A V L t r e e s ( W e i s s 4 . 4 ) B a l a n - - PowerPoint PPT Presentation

SLIDE 1

A V L t r e e s ( W e i s s 4 . 4 )

SLIDE 2

B a l a n c e d B S T s : t h e p r

• b

l e m

Tie B S T

• p

e r a t i

• n

s t a k e O ( h e i g h t

• f

t r e e ) , s

• f
• r

u n b a l a n c e d t r e e s c a n t a k e O ( n ) t i m e

SLIDE 3

B a l a n c e d B S T s : t h e s

• l

u t i

• n

T a k e B S T s a n d a d d a n e x t r a i n v a r i a n t t h a t m a k e s s u r e t h a t t h e t r e e i s b a l a n c e d

• H

e i g h t

• f

t r e e m u s t b e O ( l

• g

n )

• Tie

n a l l

• p

e r a t i

• n

s w i l l t a k e O ( l

• g

n ) t i m e

O n e p

• s

s i b l e i d e a f

• r

a n i n v a r i a n t :

• H

e i g h t

• f

l e f t c h i l d = h e i g h t

• f

r i g h t c h i l d ( f

• r

a l l n

• d

e s i n t h e t r e e )

• T

r e e w

• u

l d b e s

• r

t

• f

“ p e r f e c t l y b a l a n c e d ”

Wh a t ' s w r

• n

g w i t h t h i s i d e a ?

SLIDE 4

A t

• r

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

P e r f e c t b a l a n c e i s t

• r

e s t r i c t i v e ! N u m b e r

• f

n

• d

e s c a n

• n

l y b e 1 , 3 , 7 , 1 5 , 3 1 , . . .

• w

l

• w

l l e m u r l e m u r g

• r

i l l a g

• r

i l l a p e n g u i n p e n g u i n w

• l

f w

• l

f h a m s t e r h a m s t e r p a n d a p a n d a

SLIDE 5

A V L t r e e s – a l e s s r e s t r i c t i v e i n v a r i a n t

Tie A V L t r e e i s t h e fj r s t b a l a n c e d B S T d i s c

• v

e r e d ( f r

• m

1 9 6 2 ) – i t ' s n a m e d a f t e r A d e l s

• n
• V

e l s k y a n d L a n d i s I t ' s a B S T w i t h t h e f

• l

l

• w

i n g i n v a r i a n t :

• Tie

d i fg e r e n c e i n h e i g h t s b e t w e e n t h e l e f t a n d r i g h t c h i l d r e n

• f

a n y n

• d

e i s a t m

• s

t 1

• (

c

• m

p a r e d t

• f
• r

a p e r f e c t l y b a l a n c e d t r e e )

Tii s m a k e s t h e t r e e ' s h e i g h t O ( l

• g

n ) , s

• i

t ' s b a l a n c e d

SLIDE 6

E x a m p l e

• f

a n A V L t r e e ( f r

• m

W i k i p e d i a )

12 23 54 76 9 14 19 67 50 17 72

L e f t c h i l d h e i g h t 2 R i g h t c h i l d h e i g h t 1 L e f t c h i l d h e i g h t 2 R i g h t c h i l d h e i g h t 2 L e f t c h i l d h e i g h t 1 R i g h t c h i l d h e i g h t

SLIDE 7

W h y a r e t h e s e n

• t

A V L t r e e s ?

SLIDE 8

W h y a r e t h e s e n

• t

A V L t r e e s ?

L e f t c h i l d h e i g h t R i g h t c h i l d h e i g h t 8

SLIDE 9

W h y a r e t h e s e n

• t

A V L t r e e s ?

L e f t c h i l d h e i g h t 1 R i g h t c h i l d h e i g h t 3

SLIDE 10

R

• t

a t i

• n

R

• t

a t i

• n

r e a r r a n g e s a B S T b y m

• v

i n g a d i fg e r e n t n

• d

e t

• t

h e r

• t

, w i t h

• u

t c h a n g i n g t h e B S T ' s c

• n

t e n t s ( p i c f r

• m

W i k i p e d i a )

SLIDE 11

R

• t

a t i

• n

W e c a n s t r a t e g i c a l l y u s e r

• t

a t i

• n

s t

• r

e b a l a n c e a n u n b a l a n c e d t r e e . Tii s i s w h a t m

• s

t b a l a n c e d B S T v a r i a n t s d

• !

H e i g h t

• f

4 H e i g h t

• f

3

SLIDE 12

A V L i n s e r t i

• n

S t a r t b y d

• i

n g a B S T i n s e r t i

• n
• Tii

s m i g h t b r e a k t h e A V L ( b a l a n c e ) i n v a r i a n t

Tie n g

• u

p w a r d s f r

• m

t h e n e w l y

• i

n s e r t e d n

• d

e , l

• k

i n g f

• r

n

• d

e s t h a t b r e a k t h e i n v a r i a n t ( u n b a l a n c e d n

• d

e s ) I f y

• u

fj n d

• n

e , r

• t

a t e i t t

• fj

x t h e b a l a n c e Tie r e a r e f

• u

r c a s e s d e p e n d i n g

• n

h

• w

t h e n

• d

e b e c a m e u n b a l a n c e d

SLIDE 13

C a s e 1 : a l e f t

• l

e f t t r e e

5 c 2 5 b a

E a c h p i n k t r i a n g l e r e p r e s e n t s a n A V L t r e e w i t h h e i g h t k Tie p u r p l e r e p r e s e n t s a n i n s e r t i

• n

t h a t h a s i n c r e a s e d t h e h e i g h t

• f

t r e e a t

• k

+ 1 N

• t

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

• r

e t h e p u r p l e b i t w a s a d d e d A s s u m p t i

• n

: t h i s t r e e i s t h e d e e p e s t n

• d

e t h a t v i

• l

a t e s t h e i n v a r i a n t

SLIDE 14

C a s e 1 : a l e f t

• l

e f t t r e e

5 c 2 5 b a

H e i g h t k H e i g h t k + 2 L e f t h e i g h t m i n u s r i g h t h e i g h t = 2 : i n v a r i a n t b r

• k

e n !

SLIDE 15

C a s e 1 : a l e f t

• l

e f t t r e e

5 c 2 5 b a

Tii s i s c a l l e d a l e f t

• l

e f t t r e e b e c a u s e b

• t

h t h e r

• t

a n d t h e l e f t c h i l d a r e d e e p e r

• n

t h e l e f t T

• fj

x i t w e d

• a

r i g h t r

• t

a t i

• n

SLIDE 16

B a l a n c i n g a l e f t

• l

e f t t r e e , a f t e r w a r d s

5 c 2 5 b a

H e i g h t k + 1 H e i g h t k + 1 I n v a r i a n t r e s t

• r

e d !

SLIDE 17

C a s e 2 : a r i g h t

• r

i g h t t r e e

2 5 a 5 b c

M i r r

• r

i m a g e

• f

l e f t

• l

e f t t r e e C a n b e fj x e d w i t h l e f t r

• t

a t i

• n

SLIDE 18

C a s e 3 : a l e f t

• r

i g h t t r e e

5 c 2 5 b a

H e i g h t k H e i g h t k + 2 L e f t h e i g h t m i n u s r i g h t h e i g h t = 2 : i n v a r i a n t b r

• k

e n !

SLIDE 19

C a s e 3 : a l e f t

• r

i g h t t r e e

5 c 2 5 b a

W e c a n ' t fj x t h i s w i t h

• n

e r

• t

a t i

• n

L e t ' s l

• k

a t b ' s s u b t r e e s b

L

a n d b

R

SLIDE 20

C a s e 3 : a l e f t

• r

i g h t t r e e

5 c 2 5 a

R

• t

a t e 2 5

• s

u b t r e e t

• t

h e l e f t

4

b

R

b

L

H e i g h t k

• 1

SLIDE 21

C a s e 3 : a l e f t

• r

i g h t t r e e

5 c 2 5 a

W e n

• w

h a v e a l e f t

• l

e f t t r e e ! S

• w

e c a n fj x i t b y r

• t

a t i n g t h e w h

• l

e t r e e t

• t

h e r i g h t

4

b

R

b

L

H e i g h t k + 2 H e i g h t k H e i g h t k + 2 H e i g h t k + 1

SLIDE 22

C a s e 3 : a l e f t

• r

i g h t t r e e

5 c 2 5 a 4

b

R

b

L

B a l a n c e d ! N

• t

i c e i t w

• r

k s w h i c h e v e r

• f

b

L

a n d b

R

h a s t h e e x t r a h e i g h t

SLIDE 23

C a s e 4 : a r i g h t

• l

e f t t r e e

2 5 a 5 b c

M i r r

• r

i m a g e

• f

l e f t

• r

i g h t t r e e

SLIDE 24

H

• w

t

• i

d e n t i f y t h e c a s e s

L e f t

• l

e f t ( e x t r a h e i g h t i n l e f t

• l

e f t g r a n d c h i l d ) :

• h

e i g h t

• f

l e f t

• l

e f t g r a n d c h i l d = k + 1 h e i g h t

• f

l e f t c h i l d = k + 2 h e i g h t

• f

r i g h t c h i l d = k

• R
• t

a t e t h e w h

• l

e t r e e t

• t

h e r i g h t

L e f t

• r

i g h t ( e x t r a h e i g h t i n l e f t

• r

i g h t g r a n d c h i l d ) :

• h

e i g h t

• f

l e f t

• r

i g h t g r a n d c h i l d = k + 1 h e i g h t

• f

l e f t c h i l d = k + 2 h e i g h t

• f

r i g h t c h i l d = k

• F

i r s t r

• t

a t e t h e l e f t c h i l d t

• t

h e l e f t

• Tie

n r

• t

a t e t h e w h

• l

e t r e e t

• t

h e r i g h t

R i g h t

• l

e f t a n d r i g h t

• r

i g h t : s y m m e t r i c

A l g

• r

i t h m u s e s h e i g h t s

• f

s u b t r e e s t

• d

e t e r m i n e c a s e

SLIDE 25

T h e f

• u

r c a s e s

( p i c t u r e f r

• m

W i k i p e d i a ) Tie n u m b e r s i n t h e d i a g r a m s h

• w

t h e b a l a n c e

• f

t h e t r e e : l e f t h e i g h t m i n u s r i g h t h e i g h t T

• i

m p l e m e n t t h i s e ffj c i e n t l y , r e c

• r

d t h e b a l a n c e i n t h e n

• d

e s a n d l

• k

a t i t t

• w
• r

k

• u

t w h i c h c a s e y

• u

' r e i n

5 D 3 A 4 C B Left Right Case Right Left Case 3 A 4 5 C D B Right Right Case 5 D 4 3 B A C Left Left Case 4 5 C D Balanced 3 A B 4 5 C D Balanced 3 A B 3 A 5 D 4 B C

• 2

2 1

• 1

2

• 2

1/0

• 1/0
• 1/0

1/0

SLIDE 26

E x a m p l e : t h e q u i c k b r

• w

n f

• x

j u m p s

• v

e r a l a z y d

• g

I n s e r t “ b r

• w

n ” i n t

• “

t h e q u i c k ”

t h e q u i c k b r

• w

n L e f t

• l

e f t t r e e ! R

• t

a t e r i g h t

SLIDE 27

E x a m p l e : t h e q u i c k b r

• w

n f

• x

j u m p s

• v

e r a l a z y d

• g

I n s e r t “ b r

• w

n ” i n t

• “

t h e q u i c k ”

t h e q u i c k b r

• w

n

SLIDE 28

E x a m p l e : t h e q u i c k b r

• w

n f

• x

j u m p s

• v

e r a l a z y d

• g

I n s e r t “ j u m p s ” i n t

• “

t h e q u i c k b r

• w

n f

• x

”

t h e q u i c k b r

• w

n f

• x

j u m p s R i g h t

• r

i g h t t r e e ! ( Wh a t n

• d

e ? ) R

• t

a t e l e f t

SLIDE 29

E x a m p l e : t h e q u i c k b r

• w

n f

• x

j u m p s

• v

e r a l a z y d

• g

I n s e r t “ j u m p s ” i n t

• “

t h e q u i c k b r

• w

n f

• x

”

t h e q u i c k f

• x

j u m p s b r

• w

n

SLIDE 30

E x a m p l e : t h e q u i c k b r

• w

n f

• x

j u m p s

• v

e r a l a z y d

• g

I n s e r t “

• v

e r ” i n t

• “

t h e q u i c k b r

• w

n f

• x

j u m p s ”

t h e q u i c k f

• x

j u m p s b r

• w

n

• v

e r L e f t

• r

i g h t t r e e ! ( q u i c k → f

• x

→ j u m p s ) R

• t

a t e f

• x

l e f t . . .

SLIDE 31

E x a m p l e : t h e q u i c k b r

• w

n f

• x

j u m p s

• v

e r a l a z y d

• g

I n s e r t “

• v

e r ” i n t

• “

t h e q u i c k b r

• w

n f

• x

j u m p s ”

t h e q u i c k j u m p s

• v

e r f

• x

. . . t h e n r

• t

a t e q u i c k r i g h t b r

• w

n

SLIDE 32

E x a m p l e : t h e q u i c k b r

• w

n f

• x

j u m p s

• v

e r a l a z y d

• g

I n s e r t “

• v

e r ” i n t

• “

t h e q u i c k b r

• w

n f

• x

j u m p s ”

q u i c k j u m p s f

• x

t h e b r

• w

n

• v

e r

SLIDE 33

D e l e t i

• n

i n a n A V L t r e e

F i r s t d

• t

h e n

• r

m a l B S T d e l e t i

• n

Tie n g

• u

p t h e t r e e , fj n d i n g n

• d

e s t h a t b r e a k t h e i n v a r i a n t a n d fj x i n g t h e m u s i n g r

• t

a t i

• n

s

• S

t a r t f r

• m

t h e n

• d

e t h a t w a s r e m

• v

e d f r

• m

t h e t r e e ( r e c a l l t h a t i n t h e c a s e t h a t t h e v a l u e t

• b

e d e l e t e d w a s a n

• d

e w i t h t w

• c

h i l d r e n , t h i s w a s t h e b i g g e s t v a l u e i n t h e l e f t s u b t r e e )

Tie c a s e s y

• u

n e e d t

• c
• n

s i d e r a r e e x a c t l y t h e s a m e a s f

• r

i n s e r t i

• n

!

• Wh

e n i m p l e m e n t i n g A V L t r e e s , y

• u

n e e d

• n

l y i m p l e m e n t t h e b a l a n c i n g c

• d

e

• n

c e

• I

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

• r

i t h m w

• r

k s f

• r

a n y n

• d

e w h e r e t h e l e f t a n d r i g h t c h i l d r e n s a t i s f y t h e A V L i n v a r i a n t , b u t t h e i r h e i g h t s d i fg e r b y 2 . N

• t

h i n g s p e c i fj c t

• i

n s e r t i

• n

. . .

Tie r e i s

• n

e s u b t l e p

• i

n t , s e e n e x t s l i d e . . .

SLIDE 34

A n e x t r a c a s e i n d e l e t i

• n

5 c 2 5 b

H e i g h t k H e i g h t k + 2 Tii s c a s e d

• e

s n ’ t a p p e a r i n i n s e r t i

• n

( t h i n k a b

• u

t w h y ) . B u t i t w

• r

k s fj n e w h e n t r e a t e d a s a l e f t

• l

e f t t r e e !

a

a a n d b b

• t

h h a v e h e i g h t k + 1

SLIDE 35

C

• n

s i d e r t h e s m a l l e s t A V L t r e e w i t h h e i g h t h ( h ≥ 2 ) . I t m u s t h a v e t w

• c

h i l d r e n :

• O

n e

• f

h e i g h t h

• 1

, s

• t

h a t t h e t r e e t

• h

a v e h e i g h t h

• O

n e

• f

h e i g h t h

• 2

, s

• t

h a t t h e t r e e i s a s s m a l l a s p

• s

s i b l e

Tiu s , i f F ( h ) i s t h e s i z e

• f

t h e s m a l l e s t A V L t r e e

• f

h e i g h t h , w e h a v e :

• F

( ) = , F ( 1 ) = 1 , F ( h ) = F ( h

• 1

) + F ( h

• 2

) i f h ≥ 2

Tiu s F ( h ) i s t h e h t h F i b

• n

a c c i n u m b e r !

• F

( h ) ~ φ

h

, w h e r e i s t h e g

• l

d e n r a t i

• φ
• I

f a n A V L t r e e h a s s i z e n a n d h e i g h t h , t h e n n ≥ φ

h

• T

a k i n g l

• g

s

• f

b

• t

h s i d e s , h ≤ l

• g

φ

n = l

• g

2

n / l

• g

2

~ 1 . 4 4 l

• g

φ

2

n

S

• :

a n A V L t r e e

• f

n n

• d

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

• s

t 1 . 4 4 l

• g

2

n

H

• w

b a l a n c e d a r e A V L t r e e s ?

SLIDE 36

A V L t r e e s

U s e r

• t

a t i

• n

t

• k

e e p t h e t r e e b a l a n c e d

• W
• r

s t c a s e h e i g h t 1 . 4 4 l

• g

2

n , n

• r

m a l l y c l

• s

e t

• l
• g

2

n – s

• l
• k

u p s a r e q u i c k

I n s e r t i

• n

/ d e l e t i

• n

– B S T i n s e r t i

• n

/ d e l e t i

• n

, t h e n r

• t

a t e t

• r

e p a i r t h e i n v a r i a n t V i s u a l i s a t i

• n

:

• h

t t p : / / v i s u a l g

• .

n e t /

• h

t t p s : / / w w w . c s . u s f c a . e d u / ~ g a l l e s / v i s u a l i z a t i

• n

/ A V L t r e e . h t m l