Data Structures

1
Data Structures

• Week 7 Heap/Huffman Tree
• http//www.cs.hongik.ac.kr/rhanha/rhanha_teaching
  .html

2
Array to Store the Heap

• The same array is
• used both to compute the frequency for each char
  and to store the heap same array

A
C
B
D
E
G
F
H
I
4
3
2
9
8
7
5
6
1
D
G
E
H
A
C
F
B
I
9
7
8
6
4
3
5
2
1
3
Description

• By replacing the root with a leaf node and
  fixing the tree by swaping nodes starting from
  the root downwards (fix heap)
• a new heap obtained
• If the two subtrees of the root are heaps
• we can obtain a new heap with two heaps and a
  node of an arbitrary key value
• a heap from array by fixing the tree as in delete
  
• Thus, we can create
• recursively making subtrees heaps
• making the the first element the root of two
  subtrees(heaps)
• finally fixing the heap

4
Constructing Heap

• Step1
• Insert all elements to be sorted into a heap
  structure arbitrarily
• 4, 2, 3, 9, 8, 5, 7, 6, 1

4
3
2
7
9
8
5
arbitrary
6
1
5
Constructing Heap
construct heap
construct heap
4
3
2
7
9
8
5
6
1
6
Constructing Heap

• Step 2
• Recursively turn subtrees of root into heaps.

9
7
6
8
3
5
2
1
construct heap
construct heap
7
Constructing Heap

• Step 3
• Use Fix Heap to insert label of root

4
not max heap
fix heap
7
9
3
6
8
5
2
1
8
Fix Heap
9
7
4
3
6
8
5
2
1
9
Fix Heap
9
7
8
3
6
4
5
2
1
10
Resulting Heap
9
7
8
3
6
4
5
2
1
11
Building The Huffman Tree

• Build a minimum heap which contains the nodes of
  all symbols with the frequency values as the keys
  in the message
• Repeat until the heap is empty
• a) Delete two nodes from the heap
• concatenate the two symbols
• add their frequencies
• insert the new node into the heap
• b) Insert the new node into the Huffman tree
• the two nodes become the two children of the node
  for the concatenate symbol

12
Example of The Huffman Tree
index
1
3
2
4
5
7
6
8
9
0
symbol
A
C
B
D
E
G
F
H
I
4
3
2
9
8
7
5
6
1
frequency
index
1
3
2
4
5
7
6
8
9
0
symbol
I
C
B
A
E
G
F
H
D
1
3
2
4
8
7
5
6
9
frequency
13
index
1
3
2
4
5
7
6
8
9
0
symbol
I
C
B
A
E
G
F
H
D
1
3
2
4
8
7
5
6
9
frequency
IB3
I1
B2
14
index
3
2
4
5
7
6
8
9
1
0
symbol
C
IB
F
A
D
E
G
H
3
3
5
4
9
8
7
6
frequency
CIB6
C3
IB3
I1
B2
15
index
3
2
4
5
7
6
8
9
1
0
symbol
A
H
F
G
D
E
CIB
4
6
5
7
9
8
6
frequency
CIB6
C3
AF9
IB3
A4
F5
I1
B2
16
index
3
2
4
5
7
6
8
9
1
0
symbol
H
G
CIB
D
AF
E
6
7
6
9
9
8
frequency

HCIB12
H6
CIB6
C3
AF9
IB3
A4
F5
I1
B2
17
index
3
2
4
5
7
6
8
9
1
0
symbol
G
E
AF
D
HCIB
7
8
9
9
12
frequency

HCIB12
H6
CIB6
GE15
C3
AF9
IB3
G7
E8
A4
F5
I1
B2
18
index
3
2
4
5
7
6
8
9
1
0
symbol
D
HCIB
AF
GE
9
12
9
15
frequency

HCIB12
H6
CIB6
DAF18
C3
D9
IB3
AF9
GE15
I1
B2
A4
G7
F5
E8
19
index
3
2
4
5
7
6
8
9
1
0
symbol
HCIB
GE
DAF
12
15
18
frequency

GEHCIB27

HCIB12
GE15
H6
G7
CIB6
E8
DAF18
C3
D9
IB3
AF9
I1
B2
A4
F5
20
index
3
4
5
7
6
8
9
1
2
0
symbol
DAF
GEHCIB
27
18
frequency

DAFGEHCIB45

GEHCIB27
DAF18

D9
HCIB12
AF9
GE15
A4
G7
H6
F5
CIB6
E8
C3
IB3
I1
B2
21

DAFGEHCIB45
0
1

GEHCIB27
DAF18
0
1
0
1

HCIB12
AF9
D9
GE15
0
1
0
0
1
1
A4
G7
H6
F5
CIB6
E8
0
1
C3
IB3
1
0
I1
B2
22
Huffman Code Table