Polylog Space Compression, Pushdown Compression, and Lempel-Ziv Are Incomparable

The pressing need for efficient compression schemes for XML documents has recently been focused on stack computation (Hariharan, S., & Shankar, P. in: Proceedings of the 2006 IEEE data compression conference, p. 453, 2006; League, C., & Eng, K. in: Proceedings of the 2007 IEEE data compression conference, pp. 272-282, 2007), and in particular calls for a formulation of information-lossless stack or pushdown compressors that allows a formal analysis of their performance and a more ambitious use of the stack in XML compression, where so far it is mainly connected to parsing mechanisms. In this paper we introduce the model of pushdown compressor, based on pushdown transducers that compute a single injective function while keeping the widest generality regarding stack computation. We also consider online compression algorithms that use at most polylogarithmic space (plogon). These algorithms correspond to compressors in the data stream model. We compare the performance of these two families of compressors with each other and with the general purpose Lempel-Ziv algorithm. This comparison is made without any a priori assumption on the data's source and considering the asymptotic compression ratio for infinite sequences. We prove that in all cases they are incomparable.