E. Lehman and A. Shelat. Approximations algorithms for grammar-based compression. In Thirteenth Annual Symposium on Discrete Algorithms (SODA'02), 2002.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....time even if the size of alphabet is unbounded. 1 Introduction The grammar based compression is an optimization problem, given an input string, to nd a small context free grammar which generates the single string. This problem is known to be NP hard and not approximable within a constant factor [9], and due to a relation with an algebraic problem [6] it is unlikely to found an algorithm approximating this problem within O(log n= log log n) The framework of the grammar based compression can uniformly describe the dictionarybased coding schemes which are widely presented for real world ....

....uniformly describe the dictionarybased coding schemes which are widely presented for real world text compression. For example, LZ78 [16] including LZW [13] and BISECTION [5] encodings are considered as algorithms to nd a straight line program, which is a very restricted CFG. Lehman and Shelat [9] also showed the lower bounds of the approximation ratio of almost dictionarybased encodings to the smallest CFG, and unfortunately, these lower bounds are relatively large to O(log n) ratio. The rst polynomial time algorithms which guarantee a small approximation ratio were produced by ....

E. Lehman and A. Shelat. Approximation Algorithms for Grammar-Based Compression. In Proc. 20th Ann. ACM-SIAM Sympo. on Discrete Algorithms, 205-212, 2002.

....like ab in an input string according to the frequency. This Department of Informatics, Kyushu University, Fukuoka 812 8581, Japan encoding scheme is also included in the framework of the grammar based compression, while only the lower bound O( p log n) of its approximation ratio is known [8]. Thus, a nontrivial upper bound of the approximation ratio of Re pair is still an important open problem. Our algorithm is not Re pair in itself but is based on the strategy of the recursive replacement of pairs. Consider a situation that a string contains nonoverlapping intervals X and Y which ....

E. Lehman. Approximation Algorithms for Grammar-Based Compression. PhD thesis, MIT, 2002.

....to be NP complete. The basic novel tool is the AVL grammar. Keywords: LZ compression, minimal grammar, AVL tree, AVL grammar 1 Introduction Text compression based on context free grammars, or equivalently, on straight line programs, has recently attracted much attention, see [1,8,9] and [11,12,15,16]. The grammars give a more structured type of compression. In a grammar based compression a single text w of length n is generated by a context free grammar G. Assume we deal with grammars generating single words. In the paper, using ideas similar to unwinding from [4] and balanced grammars from ....

....single words. In the paper, using ideas similar to unwinding from [4] and balanced grammars from [7] we show a logarithmic relation between LZ factorizations and minimal grammars. Recently, approximation ratios of several grammar based compression have been investigated by Lehman and Shelat in [12]. In this paper we propose a new grammar based compression algorithm based on Lempel Ziv factorization (denoted here by LZ) which is a version of LZ77 encoding [13] For a string w of length n denote by LZ(w) the Lempel Ziv factorization of w. We show: 1. For each string w and its grammar based ....

E. Lehman, A. Shelat, Approximation algorithms for grammar-based compression, SODA 2OO2 2O la. J. Ziv and A. Lempel, A Universal algorithm for sequential data compression, IEEE Transactions on Info,'marion Theory IT-23 (1977), pp. 337-343 20

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E. Lehman and A. Shelat. Approximations algorithms for grammar-based compression. In Thirteenth Annual Symposium on Discrete Algorithms (SODA'02), 2002.

....complexity. The problem is of practical importance in areas such as data compression and pattern extraction. The smallest grammar is known to be hard to approximate to within a constant factor, and an o(log n=log log n) approximation would require progress on a long standing algebraic problem [10]. Previously, the best proved approximation ratio was O(n 1=2 ) for the Bisection algorithm [8] Our main result is an exponential improvement of this ratio; we give an O(log(n=g ) approximation algorithm, where g is the size of the smallest grammar. We then consider other computable ....

....of all rules. The smallest grammar is known to be hard to approximate within a small constant factor. Further, even for a very restricted class of strings, approximating the smallest grammar within a factor of o log n log log n would require progress on a well studied algebraic problem [10]. Previously, the best proven approximation ratio was O(n 1=2 ) for the Bisection algorithm [8] Our main result is an exponential improvement of this ratio. Several rich lines of research connect this elegant problem to elds such as Kolmogorov complexity, data compression, pattern identi ....

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E. Lehman and A. Shelat. Approximations algorithms for grammar-based compression. In Thirteenth Annual Symposium on Discrete Algorithms (SODA'02), 2002.

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E. Lehman and A. Shelat. Approximation algorithms for grammar-based compression. In 13th Symp. on Discrete Algorithms, pages 205-212, 2002.

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E. Lehman and A. Shelat. Approximation algorithms for grammar-based compression. In Proc. 13th Ann. ACM-SIAM Sympo. on Discrete Algorithms, 2002. (to appear).

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E. Lehman and A. Shelat. Approximation algorithms for grammar-based compression. In Proceedings of the 13th ACM-SIAM Symposium on Discrete Algorithms, pages 205-212, 2002.

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