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Data Compressor---Huffman

Encoding and Decoding

Huffman Encoding

• Compression
  • Typically, in files and messages,
    • Each character requires 1 byte or 8 bits
    • Already wasting 1 bit for most purposes!
• Question
  • What’s the smallest number of bits that can be used to store an arbitrary piece of text?
• Idea
  • Find the frequency of occurrence of each character
  • Encode Frequent characters short bit strings
  • Rarer characters longer bit strings

Huffman's Algorithm

• Repeatedly merges trees - maintains a forest

• Tree weight - the sum of its leaves frequencies

• For C characters to code, start with C single

node trees

• Select two trees, T 1 and T 2 , of smallest weights

and merge them

• C - 1 merge operations

Huffman Encoding

• Encoding
  • Use a tree
    • Inefficient in practice
  • Use a direct-addressed lookup table

? Finding the optimal encoding

• Smallest number of bits to represent arbitrary text

A 010

E 00

B : : N : S T

• A divide-and-conquer approach might have us

asking which characters should appear in the

left and right subtrees and trying to build the

tree from the top down.

• A greedy approach places our n characters in

n sub-trees and starts by combining the two

least weight nodes into a tree which is

assigned the sum of the two leaf node weights

as the weight for its root node.

Standard Coding Scheme

Binary Tree Representation

• For the character set of C characters, the

standard fixed-length coding needs ┌log C┐^ bits

• Fixed-length code can be represented by a

binary tree where characters are stored only

in leaf nodes - binary trie

• Each character path - start at the root, follow

the branches, record 0 for the left branch and

1 for the right branch

• Optimal code is always a full tree - all nodes

are either leaves or have two children

Improved Binary Trie

Prefix Code

• The fixed-length character code that has

characters places only at the leaves

guarantees that any bit sequence can be

decoded unambiguously

• Prefix code - characters may have varying

lengths as long as no character code is a prefix

of another code

• That means that characters can be only in

leafs

Optimal Prefix Code Tree

Optimal Prefix Code Cost