Performance analysis of grammar-based codes revisited

The compression performance of grammar-based codes is revisited from a new perspective. Previously, the compression performance of grammar-based codes was evaluated against that of the best arithmetic coding algorithm with finite contexts. In this correspondence, we first define semifinite-state sources and finite-order semi-Markov sources. Based on the definitions of semifinite-state sources and finite-order semi-Markov sources, and the idea of run-length encoding (RLE), we then extend traditional RLE algorithms to context-based RLE algorithms: RLE algorithms with k contexts and RLE algorithms of order k, where k is a nonnegative integer. For each individual sequence x, let r^*_sr,k(x) and r^*_sr|k(x) be the best compression rate given by RLE algorithms with k contexts and by RLE algorithms of order k, respectively. It is proved that for any x, r^*_sr,k is no greater than the best compression rate among all arithmetic coding algorithms with k contexts. Furthermore, it is shown that there exist stationary, ergodic semi-Markov sources for which the best RLE algorithms without any context outperform the best arithmetic coding algorithms with any finite number of contexts. Finally, we show that the worst case redundancies of grammar-based codes against r^*_sr,k(x) and r^*_sr|k(x) among all length- n individual sequences x from a finite alphabet are upper-bounded by d₁loglogn/logn and d₂loglogn/logn, respectively, where d₁ and d₂ are constants. This redundancy result is stronger than all previous corresponding results.