Optimal Time Random Access to Grammar-Compressed Strings in Small Space

The random access problem for compressed strings is to build a data structure that efficiently supports accessing the character in position $i$ of a string given in compressed form. Given a grammar of size $n$ compressing a string of size $N$, we present a data structure using $O(n\Delta \log_\Delta \frac N n \log N)$ bits of space that supports accessing position $i$ in $O(\log_\Delta N)$ time for $\Delta \leq \log^{O(1)} N$. The query time is optimal for polynomially compressible strings, i.e., when $n=O(N^{1-\epsilon})$.