What is Union-Find (Disjoint Set Union) and its optimizations?

Union-Find tracks elements partitioned into disjoint sets, supporting union (merge sets) and find (determine set membership). Basic implementation uses parent array with tree structure. Optimizations: Path Compression (during find, make all nodes point directly to root) and Union by Rank (attach smaller tree under root of larger tree). Together, they achieve near-O(1) amortized time - specifically O(alpha(n)) where alpha is inverse Ackermann function.