Application of Lempel-Ziv factorization to the approximation of grammar-based compression

We introduce new type of context-free grammars, A VL-grammars, and show their applicability to grammar-based compression. Using this type of grammars we present 0(n log \Sigma\) time and 0(log n)-ratio approximation of minimal grammar-based compression of a given string of length it over an alphabet Sigma and 0(k log n) time transformation of LZ77 encoding of size k into a grammar-based encoding of size 0(k log n). A preliminary version of this paper has been presented in Rytter (Combinatorial Pattern Matching, Lecture Notes in Computer Science, vol. 2373, Springer, Berlin, June 2000, pp. 20-31), independently of Charikar et al. (STOC, 2002), where grammar-based approximation has been attacked with different construction and a more complicated type of grammars (alpha-balanced grammars for alpha less than or equal to 1 - 1/2 root2). The AVL-grammar is a very natural and simple tool for grammar based compression, it is a straightforward extension of the classical AVL-tree. (C) 2002 Elsevier Science B.V. All rights reserved.