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CPSC 221: Algorithms and
Data Structures
Lecture #6
Balancing Act
Steve Wolfman
2011W2
Today’s Outline
• Addressing one of our problems
• Single and Double Rotations
• AVL Tree Implementation
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 one
branch is much longer
than the other!
Balance
• Balance
t
5
– height(left subtree) - height(right subtree)
– zero everywhere  perfectly balanced
– small everywhere  balanced enough
Balance between -1 and 1 everywhere 
maximum height of 1.44 lg n
7
AVL Tree
Dictionary Data Structure
• Binary search tree
properties
8
– binary tree property
– search tree property
5
11
• Balance property
– balance of every node is:
-1 b  1
– result:
• depth is (log n)
2
6
4
10
7
9
12
13 14
15
Testing the Balance Property
10
5
15
2
9
7
NULLs have
height -1
20
17 30
An AVL Tree
3
10
2
20
1
0
2
9
0
1
0
15
30
0
7
17
data
3
height
children
2
5
10
Today’s Outline
• Addressing one of our problems
• Single and Double Rotations
• AVL Tree Implementation
But, How Do We Stay Balanced?
• Need: three volunteers proud of their very diverse
height.
Beautiful Balance
(SIMPLEST version)
Insert(middle)
Insert(small)
Insert(tall)
1
0
0
Bad Case #1
(SIMPLEST version)
Insert(small)
Insert(middle)
Insert(tall)
2
1
0
Single Rotation
(SIMPLEST version)
2
1
1
0
0
0
General Single Rotation
h+1
h+2
a
b
h
h-1
h+1 b
h
a
X
h
h-1
Y
Z
h-1
h-1
Z
Y
 An insert made this BAD!
• After rotation, subtree’s height same as before insert!
• Height of all ancestors unchanged.
So?
X
Bad Case #2
(SIMPLEST version)
Insert(small)
Insert(tall)
Insert(middle)
2
1
0
Double Rotation
(SIMPLEST version)
2
2
1
1
0
1
0
0
0
General Double Rotation
a
h+2
h+1
h+1
b
h
c
h - 1?
h
b
h-1
W
c
h-1
h
Z
a
h-1
h-1
X
X
Y
Y
W
Z
h - 1?
• Height of subtree still the same as it was before insert!
• Height of all ancestors unchanged.
Today’s Outline
• Addressing one of our problems
• Single and Double Rotations
• AVL Tree Implementation
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
Mirrored cases also possible
Easy Insert
3
10
Insert(3)
1
2
5
15
0
0
2
9
0
1
12
20
0
0
17 30
Hard Insert (Bad Case #1)
3
10
Insert(33)
2
2
5
15
1
0
2
9
0
0
1
12
20
0
3
0
17 30
Single Rotation
3
3
10
10
2
3
5
15
1
0
2
9
0
0
12
2
5
20
1
2
20
0
3
2
0
2
17 30
0
3
0
33
1
15
9
0
1
1
30
0
12
17
0
33
Hard Insert (Bad Case #2)
3
10
Insert(18)
2
2
5
15
1
0
2
9
0
0
1
12
20
0
3
0
17 30
Single Rotation (oops!)
3
3
10
10
2
3
5
15
1
0
2
9
0
2
5
0
20
3
17 30
0
18
0
2
0
1
20
1
2
12
3
2
15
9
0
0
30
0
3
1
12
17
0
18
Double Rotation (Step #1)
3
3
10
10
2
3
5
15
1
0
2
9
0
2
5
0
20
3
0
2
0
1
15
1
2
12
3
17 30
9
0
2
12
17
0
1
3
20
0
0
18
18
Look familiar?
0
30
Double Rotation (Step #2)
3
3
10
10
2
3
5
15
1
0
2
2
9
2
5
0
1
2
12
17
0
18
30
1
15
0
3
0
1
9
0
20
0
0
2
1
3
17
20
0
12
0
18 30
AVL Algorithm Revisited
• Recursive
• Iterative
1. Search downward for
spot
2. Insert node
3. Unwind stack,
correcting heights
a. If imbalance #1,
single rotate
b. If imbalance #2,
double rotate
1. 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 Code
void RotateLeft(Node *& root) {
Node * temp = root->right;
root->right = temp->left;
X
temp->left = root;
root->height = max(height(root->right),
height(root->left)) + 1;
temp->height = max(height(temp->right),
height(temp->left) + 1;
root = temp;
}
root
temp
Y
Z
(The “height” function returns -1 for a NULL subtree
or the “height” field of a non-NULL subtree.)
Double Rotation Code
void DoubleRotateLeft(Node *& root) {
RotateRight(root->right);
RotateLeft(root);
}
First
a
a
b
Z
c
Z
c
Y
Rotation
X
W
b
Y
X
W
Double Rotation Completed
Second Rotation
First Rotation
c
a
c
b
a
Z
b
Y
Y
Z
X
W
X
W
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
To Do
• Read KW 11.1-11.2, 11.5
Coming Up
• Another approach to balancing, which leads to…
• Huge Search Tree Data Structure