Computer notes - Huffman Encoding

Class No.24

Data Structures

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Huffman Encoding

Huffman code is method for the compression for standard text documents.

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

Huffman code is also part of the JPEG image compression scheme.

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

Huffman Encoding

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

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

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

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

Huffman Encoding

List all the letters used, including the "space" character, along with the frequency with which they 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 amongst those with equal frequencies.

Huffman Encoding

Make a new node out of these two, and make the 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.

Huffman Encoding

Original text: traversing threaded binary trees

size: 33 characters (space and newline)

NL : 1SP : 3a : 3b : 1d : 2e : 5g : 1h : 1

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

Huffman Encoding

2 is equal to sum of the frequencies of the two children nodes.

Huffman Encoding

There a number of ways to combine nodes. We have chosen just one such way.

Huffman Encoding

869 10

Huffman Encoding

List all the letters used, including the "space" character, along with the frequency with which they 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 amongst those with equal frequencies.

Huffman Encoding

Make a new node out of these two, and make the 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.

Huffman Encoding

Start at the root. Assign 0 to left branch and 1 to the 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 read off the 0 and 1 along the path.

Huffman Encoding

10 10 1 0 10

Huffman Encoding

10 10 1 0 10

1 0 10 10

Huffman Encoding

Huffman character codes

NL 10000SP 1111 a 000 b 10001 d 0100 e 101 g 10010 h 10011 i 0101 n 0110 r 110 s 0111 t 001 v 11100 y 11101

• Notice that the code is variable length.

• Letters with higher frequencies have shorter codes.

• The tree could have been built in a number of ways; each would yielded different codes but the code would still be minimal.

Huffman Encoding

Original: traversing threaded binary trees

Encoded:

001110000111001011100111010101101001011110011001111010100001001010100111110000101011000011011101111100111010110101110000

t r a v e

Huffman Encoding

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

Encoded:001110000111001011100111010101101001011110011001111010100001001010100111110000101011000011011101111100111010110101110000

Compressed into 122 bits, 54% reduction.

Mathematical Properties of Binary Trees

Properties of Binary Tree

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

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

internal nodes: 9external nodes: 10

external node

internal node

Property: A binary tree with N internal nodes has 2N links: N-1 links to internal nodes and N+1 links to external nodes.

Threaded Binary TreeProperty: A binary tree with N internal nodes has 2N links: N-1 links to internal nodes and N+1 links to external nodes.

Internal links: 8External links: 10

external link

internal link

Property: A binary tree with N internal nodes has 2N links: N-1 links to internal nodes and N+1 links to external nodes.

• In every rooted tree, each node, except the root, has a unique parent.

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

• Similarly, each of the N+1 external nodes has one link to its parent.

• Thus N-1+N+1=2N links.http://ecomputernotes.com

Computer notes - Huffman Encoding

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