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:heavy_check_mark: graph/directed_mst.hpp

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#include "graph/base.hpp"
#include "ds/unionfind/unionfind.hpp"

template <typename GT, int NODES>
struct Directed_MST_Solver {
  using T = typename GT::cost_type;
  GT &G;

  Directed_MST_Solver(GT &G) : G(G), pid(0) {
    pool = new Node[NODES];
    assert(G.N + G.M <= NODES);
  }

  vc<int> calc(int root) {
    int N = G.N, M = G.M;
    vc<np> que(N);
    for (auto &e: G.edges) {
      que[e.to] = meld(que[e.to], new_node(e.frm, e.cost, e.id));
    }
    vc<char> used(N + M);
    used[root] = 2;
    vc<Edge> best_edge(N + M);
    vc<int> par(N + M, -1); // merge 過程の木
    vc<int> rt(N + M);
    FOR(i, N) rt[i] = i;
    UnionFind uf(N + M);
    int nxt = N;
    for (int s = 0; s < N; ++s) {
      if (used[s] != 0) continue;
      vc<int> path = {s};
      while (1) {
        int a = path.back();
        assert(used[a] == 0);
        used[a] = 1;
        if (!que[a]) { return {}; }
        best_edge[a] = pop(que[a]);
        int to = rt[uf[best_edge[a].to]];
        if (used[to] == 0) {
          path.eb(to);
          continue;
        }
        if (used[to] == 2) break;
        // cycle 発見
        int v = nxt++;
        que.eb(nullptr);
        while (1) {
          int w = POP(path);
          T sub = best_edge[w].cost;
          que[v] = meld(que[v], add(que[w], -sub));
          uf.merge(v, w), par[w] = v;
          used[w] = 2;
          if (w == to) break;
        }
        rt[uf[v]] = v;
        path.eb(v);
      }
      for (auto &v: path) used[v] = 2;
    }

    vc<int> res;
    vc<bool> done(nxt);
    done[root] = 1;
    FOR_R(v, nxt) {
      if (done[v]) continue;
      int id = best_edge[v].id;
      res.eb(id);
      int x = G.edges[id].to;
      while (x != -1 && !done[x]) { done[x] = 1, x = par[x]; }
    }
    return res;
  }

private:
  struct Edge {
    int to, id;
    T cost;
  };

  struct Node {
    Node *l, *r;
    Edge e;
    T lazy;
    int s;
  };

  Node *pool;
  using np = Node *;
  int pid;

  np new_node(int to, T cost, int id) {
    pool[pid].l = pool[pid].r = nullptr;
    pool[pid].s = 1;
    pool[pid].e = Edge{to, id, cost};
    pool[pid].lazy = 0;
    return &(pool[pid++]);
  }

  np add(np a, T x) {
    if (a) a->e.cost += x, a->lazy += x;
    return a;
  }

  np meld(np a, np b) {
    if (!a) return b;
    if (!b) return a;
    if ((a->e.cost) > (b->e.cost)) swap(a, b);
    b = add(b, -(a->lazy));
    a->r = (a->r ? meld(a->r, b) : b);
    if (!(a->l) || (a->l->s < a->r->s)) swap(a->l, a->r);
    a->s = (a->r ? a->r->s : 0) + 1;
    return a;
  }

  Edge pop(np &a) {
    Edge e = a->e;
    a = meld(add(a->l, a->lazy), add(a->r, a->lazy));
    return e;
  }
};

template <typename GT, int MAX_N>
pair<typename GT::cost_type, vc<int>> directed_mst(GT &G, int root) {
  Directed_MST_Solver<GT, 2 * MAX_N> D(G);
  using T = typename GT::cost_type;
  auto I = D.calc(root);
  T cost = 0;
  for (auto &i: I) cost += G.edges[i].cost;
  return {cost, I};
};
#line 2 "graph/base.hpp"

template <typename T>
struct Edge {
  int frm, to;
  T cost;
  int id;
};

template <typename T = int, bool directed = false>
struct Graph {
  static constexpr bool is_directed = directed;
  int N, M;
  using cost_type = T;
  using edge_type = Edge<T>;
  vector<edge_type> edges;
  vector<int> indptr;
  vector<edge_type> csr_edges;
  vc<int> vc_deg, vc_indeg, vc_outdeg;
  bool prepared;

  class OutgoingEdges {
  public:
    OutgoingEdges(const Graph* G, int l, int r) : G(G), l(l), r(r) {}

    const edge_type* begin() const {
      if (l == r) { return 0; }
      return &G->csr_edges[l];
    }

    const edge_type* end() const {
      if (l == r) { return 0; }
      return &G->csr_edges[r];
    }

  private:
    const Graph* G;
    int l, r;
  };

  bool is_prepared() { return prepared; }

  Graph() : N(0), M(0), prepared(0) {}
  Graph(int N) : N(N), M(0), prepared(0) {}

  void build(int n) {
    N = n, M = 0;
    prepared = 0;
    edges.clear();
    indptr.clear();
    csr_edges.clear();
    vc_deg.clear();
    vc_indeg.clear();
    vc_outdeg.clear();
  }

  void add(int frm, int to, T cost = 1, int i = -1) {
    assert(!prepared);
    assert(0 <= frm && 0 <= to && to < N);
    if (i == -1) i = M;
    auto e = edge_type({frm, to, cost, i});
    edges.eb(e);
    ++M;
  }

#ifdef FASTIO
  // wt, off
  void read_tree(bool wt = false, int off = 1) { read_graph(N - 1, wt, off); }

  void read_graph(int M, bool wt = false, int off = 1) {
    for (int m = 0; m < M; ++m) {
      INT(a, b);
      a -= off, b -= off;
      if (!wt) {
        add(a, b);
      } else {
        T c;
        read(c);
        add(a, b, c);
      }
    }
    build();
  }
#endif

  void build() {
    assert(!prepared);
    prepared = true;
    indptr.assign(N + 1, 0);
    for (auto&& e: edges) {
      indptr[e.frm + 1]++;
      if (!directed) indptr[e.to + 1]++;
    }
    for (int v = 0; v < N; ++v) { indptr[v + 1] += indptr[v]; }
    auto counter = indptr;
    csr_edges.resize(indptr.back() + 1);
    for (auto&& e: edges) {
      csr_edges[counter[e.frm]++] = e;
      if (!directed)
        csr_edges[counter[e.to]++] = edge_type({e.to, e.frm, e.cost, e.id});
    }
  }

  OutgoingEdges operator[](int v) const {
    assert(prepared);
    return {this, indptr[v], indptr[v + 1]};
  }

  vc<int> deg_array() {
    if (vc_deg.empty()) calc_deg();
    return vc_deg;
  }

  pair<vc<int>, vc<int>> deg_array_inout() {
    if (vc_indeg.empty()) calc_deg_inout();
    return {vc_indeg, vc_outdeg};
  }

  int deg(int v) {
    if (vc_deg.empty()) calc_deg();
    return vc_deg[v];
  }

  int in_deg(int v) {
    if (vc_indeg.empty()) calc_deg_inout();
    return vc_indeg[v];
  }

  int out_deg(int v) {
    if (vc_outdeg.empty()) calc_deg_inout();
    return vc_outdeg[v];
  }

#ifdef FASTIO
  void debug() {
    print("Graph");
    if (!prepared) {
      print("frm to cost id");
      for (auto&& e: edges) print(e.frm, e.to, e.cost, e.id);
    } else {
      print("indptr", indptr);
      print("frm to cost id");
      FOR(v, N) for (auto&& e: (*this)[v]) print(e.frm, e.to, e.cost, e.id);
    }
  }
#endif

  vc<int> new_idx;
  vc<bool> used_e;

  // G における頂点 V[i] が、新しいグラフで i になるようにする
  // {G, es}
  Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
    if (len(new_idx) != N) new_idx.assign(N, -1);
    if (len(used_e) != M) used_e.assign(M, 0);
    int n = len(V);
    FOR(i, n) new_idx[V[i]] = i;
    Graph<T, directed> G(n);
    vc<int> history;
    FOR(i, n) {
      for (auto&& e: (*this)[V[i]]) {
        if (used_e[e.id]) continue;
        int a = e.frm, b = e.to;
        if (new_idx[a] != -1 && new_idx[b] != -1) {
          history.eb(e.id);
          used_e[e.id] = 1;
          int eid = (keep_eid ? e.id : -1);
          G.add(new_idx[a], new_idx[b], e.cost, eid);
        }
      }
    }
    FOR(i, n) new_idx[V[i]] = -1;
    for (auto&& eid: history) used_e[eid] = 0;
    G.build();
    return G;
  }

private:
  void calc_deg() {
    assert(vc_deg.empty());
    vc_deg.resize(N);
    for (auto&& e: edges) vc_deg[e.frm]++, vc_deg[e.to]++;
  }

  void calc_deg_inout() {
    assert(vc_indeg.empty());
    vc_indeg.resize(N);
    vc_outdeg.resize(N);
    for (auto&& e: edges) { vc_indeg[e.to]++, vc_outdeg[e.frm]++; }
  }
};
#line 2 "ds/unionfind/unionfind.hpp"

struct UnionFind {
  int n, n_comp;
  vc<int> dat; // par or (-size)
  UnionFind(int n = 0) { build(n); }

  void build(int m) {
    n = m, n_comp = m;
    dat.assign(n, -1);
  }

  void reset() { build(n); }

  int operator[](int x) {
    while (dat[x] >= 0) {
      int pp = dat[dat[x]];
      if (pp < 0) { return dat[x]; }
      x = dat[x] = pp;
    }
    return x;
  }

  ll size(int x) {
    x = (*this)[x];
    return -dat[x];
  }

  bool merge(int x, int y) {
    x = (*this)[x], y = (*this)[y];
    if (x == y) return false;
    if (-dat[x] < -dat[y]) swap(x, y);
    dat[x] += dat[y], dat[y] = x, n_comp--;
    return true;
  }
};
#line 3 "graph/directed_mst.hpp"

template <typename GT, int NODES>
struct Directed_MST_Solver {
  using T = typename GT::cost_type;
  GT &G;

  Directed_MST_Solver(GT &G) : G(G), pid(0) {
    pool = new Node[NODES];
    assert(G.N + G.M <= NODES);
  }

  vc<int> calc(int root) {
    int N = G.N, M = G.M;
    vc<np> que(N);
    for (auto &e: G.edges) {
      que[e.to] = meld(que[e.to], new_node(e.frm, e.cost, e.id));
    }
    vc<char> used(N + M);
    used[root] = 2;
    vc<Edge> best_edge(N + M);
    vc<int> par(N + M, -1); // merge 過程の木
    vc<int> rt(N + M);
    FOR(i, N) rt[i] = i;
    UnionFind uf(N + M);
    int nxt = N;
    for (int s = 0; s < N; ++s) {
      if (used[s] != 0) continue;
      vc<int> path = {s};
      while (1) {
        int a = path.back();
        assert(used[a] == 0);
        used[a] = 1;
        if (!que[a]) { return {}; }
        best_edge[a] = pop(que[a]);
        int to = rt[uf[best_edge[a].to]];
        if (used[to] == 0) {
          path.eb(to);
          continue;
        }
        if (used[to] == 2) break;
        // cycle 発見
        int v = nxt++;
        que.eb(nullptr);
        while (1) {
          int w = POP(path);
          T sub = best_edge[w].cost;
          que[v] = meld(que[v], add(que[w], -sub));
          uf.merge(v, w), par[w] = v;
          used[w] = 2;
          if (w == to) break;
        }
        rt[uf[v]] = v;
        path.eb(v);
      }
      for (auto &v: path) used[v] = 2;
    }

    vc<int> res;
    vc<bool> done(nxt);
    done[root] = 1;
    FOR_R(v, nxt) {
      if (done[v]) continue;
      int id = best_edge[v].id;
      res.eb(id);
      int x = G.edges[id].to;
      while (x != -1 && !done[x]) { done[x] = 1, x = par[x]; }
    }
    return res;
  }

private:
  struct Edge {
    int to, id;
    T cost;
  };

  struct Node {
    Node *l, *r;
    Edge e;
    T lazy;
    int s;
  };

  Node *pool;
  using np = Node *;
  int pid;

  np new_node(int to, T cost, int id) {
    pool[pid].l = pool[pid].r = nullptr;
    pool[pid].s = 1;
    pool[pid].e = Edge{to, id, cost};
    pool[pid].lazy = 0;
    return &(pool[pid++]);
  }

  np add(np a, T x) {
    if (a) a->e.cost += x, a->lazy += x;
    return a;
  }

  np meld(np a, np b) {
    if (!a) return b;
    if (!b) return a;
    if ((a->e.cost) > (b->e.cost)) swap(a, b);
    b = add(b, -(a->lazy));
    a->r = (a->r ? meld(a->r, b) : b);
    if (!(a->l) || (a->l->s < a->r->s)) swap(a->l, a->r);
    a->s = (a->r ? a->r->s : 0) + 1;
    return a;
  }

  Edge pop(np &a) {
    Edge e = a->e;
    a = meld(add(a->l, a->lazy), add(a->r, a->lazy));
    return e;
  }
};

template <typename GT, int MAX_N>
pair<typename GT::cost_type, vc<int>> directed_mst(GT &G, int root) {
  Directed_MST_Solver<GT, 2 * MAX_N> D(G);
  using T = typename GT::cost_type;
  auto I = D.calc(root);
  T cost = 0;
  for (auto &i: I) cost += G.edges[i].cost;
  return {cost, I};
};
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