Compressing big data: When the rate of convergence to the entropy matters

It is well known from a theoretical point of view that LZ78 have an asymptotic convergence to the entropy faster than LZ77. A faster rate of convergence to the theoretical compression limit should lead to a better compression ratio. In effect, early LZ78-like and LZ77-like compressors behave accordingly to the theory. On the contrary, it seems that most of the recent commercial LZ77-like compressors perform better than the other ones. Probably this is due to a strategy of optimal parsing, which is used to factorize the text and can be applied to both LZ77 and LZ78 cases, as recent results suggest. To our best knowledge there are no theoretical results concerning the rate of convergence to the entropy of both LZ77-like and LZ78-like case when a strategy of optimal parsing is used. In this paper we investigate how an optimal parsing affect the rate of convergence to the entropy of LZ78-like compressors. We discuss some experimental results on LZ78-like compressors and we consider the ratio between the speed of convergence to the entropy of a compressor with optimal parsing and the speed of convergence to the entropy of a classical LZ78-like compressor. This ratio presents a kind of wave effect that become bigger and bigger as the entropy of the memoryless source decreases but it seems always to slowly converge to one. These results suggest that for non-zero entropy sources the optimal parsing does not improve the speed of convergence to the entropy in the case of LZ78-like compressors.