geo/delaunay_triangulation_of_convex_polygon.hpp

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• Last update: 2025-02-18 22:12:10+09:00
• Include: #include "geo/delaunay_triangulation_of_convex_polygon.hpp"
• Link: https://codeforces.com/contest/1984/problem/H
• Link: https://codeforces.com/contest/549/problem/E

Depends on

• ds/fastset.hpp
• geo/base.hpp
• geo/outcircle.hpp
• geo/triangle_area.hpp
• random/base.hpp
• random/shuffle.hpp

Code

#include "geo/outcircle.hpp"
#include "ds/fastset.hpp"
#include "random/shuffle.hpp"

// 座標の 4 乗がオーバーフローしない
// return : array<int,3>, triangles
// https://codeforces.com/contest/549/problem/E
// https://codeforces.com/contest/1984/problem/H
template <typename T>
vc<array<int, 3>> delaunay_triangulation_of_convex_polygon(vc<Point<T>> A, bool farthest) {
  using P = Point<T>;
  int N = len(A);
  if (N <= 2) return {};

  FastSet FS(N);

  vc<int> I(N);
  FOR(i, N) I[i] = i;
  shuffle(I);

  sort(I.end() - 3, I.end());

  struct E {
    int a, b, nxt, rev;
  };

  int c = POP(I), b = POP(I), a = POP(I);
  vc<E> dat;
  dat.eb(a, b, 1, -1);
  dat.eb(b, c, 2, -1);
  dat.eb(c, a, 0, -1);
  vc<int> v_to_e(N, -1);
  v_to_e[a] = 0, v_to_e[b] = 1, v_to_e[c] = 2;
  FS.insert(a), FS.insert(b), FS.insert(c);

  auto dfs = [&](auto& dfs, int i) -> void {
    int j = dat[i].rev;
    if (j == -1) return;
    int i1 = dat[i].nxt;
    int i2 = dat[i1].nxt;
    int j1 = dat[j].nxt;
    int j2 = dat[j1].nxt;
    int b = dat[i].a, c = dat[i1].a, a = dat[i2].a, d = dat[j2].a;
    int side = outcircle_side(A[a], A[b], A[c], A[d]);
    bool flip = (farthest ? (side == -1) : (side == 1));
    if (!flip) return;
    dat[i] = {d, a, i2, j};
    dat[j] = {a, d, j2, i};
    dat[i1].nxt = j, dat[i2].nxt = j1, dat[j1].nxt = i, dat[j2].nxt = i1;
    dfs(dfs, i1), dfs(dfs, i2), dfs(dfs, j1), dfs(dfs, j2);
  };

  while (len(I)) {
    int v = POP(I);
    int l = FS.prev(v), r = FS.next(v);
    if (l == -1) l = FS.prev(N);
    if (r == N) r = FS.next(0);
    FS.insert(v);
    int k = v_to_e[l];
    int s = len(dat);
    v_to_e[l] = s + 1, v_to_e[v] = s + 2;
    dat[k].rev = s;
    dat.eb(r, l, s + 1, k);
    dat.eb(l, v, s + 2, -1);
    dat.eb(v, r, s, -1);
    dfs(dfs, k);
  }
  vc<array<int, 3>> ANS;
  FOR(i, len(dat)) {
    int j = dat[i].nxt;
    int k = dat[j].nxt;
    if (i > j || i > k) continue;
    ANS.eb(array<int, 3>{dat[i].a, dat[j].a, dat[k].a});
  }

  return ANS;
}

#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 1 "geo/triangle_area.hpp"
template <typename REAL, typename T>
REAL triangle_area(Point<T> A, Point<T> B, Point<T> C) {
  return abs((B - A).det(C - A)) * 0.5;
}
#line 4 "geo/outcircle.hpp"

template <typename REAL, typename T>
Circle<REAL> outcircle(Point<T> A, Point<T> B, Point<T> C) {
  REAL b1 = B.x - A.x, b2 = B.y - A.y;
  REAL c1 = C.x - A.x, c2 = C.y - A.y;
  REAL bb = (b1 * b1 + b2 * b2) / 2;
  REAL cc = (c1 * c1 + c2 * c2) / 2;

  REAL det = b1 * c2 - b2 * c1;
  REAL x = (bb * c2 - b2 * cc) / det;
  REAL y = (b1 * cc - bb * c1) / det;
  REAL r = sqrt(x * x + y * y);
  x += A.x, y += A.y;
  return Circle<REAL>(x, y, r);
}

// ABC の外接円に対して内外どちらにあるか
// 中：1, 境界：0, 外：-1
// 座標の 4 乗がオーバーフローしないようにする
template <typename T>
int outcircle_side(Point<T> A, Point<T> B, Point<T> C, Point<T> p) {
  T d = (B - A).det(C - A);
  assert(d != 0);
  if (d < 0) swap(B, C);
  array<Point<T>, 3> pts = {A, B, C};
  array<array<T, 3>, 3> mat;
  FOR(i, 3) {
    T dx = pts[i].x - p.x, dy = pts[i].y - p.y;
    mat[i][0] = dx, mat[i][1] = dy, mat[i][2] = dx * dx + dy * dy;
  }
  T det = 0;
  det += mat[0][0] * (mat[1][1] * mat[2][2] - mat[1][2] * mat[2][1]);
  det += mat[0][1] * (mat[1][2] * mat[2][0] - mat[1][0] * mat[2][2]);
  det += mat[0][2] * (mat[1][0] * mat[2][1] - mat[1][1] * mat[2][0]);
  if (det == 0) return 0;
  return (det > 0 ? 1 : -1);
}
#line 2 "ds/fastset.hpp"

// 64-ary tree

// space: (N/63) * u64

struct FastSet {
  static constexpr u32 B = 64;
  int n, log;
  vvc<u64> seg;

  FastSet() {}
  FastSet(int n) { build(n); }

  int size() { return n; }

  template <typename F>
  FastSet(int n, F f) {
    build(n, f);
  }

  void build(int m) {
    seg.clear();
    n = m;
    do {
      seg.push_back(vc<u64>((m + B - 1) / B));
      m = (m + B - 1) / B;
    } while (m > 1);
    log = len(seg);
  }
  template <typename F>
  void build(int n, F f) {
    build(n);
    FOR(i, n) { seg[0][i / B] |= u64(f(i)) << (i % B); }
    FOR(h, log - 1) {
      FOR(i, len(seg[h])) { seg[h + 1][i / B] |= u64(bool(seg[h][i])) << (i % B); }
    }
  }

  bool operator[](int i) const { return seg[0][i / B] >> (i % B) & 1; }
  void insert(int i) {
    assert(0 <= i && i < n);
    for (int h = 0; h < log; h++) { seg[h][i / B] |= u64(1) << (i % B), i /= B; }
  }
  void add(int i) { insert(i); }
  void erase(int i) {
    assert(0 <= i && i < n);
    u64 x = 0;
    for (int h = 0; h < log; h++) {
      seg[h][i / B] &= ~(u64(1) << (i % B));
      seg[h][i / B] |= x << (i % B);
      x = bool(seg[h][i / B]);
      i /= B;
    }
  }
  void remove(int i) { erase(i); }

  // min[x,n) or n

  int next(int i) {
    assert(i <= n);
    chmax(i, 0);
    for (int h = 0; h < log; h++) {
      if (i / B == seg[h].size()) break;
      u64 d = seg[h][i / B] >> (i % B);
      if (!d) {
        i = i / B + 1;
        continue;
      }
      i += lowbit(d);
      for (int g = h - 1; g >= 0; g--) {
        i *= B;
        i += lowbit(seg[g][i / B]);
      }
      return i;
    }
    return n;
  }

  // max [0,x], or -1

  int prev(int i) {
    assert(i >= -1);
    if (i >= n) i = n - 1;
    for (int h = 0; h < log; h++) {
      if (i == -1) break;
      u64 d = seg[h][i / B] << (63 - i % B);
      if (!d) {
        i = i / B - 1;
        continue;
      }
      i -= __builtin_clzll(d);
      for (int g = h - 1; g >= 0; g--) {
        i *= B;
        i += topbit(seg[g][i / B]);
      }
      return i;
    }
    return -1;
  }

  bool any(int l, int r) { return next(l) < r; }

  // [l, r)

  template <typename F>
  void enumerate(int l, int r, F f) {
    for (int x = next(l); x < r; x = next(x + 1)) f(x);
  }

  string to_string() {
    string s(n, '?');
    for (int i = 0; i < n; ++i) s[i] = ((*this)[i] ? '1' : '0');
    return s;
  }
};
#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 "random/shuffle.hpp"

template <typename T>
void shuffle(vc<T>& A) {
  FOR(i, len(A)) {
    int j = RNG(0, i + 1);
    if (i != j) swap(A[i], A[j]);
  }
}
#line 4 "geo/delaunay_triangulation_of_convex_polygon.hpp"

// 座標の 4 乗がオーバーフローしない
// return : array<int,3>, triangles
// https://codeforces.com/contest/549/problem/E
// https://codeforces.com/contest/1984/problem/H
template <typename T>
vc<array<int, 3>> delaunay_triangulation_of_convex_polygon(vc<Point<T>> A, bool farthest) {
  using P = Point<T>;
  int N = len(A);
  if (N <= 2) return {};

  FastSet FS(N);

  vc<int> I(N);
  FOR(i, N) I[i] = i;
  shuffle(I);

  sort(I.end() - 3, I.end());

  struct E {
    int a, b, nxt, rev;
  };

  int c = POP(I), b = POP(I), a = POP(I);
  vc<E> dat;
  dat.eb(a, b, 1, -1);
  dat.eb(b, c, 2, -1);
  dat.eb(c, a, 0, -1);
  vc<int> v_to_e(N, -1);
  v_to_e[a] = 0, v_to_e[b] = 1, v_to_e[c] = 2;
  FS.insert(a), FS.insert(b), FS.insert(c);

  auto dfs = [&](auto& dfs, int i) -> void {
    int j = dat[i].rev;
    if (j == -1) return;
    int i1 = dat[i].nxt;
    int i2 = dat[i1].nxt;
    int j1 = dat[j].nxt;
    int j2 = dat[j1].nxt;
    int b = dat[i].a, c = dat[i1].a, a = dat[i2].a, d = dat[j2].a;
    int side = outcircle_side(A[a], A[b], A[c], A[d]);
    bool flip = (farthest ? (side == -1) : (side == 1));
    if (!flip) return;
    dat[i] = {d, a, i2, j};
    dat[j] = {a, d, j2, i};
    dat[i1].nxt = j, dat[i2].nxt = j1, dat[j1].nxt = i, dat[j2].nxt = i1;
    dfs(dfs, i1), dfs(dfs, i2), dfs(dfs, j1), dfs(dfs, j2);
  };

  while (len(I)) {
    int v = POP(I);
    int l = FS.prev(v), r = FS.next(v);
    if (l == -1) l = FS.prev(N);
    if (r == N) r = FS.next(0);
    FS.insert(v);
    int k = v_to_e[l];
    int s = len(dat);
    v_to_e[l] = s + 1, v_to_e[v] = s + 2;
    dat[k].rev = s;
    dat.eb(r, l, s + 1, k);
    dat.eb(l, v, s + 2, -1);
    dat.eb(v, r, s, -1);
    dfs(dfs, k);
  }
  vc<array<int, 3>> ANS;
  FOR(i, len(dat)) {
    int j = dat[i].nxt;
    int k = dat[j].nxt;
    if (i > j || i > k) continue;
    ANS.eb(array<int, 3>{dat[i].a, dat[j].a, dat[k].a});
  }

  return ANS;
}

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