In a seminal paper of Charikar et al. (IEEE Transactions on Information Theory, 51( 7): 2005) 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 the gaps for LZ78 and BISECTION are closed by showing that the approximation ratio of LZ78 is Θ((n/ log n) 2/3 ), whereas the approximation ratio of BISECTION is Θ( n/ log n). In addition, the lower bound for RePair is improved from Ω( √ log n) to Ω(log n/ log log n). Finally, results of Arpe and Reischuk relating grammar-based compression for arbitrary alphabets and binary alphabets are improved.
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