x g O C 2 k 77

73. Concluding remarks I t w o u l d b e i n r s g ¯ ± c a h m p x - O C T. A v f , y 77

74. The Important key notes:1 . T h e r d u c S - s k l t o n f a 2 i m p q 77

75. About the author…
吳邦一
Associate Professor, Dept. Information Engineering & Computer Science, Shu-Te University. 77

76. More … P r o b l e m R s t i c n a O C T N g v ( ) ¸ u ; = A S + 2 M 1- , f p j h 3 77

77. More and More…
You can see the book written by Bang Ye Wu and Kun-Mao Chao 77

78. 77