union find

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

Kruskal builds a minimum spanning tree by sorting edges and adding the cheapest 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.