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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).
For the example mentioned above, the encoding becomes: (1,1,2), ('B','A','C','D') This means that the first symbol B is of length 1, then the A of length 2, and remaining 2 symbols (C and D) of length 3. Since the symbols are sorted by bit-length, we can efficiently reconstruct the codebook.
The optimal length-limited Huffman code will encode symbol i with a bit string of length h i. The canonical Huffman code can easily be constructed by a simple bottom-up greedy method, given that the h i are known, and this can be the basis for fast data compression. [2]
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 ...
A code is non-singular if each source symbol is mapped to a different non-empty bit string; that is, the mapping from source symbols to bit strings is injective.. For example, the mapping = {,,} is not non-singular because both "a" and "b" map to the same bit string "0"; any extension of this mapping will generate a lossy (non-lossless) coding.
Arithmetic coding differs from other forms of entropy encoding, such as Huffman coding, in that rather than separating the input into component symbols and replacing each with a code, arithmetic coding encodes the entire message into a single number, an arbitrary-precision fraction q, where 0.0 ≤ q < 1.0.
Unlike standard Huffman codes, the ones constructed in this way are alphabetical, meaning that the sorted order of these binary codes is the same as the input ordering of the values. If the weight of a value is its frequency in a message to be encoded, then the output of the Garsia–Wachs algorithm is the alphabetical Huffman code that ...
Modified Huffman coding is used in fax machines to encode black-on-white images . It combines the variable-length codes of Huffman coding with the coding of repetitive data in run-length encoding . The basic Huffman coding provides a way to compress files with much repeating data, like a file containing text, where the alphabet letters are the ...