
Huffman coding uses a specific method for choosing the representation for each symbol, resulting in a prefix code (sometimes called "prefixfree codes") (that is, the bit string representing some particular symbol is never a prefix of the bit string representing any other symbol) that expresses the most common characters using shorter strings of bits than are used for less common source symbols. Huffman was able to design the most efficient compression method of this type: no other mapping of individual source symbols to unique strings of bits will produce a smaller average output size when the actual symbol frequencies agree with those used to create the code. A method was later found to do this in linear time if input probabilities (also known as weights) are sorted.
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, e.g., ASCII coding. Huffman coding is such a widespread method for creating prefix codes that the term "Huffman code" is widely used as a synonym for "prefix code" even when such a code is not produced by Huffman's algorithm.
Although Huffman coding is optimal for a symbolbysymbol coding with a known input probability distribution, its optimality can sometimes accidentally be overstated. For example, arithmetic coding and LZW coding often have better compression capability. Both these methods can combine an arbitrary number of symbols for more efficient coding, and generally adapt to the actual input statistics, the latter of which is useful when input probabilities are not precisely known or vary significantly within the stream.
In 1951, David A. 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 frequencysorted 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 the suboptimal ShannonFano coding by building the tree from the bottom up instead of from the top down.
Input (A, W)  Symbol (a_{i})  a  b  c  d  e  Sum 

Weights (w_{i})  0.10  0.15  0.30  0.16  0.29  = 1  
Output C  Codewords (c_{i})  000  001  10  01  11  
Codeword length (in bits) (l_{i})  3  3  2  2  2  
Weighted path length (l_{i} w_{i} )  0.30  0.45  0.60  0.32  0.58  L(C) = 2.25  
Optimality  Probability budget (2^{li})  1/8  1/8  1/4  1/4  1/4  = 1.00 
Information content (in bits) (−log_{2} w_{i}) ≈  3.32  2.74  1.74  2.64  1.79  
Entropy (−w_{i} log_{2} w_{i})  0.332  0.411  0.521  0.423  0.518  H(A) = 2.205 
For any code that is biunique, meaning that the code is uniquely decodeable, the sum of the probability budgets across all symbols is always less than or equal to one. In this example, the sum is strictly equal to one; as a result, the code is termed a complete code. If this is not the case, you can always derive an equivalent code by adding extra symbols (with associated null probabilities), to make the code complete while keeping it biunique.
As defined by Shannon (1948), the information content h (in bits) of each symbol a_{i} with nonnull probability is
The entropy H (in bits) is the weighted sum, across all symbols a_{i} with nonzero probability w_{i}, of the information content of each symbol:
(Note: A symbol with zero probability has zero contribution to the entropy. When w = 0, $w\; log\_2\; (1/w)\; =\; 0\; cdot\; infty$ is an indefinite form; so by L'Hôpital's rule:
For simplicity, symbols with zero probability are left out of the formula above.)
As a consequence of Shannon's Source coding theorem, the entropy is a measure of the smallest codeword length that is theoretically possible for the given alphabet with associated weights. In this example, the weighted average codeword length is 2.25 bits per symbol, only slightly larger than the calculated entropy of 2.205 bits per symbol. So not only is this code optimal in the sense that no other feasible code performs better, but it is very close to the theoretical limit established by Shannon.
Note that, in general, a Huffman code need not be unique, but it is always one of the codes minimizing $L(C)$.
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 $n1$ internal nodes.
The process essentially begins with the leaf nodes containing the probabilities of the symbol they represent, then a new node whose children are the 2 nodes with smallest probability is created, such that the new node's probability is equal to the sum of the children's probability. With the previous 2 nodes merged into one node (thus not considering them anymore), and with the new node being now considered, the procedure is repeated until only one node remains, the Huffman tree.
The simplest construction algorithm uses a priority queue where the node with lowest probability is given highest priority:
Since efficient priority queue data structures require O(log n) time per deletion, and a tree with n leaves has 2n−1 nodes, this algorithm operates in O(n log n) time.
If the symbols are sorted by probability, there is a lineartime (O(n)) method to create a Huffman tree using two queues, the first one containing the initial weights (along with pointers to the associated leaves), and combined weights (along with pointers to the trees) being put in the back of the second queue. This assures that the lowest weight is always kept at the front of one of the two queues:
It is generally beneficial to minimize the variance of codeword length. For example, a communication buffer receiving Huffmanencoded data may need to be larger to deal with especially long symbols if the tree is especially unbalanced. To minimize variance, simply break ties between queues by choosing the item in the first queue. This modification will retain the mathematical optimality of the Huffman coding while both minimizing variance and minimizing the length of the longest character code.
Huffman coding is optimal when the probability of each input symbol is a negative power of two. Prefix codes tend to have slight inefficiency on small alphabets, where probabilities often fall between these optimal points. "Blocking", or expanding the alphabet size by coalescing multiple symbols into "words" of fixed or variablelength before Huffman coding, usually helps, especially when adjacent symbols are correlated (as in the case of natural language text). The worst case for Huffman coding can happen when the probability of a symbol exceeds 2^{1} = 0.5, making the upper limit of inefficiency unbounded. These situations often respond well to a form of blocking called runlength encoding.
Arithmetic coding produces slight gains over Huffman coding, but in practice these gains have seldom been large enough to offset arithmetic coding's higher computational complexity and patent royalties. (As of July 2006, IBM owns patents on many methods of arithmetic coding in several jurisdictions; see US patents on arithmetic coding.)
Huffman coding with unequal letter costs is the generalization in which this assumption is no longer assumed true: the letters of the encoding alphabet may have nonuniform lengths, due to characteristics of the transmission medium. An example is the encoding alphabet of Morse code, where a 'dash' takes longer to send than a 'dot', and therefore the cost of a dash in transmission time is higher. The goal is still to minimize the weighted average codeword length, but it is no longer sufficient just to minimize the number of symbols used by the message. No algorithm is known to solve this in the same manner or with the same efficiency as conventional Huffman coding.
If weights corresponding to the alphabetically ordered inputs are in numerical order, the Huffman code has the same lengths as the optimal alphabetic code, which can be found from calculating these lengths, rendering HuTucker coding unnecessary. The code resulting from numerically (re)ordered input is sometimes called the canonical Huffman code and is often the code used in practice, due to ease of encoding/decoding. The technique for finding this code is sometimes called HuffmanShannonFano coding, since it is optimal like Huffman coding, but alphabetic in weight probability, like ShannonFano coding. The HuffmanShannonFano code corresponding to the example is $\{000,001,01,10,11\}$, which, having the same codeword lengths as the original solution, is also optimal.
Huffman coding today is often used as a "backend" to some other compression method. DEFLATE (PKZIP's algorithm) and multimedia codecs such as JPEG and MP3 have a frontend model and quantization followed by Huffman coding.