CSE 326: Data StructuresLecture #10

Balancing Act andWhat AVL Stands For

Steve Wolfman

Winter Quarter 2000

Today’s Outline

• Double-tailed distributions and the mean• AVL trees • What AVL Stands For• Bonus! (Deletion)

Beauty is Only (log n) Deep

• Binary Search Trees are fast if they’re shallow:– perfectly complete

– perfectly complete except the one level fringe (like a heap)

– anything else?

What matters here?Problems occur when onebranch is much longer than the other!

Balance

• Balance– height(left subtree) - height(right subtree)

– zero everywhere perfectly balanced

– small everywhere balanced enough

t

57

Balance between -1 and 1 everywhere maximum height of 1.44 log n

AVL Tree Dictionary Data Structure

4

121062

115

8

14137 9

• Binary search tree properties– binary tree property

– search tree property

• Balance property– balance of every node is:

-1 b 1– result:

• depth is (log n)

15

Testing the Balance Property

2092

155

10

30177

NULLs have height -1

An AVL Tree

20

92 15

5

10

30

177

0

0 0

011

2 2

3 10

3

data

height

children

But, How Do We Stay Balanced?

• I need:– the smallest person in the class

– the tallest person in the class

– the averagest (?) person in the class

Beautiful Balance

Insert(middle)

Insert(small)

Insert(tall)

00

1

Bad Case #1

Insert(small)

Insert(middle)

Insert(tall)

0

1

2

Single Rotation

0

1

2

00

1

General Single Rotation

• Height of subtree same as it was before insert!• Height of all ancestors unchanged.

So?

a

X

Y

b

Z

a

XY

b

Zh h - 1

h + 1 h - 1

h + 2

h

h - 1

h

h - 1

h + 1

Bad Case #2

Insert(small)

Insert(tall)

Insert(middle)

0

1

2

Double Rotation

00

1

0

1

2

0

1

2

General Double Rotation

• Height of subtree still the same as it was before insert!• Height of all ancestors unchanged.

a

Z

b

W

c

X Y

a

Z

b

W

c

X Y

h

h - 1?

h - 1

h - 1

h + 2

h + 1

h - 1h - 1

h

h + 1

h

h - 1?

Insert Algorithm

• Find spot for value• Hang new node• Search back up for imbalance• If there is an imbalance:

case #1: Perform single rotation and exit

case #2: Perform double rotation and exit

Easy Insert

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10

3017

Insert(3)

120

0

100

1 2

3

0

Hard Insert (Bad Case #1)

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10

3017

Insert(33)

3

121

0

100

2 2

3

00

Single Rotation

2092

155

10

30173

12

33

1

0

200

2 3

3

10

0

3092

205

10

333

151

0

110

2 2

3

001712

0

Hard Insert (Bad Case #2)

Insert(18)

2092

155

10

30173

121

0

100

2 2

3

00

Single Rotation (oops!)

2092

155

10

30173

121

1

200

2 3

3

00

3092

205

10

3

151

1

020

2 3

3

01712

0

180

180

Double Rotation (Step #1)

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10

30173

121

1

200

2 3

3

00

180

1792

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10

203

121 200

2 3

3

10

300

Look familiar?18

0

Double Rotation (Step #2)

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10

203

121 200

2 3

3

10

300

180

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151

0

110

2 2

3

0012

018

AVL Algorithm Revisited• Recursive1. Search downward for

spot

2. Insert node

3. Unwind stack,

correcting heights

a. If imbalance #1,

single rotate

b. If imbalance #2,

double rotate

• Iterative1. Search downward for

spot, stacking

parent nodes

2. Insert node

3. Unwind stack,

correcting heights

a. If imbalance #1,

single rotate and

exit

b. If imbalance #2,

double rotate and

exit

Single Rotation Codevoid RotateRight(Node *& root) {

Node * temp = root->right;

root->right = temp->left;

temp->left = root;

root->height = max(root->right->height,

root->left->height) + 1;

temp->height = max(temp->right->height,

temp->left->height) + 1;

root = temp;

}

X

Y

Z

root

temp

Double Rotation Codevoid DoubleRotateRight(Node *& root) {

RotateLeft(root->right);

RotateRight(root);

}

a

Z

b

W

c

XY

a

Z

c

b

X

Y

W

First Rotation

Double Rotation Completed

a

Z

c

b

X

Y

W

a

Z

c

b

XY

W

First Rotation Second Rotation

AVL

• Automatically Virtually Leveled• Architecture for inVisible Leveling (the “in” is inVisible)

• All Very Low• Articulating Various Lines• Amortizing? Very Lousy!• Absolut Vodka Logarithms• Amazingly Vexing Letters

Adelson-Velskii Landis

Bonus: Deletion (Easy Case)

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30173

121

0

100

2 2

3

00

Delete(15)

Deletion (Hard Case #1)

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303

121 100

2 2

3

00

Delete(12)

Single Rotation on Deletion

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303

1 10

2 2

3

00

3092

205

10

17

3

1 00

2 1

3

0

0

Something very bad happened!

To Do

• Finish Project II• Read chapter 4 in the book• Understand Zasha’s worksheet• Prepare for the midterm

Coming Up

• A bit more AVL trees• Splay trees• Another worksheet and midterm study guide• Second project due (January 31st)• Midterm (February 4th)