Disjoint-Set forests, consisting of Union-Find trees, are data structures having a widespread practical application due to their efficiency. Despite them being well-known, no exact structural characterization of these trees is known (such a characterization exists for Union trees which are constructed without using path compression) for the case assuming union-by-rank strategy for merging. In this paper we provide such a characterization by means of a simple PUSH operation and show that the decision problems whether a (ranked or unranked) tree is a Union-Find tree are NP-complete, complementing our earlier similar result for the union-by-size strategy.

中文翻译：

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识别联合查找树是 NP 完全的，即使没有排名信息

由联合查找树组成的不相交集森林是一种数据结构，由于其效率而具有广泛的实际应用。尽管它们是众所周知的，但对于假设联合按等级策略进行合并的情况，这些树的确切结构特征是已知的（这种特征存在于不使用路径压缩构建的联合树中）。在本文中，我们通过简单的 PUSH 操作提供了这样的表征，并表明（排序或未排序）树是否为 Union-Find 树的决策问题是 NP 完全的，补充了我们之前对 union-by 的类似结果-规模战略。