P-adic arithmetic coding

A new incremental algorithm for data compression is presented. For a sequence of input symbols algorithm incrementally constructs a p-adic integer number as an output. Decoding process starts with less significant part of a p-adic integer and incrementally reconstructs a sequence of input symbols. Algorithm is based on certain features of p-adic numbers and p-adic norm. p-adic coding algorithm may be considered as of generalization a popular compression technique – arithmetic coding algorithms. It is shown that for p = 2 the algorithm works as integer variant of arithmetic coding; for a special class of models it gives exactly the same codes as Huffman’s algorithm, for another special model and a specific alphabet it gives Golomb-Rice codes. “For more than forty years I've been speaking in prose without even knowing it!” Moliere. Le Bourgeois Gentilhomme. Introduction Arithmetic coding algorithm in its modern version was published in Communications of ACM in June 1987 [Witten], but the authors, Ian Witten, Radford Neal and John Cleary, referred to [Abrahamson] as to “the first reference to what was to become the method of arithmetic coding”. So we may say that it is known “for more than forty years”. The algorithm now is a common knowledge – it was published in numerous textbooks (see for example [Salomon, Sayood]), some reviews were published [Bodden, Said], Dr. Dobb’s Journal popularized it [Nelson], wiki [wiki] contains an article about it, a lot of sources could be found on web... So why one more paper on this subject and what is this “p-adic arithmetic”? Let go back to the original idea of arithmetic coding. In arithmetic coding a message is represented as a subinterval [b, e) of union semi interval [0, 1). (We will give all definitions later) When a new symbol s comes a new subinterval [b(s), e(s)) of [b, e) is constructed. Common method to calculate a new subinterval is to divide a current interval into |A| (A is an alphabet, |A|number of symbols) subintervals, each subinterval represents a symbol from A and has length proportional to probability of this symbol. For a new symbol s corresponding subinterval [b(s), e(s)) will be return by encoder. Thus encoding is a process of narrowing intervals (we will call them message intervals) starting from the union interval: [0, 1) ≡ [b0, e0), [b1, e1), [b2, e2), ... , [bt, et) where 0 = b0, ≤ b1 ≤ b2 ≤ ... ≤ bt 1 = e0, ≥ e 1 ≥ e 2 ≥ ... ≥ e t All bi and ei are real numbers. A last constructed subinterval may be used as a final output, or any point x from last subinterval and message length. But usually a special symbol EOM (End Of Message), which does not belong to the alphabet, is used as termination symbol of a message. In this case only a point x can be used as coding result. Decoding is also a process on narrowing intervals. It starts with union interval and a point x inside it. Decoder finds a symbol by dividing current intervals into |A| subintervals and finds the one that contains point x, say [b1, e1). Corresponding to this interval symbol s1 is pushed into an output buffer; [b1, e1) is used as a new current interval. And so on until EOM symbol is received.