# 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.

## References

1. ^ Kieffer, J. C.; Yang, E.-H. (2000), "Grammar-based codes: A new class of universal lossless source codes", IEEE Trans. Inform. Theory 46 (3): 737–754, doi:10.1109/18.841160
2. ^ Charikar, M.; Lehman, E.; Liu, D.; Panigrapy, R.; Prabharakan, M.; Sahai, A.; Shelat, A. (2005), "The Smallest Grammar Problem", IEEE Trans. Inform. Theory 51 (7): 2554–2576
3. ^ Kieffer, J. C.; Yang, E.-H.; Nelson, G.; Cosman, P. (2000), "Universal lossless compression via multilevel pattern matching", IEEE Trans. Inform. Theory 46 (4): 1227–1245, doi:10.1109/18.850665
4. ^ Ziv, J.; Lempel, A. (1978), "Compression of individual sequences via variable rate coding", IEEE Trans. Inform. Theory 24 (5): 530–536, doi:10.1109/TIT.1978.1055934
5. ^ Nevill-Manning, C. G.; Witten, I. H. (1997), "Identifying Hierarchical Structure in Sequences: A linear-time algorithm", Journal of Artificial Intelligence Research 7 (4): 67–82, hdl:10289/1186
6. ^ Larsson, N. J.; Moffat, A. (2000), "Offline Dictionary-Based Compression", IEEE 88 (11): 1722–1732