Кодирование диапазонами: эффективный метод энтропийного кодирования.

Range coding (or range encoding) is an entropy coding method defined by G. Nigel N. Martin in a 1979 paper, which effectively rediscovered the FIFO arithmetic code first introduced by Richard Clark Pasco in 1976. Given a stream of symbols and their probabilities, a range coder produces a space efficient stream of bits to represent these symbols and, given the stream and the probabilities, a range decoder reverses the process. Range coding is very similar to arithmetic coding, except that coding is done with digits in any base, instead of with bits, and so it is faster when using larger bases (e. g. a byte) at small cost in compression efficiency. After the expiration of the first (1978) arithmetic coding patent, range coding appeared to clearly be free of patent encumbrances. This particularly drove interest in the technique in the open source community. Since that time, patents on various well known arithmetic coding techniques have also expired.

Range coding conceptually encodes all the symbols of the message into one number, unlike Huffman coding which assigns each symbol a bit pattern and concatenates all the bit patterns together. Thus range coding can achieve greater compression ratios than the one bit per symbol lower bound on Huffman coding and it does not suffer the inefficiencies that Huffman does when dealing with probabilities that are not an exact power of two. The central concept behind range coding is this: given a large enough range of integers, and a probability estimation for the symbols, the initial range can easily be divided into sub ranges whose sizes are proportional to the probability of the symbol they represent. Each symbol of the message can then be encoded in turn, by reducing the current range down to just that sub range which corresponds to the next symbol to be encoded. The decoder must have the same probability estimation the encoder used, which can either be sent in advance, derived from already transferred data or be part of the compressor and decompressor. When all symbols have been encoded, merely identifying the sub range is enough to communicate the entire message (presuming of course that the decoder is somehow notified when it has extracted the entire message). A single integer is actually sufficient to identify the sub range, and it may not even be necessary to transmit the entire integer; if there is a sequence of digits such that every integer beginning with that prefix falls within the sub range, then the prefix alone is all that's needed to identify the sub range and thus transmit the message.

Arithmetic coding is the same as range coding, but with the integers taken as being the numerators of fractions. These fractions have an implicit, common denominator, such that all the fractions fall in the range [0,1). Accordingly, the resulting arithmetic code is interpreted as beginning with an implicit "0". As these are just different interpretations of the same coding methods, and as the resulting arithmetic and range codes are identical, each arithmetic coder is its corresponding range encoder, and vice versa. In other words, arithmetic coding and range coding are just two, slightly different ways of understanding the same thing. In practice, though, so called range encoders tend to be implemented pretty much as described in Martin's paper, while arithmetic coders more generally tend not to be called range encoders. An often noted feature of such range encoders is the tendency to perform renormalization a byte at a time, rather than one bit at a time (as is usually the case). In other words, range encoders tend to use bytes as coding digits, rather than bits. While this does reduce the amount of compression that can be achieved by a very small amount, it is faster than when performing renormalization for each bit.