Data Structures and Algorithm - Week 5 - AVL Trees

Data Structures and
Algorithms
Week 5: AVL Trees
Ferdin Joe John Joseph, PhD
Faculty of Information Technology
Thai-Nichi Institute of Technology, Bangkok

Week 5
• AVL Trees
• Implementation in Java
Lecture series for Data Structures and
Algorithms, Data Science and Analytics,
Thai-Nichi Institute of Technology
2

AVL Trees
• Named after Adelson, Velski and EM Landis
3

Balance
Balance == 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 log n
t
5
7
4

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
15
5

An AVL Tree
15
92 12
5
10
20
17
0
0
100
1 2
3 10
3
data
height
children
30
0
6

Not AVL Trees
15
12
5
10
20
17
0
10
0 2
3
30
0
15
10
20
0
1
2 (-1)-1 = -2
0-2 = -2
Note: height(empty tree) == -1
7

Good Insert Case:
Balance Preserved
Good case: insert middle, then small,then tall
Insert(middle)
Insert(small)
Insert(tall)
M
S T
00
1
8

Bad Insert Case #1:
Left-Left or Right-Right Imbalance
Insert(small)
Insert(middle)
Insert(tall)
T
M
S
0
1
2
BC#1 Imbalance caused by either:
 Insert into left child’s left subtree
 Insert into right child’s right subtree
9

Single Rotation
T
M
S
0
1
2
M
S T
00
1
Basic operation used in AVL trees:
A right child could legally have its parent as its left child.
10

General Bad Case #1
a
X
Y
b
Z
h h - 1
h + 1 h - 1
h + 2
a
X
Y
b
Z
h-1 h - 1
h h - 1
h + 1
Note: imbalance is left-left
11

Single Rotation
Fixes Case #1 Imbalance
• Height of left subtree same as it was before insert!
• Height of all ancestors unchanged
• We can stop here!
a
X
Y
b
Z
a
XY
b
Z
h h - 1
h + 1 h - 1
h + 2
h
h - 1
h
h - 1
h + 1
12

Bad Insert Case #2:
Left-Right or Right-Left Imbalance
Insert(small)
Insert(tall)
Insert(middle)
M
T
S
0
1
2
Will a single rotation fix this?
BC#2 Imbalance caused by either:
 Insert into left child’s right subtree
 Insert into right child’s left subtree
13

Double Rotation
M
S T
00
1
M
T
S
0
1
2
T
M
S
0
1
2
14

General Bad Case #2
a
X
b
Z
h
h + 1 h - 1
h + 2
a
X
b
Z
h-1 h - 1
h h - 1
h + 1
Note: imbalance is left-right
h-1
Y
Y
15

Double Rotation
Fixes Case #2 Imbalance
Initially: insert into either X or Y unbalances tree (root balance goes to 2 or -2)
“Zig zag” to pull up c – restores root height to h+1, left subtree height to h
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?
16

AVL Insert Algorithm
• Find spot for value
• Hang new node
• Search back up looking for imbalance
• If there is an imbalance:
case #1: Perform single rotation
case #2: Perform double rotation
• Done!
(There can only be one imbalance!)
17

Easy Insert
2092
155
10
3017
Insert(3)
12
0
0
100
1 2
3
0
18

Hard Insert (Bad Case #1)
2092
155
10
3017
Insert(33)
3
12
1
0
100
2 2
3
00
19

Single Rotation
2092
155
10
30173
12
33
1
0
200
2 3
3
10
0
3092
205
10
333
15
1
0
110
2 2
3
00
1712
0
20

Hard Insert (Bad Case #2)
Insert(18)
2092
155
10
30173
12
1
0
100
2 2
3
00
21

Single Rotation (oops!)
2092
155
10
30173
12
1
1
200
2 3
3
00
3092
205
10
3
15
1
1
020
2 3
3
0
1712
0
18
0
18
0
22

Double Rotation (Step #1)
2092
155
10
30173
12
1
1
200
2 3
3
00
18
0
1792
155
10
203
12
1 200
2 3
3
10
30
0
Look familiar? 18
0
23

Double Rotation (Step #2)
1792
155
10
203
12
1 200
2 3
3
10
30
0
18
0
2092
175
10
303
15
1
0
110
2 2
3
00
12
0
18
24

AVL Insert Algorithm Revisited
• Recursive
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
• Iterative
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
25

Single Rotation Code
void 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
26

Double Rotation Code
void DoubleRotateRight(Node *& root) {
RotateLeft(root->right);
RotateRight(root);
}
a
Z
b
W
c
XY
a
Z
c
b
X
Y
W
First Rotation
27

Double Rotation Completed
a
Z
c
b
X
Y
W
a
Z
c
b
XY
W
First Rotation Second Rotation
28

Deletion: Really Easy Case
2092
155
10
30173
12
1
0
100
2 2
3
00
Delete(17)
29

Deletion: Pretty Easy Case
2092
155
10
30173
12
1
0
100
2 2
3
00
Delete(15)
30

Deletion: Pretty Easy Case (cont.)
2092
175
10
303
12
1 100
2 2
3
00
Delete(15)
31

Deletion (Hard Case #1)
2092
175
10
303
12
1 100
2 2
3
00
Delete(12)
32

Single Rotation on Deletion
2092
175
10
303
1 10
2 2
3
00
3092
205
10
17
3
1 00
2 1
3
0
0
Deletion can differ from insertion – How?
33

Deletion (Hard Case)
Delete(9)
2092
175
10
303
12
1 220
2 3
4
0
33
15
13
0 0
1
0
20
30
12
33
15
13
1
0 0
11
0
18
0
34

Double Rotation on Deletion
2
3
0
202
175
10
30
12
1 22
2 3
4
33
15
13
1
0 0
1
11
0
18
0
2052
173
10
30
12
0 220
1 3
4
33
15
13
1
0 0
1
11
0
18
00
Not
finished!
35

Deletion with Propagation
We get to choose whether
to single or double rotate!
2052
173
10
30
12
0 220
1 3
4
33
15
13
1
0 0
1
11
0
18
0
What different about this case?
36

Propagated Single Rotation
0
30
20
17
33
12
15
13
1
0
52
3
10
4
3 2
1 2 1
0 0 0
11
0
2052
173
10
30
12
0 220
1 3
4
33
15
13
1
0
1
11
0
18
0
18
0
37

Propagated Double Rotation
0
17
12
11
52
3
10
4
2 3
1 0
0 0
2052
173
10
30
12
0 220
1 3
4
33
15
13
1
0
1
11
0
18
0
15
1
0
20
30
33
1
18
0
13
0
2
38

AVL Deletion Algorithm
• Recursive
node
2. Delete node
3. Unwind, correcting
heights as we go
a. If imbalance #1,
single rotate
b. If imbalance #2
(or don’t care),
double rotate
• Iterative
node, stacking
parent nodes
2. Delete node
3. Unwind stack,
correcting heights
a. If imbalance #1,
single rotate
b. If imbalance #2
(or don’t care)
double rotate
39

Assignment
• Write Java code to perform AVL tree based on the
Fibonacci series of length and elements as user
input.
40

Next Week
Red Black Trees
Graph Theory
41

Data Structures and Algorithm - Week 5 - AVL Trees

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