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Huffman Coding

Module by: Behnaam Aazhang. E-mail the author

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Summary: A description of the Huffman source encoding algorithm.

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

Huffman encoding algorithm

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

Example 1

X∈ABCD X A B C D with probabilities { 12 1 2 , 14 1 4 , 18 1 8 , 18 1 8 }

Figure 1
Figure 1 (Figure7-24.png)

Average length=121+142+183+183=148 Average length 1 2 1 1 4 2 1 8 3 1 8 3 14 8 . As you may recall, the entropy of the source was also HX=148 H X 14 8 . In this case, the Huffman code achieves the lower bound of 148bitsoutput 14 8 bits output .

In general, we can define average code length as

ℓ¯=∑x∈ X ¯ pXxℓx ℓ x x X ¯ p X x ℓ x (1)
where X ¯ X ¯ is the set of possible values of xx.

It is not very hard to show that

HX≥ℓ¯>HX+1 H X ℓ H X 1 (2)
For compressing single source output at a time, Huffman codes provide nearly optimum code lengths.

The drawbacks of Huffman coding

  1. Codes are variable length.
  2. The algorithm requires the knowledge of the probabilities, pXx p X x for all x∈ X ¯ x X ¯ .
Another powerful source coder that does not have the above shortcomings is Lempel and Ziv.

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