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 1 4 3 ( 2 ) - B n g Y W u If you can not see some characters, please install TEX4PPT for the url. http://www.ecs.soton.ac.uk/~srg/softwaretools/presentation/TeX4PPT/ If the problem exists, please install a full MikTex software. S p e a k r : C h n g - u L i 2 4 / 1 9

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

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

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

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

Some special cases of the problem-1 q 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

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

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

Examples for PROCT, SROCT 1 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

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

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 ¸ ; = ½ 1 2 H V 77

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 O y m b . H w , 2 N P f c g A q u 77

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

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 g y ¯ x ¸ 2 . O u W F 3 77

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

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

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

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 ½ A O j + 77

Some Definitions-3 D e ¯ n i t o 3 . L T b a s p g r f h G d S = ; , V 1 ; 2 , V ( ) v u c F y x m P w q 4 - l O C 77

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

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 ¯ a g 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

Some Definitions F L e t U = f u ; : g b a s o B l v r i - X c h m , d 1 ; 2 : n g b a s o B l v r i - X c h m , d y ¯ . A ® x ( _ ¹ 3 5 ) | { z } ^ 7 9 4 ¡ T j 77

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 T Y ( ) p m g y - 6 = V E w 1 2 ® ¸ M R C ; P + 77

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 ¹ j · [ y < + 3 8 ¡ 77

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 a ai ai+1 an E2 E1 E3 s1 … … s2 ¹ a 1 ¹ a 2 ¹ a 3 ¹ a i ¹ a i + 1 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

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

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 : · ¹ ¡ O S 77

Example: ( u _ ¹ ) ^ a ¹ s1 s2 a1 a5 a3 a4 a2 b1 b2 b3 b4 1 3 5 2 4 1 77

( u _ ¹ ) ^ t : f T , F g a ¹ s1 s2 a1 a5 a3 a4 a2 b1 b2 b3 b4 Y 1 3 5 77

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

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

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 ¹ g F S d > ¡ y x A + ¸ B = 4 77

If PT contains b2 ¹ a ¹ a ¹ a ¹ a ¹ a a1 a2 a3 a4 a5 s1 s2 b1 b2 b3 b4 77

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

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 1 + ¡ F Y v V ¸ w 77

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 X i , 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

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

The 2-MRCT() problem is NP-hard even for metric graphs y 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 ¹ = V ; £ w 3 77

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

Finally, p-OCT is NP-hard e 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

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

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 l f w d b n y v - . T x 2 O C , u ( 1 ; ) 77

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 x 2 d q ( v ) f x V . O T y ; F / * C + · 77

Algorithm A1 s2 s1 77

The important definition ¯ n 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

The ratio and running time of A1 6 . 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

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

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

S i n c e t h m u o f w b r s a g d , v ( ) + 2 ; · = £ : ¡ ¸ y l 5 W 1 ; · = £ : ¡ ¸ T y l 5 W x . - P V p 77

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

The reduced skeleton of a tree ¯ n i t o 8 . L T b a s p g r f m c h G d S ½ V ( ) u - k l Y y ; v 2 P X = E w : F , x 77

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

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

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

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 x b y p . B w ? I 2 j ¡ = ( ) 77

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

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

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 ½ f d q v . O T F 1 - X x [ y / * C P + ¤ z k b 77

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 l s - x u O C ( ¡ 1 ) w ¸ P X v b ¢ £ 4 V k ¤ ; S 77

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 ( ) · 2 ¡ F w x < g A = ± 77

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 B 1 ; 2 ) X a P = Y . R v g h f r , c l - p b y B V [ A k T E x 77

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

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

Let S1 and S2 as “Big” sources F 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

S i n c e w ( v ; u ) · d P + f a , R . B y ¯ t o h s b m p l : = r q 1 ) · d Y P + f a 2 , R . B y ¯ t o h s b m p l : = r T q k g - x 77

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 ½ v z I = 77

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

The spirit of Algorithm A3 u r a p o x i m t n l g h c s ¯ d e b w - f v . T s1 s2 77

Algorithm A3 I n p u t : A g r a h G = ( V ; E w ) , o s c e d q i m v 1 2 d q i m v O T f . F X b l L 77

The ratio of the Algorithm A3 9 . 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

The result of Algorithm A3 1 . 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

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

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

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

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

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

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

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