random/random_polygon.hpp

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• Last update: 2024-12-25 20:50:37+09:00
• Include: #include "random/random_polygon.hpp"

Depends on

• alg/monoid/add.hpp
• ds/fenwicktree/fenwicktree.hpp
• ds/fenwicktree/fenwicktree_01.hpp
• geo/angle_sort.hpp
• geo/base.hpp
• geo/convex_hull.hpp
• geo/count_points_in_triangles.hpp
• geo/cross_point.hpp
• random/base.hpp

Verified with

• test/1_mytest/polygon_triangulation.test.cpp

Code

#include "random/base.hpp"
#include "geo/base.hpp"
#include "geo/convex_hull.hpp"
#include "geo/cross_point.hpp"
#include "geo/count_points_in_triangles.hpp"

vc<Point<ll>> random_polygon(int N, int XY_ABS_MAX = 10) {
  assert(N >= 3);
  using P = Point<ll>;
  auto trial = [&]() -> vc<P> {
    set<Point<ll>> S;
    while (len(S) < N) {
      int x = RNG(-XY_ABS_MAX, XY_ABS_MAX + 1);
      int y = RNG(-XY_ABS_MAX, XY_ABS_MAX + 1);
      S.insert(Point<ll>(x, y));
    }
    vc<P> point(all(S));
    auto I = ConvexHull<ll, true>(point);
    Count_Points_In_Triangles CT(point, point);
    vc<int> other;
    vc<int> done(N);
    for (auto& i: I) done[i]++;
    if (MAX(done) >= 2) return {};
    FOR(i, N) if (!done[i]) other.eb(i);
    int fail = 0;
    while (len(other)) {
      if (fail > 1000) return {};
      ++fail;
      int i = RNG(0, len(I)), j = RNG(0, len(other));
      swap(other[j], other.back());
      int a = I[i], b = I[(i + 1) % len(I)], c = other.back();
      if ((point[b] - point[a]).det(point[c] - point[a]) < 0) continue;
      if (CT.count3(a, b, c)) continue;
      if (CT.count2(a, c) + CT.count2(b, c)) continue;
      bool ok = 1;
      for (auto& v: {a, b}) {
        FOR(i, len(I)) {
          Segment<ll> S1(point[v], point[c]);
          Segment<ll> S2(point[I[i]], point[I[(i + 1) % len(I)]]);
          if (count_cross(S1, S2, false)) ok = 0;
        }
      }
      if (!ok) continue;
      fail = 0;
      I.insert(I.begin() + i + 1, POP(other));
    }
    point = rearrange(point, I);
    FOR(i, N) {
      if ((point[(i + 2) % N] - point[i]).det(point[(i + 1) % N] - point[i]) == 0) return {};
    }
    return point;
  };
  while (1) {
    vc<P> ANS = trial();
    if (ANS.empty()) continue;
    int k = RNG(0, len(ANS));
    rotate(ANS.begin(), ANS.begin() + k, ANS.end());
    return ANS;
  }
}

#line 1 "random/random_polygon.hpp"

#line 2 "random/base.hpp"

u64 RNG_64() {
  static u64 x_ = u64(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 "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+=(const Point p) {
    x += p.x, y += p.y;
    return *this;
  }
  Point operator-=(const Point p) {
    x -= p.x, y -= p.y;
    return *this;
  }
  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(const Point& other) const { return x * other.x + y * other.y; }
  T det(const Point& other) const { 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};
  }
  Point rot90(bool ccw) { return (ccw ? Point{-y, x} : Point{y, -x}); }
};

#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) {
    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;
  }
};
#line 2 "geo/convex_hull.hpp"

#line 4 "geo/convex_hull.hpp"

// allow_180=true で同一座標点があるとこわれる
// full なら I[0] が sorted で min になる
template <typename T, bool allow_180 = false>
vector<int> ConvexHull(vector<Point<T>>& XY, string mode = "full", bool sorted = false) {
  assert(mode == "full" || mode == "lower" || mode == "upper");
  ll N = XY.size();
  if (N == 1) return {0};
  if (N == 2) {
    if (XY[0] < XY[1]) return {0, 1};
    if (XY[1] < XY[0]) return {1, 0};
    return {0};
  }
  vc<int> I(N);
  if (sorted) {
    FOR(i, N) I[i] = i;
  } else {
    I = argsort(XY);
  }
  if constexpr (allow_180) { FOR(i, N - 1) assert(XY[i] != XY[i + 1]); }

  auto check = [&](ll i, ll j, ll k) -> bool {
    T det = (XY[j] - XY[i]).det(XY[k] - XY[i]);
    if constexpr (allow_180) return det >= 0;
    return det > T(0);
  };

  auto calc = [&]() {
    vector<int> P;
    for (auto&& k: I) {
      while (P.size() > 1) {
        auto i = P[P.size() - 2];
        auto j = P[P.size() - 1];
        if (check(i, j, k)) break;
        P.pop_back();
      }
      P.eb(k);
    }
    return P;
  };

  vc<int> P;
  if (mode == "full" || mode == "lower") {
    vc<int> Q = calc();
    P.insert(P.end(), all(Q));
  }
  if (mode == "full" || mode == "upper") {
    if (!P.empty()) P.pop_back();
    reverse(all(I));
    vc<int> Q = calc();
    P.insert(P.end(), all(Q));
  }
  if (mode == "upper") reverse(all(P));
  while (len(P) >= 2 && XY[P[0]] == XY[P.back()]) P.pop_back();
  return P;
}
#line 2 "geo/cross_point.hpp"

#line 4 "geo/cross_point.hpp"

// 平行でないことを仮定
template <typename REAL, typename T>
Point<REAL> cross_point(const Line<T> L1, const Line<T> L2) {
  T det = L1.a * L2.b - L1.b * L2.a;
  assert(det != 0);
  REAL x = -REAL(L1.c) * L2.b + REAL(L1.b) * L2.c;
  REAL y = -REAL(L1.a) * L2.c + REAL(L1.c) * L2.a;
  return Point<REAL>(x / det, y / det);
}

// 浮動小数点数はエラー
// 0: 交点なし
// 1: 一意な交点
// 2：2 つ以上の交点（整数型を利用して厳密にやる）
template <typename T>
int count_cross(Segment<T> S1, Segment<T> S2, bool include_ends) {
  static_assert(!std::is_floating_point<T>::value);
  Line<T> L1 = S1.to_Line();
  Line<T> L2 = S2.to_Line();
  if (L1.is_parallel(L2)) {
    if (L1.eval(S2.A) != 0) return 0;
    // 4 点とも同一直線上にある
    T a1 = S1.A.x, b1 = S1.B.x;
    T a2 = S2.A.x, b2 = S2.B.x;
    if (a1 == b1) {
      a1 = S1.A.y, b1 = S1.B.y;
      a2 = S2.A.y, b2 = S2.B.y;
    }
    if (a1 > b1) swap(a1, b1);
    if (a2 > b2) swap(a2, b2);
    T a = max(a1, a2);
    T b = min(b1, b2);
    if (a < b) return 2;
    if (a > b) return 0;
    return (include_ends ? 1 : 0);
  }
  // 平行でない場合
  T a1 = L2.eval(S1.A), b1 = L2.eval(S1.B);
  T a2 = L1.eval(S2.A), b2 = L1.eval(S2.B);
  if (a1 > b1) swap(a1, b1);
  if (a2 > b2) swap(a2, b2);
  bool ok1 = 0, ok2 = 0;

  if (include_ends) {
    ok1 = (a1 <= T(0)) && (T(0) <= b1);
    ok2 = (a2 <= T(0)) && (T(0) <= b2);
  } else {
    ok1 = (a1 < T(0)) && (T(0) < b1);
    ok2 = (a2 < T(0)) && (T(0) < b2);
  }
  return (ok1 && ok2 ? 1 : 0);
}

// 4 次式まで登場している、オーバーフロー注意！
// https://codeforces.com/contest/607/problem/E
template <typename REAL, typename T>
vc<Point<REAL>> cross_point(const Circle<T> C, const Line<T> L) {
  T a = L.a, b = L.b, c = L.a * (C.O.x) + L.b * (C.O.y) + L.c;
  T r = C.r;
  bool SW = 0;
  T abs_a = (a < 0 ? -a : a);
  T abs_b = (b < 0 ? -b : b);
  if (abs_a < abs_b) {
    swap(a, b);
    SW = 1;
  }
  // ax+by+c=0, x^2+y^2=r^2
  T D = 4 * c * c * b * b - 4 * (a * a + b * b) * (c * c - a * a * r * r);
  if (D < 0) return {};
  REAL sqD = sqrtl(D);
  REAL y1 = (-2 * b * c + sqD) / (2 * (a * a + b * b));
  REAL y2 = (-2 * b * c - sqD) / (2 * (a * a + b * b));
  REAL x1 = (-b * y1 - c) / a;
  REAL x2 = (-b * y2 - c) / a;
  if (SW) swap(x1, y1), swap(x2, y2);
  x1 += C.O.x, x2 += C.O.x;
  y1 += C.O.y, y2 += C.O.y;
  if (D == 0) return {Point<REAL>(x1, y1)};
  return {Point<REAL>(x1, y1), Point<REAL>(x2, y2)};
}

// https://codeforces.com/contest/2/problem/C
template <typename REAL, typename T>
tuple<bool, Point<T>, Point<T>> cross_point_circle(Circle<T> C1, Circle<T> C2) {
  using P = Point<T>;
  P O{0, 0};
  P A = C1.O, B = C2.O;
  if (A == B) return {false, O, O};
  T d = (B - A).norm();
  REAL cos_val = (C1.r * C1.r + d * d - C2.r * C2.r) / (2 * C1.r * d);
  if (cos_val < -1 || 1 < cos_val) return {false, O, O};
  REAL t = acos(cos_val);
  REAL u = (B - A).angle();
  P X = A + P{C1.r * cos(u + t), C1.r * sin(u + t)};
  P Y = A + P{C1.r * cos(u - t), C1.r * sin(u - t)};
  return {true, X, Y};
}
#line 1 "geo/count_points_in_triangles.hpp"

#line 2 "geo/angle_sort.hpp"

#line 4 "geo/angle_sort.hpp"

// lower: -1, origin: 0, upper: 1, (-pi,pi]

template <typename T> int lower_or_upper(const Point<T> &p) {
  if (p.y != 0)
    return (p.y > 0 ? 1 : -1);
  if (p.x > 0)
    return -1;
  if (p.x < 0)
    return 1;
  return 0;
}

// L<R:-1, L==R:0, L>R:1, (-pi,pi]

template <typename T> int angle_comp_3(const Point<T> &L, const Point<T> &R) {
  int a = lower_or_upper(L), b = lower_or_upper(R);
  if (a != b)
    return (a < b ? -1 : +1);
  T det = L.det(R);
  if (det > 0)
    return -1;
  if (det < 0)
    return 1;
  return 0;
}

// 偏角ソートに対する argsort, (-pi,pi]

template <typename T> vector<int> angle_sort(vector<Point<T>> &P) {
  vc<int> I(len(P));
  FOR(i, len(P)) I[i] = i;
  sort(all(I), [&](auto &L, auto &R) -> bool {
    return angle_comp_3(P[L], P[R]) == -1;
  });
  return I;
}

// 偏角ソートに対する argsort, (-pi,pi]

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 "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 MX = 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);
  }
  void set(int k, E x) { add(k, G::op(G::inverse(prod(k, 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);
    elif (x == -1) remove(k);
    else assert(0);
  }

  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 count3(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];
  }

  // segment
  int count2(int i, int j) {
    i = new_idx[i], j = new_idx[j];
    if (i > j) swap(i, j);
    return seg[i][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 7 "random/random_polygon.hpp"

vc<Point<ll>> random_polygon(int N, int XY_ABS_MAX = 10) {
  assert(N >= 3);
  using P = Point<ll>;
  auto trial = [&]() -> vc<P> {
    set<Point<ll>> S;
    while (len(S) < N) {
      int x = RNG(-XY_ABS_MAX, XY_ABS_MAX + 1);
      int y = RNG(-XY_ABS_MAX, XY_ABS_MAX + 1);
      S.insert(Point<ll>(x, y));
    }
    vc<P> point(all(S));
    auto I = ConvexHull<ll, true>(point);
    Count_Points_In_Triangles CT(point, point);
    vc<int> other;
    vc<int> done(N);
    for (auto& i: I) done[i]++;
    if (MAX(done) >= 2) return {};
    FOR(i, N) if (!done[i]) other.eb(i);
    int fail = 0;
    while (len(other)) {
      if (fail > 1000) return {};
      ++fail;
      int i = RNG(0, len(I)), j = RNG(0, len(other));
      swap(other[j], other.back());
      int a = I[i], b = I[(i + 1) % len(I)], c = other.back();
      if ((point[b] - point[a]).det(point[c] - point[a]) < 0) continue;
      if (CT.count3(a, b, c)) continue;
      if (CT.count2(a, c) + CT.count2(b, c)) continue;
      bool ok = 1;
      for (auto& v: {a, b}) {
        FOR(i, len(I)) {
          Segment<ll> S1(point[v], point[c]);
          Segment<ll> S2(point[I[i]], point[I[(i + 1) % len(I)]]);
          if (count_cross(S1, S2, false)) ok = 0;
        }
      }
      if (!ok) continue;
      fail = 0;
      I.insert(I.begin() + i + 1, POP(other));
    }
    point = rearrange(point, I);
    FOR(i, N) {
      if ((point[(i + 2) % N] - point[i]).det(point[(i + 1) % N] - point[i]) == 0) return {};
    }
    return point;
  };
  while (1) {
    vc<P> ANS = trial();
    if (ANS.empty()) continue;
    int k = RNG(0, len(ANS));
    rotate(ANS.begin(), ANS.begin() + k, ANS.end());
    return ANS;
  }
}

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