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Straight-line grammar (with start symbol ß) for the second sentence of the United States Declaration of Independence. Each blue character denotes a nonterminal symbol; they were obtained from a gzip-compression of the sentence.

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] To compress a data sequence $x=x_{1}\cdots x_{n}$ , a grammar-based code transforms $x$ into a context-free grammar $G$ . The problem of finding a smallest grammar for an input sequence (smallest grammar problem) 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 the 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 using the strategy of most-frequent-first substitution. The compressive performance is powerful, although the main memory space requirement is very large.
GLZA,^[7] which constructs a grammar that may be reducible, i.e., contain repeats, where the entropy-coding cost of "spelling out" the repeats is less than the cost creating and entropy-coding a rule to capture them. (In general, the compression-optimal SLG is not irreducible, and the Smallest Grammar Problem is different from the actual SLG compression problem.)

References

^ Kieffer, J. C.; Yang, E.-H. (2000), "Grammar-based codes: A new class of universal lossless source codes", IEEE Trans. Inf. Theory, 46 (3): 737–754, doi:10.1109/18.841160
^ Charikar, M.; Lehman, E.; Liu, D.; Panigrahy, R.; Prabharakan, M.; Sahai, A.; Shelat, A. (2005), "The Smallest Grammar Problem", IEEE Trans. Inf. Theory, 51 (7): 2554–2576, doi:10.1109/tit.2005.850116
^ Kieffer, J. C.; Yang, E.-H.; Nelson, G.; Cosman, P. (2000), "Universal lossless compression via multilevel pattern matching", IEEE Trans. Inf. Theory, 46 (4): 1227–1245, doi:10.1109/18.850665
^ Ziv, J.; Lempel, A. (1978), "Compression of individual sequences via variable rate coding", IEEE Trans. Inf. Theory, 24 (5): 530–536, doi:10.1109/TIT.1978.1055934, hdl:10338.dmlcz/142945
^ 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, arXiv:cs/9709102, doi:10.1613/jair.374, hdl:10289/1186
^ Larsson, N. J.; Moffat, A. (2000), "Offline Dictionary-Based Compression" (PDF), Proceedings of the IEEE, 88 (11): 1722–1732, doi:10.1109/5.892708
^ Conrad, Kennon J.; Wilson, Paul R. (2016), "Grammatical Ziv-Lempel Compression: Achieving PPM-Class Text Compression Ratios with LZ-Class Decompression Speed", IEEE Data Compression Conference: 586, doi:10.1109/DCC.2016.119, ISBN 978-1-5090-1853-6

External links

GLZA discussion and paper
Description of grammar-based codes with example
Sequitur codes
Re-Pair codes
Re-Pair codes a version of Gonzalo Navarro.
GrammarViz 2.0 - implementation of Sequitur, Re-Pair, and parallel Re-Pair in Java.

[1] Kieffer, J. C.; Yang, E.-H. (2000), "Grammar-based codes: A new class of universal lossless source codes", IEEE Trans. Inf. Theory, 46 (3): 737–754, doi:10.1109/18.841160

[2] Charikar, M.; Lehman, E.; Liu, D.; Panigrahy, R.; Prabharakan, M.; Sahai, A.; Shelat, A. (2005), "The Smallest Grammar Problem", IEEE Trans. Inf. Theory, 51 (7): 2554–2576, doi:10.1109/tit.2005.850116

[3] Kieffer, J. C.; Yang, E.-H.; Nelson, G.; Cosman, P. (2000), "Universal lossless compression via multilevel pattern matching", IEEE Trans. Inf. 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. Inf. Theory, 24 (5): 530–536, doi:10.1109/TIT.1978.1055934, hdl:10338.dmlcz/142945

[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, arXiv:cs/9709102, doi:10.1613/jair.374, hdl:10289/1186

[6] Larsson, N. J.; Moffat, A. (2000), "Offline Dictionary-Based Compression" (PDF), Proceedings of the IEEE, 88 (11): 1722–1732, doi:10.1109/5.892708

[7] Conrad, Kennon J.; Wilson, Paul R. (2016), "Grammatical Ziv-Lempel Compression: Achieving PPM-Class Text Compression Ratios with LZ-Class Decompression Speed", IEEE Data Compression Conference: 586, doi:10.1109/DCC.2016.119, ISBN 978-1-5090-1853-6

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 == Examples and characteristics ==
-The class of grammar-based codes is very broad. It includes [[block code]]s, variations of the incremental parsing [[LZ77 and LZ78|Lempel-Ziv code]],<ref>{{Citation | last = Kieffer | first = J. C. | last2 = Yang | first2 = E.-H.  | last3 = Nelson | first3 = G. | last4 = Cosman | first4 = P. | title = Universal lossless compression via multilevel pattern matching | journal = IEEE Trans. Inf. Theory | volume = 46 | pages = 1227–1245 | year = 2000 | doi = 10.1109/18.850665 | issue = 4 | url = https://escholarship.org/uc/item/39k54514 }}</ref> the multilevel pattern matching (MPM) algorithm,<ref>{{Citation | last = Ziv | first = J. | last2 = Lempel | first2 = A. | title = Compression of individual sequences via variable rate coding | journal = IEEE Trans. Inf. Theory | volume = 24 | pages = 530–536 | year = 1978 | doi = 10.1109/TIT.1978.1055934 | issue = 5 | hdl = 10338.dmlcz/142945 }}</ref> and many other new universal lossless compression algorithms.
+The class of grammar-based codes is very broad. It includes [[block code]]s, variations of the incremental parsing [[LZ77 and LZ78|Lempel-Ziv code]],<ref>{{Citation | last = Kieffer | first = J. C. | last2 = Yang | first2 = E.-H.  | last3 = Nelson | first3 = G. | last4 = Cosman | first4 = P. | title = Universal lossless compression via multilevel pattern matching | journal = IEEE Trans. Inf. Theory | volume = 46 | pages = 1227–1245 | year = 2000 | doi = 10.1109/18.850665 | issue = 4 | url = https://escholarship.org/uc/item/39k54514 }}</ref> the multilevel pattern matching (MPM) algorithm,<ref>{{Citation | last = Ziv | first = J. | last2 = Lempel | first2 = A. | title = Compression of individual sequences via variable rate coding | journal = IEEE Trans. Inf. Theory | volume = 24 | pages = 530–536 | year = 1978 | doi = 10.1109/TIT.1978.1055934 | issue = 5 | hdl = 10338.dmlcz/142945 | hdl-access = free }}</ref> 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.

Revision as of 11:59, 13 April 2020

Examples and characteristics

Practical algorithms

See also

References

External links