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:heavy_check_mark: geo/count_points_in_triangles.hpp

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Code

#include "geo/angle_sort.hpp"
#include "geo/base.hpp"
#include "random/base.hpp"
#include "ds/fenwicktree/fenwicktree_01.hpp"

// 点群 A, B を入力 (Point<ll>)
// query(i,j,k):三角形 AiAjAk 内部の Bl の個数(非負)を返す
// 前計算 O(NMlogM)、クエリ O(1)
// https://codeforces.com/contest/13/problem/D
// https://codeforces.com/contest/852/problem/H
struct Count_Points_In_Triangles {
  using P = Point<ll>;
  const int LIM = 1'000'000'000 + 10;
  vc<P> A, B;
  vc<int> new_idx; // O から見た偏角ソート順を管理
  vc<int> point;   // A[i] と一致する B[j] の数え上げ
  vvc<int> seg;    // 線分 A[i]A[j] 内にある B[k] の数え上げ
  vvc<int> tri;    // OA[i]A[j] 内部にある B[k] の数え上げ
  Count_Points_In_Triangles(const vc<P>& A, const vc<P>& B) : A(A), B(B) {
    for (auto&& p: A) assert(max(abs(p.x), abs(p.y)) < LIM);
    for (auto&& p: B) assert(max(abs(p.x), abs(p.y)) < LIM);
    build();
  }

  int query(int i, int j, int k) {
    i = new_idx[i], j = new_idx[j], k = new_idx[k];
    if (i > j) swap(i, j);
    if (j > k) swap(j, k);
    if (i > j) swap(i, j);
    assert(i <= j && j <= k);
    ll d = (A[j] - A[i]).det(A[k] - A[i]);
    if (d == 0) return 0;
    if (d > 0) { return tri[i][j] + tri[j][k] - tri[i][k] - seg[i][k]; }
    int x = tri[i][k] - tri[i][j] - tri[j][k];
    return x - seg[i][j] - seg[j][k] - point[j];
  }

private:
  P take_origin() {
    // OAiAj, OAiBj が同一直線上にならないようにする
    // fail prob: at most N(N+M)/LIM
    return P{-LIM, RNG(-LIM, LIM)};
  }

  void build() {
    P O = take_origin();
    for (auto&& p: A) p = p - O;
    for (auto&& p: B) p = p - O;
    int N = len(A), M = len(B);
    vc<int> I = angle_sort(A);
    A = rearrange(A, I);
    new_idx.resize(N);
    FOR(i, N) new_idx[I[i]] = i;

    I = angle_sort(B);
    B = rearrange(B, I);

    point.assign(N, 0);
    seg.assign(N, vc<int>(N));
    tri.assign(N, vc<int>(N));

    // point
    FOR(i, N) FOR(j, M) if (A[i] == B[j])++ point[i];

    int m = 0;
    FOR(j, N) {
      // OA[i]A[j], B[k]
      while (m < M && A[j].det(B[m]) < 0) ++m;
      vc<P> C(m);
      FOR(k, m) C[k] = B[k] - A[j];
      vc<int> I(m);
      FOR(i, m) I[i] = i;
      sort(all(I),
           [&](auto& a, auto& b) -> bool { return C[a].det(C[b]) > 0; });
      C = rearrange(C, I);
      vc<int> rk(m);
      FOR(k, m) rk[I[k]] = k;
      FenwickTree_01 bit(m);

      int k = m;
      FOR_R(i, j) {
        while (k > 0 && A[i].det(B[k - 1]) > 0) { bit.add(rk[--k], 1); }
        P p = A[i] - A[j];
        int lb = binary_search(
            [&](int n) -> bool {
              return (n == 0 ? true : C[n - 1].det(p) > 0);
            },
            0, m + 1);
        int ub = binary_search(
            [&](int n) -> bool {
              return (n == 0 ? true : C[n - 1].det(p) >= 0);
            },
            0, m + 1);
        seg[i][j] += bit.sum(lb, ub), tri[i][j] += bit.sum(lb);
      }
    }
  }
};
#line 1 "geo/count_points_in_triangles.hpp"

#line 2 "geo/angle_sort.hpp"

#line 2 "geo/base.hpp"
template <typename T>
struct Point {
  T x, y;

  Point() : x(0), y(0) {}

  template <typename A, typename B>
  Point(A x, B y) : x(x), y(y) {}

  template <typename A, typename B>
  Point(pair<A, B> p) : x(p.fi), y(p.se) {}

  Point operator+(Point p) const { return {x + p.x, y + p.y}; }
  Point operator-(Point p) const { return {x - p.x, y - p.y}; }
  bool operator==(Point p) const { return x == p.x && y == p.y; }
  bool operator!=(Point p) const { return x != p.x || y != p.y; }
  Point operator-() const { return {-x, -y}; }
  Point operator*(T t) const { return {x * t, y * t}; }
  Point operator/(T t) const { return {x / t, y / t}; }

  bool operator<(Point p) const {
    if (x != p.x) return x < p.x;
    return y < p.y;
  }
  T dot(Point other) { return x * other.x + y * other.y; }
  T det(Point other) { return x * other.y - y * other.x; }

  double norm() { return sqrtl(x * x + y * y); }
  double angle() { return atan2(y, x); }

  Point rotate(double theta) {
    static_assert(!is_integral<T>::value);
    double c = cos(theta), s = sin(theta);
    return Point{c * x - s * y, s * x + c * y};
  }
};

#ifdef FASTIO
template <typename T>
void rd(Point<T>& p) {
  fastio::rd(p.x), fastio::rd(p.y);
}
template <typename T>
void wt(Point<T>& p) {
  fastio::wt(p.x);
  fastio::wt(' ');
  fastio::wt(p.y);
}
#endif

// A -> B -> C と進むときに、左に曲がるならば +1、右に曲がるならば -1
template <typename T>
int ccw(Point<T> A, Point<T> B, Point<T> C) {
  T x = (B - A).det(C - A);
  if (x > 0) return 1;
  if (x < 0) return -1;
  return 0;
}

template <typename REAL, typename T, typename U>
REAL dist(Point<T> A, Point<U> B) {
  REAL dx = REAL(A.x) - REAL(B.x);
  REAL dy = REAL(A.y) - REAL(B.y);
  return sqrt(dx * dx + dy * dy);
}

// ax+by+c
template <typename T>
struct Line {
  T a, b, c;

  Line(T a, T b, T c) : a(a), b(b), c(c) {}
  Line(Point<T> A, Point<T> B) {
    a = A.y - B.y, b = B.x - A.x, c = A.x * B.y - A.y * B.x;
  }
  Line(T x1, T y1, T x2, T y2) : Line(Point<T>(x1, y1), Point<T>(x2, y2)) {}

  template <typename U>
  U eval(Point<U> P) {
    return a * P.x + b * P.y + c;
  }

  template <typename U>
  T eval(U x, U y) {
    return a * x + b * y + c;
  }

  // 同じ直線が同じ a,b,c で表現されるようにする
  void normalize() {
    static_assert(is_same_v<T, int> || is_same_v<T, long long>);
    T g = gcd(gcd(abs(a), abs(b)), abs(c));
    a /= g, b /= g, c /= g;
    if (b < 0) { a = -a, b = -b, c = -c; }
    if (b == 0 && a < 0) { a = -a, b = -b, c = -c; }
  }

  bool is_parallel(Line other) { return a * other.b - b * other.a == 0; }
  bool is_orthogonal(Line other) { return a * other.a + b * other.b == 0; }
};

template <typename T>
struct Segment {
  Point<T> A, B;

  Segment(Point<T> A, Point<T> B) : A(A), B(B) {}
  Segment(T x1, T y1, T x2, T y2)
      : Segment(Point<T>(x1, y1), Point<T>(x2, y2)) {}

  bool contain(Point<T> C) {
    static_assert(is_integral<T>::value);
    T det = (C - A).det(B - A);
    if (det != 0) return 0;
    return (C - A).dot(B - A) >= 0 && (C - B).dot(A - B) >= 0;
  }

  Line<T> to_Line() { return Line(A, B); }
};

template <typename REAL>
struct Circle {
  Point<REAL> O;
  REAL r;
  Circle(Point<REAL> O, REAL r) : O(O), r(r) {}
  Circle(REAL x, REAL y, REAL r) : O(x, y), r(r) {}
  template <typename T>
  bool contain(Point<T> p) {
    REAL dx = p.x - O.x, dy = p.y - O.y;
    return dx * dx + dy * dy <= r * r;
  }
};

template <typename T>
struct Polygon {
  vc<Point<T>> points;
  T a;

  template <typename A, typename B>
  Polygon(vc<pair<A, B>> pairs) {
    for (auto&& [a, b]: pairs) points.eb(Point<T>(a, b));
    build();
  }
  Polygon(vc<Point<T>> points) : points(points) { build(); }

  int size() { return len(points); }

  template <typename REAL>
  REAL area() {
    return a * 0.5;
  }

  template <enable_if_t<is_integral<T>::value, int> = 0>
  T area_2() {
    return a;
  }

  bool is_convex() {
    FOR(j, len(points)) {
      int i = (j == 0 ? len(points) - 1 : j - 1);
      int k = (j == len(points) - 1 ? 0 : j + 1);
      if ((points[j] - points[i]).det(points[k] - points[j]) < 0) return false;
    }
    return true;
  }

private:
  void build() {
    a = 0;
    FOR(i, len(points)) {
      int j = (i + 1 == len(points) ? 0 : i + 1);
      a += points[i].det(points[j]);
    }
    if (a < 0) {
      a = -a;
      reverse(all(points));
    }
  }
};
#line 4 "geo/angle_sort.hpp"

// 偏角ソートに対する argsort

template <typename T>
vector<int> angle_sort(vector<Point<T>>& P) {
  vector<int> lower, origin, upper;
  const Point<T> O = {0, 0};
  FOR(i, len(P)) {
    if (P[i] == O) origin.eb(i);
    elif ((P[i].y < 0) || (P[i].y == 0 && P[i].x > 0)) lower.eb(i);
    else upper.eb(i);
  }
  sort(all(lower), [&](auto& i, auto& j) { return P[i].det(P[j]) > 0; });
  sort(all(upper), [&](auto& i, auto& j) { return P[i].det(P[j]) > 0; });
  auto& I = lower;
  I.insert(I.end(), all(origin));
  I.insert(I.end(), all(upper));
  return I;
}

// 偏角ソートに対する argsort

template <typename T>
vector<int> angle_sort(vector<pair<T, T>>& P) {
  vc<Point<T>> tmp(len(P));
  FOR(i, len(P)) tmp[i] = Point<T>(P[i]);
  return angle_sort<T>(tmp);
}
#line 2 "random/base.hpp"

u64 RNG_64() {
  static uint64_t x_
      = uint64_t(chrono::duration_cast<chrono::nanoseconds>(
                     chrono::high_resolution_clock::now().time_since_epoch())
                     .count())
        * 10150724397891781847ULL;
  x_ ^= x_ << 7;
  return x_ ^= x_ >> 9;
}

u64 RNG(u64 lim) { return RNG_64() % lim; }

ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); }
#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 6 "geo/count_points_in_triangles.hpp"

// 点群 A, B を入力 (Point<ll>)
// query(i,j,k):三角形 AiAjAk 内部の Bl の個数(非負)を返す
// 前計算 O(NMlogM)、クエリ O(1)
// https://codeforces.com/contest/13/problem/D
// https://codeforces.com/contest/852/problem/H
struct Count_Points_In_Triangles {
  using P = Point<ll>;
  const int LIM = 1'000'000'000 + 10;
  vc<P> A, B;
  vc<int> new_idx; // O から見た偏角ソート順を管理
  vc<int> point;   // A[i] と一致する B[j] の数え上げ
  vvc<int> seg;    // 線分 A[i]A[j] 内にある B[k] の数え上げ
  vvc<int> tri;    // OA[i]A[j] 内部にある B[k] の数え上げ
  Count_Points_In_Triangles(const vc<P>& A, const vc<P>& B) : A(A), B(B) {
    for (auto&& p: A) assert(max(abs(p.x), abs(p.y)) < LIM);
    for (auto&& p: B) assert(max(abs(p.x), abs(p.y)) < LIM);
    build();
  }

  int query(int i, int j, int k) {
    i = new_idx[i], j = new_idx[j], k = new_idx[k];
    if (i > j) swap(i, j);
    if (j > k) swap(j, k);
    if (i > j) swap(i, j);
    assert(i <= j && j <= k);
    ll d = (A[j] - A[i]).det(A[k] - A[i]);
    if (d == 0) return 0;
    if (d > 0) { return tri[i][j] + tri[j][k] - tri[i][k] - seg[i][k]; }
    int x = tri[i][k] - tri[i][j] - tri[j][k];
    return x - seg[i][j] - seg[j][k] - point[j];
  }

private:
  P take_origin() {
    // OAiAj, OAiBj が同一直線上にならないようにする
    // fail prob: at most N(N+M)/LIM
    return P{-LIM, RNG(-LIM, LIM)};
  }

  void build() {
    P O = take_origin();
    for (auto&& p: A) p = p - O;
    for (auto&& p: B) p = p - O;
    int N = len(A), M = len(B);
    vc<int> I = angle_sort(A);
    A = rearrange(A, I);
    new_idx.resize(N);
    FOR(i, N) new_idx[I[i]] = i;

    I = angle_sort(B);
    B = rearrange(B, I);

    point.assign(N, 0);
    seg.assign(N, vc<int>(N));
    tri.assign(N, vc<int>(N));

    // point
    FOR(i, N) FOR(j, M) if (A[i] == B[j])++ point[i];

    int m = 0;
    FOR(j, N) {
      // OA[i]A[j], B[k]
      while (m < M && A[j].det(B[m]) < 0) ++m;
      vc<P> C(m);
      FOR(k, m) C[k] = B[k] - A[j];
      vc<int> I(m);
      FOR(i, m) I[i] = i;
      sort(all(I),
           [&](auto& a, auto& b) -> bool { return C[a].det(C[b]) > 0; });
      C = rearrange(C, I);
      vc<int> rk(m);
      FOR(k, m) rk[I[k]] = k;
      FenwickTree_01 bit(m);

      int k = m;
      FOR_R(i, j) {
        while (k > 0 && A[i].det(B[k - 1]) > 0) { bit.add(rk[--k], 1); }
        P p = A[i] - A[j];
        int lb = binary_search(
            [&](int n) -> bool {
              return (n == 0 ? true : C[n - 1].det(p) > 0);
            },
            0, m + 1);
        int ub = binary_search(
            [&](int n) -> bool {
              return (n == 0 ? true : C[n - 1].det(p) >= 0);
            },
            0, m + 1);
        seg[i][j] += bit.sum(lb, ub), tri[i][j] += bit.sum(lb);
      }
    }
  }
};
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