Quasi-distinct parsing and optimal compression methods

In this paper, the optimality proof of Lempel-Ziv coding is re-studied, and a much more general compression optimality theorem is derived. In particular, the property of quasi-distinct parsing is defined. This property is much weaker than distinct parsing required in the original proof, yet we show that the theorem holds with this weaker property as well. This provides a better understanding of the optimality proof of Lempel-Ziv coding, together with a new tool for proving optimality of other compression schemes. To demonstrate the possible use of this generalization, a new coding method - the APT coding - is presented. This new coding method is based on a principle that is very different from Lempel-Ziv's coding. Moreover, it does not directly define any parsing technique. Nevertheless, APT coding is analyzed in this paper and using the generalized theorem shown to be asymptotically optimal up to a constant factor, if APT quasi-distinctness hypothesis holds. An empirical evidence that this hypothesis holds is also given.