In a seminal paper, Charikar et al. 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 $\Theta ((\text {n}/\log \text {n})^{2/3})$ , whereas the approximation ratio of BISECTION is $\Theta (\sqrt {\text {n}/\log \text {n}})$ . In addition, the lower bound for RePair is improved from $\Omega (\sqrt {\log \text {n}})$ to $\Omega (\log \text {n}/\log \log \text {n})$ . Finally, results of Arpe and Reischuk relating grammar-based compression for arbitrary alphabets and binary alphabets are improved.

中文翻译：

---

重温最小的语法问题

在一篇开创性的论文中，Charikar 等。导出几个基于语法的压缩器的近似比率的上限和下限，但在所有情况下，下限和上限之间都存在差距。这里的差距为LZ78 和 平分 通过显示近似比率来关闭 LZ78 是 $\Theta ((\text {n}/\log \text {n})^{2/3})$ ，而近似比为 平分 是 $\Theta (\sqrt {\text {n}/\log \text {n}})$ . 此外，下限为修理 改进自 $\Omega (\sqrt {\log \text {n}})$ 至 $\Omega (\log \text {n}/\log \log \text {n})$ . 最后，改进了 Arpe 和 Reischuk 对任意字母和二进制字母的基于语法的压缩的结果。