（块）图中的字符串匹配：基于游走长度的完整分类
String Matching in (Block) Graphs: A Full Classification by Walk Length
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中文总结 AI 辅助
该文研究节点带字符串标签的图中的字符串匹配问题，针对游走长度限制的 $b$-SMBG 问题，给出不同 $b$ 取值下的复杂度分类与对应算法，完善了相关问题的复杂度理论。
中文摘要 AI 辅助
我们研究节点带有字符串标签的有向图，这类图中的一条游走自然对应于所访问节点标签的拼接，在生物信息学中被广泛用于紧凑描述大量高度相似的基因组集合。给定这样的图 $G=(V,E)$ 和长度为 $m$ 的模式，我们寻找一条其对应字符串包含该模式出现的游走，将此问题称为 SMLG 问题。Amir 等人（《J. Algorithms》，2000）证明 SMLG 可在 $\boldsymbol{O}(m|E| + N)$ 时间内求解，其中 $N$ 是所有节点标签的总长度；Equi 等人（《ACM Trans. Algorithms》，2023）证明该时间复杂度在 SETH 假设下是本质最优的。现有下界假设所求游走长度为 $\boldsymbol{\theta}(|V|)$，因此我们可通过将游走长度限制为 $b-1$ 来绕开该下界，这自然对应于输入节点被划分为 $b$ 个块的有向图，我们需要寻找一条从第一个块出发、到最后一个块结束的游走，将此问题称为 $b$-SMBG 问题。我们提供了更精细的分类，本质上解决了以 $b$ 为参数的 $b$-SMBG 的复杂度：（1）给出 $b=3$ 时的近线性时间算法；（2）证明对于任意 $b\boldsymbol{\ue451}4$，不存在组合算法能优于现有 $\boldsymbol{O}(m|E| + N)$ 复杂度；（3）提出基于快速矩阵乘法的算法，可在 $b \boldsymbol{\ue451}\boldsymbol{O}(1)$ 时实现改进，且该改进在条件意义下是最优的；（4）最后证明，在 SETH 假设下，对于任意 $b \boldsymbol{\ue451}\boldsymbol{\ue451}(\boldsymbol{\ue451}|V|)$，不存在算法能优于现有复杂度。
英文摘要
We consider directed graphs in which the nodes are labeled with strings. A walk in such a graph naturally corresponds to the concatenation of the visited nodes' labels. These graphs are widely used in bioinformatics to compactly describe large collections of highly similar genomes. Given such a graph $G=(V,E)$ and a pattern of length $m$, we seek a walk whose corresponding string has an occurrence of the pattern. We call this the SMLG problem. Amir et al. [J. Algorithms, 2000] showed that SMLG can be solved in $\mathcal{O}(m|E| + N)$ time, where $N$ is the total length of all node labels. Equi et al. [ACM Trans. Algorithms, 2023] showed that this is essentially optimal (under SETH). The existing lower bound assumes that the sought walk is of length $Θ(|V|)$. Thus, we might be able to bypass this lower bound by restricting the walk length to $b-1$, which naturally reduces to having as input a directed graph whose set of nodes is partitioned into $b$ blocks. Then, we seek a walk in this graph that starts in the first block and ends in the last block. We call this the $b$-SMBG problem. We provide a more fine-grained classification that essentially settles the complexity of $b$-SMBG parameterized by $b$: (1) We give a near-linear-time algorithm for $b=3$. (2) We show that there is no combinatorial algorithm improving over the state-of-the-art $\mathcal{O}(m|E| + N)$ bound for any $b\ge 4$. (3) We also present a fast matrix multiplication-based algorithm yielding an improvement for $b \in \mathcal{O}(1)$, which is conditionally optimal. (4) Finally, we show that under SETH, for any $b \in ω(\log |V|)$, no algorithm can improve over the state of the art.
