Computing convolution on grammar-compressed text

The convolution between a text string $S$ of length $N$ and a pattern string $P$ of length $m$ can be computed in $O(N \log m)$ time by FFT. It is known that various types of approximate string matching problems are reducible to convolution. In this paper, we assume that the input text string is given in a compressed form, as a \emph{straight-line program (SLP)}, which is a context free grammar in the Chomsky normal form that derives a single string. Given an SLP $\mathcal{S}$ of size $n$ describing a text $S$ of length $N$, and an uncompressed pattern $P$ of length $m$, we present a simple $O(nm \log m)$-time algorithm to compute the convolution between $S$ and $P$. We then show that this can be improved to $O(\min\{nm, N-\alpha\} \log m)$ time, where $\alpha \geq 0$ is a value that represents the amount of redundancy that the SLP captures with respect to the length-$m$ substrings. The key of the improvement is our new algorithm that computes the convolution between a trie of size $r$ and a pattern string $P$ of length $m$ in $O(r \log m)$ time.