Computer Notes - Data Structures - 24

8/3/2019 Computer Notes - Data Structures - 24

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Class No.24

Data Structures

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Huffman Encoding

Huffman code is method for the compression forstandard text documents.

It makes use of a binary tree to develop codes ofvarying lengths for the letters used in the original

message.

Huffman code is also part of the JPEG imagecompression scheme.

The algorithm was introduced by David Huffmanin 1952 as part of a course assignment at MIT.

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Huffman Encoding

To understand Huffman encoding, it isbest to use a simple example.

Encoding the 32-character phrase:"traversing threaded binary trees",

If we send the phrase as a message in anetwork using standard 8-bit ASCII codes,we would have to send 8*32= 256 bits.

Using the Huffman algorithm, we can sendthe message with only 116 bits.

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Huffman Encoding

List all the letters used, including the "space"character, along with the frequency with whichthey occur in the message.

Consider each of these (character,frequency)

pairs to be nodes; they are actually leaf nodes,as we will see.

Pick the two nodes with the lowest frequency,

and if there is a tie, pick randomly amongstthose with equal frequencies.



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Huffman Encoding

Make a new node out of these two, and makethe two nodes its children.

This new node is assigned the sum of thefrequencies of its children.

Continue the process of combining the twonodes of lowest frequency until only one node,the root, remains.



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Huffman Encoding

Original text:traversing threaded binary treessize: 33 characters (space and newline)

NL : 1SP : 3a : 3b : 1

d : 2e : 5g : 1h : 1

i : 2n : 2r : 5s : 2t : 3v : 1y : 1



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Huffman Encoding

v

1

y

1

SP

3

r

5

h

1

e

5

g

1

b

1

NL

1

s

2

n

2

i

2

d

2

t3

a

3

2

2 is equal to sumof the frequencies ofthe two children nodes.

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Huffman Encoding

v

1

y

1

SP

3

r

5

h

1

e

5

g

1

b

1

NL

1

s

2

n

2

i

2

d

2

t3

a

3

2 2

There a number of ways to combinenodes. We have chosen just one suchway.



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Huffman Encoding

v

1

y

1

SP

3

r

5

h

1

e

5

g

1

b

1

NL

1

s

2

n

2

i

2

d

2

t3

a

3

2 2 2



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Huffman Encoding

v

1

y

1

SP

3

r

5

h

1

e

5

g

1

b

1

NL

1

s

2

n

2

i

2

d

2

t3

a

3

2 2 2

4 4



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Huffman Encoding

v

1

y

1

SP

3

r

5

h

1

e

5

g

1

b

1

NL

1

s

2

n

2

i

2

d

2

t3

a

3

2 2 2

5444

6



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Huffman Encoding

v

1

y

1

SP

3

r

5

h

1

e

5

g

1

b

1

NL

1

s

2

n

2

i

2

d

2

t3

a

3

2 2 2

5444

86 9 10



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Huffman Encoding

v

1

y

1

SP

3

r

5

h

1

e

5

g

1

b

1

NL

1

s

2

n

2

i

2

d

2

t3

a

3

2 2 2

5444

86

14

9

19

10



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Huffman Encoding

v

1

y

1

SP

3

r

5

h

1

e

5

g

1

b

1

NL

1

s

2

n

2

i

2

d

2

t3

a

3

2 2 2

5444

86

14

9

19

10

33



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Huffman Encoding

List all the letters used, including the "space"character, along with the frequency with whichthey occur in the message.

Consider each of these (character,frequency)pairs to be nodes; they are actually leaf nodes,as we will see.

Pick the two nodes with the lowest frequency,and if there is a tie, pick randomly amongstthose with equal frequencies.



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Huffman Encoding

Make a new node out of these two, and makethe two nodes its children.

This new node is assigned the sum of the

frequencies of its children. Continue the process of combining the two

nodes of lowest frequency until only one node,the root, remains.



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Huffman Encoding

Start at the root. Assign 0 to left branch and 1 tothe right branch.

Repeat the process down the left and right

subtrees. To get the code for a character, traverse the tree

from the root to the character leaf node and readoff the 0 and 1 along the path.



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Huffman Encoding

v

1

y

1

SP

3

r

5

h

1

e

5

g

1

b

1

NL

1

s

2

n

2

i

2

d

2

t3

a

3

2 2 2

5444

86

14

9

19

10

3310



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Huffman Encoding

v

1

y

1

SP

3

r

5

h

1

e

5

g

1

b

1

NL

1

s

2

n

2

i

2

d

2

t3

a

3

2 2 2

5444

86

14

9

19

10

3310

1010



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Huffman Encoding

v

1

y

1

SP

3

r

5

h

1

e

5

g

1

b

1

NL

1

s

2

n

2

i

2

d

2

t3

a

3

2 2 2

5444

86

14

9

19

10

3310

1010

1 010

1010

10 10 1 0 10



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Huffman Encoding

v

1

y

1

SP

3

r

5

h

1

e

5

g

1

b

1

NL

1

s

2

n

2

i

2

d

2

t3

a

3

2 2 2

5444

86

14

9

19

10

3310

1010

1 010

1010

10 10 1 0 10

1 0 10 10



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Huffman Encoding

Huffman character codes

NL 10000SP 1111a 000

b 10001d 0100e 101g 10010h 10011i 0101

n 0110r 110s 0111t 001v 11100y 11101

Notice that the code isvariable length.

Letters with higherfrequencies have shortercodes.

The tree could have beenbuilt in a number of ways;

each would yieldeddifferent codes but thecode would still beminimal.

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Huffman Encoding

Original: traversing threaded binary trees

Encoded:

0011100001110010111001110101011010010111100110011110101000010010101001

11110000101011000011011101111100111010110101110000

t r a v e



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Huffman Encoding

Original: traversing threaded binary treesWith 8 bits per character, length is 264.

Encoded:001110000111001011100111010101101001011110011001111010100001001010100111110000101011000011011101111100111

010110101110000

Compressed into 122 bits, 54% reduction.



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Mathematical Properties of

Binary Trees



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Properties of Binary Tree

Property: A binary tree with N internal nodeshas N+1 external nodes.



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A binary tree with N internal nodes has N+1 external nodes.

D F

B C

G

A

E

FE

internal nodes: 9external nodes: 10

external node

internal node



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Property: A binary tree with N internal nodes has2N links: N-1 links to internal nodes and N+1 links

to external nodes.



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Threaded Binary Tree

Property: A binary tree with N internal nodes has 2N links: N-1 links to

internal nodes and N+1 links to external nodes.

D F

B C

G

A

E

FE

Internal links: 8External links: 10

external link

internal link



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Property: A binary tree with N internal nodes has2N links: N-1 links to internal nodes and N+1 links

to external nodes.

In every rooted tree, each node, except theroot, has a unique parent.

Every link connects a node to its parent, sothere are N-1 links connecting internal nodes.

Similarly, each of the N+1 external nodes hasone link to its parent.

Thus N-1+N+1=2Nlinks.

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Computer Notes - Data Structures - 24

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