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Huffman tree generated from the exact frequencies of the text "this is an example of a huffman tree". Encoding the sentence with this code requires 135 (or 147) bits, as opposed to 288 (or 180) bits if 36 characters of 8 (or 5) bits were used (This assumes that the code tree structure is known to the decoder and thus does not need to be counted as part of the transmitted information).
The picture is an example of Huffman coding. Colors make it clearer, but they are not necessary to understand it (according to Wikipedia's guidelines): probability is shown in red, binary code is shown in blue inside a yellow frame. For a more detailed description see below (I couldn't insert a table here). Date: 18 May 2007: Source: self-made
The normal Huffman coding algorithm assigns a variable length code to every symbol in the alphabet. More frequently used symbols will be assigned a shorter code. For example, suppose we have the following non-canonical codebook: A = 11 B = 0 C = 101 D = 100 Here the letter A has been assigned 2 bits, B has 1 bit, and C and D both have 3 bits.
A Huffman code must distinguish the 3 cases. The length of each code is based on the frequency of each type of sub expressions. Initially constants are all assigned the same length/probability. Later constants may be assigned a probability using the Huffman code based on the number of uses of the function id in all expressions recorded so far.
More precisely, the source coding theorem states that for any source distribution, the expected code length satisfies [(())] [ (())], where is the number of symbols in a code word, is the coding function, is the number of symbols used to make output codes and is the probability of the source symbol. An entropy coding attempts to ...
If symbols are assigned in ranges of lengths being powers of 2, we would get Huffman coding. For example, a->0, b->100, c->101, d->11 prefix code would be obtained for tANS with "aaaabcdd" symbol assignment. Example of generation of tANS tables for m = 3 size alphabet and L = 16 states, then applying them for stream decoding.
When naively Huffman coding binary strings, no compression is possible, even if entropy is low (e.g. ({0, 1}) has probabilities {0.95, 0.05}). Huffman encoding assigns 1 bit to each value, resulting in a code of the same length as the input. By contrast, arithmetic coding compresses bits well, approaching the optimal compression ratio of
Huffman coding and arithmetic coding (when they can be used) give at least as good, and often better compression than any universal code.. However, universal codes are useful when Huffman coding cannot be used — for example, when one does not know the exact probability of each message, but only knows the rankings of their probabilities.