[PPT] - C o mp l e x i t y ( W e i s s c h a p t e r 5 PowerPoint Presentation

SLIDE 1

C

mp

l e x i t y ( W e i s s c h a p t e r 5 )

SLIDE 2

C

m

p l e x i t y

Tii s l e c t u r e i s a l l a b

u

t h

w

t

d

e s c r i b e t h e p e r f

r

m a n c e

f

a n a l g

r

i t h m L a s t t i m e w e h a d t h r e e v e r s i

n

s

f

t h e fj l e

r

e a d i n g p r

g

r a m . F

r

a fj l e

f

s i z e n :

Tie

fj r s t

n

e n e e d e d t

c
p

y n

2

/ 2 c h a r a c t e r s

Tie

s e c

n

d

n

e n e e d e d t

c
p

y n

2

/ 2 c h a r a c t e r s

Tie

t h i r d n e e d e d t

c
p

y 2 n c h a r a c t e r s

W e w

r

k e d

u

t t h e s e f

r

m u l a s , b u t i t w a s a b i t

f

w

r

k – n

w

w e ' l l s e e a n e a s i e r w a y

SLIDE 3

B i g i d e a : i g n

r

e c

n

s t a n t f a c t

r

s !

SLIDE 4

W h y d

w

e i g n

r

e c

n

s t a n t f a c t

r

s ?

W e l l , w h e n n i s 1 . . .

l
g

2

n i s 2

n

i s 1

n

2

i s 1

2

n

i s a n u m b e r w i t h 3 , d i g i t s . . .

G i v e n t w

a

l g

r

i t h m s :

Tie

fj r s t t a k e s 1 l

g

2

n s t e p s t

r

u n

Tie

s e c

n

d t a k e s . 1 × 2

n

Tie fj r s t i s m i l e s b e t t e r ! C

n

s t a n t f a c t

r

s n

r

m a l l y d

n

' t m a t t e r

SLIDE 5

B i g O n

t

a t i

n

I n s t e a d

f

s a y i n g . . .

Tie

fj r s t i m p l e m e n t a t i

n

c

p

i e s n

2

/ 2 c h a r a c t e r s

Tie

s e c

n

d c

p

i e s n

2

/ 2 c h a r a c t e r s

Tie

t h i r d c

p

i e s 2 n c h a r a c t e r s

W e w i l l j u s t s a y . . .

Tie

fj r s t i m p l e m e n t a t i

n

c

p

i e s O ( n

2

) c h a r a c t e r s

Tie

s e c

n

d c

p

i e s O ( n

2

) c h a r a c t e r s

Tie

t h i r d c

p

i e s O ( n ) c h a r a c t e r s

O ( n

2

) me a n s “ p r

p
r

t i

n

a l t

n

2

” ( a l mo s t )

SLIDE 6

T i m e c

m

p l e x i t y

S u p p

s

e a n a l g

r

i t h m t a k e s n

2

/ 2 s t e p s , a n d e a c h s t e p t a k e s 1 n s t

r

u n

Tie

t

t

a l t i m e t a k e n i s 5 n

2

n s

Tii

s i s O ( n

2

)

Tie

n u m b e r

f

s t e p s t a k e n i s a l s

O

( n

2

)

I t d

e

s n ' t m a t t e r w h e t h e r w e c

u

n t s t e p s

r

t i m e ! W e s a y t h a t t h e a l g

r

i t h m h a s O ( n

2

) t i m e c

m

p l e x i t y

r

s i m p l y c

m

p l e x i t y

SLIDE 7

W h y i g n

r

e c

n

s t a n t f a c t

r

s ?

B i g O r e a l l y s i m p l i fj e s t h i n g s :

A

s m a l l p h r a s e l i k e O ( n

2

) t e l l s y

u

a l

t
I

t ' s e a s i e r t

c

a l c u l a t e t h a n a p r e c i s e f

r

m u l a

W

e g e t t h e s a m e a n s w e r w h e t h e r w e c

u

n t n u m b e r

f

s t a t e m e n t s e x e c u t e d

r

t i m e t a k e n (

r

i n t h i s c a s e n u m b e r

f

e l e m e n t s c

p

i e d ) – s

w

e c a n b e a b i t c a r e l e s s w h a t w e c

u

n t

O n t h e

t

h e r h a n d :

S
m

e t i m e s w e d

c

a r e a b

u

t c

n

s t a n t f a c t

r

s !

B i g O i s n

r

m a l l y a g

d

c

m

p r

m

i s e

SLIDE 8

W h a t h a p p e n s w i t h

u

t b i g O ?

H

w

m a n y s t e p s d

e

s t h i s f u n c t i

n

t a k e

n

a n a r r a y

f

l e n g t h n ( i n t h e w

r

s t c a s e ) ?

O b j e c t s e a r c h ( O b j e c t [ ] a , O b j e c t x ) { f

r

( i n t i = ; i < a . l e n g t h ; i + + ) { i f ( a [ i ] . e q u a l s ( t a r g e t ) ) r e t u r n a [ i ] ; } r e t u r n n u l l ; }

A s s u m e t h a t l

p

b

d

y t a k e s 1 s t e p

SLIDE 9

W h a t h a p p e n s w i t h

u

t b i g O ?

H

w

m a n y s t e p s d

e

s t h i s f u n c t i

n

t a k e

n

a n a r r a y

f

l e n g t h n ( i n t h e w

r

s t c a s e ) ?

O b j e c t s e a r c h ( O b j e c t [ ] a , O b j e c t x ) { f

r

( i n t i = ; i < a . l e n g t h ; i + + ) { i f ( a [ i ] . e q u a l s ( t a r g e t ) ) r e t u r n a [ i ] ; } r e t u r n n u l l ; }

A n s w e r : n

SLIDE 10

W h a t a b

u

t t h i s

n

e ?

b

l

e a n u n i q u e ( O b j e c t [ ] a ) { f

r

( i n t i = ; i < a . l e n g t h ; i + + ) f

r

( i n t j = ; j < a . l e n g t h ; j + + ) i f ( a [ i ] . e q u a l s ( a [ j ] ) & & i ! = j ) r e t u r n f a l s e ; r e t u r n t r u e ; }

SLIDE 11

W h a t a b

u

t t h i s

n

e ?

b

l

e a n u n i q u e ( O b j e c t [ ] a ) { f

r

( i n t i = ; i < a . l e n g t h ; i + + ) f

r

( i n t j = ; j < a . l e n g t h ; j + + ) i f ( a [ i ] . e q u a l s ( a [ j ] ) & & i ! = j ) r e t u r n f a l s e ; r e t u r n t r u e ; }

O u t e r l

p

r u n s n t i m e s E a c h t i m e , i n n e r l

p

r u n s n t i m e s T

t

a l : n × n = n

2

SLIDE 12

W h a t a b

u

t t h i s

n

e ?

b

l

e a n u n i q u e ( O b j e c t [ ] a ) { f

r

( i n t i = ; i < a . l e n g t h ; i + + ) f

r

( i n t j = ; j < i ; j + + ) i f ( a [ i ] . e q u a l s ( a [ j ] ) ) r e t u r n f a l s e ; r e t u r n t r u e ; }

L

p

r u n s t

i

i n s t e a d

f

n

SLIDE 13

S

m

e h a r d s u m s

Wh e n i = , i n n e r l

p

r u n s t i m e s Wh e n i = 1 , i n n e r l

p

r u n s 1 t i m e … Wh e n i = n

1

, i n n e r l

p

r u n s n

1

t i m e s T

t

a l :

=

+ 1 + 2 + … + n

1

w h i c h i s n ( n

1

) / 2

∑

i=0 n−1

i

SLIDE 14

W h a t a b

u

t t h i s

n

e ?

b

l

e a n u n i q u e ( O b j e c t [ ] a ) { f

r

( i n t i = ; i < a . l e n g t h ; i + + ) f

r

( i n t j = ; j < i ; j + + ) i f ( a [ i ] . e q u a l s ( a [ j ] ) ) r e t u r n f a l s e ; r e t u r n t r u e ; }

A n s w e r : n ( n

1

) / 2

SLIDE 15

W h a t a b

u

t t h i s

n

e ?

b

l

e a n u n i q u e ( O b j e c t [ ] a ) { f

r

( i n t i = ; i < a . l e n g t h ; i + + ) f

r

( i n t j = ; j < i ; j + + ) f

r

( i n t k = ; k < j ; k + + ) “ s

m

e t h i n g t h a t t a k e s 1 s t e p ” }

SLIDE 16

M

r

e h a r d s u m s

O u t e r l

p

: i g

e

s f r

m

t

n
1

Mi d d l e l

p

: j g

e

s f r

m

t

i
1

I n n e r l

p

: k g

e

s f r

m

t

j
1

C

u

n t s : h

w

m a n y v a l u e s i , j , k w h e r e ≤ i < n , ≤ j < i , ≤ k ≤ j

∑

i=0 n−1

∑

j=0 i−1

∑

k=0 i−1

1

SLIDE 17

M

r

e h a r d s u m s

C

u

n t s : h

w

m a n y v a l u e s i , j , k w h e r e ≤ i < n , ≤ j < i , ≤ k ≤ j

∑

i=0 n−1

∑

j=0 i−1

∑

k=0 i−1

1 I h a v e n

i

d e a h

w

t

s
l

v e t h i s ! W

l

f r a m A l p h a s a y s i t ' s n ( n

1

) ( n

2

) / 6

SLIDE 18

W h a t a b

u

t t h i s

n

e ?

b

l

e a n u n i q u e ( O b j e c t [ ] a ) { f

r

( i n t i = ; i < a . l e n g t h ; i + + ) f

r

( i n t j = ; j < i ; j + + ) f

r

( i n t k = ; k < j ; k + + ) “ s

m

e t h i n g t h a t t a k e s 1 s t e p ” }

A n s w e r : n ( n

1

) ( n

2

) / 6 , a p p a r e n t l y

SLIDE 19

T h i s i s j u s t h

r

r i b l e ! I s n ' t t h e r e a b e t t e r w a y ?

SLIDE 20

U s i n g b i g O c

m

p l e x i t y

b

l

e a n u n i q u e ( O b j e c t [ ] a ) { f

r

( i n t i = ; i < a . l e n g t h ; i + + ) f

r

( i n t j = ; j < i ; j + + ) f

r

( i n t k = ; k < j ; k + + ) “ s

m

e t h i n g t h a t t a k e s 1 s t e p ” }

Tir e e n e s t e d l

p

s , a l l r u n n i n g f r

m

t

n

. . . A n s w e r : O ( n

3

) !

SLIDE 21

W h y i g n

r

e c

n

s t a n t f a c t

r

s ? ( a g a i n )

B i g O r e a l l y s i m p l i fj e s t h i n g s :

A

s m a l l p h r a s e l i k e O ( n

2

) t e l l s y

u

a l

t
I

t ' s e a s i e r t

c

a l c u l a t e t h a n a p r e c i s e f

r

m u l a

W

e g e t t h e s a m e a n s w e r w h e t h e r w e c

u

n t n u m b e r

f

s t a t e m e n t s e x e c u t e d

r

t i m e t a k e n (

r

i n t h i s c a s e n u m b e r

f

e l e m e n t s c

p

i e d ) – s

w

e c a n b e a b i t c a r e l e s s w h a t w e c

u

n t

O n t h e

t

h e r h a n d :

S
m

e t i m e s w e d

c

a r e a b

u

t c

n

s t a n t f a c t

r

s !

B i g O i s n

r

m a l l y a g

d

c

m

p r

m

i s e

SLIDE 22

W h y i g n

r

e c

n

s t a n t f a c t

r

s ? ( a g a i n )

B i g O r e a l l y s i m p l i fj e s t h i n g s :

A

s m a l l p h r a s e l i k e O ( n

2

) t e l l s y

u

a l

t
I

t ' s e a s i e r t

c

a l c u l a t e t h a n a p r e c i s e f

r

m u l a

W

e g e t t h e s a m e a n s w e r w h e t h e r w e c

u

n t n u m b e r

f

s t a t e m e n t s e x e c u t e d

r

t i m e t a k e n (

r

i n t h i s c a s e n u m b e r

f

e l e m e n t s c

p

i e d ) – s

w

e c a n b e a b i t c a r e l e s s w h a t w e c

u

n t

O n t h e

t

h e r h a n d :

S
m

e t i m e s w e d

c

a r e a b

u

t c

n

s t a n t f a c t

r

s !

B i g O i s n

r

m a l l y a g

d

c

m

p r

m

i s e

I s n ' t i t ! O u r l

n

g c a l c u l a t i

n
n

l y t

l

d u s h

w

m a n y s t e p s t h e a l g

r

i t h m t a k e s , n

t

h

w

m u c h t i m e ! B u t n

r

m a l l y n

t

e n

u

g h t

g
t
a

l l t h i s t r

u

b l e !

SLIDE 23

T h e r e s t

f

t h e l e c t u r e

H

w

t

c

a l c u l a t e b i g

O

c

m

p l e x i t y :

W

e w i l l fj r s t h a v e t

d

e fj n e f

r

m a l l y w h a t i t m e a n s f

r

a n a l g

r

i t h m t

h

a v e a c e r t a i n c

m

p l e x i t y

W

e w i l l t h e n c

m

e u p w i t h s

m

e r u l e s f

r

c a l c u l a t i n g c

m

p l e x i t y

T
c
m

e u p w i t h t h

s

e r u l e s , w e w i l l h a v e t

d
“

h a r d s u m s ” , b u t

n

c e w e h a v e t h e r u l e s w e c a n f

r

g e t t h e s u m s

(

v e r y

c

c a s i

n

a l l y , y

u

m i g h t s t i l l h a v e t

d
t

h e s u m s y

u

r s e l f )

SLIDE 24

B i g O , f

r

m a l l y

B i g O m e a s u r e s t h e g r

w

t h

f

a m a t h e m a t i c a l f u n c t i

n
T

y p i c a l l y a f u n c t i

n

T ( n ) g i v i n g t h e n u m b e r

f

s t e p s t a k e n b y a n a l g

r

i t h m

n

i n p u t

f

s i z e n

B

u t c a n a l s

b

e u s e d t

m

e a s u r e s p a c e c

m

p l e x i t y ( m e m

r

y u s a g e )

r

a n y t h i n g e l s e

F

r

m a l l y , w e s a y “ T ( n ) i s O ( f ( n ) ) ”

E

. g . , “ T ( n ) i s O ( n

2

) ”

Tii s m e a n s :

T

( n ) ≤ a × f ( n ) , f

r

s

m

e c

n

s t a n t a ( i . e . , T ( n ) i s p r

p
r

t i

n

a l t

f

( n )

r

s ma l l e r )

B

u t t h i s n e e d

n

l y h

l

d f

r

a l l n a b

v

e s

m

e t h r e s h

l

d n

SLIDE 25

A n e x a m p l e : n

2

+ 2 n + 3 i s O ( n

2

)

n

2

+ 2 n + 3 ≤ 2 n

2

f

r

n ≥ 3 n = 3 c = 2

SLIDE 26

E x e r c i s e s

I

s n

2

+ 2 n + 3 i n O ( n

3

) ?

I

s 3 n + 5 i n O ( n ) ?

Wh

y d

w

e n e e d t h e “ t h r e s h

l

d ” n ?

SLIDE 27

SLIDE 28

G r

w

t h r a t e s

I m a g i n e t h a t w e d

u

b l e t h e i n p u t s i z e f r

m

n t

2

n . I f a n a l g

r

i t h m i s . . .

O

( 1 ) , t h e n i t t a k e s t h e s a m e t i m e a s b e f

r

e

O

( l

g

n ) , t h e n i t t a k e s a c

n

s t a n t a m

u

n t m

r

e

O

( n ) , t h e n i t t a k e s t w i c e a s l

n

g

O

( n l

g

n ) , t h e n i t t a k e s t w i c e a s l

n

g p l u s a l i t t l e b i t m

r

e

O

( n

2

) , t h e n i t t a k e s f

u

r t i m e s a s l

n

g

I f a n a l g

r

i t h m i s O ( 2

n

) , t h e n a d d i n g

n

e e l e m e n t m a k e s i t t a k e t w i c e a s l

n

g

SLIDE 29

SLIDE 30

A d d i n g b i g O ( a h i e r a r c h y )

O ( 1 ) < O ( l

g

n ) < O ( n ) < O ( n l

g

n ) < O ( n

2

) < O ( n

3

) < O ( 2

n

) Wh e n a d d i n g a t e r m l

w

e r i n t h e h i e r a r c h y t

n

e h i g h e r i n t h e h i e r a r c h y , t h e l

w

e r

c
m

p l e x i t y t e r m d i s a p p e a r s :

O ( 1 ) + O ( l

g

n ) = O ( l

g

n ) O ( l

g

n ) + O ( n

k

) = O ( n

k

) ( i f k ≥ ) O ( n

j

) + O ( n

k

) = O ( n

k

) , i f j ≤ k O ( n

k

) + O ( 2

n

) = O ( 2

n

)

SLIDE 31

A n e x a m p l e : n

2

+ 2 n + 3 i s O ( n

2

)

U s e h i e r a r c h y : n

2

+ 2 n + 3 = O ( n

2

) + O ( n ) + O ( 1 ) = O ( n

2

)

SLIDE 32

Q u i z

Wh a t a r e t h e s e i n B i g O n

t

a t i

n

?

n

2

+ 1 1

2

n

3

+ 3 n – 1

n

4

+ 2

n

SLIDE 33

J u s t u s e h i e r a r c h y !

n

2

+ 1 1 = O ( n

2

) + O ( 1 ) = O ( n

2

) 2 n

3

+ 3 n – 1 = O ( n

3

) + O ( n ) + O ( 1 ) = O ( n

3

) n

4

+ 2

n

= O ( n

4

) + O ( 2

n

) = O ( 2

n