Introduction

This pearl, and the one following, is all about arithmetic coding, a way of doing data compression. Unlike other methods, arithmetic coding does not represent each individual symbol of the text as an integral number of bits; instead, the text as a whole is encoded as a binary fraction in the unit interval. Although the idea can be traced back much earlier, it was not until the publication of an “accessible implementation” by Witten, Neal and Cleary in 1987 that arithmetic coding became a serious competitor in the world of data compression. Over the past two decades the method has been refined and its advantages and disadvantages over rival schemes have been elucidated. Arithmetic coding can be more effective at compression than rivals such as Huffman coding, or Shannon–Fano coding, and is well suited to take account of the statistical properties of the symbols in a text. On the other hand, coding and decoding times are longer than with other methods.

Arithmetic coding has a well-deserved reputation for being tricky to implement; nevertheless, our aim in these two pearls is to give a formal development of the basic algorithms. In the present pearl, coding and decoding are implemented in terms of arbitrary-precision rational arithmetic. This implementation is simple and elegant, though expensive in time and space. In the following pearl, coding and decoding are reimplemented in terms of finite-precision integers. This is where most of the subtleties of the problem reside.