Disjoint Set Union

Start by clarifying the exact contract for Disjoint Set Union: input representation, mutation rules, duplicate behavior, and what should happen on invalid or empty input. The core idea is amortized connectivity operations with structural invariants. Work a tiny example aloud and name the invariant after each step, because that is the proof that the approach stays correct as the input grows.

A dependable implementation plan is to initialize each node as its own root; path-compress find; union roots by size or rank; compare roots for connectivity. DSU supports additions well but not ordinary deletions. Keep the code aligned with that invariant instead of adding special cases that silently change the contract. Test the smallest input, a normal case, duplicates or ties where relevant, an adversarial boundary, and a case that exercises the failure path. The expected complexity is O(alpha(n)) amortized per operation and O(n) space. Explain both time and auxiliary memory, including recursion or helper structures; do not hide sorting, copying, or repeated scans. If the title omits critical constraints, say what you need to know before committing to a more specialized solution.