Lempel-Zip Complexity Reference

The aim of this note is to provide some reference facts for LZW---mostly from Thomas and Cover \cite{Cover:2006aa} and provide a reference for some metrics that can be derived from it. LZW is an algorithm to compute a Kolmogorov Complexity estimate derived from a limited programming language that only allows copy and insertion in strings (not Turing complete set). Despite its delightful simplicity, it is rather powerful and fast. We then focus on definitions of LZW derived complexity metrics consistent with the notion of descriptive length, and discuss different normalizations, which result in a set of metrics we call $\rho_0$, $\rho_1$ and $\rho_2$, in addition to the Description Length $l_{LZW}$ and the Entropy Rate.