Approximation ratio of RePair

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. Here we improve the lower bound for the famous {\sf RePair} algorithm from $\Omega (\sqrt{\log n})$ to $\Omega (\log n/\log \log n)$. The family of words used in our proof is defined over a binary alphabet, while the lower bound from Charikar et al. needs an alphabet of logarithmic size in the length of the provided words.