geo/count_points_in_triangles.hpp

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• Last update: 2023-11-07 01:10:45+09:00
• Include: #include "geo/count_points_in_triangles.hpp"
• Link: https://codeforces.com/contest/13/problem/D

Depends on

• geo/angle_sort.hpp
• geo/base.hpp
• random/base.hpp

Verified with

• test/mytest/count_points_in_triangles.test.cpp
• test_atcoder/abc202_f.test.cpp

Code

#include "geo/angle_sort.hpp"
#include "geo/base.hpp"
#include "random/base.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(N^2M)、クエリ O(1)
// https://codeforces.com/contest/13/problem/D
struct Count_Points_In_Triangles {
  using P = Point<ll>;
  const int LIM = 1'000'000'000 + 10;
  vc<P> A, B;
  vc<int> I, rk; // 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(vc<P> A, vc<P> B) : A(A), B(B) {
    for (auto&& p: A) assert(-LIM < min(p.x, p.y) && max(p.x, p.y) < LIM);
    for (auto&& p: B) assert(-LIM < min(p.x, p.y) && max(p.x, p.y) < LIM);
    build();
  }

  int query(int i, int j, int k) {
    i = rk[i], j = rk[j], k = rk[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() {
    int N = len(A), M = len(B);
    while (1) {
      P O = P{-LIM, RNG(-LIM, LIM)};
      bool ok = 1;
      FOR(i, N) FOR(j, N) {
        if (A[i] == A[j]) continue;
        if ((A[i] - O).det(A[j] - O) == 0) ok = 0;
      }
      FOR(i, N) FOR(j, M) {
        if (A[i] == B[j]) continue;
        if ((A[i] - O).det(B[j] - O) == 0) ok = 0;
      }
      if (ok) return O;
    }
    return P{};
  }

  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);
    I.resize(N), rk.resize(N);
    iota(all(I), 0);
    sort(all(I), [&](auto& a, auto& b) -> bool { return A[a].det(A[b]) > 0; });
    FOR(i, N) rk[I[i]] = i;
    A = rearrange(A, I);
    point.assign(N, 0);
    seg.assign(N, vc<int>(N));
    tri.assign(N, vc<int>(N));

    FOR(i, N) FOR(j, M) if (A[i] == B[j])++ point[i];
    FOR(i, N) FOR(j, i + 1, N) {
      FOR(k, M) {
        if (A[i].det(B[k]) <= 0) continue;
        if (A[j].det(B[k]) >= 0) continue;
        ll d = (B[k] - A[i]).det(A[j] - A[i]);
        if (d == 0) ++seg[i][j];
        if (d < 0) ++tri[i][j];
      }
    }
  }
};

#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>
REAL dist(Point<T> A, Point<T> B) {
  A = A - B;
  T p = A.dot(A);
  return sqrt(REAL(p));
}

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 5 "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(N^2M)、クエリ O(1)
// https://codeforces.com/contest/13/problem/D
struct Count_Points_In_Triangles {
  using P = Point<ll>;
  const int LIM = 1'000'000'000 + 10;
  vc<P> A, B;
  vc<int> I, rk; // 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(vc<P> A, vc<P> B) : A(A), B(B) {
    for (auto&& p: A) assert(-LIM < min(p.x, p.y) && max(p.x, p.y) < LIM);
    for (auto&& p: B) assert(-LIM < min(p.x, p.y) && max(p.x, p.y) < LIM);
    build();
  }

  int query(int i, int j, int k) {
    i = rk[i], j = rk[j], k = rk[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() {
    int N = len(A), M = len(B);
    while (1) {
      P O = P{-LIM, RNG(-LIM, LIM)};
      bool ok = 1;
      FOR(i, N) FOR(j, N) {
        if (A[i] == A[j]) continue;
        if ((A[i] - O).det(A[j] - O) == 0) ok = 0;
      }
      FOR(i, N) FOR(j, M) {
        if (A[i] == B[j]) continue;
        if ((A[i] - O).det(B[j] - O) == 0) ok = 0;
      }
      if (ok) return O;
    }
    return P{};
  }

  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);
    I.resize(N), rk.resize(N);
    iota(all(I), 0);
    sort(all(I), [&](auto& a, auto& b) -> bool { return A[a].det(A[b]) > 0; });
    FOR(i, N) rk[I[i]] = i;
    A = rearrange(A, I);
    point.assign(N, 0);
    seg.assign(N, vc<int>(N));
    tri.assign(N, vc<int>(N));

    FOR(i, N) FOR(j, M) if (A[i] == B[j])++ point[i];
    FOR(i, N) FOR(j, i + 1, N) {
      FOR(k, M) {
        if (A[i].det(B[k]) <= 0) continue;
        if (A[j].det(B[k]) >= 0) continue;
        ll d = (B[k] - A[i]).det(A[j] - A[i]);
        if (d == 0) ++seg[i][j];
        if (d < 0) ++tri[i][j];
      }
    }
  }
};

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