# Grammar-based code

(Redirected from Grammar-based codes)

Grammar-based codes or Grammar-based compression are compression algorithms based on the idea of constructing a context-free grammar (CFG) for the string to be compressed. Examples include universal lossless data compression algorithms [1] and SEQUITUR, among others. To compress a data sequence $x = x_1 \cdots x_n$, a grammar-based code transforms $x$ into a context-free CFG $G$. The problem to find a smallest grammar for an input sequence is known to be NP-hard,[2] so many grammar-transform algorithms are proposed from theoretical and practical viewpoints. Generally, the produced grammar $G$ is further compressed by statistical encoders like arithmetic coding.

## Examples and characteristics

The class of grammar-based codes is very broad. It includes block codes, variations of the incremental parsing Lempel-Ziv code,[3] the multilevel pattern matching (MPM) algorithm,[4] and many other new universal lossless compression algorithms. Grammar-based codes are universal in the sense that they can achieve asymptotically the entropy rate of any stationary, ergodic source with a finite alphabet.

## Practical algorithms

The compression programs of following are available from external links.

• Sequitur[5] is a classical grammar compression algorithm that sequentially translates an input text into a CFG, and then the produced CFG is encoded by an arithmetic coder.
• Re-Pair[6] is a greedy algorithm by the strategy of most-frequent-first substitution. The compressive performance is powerful, although the main memory space is very large.