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In the field of [[data compression]], '''Shannon–Fano coding''', named after [[Claude Shannon]] and [[Robert Fano]], is
* '''Shannon's method''' chooses a prefix code where a source symbol <math>i</math> is given the codeword length <math>l_i = \lceil - \log_2 p_i\rceil</math>. One common way of choosing the codewords uses the binary expansion of the cumulative probabilities. This method was proposed in Shannon's "[[A Mathematical Theory of Communication]]" (1948), his article introducing the field of [[information theory]].
* '''Fano's method''' divides the source symbols into two sets ("0" and "1") with probabilities as close to 1/2 as possible. Then those sets are themselves divided in two, and so on, until each set contains only one symbol. The codeword for that symbol is the string of "0"s and "1"s that records which half of the divides it fell on. This method was proposed in a later (in print) [[technical report]] by Fano (1949).
Shannon–Fano codes are [[Optimization (mathematics)|suboptimal]] in the sense that they do not always achieve the lowest possible expected codeword length, as [[Huffman coding]] does.<ref name="Kaur">{{cite journal |last1=Kaur |first1=Sandeep |last2=Singh |first2=Sukhjeet |title=Entropy Coding and Different Coding Techniques |journal=Journal of Network Communications and Emerging Technologies |date=May 2016 |volume=6 |issue=5 |page=5 |s2cid=212439287 |url=https://pdfs.semanticscholar.org/4253/7898a836d0384c6689a3c098b823309ab723.pdf |archive-url=https://web.archive.org/web/20191203151816/https://pdfs.semanticscholar.org/4253/7898a836d0384c6689a3c098b823309ab723.pdf |url-status=dead |archive-date=2019-12-03 |access-date=3 December 2019}}</ref> However, Shannon–Fano codes have an expected codeword length within 1 bit of optimal. Fano's method usually produces encoding with shorter expected lengths than Shannon's method. However, Shannon's method is easier to analyse theoretically.
Shannon–Fano coding should not be confused with [[Shannon–Fano–Elias coding]] (also known as Elias coding), the precursor to [[arithmetic coding]].
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==Naming==
Regarding the confusion in the two different codes being referred to by the same name, Krajči et al. write:<ref name="Kraj">Stanislav Krajči, Chin-Fu Liu, Ladislav Mikeš and Stefan M. Moser (2015), "Performance analysis of Fano coding", ''2015 IEEE International Symposium on Information Theory (ISIT)''.</ref>
<blockquote>
Around 1948, both Claude E. Shannon (1948) and Robert M. Fano (1949) independently proposed two different source coding algorithms for an efficient description of a discrete memoryless source. Unfortunately, in spite of being different, both schemes became known under the same name ''Shannon–Fano coding''.
There are several reasons for this mixup. For one thing, in the discussion of his coding scheme, Shannon mentions Fano’s scheme and calls it “substantially the same” (Shannon, 1948, p. 17 [reprint]).<ref>{{Cite book |url=https://archive.org/details/sim_att-technical-journal_1948-07_27_3/page/402/ |title=The Bell System Technical Journal 1948-07: Vol 27 Iss 3 |date=1948-07-01 |publisher=AT & T Bell Laboratories |pages=403 |language=en}}</ref> For another, both Shannon’s and Fano’s coding schemes are similar in the sense that they both are efficient, but ''suboptimal'' prefix-free coding schemes with a similar performance.
</blockquote>
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A second method makes use of cumulative probabilities. First, the probabilities are written in decreasing order <math>p_1 \geq p_2 \geq \cdots \geq p_n</math>. Then, the cumulative probabilities are defined as
:<math>c_1 = 0, \qquad c_i = \sum_{j=1}^{i-1}
so <math>c_1 = 0, c_2 = p_1, c_3 = p_1 + p_2</math> and so on.
The codeword for symbol <math>i</math> is chosen to be the first <math>l_i</math> binary digits in the [[binary number|binary expansion]] of <math>c_i</math>.
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===Example===
This example shows the construction of a Shannon–Fano code for a small alphabet. There are 5 different source symbols. Suppose 39 total symbols have been observed with the following frequencies, from which we can estimate the symbol probabilities.
:{| class="wikitable" style="text-align: center;"
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Note that although the codewords under the two methods are different, the word lengths are the same. We have lengths of 2 bits for A, and 3 bits for B, C, D and E, giving an average length of
:<math display="block">\frac{2\,\text{bits}\cdot(15) + 3\,\text{bits} \cdot (7+6+6+5)}{39\, \text{symbols}} \approx 2.62\,\text{bits per symbol,}</math>
which is within one bit of the entropy.
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Hence the expected word length satisfies
:<math display="block">\mathbb E L = \sum_{i=1}^n p_il_i \leq \sum_{i=1}^n p_i (-\log_2 p_i + 1) = -\sum_{i=1}^n p_i \log_2 p_i + \sum_{i=1}^n p_i = H(X) + 1.</math>
Here, <math>H(X) = - \textstyle\sum_{i=1}^n p_i \log_2 p_i</math> is the [[Entropy (information theory)|entropy]], and [[Shannon's source coding theorem]] says that any code must have an average length of at least <math>H(X)</math>. Hence we see that the Shannon–Fano code is always within one bit of the optimal expected word length.
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The algorithm 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 coding 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.
Fano's version of Shannon–Fano coding is used in the <
| url = http://www.pkware.com/documents/casestudies/APPNOTE.TXT
| title = <
| access-date = 2008-01-06
| publisher = PKWARE Inc
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This results in lengths of 2 bits for A, B and C and per 3 bits for D and E, giving an average length of
:<math display="block">\frac{2\,\text{bits}\cdot(15+7+6) + 3\,\text{bits} \cdot (6+5)}{39\, \text{symbols}} \approx 2.28\,\text{bits per symbol.}</math>
We see that Fano's method, with an average length of 2.28, has outperformed Shannon's method, with an average length of 2.62.
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{{Main|Huffman coding}}
A few years later, [[David A. Huffman]] (
# Create a leaf node for each symbol and add it to a [[priority queue]], using its frequency of occurrence as the priority.
# While there is more than one node in the queue:
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This results in the lengths of 1 bit for A and per 3 bits for B, C, D and E, giving an average length of
:<math display="block">\frac{1\,\text{bit}\cdot 15 + 3\,\text{bits} \cdot (7+6+6+5)}{39\, \text{symbols}} \approx 2.23\,\text{bits per symbol.}</math>
We see that the Huffman code has outperformed both types of Shannon–Fano code, which had expected lengths of 2.62 and 2.28.
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==References==
* {{cite journal | first = R.M. | last = Fano | title = The transmission of information |
* {{cite journal |
{{Compression methods}}
{{DEFAULTSORT:Shannon-Fano coding}}
▲[[Category:Lossless compression algorithms]]
[[Category:Claude Shannon]]
[[Category:Entropy coding]]
[[Category:Data compression]]
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