Interesting applications of union-find

Question

I've been trying to find interesting applications of union-find that are lesser known. Here are some popular algorithms based on union-find that I know:

• Kruskal's algorithm for MST
• Tarjan's off-line lowerst common ancestors (LCA) algorithm 1
• Solving range minimum queries using union-find by reduction to LCA (with $O(m+n)$ bound given by [Tarjan & Gabow '83, see 1]
• Finding connected components
• Various textbook examples

Question: What are some interesting use-cases of union-find inside a larger, more complicated algorithm that are lesser-known?

Answer 1

Here are two additional interesting uses of union find in formal methods and programming languages, that are perhaps lesser-known at an undergraduate level.

• Decision procedure for the quantifier free fragment of theory of equality for uninterpreted functions in Satisfiability Modulo Theory (SMT) solving (https://www.cs.utexas.edu/~isil/cs389L/lecture11-6up.pdf). In this particular case, we are given (quantifier-free) logical formulas that range over constant symbols, (possibly uninterpreted) function symbols, and equality. We are asked to decide whether such a formula is satisfiable.
• Tarjan's algorithm for dominator tree construction and path expression computation over a directed graph (https://dl.acm.org/doi/pdf/10.1145/322261.322273) all use ideas or variants of the union-find data structure.

The bottom line is, in each case above we seek to partition some particular set of elements into equivalence classes. Union-find is very effective in keeping track of the equivalence class each object belongs to and updating such information.

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EDIT May 2024: Here are two partition refinement-type algorithms from applications in logic:

• Algorithm for testing whether a first-order logic formula lies in a decidable fragment by Ge and de Moura, CAV 2009: The union-find data structure is used to maintain equivalence classes in the quantifier stratification graph. An implementation may be found in the Ivy tool for decidable verification.
• Hopcroft-Karp Algorithm for testing equivalence of two nondeterministic finite automata. This is a very elegant algorithm and the idea is to use union-find to maintain partitions of automata states that are equivalent.