Improved redundancy of a version of the Lempel-Ziv algorithm

The fixed-database Lempel-Ziv algorithm (FDLZ) closely resembles practical versions of the LZ algorithm that are widely in use. Bender and Wolf (1991) suggested a variant of LZ which empirically appears to perform well. We suggest a finite memory version of their scheme, and show that it has redundancy ρ_n=O(1/log n) where n is the memory size. We are concerned with a data source which is a stationary, finite-memory random sequence that takes values in an alphabet of finite size A. The data source can be losslessly encoded using (H+ρ_n ) bits per source symbol, where n is a measure of the complexity of the code, and ρ_n→0, as n→∞. The LZ algorithm is a universal procedure (which does not depend on the source statistics) for encoding the source at a rate close to the entropy