a textbook example of an algorithm that works very well with union-find is the kruskal's algorithm to find the minimum weight spanning tree given a graph. Using union-find improves the time complexity of the algorithm from O(V²) to O(ElogV).
This happens because kruskal's algorithm essentially selects the cheapest edge not already included in our spanning tree that won't cause a cycle. So union-find is able to speed up this potential cycle check which would otherwise be naively quadratic.
This happens because kruskal's algorithm essentially selects the cheapest edge not already included in our spanning tree that won't cause a cycle. So union-find is able to speed up this potential cycle check which would otherwise be naively quadratic.