The Hopcroft-Karp algorithm is a powerful algorithm used to find the maximum cardinality matching in a bipartite graph. A matching in a graph is a subset of edges such that no two edges share a common vertex. The maximum cardinality matching is the matching with the largest number of edges. The Hopcroft-Karp algorithm is a polynomial-time algorithm that finds the maximum cardinality matching in a bipartite graph. It was first proposed by John Hopcroft and Richard Karp in 1973.
The Hopcroft-Karp algorithm is used in many applications such as network flow, scheduling, and graph coloring. It is also used in the analysis of social networks, where it can be used to find the maximum number of edges that can be added to a graph without creating a cycle.
In this blog post, we will discuss the Hopcroft-Karp algorithm in detail and provide an example of how it can be used to solve a maximum cardinality matching problem.
A bipartite graph is a graph in which the vertices can be divided into two disjoint sets, U and V, such that every edge in the graph has one endpoint in U and one endpoint in V. In other words, a bipartite graph is a graph in which the vertices can be divided into two sets such that no two edges share a common vertex.
A matching in a graph is a subset of edges such that no two edges share a common vertex. In a bipartite graph, a matching is a subset of edges such that each edge has one endpoint in U and one endpoint in V. A maximum cardinality matching is a matching with the largest number of edges.
The Hopcroft-Karp algorithm is a polynomial-time algorithm that finds the maximum cardinality matching in a bipartite graph. It was first proposed by John Hopcroft and Richard Karp in 1973. The algorithm works by repeatedly finding augmenting paths in the graph and using them to increase the size of the matching.
The algorithm works as follows:
An augmenting path is a path in the graph that starts and ends at unmatched vertices and alternates between matched and unmatched edges. Finding an augmenting path can be done using a breadth-first search.
The following pseudocode describes the Hopcroft-Karp algorithm:
HOPCROFT-KARP(G, U, V):
// G is the bipartite graph, U and V are the two sets of vertices
// Initialize the matching to be empty
M = {}
while there is an augmenting path in G:
// Find an augmenting path in G
P = FIND-AUGMENTING-PATH(G, M)
// Augment the matching by adding the edges of the augmenting path
for each edge (u, v) in P:
if (u, v) is not in M:
M = M U {(u, v)}
else:
M = M \ {(u, v)}
// Return the matching
return M
The maximum cardinality matching in this graph is {(1, 4), (2, 5), (3, 6)}. The Hopcroft-Karp algorithm can be used to find this matching. The algorithm starts by initializing the matching to be empty. It then finds an augmenting path in the graph and augments the matching by adding the edges of the augmenting path. In this example, the algorithm finds the augmenting path (1, 4), (2, 5), (3, 6) and adds these edges to the matching. The algorithm then returns the matching, which is the maximum cardinality matching in the graph.
The Hopcroft-Karp algorithm is a powerful algorithm used to find the maximum cardinality matching in a bipartite graph. It is a polynomial-time algorithm that finds an augmenting path in the graph and uses it to increase the size of the matching. The algorithm has many applications in network flow, scheduling, and graph coloring. In this blog post, we discussed the Hopcroft-Karp algorithm in detail and provided an example of how it can be used to solve a maximum cardinality matching problem.