使用前向稳定划分对有限状态自动机进行索引
Indexing Finite-State Automata Using Forward-Stable Partitions
AI总结：
本文提出了一类新的偏预序用于有限状态自动机索引，可在多项式时间内计算，其宽度不大于现有方法，且在无限类自动机上能实现线性级别的宽度缩减。
摘要：
有限状态自动机上的索引是一种数据结构，能够定位自动机路径上的特定模式，从而定位自动机本身所接受的正则语言中的模式。Cotumaccio 和 Prezza [SODA '21] 提出了一种能够在自动机上解决模式匹配查询的数据结构，推广了 Ferragina 和 Manzini [FOCS '00] 著名的字符串 FM-index。其索引的效率取决于自动机状态的特定偏序的宽度，偏序的宽度越小，索引的速度越快。然而，计算最小宽度的偏序是 NP-hard 的。Cotumaccio [DCC '22] 缓解了这一问题，他放宽了对偏序的条件，允许其成为偏预序。这种放宽使得存在一个唯一的最小宽度偏预序，且可以在多项式时间内计算。在本文中，我们提出了一类新的偏预序，并证明它们具有以下有用性质：(i) 它们可以在多项式时间内计算，(ii) 它们的宽度永远不会大于 Cotumaccio 的预序的宽度，以及 (iii) 存在无限类自动机，在这些自动机上，Cotumaccio 的预序的宽度线性大于我们预序的宽度。
英文摘要：
An index on a finite-state automaton is a data structure able to locate specific patterns on the automaton's paths and consequently on the regular language accepted by the automaton itself. Cotumaccio and Prezza [SODA '21], introduced a data structure able to solve pattern matching queries on automata, generalizing the famous FM-index for strings of Ferragina and Manzini [FOCS '00]. The efficiency of their index depends on the width of a particular partial order of the automaton's states, the smaller the width of the partial order, the faster is the index. However, computing the partial order of minimal width is NP-hard. This problem was mitigated by Cotumaccio [DCC '22], who relaxed the conditions on the partial order, allowing it to be a partial preorder. This relaxation yields the existence of a unique partial preorder of minimal width that can be computed in polynomial time. In the paper at hand, we present a new class of partial preorders and show that they have the following useful properties: (i) they can be computed in polynomial time, (ii) their width is never larger than the width of Cotumaccio's preorders, and (iii) there exist infinite classes of automata on which the width of Cotumaccio's pre-order is linearly larger than the width of our preorder.
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