Huffman coding
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.
For a set of symbols whose cardinality is a power of two and a uniform probability distribution, 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.
The technique works by creating a binary tree of symbols:
Huffman coding is optimal when the probability of each input symbol is a power of two. Prefix-free codes tend to have slight inefficiency on small alphabets, where probabilities often fall between powers of two. Expanding the alphabet size by coalescing multiple symbols into "words" before Huffman coding can help a bit.
Extreme cases of Huffman codes are connected with Fibonacci numbers.
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.
History
Basic technique
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 to use non-numerical weights (costs, frequences).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
External links