具有$k$不匹配的长LCF的线性时间算法
Linear-Time Algorithm for Long LCF with $k$ Mismatches
摘要：
在具有$k$不匹配的最长公因子（LCF $_k$）问题中，给定两个字符串$X$和$Y$，它们的总长度为$n$，我们被要求找到一对最大长度因子，其中一个是$X$，另一个是$Y$，使得它们的汉明距离至多为$k$ . Thankachan等人表明，对于恒定的$k$，该问题可以在$\mathcal{O}(n \log^k n)$时间和$\mathcal{O}(n)$空间中解决。我们考虑LCF $_k$（$\ell$）问题，其中我们假设所寻求的因子的长度至少为$\ell$，以及$\ell=Ω(\log^{2k+2} n)$的LCF $_k$（$\ell$）问题，我们称之为长LCF $_k$问题。我们使用差异覆盖减少长LCF $_k$问题的任务，涉及$m=\mathcal{O}(n/\log^{k+1}n)$同步因素。后者可以在$\mathcal{O}(m \log^{k+1}m)$时间内求解，这导致长LCF $_k$的线性时间算法。一般来说，我们对任意$\ell$的LCF $_k$（$\ell$）的解决方案需要$\mathcal{O}(n + n \log^{k+1} n/\sqrt{\ell})$时间。
英文摘要：
In the Longest Common Factor with $k$ Mismatches (LCF$_k$) problem, we are given two strings $X$ and $Y$ of total length $n$, and we are asked to find a pair of maximal-length factors, one of $X$ and the other of $Y$, such that their Hamming distance is at most $k$. Thankachan et al. show that this problem can be solved in $\mathcal{O}(n \log^k n)$ time and $\mathcal{O}(n)$ space for constant $k$. We consider the LCF$_k$($\ell$) problem in which we assume that the sought factors have length at least $\ell$, and the LCF$_k$($\ell$) problem for $\ell=Ω(\log^{2k+2} n)$, which we call the Long LCF$_k$ problem. We use difference covers to reduce the Long LCF$_k$ problem to a task involving $m=\mathcal{O}(n/\log^{k+1}n)$ synchronized factors. The latter can be solved in $\mathcal{O}(m \log^{k+1}m)$ time, which results in a linear-time algorithm for Long LCF$_k$. In general, our solution to LCF$_k$($\ell$) for arbitrary $\ell$ takes $\mathcal{O}(n + n \log^{k+1} n/\sqrt{\ell})$ time.
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