union find

Union-Find answers 'are these in the same group?' almost instantly and drives connected-components, cycle detection, and clustering. Interviewers look for path compression plus union by rank for near-O(1) operations. The answer follows.

Kruskal builds a minimum spanning tree by sorting edges and adding the cheapest one that does not form a cycle, using union-find for the cycle check. The signal is the greedy cut property and why disjoint-set is the right tool. Here is the answer.