# Recognizing Union-Find trees built up using union-by-rank strategy is NP-complete

## TL;DR

This paper proves that recognizing whether a given tree with rank info is a Union-Find tree built with union-by-rank is NP-complete, providing a new structural characterization and extending previous results.

## Contribution

It introduces a simple push operation for characterizing Union-Find trees with union-by-rank and proves the recognition problem is NP-complete.

## Key findings

- Recognition of Union-Find trees with union-by-rank is NP-complete.
- Provides a structural characterization using a push operation.
- Extends previous NP-completeness results for union-by-size strategy.

## 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 problem whether a given tree (along with the rank info of its nodes) is a Union-Find tree is NP-complete, complementing our earlier similar result for the union-by-size strategy.