Pattern Matching with Mismatches and Wildcards

In this work, we address the problem of approximate pattern matching with wildcards. Given a pattern $P$ of length $m$ containing $D$ wildcards, a text $T$ of length $n$, and an integer $k$, our objective is to identify all fragments of $T$ within Hamming distance $k$ from $P$. Our primary contribution is an algorithm with runtime $O(n+(D+k)(G+k)\cdot n/m)$ for this problem. Here, $G \le D$ represents the number of maximal wildcard fragments in $P$. We derive this algorithm by elaborating in a non-trivial way on the ideas presented by [Charalampopoulos et al., FOCS'20] for pattern matching with mismatches (without wildcards). Our algorithm improves over the state of the art when $D$, $G$, and $k$ are small relative to $n$. For instance, if $m = n/2$, $k=G=n^{2/5}$, and $D=n^{3/5}$, our algorithm operates in $O(n)$ time, surpassing the $Ω(n^{6/5})$ time requirement of all previously known algorithms. In the case of exact pattern matching with wildcards ($k=0$), we present a much simpler algorithm with runtime $O(n+DG\cdot n/m)$ that clearly illustrates our main technical innovation: the utilisation of positions of $P$ that do not belong to any fragment of $P$ with a density of wildcards much larger than $D/m$ as anchors for the sought (approximate) occurrences. Notably, our algorithm outperforms the best-known $O(n\log m)$-time FFT-based algorithms of [Cole and Hariharan, STOC'02] and [Clifford and Clifford, IPL'04] if $DG = o(m\log m)$. We complement our algorithmic results with a structural characterization of the $k$-mismatch occurrences of $P$. We demonstrate that in a text of length $O(m)$, these occurrences can be partitioned into $O((D+k)(G+k))$ arithmetic progressions. Additionally, we construct an infinite family of examples with $Ω((D+k)k)$ arithmetic progressions of occurrences, leveraging a combinatorial result on progression-free sets [Elkin, SODA'10].