【图论】二分图最大匹配

验证链接：[洛谷P3386 - 二分图最大匹配]

最大流

int N = n1 + n2, S = ++N, T = ++N;
dinic.Init(N, S, T);
for(int i = 1; i <= n1; ++i) {
    int V1i = i;
    dinic.AddEdge(S, V1i, 1);
}
for(int i = 1; i <= n2; ++i) {
    int V2i = n1 + i;
    dinic.AddEdge(V2i, T, 1);
}
while(m--) {
    int u = ?, v = ?;
    int V1i = u, V2i = n1 + v;
    dinic.AddEdge(V1i, V2i, 1);
}
int ans = dinic.Maxflow();

Dinic模板

求解二分图等单位网络时，复杂度为 \(O(m\sqrt{n})\) 。
除了超级点外，所有点的“入度和出度的最小值”不超过1的，为单位网络。

struct Dinic {

    static const int MAXN = 2e5 + 10;
    static const int MAXM = 4e6 + 10;

    int n, m, s, t;

    struct Edge {
        int v, nxt;
        ll flow;
    } e[MAXM];
    int h[MAXN];

    int que[MAXN], dep[MAXN], cur[MAXN];

    bool BFS() {
        memset(dep, 0, sizeof(dep[0]) * (n + 1));
        memcpy(cur, h, sizeof(cur[0]) * (n + 1));
        int l = 1, r = 0;
        que[++r] = s, dep[s] = 1;
        while(l <= r) {
            int u = que[l++];
            for(int i = h[u]; i; i = e[i].nxt) {
                if(e[i].flow && !dep[e[i].v])
                    dep[e[i].v] = dep[u] + 1, que[++r] = e[i].v;
            }
        }
        return dep[t];
    }

    ll DFS(int u, ll in) {
        if(u == t)
            return in;
        ll out = 0;
        for(int &i = cur[u]; i; i = e[i].nxt) {
            if(e[i].flow && dep[u] + 1 == dep[e[i].v]) {
                ll res = DFS(e[i].v, min(in, e[i].flow));
                e[i].flow -= res, in -= res;
                e[i ^ 1].flow += res, out += res;
                if(in == 0)
                    break;
            }
        }
        if(out == 0)
            dep[u] = 0;
        return out;
    }

    void Init(int n, int s, int t) {
        this->n = n, m = 1, this->s = s, this->t = t;
        for(int i = 1; i <= n; ++i)
            h[i] = 0;
    }

    void AddEdge(int u, int v, ll w) {
        e[++m] = {v, h[u], w}, h[u] = m;
        e[++m] = {u, h[v], 0}, h[v] = m;
    }

    ll Maxflow() {
        ll flow = 0;
        while(BFS())
            flow += DFS(s, LINF);
        return flow;
    }

} dinic;