Streaming dictionary matching with mismatches

Abstract: In the $k$-mismatch problem we are given a pattern of length $n$ and a text and must find all locations where the Hamming distance between the pattern and the text is at most $k$. A series of recent breakthroughs have resulted in an ultra-efficient streaming algorithm for this problem that requires only $O(k \log \frac{n}{k})$ space and $$O(\log \frac{n}{k} (\sqrt{k \log k} + \log³ n))$$ time per letter [Clifford, Kociumaka, Porat, SODA 2019]. In this work, we consider a strictly harder problem called dictionary matching with $k$ mismatches. In this problem, we are given a dictionary of $d$ patterns, where the length of each pattern is at most $n$, and must find all substrings of the text that are within Hamming distance $k$ from one of the patterns. We develop a streaming algorithm for this problem with $O(k d \log^k d \mathrm{polylog}(n))$ space and $$O(k \log^{k} d \mathrm{polylog}(n) + |\mathrm{occ}|)$$ time per position of the text. The algorithm is randomised and outputs correct answers with high probability. On the lower bound side, we show that any streaming algorithm for dictionary matching with $k$ mismatches requires $\Omega(k d)$ bits of space.