We are interested in noiseless (lossless) coding, i.e. in efficient codes that can be used to send data over a communication line that does not introduce noise (cause loss). Such codes can also be used to compress data for storage purposes. Here 'efficiency' refers only to the code itself, not to the amount of time or space required to do the encoding and decoding processes.

source symbols

---------> encoding

(0|1)*

---------> decoding

source symbols

The average code-word length (message length), for events v from S, is

∑_{v in S} {P(v) . |code(v)|}

Decoding

A code is uniquely decodable if every finite string of bits represents at most one sequence of source text. i.e. There is no possibility of two different source texts giving the same coded string.

Synchronizable Codes

A code is comma-free if it is impossible to get one out-of-frame code-word in the middle of an encoded string. In the early days of molecular biology it was briefly suspected that the genetic code (DNA³->amino acids) was comma-free, but this turned out not to be the case. A code has a stagger limit of 'k' if the synchronisation of all windows of length k+1 or greater is apparent without any more context.
B. Neveln, Comma-Free and Synchnronizable Codes, J. Theor. Biol., 144, pp.209-212, 1990, suggests synchronizability, not comma free ness, was the requirement of an early genetic code.

The genetic code is very nearly a Gray code, i.e. an error in the (en | de) coding (DNA) will probably either be silent (code for the same amino acid), or code for a chemically similar amino acid. The code seems to have evolved to minimise the effect of mutations and misreading:
R. Swanson, A Unifying Concept for the Amino Acid Code, Bull. Math. Biol., 46(2), pp.187-203, 1984.

Prefix Property

Another desirable property for a code to have is the prefix property: No code word is a prefix of any other code word. A code having the prefix property is uniquely decodable, but not necessarily v.v..

The prefix property allows us to identify the end of the first (and second etc.) code word in a data stream immediately, and is required to be able to decode a stream of data unambiguously without look-ahead.

While the prefix property seems to be quite strict, some fundamental results show that such codes are all that we need to consider for the purposes of inference.

Example

Recalling the biased four sided dice, P(a)=1/2, P(c)=1/4, P(g)=P(t)=1/8, the following is an efficient code for data generated by the dice:

a:'0'	1?
	c:'10'	11?
		g:'110'	t:'111'
1/2	1/4	1/8	1/8

Codes and Entropy

Suppose we have events with probabilites p₁, p₂ ..., p_N and there is a code for these events with code-word length l₁, l₂ ..., l_N, where ∑_i=1..N{2^-l_i}≤1. The average code-word length will be

∑_i=1..N  p_i l_i

Kraft Inequality

There is a prefix code with code-words of lengths l₁, l₂, ..., l_N iff ∑_i=l..N{2^-l_i} ≤ 1.
- Kraft (1949).

Proof (a) code exists => inequality:

	1	2	3
root	1	11	111
			110
		10	101
			100
	0	01	011
			010
		00	001
			000

Every code-word must appear somewhere in the tree and because of the prefix property no other code-word can be a descendant or an ancestor of it in the tree. There are 2^k bit-strings of length k. A bit-string of length max, has probability 2^-max. The probability of a shorter bit-string is the sum of the probabilities of the max-bit strings below it. The sum of the probabilities of the code-words cannot exceed 1.0.

Proof (b) inequality => code exits.

Sort the lengths so that l₁≤l₂≤ ... ≤l_N.

1. If l_N = 1 then N=1 or N=2 and in the latter case {'0', '1'} will do as our prefix code.

2. Otherwise l_N > 1.
   Let S = ∑_i=1..N{2^-l_i}.
   
   1. If S ≤ 1/2, note that either
      1. l₁=1, and in that case N=1, and {'0'} is a satisfactory code.
      2. l₁≥2
         Now S ≤ 1/2, so 2S = ∑_i=1..N{2^-(l_i-1)} ≤ 1
         We have shrunk the max' length, so by induction there is a code with word lengths l₁-1, ..., l_N-1. Prepend '0' to each code-word to get one with the required word lengths.
   2. Otherwise S > 1/2
      Remember l₁≤l₂≤ ... ≤l_N.
      So 2^-l₁ ≥ 2^-l₂ ≥ ...
      Let m be the smallest value for which S_m = ∑_i=1..m{2^-l_i} ≥ 1/2.
      In fact S_m = 1/2, because the "gap", 1/2-S_k for k<m, must be a multiple of 2^-l_m.
      Now deal with {l₁, ..., l_m} and {l_m+1, ..., l_N} separately, prefixing the code words for the former with '0' and the code words for the latter with '1', ... done.

Theorem

There exists a prefix code with code-words of length l₁, l₂, ..., l_N, iff there is a uniquely decodable code with these lengths

G.F.'s notes give Welsh Codes and Cryptography, OUP, 1988, as a reference.

So it is reasonable to insist on the use of prefix codes because if there is any uniquely decodable code then there is a prefix code.

Shannon (1948)

Noiseless coding theorem for sources without memory:

Let S be a source with entropy H.

1. Any uniquely decodable code for S has average length ≥ H.
2. There exists a uniquely decodable code (in fact a prefix code) for S with average length ≤ H+1.

See G.Farr (1999) p9 for (1).

Proof (2):
Let l_i = ceiling(-log₂(p_i)).
Clearly ∑_i=1..N{2^-l_i} ≤ 1.
By the Kraft inequality there is a prefix code with these code-word lengths.

∑i=1..N{pi li}

≤ ∑i=1..N{pi(-log2(pi) + 1)}

= H+1

Huffman (1952)

Algorithm for the construction of minimum redundancy codes.

Huffman's algorithm starts with a list of probabilities for events, p₁, p₂, .., p_N. It forms a binary tree with these probabilities at leaves of the tree:

1. Part 1:
   1. Find the two smallest values, x, y, in the list.
   2. Replace them by the value x+y associated with a node in the tree fork(x,y).
   3. Repeat until the list contains one entry.
2. Now allocate code-words from the top of the tree downwards, as implied by the example at the top of this web page.

Huffman's algorithm does give an optimal code if one is restricted to transmiting symbols one at a time. It may be possible to do better by grouping symbols together. (Arithmetic coding effectively takes this to the extreme.)

(Huffman's algorithm is bottom-up - in the initial phase. The other strategy that one might think of is "top-down" but this does not lead to efficient codes in general.)

Arithmetic Coding

Arithmetic coding works by dividing the interval [0, 1), i.e. {x | x≥0 & x <1}, into smaller and smaller intervals which represent the successive symbols of an input sequence. It can be thought of as an extension of prefix coding where the interval boundaries lie neatly between adjacent powers of 1/2. e.g. The (trivial) prefix code for {F,T}^* where P(F)=P(T)=1/2:

e.g. TTFT... where P(F)=1/2, P(T)=1/2

1/2 F								1/2 T								1
								1/4 F				1/4 T				2
												1/8 F		1/8 T		3
												1/16 F	1/16 T			4
0								1/2							1

Note that 1/2 = 0.5₁₀ = 0.1₂, 1/4 = 0.25₁₀ = 0.01₂, 1/8 = 0.125₁₀ = 0.001₂, etc.

Langdon (1984) gives an excellent tutorial introduction to arithmetic coding [But beware: fig.3 seems not to correspond to table 3 and the text: 'a' and 'b' seem to have got switched.], crediting Pasco (1976) and independently Martin (1979), and Jones (1981) with discovering a successful FIFO arithmetic code.

In arithmetic coding, probabilities of events need not be neat powers of 1/2. e.g. he gives a 2-symbol example where P(F)=1/8, P(T)=7/8 in position one and P(F)=P(T)=1/2 thereafter. (The probabilities are allowed to vary with position in the sequence, so long as encoder and decoder both know them or can calculate them.)

For input "TTFT..." the successive interval widths would ideally be 7/8 (T), 7/16 (T), 7/32 (F), 7/64 (T). In general, the precision required to state an interval width could increase without limit. Arithmetic coding places a limit on the accuracy required; the price is a small loss in coding efficiency and Langdon (1984) states "less than 4%" for the choices in that paper.

e.g. TTFT... where 1: P(F)=1/8, P(T)=7/8, and then 2, 3, 4: P(F)=P(T)=1/2

1/8 F

7/8 T

P(F)=1/8

1/4 F

5/8 T

P(F)=1/2

1/4 F

3/8 T

P(F)=1/2

1/8 F

1/8 T

P(F)=1/2

0

1/2

1

Note that the second-level partition is 1/4:5/8 not the ideal 7/16:7/16.

Successive intervals [C, C+A), of left-hand end 'C' and width 'A', are encoded by C: "0...", "0...", "0...", "1...", ... . The end of the output is padded out by sufficient zeros.

Decoding

The decoder follows a similar set of actions to the encoder which enables the input source string to be recovered.

Precision

The left-hand end, C, of the current interval is a binary number. The method guarantees that it (and, A, the interval width) can be represented with bounded precision, i.e. of the form 0.000...000{0 | 1}^k, and can be represented by a fixed-point number 1.{0 | 1}^k-1 (and exponent). This removes any problems of numerical accuracy.

It is required that P(F)≤P(T) and P(F) is approximated by the smallest power of 1/2 no larger than it. P(T) is approximated by "what's left" of the current interval. If P(F)>P(T), and note that both the encoder and decoder must know this, they can switch their roles and carry on as before.

Carry-Over

It can happen that when additions are made to C, to move the left-hand end of the interval, a carry can propagate. This requires keeping the output string in a FIFO buffer, potentially of arbitrary length.

The length of the buffer can be limited (e.g. to 16 bits Langdon (1984)) in a technique called bit-stuffing (Langdon & Rissanen 1981) by forcing a '0' out if 16 '1's have been output. The decoder can detect this situation.

Multi-symbol Alphabets

Larger input symbol-sets can be handled either by converting the symbols to binary strings which are then encoded as above, or by more direct extensions of the method to deal with multiple symbols and their individual probabilities.

Entropy

As described, arithmetic coding is very efficient and extensions of it can get arbitrarily close to the entropy of the data. In some sense it has "solved" the coding problem, making it clear that compression = data modelling + coding. The (predicted) probabilities of the events can come from anywhere provided that the encoder and decoder agree on them; they can for example be fixed, or come from an order-k Markov model, or be estimated from statistics gathered from the data previously (en|de)coded, etc.

... probability ... predict ... model ... inference ...

Notes

• S. E. Shannon. A Mathematical Theory of Communication. Bell Syst. Technical Jrnl. 27 pp.379-423, pp.623-656, 1948
• S. E. Shannon & W. Weaver. A Mathematical Theory of Communication. U. of Illinois Press, 1949
• D. A. Huffman. A Method for the Construction of Minimum-Redundancy Codes. Proc. Inst. Radio Eng. 40(9) pp.1098-1101, Sept. 1952
• R. Pasco. Source Coding Algorithms for Fast Data Compression. PhD Thesis, Dept. Elec. Eng., Stanford University, 1976
  Arithmetic coding
• G. N. N. Martin. Range Encoding: an Algorithm for Removing Redundancy from a Digitized Message. Video and Data Recording Conf., Southampton, 1979
  Arithmetic coding
• C. B. Jones. An Efficient Coding System for Long Source Sequences. IEEE Trans. Info Theory. IT-27 pp.280-291, 1981
  Arithmetic coding
• G. G. Langdon jr. & J. Rissanen. A Simple general Binary Source Code. IEEE Trans. Inf. Theory IT-28 pp.800-803, 1982
  Bit stuffing
• G. G. Langdon jr. An Introduction to Arithmetic Coding. IBM J. Res. and Dev. 28(2) pp.135-149, 1984
• I. H. Witten, R. M. Neal & J. G. Cleary. Arithmetic Coding for Data Compression. CACM 30(6) pp.520-540, 1987