Data Structures

Data Structures

Week 7: Heap/Huffman Tree

http://www.cs.hongik.ac.kr/~rhanha/rhanha_teaching.html

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

4 32 9

A CB D

8 75 6

E GF H

1

I

9 78 6

D GE H

4 35 2

A CF B

1

I

DescriptionBy 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

Constructing HeapStep1Insert all elements to be sorted into a

heap structure arbitrarily[4, 2, 3, 9, 8, 5, 7, 6, 1]

4

2 3

9 8

6

5

1

7

arbitrary

Constructing Heap

4

2 3

9 8

6

5

1

7

construct heap construct heap

Constructing Heap

9

7 6 8

2 5

1 3

construct heap construct heap

Step 2Recursively turn subtrees of root into

heaps.

Constructing Heap

Step 3Use Fix Heap to insert label of root

4

9 7

6 8

2

5

1

3

not max heap fix heap

Fix Heap

9

4 7

6 8

2

5

1

3

Fix Heap

9

8 7

6 4

2

5

1

3

Resulting Heap

9

8 7

6 4

2

5

1

3

Building The Huffman Tree1. Build a minimum heap which contains the

nodes of all symbols with the frequency values as the keys in the message

2. Repeat until the heap is emptya) 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

Example of The Huffman Tree

4 32 9

A CB D

8 75 6

E GF H

1

I

1 32 4 5 76 8 9index

symbol

frequency

1 32 4

I CB A

8 75 6

E GF H

9

D

1 32 4 5 76 8 9index

symbol

frequency

0

0

1 32 4

I CB A

8 75 6

E GF H

9

D

1 32 4 5 76 8 9index

symbol

frequency

0

I1 B2

IB3

3 3 5

C IB F

4 98 7

A DE G

6

H

1 32 4 5 76 8 9index

symbol

frequency

0

I1 B2

IB3C3

CIB6

4 6 5

A H F

7 98 6

G DE CIB

1 32 4 5 76 8 9index

symbol

frequency

0

I1 B2

IB3C3

CIB6

A4 F5

AF9

6 7 6

H G CIB

9 98

D AFE

1 32 4 5 76 8 9index

symbol

frequency

0

I1 B2

IB3C3

CIB6

A4 F5

AF9

H6

`HCIB12

7 8 9

G E AF

9 12

D HCIB

1 32 4 5 76 8 9index

symbol

frequency

0

I1 B2

IB3C3

CIB6

A4 F5

AF9

H6

`HCIB12

G7 E8

GE15

9 12 9

D HCIB AF

15

GE

1 32 4 5 76 8 9index

symbol

frequency

0

I1 B2

IB3C3

CIB6

A4 F5

AF9

H6

`HCIB12

G7 E8

GE15 D9

DAF18

12 15 18

HCIB GE DAF

1 32 4 5 76 8 9index

symbol

frequency

0

I1 B2

IB3C3

CIB6H6

`HCIB12

G7 E8

GE15

A4 F5

AF9D9

DAF18

`GEHCIB27

2718

GEHCIBDAF

1 32 4 5 76 8 9index

symbol

frequency

0

I1 B2

IB3C3

CIB6H6

`HCIB12

G7 E8

GE15

A4 F5

AF9D9

DAF18 `GEHCIB27

`DAFGEHCIB45

I1 B2

IB3C3

CIB6H6

`HCIB12

G7 E8

GE15

A4 F5

AF9D9

DAF18 `GEHCIB27

`DAFGEHCIB45

0

0

0 0

0

0

0

0

1

1 1 1

1

1

1

1

Huffman Code Table

Symbol Frequency Huffman Code

D 9 00

E 8 101

G 7 100

H 6 110

F 5 011

A 4 010

C 3 1110

B 2 11111

I 1 11110