Recognizing Union-Find trees is NP-complete

TL;DR

This paper proves that recognizing whether a given tree is a Union-Find tree is an NP-complete problem, highlighting the computational difficulty of structural characterization of these data structures.

Contribution

It provides the first exact structural characterization of Union-Find trees and establishes the NP-completeness of recognizing such trees.

Findings

Structural characterization of Union-Find trees provided

Recognition problem is NP-complete

Highlights computational complexity of structural analysis

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). In this paper we provide such a characterization and show that the decision problem whether a given tree is a Union-Find tree is $\NP$-complete.