Abstract
In a seminal paper of Charikar et al. on the smallest grammar problem, the authors derive upper and lower bounds on the approximation ratios for several grammar-based compressors, but in all cases there is a gap between the lower and upper bound. Here we close the gaps for LZ78 and BISECTION by showing that the approximation ratio of LZ78 is \(\varTheta ( (n/\log n)^{2/3})\), whereas the approximation ratio of BISECTION is \(\varTheta ( (n/\log n)^{1/2})\). We also derive a lower bound for a smallest grammar for a word in terms of its number of LZ77-factors, which refines existing bounds of Rytter. Finally, we improve results of Arpe and Reischuk relating grammar-based compression for arbitrary alphabets and binary alphabets.
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Notes
- 1.
It is shown in [14] that every SLP in Chomsky normal form for w has at least \(g_{\mathsf {LZ77}}(w)\) many nonterminals. But the number of nonterminals in a smallest Chomsky normal form SLP for w is bounded by g(w).
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The work in this paper was supported by the DFG grant LO 748/10-1.
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Hucke, D., Lohrey, M., Reh, C.P. (2016). The Smallest Grammar Problem Revisited. In: Inenaga, S., Sadakane, K., Sakai, T. (eds) String Processing and Information Retrieval. SPIRE 2016. Lecture Notes in Computer Science(), vol 9954. Springer, Cham. https://doi.org/10.1007/978-3-319-46049-9_4
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