Use parentheses and white-space to make code readable.

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 Coding paper 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 bytes assuming an 8-bit alphabet.

Many other techniques are possible Coding paper 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 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.

Main properties[ edit ] 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 requires that a frequency table must be stored with the compressed text. See the Decompression section above for more information about the various techniques employed for this purpose. See also Arithmetic coding Huffman coding Huffman's original algorithm is optimal for a symbol-by-symbol coding with a known input probability distribution, i.

However, it is not optimal when the symbol-by-symbol restriction is dropped, or when the probability mass functions are unknown. Also, if symbols are not independent and identically distributeda single code may be insufficient for optimality. Other methods such as arithmetic coding often have better compression capability.

Although both aforementioned methods can combine an arbitrary number of symbols for more efficient coding and generally adapt to the actual input statistics, arithmetic coding does so without significantly increasing its computational or algorithmic complexities though the simplest version is slower and more complex than Huffman coding.

Such flexibility is especially useful when input probabilities are not precisely known or vary significantly within the stream. However, Huffman coding is usually faster and arithmetic coding was historically a subject of some concern over patent issues.

Thus many technologies have historically avoided arithmetic coding in favor of Huffman and other prefix coding techniques. As of mid, the most commonly used techniques for this alternative to Huffman coding have passed into the public domain as the early patents have expired.

For a set of symbols with a uniform probability distribution and a number of members which is a power of twoHuffman coding is equivalent to simple binary block encodinge. This reflects the fact that compression is not possible with such an input, no matter what the compression method, i.

Huffman coding is optimal among all methods in any case where each input symbol is a known independent and identically distributed random variable having a probability that is dyadic, i. Prefix codes, and thus Huffman coding in particular, tend to have inefficiency on small alphabets, where probabilities often fall between these optimal dyadic points.

There are two related approaches for getting around this particular inefficiency while still using Huffman coding. Combining a fixed number of symbols together "blocking" often increases and never decreases compression.

As the size of the block approaches infinity, Huffman coding theoretically approaches the entropy limit, i. However, blocking arbitrarily large groups of symbols is impractical, as the complexity of a Huffman code is linear in the number of possibilities to be encoded, a number that is exponential in the size of a block.

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Coding Style and Good Computing Practices