Huffman coding Guide, Meaning , Facts, Information and Description
In computer science, Huffman coding is an entropy encoding algorithm used for data compression that finds the optimal system of encoding strings based on the relative frequency of each character. It was developed by David A. Huffman as a PhD student at MIT in 1952, and published in A Method for the Construction of Minimum-Redundancy Codes.Huffman coding uses a specific method for choosing the representations for each symbol, resulting in a prefix-free code (that is, no bit string of any symbol is a prefix of the bit string of any other symbol) that expresses the most common characters in the shortest way possible. It has been proven that Huffman coding is the most effective compression method of this type: no other mapping of source symbols to strings of bits will produce a smaller output when the actual symbol frequencies agree with those used to create the code. (Huffman coding is such a widespread method for creating prefix-free codes that the term "Huffman code" is widely used as a synonym for "prefix-free code" even when such a code was not produced by Huffman's algorithm.)
For a set of symbols with a uniform probability distribution and a number of members which is a power of two, Huffman coding is equivalent to simple binary block encoding.
In 1951, David Huffman and his MIT information theory classmates were given the choice of a term paper or a final exam. The professor, Robert M. Fano, assigned a term paper on the problem of finding the most efficient binary code. Huffman, unable to prove any codes were the most efficient, was about to give up and start studying for the final when he hit upon the idea of using a frequency-sorted binary tree, and quickly proved this method the most efficient.
In doing so, the student outdid his professor, who had worked with information theory inventor Claude Shannon to develop a similar code. Huffman avoided the major flaw of Shannon-Fano coding by building the tree from the bottom up instead of from the top down.
Input.
Output.
The technique works by creating a binary tree of nodes. These can be stored in a regular array, the size of which depends on the number of symbols(N). A node can be either a leaf node or an internal node. Initially, all nodes are leaf nodes, which contain the symbol itself, the weight (frequency of appearance) of the symbol and optionally, a link to a parent node which makes it easy to read the code (in reverse) starting from a leaf node. Internal nodes contain symbol weight, links to two child nodes and the optional link to a parent node. As a common convention, bit '0' represents following the left child and bit '1' represents following the right child. A finished tree has N leaf nodes and N−1 internal nodes.
A fast way to create a Huffman tree is to use the heap data structure, which keeps the nodes in partially sorted order according to a predetermined criterion. In this case, the node with the lowest weight is always kept at the root of the heap for easy access.
Creating the tree:
Huffman coding is optimal when the probability of each input symbol is a negative power of two. Prefix-free codes tend to have slight inefficiency on small alphabets, where probabilities often fall between these optimal points. Expanding the alphabet size by coalescing multiple symbols into "words" before Huffman coding can help a bit. The worst case for Huffman coding can happen when the probability of a symbol exceeds 2-1 making the upper limit of inefficiency unbounded. To prevent this, run-length encoding can be used to preprocess the symbols.
Extreme cases of Huffman codes are connected with Fibonacci and Lucas numbers and Wythoff array.
Arithmetic coding produces slight gains over Huffman coding, but in practice these gains have not been large enough to offset arithmetic coding's higher computational complexity and patent royalties (As of November 2001, IBM owns patents on the core concepts of arithmetic coding in several jurisdictions.)
Huffman coding today is often used as a "back-end" to some other compression method.
DEFLATE (PKZIP's algorithm) and multimedia codecs such as JPEG and MP3 have a front-end model and quantization followed by Huffman coding.
This is an Article on Huffman coding. Page Contains Information, Facts Details or Explanation Guide About Huffman coding History
Problem definition
Informal description
Given. A set of symbols and their costs..
Find. A prefix binary character code (a sets of codewords) with minimum weighted path length..
Note-1. A code wherein each character is represented by a unique binary string (codeword) is called a binary character code..
Note-2. A prefix code is a code having the property that no codeword is a prefix of any other codeword.Formalized description
Input.
Alphabet A = {a[1], a[2], ..., a[n]} which is the symbol alphabet of size n.
Set C = {c[1], c[2], ..., c[n]} which is the set of the symbol costs, i.e., c[i] = cost (a[i]), 1 <= i <= n.
Output.
Code H(A,C) = {h[1], h[2], ..., h[n]} which is the set of (binary) codewords, where h[i] is codeword of a[i], 1 <= i <= n.
Goal.
Let S(H) = sum (c[i] * length (h[i])) (1 <= i <= n) be weighted path length of code H. Must be: S(H) <= S(T) for any code T(A,C).Samples
Sample-1
Input.
Symbol alphabet A = {a, b, c, d, e, f, g, h, i },
Set of costs C = {10, 15, 5, 15, 20, 5, 15, 30, 5 }.
Output.
Code H(A,C) = {001, 010, 00000, 011, 101, 00001, 100, 11, 0001}.
Sum S(H) = 10*3 + 15*3 + 5*5 + 15*3 + 20*3 + 5*4 + 15*3 + 30*2 + 5*4 = 355.Sample-2
Symbol alphabet A = {a, b, c, d, e, f, g, h, i),
Set of costs C = {3, 21, 2, 1, 8, 34, 1, 13, 5}.
Code H(A,C) = {000001, 01, 0000001, 00000000, 0001, 1, 00000001, 001, 00001}.
Sum S(H) = 3*6 + 21*2 + 2*7 + 1*8 + 8*4 + 34*1 + 1*8 + 13*3 + 5*5 = 220.Basic technique
Note: it may be beneficial to generate codes with minimum length variance, since in the worst case when several long codewords have to be transmitted in a row it may lead to unwanted side effects(for example if we use a buffer and constant rate transmitter, it increases the mimimum buffer size).
To reduce variance every newly generated node must be favored among same weight nodes and placed as high as possible. This will balance the length, keeping same average rate.Main properties
The frequencies used can be generic ones for the application domain that are based on average experience, or they can be the actual frequencies found in the text being compressed.
(This variation requires that a frequency table or other hint as to the encoding must be stored with the compressed text; implementations employ various tricks to store these tables efficiently.)Variations
Adaptive Huffman coding
A variation called adaptive Huffman coding calculates the frequencies dynamically based on recent actual frequencies in the source string. This is somewhat related to the LZ family of algorithms.Huffman Template algorithm
Most often, the weights used in implementations of Huffman coding represent numeric probabilities, but the algorithm given above does not require this; it requires only a way to order weights and to add them. Huffman Template algorithm enables one to use non-numerical weights (costs, frequencies).n-ary Huffman coding
The n-ary Huffman algorithm uses the {0, 1, ..., n-1} alphabet to encode message and build an n-ary tree.Huffman coding with unequal letter costs
In the standard Huffman coding problem, one is given a set of words and for each word a positive frequency. The goal is to encode each word w as a codeword c(w) over a given alphabet.
Huffman coding with unequal letter costs is the generalization in which the letters of the encoding alphabet may have non-uniform lengths. The goal is to minimize the weighted average codeword length. Applications
Arithmetic coding can be viewed as a generalization of Huffman coding.
Although arithmetic coding offers better performance than Huffman coding, Huffman coding is still in wide use because of its simplicity and lack of encumbrance by patents.External links