Optimal Rank and Select Queries on Dictionary-Compressed Text

We study the problem of supporting queries on a string $S$ of length $n$ within a space bounded by the size $\gamma$ of a string attractor for $S$. Recent works showed that random access on $S$ can be supported in optimal $O(\log(n/\gamma)/\log\log n)$ time within $O\left (\gamma\ \rm{polylog}\ n \right)$ space. In this paper, we extend this result to \emph{rank} and \emph{select} queries and provide lower bounds matching our upper bounds on alphabets of polylogarithmic size. Our solutions are given in the form of a space-time trade-off that is more general than the one previously known for grammars and that improves existing bounds on LZ77-compressed text by a $\log\log n$ time-factor in \emph{select} queries. We also provide matching lower and upper bounds for \emph{partial sum} and \emph{predecessor} queries within attractor-bounded space, and extend our lower bounds to encompass navigation of dictionary-compressed tree representations.