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:question: graph/ds/static_toptree.hpp

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#include "graph/tree.hpp"

/*
参考 joitour tatyam
クラスタは根が virtual なもののみであるような簡易版
N 個の (頂+辺) をマージしていって，木全体＋根から親への辺とする．
single(v) : v とその親辺を合わせたクラスタ
rake(L,R) : L の boundary を維持
compress(L,R)  (top-down) 順に x,y
*/
template <typename TREE>
struct Static_TopTree {
  int N;
  TREE &tree;
  vc<int> par, lch, rch, A, B; // A, B boundary (top-down)
  vc<bool> is_compress;

  Static_TopTree(TREE &tree) : tree(tree) { build(); }

  void build() {
    N = tree.N;
    par.assign(N, -1), lch.assign(N, -1), rch.assign(N, -1), A.assign(N, -1), B.assign(N, -1), is_compress.assign(N, 0);
    FOR(v, N) { A[v] = tree.parent[v], B[v] = v; }
    build_dfs(tree.V[0]);
    assert(len(par) == 2 * N - 1);
  }

  // 木全体での集約値を得る
  // single(v) : v とその親辺を合わせたクラスタ
  // rake(x, y, u, v) uv(top down) が boundary になるように rake (maybe v=-1)
  // compress(x,y,a,b,c)  (top-down) 順に (a,b] + (b,c]
  template <typename TREE_DP, typename F>
  typename TREE_DP::value_type tree_dp(F single) {
    using Data = typename TREE_DP::value_type;
    auto dfs = [&](auto &dfs, int k) -> Data {
      if (0 <= k && k < N) return single(k);
      Data x = dfs(dfs, lch[k]), y = dfs(dfs, rch[k]);
      if (is_compress[k]) {
        assert(B[lch[k]] == A[rch[k]]);
        return TREE_DP::compress(x, y);
      }
      return TREE_DP::rake(x, y);
    };
    return dfs(dfs, 2 * N - 2);
  }

private:
  int new_node(int l, int r, int a, int b, bool c) {
    int v = len(par);
    par.eb(-1), lch.eb(l), rch.eb(r), A.eb(a), B.eb(b), is_compress.eb(c);
    par[l] = par[r] = v;
    return v;
  }

  // height, node idx
  // compress 参考：https://atcoder.jp/contests/abc351/editorial/9910
  // ただし heavy path の選び方までは考慮しない
  pair<int, int> build_dfs(int v) {
    assert(tree.head[v] == v);
    auto path = tree.heavy_path_at(v);
    vc<pair<int, int>> stack;
    stack.eb(0, path[0]);
    auto merge_last_two = [&]() -> void {
      auto [h2, k2] = POP(stack);
      auto [h1, k1] = POP(stack);
      stack.eb(max(h1, h2) + 1, new_node(k1, k2, A[k1], B[k2], true));
    };

    FOR(i, 1, len(path)) {
      pqg<pair<int, int>> que;
      int k = path[i];
      que.emplace(0, k);
      for (auto &c: tree.collect_light(path[i - 1])) { que.emplace(build_dfs(c)); }
      while (len(que) >= 2) {
        auto [h1, i1] = POP(que);
        auto [h2, i2] = POP(que);
        if (i2 == k) swap(i1, i2);
        int i3 = new_node(i1, i2, A[i1], B[i1], false);
        if (k == i1) k = i3;
        que.emplace(max(h1, h2) + 1, i3);
      }
      stack.eb(POP(que));

      while (1) {
        int n = len(stack);
        if (n >= 3 && (stack[n - 3].fi == stack[n - 2].fi || stack[n - 3].fi <= stack[n - 1].fi)) {
          auto [h3, k3] = POP(stack);
          merge_last_two(), stack.eb(h3, k3);
        }
        elif (n >= 2 && stack[n - 2].fi <= stack[n - 1].fi) { merge_last_two(); }
        else break;
      }
    }
    while (len(stack) >= 2) { merge_last_two(); }
    return POP(stack);
  }
};
#line 2 "graph/tree.hpp"

#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}
  // sum(deg(v)) の計算量になっていて、
  // 新しいグラフの n+m より大きい可能性があるので注意
  Graph<T, directed> rearrange(vc<int> V, bool keep_eid = 0) {
    if (len(new_idx) != N) new_idx.assign(N, -1);
    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 (len(used_e) <= e.id) used_e.resize(e.id + 1);
        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;
  }

  Graph<T, true> to_directed_tree(int root = -1) {
    if (root == -1) root = 0;
    assert(!is_directed && prepared && M == N - 1);
    Graph<T, true> G1(N);
    vc<int> par(N, -1);
    auto dfs = [&](auto& dfs, int v) -> void {
      for (auto& e: (*this)[v]) {
        if (e.to == par[v]) continue;
        par[e.to] = v, dfs(dfs, e.to);
      }
    };
    dfs(dfs, root);
    for (auto& e: edges) {
      int a = e.frm, b = e.to;
      if (par[a] == b) swap(a, b);
      assert(par[b] == a);
      G1.add(a, b, e.cost);
    }
    G1.build();
    return G1;
  }

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 4 "graph/tree.hpp"

// HLD euler tour をとっていろいろ。

template <typename GT>
struct Tree {
  using Graph_type = GT;
  GT &G;
  using WT = typename GT::cost_type;
  int N;
  vector<int> LID, RID, head, V, parent, VtoE;
  vc<int> depth;
  vc<WT> depth_weighted;

  Tree(GT &G, int r = 0, bool hld = 1) : G(G) { build(r, hld); }

  void build(int r = 0, bool hld = 1) {
    if (r == -1) return; // build を遅延したいとき

    N = G.N;
    LID.assign(N, -1), RID.assign(N, -1), head.assign(N, r);
    V.assign(N, -1), parent.assign(N, -1), VtoE.assign(N, -1);
    depth.assign(N, -1), depth_weighted.assign(N, 0);
    assert(G.is_prepared());
    int t1 = 0;
    dfs_sz(r, -1, hld);
    dfs_hld(r, t1);
  }

  void dfs_sz(int v, int p, bool hld) {
    auto &sz = RID;
    parent[v] = p;
    depth[v] = (p == -1 ? 0 : depth[p] + 1);
    sz[v] = 1;
    int l = G.indptr[v], r = G.indptr[v + 1];
    auto &csr = G.csr_edges;
    // 使う辺があれば先頭にする

    for (int i = r - 2; i >= l; --i) {
      if (hld && depth[csr[i + 1].to] == -1) swap(csr[i], csr[i + 1]);
    }
    int hld_sz = 0;
    for (int i = l; i < r; ++i) {
      auto e = csr[i];
      if (depth[e.to] != -1) continue;
      depth_weighted[e.to] = depth_weighted[v] + e.cost;
      VtoE[e.to] = e.id;
      dfs_sz(e.to, v, hld);
      sz[v] += sz[e.to];
      if (hld && chmax(hld_sz, sz[e.to]) && l < i) { swap(csr[l], csr[i]); }
    }
  }

  void dfs_hld(int v, int &times) {
    LID[v] = times++;
    RID[v] += LID[v];
    V[LID[v]] = v;
    bool heavy = true;
    for (auto &&e: G[v]) {
      if (depth[e.to] <= depth[v]) continue;
      head[e.to] = (heavy ? head[v] : e.to);
      heavy = false;
      dfs_hld(e.to, times);
    }
  }

  vc<int> heavy_path_at(int v) {
    vc<int> P = {v};
    while (1) {
      int a = P.back();
      for (auto &&e: G[a]) {
        if (e.to != parent[a] && head[e.to] == v) {
          P.eb(e.to);
          break;
        }
      }
      if (P.back() == a) break;
    }
    return P;
  }

  int heavy_child(int v) {
    int k = LID[v] + 1;
    if (k == N) return -1;
    int w = V[k];
    return (parent[w] == v ? w : -1);
  }

  int e_to_v(int eid) {
    auto e = G.edges[eid];
    return (parent[e.frm] == e.to ? e.frm : e.to);
  }
  int v_to_e(int v) { return VtoE[v]; }
  int get_eid(int u, int v) {
    if (parent[u] != v) swap(u, v);
    assert(parent[u] == v);
    return VtoE[u];
  }

  int ELID(int v) { return 2 * LID[v] - depth[v]; }
  int ERID(int v) { return 2 * RID[v] - depth[v] - 1; }

  // 目標地点へ進む個数が k

  int LA(int v, int k) {
    assert(k <= depth[v]);
    while (1) {
      int u = head[v];
      if (LID[v] - k >= LID[u]) return V[LID[v] - k];
      k -= LID[v] - LID[u] + 1;
      v = parent[u];
    }
  }
  int la(int u, int v) { return LA(u, v); }

  int LCA(int u, int v) {
    for (;; v = parent[head[v]]) {
      if (LID[u] > LID[v]) swap(u, v);
      if (head[u] == head[v]) return u;
    }
  }

  int meet(int a, int b, int c) { return LCA(a, b) ^ LCA(a, c) ^ LCA(b, c); }
  int lca(int u, int v) { return LCA(u, v); }

  int subtree_size(int v, int root = -1) {
    if (root == -1) return RID[v] - LID[v];
    if (v == root) return N;
    int x = jump(v, root, 1);
    if (in_subtree(v, x)) return RID[v] - LID[v];
    return N - RID[x] + LID[x];
  }

  int dist(int a, int b) {
    int c = LCA(a, b);
    return depth[a] + depth[b] - 2 * depth[c];
  }

  WT dist_weighted(int a, int b) {
    int c = LCA(a, b);
    return depth_weighted[a] + depth_weighted[b] - WT(2) * depth_weighted[c];
  }

  // a is in b

  bool in_subtree(int a, int b) { return LID[b] <= LID[a] && LID[a] < RID[b]; }

  int jump(int a, int b, ll k) {
    if (k == 1) {
      if (a == b) return -1;
      return (in_subtree(b, a) ? LA(b, depth[b] - depth[a] - 1) : parent[a]);
    }
    int c = LCA(a, b);
    int d_ac = depth[a] - depth[c];
    int d_bc = depth[b] - depth[c];
    if (k > d_ac + d_bc) return -1;
    if (k <= d_ac) return LA(a, k);
    return LA(b, d_ac + d_bc - k);
  }

  vc<int> collect_child(int v) {
    vc<int> res;
    for (auto &&e: G[v])
      if (e.to != parent[v]) res.eb(e.to);
    return res;
  }

  vc<int> collect_light(int v) {
    vc<int> res;
    bool skip = true;
    for (auto &&e: G[v])
      if (e.to != parent[v]) {
        if (!skip) res.eb(e.to);
        skip = false;
      }
    return res;
  }

  vc<pair<int, int>> get_path_decomposition(int u, int v, bool edge) {
    // [始点, 終点] の"閉"区間列。

    vc<pair<int, int>> up, down;
    while (1) {
      if (head[u] == head[v]) break;
      if (LID[u] < LID[v]) {
        down.eb(LID[head[v]], LID[v]);
        v = parent[head[v]];
      } else {
        up.eb(LID[u], LID[head[u]]);
        u = parent[head[u]];
      }
    }
    if (LID[u] < LID[v]) down.eb(LID[u] + edge, LID[v]);
    elif (LID[v] + edge <= LID[u]) up.eb(LID[u], LID[v] + edge);
    reverse(all(down));
    up.insert(up.end(), all(down));
    return up;
  }

  // 辺の列の情報 (frm,to,str)

  // str = "heavy_up", "heavy_down", "light_up", "light_down"

  vc<tuple<int, int, string>> get_path_decomposition_detail(int u, int v) {
    vc<tuple<int, int, string>> up, down;
    while (1) {
      if (head[u] == head[v]) break;
      if (LID[u] < LID[v]) {
        if (v != head[v]) down.eb(head[v], v, "heavy_down"), v = head[v];
        down.eb(parent[v], v, "light_down"), v = parent[v];
      } else {
        if (u != head[u]) up.eb(u, head[u], "heavy_up"), u = head[u];
        up.eb(u, parent[u], "light_up"), u = parent[u];
      }
    }
    if (LID[u] < LID[v]) down.eb(u, v, "heavy_down");
    elif (LID[v] < LID[u]) up.eb(u, v, "heavy_up");
    reverse(all(down));
    concat(up, down);
    return up;
  }

  vc<int> restore_path(int u, int v) {
    vc<int> P;
    for (auto &&[a, b]: get_path_decomposition(u, v, 0)) {
      if (a <= b) {
        FOR(i, a, b + 1) P.eb(V[i]);
      } else {
        FOR_R(i, b, a + 1) P.eb(V[i]);
      }
    }
    return P;
  }

  // path [a,b] と [c,d] の交わり. 空ならば {-1,-1}.

  // https://codeforces.com/problemset/problem/500/G

  pair<int, int> path_intersection(int a, int b, int c, int d) {
    int ab = lca(a, b), ac = lca(a, c), ad = lca(a, d);
    int bc = lca(b, c), bd = lca(b, d), cd = lca(c, d);
    int x = ab ^ ac ^ bc, y = ab ^ ad ^ bd; // meet(a,b,c), meet(a,b,d)

    if (x != y) return {x, y};
    int z = ac ^ ad ^ cd;
    if (x != z) x = -1;
    return {x, x};
  }

  // uv path 上で check(v) を満たす最後の v

  // なければ （つまり check(v) が ng ）-1

  template <class F>
  int max_path(F check, int u, int v) {
    if (!check(u)) return -1;
    auto pd = get_path_decomposition(u, v, false);
    for (auto [a, b]: pd) {
      if (!check(V[a])) return u;
      if (check(V[b])) {
        u = V[b];
        continue;
      }
      int c = binary_search([&](int c) -> bool { return check(V[c]); }, a, b, 0);
      return V[c];
    }
    return u;
  }
};
#line 2 "graph/ds/static_toptree.hpp"

/*
参考 joitour tatyam
クラスタは根が virtual なもののみであるような簡易版
N 個の (頂+辺) をマージしていって，木全体＋根から親への辺とする．
single(v) : v とその親辺を合わせたクラスタ
rake(L,R) : L の boundary を維持
compress(L,R)  (top-down) 順に x,y
*/
template <typename TREE>
struct Static_TopTree {
  int N;
  TREE &tree;
  vc<int> par, lch, rch, A, B; // A, B boundary (top-down)
  vc<bool> is_compress;

  Static_TopTree(TREE &tree) : tree(tree) { build(); }

  void build() {
    N = tree.N;
    par.assign(N, -1), lch.assign(N, -1), rch.assign(N, -1), A.assign(N, -1), B.assign(N, -1), is_compress.assign(N, 0);
    FOR(v, N) { A[v] = tree.parent[v], B[v] = v; }
    build_dfs(tree.V[0]);
    assert(len(par) == 2 * N - 1);
  }

  // 木全体での集約値を得る
  // single(v) : v とその親辺を合わせたクラスタ
  // rake(x, y, u, v) uv(top down) が boundary になるように rake (maybe v=-1)
  // compress(x,y,a,b,c)  (top-down) 順に (a,b] + (b,c]
  template <typename TREE_DP, typename F>
  typename TREE_DP::value_type tree_dp(F single) {
    using Data = typename TREE_DP::value_type;
    auto dfs = [&](auto &dfs, int k) -> Data {
      if (0 <= k && k < N) return single(k);
      Data x = dfs(dfs, lch[k]), y = dfs(dfs, rch[k]);
      if (is_compress[k]) {
        assert(B[lch[k]] == A[rch[k]]);
        return TREE_DP::compress(x, y);
      }
      return TREE_DP::rake(x, y);
    };
    return dfs(dfs, 2 * N - 2);
  }

private:
  int new_node(int l, int r, int a, int b, bool c) {
    int v = len(par);
    par.eb(-1), lch.eb(l), rch.eb(r), A.eb(a), B.eb(b), is_compress.eb(c);
    par[l] = par[r] = v;
    return v;
  }

  // height, node idx
  // compress 参考：https://atcoder.jp/contests/abc351/editorial/9910
  // ただし heavy path の選び方までは考慮しない
  pair<int, int> build_dfs(int v) {
    assert(tree.head[v] == v);
    auto path = tree.heavy_path_at(v);
    vc<pair<int, int>> stack;
    stack.eb(0, path[0]);
    auto merge_last_two = [&]() -> void {
      auto [h2, k2] = POP(stack);
      auto [h1, k1] = POP(stack);
      stack.eb(max(h1, h2) + 1, new_node(k1, k2, A[k1], B[k2], true));
    };

    FOR(i, 1, len(path)) {
      pqg<pair<int, int>> que;
      int k = path[i];
      que.emplace(0, k);
      for (auto &c: tree.collect_light(path[i - 1])) { que.emplace(build_dfs(c)); }
      while (len(que) >= 2) {
        auto [h1, i1] = POP(que);
        auto [h2, i2] = POP(que);
        if (i2 == k) swap(i1, i2);
        int i3 = new_node(i1, i2, A[i1], B[i1], false);
        if (k == i1) k = i3;
        que.emplace(max(h1, h2) + 1, i3);
      }
      stack.eb(POP(que));

      while (1) {
        int n = len(stack);
        if (n >= 3 && (stack[n - 3].fi == stack[n - 2].fi || stack[n - 3].fi <= stack[n - 1].fi)) {
          auto [h3, k3] = POP(stack);
          merge_last_two(), stack.eb(h3, k3);
        }
        elif (n >= 2 && stack[n - 2].fi <= stack[n - 1].fi) { merge_last_two(); }
        else break;
      }
    }
    while (len(stack) >= 2) { merge_last_two(); }
    return POP(stack);
  }
};
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