Shannon–Fano coding

related topics
{math, number, function}
{rate, high, increase}
{system, computer, user}
{area, part, region}

In the field of data compression, Shannon–Fano coding, named after Claude Elwood Shannon and Robert Fano, is a technique for constructing a prefix code based on a set of symbols and their probabilities (estimated or measured). It is suboptimal in the sense that it does not achieve the lowest possible expected code word length like Huffman coding; however unlike Huffman coding, it does guarantee that all code word lengths are within one bit of their theoretical ideal − logP(x). The technique was proposed in Shannon's "A Mathematical Theory of Communication", his 1948 article introducing the field of information theory. The method was attributed to Fano, who later published it as a technical report.[1] Shannon–Fano coding should not be confused with Shannon coding, the coding method used to prove Shannon's noiseless coding theorem, or with Shannon-Fano-Elias coding (also known as Elias coding), the precursor to arithmetic coding.

In Shannon–Fano coding, the symbols are arranged in order from most probable to least probable, and then divided into two sets whose total probabilities are as close as possible to being equal. All symbols then have the first digits of their codes assigned; symbols in the first set receive "0" and symbols in the second set receive "1". As long as any sets with more than one member remain, the same process is repeated on those sets, to determine successive digits of their codes. When a set has been reduced to one symbol, of course, this means the symbol's code is complete and will not form the prefix of any other symbol's code.

The algorithm works, and it produces fairly efficient variable-length encodings; when the two smaller sets produced by a partitioning are in fact of equal probability, the one bit of information used to distinguish them is used most efficiently. Unfortunately, Shannon–Fano does not always produce optimal prefix codes; the set of probabilities {0.35, 0.17, 0.17, 0.16, 0.15} is an example of one that will be assigned non-optimal codes by Shannon–Fano coding.

For this reason, Shannon–Fano is almost never used; Huffman coding is almost as computationally simple and produces prefix codes that always achieve the lowest expected code word length, under the constraints that each symbol is represented by a code formed of an integral number of bits. This is a constraint that is often unneeded, since the codes will be packed end-to-end in long sequences. If we consider groups of codes at a time, symbol-by-symbol Huffman coding is only optimal if the probabilities of the symbols are independent and are some power of a half, i.e., \textstyle \frac{1}{2^n}. In most situations, arithmetic coding can produce greater overall compression than either Huffman or Shannon–Fano, since it can encode in fractional numbers of bits which more closely approximate the actual information content of the symbol. However, arithmetic coding has not superseded Huffman the way that Huffman supersedes Shannon–Fano, both because arithmetic coding is more computationally expensive and because it is covered by multiple patents.

Shannon–Fano coding is used in the IMPLODE compression method, which is part of the ZIP file format.[2]

Full article ▸

related documents
Upper and lower bounds
Lambert W function
Conjugacy class
Pointless topology
Cofinality
Semi-continuity
ZPP
Iterative method
Ternary numeral system
Counting sort
Five lemma
Elliptic function
Distributivity
Division ring
Homomorphism
Intermediate value theorem
Dedekind cut
Arithmetic function
Infimum
Principal ideal domain
Linear cryptanalysis
Group isomorphism
Arithmetic shift
Soundness
Zorn's lemma
Multiplication table
Histogram
Local field
Cyclone (programming language)
SECD machine