The Efficiency of the ANS Entropy Encoding

The Asymmetric Numeral Systems (ANS) is a class of entropy encoders by Duda that had an immense impact on the data compression, substituting arithmetic and Huffman coding. The optimality of ANS was studied by Duda et al. but the precise asymptotic behaviour of its redundancy (in comparison to the entropy) was not completely understood. In this paper we establish an optimal bound on the redundancy for the tabled ANS (tANS), the most popular ANS variant. Given a sequence $a_1,\ldots,a_n$ of letters from an alphabet $\{0,\ldots,\sigma-1\}$ such that each letter $a$ occurs in it $f_a$ times and $n=2^r$, the tANS encoder using Duda's ``precise initialization'' to fill tANS tables transforms this sequence into a bit string of length (frequencies are not included in the encoding size): $$ \sum\limits_{a\in [0..\sigma)}f_a\cdot\log\frac{n}{f_a}+O(\sigma+r), $$ where $O(\sigma + r)$ can be bounded by $\sigma\log e+r$. The $r$-bit term is an encoder artifact indispensable to ANS; the rest incurs a redundancy of $O(\frac{\sigma}{n})$ bits per letter. We complement this bound by a series of examples showing that an $\Omega(\sigma+r)$ redundancy is necessary when $\sigma > n/3$, where $\Omega(\sigma + r)$ is at least $\frac{\sigma-1}{4}+r-2$. We argue that similar examples exist for any methods that distribute letters in tANS tables using only the knowledge about frequencies. Thus, we refute Duda's conjecture that the redundancy is $O(\frac{\sigma}{n^2})$ bits per letter. We also propose a new variant of range ANS (rANS), called rANS with fixed accuracy, that is parameterized by $k \ge 1$. In this variant the integer division, which is unavoidable in rANS, is performed only in cases when its result belongs to $[2^k..2^{k+1})$. Hence, the division can be computed by faster methods provided $k$ is small. We bound the redundancy for the rANS with fixed accuracy $k$ by $\frac{n}{2^k-1}\log e+r$.