Approximate pattern matching with k-mismatches in packed text

Abstract

Given strings $P$ of length $m$ and $T$ of length $n$ over an alphabet of size $σ$, the string matching with $k$-mismatches problem is to find the positions of all the substrings in $T$ that are at Hamming distance at most $k$ from $P$. If $T$ can be read only one character at the time the best known bounds are $O(n\sqrt{k\log k})$ and $O(n + n\sqrt{k/w}\log k)$ in the word-RAM model with word length $w$. In the RAM models (including $AC^0$ and word-RAM) it is possible to read up to $\floor{w / \log σ}$ characters in constant time if the characters of $T$ are encoded using $\ceil{\log σ}$ bits. The only solution for $k$-mismatches in packed text works in $O((n \logσ/\log n)\ceil{m \log (k + \log n / \logσ) / w} + n^{\varepsilon})$ time, for any $\varepsilon > 0$. We present an algorithm that runs in time $O(\frac{n}{\floor{w/(m\logσ)}} (1 + \log \min(k,σ) \log m / \logσ))$ in the $AC^0$ model if $m=O(w / \logσ)$ and $T$ is given packed. We also describe a simpler variant that runs in time $O(\frac{n}{\floor{w/(m\logσ)}}\log \min(m, \log w / \logσ))$ in the word-RAM model. The algorithms improve the existing bound for $w = Ω(\log^{1+ε}n)$, for any $ε> 0$. Based on the introduced technique, we present algorithms for several other approximate matching problems.