The redundancy and distribution of the phrase lengths of the fixed-database Lempel-Ziv algorithm

Abstract

The fixed-database version of the Lempel-Ziv algorithm closely resembles many versions that appear in practice. We ascertain several key asymptotic properties of the algorithm as applied to sources with finite memory. First, we determine that for a dictionary of size n, the algorithm achieves a redundancy /spl rho//sub n/=Hlog log n/log n+0(log log n/log n) where H is the entropy of the process. This is the first, nontrivial, lower bound on any Lempel-Ziv-type compression scheme. We then find the limiting distribution and all moments of the lengths of the phrases by comparing them to a random-walk-like variable with well-known behavior.