Huffman coding

In computer science and information theory, a Huffman code is a particular type of optimal prefix code that is commonly used for lossless data compression. The process of finding or using such a code is Huffman coding, an algorithm developed by David A. Huffman while he was a Sc.D. student at MIT, and published in the 1952 paper "A Method for the Construction of Minimum-Redundancy Codes".
The output from Huffman's algorithm can be viewed as a variable-length code table for encoding a source symbol. The algorithm derives this table from the estimated probability or frequency of occurrence for each possible value of the source symbol. As in other entropy encoding methods, more common symbols are generally represented using fewer bits than less common symbols. Huffman's method can be efficiently implemented, finding a code in time linear to the number of input weights if these weights are sorted. However, although optimal among methods encoding symbols separately, Huffman coding [|is not always optimal] among all compression methods – it is replaced with arithmetic coding if a better compression ratio is required.

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, Huffman outdid Fano, who had worked with Claude Shannon to develop a similar code. Building the tree from the bottom up guaranteed optimality, unlike the top-down approach of Shannon–Fano coding.

Terminology

Huffman coding uses a specific method for choosing the representation for each symbol, resulting in a prefix code. 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.

Problem definition

Informal description

;Given: A set of symbols and for each symbol, the frequency representing the fraction of symbols in the text that are equal to.
;Find: A prefix-free binary code with minimum expected codeword length.

Formalized description

Input.

Alphabet, which is the symbol alphabet of size.

Tuple, which is the tuple of the symbol weights, i.e..

Output.

Code, which is the tuple of codewords, where is the codeword for.

Goal.

Let be the weighted path length of code. Condition: for any code.

Example

We give an example of the result of Huffman coding for a code with five characters and given weights. We will not verify that it minimizes L over all codes, but we will compute L and compare it to the Shannon entropy H of the given set of weights; the result is nearly optimal.

Input	Symbol	a	b	c	d	e	Sum
Input	Weights	0.10	0.15	0.30	0.16	0.29	= 1
Output C	Codewords	`010`	`011`	`11`	`00`	`10`
Output C	Codeword length	3	3	2	2	2
Output C	Contribution to weighted path length	0.30	0.45	0.60	0.32	0.58	L = 2.25
Optimality	Probability budget	1/8	1/8	1/4	1/4	1/4	= 1.00
Optimality	Information content ≈	3.32	2.74	1.74	2.64	1.79
Optimality	Contribution to entropy	0.332	0.411	0.521	0.423	0.518	H = 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, one can always derive an equivalent code by adding extra symbols, to make the code complete while keeping it biunique.
As defined by Shannon, the information content h of each symbol a_i with non-null probability is
The entropy H is the weighted sum, across all symbols with non-zero probability, of the information content of each symbol:
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.
In general, a Huffman code need not be unique. Thus the set of Huffman codes for a given probability distribution is a non-empty subset of the codes minimizing for that probability distribution.

Basic technique

Compression

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,. 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 of the symbol and optionally, a link to a parent node which makes it easy to read the code starting from a leaf node. Internal nodes contain a weight, links to two child nodes and an 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 leaf nodes and internal nodes. A Huffman tree that omits unused symbols produces the most optimal code lengths.
The process begins with the leaf nodes containing the probabilities of the symbol they represent. Then, the process takes the two nodes with smallest probability, and creates a new internal node having these two nodes as children. The weight of the new node is set to the sum of the weight of the children. We then apply the process again, on the new internal node and on the remaining nodes, we repeat this process until only one node remains, which is the root of 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:
# Remove the two nodes of highest priority from the queue
# Create a new internal node with these two nodes as children and with probability equal to the sum of the two nodes' probabilities.
# 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 time per insertion, and a tree with n leaves has 2n−1 nodes, this algorithm operates in O time, where n is the number of symbols.
If the symbols are sorted by probability, there is a linear-time method to create a Huffman tree using two queues, the first one containing the initial weights, and combined weights 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.
While there is more than one node in the queues:
#Dequeue the two nodes with the lowest weight by examining the fronts of both queues.
#Create a new internal node, with the two just-removed nodes as children and the sum of their weights as the new weight.
#Enqueue the new node into the rear of the second queue.
The remaining node is the root node; the tree has now been generated.

Once the Huffman tree has been generated, it is traversed to generate a dictionary which maps the symbols to binary codes as follows:

Start with current node set to the root.
If node is not a leaf node, label the edge to the left child as 0 and the edge to the right child as 1. Repeat the process at both the left child and the right child.

The final encoding of any symbol is then read by a concatenation of the labels on the edges along the path from the root node to the symbol.
In many cases, time complexity is not very important in the choice of algorithm here, since n here is the number of symbols in the alphabet, which is typically a very small number ; whereas complexity analysis concerns the behavior when n grows to be very large.
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.

Decompression

Generally speaking, the process of decompression is simply a matter of translating the stream of prefix codes to individual byte values, usually by traversing the Huffman tree node by node as each bit is read from the input stream. Before this can take place, however, the Huffman tree must be somehow reconstructed. In the simplest case, where character frequencies are fairly predictable, the tree can be preconstructed and thus reused every time, at the expense of at least some measure of compression efficiency. Otherwise, the information to reconstruct the tree must be sent a priori. A naive approach might be to prepend the frequency count of each character to the compression stream. Unfortunately, the overhead in such a case could amount to several kilobytes, so this method has little practical use. If the data is compressed using canonical encoding, the compression model can be precisely reconstructed with just bits of information. Another method is to simply prepend the Huffman tree, bit by bit, to the output stream. For example, assuming that the value of 0 represents a parent node and 1 a leaf node, whenever the latter is encountered the tree building routine simply reads the next 8 bits to determine the character value of that particular leaf. The process continues recursively until the last leaf node is reached; at that point, the Huffman tree will thus be faithfully reconstructed. The overhead using such a method ranges from roughly 2 to 320 bytes. Many other techniques are possible as well. In any case, since 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 compressed data along with the compression model or by defining a special code symbol to signify the end of input.