标签树索引的完整视角：新的大小界限，有效的构造及其他
Towards a complete perspective on labeled tree indexing: new size bounds, efficient constructions, and beyond
摘要：
标记树（或trie）是字符串的自然推广，也可以被视为一组字符串的紧凑表示。本文考虑了标记树索引问题，并在空间边界分析、有效构造算法和模式匹配查询等方面提供了一些新的结果。Kakhaju [FOCS 1989]是第一个考虑标记树索引问题的人，他提出了向后trie的后缀树，其中trie中的字符串在叶到根的方向上读取。与向后trie相比，我们称普通trie为向前trie。尽管在Komaju的论文之后有一些后续工作，但索引向前/向后尝试还没有得到很好的理解。在本文中，我们展示了一个完整的角度对索引结构的大小，如后缀树，DAWG，CDAWGs，后缀数组，词缀树，词缀数组的向前和向后尝试。它们中的一些以输入trie的大小$n$占用$O(n)$空间，而另一些在最坏的情况下可以占用$O(n^2)$空间。特别是，我们表明，与$n$节点的正向trie的DAWG的大小是$Ω(σn)$，其中$σ$是在trie中的不同字符的数量。对于大小为$σ= Θ(n)$的字母表，这将成为$Ω(n^2)$。尽管如此，我们表明，有一个紧凑的$O(n)$空间隐式表示的DAWG的前向特里，其空间要求是独立的字母表的大小。该紧凑表示允许在$O(\log σ)$时间上模拟每个DAWG边遍历，并且可以在$O(n)$大小的任何整数字母表上在$O(n)$时间和空间上构造。此外，这容易扩展到第一索引结构，其允许在输入trie树大小的线性空间内的trie树上进行双向模式搜索。
英文摘要：
A labeled tree (or a trie) is a natural generalization of a string, which can also be seen as a compact representation of a set of strings. This paper considers the labeled tree indexing problem, and provides a number of new results on space bound analysis, and on algorithms for efficient construction and pattern matching queries. Kosaraju [FOCS 1989] was the first to consider the labeled tree indexing problem, and he proposed the suffix tree for a backward trie, where the strings in the trie are read in the leaf-to-root direction. In contrast to a backward trie, we call a usual trie as a forward trie. Despite a few follow-up works after Kosaraju's paper, indexing forward/backward tries is not well understood yet. In this paper, we show a full perspective on the sizes of indexing structures such as suffix trees, DAWGs, CDAWGs, suffix arrays, affix trees, affix arrays for forward and backward tries. Some of them take $O(n)$ space in the size $n$ of the input trie, while the others can occupy $O(n^2)$ space in the worst case. In particular, we show that the size of the DAWG for a forward trie with $n$ nodes is $Ω(σn)$, where $σ$ is the number of distinct characters in the trie. This becomes $Ω(n^2)$ for an alphabet of size $σ= Θ(n)$. Still, we show that there is a compact $O(n)$-space implicit representation of the DAWG for a forward trie, whose space requirement is independent of the alphabet size. This compact representation allows for simulating each DAWG edge traversal in $O(\log σ)$ time, and can be constructed in $O(n)$ time and space over any integer alphabet of size $O(n)$. In addition, this readily extends to the first indexing structure that permits bidirectional pattern searches over a trie within linear space in the input trie size.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
