Hardness of Frequency-Related Queries on Compressed Strings

Compressed indexing aims to support fundamental string queries in space proportional to compressed input size. For grammar compression, a length-$n$ string $T \in \Sigma^n$ represented by a grammar of size $|G|$ can support random access in $O(|G|\log^{O(1)} n)$ space and $O(\log^{O(1)} n)$ time, and the same bounds are known for many other queries, including pattern matching, longest common extension, lexicographic predecessor/successor, the Burrows-Wheeler transform, suffix arrays, and suffix trees. Frequency-related queries remain less understood. These include rank queries, which report the number of occurrences of a symbol $c \in \Sigma$ in a substring $T(b..e]$, and symbol occurrence queries, which ask whether $c$ occurs in $T(b..e]$. No fully general data structure is known for these queries with $O(|G|\log^{O(1)} n)$ space and $O(\log^{O(1)} n)$ query time. We establish new conditional lower bounds for such problems. First, we show that answering rank and symbol occurrence queries on grammar-compressed texts in polylogarithmic time using an $O(|G|\log^{O(1)} n)$-space structure constructible in $O(|G|\log^{O(1)} n)$ time would imply an $O(n^2\log^{O(1)} n)$-time algorithm for Boolean Matrix Multiplication. The proof uses a more general lower bound for efficiently answering a batch of such queries. Second, we extend the exact lower bounds from straight-line programs to LZ78-compressed strings, a weaker compression model. Third, independently, we show that even additive approximations of rank queries on straight-line grammars would imply faster Boolean Matrix Multiplication algorithms. Finally, assuming the Orthogonal Vectors conjecture, we show that other frequency-related problems, including range distinct counting and range mode frequency, also cannot be efficiently supported in compressed space.