Hopkroft Karp Bipartite Matching.cpp

// Maximum bipartite matching
// Index from 1
// Find max independent set:
// for(i = 1 → M) if (mat.matchL[i] > 0) {
//   if (mat.dist[i] < inf) {
//     for(j = 1 → N) if (ke[i][j]) right.erase(j); }
//   else left.erase(i);
// }
// Find vertices that belong to all maximum matching:
// - L = vertices not matched on left side --> BFS from these vertices
//   (left --> right: unmatched edges, right --> left: matched edges)
//   reachable vertices on left side --> not belong to some maximum matching
// - Do similar for right side

//  Find minimum vertex cover:
//   L = vertices not matched on left side --> BFS from these vertices
//   (left --> right: unmatched edges, right --> left: matched edges)
//   reachable vertices on left side --> not belong to some maximum matching
//   Let U be all visited vertices. from left side, take vertices that are not in U, from right side , take vertices that are in U;
// all taken vertices must have a match
struct Matching {
    int n;
    vector<int> matchL, matchR, dist;
    vector<bool> seen;
    vector< vector<int> > ke;

    Matching(int n) : n(n), matchL(n+1), matchR(n+1), dist(n+1), seen(n+1, false), ke(n+1) {}

    void addEdge(int u, int v) {
        ke[u].push_back(v);
    }

    bool bfs() {
        queue<int> qu;
        for(int u = 1; u <= n; ++u)
            if (!matchL[u]) {
                dist[u] = 0;
                qu.push(u);
            } else dist[u] = inf;
        dist[0] = inf;

        while (!qu.empty()) {
            int u = qu.front(); qu.pop();
            for(__typeof(ke[u].begin()) v = ke[u].begin(); v != ke[u].end(); ++v) {
                if (dist[matchR[*v]] == inf) {
                    dist[matchR[*v]] = dist[u] + 1;
                    qu.push(matchR[*v]);
                }
            }
        }
        return dist[0] != inf;
    }

    bool dfs(int u) {
        if (u) {
            for(__typeof(ke[u].begin()) v = ke[u].begin(); v != ke[u].end(); ++v)
                if (dist[matchR[*v]] == dist[u] + 1 && dfs(matchR[*v])) {
                    matchL[u] = *v;
                    matchR[*v] = u;
                    return true;
                }
            dist[u] = inf;
            return false;
        }
        return true;
    }

    int match() {
        int res = 0;
        while (bfs()) {
            for(int u = 1; u <= n; ++u)
                if (!matchL[u])
                    if (dfs(u)) ++res;
        }
        return res;
    }
};