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:heavy_check_mark: graph/ds/static_toptree.hpp

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#include "graph/tree.hpp"

/*
tute さんの実装 https://yukicoder.me/submissions/838092 を参考にしている.
いわゆる toptree (辺からはじめてマージ過程を木にする)とは少し異なるはず.
木を「heavy path 上の辺で分割」「根を virtual にする」
「light edges の分割」「light edge を消す」で頂点に分割していく.
逆にたどれば,1 頂点からはじめて木全体を作る高さ O(logN) の木になる.
高さについて:https://www.mathenachia.blog/mergetech-and-logn/
・lch == rch == -1:頂点
・rch == -1:
  ・heavy なら light の集約に頂点を付加したもの
  ・light なら 根付き木に light edge を付加したもの
・子が 2 つ
  ・heavy なら heavy path を辺で結合したもの
  ・light なら light edge たちのマージ
*/
template <typename TREE>
struct Static_TopTree {
  TREE &tree;

  vc<int> par, lch, rch, A, B;
  vc<bool> heavy;

  Static_TopTree(TREE &tree) : tree(tree) {
    int root = tree.V[0];
    build(root);
    // relabel
    int n = len(par);
    reverse(all(par)), reverse(all(lch)), reverse(all(rch)), reverse(all(A)),
        reverse(all(B)), reverse(all(heavy));
    for (auto &x: par) x = (x == -1 ? -1 : n - 1 - x);
    for (auto &x: lch) x = (x == -1 ? -1 : n - 1 - x);
    for (auto &x: rch) x = (x == -1 ? -1 : n - 1 - x);
  }

  // 木全体での集約値を得る
  // from_vertex(v)
  // add_vertex(x, v)
  // add_edge(x, u, v)  : u が親
  // merge_light(x, y)
  // merge_heavy(x, y, a, b, c, d)  : [a,b] + [c,d] = [a,d]
  template <typename Data, typename F1, typename F2, typename F3, typename F4,
            typename F5>
  Data tree_dp(F1 from_vertex, F2 add_vertex, F3 add_edge, F4 merge_light,
               F5 merge_heavy) {
    auto dfs = [&](auto &dfs, int k) -> Data {
      if (lch[k] == -1 && rch[k] == -1) { return from_vertex(A[k]); }
      if (rch[k] == -1) {
        Data x = dfs(dfs, lch[k]);
        if (heavy[k]) {
          return add_vertex(x, A[k]);
        } else {
          return add_edge(x, A[k], B[lch[k]]);
        }
      }
      Data x = dfs(dfs, lch[k]);
      Data y = dfs(dfs, rch[k]);
      if (heavy[k]) {
        return merge_heavy(x, y, A[lch[k]], B[lch[k]], A[rch[k]], B[rch[k]]);
      }
      return merge_light(x, y);
    };
    return dfs(dfs, 0);
  }

private:
  int add_node(int l, int r, int a, int b, bool h) {
    int ret = len(par);
    par.eb(-1), lch.eb(l), rch.eb(r), A.eb(a), B.eb(b), heavy.eb(h);
    if (l != -1) par[l] = ret;
    if (r != -1) par[r] = ret;
    return ret;
  }

  int build(int v) {
    // v は heavy path の根なので v を根とする部分木に対応するノードを作る
    assert(tree.head[v] == v);
    auto path = tree.heavy_path_at(v);
    reverse(all(path));

    auto dfs = [&](auto &dfs, int l, int r) -> int {
      // path[l:r)
      if (l + 1 < r) {
        int m = (l + r) / 2;
        int x = dfs(dfs, l, m);
        int y = dfs(dfs, m, r);
        return add_node(x, y, path[l], path[r - 1], true);
      }
      assert(r == l + 1);
      int me = path[l];
      // sz, idx
      pqg<pair<int, int>> que;
      for (auto &to: tree.collect_light(me)) {
        int x = build(to);
        int y = add_node(x, -1, me, me, false);
        que.emplace(tree.subtree_size(to), y);
      }
      if (que.empty()) { return add_node(-1, -1, me, me, true); }
      while (len(que) >= 2) {
        auto [s1, x] = POP(que);
        auto [s2, y] = POP(que);
        int z = add_node(x, y, me, me, false);
        que.emplace(s1 + s2, z);
      }
      auto [s, x] = POP(que);
      return add_node(x, -1, me, me, true);
    };
    return dfs(dfs, 0, len(path));
  }
};
#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}
  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 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 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;
    }
  }
  // root を根とした場合の lca

  int LCA_root(int u, int v, int root) {
    return LCA(u, v) ^ LCA(u, root) ^ LCA(v, root);
  }
  int lca(int u, int v) { return LCA(u, v); }
  int lca_root(int u, int v, int root) { return LCA_root(u, v, root); }

  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;
  }

  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;
  }
};
#line 2 "graph/ds/static_toptree.hpp"

/*
tute さんの実装 https://yukicoder.me/submissions/838092 を参考にしている.
いわゆる toptree (辺からはじめてマージ過程を木にする)とは少し異なるはず.
木を「heavy path 上の辺で分割」「根を virtual にする」
「light edges の分割」「light edge を消す」で頂点に分割していく.
逆にたどれば,1 頂点からはじめて木全体を作る高さ O(logN) の木になる.
高さについて:https://www.mathenachia.blog/mergetech-and-logn/
・lch == rch == -1:頂点
・rch == -1:
  ・heavy なら light の集約に頂点を付加したもの
  ・light なら 根付き木に light edge を付加したもの
・子が 2 つ
  ・heavy なら heavy path を辺で結合したもの
  ・light なら light edge たちのマージ
*/
template <typename TREE>
struct Static_TopTree {
  TREE &tree;

  vc<int> par, lch, rch, A, B;
  vc<bool> heavy;

  Static_TopTree(TREE &tree) : tree(tree) {
    int root = tree.V[0];
    build(root);
    // relabel
    int n = len(par);
    reverse(all(par)), reverse(all(lch)), reverse(all(rch)), reverse(all(A)),
        reverse(all(B)), reverse(all(heavy));
    for (auto &x: par) x = (x == -1 ? -1 : n - 1 - x);
    for (auto &x: lch) x = (x == -1 ? -1 : n - 1 - x);
    for (auto &x: rch) x = (x == -1 ? -1 : n - 1 - x);
  }

  // 木全体での集約値を得る
  // from_vertex(v)
  // add_vertex(x, v)
  // add_edge(x, u, v)  : u が親
  // merge_light(x, y)
  // merge_heavy(x, y, a, b, c, d)  : [a,b] + [c,d] = [a,d]
  template <typename Data, typename F1, typename F2, typename F3, typename F4,
            typename F5>
  Data tree_dp(F1 from_vertex, F2 add_vertex, F3 add_edge, F4 merge_light,
               F5 merge_heavy) {
    auto dfs = [&](auto &dfs, int k) -> Data {
      if (lch[k] == -1 && rch[k] == -1) { return from_vertex(A[k]); }
      if (rch[k] == -1) {
        Data x = dfs(dfs, lch[k]);
        if (heavy[k]) {
          return add_vertex(x, A[k]);
        } else {
          return add_edge(x, A[k], B[lch[k]]);
        }
      }
      Data x = dfs(dfs, lch[k]);
      Data y = dfs(dfs, rch[k]);
      if (heavy[k]) {
        return merge_heavy(x, y, A[lch[k]], B[lch[k]], A[rch[k]], B[rch[k]]);
      }
      return merge_light(x, y);
    };
    return dfs(dfs, 0);
  }

private:
  int add_node(int l, int r, int a, int b, bool h) {
    int ret = len(par);
    par.eb(-1), lch.eb(l), rch.eb(r), A.eb(a), B.eb(b), heavy.eb(h);
    if (l != -1) par[l] = ret;
    if (r != -1) par[r] = ret;
    return ret;
  }

  int build(int v) {
    // v は heavy path の根なので v を根とする部分木に対応するノードを作る
    assert(tree.head[v] == v);
    auto path = tree.heavy_path_at(v);
    reverse(all(path));

    auto dfs = [&](auto &dfs, int l, int r) -> int {
      // path[l:r)
      if (l + 1 < r) {
        int m = (l + r) / 2;
        int x = dfs(dfs, l, m);
        int y = dfs(dfs, m, r);
        return add_node(x, y, path[l], path[r - 1], true);
      }
      assert(r == l + 1);
      int me = path[l];
      // sz, idx
      pqg<pair<int, int>> que;
      for (auto &to: tree.collect_light(me)) {
        int x = build(to);
        int y = add_node(x, -1, me, me, false);
        que.emplace(tree.subtree_size(to), y);
      }
      if (que.empty()) { return add_node(-1, -1, me, me, true); }
      while (len(que) >= 2) {
        auto [s1, x] = POP(que);
        auto [s2, y] = POP(que);
        int z = add_node(x, y, me, me, false);
        que.emplace(s1 + s2, z);
      }
      auto [s, x] = POP(que);
      return add_node(x, -1, me, me, true);
    };
    return dfs(dfs, 0, len(path));
  }
};
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