Simple Worst-Case Optimal Adaptive Prefix-Free Coding

Suppose we want to store a string
$S[1..n]$
over an alphabet of size
$\sigma$
using adaptive prefix-free coding with fast encoding and decoding. If we are not too concerned about compression, we can process
$/\mathrm{S}$
in blocks of
$\sigma$
characters as follows: we encode or decode the first block with a Shannon code for the uniform distribution; to encode or decode the
$ith$
block, for
$i > 1$
, we build a Shannon code for the average of the distribution of characters we have seen so far and of the uniform distribution; since this averaged distribution assigns each character probability at least
$1/(2\sigma)$
(even when we have not seen it), the code-tree has height at most
$[\text{lg}(2\sigma)] < \text{lg}\sigma+2$
and can be represented with an
$O(\sigma)$
-space table that takes
$O(\sigma)$
time to build and allows constant-time encoding and decoding of characters - so we process the block in constant time per character. We store the first
$\sigma$
copies of each distinct character using fewer than
$\text{lg}\sigma+2$
bits each and, for
$j > \sigma$
, we store the
$j\text{th}$
copy
$S[k]$
using at most
$\left[\text{lg}(\frac{1}{2}(\frac{j-\sigma+1}{\sigma\lfloor(k-1)/\sigma\rfloor}+\frac{1}{\sigma}))^{-1}\right]< \text{lg}\frac{k}{j-\sigma+1}+2$
bits. Fairly straight forward calculation shows that when
$\sigma\ll n$
, we store at most
$n(H+2)+o(n)$
bits overall, where
$H$
is the entropy of the distribution of characters in
$S$
.