Devide and Conquar

8/9/2019 Devide and Conquar

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2 S T R A T E G Y D I V I D E - A N D - C O N Q U E R

D i v i d e P r o b l e m P i n t o s m a l l e r p r o b l e m

P

1

P

2

: : : P

k

S o l v e p r o b l e m s P

1

P

2

: : : P

k

t o o b t a i n s o l u -

t i o n s S

1

S

2

: : : S

k

C o m b i n e s o l u t i o n S

1

S

2

: : : S

k

t o g e t t h e n a l

s o l u t i o n .

S u b p r o b l e m s P

1

P

2

: : : P

k

a r e s o l v e d r e c u r s i v e l y

u s i n g d i v i d e - a n d - c o n q u e r .

E x a m p l e s : Q u i c k s o r t a n d m e r g e s o r t .

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3 S T R A T E G Y G R E E D Y

S o l u t i o n

f o r i 1 t o n d o

S E L E C T t h e n e x t i n p u t x

I f f x g S o l u t i o n i s F E A S I B L E t h e n

s o l u t i o n C O M B I N E ( S o l u t i o n , x )

S E L E C T a p p r o p r i a t e l y n d s t h e n e x t i n p u t t o

b e c o n s i d e r e d .

A F E A S I B L E s o l u t i o n s a t i s e s t h e c o n s t r a i n t s

r e q u i r e d f o r t h e o u t p u t .

C O M B I N E e n l a r g e s t h e c u r r e n t s o l u t i o n t o

i n c l u d e a n e w i n p u t .

E x a m p l e s : M a x n d i n g , S e l e c t i o n S o r t , a n d

K r u s k a l ' s S m a l l e s t E d g e F i r s t a l g o r i t h m f o r M i n -

i m u m S p a n n i n g T r e e .

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4 S T R A T E G Y D Y N A M I C P R O G R A M M I N G

F i b o n a c c i N u m b e r s :

F

n

= F

n ; 1

+ F

n ; 2

F

1

= F

0

= 1

| R e c u r s i v e s o l u t i o n r e q u i r e s e x p o n e n t i a l t i m e :

h a s o v e r l a p p i n g s u b p r o b l e m s .

| B o t t o m - u p i t e r a t i v e s o l u t i o n i s l i n e a r {

c o m p u t e o n c e , s t o r e , a n d u s e m a n y

t i m e s .

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4 . 1 M a t r i x S e q u e n c e M u l t i p l i c a t i o n

e g . 1 :

A

3 0 x 1

X B

1 x 4 0

X C

4 0 x 1 0

X D

1 0 x 2 5

X E

2 5 x 1

| L e f t t o r i g h t e v a l u a t i o n r e q u i r e s m o r e t h a n

1 2 K m u l t i p l i c a t i o n s .

| ( A X ( ( B X C ) X ( D X E ) ) ) n e e d s o n l y

6 9 0 m u l t i p l i c a t i o n s ( m i n i m u m n e e d e d ) .

| G r e e d y A l g o r i t h m : L a r g e s t C o m m o n

D i m e n s i o n F i r s t

e g . 2 :

A

1 x 2

X B

2 x 3

X C

3 x 4

X D

4 x 5

X E

5 x 6

| L a r g e s t C o m m o n D i m e n s i o n F i r s t i m p o s e s

f o l l o w i n g o r d e r :

( A X ( B X ( C X ( D X E ) ) ) )

w h i c h n e e d s 2 4 0 m u l t i p l i c a t i o n s .

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| B e s t o r d e r :

( ( ( ( A X B ) X C ) X D ) X E )

w h i c h n e e d s 6 8 m u l t i p l i c a t i o n s .

| A n o t h e r G r e e d y A l g o r i t h m : S m a l l -

e s t C o m m o n D i m e n s i o n F i r s t b u t d i d

n o t w o r k f o r e g . 1 .

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4 . 2 D i v i d e a n d C o n q u e r S o l u t i o n

I n p u t :

A

1

* A

2

: : : * A

n

d

0

d

1

d

1

d

2

: : : d

n ; 1

d

n

O u t p u t : A p a r a n t h e s i z a t i o n o f t h e i n p u t s e q u e n c e

r e s u l t i n g i n m i n i m u m n u m b e r o f m u l t i p l i c a t i o n s

n e e d e d t o m u l t i p l y t h e n m a t r i c e s .

S u b g o a l : I g n o r e S t r u c t u r e o f O u t p u t ( o r d e r

o f p a r e n t h e s i z a t i o n ) , f o c u s o n o b t a i n i n g a n u -

m e r i c a l s o l u t i o n ( m i n i m u m n u m b e r o f m u l t i p l i -

c a t i o n s )

D e n e M i , j ] = t h e m i n i m u m n u m b e r o f m u l t i -

p l i c a t i o n s n e e d e d t o c o m p u t e

A

i

A

i + 1

A

j

f o r i j n

S u b g o a l i s t o o b t a i n M 1 n ]

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e . g . F o r ,

A 1

3 0 x 1

X A 2

1 x 4 0

X A 3

4 0 x 1 0

X A 4

1 0 x 2 5

X A 5

2 5 x 1

M 1 1 ] = 0 M 1 2 ] = 1 2 0 0 M 1 5 ] = 6 9 0

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4 . 3 R e c u r s i v e F o r m u l a t i o n o f M i j ]

A 1 X A 2 = ( A 1 ) X ( A 2 )

| P a r t i t i o n a t k = 1 : S u b p r o b l e m s ( A 1 ) a n d ( A 2 )

| c o s t o f ( A 1 ) i s M 1 1 ] a n d t h a t o f ( A 2 ) i s

M 2 2 ]

| c o s t o f c o m b i n i n g ( A 1 ) a n d ( A 2 ) i n t o o n e i s

d

0

d

1

d 2

| M 1 2 ] = M 1 1 ] + M 2 2 ] + d

0

d

1

d 2

| M 1 2 ] = 0 + 0 + 1 2 0 0 = 1 2 0 0 .

A 2 X A 3 X A 4 = ( A 2 ) X ( A 3 X A 4 ) ( k = 2 )

O r , = ( A 2 X A 3 ) X A 4 ( k = 3 )

| k = 2 : c o s t = M 2 2 ] + M 3 4 ] + d 1 d 2 d 4

| k = 3 : c o s t = M 2 3 ] + M 4 4 ] + d 1 d 3 d 4

| M 2 4 ] = m i n ( M 2 2 ] + M 3 4 ] + d 1 d 2

d 4 M 2 3 ] + M 4 4 ] + d 1 d 3 d 4 )

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I n s h o r t , M 2 4 ] =

m i n

2 k 3

( M 2 k ] + M k 4 ] + d

1

d

k

d

4

)

A 2 X A 3 X A 4 X A 5

= ( A 2 ) X ( A 3 X A 4 X A 5 ) ( k = 2 )

O r , = ( A 2 X A 3 ) X ( A 4 X A 5 ) ( k = 3 )

O r , = ( A 2 X A 3 X A 4 ) X ( A 5 ) ( k = 4 )

M 2 5 ] = ( M 2 2 ] + M 3 5 ] + d

1

d

2

d

5

M 2 3 ] +

M 4 5 ] + d

1

d

3

d

5

M 2 4 ] + M 5 5 ] + d

1

d

4

d

5

)

I n g e n e r a l , b y f a c t o r i n g ( A

i

A

i + 1

A

j

)

a t k t h i n d e x p o s i t i o n i n t o ( A

i

A

i + 1

A

k

)

a n d ( A

k + 1

A

j

) n e e d M i k ] + M k + 1 j ]

m u l t i p l i c a t i o n s a n d c r e a t e s m a t r i c e s o f d i m e n -

s i o n s d

i ; 1

d

k

a n d d

k

d

j

. T h e s e t w o m a t r i c e s

n e e d a d d i t i o n a l d

i ; 1

d

k

d

j

m u l t i p l i c a t i o n s t o

c o m b i n e .

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4 . 4 R e c u r s i v e F o r m u l a a n d T i m e T a k e n

R e c u r s i v e l y ,

M i j ] =

m i n

i k j ; 1

M i k ] + M k + 1 j ] + d

i ; 1

d

k

d

j

!

M ( i i ) = 0

O p t i m a l S u b s t r u c t u r e

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W e c a n r e c u r s i v e l y s o l v e f o r

M 1 n ] =

m i n

1 k n ; 1

( M 1 k ] + M k + 1 n ] + d

0

d

k

d

n

)

= m i n M 1 1 ] + M 2 n ] + d

0

d

1

d

n

,

M 1 2 ] + M 3 n ] + d

0

d

2

d

n

,

M 1 3 ] + M 4 n ] + d

0

d

3

d

n

,

M 1 n ; 1 ] + M n n ] + d

0

d

n ; 1

d

n

T i m e C o m p l e x i t y :

T

n

= n + T

1

+ T

n ; 1

+ T

2

+ T

n ; 2

+ T

3

+ T

n ; 3

+ T

n ; 2

+ T

2

+ T

n ; 1

+ T

1

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T

n

= n + 2 T

1

+ 2 T

2

+ + 2 T

n ; 1

( 1 )

T

n ; 1

= n ; 1 + 2 T

1

+ 2 T

2

+ + 2 T

n ; 2

( 2 )

S u b t r a c t i n g ( I ) - ( I I ) y i e l d s

T

n

; T

n ; 1

= 1 + 2 T

n ; 1

T

n

= 1 + 3 T

n ; 1

= 1 + 3 ( 1 + 3 T

n ; 2

)

T

n

= 1 + 3 + 3

2

+ 3

3

+ + 3

n ; 1

T

1

= 1 + 3 + 3

2

+ 3

3

+ + 3

n ; 2

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R e c u r s i v e S o l u t i o n i s e x p o n e n t i a l t i m e ( 3

n ; 2

)

S p a c e O ( n ) s t a c k d e p t h .

O v e r l a p p i n g s u b p r o b l e m s : e . g . R e c u r s i o n

t r e e f o r M 1 4 ]

2 6 r e c u r s i v e c a l l s f o r j u s t 1 0 s u b p r o b l e m s

M 1 1 ] M 2 2 ] M 3 3 ] M 4 4 ] M 1 2 ] M 2 3

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S o w e t u r n t o d y n a m i c P r o g r a m m i n g ,

t h e s a m e f o r m u l a t i o n

a p p r o a c h t h e p r o b l e m b o t t o m - t o - t o p

n d a s u i t a b l e t a b l e t o s t o r e t h e s u b - s o l u t i o n s .

H o w m a n y s u b - s o l u t i o n s d o w e h a v e ?

M 1 1 ] M 2 2 ] M 3 3 ] M n n ] n

M 1 2 ] M 2 3 ] M n ; 1 n ] n ; 1

M 1 3 ] M 2 4 ] M n ; 2 n ] n ; 2

M 1 n ; 1 ] M 2 n ] 2

M 1 n ] 1

n ( n ; 1 )

2

) w e n e e d O ( n

2

) s p a c e

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4 . 5 M a t r i x P a r e n t h e s i z a t i o n O r d e r

M , F a c t o r : M a t r i x

f o r i 1 t o n d o M i i ] 0

/ * m a i n d i a g o n a l * /

f o r d i a g o n a l 1 t o n ; 1 d o

f o r i 1 t o n ; d i a g o n a l d o

j = i + d i a g o n a l

M i j ] = m i n

i k j ; 1

( M i k ] + M k + 1 j ]

+ d

i ; 1

d

k

d

j

)

F a c t o r i j ] = k t h a t g a v e t h e m i n i m u m v a l u e

f o r M i j ]

e n d f o r

e n d f o r

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4 . 6 W o r k o u t

A 1

3 0 x 1

X A 2

1 x 4 0

X A 3

4 0 x 1 0

X A 4

1 0 x 2 5

X A 5

2 5 x 1

M 1 2 ] = m i n

1 k 1

M i k ] + M k + 1 j ] +

d

i ; 1

d

k

d

j

]

= m i n M 1 1 ] + M 2 2 ] + d

0

d

1

d

2

]

= 0 + 0 + 3 0 1 4 0

= 1 2 0 0

M 1 3 ] = m i n

1 k 3 ; 1

M i k ] + M k + 1 j ] +

d

i ; 1

d

k

d

j

]

= m i n M 1 1 ] + M 2 3 ] + d

0

d

1

d

3

M 1 2 ] + M 3 3 ] + d

0

d

2

d

3

]

= m i n 0 + 4 0 0 + 3 0 1 1 0

1 2 0 0 + 0 + 3 0 4 0 1 0 ]

= m i n 7 0 0 1 2 0 0 0 + 1 2 0 0 ]

= 7 0 0

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M 2 4 ] = m i n M i k ] + M k + 1 j ] + d

i ; 1

d

k

d

j

]

2 k 3

= m i n M 2 2 ] + M 3 4 ] + d

1

d

2

d

4

M 2 3 ] + M 4 4 ] + d

1

d

3

d

4

]

= m i n 0 + 1 0 0 0 0 + 1 4 0 2 5 4 0 0 + 0 + 1 1 0 2 5 ]

= m i n 1 0 1 0 0 6 5 0 ] = 6 5 0

M 3 5 ] = m i n M i k ] + M k + 1 j ] + d

i ; 1

d

k

d

j

]

3 k 4

= m i n M 3 3 ] + M 4 5 ] + d

2

d

3

d

5

M 3 4 ] + M 5 5 ] + d

2

d

4

d

5

]

= m i n 0 + 2 5 0 + 4 0 1 0 1 1 0 0 0 0 + 0 + 4 0 2 5 1 ]

= m i n 6 5 0 ; ] = 6 5 0

M 1 4 ] = m i n M 1 1 ] + M 2 4 ] + d

0

d

1

d

4

M 1 2 ] + M 3 4 ] + d

0

d

2

d

4

M 1 3 ] + M 4 4 ] + d

0

d

3

d

4

]

= m i n 0 + 6 5 0 + 3 0 1 2 5

1 2 0 0 + 1 0 0 0 0 + 3 0 4 0 2 5

7 0 0 + 0 + 3 0 1 0 2 5 ]

= m i n 1 4 0 0 ; ; ] = 1 4 0 0

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5 D Y N A M I C P R O G R A M M I N G R E Q U I R E M E N T S

R e q u i r e m e n t s : a ) O p t i m a l S u b s t r u c t u r e

b ) O v e r l a p p i n g s u b p r o b l e m

S t e p s : 1 ) C h a r a c t e r i z e t h e s t r u c t u r e o f a n o p t i -

m a l s o l u t i o n

2 ) F o r m u l a t e a r e c u r s i v e s o l u t i o n

3 ) C o m p u t e t h e v a l u e o f a n o p t . s o l u t i o n

b o t t o m - u p . ( g e t v a l u e r a t h e r t h a n t h e s t r u c -

t u r e )

4 ) C o n s t r u c t a n o p t i m a l s o l u t i o n ( s t r u c t u r e )

f r o m c o m p u t e d i n f o m a t i o n .

M e m o i z a t i o n : T o p - d o w n , c o m p u t e a n d s t o r e r s t

t i m e , r e u s e s u b s e q u e n t t i m e s .

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5 . 1 O b s e r v a t i o n 1 : O p t i m a l S u b s t r u c t u r e

T h e o p t i m a l s o l u t i o n c o n t a i n i n g s o p t i m a l s u b s o l u -

t i o n s .

R e c u r s i o n T r e e ( d o n o t w i d e y e t )

D e p t h ? ( m + n )

o u t d e g r e e 3 ) n u m b e r o f n o d e s a m o u n t o f w o r k

i n r e c u r s i v e c a l l s i s ( 3

m + n

)

2 0


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5 . 2

O b s e r v a t i o n 2 : O v e r l a p p i n g S u b p r o b l e m s

w i d e s o m e r e p e a t e d p r o b l e m s , a s a b o v e .

a f e w p r o b l e m s , b u t m a n y r e c u r s i v e i n s t a n c e s

u n l i k e g o o d d i v i d e - a n d - c o n q u e r w h e r e p r o b l e m s

a r e i n d e p e n d e n t .

L C S h a s a n m n d i s t i n c t p r o b l e m s .

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5 . 3 M e m o r i z e

( t o d e a l w i t h o v e r l a p p i n g p r o b l e m s )

a f t e r c o m p u t i n g s o l u t i o n t o a s u b p r o b l e m , s o r t

i n a t a b l e . S u b s e q u e n t c a l l - d o t a b l e l o o k u p .

T i m e O ( m n )

| e a c h p r o b l e m i s s o l v e d o n c e a n d u s e d t w i c e .

| s e e i n g u r e f o r 1 2 o r 2 3

| m i g h t n o t n e e d t o s o l v e a l l s u b p r o b l e m , o n l y

t h o s e n e e d e d .

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5 . 4 D y n a m i c P r o g r a m m i n g I m p l i m e n t a t i o n

C o m p u t e t a b l e b o t t o m - u p i n s t e a d o f s t a r t i n g a t

( m n ) , s t a r t a t ( 1 1 )

D e m o n s t r a t e a l g o r i t h m

| t i m e = ( m n )

| s p a c e = ( m i n ( m n ) )

I n i t i a l i z e t o p r o w & l e f t c o l u m n t o 0

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P r o d u c e f r o m t o p r o w , l e f t t o r i g h t , x i ] = y j ]

l l d i a g o n a l n e i g h b o r + 1 & c h a w . s l r e l l m a x

o f t h e o t h e r n e i g h b o r s .

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