Low-Memory Adaptive Prefix Coding

In this paper we study the adaptive prefix coding problem in  cases where the size of the input alphabet is large.  We present an online prefix coding algorithm that uses  $O(\sigma^{1 / \lambda + \epsilon}) $ bits of space for any constants $\eps≫0$, $\lambda≫1$, and encodes the string of symbols in  $O(\log \log \sigma)$ time per symbol  \emph{in the worst case}, where $\sigma$ is  the size of the alphabet.  The upper bound on the encoding length is  $\lambda n H (s) +(\lambda / \ln 2 + 2 + \epsilon) n +  O (\sigma^{1 / \lambda} \log^2 \sigma)$ bits.