Disjoint Set Union (Union-Find) Implementation

To solve this efficiently, I would implement the Disjoint Set Union (DSU) structure using two arrays: one for parent pointers and another for tracking the rank of each tree root. First, I initialize each element as its own parent with a rank of zero. For the find operation, I apply path compression by making every node on the path point directly to the root, which drastically reduces future query times. When performing union, I use union by rank: I compare the ranks of the two roots and attach the smaller tree to the larger one, incrementing the rank only if they were equal. This ensures the tree height remains logarithmic. The time complexity for both operations becomes nearly constant, specifically O(alpha(n)), where alpha is the inverse Ackermann function, which grows so slowly it is effectively constant for all practical inputs. This approach is critical for systems like Adobe's graphics rendering engines, where we frequently check connectivity between pixels or vector paths. By optimizing these two aspects, we prevent stack overflow issues and ensure high performance even with millions of elements, demonstrating the robustness required for large-scale applications.