Recognizing Union-Find Trees is NP-Complete, Even Without Rank Info

Abstract

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.