We consider compact representations of collections of similar strings that support random access queries. The collection of strings is given by a rooted tree where edges are labeled by an edit operation (inserting, deleting, or replacing a character) and a node represents the string obtained by applying the sequence of edit operations on the path from the root to the node. The goal is to compactly represent the entire collection while supporting fast random access to any part of a string in the collection. This problem captures natural scenarios such as representing the past history of a edited document or representing highly-repetitive collections. Given a tree with $n$ nodes, we show how to represent the corresponding collection in $O(n)$ space and optimal $O(\log n/ \log \log n)$ query time. This improves the previous time-space trade-offs for the problem. To obtain our results, we introduce new techniques and ideas, including a reduction to a new geometric line segment selection together with an efficient solution.

中文翻译：

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持久字符串中的随机访问

我们考虑支持随机访问查询的相似字符串集合的紧凑表示。字符串集合由有根树给出，其中边由编辑操作（插入、删除或替换字符）标记，节点表示通过在从根到节点的路径上应用编辑操作序列获得的字符串。节点。目标是紧凑地表示整个集合，同时支持对集合中字符串的任何部分的快速随机访问。此问题捕获自然场景，例如表示已编辑文档的过去历史或表示高度重复的集合。给定具有 $n$ 个节点的树，我们展示了如何在 $O(n)$ 空间和最优 $O(\log n/ \log \log n)$ 查询时间中表示相应的集合。这改进了之前针对该问题的时空权衡。为了获得我们的结果，我们引入了新的技术和想法，包括减少新的几何线段选择以及有效的解决方案。