Hopcroft–Karp Algorithm

The Hopcroft–Karp algorithm finds the maximum matching in a bipartite graph efficiently. Imagine two sets of points, with lines connecting some pairs. The goal is to match as many points as possible without overlaps. The algorithm repeatedly searches for shortest alternating paths to improve the matches, using a process called Breadth-First Search (BFS) to identify potential improvements, then Depth-First Search (DFS) to update the matches along these paths. This approach significantly reduces the number of steps needed compared to naive methods, allowing for faster solutions in large graphs.