Compression Optimality of Asymmetric Numeral Systems

Compression also known as entropy coding has a rich and long history. However, a recent explosion of multimedia Internet applications (such as teleconferencing and video streaming for instance) renews an interest in fast compression that also squeezes out as much redundancy as possible. In 2009 Jarek Duda invented his asymmetric numeral system (ANS). Apart from a beautiful mathematical structure, it is very efficient and offers compression with a very low residual redundancy. ANS works well for any symbol source statistics. Besides, ANS has become a preferred compression algorithm in the IT industry. However, designing ANS instance requires a random selection of its symbol spread function. Consequently, each ANS instance offers compression with a slightly different compression rate. The paper investigates compression optimality of ANS. It shows that ANS is optimal (i.e. the entropies of encoding and source are equal) for any symbol sources whose probability distribution is described by natural powers of 1/2. We use Markov chains to calculate ANS state probabilities. This allows us to determine ANS compression rate precisely. We present two algorithms for finding ANS instances with high compression rates. The first explores state probability approximations in order to choose ANS instances with better compression rates. The second algorithm is a probabilistic one. It finds ANS instances, whose compression rate can be made as close to the best rate as required. This is done at the expense of the number $\theta$ of internal random ``coin'' tosses. The algorithm complexity is ${\cal O}(\theta L^3)$, where $L$ is the number of ANS states. The complexity can be reduced to ${\cal O}(\theta L\log{L})$ if we use a fast matrix inversion. If the algorithm is implemented on quantum computer, its complexity becomes ${\cal O}(\theta (\log{L})^3)$.