# 5.5 Huffman coding

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A description of the Huffman source encoding algorithm.

One particular source coding algorithm is the Huffman encoding algorithm. It is a source coding algorithm which approaches, and sometimes achieves, Shannon's bound forsource compression. A brief discussion of the algorithm is also given in another module .

## Huffman encoding algorithm

• Sort source outputs in decreasing order of their probabilities
• Merge the two least-probable outputs into a single output whose probability is the sum of the corresponding probabilities.
• If the number of remaining outputs is more than 2, then go to step 1.
• Arbitrarily assign 0 and 1 as codewords for the two remaining outputs.
• If an output is the result of the merger of two outputs in a preceding step, append the current codeword with a 0 and a 1 toobtain the codeword the the preceding outputs and repeat step 5. If no output is preceded by another output in a preceding step, thenstop.

$X\in \{A, B, C, D\}$ with probabilities { $\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{1}{8}$ , $\frac{1}{8}$ }

$\mathrm{Average length}=\frac{1}{2}\times 1+\frac{1}{4}\times 2+\frac{1}{8}\times 3+\frac{1}{8}\times 3=\frac{14}{8}$ . As you may recall, the entropy of the source was also $H(X)=\frac{14}{8}$ . In this case, the Huffman code achieves the lower bound of $\frac{14}{8}\frac{\mathrm{bits}}{\mathrm{output}}$ .

In general, we can define average code length as

$\langle \rangle =\sum_{x\in \stackrel{}{X}} p(X, x)(x)$
where $\stackrel{}{X}$ is the set of possible values of $x$ .

It is not very hard to show that

$H(X)\ge \langle \rangle > H(X)+1$
For compressing single source output at a time, Huffman codes provide nearly optimum code lengths.

The drawbacks of Huffman coding

• Codes are variable length.
• The algorithm requires the knowledge of the probabilities, $p(X, x)$ for all $x\in \stackrel{}{X}$ .
Another powerful source coder that does not have the aboveshortcomings is Lempel and Ziv.

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