How to use Hopcroft-Karp algorithm for maximum bipartite matching?

The Hopcroft-Karp algorithm is used to find the maximum matching in a bipartite graph.

To use the Hopcroft-Karp algorithm, we first need to represent the bipartite graph as an adjacency list. We can then initialise an empty matching and a distance array. The distance array will be used to keep track of the distance from each vertex in the left partition to the nearest unmatched vertex in the right partition.

We then repeat the following steps until no augmenting path can be found:
1. Set the distance array to infinity.
2. For each vertex in the left partition that is currently unmatched, perform a breadth-first search to find the shortest augmenting path to an unmatched vertex in the right partition. If such a path is found, update the matching and distance arrays accordingly.
3. Return the size of the matching.

The time complexity of the Hopcroft-Karp algorithm is O(E * sqrt(V)), where E is the number of edges and V is the number of vertices in the bipartite graph. This makes it more efficient than the naive algorithm of trying all possible matchings, which has a time complexity of O(2^V).

Overall, the Hopcroft-Karp algorithm is a useful tool for finding the maximum matching in a bipartite graph.