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template <typename T>
class FlowGraph {
  public:
    struct Edge {
        std::size_t to;
        T cap; // flow, capacity
    };
    std::vector<int> dist;
    std::vector<std::size_t> cur;
    std::vector<std::vector<std::size_t>> adj;
    std::vector<Edge> edges;
    std::size_t s, t;
    std::size_t vtot, etot;

    // ! ===== functionality =====
    FlowGraph() : vtot(0), etot(0) {}

    void init(std::size_t s, std::size_t t, std::size_t vtot) {
        this->s = s, this->t = t, this->vtot = vtot;
        adj.resize(vtot), dist.resize(vtot), cur.resize(vtot);
    }

    void add(std::size_t u, std::size_t v, T f) {
        adj.at(u).push_back(edges.size());
        edges.emplace_back(v, f);
        adj.at(v).push_back(edges.size());
        edges.emplace_back(u, 0);
    }
};


template <typename T>
class PushRelabel : public FlowGraph<T> {
  public:
    std::vector<T> height;
    std::vector<T> excess;
    std::vector<std::size_t> gap;
    std::vector<std::vector<std::size_t>> bucket;
    T level{0};
    static constexpr T inf = std::numeric_limits<T>::max();

    bool push(std::size_t u) {
        bool init = u == this->s;

        for (auto e : this->adj.at(u)) {
            auto [v, cap] = this->edges.at(e);
            if (cap == 0 or this->height.at(v) == inf) continue;
            if (!init and this->height.at(u) != this->height.at(v) + 1) continue;

            T k = init ? cap : std::min(cap, this->excess.at(u));
            if (v != this->s and v != this->t and this->excess.at(v) == 0) {
                this->bucket.at(this->height.at(v)).push_back(v);
                this->level = std::max(this->level, this->height.at(v));
            }
            // push
            this->excess.at(u) -= k;
            this->excess.at(v) += k;
            this->edges.at(e).cap -= k;
            this->edges.at(e ^ 1).cap += k;
            if (this->excess.at(u) == 0) return false; // finish pushing
        }
        return true;
    }

    void relabel(std::size_t u) {
        this->height.at(u) = inf;
        for (auto e : this->adj.at(u)) {
            auto [v, cap] = this->edges.at(e);
            if (cap > 0) this->height.at(u) = std::min(this->height.at(u), this->height.at(v));
        }
        this->height.at(u)++;
        if (this->height.at(u) < static_cast<T>(this->vtot)) {
            this->bucket.at(this->height.at(u)).push_back(u);
            level = std::max(level, this->height.at(u));
            ++this->gap.at(this->height.at(u));
        }
    }

    bool bfs_init() {
        this->height.assign(this->vtot, inf);
        std::queue<std::size_t> q;
        q.push(this->t);
        this->height.at(this->t) = 0;
        while (!q.empty()) {
            auto u = q.front();
            q.pop();
            for (auto e : this->adj.at(u)) {
                auto [v, cap]   = this->edges.at(e);
                auto [rv, rcap] = this->edges.at(e ^ 1);
                if (rcap > 0 and this->height.at(v) > this->height.at(u) + 1) {
                    this->height.at(v) = this->height.at(u) + 1;
                    q.push(v);
                }
            }
        }
        return this->height.at(this->s) != inf;
    }

    std::size_t select() {
        while (this->level > -1 and this->bucket.at(this->level).size() == 0) this->level--;
        return this->level == -1 ? 114514 : this->bucket.at(this->level).back();
    }

  public:
    void init(std::size_t s, std::size_t t, std::size_t n) {
        FlowGraph<T>::init(s, t, n);
        this->height.assign(n, inf);
        this->excess.assign(n, 0);
        this->gap.assign(n + 1, 0);
        this->bucket.assign(n + 1, {});
        this->level = 0;
    }
    T max_flow() {
        if (not this->bfs_init()) return 0;

        this->gap.assign(this->vtot, 0);
        for (std::size_t i = 0; i < this->vtot; i++)
            if (this->height.at(i) != inf) this->gap.at(this->height.at(i))++;
        this->height.at(this->s) = this->vtot;
        this->push(this->s);

        for (std::size_t u = select(); u != 114514; u = select()) {
            this->bucket.at(this->level).pop_back();
            if (this->push(u)) {
                if (not --this->gap.at(this->height.at(u))) {
                    for (std::size_t i = 0; i < this->vtot; i++) {
                        if (i == this->s) continue;
                        if (this->height.at(i) >= static_cast<T>(this->vtot + 1)) continue;
                        if (this->height.at(i) <= this->height.at(u)) continue;
                        this->height.at(i) = static_cast<T>(this->vtot + 1);
                    }
                }
                this->relabel(u);
            }
        }
        return this->excess.at(this->t);
    }
};