Graph Theory Questions Medium
The Hopcroft-Karp algorithm is an efficient algorithm used to solve the bipartite matching problem in a graph. The bipartite matching problem involves finding the maximum cardinality matching in a bipartite graph, where a matching is a set of edges in which no two edges share a common vertex.
The algorithm starts by initializing an empty matching and a distance array. It then iteratively performs breadth-first search (BFS) from unmatched vertices in the first partition of the bipartite graph. During each BFS, the algorithm updates the distance array to store the shortest path distance from unmatched vertices in the first partition to unmatched vertices in the second partition.
Once a shortest augmenting path is found, the algorithm attempts to augment the matching by flipping the matching status of the edges along the augmenting path. This process is repeated until no more augmenting paths can be found.
The Hopcroft-Karp algorithm guarantees to find the maximum cardinality matching in a bipartite graph in O(sqrt(V) * E) time complexity, where V is the number of vertices and E is the number of edges in the graph. This makes it one of the most efficient algorithms for solving the bipartite matching problem.
In summary, the Hopcroft-Karp algorithm is a powerful algorithm that efficiently solves the bipartite matching problem by iteratively finding augmenting paths and updating the matching.