AI 中文总结
针对Wheeler NFA确定化为Wheeler DFA的问题，提出在给定Wheeler序下时间复杂度为O(n_A+m_A+n_D+m_D)的算法，比现有最优算法快n_A^2/σ倍，σ=O(1)时达到线性时间，并证明界是紧的。
AI 中文摘要
给定一个Wheeler NFA $\mathcal{A}$，Wheeler确定化问题是构造一个接受与$\mathcal{A}$相同语言的Wheeler DFA $\mathcal{D}$。我们用$n_{\mathcal{A}},m_{\mathcal{A}}$表示$\mathcal{A}$的顶点数和边数，类似地用$n_{\mathcal{D}},m_{\mathcal{D}}$表示$\mathcal{D}$的顶点数和边数。Alanko等人[SODA 2020, Inf. Comp. 2021]表明我们可以在$O(n_{\mathcal{A}}^3)$时间内解决该问题。在本文中，我们展示了在给定$\mathcal{A}$的Wheeler序（可通过Becker等人[ESA 2023]的算法在$O(m_{\mathcal{A}}\log n_{\mathcal{A}})$时间内计算）的情况下，如何将运行时间改进到$O(n_{\mathcal{A}} + m_{\mathcal{A}} + n_{\mathcal{D}} + m_{\mathcal{D}})$。我们的运行时间比现有最优算法快$n_{\mathcal{A}}^2/\sigma$倍，其中$\sigma$是字母表大小。此外，对于$\sigma=O(1)$，我们得到了该问题的第一个线性时间算法。我们通过给出一族输入（其中输出$\mathcal{D}$是最小的且最大尺寸为$\Theta(n\sigma)$）证明了对于$n$和$\sigma$的任何组合，我们的界对于排序输入是紧的。
英文摘要
Given a Wheeler NFA $\mathcal{A}$, the Wheeler determinization problem is to construct a Wheeler DFA $\mathcal{D}$ that accepts the same language as $\mathcal{A}$. We use the notation $n_{\mathcal{A}},m_{\mathcal{A}}$ for the number of vertices and edges of $\mathcal{A}$, and equivalently $n_{\mathcal{D}},m_{\mathcal{D}}$ for $\mathcal{D}$. Alanko et al. [SODA 2020, Inf. Comp. 2021] solve this problem in $O(n_{\mathcal{A}}^3)$ time, by constructing a $\\mathcal{D}$ that always satisfies $n_\mathcal{D}\leq 2n_\mathcal{A} - 1$. In this paper, we show how to improve the running time to $O(n_{\mathcal{A}} + m_{\mathcal{A}} + n_{\mathcal{D}} + m_{\mathcal{D}})$ when the Wheeler order of $\mathcal{A}$ is given. If the Wheeler order is not present, we achieve $O(n_\mathcal{A} + m_\mathcal{A}\log n_\mathcal{A} + n_\mathcal{D} + m_\mathcal{D})$ time by using an algorithm of Becker et al. [ESA 2023]). Our running time is a factor $n_{\mathcal{A}}^2/σ$ faster than the state of the art for sorted inputs, where $σ$ is the size of the alphabet. Furthermore, for $σ=O(1)$ we have the first linear time algorithm for this problem. We show that our bound is tight with any combination of $n_\mathcal{A}$ and $σ$, by giving a family of inputs for which our output $\mathcal{D}$ is minimum, and of maximum size $Θ(n_\mathcal{A}σ)$.
Comments6 pages main body, 1 figure
