The weighted ancestor problem is a well-known generalization of the predecessor problem to trees. It is known that it requires $\Omega(\log\log n)$ time for queries provided $O(n\mathop{\mathrm{polylog}} n)$ space is available and weights are from $[0..n]$, where $n$ is the number of tree nodes. However, when applied to suffix trees, the problem, surprisingly, admits an $O(n)$-space solution with constant query time as was shown by Gawrychowski, Lewenstein, and Nicholson. This variant of the problem can be reformulated as follows: given the suffix tree of a string $s$, we need a data structure that can locate in the tree any substring $s[p..q]$ of $s$ in $O(1)$ time (as if one descended from the root reading $s[p..q]$ along the way). Unfortunately, the data structure of Gawrychowski et al. has no efficient construction algorithm, which apparently prevents its wide usage. In this paper we resolve this issue describing a data structure for weighted ancestors in suffix trees with constant query time and a linear construction algorithm. Our solution is based on a novel approach using so-called irreducible LCP values.

中文翻译：

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重新审视后缀树中的加权祖先

加权祖先问题是前身问题到树的众所周知的概括。众所周知，如果$ O（n \ mathop {\ mathrm {polylog}} n）$空间可用并且权重从$ [0..n起，这需要$ \ Omega（\ log \ log n）$来进行查询。 ] $，其中$ n $是树节点的数量。但是，当应用于后缀树时，该问题出人意料地接受了具有不变查询时间的$ O（n）$-空间解，如Gawrychowski，Lewenstein和Nicholson所示。此问题的变体可以重新构造为：给定字符串$ s $的后缀树，我们需要一个数据结构，该数据结构可以在树中找到$ s $中$ s $的任何子字符串$ s [p..q] $ O（1）$时间（好像一个从根向下沿读$ s [p..q] $）。不幸的是，Gawrychowski等人的数据结构。没有有效的构造算法，显然阻止了它的广泛使用。在本文中，我们解决了这个问题，该问题描述了具有恒定查询时间和线性构造算法的后缀树中加权祖先的数据结构。我们的解决方案基于一种使用不可约LCP值的新颖方法。