graph/ds/range_edge_connected_component_query.hpp

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• Last update: 2024-04-27 11:55:26+09:00
• Include: #include "graph/ds/range_edge_connected_component_query.hpp"
• Link: https://codeforces.com/problemset/problem/1386/C

Depends on

• alg/monoid/add.hpp
• alg/monoid/min.hpp
• ds/fenwicktree/fenwicktree.hpp
• ds/fenwicktree/fenwicktree_01.hpp
• graph/base.hpp
• graph/ds/lct_node_commutative_monoid.hpp
• graph/ds/link_cut_tree.hpp

Code

#include "graph/ds/link_cut_tree.hpp"
#include "graph/ds/lct_node_commutative_monoid.hpp"
#include "alg/monoid/min.hpp"
#include "graph/base.hpp"
#include "ds/fenwicktree/fenwicktree_01.hpp"

// https://codeforces.com/problemset/problem/1386/C (TLE)
// query(L,R) = # of component if edge L,...,R-1 are used.
struct Range_Edge_Conneced_Component_Query {
  Graph<int, 0>& G;
  vc<pair<int, int>> query;

  Range_Edge_Conneced_Component_Query(Graph<int, 0>& G) : G(G) {}

  void add_query(int l, int r) { query.eb(l, r); }

  using Mono = Monoid_Min<int>;
  using Node = LCT_Node_Commutative_Monoid<Mono>;

  vc<int> calc() {
    int N = G.N, M = G.M;
    Link_Cut_Tree<Node> LCT(N + M);
    int Q = len(query);
    vc<int> ANS(Q);
    vvc<int> QID(M);
    FOR(q, Q) {
      auto [l, r] = query[q];
      assert(0 <= l && l <= r && r <= M);
      if (r) QID[r - 1].eb(q);
    }

    FenwickTree_01 bit(M);
    FOR(i, M) {
      int a = G.edges[i].frm, b = G.edges[i].to;
      if (a != b && LCT.get_root(a) == LCT.get_root(b)) {
        int k = LCT.prod_path(a, b);
        int c = G.edges[k].frm, d = G.edges[k].to;
        bit.add(k, -1);
        LCT.cut(c, N + k), LCT.cut(d, N + k);
      }
      if (a != b) {
        LCT.set(N + i, i);
        LCT.link(a, N + i), LCT.link(b, N + i);
        bit.add(i, 1);
      }
      for (auto& q: QID[i]) {
        auto [l, r] = query[q];
        ANS[q] = N - bit.sum(l, r);
      }
    }
    return ANS;
  }
};

#line 1 "graph/ds/range_edge_connected_component_query.hpp"

#line 1 "graph/ds/link_cut_tree.hpp"
/*
各 heavy path を head が左, tail が右となるように splay tree で持つ.
ユーザーが直接呼ぶ可能性があるものだけ int でも実装.
LCT 外で探索するときなど，push を忘れないように注意.
*/

template <typename Node>
struct Link_Cut_Tree {
  using np = Node *;
  int n;
  vc<Node> nodes;

  Link_Cut_Tree(int n = 0) : n(n), nodes(n) { FOR(i, n) nodes[i] = Node(i); }

  Node *operator[](int v) { return &nodes[v]; }

  // underlying tree の根
  Node *get_root(Node *c) {
    expose(c);
    c->push();
    while (c->l) {
      c = c->l;
      c->push();
    }
    splay(c);
    return c;
  }

  // underlying tree の根
  int get_root(int c) { return get_root(&nodes[c])->idx; }

  // parent(c)==p となるように link.
  void link(Node *c, Node *p) {
    evert(c);
    expose(p);
    p->push();
    // no edge -> heavy edge
    assert(!(c->p));
    assert(!(p->r));
    c->p = p;
    p->r = c;
    p->update();
  }

  // parent(c)==p となるように link.
  void link(int c, int p) { return link(&nodes[c], &nodes[p]); }

  void cut(Node *a, Node *b) {
    evert(a);
    expose(b);
    assert(!b->p);
    assert((b->l) == a);
    // heavy edge -> no edge
    b->l->p = nullptr;
    b->l = nullptr;
    b->update();
  }

  void cut(int a, int b) { return cut(&nodes[a], &nodes[b]); }

  // c を underlying tree の根とする.
  // c は splay tree の根にもなる.
  // c は push 済になる
  void evert(Node *c) {
    expose(c);
    c->reverse();
    c->push();
  }

  // c を underlying tree の根とする.
  // c は splay tree の根にもなる.
  void evert(int c) { evert(&nodes[c]); }

  Node *lca(Node *u, Node *v) {
    assert(get_root(u) == get_root(v));
    expose(u);
    return expose(v);
  }

  int lca(int u, int v) { return lca(&nodes[u], &nodes[v])->idx; }

  // 辺の個数
  int dist(int u, int v) {
    evert(u), expose(v);
    return ((*this)[v]->size) - 1;
  }

  Node *jump(Node *u, Node *v, int k) {
    evert(v);
    expose(u);
    assert(0 <= k && k < (u->size));
    while (1) {
      u->push();
      int rs = (u->r ? u->r->size : 0);
      if (k < rs) {
        u = u->r;
        continue;
      }
      if (k == rs) { break; }
      k -= rs + 1;
      u = u->l;
    }
    splay(u);
    return u;
  }

  int jump(int u, int v, int k) {
    auto c = jump((*this)[u], (*this)[v], k);
    return c->idx;
  }

  // [root, c] がひとつの splay tree になるように変更する.
  // c が右端で splay tree の根という状態になる.
  // path query はこの状態で c の data を見る.
  // c は push 済になる
  virtual Node *expose(Node *c) {
    Node *now = c;
    Node *rp = nullptr; // 今まで作ったパス
    while (now) {
      splay(now);
      // heavy -> light, light -> heavy.
      if (now->r) { now->add_light(now->r); }
      if (rp) { now->erase_light(rp); }
      now->r = rp;
      now->update();
      rp = now;
      now = now->p;
    }
    splay(c);
    return rp;
  }

  // [root, c] がひとつの splay tree になるように変更する.
  // c が右端で splay tree の根という状態になる.
  // path query はこの状態で c の data を見る.
  int expose(int c) {
    Node *x = expose(&nodes[c]);
    if (!x) return -1;
    return x->idx;
  }

  Node *get_parent(Node *x) {
    expose(x);
    if (!x->l) return nullptr;
    x = x->l;
    while (x->r) x = x->r;
    return x;
  }

  int get_parent(int x) {
    Node *p = get_parent((*this)[x]);
    return (p ? p->idx : -1);
  }

  void set(Node *c, typename Node::VX x) {
    evert(c);
    c->set(x);
  }

  void set(int c, typename Node::VX x) { set((*this)[c], x); }

  typename Node::X prod_path(int a, int b) {
    evert(a), expose(b);
    return (*this)[b]->x;
  }

  // subtree 用の node を使う
  typename Node::X prod_subtree(int v, int root) {
    static_assert(Node::NODE_FOR_SUBTREE);
    if (v == root) {
      evert(root);
      return (*this)[root]->x;
    }
    root = jump(v, root, 1);
    cut(v, root);
    typename Node::X res = (*this)[v]->x;
    link(v, root);
    return res;
  }

  vc<int> collect_heavy_path(int v) {
    np c = (*this)[v];
    while (!is_root(c)) c = c->p;
    vc<int> res;
    auto dfs = [&](auto &dfs, np c, bool rev) -> void {
      if (!rev) {
        if (c->l) dfs(dfs, c->l, rev ^ c->rev);
        res.eb(c->idx);
        if (c->r) dfs(dfs, c->r, rev ^ c->rev);
      } else {
        if (c->r) dfs(dfs, c->r, rev ^ c->rev);
        res.eb(c->idx);
        if (c->l) dfs(dfs, c->l, rev ^ c->rev);
      }
    };
    dfs(dfs, c, false);
    return res;
  }

  void debug() {
    print("p, l, r, rev");
    auto f = [&](np c) -> int { return (c ? c->idx : -1); };
    FOR(i, len(nodes)) {
      print(i, ",", f((*this)[i]->p), f((*this)[i]->l), f((*this)[i]->r),
            (*this)[i]->rev);
    }
    FOR(i, len(nodes)) {
      np c = (*this)[i];
      if (c->l) assert(c->l->p == c);
      if (c->r) assert(c->r->p == c);
    }
  }

private:
  // splay tree 内で完結する操作. 特に heavy, light 構造は変わらない.
  // light pointer は rotate 内でケア
  // c は push 済になる
  void splay(Node *c) {
    c->push();
    while (!is_root(c)) {
      Node *p = c->p;
      Node *pp = (p ? p->p : nullptr);
      if (state(p) == 0) {
        p->push(), c->push();
        rotate(c);
      }
      elif (state(c) == state(p)) {
        pp->push(), p->push(), c->push();
        rotate(p);
        rotate(c);
      }
      else {
        pp->push(), p->push(), c->push();
        rotate(c);
        rotate(c);
      }
    }
  }

  // パスを表す splay tree の根になっているかどうか
  // underlying tree ではない
  bool is_root(Node *c) { return state(c) == 0; }

  // splay tree 内で完結する操作. 特に heavy, light 構造は変わらない.
  // light edge のポインタは変更されうる
  void rotate(Node *n) {
    // n を根に近づける
    Node *pp, *p, *c;
    p = n->p;
    pp = p->p;
    if (p->l == n) {
      c = n->r;
      n->r = p;
      p->l = c;
    } else {
      c = n->l;
      n->l = p;
      p->r = c;
    }
    p->update(), n->update();

    if (pp) {
      if (pp->l == p) pp->l = n;
      elif (pp->r == p) pp->r = n;
      else {
        // light edge pointer が (pp-p) から (pp-n) に変わる
        pp->change_light(p, n);
      }
    }
    n->p = pp;
    p->p = n;
    if (c) c->p = p;
  }

  inline int state(Node *n) {
    if (!n->p) return 0;
    if (n->p->l == n) return 1;
    if (n->p->r == n) return -1;
    return 0;
  }
};
#line 1 "graph/ds/lct_node_commutative_monoid.hpp"

// SUBTREE : cluster が subtree 情報を持つ場合
template <typename Monoid, bool SUBTREE = false>
struct LCT_Node_Commutative_Monoid {
  static_assert(Monoid::commute);
  static constexpr bool NODE_FOR_SUBTREE = SUBTREE;
  using np = LCT_Node_Commutative_Monoid *;
  // デフォルト
  np l, r, p;
  int idx, size; // size は heavy path の頂点数
  bool rev;
  // 目的ごとに定義する.
  using MX = Monoid;
  using X = typename MX::value_type;
  using VX = X;
  X x, vx, mid;

  LCT_Node_Commutative_Monoid(int i = 0)
      : l(nullptr),
        r(nullptr),
        p(nullptr),
        idx(i),
        size(1),
        rev(0),
        x(MX::unit()),
        vx(MX::unit()),
        mid(MX::unit()) {}

  void update() {
    size = 1;
    x = vx;
    if constexpr (SUBTREE) x = MX::op(x, mid);
    if (l) { size += l->size, x = Monoid::op(l->x, x); }
    if (r) { size += r->size, x = Monoid::op(x, r->x); }
  }

  void push() {
    if (rev) {
      if (l) l->reverse();
      if (r) r->reverse();
      rev = 0;
    }
  }

  // data の reverse も行う
  void reverse() {
    rev ^= 1;
    swap(l, r);
  }

  // LCT 内で expose, update を行うのでここは変更だけ
  void set(VX x) { vx = x; }

  void add_light(np c) {
    if constexpr (SUBTREE) mid = MX::op(mid, c->x);
  }
  void erase_light(np c) {
    if constexpr (SUBTREE) mid = MX::op(mid, MX::inverse(c->x));
  }

  // b->x に subtree value が入っている.
  void change_light(np a, np b) {}
};
#line 2 "alg/monoid/min.hpp"

template <typename E>
struct Monoid_Min {
  using X = E;
  using value_type = X;
  static constexpr X op(const X &x, const X &y) noexcept { return min(x, y); }
  static constexpr X unit() { return infty<E>; }
  static constexpr bool commute = true;
};
#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);
    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;
  }

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/fenwicktree/fenwicktree_01.hpp"

#line 2 "alg/monoid/add.hpp"

template <typename E>
struct Monoid_Add {
  using X = E;
  using value_type = X;
  static constexpr X op(const X &x, const X &y) noexcept { return x + y; }
  static constexpr X inverse(const X &x) noexcept { return -x; }
  static constexpr X power(const X &x, ll n) noexcept { return X(n) * x; }
  static constexpr X unit() { return X(0); }
  static constexpr bool commute = true;
};
#line 3 "ds/fenwicktree/fenwicktree.hpp"

template <typename Monoid>
struct FenwickTree {
  using G = Monoid;
  using E = typename G::value_type;
  int n;
  vector<E> dat;
  E total;

  FenwickTree() {}
  FenwickTree(int n) { build(n); }
  template <typename F>
  FenwickTree(int n, F f) {
    build(n, f);
  }
  FenwickTree(const vc<E>& v) { build(v); }

  void build(int m) {
    n = m;
    dat.assign(m, G::unit());
    total = G::unit();
  }
  void build(const vc<E>& v) {
    build(len(v), [&](int i) -> E { return v[i]; });
  }
  template <typename F>
  void build(int m, F f) {
    n = m;
    dat.clear();
    dat.reserve(n);
    total = G::unit();
    FOR(i, n) { dat.eb(f(i)); }
    for (int i = 1; i <= n; ++i) {
      int j = i + (i & -i);
      if (j <= n) dat[j - 1] = G::op(dat[i - 1], dat[j - 1]);
    }
    total = prefix_sum(m);
  }

  E prod_all() { return total; }
  E sum_all() { return total; }
  E sum(int k) { return prefix_sum(k); }
  E prod(int k) { return prefix_prod(k); }
  E prefix_sum(int k) { return prefix_prod(k); }
  E prefix_prod(int k) {
    chmin(k, n);
    E ret = G::unit();
    for (; k > 0; k -= k & -k) ret = G::op(ret, dat[k - 1]);
    return ret;
  }
  E sum(int L, int R) { return prod(L, R); }
  E prod(int L, int R) {
    chmax(L, 0), chmin(R, n);
    if (L == 0) return prefix_prod(R);
    assert(0 <= L && L <= R && R <= n);
    E pos = G::unit(), neg = G::unit();
    while (L < R) { pos = G::op(pos, dat[R - 1]), R -= R & -R; }
    while (R < L) { neg = G::op(neg, dat[L - 1]), L -= L & -L; }
    return G::op(pos, G::inverse(neg));
  }

  vc<E> get_all() {
    vc<E> res(n);
    FOR(i, n) res[i] = prod(i, i + 1);
    return res;
  }

  void add(int k, E x) { multiply(k, x); }
  void multiply(int k, E x) {
    static_assert(G::commute);
    total = G::op(total, x);
    for (++k; k <= n; k += k & -k) dat[k - 1] = G::op(dat[k - 1], x);
  }

  template <class F>
  int max_right(const F check, int L = 0) {
    assert(check(G::unit()));
    E s = G::unit();
    int i = L;
    // 2^k 進むとダメ
    int k = [&]() {
      while (1) {
        if (i % 2 == 1) { s = G::op(s, G::inverse(dat[i - 1])), i -= 1; }
        if (i == 0) { return topbit(n) + 1; }
        int k = lowbit(i) - 1;
        if (i + (1 << k) > n) return k;
        E t = G::op(s, dat[i + (1 << k) - 1]);
        if (!check(t)) { return k; }
        s = G::op(s, G::inverse(dat[i - 1])), i -= i & -i;
      }
    }();
    while (k) {
      --k;
      if (i + (1 << k) - 1 < len(dat)) {
        E t = G::op(s, dat[i + (1 << k) - 1]);
        if (check(t)) { i += (1 << k), s = t; }
      }
    }
    return i;
  }

  // check(i, x)
  template <class F>
  int max_right_with_index(const F check, int L = 0) {
    assert(check(L, G::unit()));
    E s = G::unit();
    int i = L;
    // 2^k 進むとダメ
    int k = [&]() {
      while (1) {
        if (i % 2 == 1) { s = G::op(s, G::inverse(dat[i - 1])), i -= 1; }
        if (i == 0) { return topbit(n) + 1; }
        int k = lowbit(i) - 1;
        if (i + (1 << k) > n) return k;
        E t = G::op(s, dat[i + (1 << k) - 1]);
        if (!check(i + (1 << k), t)) { return k; }
        s = G::op(s, G::inverse(dat[i - 1])), i -= i & -i;
      }
    }();
    while (k) {
      --k;
      if (i + (1 << k) - 1 < len(dat)) {
        E t = G::op(s, dat[i + (1 << k) - 1]);
        if (check(i + (1 << k), t)) { i += (1 << k), s = t; }
      }
    }
    return i;
  }

  template <class F>
  int min_left(const F check, int R) {
    assert(check(G::unit()));
    E s = G::unit();
    int i = R;
    // false になるところまで戻る
    int k = 0;
    while (i > 0 && check(s)) {
      s = G::op(s, dat[i - 1]);
      k = lowbit(i);
      i -= i & -i;
    }
    if (check(s)) {
      assert(i == 0);
      return 0;
    }
    // 2^k 進むと ok になる
    // false を維持して進む
    while (k) {
      --k;
      E t = G::op(s, G::inverse(dat[i + (1 << k) - 1]));
      if (!check(t)) { i += (1 << k), s = t; }
    }
    return i + 1;
  }

  int kth(E k, int L = 0) {
    return max_right([&k](E x) -> bool { return x <= k; }, L);
  }
};
#line 4 "ds/fenwicktree/fenwicktree_01.hpp"

struct FenwickTree_01 {
  int N, n;
  vc<u64> dat;
  FenwickTree<Monoid_Add<int>> bit;
  FenwickTree_01() {}
  FenwickTree_01(int n) { build(n); }
  template <typename F>
  FenwickTree_01(int n, F f) {
    build(n, f);
  }

  void build(int m) {
    N = m;
    n = ceil<int>(N + 1, 64);
    dat.assign(n, u64(0));
    bit.build(n);
  }

  template <typename F>
  void build(int m, F f) {
    N = m;
    n = ceil<int>(N + 1, 64);
    dat.assign(n, u64(0));
    FOR(i, N) { dat[i / 64] |= u64(f(i)) << (i % 64); }
    bit.build(n, [&](int i) -> int { return popcnt(dat[i]); });
  }

  int sum_all() { return bit.sum_all(); }
  int sum(int k) { return prefix_sum(k); }
  int prefix_sum(int k) {
    int ans = bit.sum(k / 64);
    ans += popcnt(dat[k / 64] & ((u64(1) << (k % 64)) - 1));
    return ans;
  }
  int sum(int L, int R) {
    if (L == 0) return prefix_sum(R);
    int ans = 0;
    ans -= popcnt(dat[L / 64] & ((u64(1) << (L % 64)) - 1));
    ans += popcnt(dat[R / 64] & ((u64(1) << (R % 64)) - 1));
    ans += bit.sum(L / 64, R / 64);
    return ans;
  }

  void add(int k, int x) {
    if (x == 1) add(k);
    if (x == -1) remove(k);
  }

  void add(int k) {
    dat[k / 64] |= u64(1) << (k % 64);
    bit.add(k / 64, 1);
  }
  void remove(int k) {
    dat[k / 64] &= ~(u64(1) << (k % 64));
    bit.add(k / 64, -1);
  }

  int kth(int k, int L = 0) {
    if (k >= sum_all()) return N;
    k += popcnt(dat[L / 64] & ((u64(1) << (L % 64)) - 1));
    L /= 64;
    int mid = 0;
    auto check = [&](auto e) -> bool {
      if (e <= k) chmax(mid, e);
      return e <= k;
    };
    int idx = bit.max_right(check, L);
    if (idx == n) return N;
    k -= mid;
    u64 x = dat[idx];
    int p = popcnt(x);
    if (p <= k) return N;
    k = binary_search([&](int n) -> bool { return (p - popcnt(x >> n)) <= k; },
                      0, 64, 0);
    return 64 * idx + k;
  }

  int next(int k) {
    int idx = k / 64;
    k %= 64;
    u64 x = dat[idx] & ~((u64(1) << k) - 1);
    if (x) return 64 * idx + lowbit(x);
    idx = bit.kth(0, idx + 1);
    if (idx == n || !dat[idx]) return N;
    return 64 * idx + lowbit(dat[idx]);
  }

  int prev(int k) {
    if (k == N) --k;
    int idx = k / 64;
    k %= 64;
    u64 x = dat[idx];
    if (k < 63) x &= (u64(1) << (k + 1)) - 1;
    if (x) return 64 * idx + topbit(x);
    idx = bit.min_left([&](auto e) -> bool { return e <= 0; }, idx) - 1;
    if (idx == -1) return -1;
    return 64 * idx + topbit(dat[idx]);
  }
};
#line 7 "graph/ds/range_edge_connected_component_query.hpp"

// https://codeforces.com/problemset/problem/1386/C (TLE)
// query(L,R) = # of component if edge L,...,R-1 are used.
struct Range_Edge_Conneced_Component_Query {
  Graph<int, 0>& G;
  vc<pair<int, int>> query;

  Range_Edge_Conneced_Component_Query(Graph<int, 0>& G) : G(G) {}

  void add_query(int l, int r) { query.eb(l, r); }

  using Mono = Monoid_Min<int>;
  using Node = LCT_Node_Commutative_Monoid<Mono>;

  vc<int> calc() {
    int N = G.N, M = G.M;
    Link_Cut_Tree<Node> LCT(N + M);
    int Q = len(query);
    vc<int> ANS(Q);
    vvc<int> QID(M);
    FOR(q, Q) {
      auto [l, r] = query[q];
      assert(0 <= l && l <= r && r <= M);
      if (r) QID[r - 1].eb(q);
    }

    FenwickTree_01 bit(M);
    FOR(i, M) {
      int a = G.edges[i].frm, b = G.edges[i].to;
      if (a != b && LCT.get_root(a) == LCT.get_root(b)) {
        int k = LCT.prod_path(a, b);
        int c = G.edges[k].frm, d = G.edges[k].to;
        bit.add(k, -1);
        LCT.cut(c, N + k), LCT.cut(d, N + k);
      }
      if (a != b) {
        LCT.set(N + i, i);
        LCT.link(a, N + i), LCT.link(b, N + i);
        bit.add(i, 1);
      }
      for (auto& q: QID[i]) {
        auto [l, r] = query[q];
        ANS[q] = N - bit.sum(l, r);
      }
    }
    return ANS;
  }
};

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