16 June 2026

One thought that i should have capitalised on in the "Linear-time KL-optimal frequency normalisation" paper:

The core of Huffman's algorithm is a KL-optimal frequency normaliser from any distribution to the dyadic distribution. once any PMF is turned dyadic ( $1/2^{-x}$ ), lengths are obvious and thus codes are also obvious via canonical coding.

My paper is a KL-optimal frequency normaliser from (after minor adjustments) any distribution to any distribution. So in principle we can turn the result from my paper into an optimal prefix code as follows:

ask the O(n) algorithm for quantisation to $1/2^{-x_i}$
$-\log_2 x_i$ are integer; computable via __builtin_clz. standard canonical code algorithm provides an onto-mapping from lengths to codes.

so the paper is in fact the unifying theory of huffman ( $1/2^{-x_i}$ ), ANS ( $x_i/2^N$ ) and arithmetic coding ( $x_i/\sum x_i$ ).