The Smallest Grammar Problem Revisited

Hideo Bannai, Momoko Hirayama, Danny Hucke, Shunsuke Inenaga, Artur Jez, Markus Lohrey, Carl Philipp Reh

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


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.

Original languageEnglish
Article number9259056
Pages (from-to)317-328
Number of pages12
JournalIEEE Transactions on Information Theory
Issue number1
Publication statusPublished - Jan 2021

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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