AVL Trees - Horowitz Sahni CPP - Lec43

7/23/2019 AVL Trees - Horowitz Sahni CPP - Lec43

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Dynamic Dictionaries

Primary Operations: Get(key) =>search

Insert(key, element) =>insert

Delete(key) =>delete

Additional operations: Ascend()

Get(index)

Delete(index)


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omplexity O! Dictionary Operations

Get(), Insert() and Delete()

Data "tr#ct#re $orst ase %xpected

&ash 'ale O(n) O()

*inary "earch

'ree

O(n) O(lo+ n)

*alanced

*inary "earch'ree

O(lo+ n) O(lo+ n)

nis n#mer o! elements in dictionary


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omplexity O! Other Operations

Ascend(), Get(index), Delete(index)

Data "tr#ct#re Ascend Get and Delete

&ash 'ale O(D n lo+ n) O(D n lo+ n)

Indexed *"' O(n) O(n)

Indexed

*alanced *"'

O(n) O(lo+ n)

Dis n#mer o! #ckets


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A-. 'ree

inary tree

!or e/ery node x, de!ine its alance !actor

alance !actor o! x = hei+ht o! le!t s#tree o! x

0 hei+ht o! ri+ht s#tree o! x

alance !actor o! e/ery nodexis0 ,1, or


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*alance 2actors

1 1

11

1

3 1

1

3

3

'his is an A-. tree4


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&ei+ht O! An A-. 'ree

'he hei+ht o! an A-. tree that has nnodes is atmost 455 lo+6(n6)4

'he hei+ht o! e/ery nnode inary tree is at least

lo+6(n)4

lo+6(n) 7= hei+ht 7=455 lo+6(n6)


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Proo! O! 8pper *o#nd On &ei+ht

.et9h=min o! nodes in an A-. tree

;hose hei+ht is h4

91= 14

9)= 4


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9h, h >

*oth L andR are A-. trees4 'he hei+ht o! one is h34

'he hei+ht o! the other is h364

'he s#tree ;hose hei+ht is h3 has9h3nodes4

'he s#tree ;hose hei+ht is h36 has9h36nodes4

"o,9h=9h3 9h36 4

L R


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2ionacci 9#mers

21= 1, 2= 4

2i=2i3 2i36, i > 4

91= 1, 9= 4 9h=9h3 9h36 , i > 4

9h=2h60 4

2i< isrt(?)4

= (1 + srt(5))/2.


10/31

A-. "earch 'ree

1 1

1 1

1

3 1

1

3

3

1

@

B

?

B1

51

61

6?

B?

5?

C1


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Insert()

1 1

1 1

1

3 1

1

3

3

1

3

1

1

@

B

?

B1

51

61

6?

B?

5?

C1


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Insert(6)

1 1

1 1

1

1

1

3

3

1

@

B

?

B1

51

61

6?

B?

5?

C1

6

1

3

336

EE imalance=>ne; node is inri+ht s#tree o! ri+ht s#tree o!

;hite node (node ;ith ! =06)


13/31

1 1

11

1

1

1

3

3

1

@

B

?

B1

51

6?B?

5?

C11

EE rotation4

611

6

Insert(6)


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Insert

2ollo;in+ insert, retrace path to;ards root andadF#st alance !actors as needed4

"top ;hen yo# reach a node ;hose alance

!actor ecomes 1, 6, or06, or ;hen yo# reachthe root4

'he ne; tree is not an A-. tree only i! yo#reach a node ;hose alance !actor is either 6or064

In this case, ;e say the tree has ecome#nalanced4


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A39ode

.et Ae the nearest ancestor o! the ne;ly

inserted node ;hose alance !actor

ecomes 6or06!ollo;in+ the insert4 *alance !actor o! nodes et;een ne; node

and Ais 1e!ore insertion4


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Imalance 'ypes

EE ne;ly inserted node is in the ri+ht

s#tree o! the ri+ht s#tree o! A4

.. le!t s#tree o! le!t s#tree o! A4

E.le!t s#tree o! ri+ht s#tree o! A4

.Eri+ht s#tree o! le!t s#tree o! A4


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.. Eotation

"#tree hei+ht is #nchan+ed4

9o !#rther adF#stments to e done4

*e!ore insertion4

1

A

B

BL BR

AR

h h

h

A

B

BL BR

AR

A!ter insertion4

h h

h

B

A

A!ter rotation4

BRh

ARh

BL

h

1

1

6


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.E Eotation (case )



*e!ore insertion4

1

A

B

A

B

A!ter insertion4

C

C

A

A!ter rotation4

B

1

3

6

1 1

1


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.E Eotation (case 6)



C

A

CRh3

ARh

1

A

B

BL

CR

AR

h

h3

h

1

CL

h3

C

A

B

BL

CR

AR

h

h3

h

CLh

C

B

BLh

CLh

3

6

1 3

1


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.E Eotation (case B)



1

A

B

BL

CR

AR

h

h3

h

1

CL

h3

C

A

B

BL

CR

AR

h

h

h

CLh3

C3

3

6 C

A

CRh

ARh

B

BLh

CLh3

1

1


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"in+le H Do#le Eotations

"in+le

.. and EE

Do#le

.E and E.

.E is EE !ollo;ed y ..

E. is .. !ollo;ed y EE


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.E Is EE ..

A

B

BL

CR

AR

h

h

h

CLh3

C3

3

6

A!ter insertion4

A

C

CL

CR

AR

h

h

BLh

B

6

A!ter EE rotation4

h3

C

A

CRh

ARh

B

BLh

CLh3

1

1

A!ter .. rotation4


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Delete An %lement

1 1

11

1

3 1

1

3

3

1

@

B

?

B1

51

61

6?

B?

5?

C1

Delete 4


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Delete An %lement

1 1

1

6

1

3 1

1

3

3

1

@

B

?

B1

51

61

6?

B?

5?

C1

.et e parent o! deleted node4

Eetrace path !rom to;ards root4


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9e; *alance 2actor O!

Deletion !rom le!t s#tree o! =>!334 Deletion !rom ri+ht s#tree o! =>!4

9e; alance !actor = or0 =>no chan+e in hei+ht o!

s#tree rooted at 4

9e; alance !actor = 1=>hei+ht o! s#tree rooted at

has decreased y4

9e; alance !actor = 6or06 =>tree is #nalanced at 4

q


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Imalance lassi!ication

.et Ae the nearest ancestor o! the deleted

node ;hose alance !actor has ecome 6or06!ollo;in+ a deletion4

Deletion !rom le!t s#tree o! A =>type .4

Deletion !rom ri+ht s#tree o! A =>type E4 'ype E =>ne;!(A) = 64

"o, old!(A) = 4

"o, Ahas a le!t child *4 !(*) = 1 =>E14

!(*) = =>E4

!(*) = 0 =>E34


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E1 Eotation



"imilar to ..rotation4

*e!ore deletion4

1

A

B

BL BR

AR

h h

h

B

A

A!ter rotation4

BRh

ARh3

BL

h

3A

B

BL BR

AR

A!ter deletion4

h h

h3

1

6


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E Eotation

"#tree hei+ht is red#ced y 4

#st contin#e on path to root4

"imilar to ..and E1rotations4

*e!ore deletion4

A

B

BL BR

AR

h h3

h

B

A

A!ter rotation4

BRh3

ARh3

BL

h

1

1A

B

BL BR

AR

A!ter deletion4

h h3

h3

6


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E3 Eotation

9e; alance !actor o! Aand *depends on4

"#tree hei+ht is red#ced y 4

#st contin#e on path to root4

"imilar to .E4

3

A

B

BL

CR

AR

h3

h

CL

C

A

B

BL

CR

AR

h3

h3

CL

C

3

6 C

A

CR ARh3

B

BLh3

CL

1


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9#mer O! Eealancin+ Eotations

At most !or an insert4

O(lo+ n)!or a delete4


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Eotation 2re#ency

Insert random n#mers4

9o rotation ?B45J (approx)4

..EE 6B4BJ (approx)4

.EE. 6B46J (approx)4

AVL Trees - Horowitz Sahni CPP - Lec43

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