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75. Huffman codes are always optimal (the best possible), but this particular code has average code length equal to the entropy, and it is never possible to create a code with shorter average length. Most Huffman codes have average code length greater than the entropy (unless all frequencies are a fraction with numerator and denominator a power of 2). The next simple example shows this. Example 2. Start with the file aaaaaabc. 25. 75 5. 4 Huffman Tree: Entropy less than average code length. 5gives the corresponding Huffman tree, along with the code words: With this product code, the entropy and average code length work out to be and , but each new symbol (upper-case letter) represents two of the original symbols.
5. The Huffman Code for Compression 43 with the data, so that decoding is possible at the other end. Exercises 1. Devise a simple example where there are different choices for the least trees and where the Huffman algorithm yields different answers. Get an example where there are even two different distributions for the lengths of the codewords. Verify that the average code lengths are the same for the two examples. ] 2. After looking at the answer to the previous problem, see if you can create the simplest possible example and argue that there is no simpler example.
This is the concept of algorithmic information theory, invented by Chaitin and Kolmogoroff, which is beyond the scope of this discussion. Still speaking intuitively, the result of a good compression algorithm is a file that appears random. ) Also, a good compression algorithm will end up relatively close to the entropy, so one knows that no further compression is possible. 36 II. Coding and Information Theory Law COMPRESSION-1: Just like Niven’s law (never fire a laser at a mirror), one says: never compress a message that’s already compressed.