Huffman coding

Huffman tree generated from the exact frequencies of the text "this is an example of a huffman tree". The frequencies and codes of each character are below. Encoding the sentence with this code requires 135 bits. (This assumes that the code tree structure is known to the decoder and thus does not need to be counted as part of the transmitted information.)

Char	Freq	Code
space	7	111
a	4	010
e	4	000
f	3	1101
h	2	1010
i	2	1000
m	2	0111
n	2	0010
s	2	1011
t	2	0110
l	1	11001
o	1	00110
p	1	10011
r	1	11000
u	1	00111
x	1	10010

In computer science and information theory, Huffman coding is an entropy encoding algorithm used for lossless data compression. The term refers to the use of a variable-length code table for encoding a source symbol (such as a character in a file) where the variable-length code table has been derived in a particular way based on the estimated probability of occurrence for each possible value of the source symbol. It was developed by David A. Huffman while he was a Ph.D. student at MIT, and published in the 1952 paper "A Method for the Construction of Minimum-Redundancy Codes."

Huffman coding uses a specific method for choosing the representation for each symbol, resulting in a prefix code (sometimes called "prefix-free 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 source symbols 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 design a Huffman code 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's original algorithm is optimal for a symbol-by-symbol coding (i.e. a stream of unrelated symbols) with a known input probability distribution, it is not optimal when the symbol-by-symbol restriction is dropped, or when the probability mass functions are unknown, not identically distributed, or not independent (e.g., "cat" is more common than "cta"). Other methods such as arithmetic coding and LZW coding often have better compression capability: both of 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. However, the limitations of Huffman coding should not be overstated; it can be used adaptively, accommodating unknown, changing, or context-dependent probabilities. In the case of known independent and identically-distributed random variables, combining symbols together reduces inefficiency in a way that approaches optimality as the number of symbols combined increases.

History

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 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 the suboptimal Shannon-Fano coding by building the tree from the bottom up instead of from the top down.

Problem definition

Informal description

Given: A set of symbols and their weights (usually proportional to probabilities).
Find: A prefix-free binary code (a set of codewords) with minimum expected codeword length (equivalently, a tree with minimum weighted path length from the root).

Formalized description

Input.
Alphabet $A = \left\{a_{1},a_{2},\cdots,a_{n}\right\}$ , which is the symbol alphabet of size $n$ .
Set $W = \left\{w_{1},w_{2},\cdots,w_{n}\right\}$ , which is the set of the (positive) symbol weights (usually proportional to probabilities), i.e. $w_{i} = \mathrm{weight}\left(a_{i}\right), 1\leq i \leq n$ .

Output.
Code $C \left(A,W\right) = \left\{c_{1},c_{2},\cdots,c_{n}\right\}$ , which is the set of (binary) codewords, where $c_{i}$ is the codeword for $a_{i}, 1 \leq i \leq n$ .

Goal.
Let $L\left(C\right) = \sum_{i=1}^{n}{w_{i}\times\mathrm{length}\left(c_{i}\right)}$ be the weighted path length of code $C$ . Condition: $L\left(C\right) \leq L\left(T\right)$ for any code $T\left(A,W\right)$ .

Samples

Input (A, W)	Symbol (a_i)	a	b	c	d	e	Sum
Input (A, W)	Weights (w_i)	0.10	0.15	0.30	0.16	0.29	= 1
Output C	Codewords (c_i)	`010`	`011`	`11`	`00`	`10`
	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^-l_i)	1/8	1/8	1/4	1/4	1/4	= 1.00
	Information content (in bits) (−log₂ w_i) ≈	3.32	2.74	1.74	2.64	1.79
	Entropy (−w_i log₂ 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 non-null probability is

$h(a_i) = \log_2{1 \over w_i}.$

The entropy H (in bits) is the weighted sum, across all symbols a_i with non-zero probability w_i, of the information content of each symbol:

$H(A) = \sum_{w_i > 0} w_i h(a_i) = \sum_{w_i > 0} w_i \log_2{1 \over w_i} = - \sum_{w_i > 0} w_i \log_2{w_i}.$

(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:

$\lim_{w \to 0^+} \frac{\log_2 \frac{1}{w}}{\frac{1}{w}} = \lim_{w \to 0^+} \frac{-\frac{1}{w \ln 2}}{-\frac{1}{w^2}} = \lim_{w \to 0^+} \frac{w}{\ln 2} = 0$ .

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 $L(C)$ .

Basic technique

A source generates 4 different symbols $\{a_1 , a_2 , a_3 , a_4 \}$ with probability $\{0.4�; 0.35�; 0.2�; 0.05 \}$ . A binary tree is generated from left to right taking the two less probable symbols, putting them together to form another equivalent symbol having a probability that equals the sum of the two symbols. The process is repeated until there is just one symbol. The tree can then be read backwards, from right to left, assigning different bits to different branches. The final Huffman code is:

Symbol	Code
a1	0
a2	10
a3	110
a4	111

The standard way to represent a signal made of 4 symbols is by using 2 bits/symbol, but the entropy of the source is 1.73 bits/symbol. If this Huffman code is used to represent the signal, then the average length is lowered to 1.85 bits/symbol; it is still far from the theoretical limit because the probabilities of the symbols are different from negative powers of two.

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 up to $n$ leaf nodes and $n-1$ internal nodes. A Huffman tree that omits unused symbols produces the most optimal code lengths.

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:

Create a leaf node for each symbol and add it to the priority queue.
While there is more than one node in the queue:
1. Remove the two nodes of highest priority (lowest probability) from the queue
2. Create a new internal node with these two nodes as children and with probability equal to the sum of the two nodes' probabilities.
3. Add the new node to the queue.
The remaining node is the root node and the tree is complete.

Since efficient priority queue data structures require O(log n) time per insertion, 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 linear-time (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:

Start with as many leaves as there are symbols.
Enqueue all leaf nodes into the first queue (by probability in increasing order so that the least likely item is in the head of the queue).
While there is more than one node in the queues:
1. Dequeue the two nodes with the lowest weight by examining the fronts of both queues.
2. Create a new internal node, with the two just-removed nodes as children (either node can be either child) and the sum of their weights as the new weight.
3. Enqueue the new node into the rear of the second queue.
The remaining node is the root node; the tree has now been generated.

It is generally beneficial to minimize the variance of codeword length. For example, a communication buffer receiving Huffman-encoded 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.

Here's an example using the French subject string "j'aime aller sur le bord de l'eau les jeudis ou les jours impairs":

Main properties

The probabilities 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 tables efficiently.)

Huffman coding is optimal when the probability of each input symbol is a negative power of two. Prefix codes tend to have inefficiency on small alphabets, where probabilities often fall between these optimal points. "Blocking", or expanding the alphabet size by grouping multiple symbols into "words" of fixed or variable-length before Huffman coding helps both to reduce that inefficiency and to take advantage of statistical dependencies between input symbols within the group (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 run-length encoding; for the simple case of Bernoulli processes, Golomb coding is a provably optimal run-length code.

Arithmetic coding produces some gains over Huffman coding, although arithmetic coding has higher computational complexity. Also, arithmetic coding was historically a subject of some concern over patent issues. However, as of mid-2010, various well-known effective techniques for arithmetic coding have passed into the public domain as the early patents have expired.

Variations

Many variations of Huffman coding exist, some of which use a Huffman-like algorithm, and others of which find optimal prefix codes (while, for example, putting different restrictions on the output). Note that, in the latter case, the method need not be Huffman-like, and, indeed, need not even be polynomial time. An exhaustive list of papers on Huffman coding and its variations is given by "Code and Parse Trees for Lossless Source Encoding"[1].

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. This approach was considered by Huffman in his original paper. The same algorithm applies as for binary (n equals 2) codes, except that the n least probable symbols are taken together, instead of just the 2 least probable. Note that for n greater than 2, not all sets of source words can properly form an n-ary tree for Huffman coding. In this case, additional 0-probability place holders must be added. This is because the tree must form an n to 1 contractor; for binary coding, this is a 2 to 1 contractor, and any sized set can form such a contractor. If the number of source words is congruent to 1 modulo n-1, then the set of source words will form a proper Huffman tree.

Adaptive Huffman coding

A variation called adaptive Huffman coding involves calculating the probabilities dynamically based on recent actual frequencies in the sequence of source symbols, and changing the coding tree structure to match the updated probability estimates.

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. The Huffman template algorithm enables one to use any kind of weights (costs, frequencies, pairs of weights, non-numerical weights) and one of many combining methods (not just addition). Such algorithms can solve other minimization problems, such as minimizing $\max_i\left[w_{i}+\mathrm{length}\left(c_{i}\right)\right]$ , a problem first applied to circuit design [2].

Length-limited Huffman coding

Length-limited Huffman coding is a variant where the goal is still to achieve a minimum weighted path length, but there is an additional restriction that the length of each codeword must be less than a given constant. The package-merge algorithm solves this problem with a simple greedy approach very similar to that used by Huffman's algorithm. Its time complexity is $O(nL)$ , where $L$ is the maximum length of a codeword. No algorithm is known to solve this problem in linear or linearithmic time, unlike the presorted and unsorted conventional Huffman problems, respectively.

Huffman coding with unequal letter costs

In the standard Huffman coding problem, it is assumed that each symbol in the set that the code words are constructed from has an equal cost to transmit: a code word whose length is N digits will always have a cost of N, no matter how many of those digits are 0s, how many are 1s, etc. When working under this assumption, minimizing the total cost of the message and minimizing the total number of digits are the same thing.

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 non-uniform 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.

Optimal alphabetic binary trees (Hu-Tucker coding)

In the standard Huffman coding problem, it is assumed that any codeword can correspond to any input symbol. In the alphabetic version, the alphabetic order of inputs and outputs must be identical. Thus, for example, $A = \left\{a,b,c\right\}$ could not be assigned code $H\left(A,C\right) = \left\{00,1,01\right\}$ , but instead should be assigned either $H\left(A,C\right) =\left\{00,01,1\right\}$ or $H\left(A,C\right) = \left\{0,10,11\right\}$ . This is also known as the Hu-Tucker problem, after the authors of the paper presenting the first linearithmic solution to this optimal binary alphabetic problem, which has some similarities to Huffman algorithm, but is not a variation of this algorithm. These optimal alphabetic binary trees are often used as binary search trees.

The canonical Huffman code

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 Hu-Tucker 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 Huffman-Shannon-Fano coding, since it is optimal like Huffman coding, but alphabetic in weight probability, like Shannon-Fano coding. The Huffman-Shannon-Fano code corresponding to the example is $\{000,001,01,10,11\}$ , which, having the same codeword lengths as the original solution, is also optimal.

Model reconstruction

Decompression generally requires transmission of information in order to reconstruct the compression model (methods such as adaptive Huffman do not, although they typically produce less than optimal code lengths). Originally, symbol frequencies were passed along to the decompressor, but this method is very inefficient, as it can produce an unacceptable level of overhead. The most common technique utilizes canonical Huffman encoding, which only requires $Bn$ bits of information (where $B$ is the number of bits per symbol and $n$ is the size of the symbol's alphabet). Other methods, such as the "direct transmission" of the Huffman tree, produce variable-length encoding of the model which can reduce the overhead to just a few bytes, in some cases.

Because the compressed data can include unused "trailing bits", the decompressor must be able to determine when to stop producing output. This can be accomplished by either transmitting the length of the decompressed data along with the compression model or by defining a special code symbol to signify the end of input (the latter method can adversely affect code length optimality, however).

Applications

Arithmetic coding can be viewed as a generalization of Huffman coding; indeed, in practice arithmetic coding is often preceded by Huffman coding, as it is easier to find an arithmetic code for a binary input than for a nonbinary input. Also, although arithmetic coding offers better compression performance than Huffman coding, Huffman coding is still in wide use because of its simplicity and high speed.

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 (or variable-length prefix-free codes with a similar structure, although perhaps not necessarily designed by using Huffman's algorithm).

References

Huffman's original article: D.A. Huffman, "A Method for the Construction of Minimum-Redundancy Codes", Proceedings of the I.R.E., September 1952, pp 1098–1102
Background story: Profile: David A. Huffman, Scientific American, September 1991, pp. 54–58
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 16.3, pp. 385–392.

External links

Program for explaining the Huffman Coding procedure.
n-ary Huffman Template Algorithm
Sloane A098950 Minimizing k-ordered sequences of maximum height Huffman tree
Mordecai J. Golin, Claire Kenyon, Neal E. Young "Huffman coding with unequal letter costs" (PDF), STOC 2002: 785-791
Huffman Coding: A CS2 Assignment a good introduction to Huffman coding
A quick tutorial on generating a Huffman tree
Pointers to Huffman coding visualizations
Huffman in C
Description of an implementation in Python
Explanation of Huffman coding with examples in several languages
Interactive Huffman Tree Construction
A C program doing basic huffman coding on binary and text files

Data compression methods

Lossless

Theory	Entropy · Complexity · Redundancy · Lossy

Entropy encoding	Shannon–Fano · Shannon–Fano–Elias · Huffman · Adaptive Huffman · Arithmetic · Range · Golomb · Universal (Gamma · Exp-Golomb · Fibonacci · Levenshtein)

Dictionary	RLE · Byte pair encoding · DEFLATE · Lempel–Ziv (LZ77/78 · LZSS · LZW · LZWL · LZO · LZMA · LZX · LZRW · LZJB · LZS · LZT · ROLZ)

Others	CTW · BWT · PPM · DMC · Delta

Audio

Theory	Companding · Convolution · Dynamic range · Latency · Sampling · Nyquist–Shannon theorem · Sound quality

Audio codec parts	LPC (LAR · LSP) · WLPC · CELP · ACELP · A-law · μ-law · ADPCM · DPCM · MDCT · Fourier transform · Psychoacoustic model

Others	Bit rate (CBR · ABR · VBR) · Speech compression · Sub-band coding

Image

Terms	Color space · Pixel · Chroma subsampling · Compression artifact · Image resolution

Methods	RLE · Fractal · Wavelet · EZW · SPIHT · LP · DCT · Chain code · KLT

Others	Test images · PSNR quality measure · Quantization

Video

Terms	Video characteristics · Frame · Frame rate · Interlace · Frame types · Video quality · Video resolution

Video codec parts	Motion compensation · DCT · Quantization

Others	Video codecs · Rate distortion theory · Bit rate (CBR · ABR · VBR)

Timeline of information theory, data compression, and error-correcting codes

See Compression formats for formats and Compression software implementations for codecs