C-Plus-Plus/graph/hopcroft_karp.cpp

/**
 * @file 
 * @brief  Implementation of [Hopcroft–Karp](https://en.wikipedia.org/wiki/Hopcroft%E2%80%93Karp_algorithm) algorithm.
 * @details 
 * The Hopcroft–Karp algorithm is an algorithm that takes as input a bipartite graph 
 * and produces as output a maximum cardinality matching, it runs in O(E√V) time in worst case.
 * 
 * ###Bipartite graph :
 * A bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint 
 * and independent sets U and V such that every edge connects a vertex in U to one in V. 
 * Vertex sets U and V are usually called the parts of the graph. 
 * Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.
 * 
 * ###Matching and Not-Matching edges :
 * Given a matching M, edges that are part of matching are called Matching edges and edges that are not part 
 * of M (or connect free nodes) are called Not-Matching edges.
 * 
 * ###Maximum cardinality matching :
 * Given a bipartite graphs G = ( V = ( X , Y ) , E ) whose partition has the parts X and Y, 
 * with E denoting the edges of the graph, the goal is to find a matching with as many edges as possible. 
 * Equivalently, a matching that covers as many vertices as possible.
 * 
 * ###Augmenting paths :
 * Given a matching M, an augmenting path is an alternating path that starts from and ends on free vertices. 
 * All single edge paths that start and end with free vertices are augmenting paths.
 * 
 * 
 * ###Concept :
 * A matching M is not maximum if there exists an augmenting path. It is also true other way,
 * i.e, a matching is maximum if no augmenting path exists.
 * 
 * 
 * ###Algorithm :
 * 1) Initialize the Maximal Matching M as empty.
 * 2) While there exists an Augmenting Path P
 *   Remove matching edges of P from M and add not-matching edges of P to M
 *   (This increases size of M by 1 as P starts and ends with a free vertex
 *   i.e. a node that is not part of matching.)
 * 3) Return M. 
 * 
 * 
 *
 * @author [Krishna Pal Deora](https://github.com/Krishnapal4050)
 * 
 */


// C++ implementation of Hopcroft Karp algorithm for
// maximum matching
#include <iostream>
#include <cstdlib> 
#include <queue>
#include <list>
#include <climits>
const int NIL = 0;
const int INF = INT_MAX;

// A class to represent Bipartite graph for
// Hopcroft Karp implementation
class BGraph
{
    // m and n are number of vertices on left
    // and right sides of Bipartite Graph
    int m, n;

    // adj[u] stores adjacents of left side
    // vertex 'u'. The value of u ranges from 1 to m.
    // 0 is used for dummy vertex
    std::list<int> *adj;

    // pointers for hopcroftKarp()
    int *pair_u, *pair_v, *dist;

public:
    BGraph(int m, int n);     // Constructor
	~BGraph();				  // destructor
    void addEdge(int u, int v); // To add edge

    // Returns true if there is an augmenting path
    bool bfs();

    // Adds augmenting path if there is one beginning
    // with u
    bool dfs(int u);

    // Returns size of maximum matching
    int hopcroftKarpAlgorithm();
};

// Returns size of maximum matching
int BGraph::hopcroftKarpAlgorithm()
{
    // pair_u[u] stores pair of u in matching on left side of Bipartite Graph.
    // If u doesn't have any pair, then pair_u[u] is NIL
    pair_u = new int[m + 1];

    // pair_v[v] stores pair of v in matching on right side of Biparite Graph.
    // If v doesn't have any pair, then pair_u[v] is NIL
    pair_v = new int[n + 1];

    // dist[u] stores distance of left side vertices
    dist = new int[m + 1];

    // Initialize NIL as pair of all vertices
    for (int u = 0; u <= m; u++){
        pair_u[u] = NIL;
	}
    for (int v = 0; v <= n; v++){
        pair_v[v] = NIL;
	}

    // Initialize result
    int result = 0;

    // Keep updating the result while there is an
    // augmenting path possible.
    while (bfs())
    {
        // Find a free vertex to check for a matching
        for (int u = 1; u <= m; u++){

            // If current vertex is free and there is
            // an augmenting path from current vertex
            // then increment the result
            if (pair_u[u] == NIL && dfs(u)){
                result++;
			}
		}
    }
    return result;
}

// Returns true if there is an augmenting path available, else returns false
bool BGraph::bfs()
{
    std::queue<int> q; //an integer queue for bfs

    // First layer of vertices (set distance as 0)
    for (int u = 1; u <= m; u++)
    {
        // If this is a free vertex, add it to queue
        if (pair_u[u] == NIL){
            // u is not matched so distance is 0
            dist[u] = 0;
            q.push(u);
        }

        // Else set distance as infinite so that this vertex is considered next time for availibility
        else{
            dist[u] = INF;
		}
    }

    // Initialize distance to NIL as infinite
    dist[NIL] = INF;

    // q is going to contain vertices of left side only.
    while (!q.empty())
    {
        // dequeue a vertex
        int u = q.front();
        q.pop();

        // If this node is not NIL and can provide a shorter path to NIL then
        if (dist[u] < dist[NIL])
        {
            // Get all the adjacent vertices of the dequeued vertex u
            std::list<int>::iterator it;
            for (it = adj[u].begin(); it != adj[u].end(); ++it)
            {
                int v = *it;

                // If pair of v is not considered so far
                // i.e. (v, pair_v[v]) is not yet explored edge.
                if (dist[pair_v[v]] == INF)
                {
                    // Consider the pair and push it to queue
                    dist[pair_v[v]] = dist[u] + 1;
                    q.push(pair_v[v]);
                }
            }
        }
    }

    // If we could come back to NIL using alternating path of distinct
    // vertices then there is an augmenting path available
    return (dist[NIL] != INF);
}

// Returns true if there is an augmenting path beginning with free vertex u
bool BGraph::dfs(int u)
{
    if (u != NIL)
    {
        std::list<int>::iterator it;
        for (it = adj[u].begin(); it != adj[u].end(); ++it)
        {
            // Adjacent vertex of u
            int v = *it;

            // Follow the distances set by BFS search
            if (dist[pair_v[v]] == dist[u] + 1)
            {
                // If dfs for pair of v also returnn true then
                if (dfs(pair_v[v]) == true)
                {   // new matching possible, store the matching
                    pair_v[v] = u;
                    pair_u[u] = v;
                    return true;
                }
            }
        }

        // If there is no augmenting path beginning with u then.
        dist[u] = INF;
        return false;
    }
    return true;
}

// Constructor for initialization
BGraph::BGraph(int m, int n)
{
    this->m = m;
    this->n = n;
    adj = new std::list<int>[m + 1];
}

// destructor 
BGraph::~BGraph()
{
	delete[] pair_u;
	delete[] pair_v;
	delete[] dist;
	delete[] adj;
}

// function to add edge from u to v
void BGraph::addEdge(int u, int v)
{
    adj[u].push_back(v); // Add v to u’s list.
}

/*
 Test case :
 Input:
// vertices of left and right side and total edges
// B-Graph shown in the above example
4 4 6
1 1
1 3
2 3
3 4
4 3
4 2
Output:
// size of maximum matching
Maximum matching is 4
*/

int main()
{
    int v1 = 0, v2 = 0, e = 0;
    std::cin >> v1 >> v2 >> e; // vertices of left side, right side and edges
    BGraph g(v1, v2); // 
    int u = 0, v = 0;
    for (int i = 0; i < e; ++i)
    {
        std::cin >> u >> v;
        g.addEdge(u, v);
    }

    int res = g.hopcroftKarpAlgorithm();
    std::cout << "Maximum matching is " << res <<"\n";

    return 0;

}