geo/minimum_enclosing_circle.hpp

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• Last update: 2024-10-25 19:29:47+09:00
• Include: #include "geo/minimum_enclosing_circle.hpp"
• Link: https://codeforces.com/problemset/problem/119/E
• Link: https://qoj.ac/contest/1452/problem/7934

Depends on

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

Code

#include "geo/base.hpp"
#include "random/shuffle.hpp"
#include "geo/outcircle.hpp"

// randomize を利用した expected O(N) アルゴリズム
// 座標の 4 乗が登場！オーバーフロー注意！
// return: {C,0,-1,-1} or {C,i,j,-1} or {C,i,j,k}
// https://codeforces.com/problemset/problem/119/E
// https://qoj.ac/contest/1452/problem/7934
template <typename REAL, typename T>
tuple<Circle<REAL>, int, int, int> minimum_enclosing_circle(vc<Point<T>> points) {
  const int n = len(points);
  assert(n >= 1);
  if (n == 1) {
    Circle<REAL> C(points[0].x, points[0].y, 0);
    return {C, 0, -1, -1};
  }
  vc<int> I(n);
  FOR(i, n) I[i] = i;
  shuffle(I);

  points = rearrange(points, I);

  tuple<int, int, int> c = {0, -1, -1};
  auto contain = [&](Point<T> p) -> bool {
    auto [i, j, k] = c;
    if (j == -1) { return p == points[i]; }
    if (k == -1) { return (points[i] - p).dot(points[j] - p) <= 0; }
    return outcircle_side(points[i], points[j], points[k], p) >= 0;
  };

  FOR(i, 1, n) {
    if (contain(points[i])) continue;
    c = {0, i, -1};
    FOR(j, 1, i) {
      if (contain(points[j])) continue;
      c = {i, j, -1};
      FOR(k, j) {
        if (contain(points[k])) continue;
        c = {i, j, k};
      }
    }
  }
  auto [i, j, k] = c;
  if (k == -1) {
    REAL x1 = points[i].x;
    REAL y1 = points[i].y;
    REAL x2 = points[j].x;
    REAL y2 = points[j].y;
    Point<REAL> O = {0.5 * (x1 + x2), 0.5 * (y1 + y2)};
    REAL r = sqrtl((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2)) / 2;
    Circle<REAL> C(O, r);
    return {C, I[i], I[j], -1};
  }
  Circle<REAL> C = outcircle<REAL>(points[i], points[j], points[k]);
  return {C, I[i], I[j], I[k]};
}

#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 "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 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 4 "geo/minimum_enclosing_circle.hpp"

// randomize を利用した expected O(N) アルゴリズム
// 座標の 4 乗が登場！オーバーフロー注意！
// return: {C,0,-1,-1} or {C,i,j,-1} or {C,i,j,k}
// https://codeforces.com/problemset/problem/119/E
// https://qoj.ac/contest/1452/problem/7934
template <typename REAL, typename T>
tuple<Circle<REAL>, int, int, int> minimum_enclosing_circle(vc<Point<T>> points) {
  const int n = len(points);
  assert(n >= 1);
  if (n == 1) {
    Circle<REAL> C(points[0].x, points[0].y, 0);
    return {C, 0, -1, -1};
  }
  vc<int> I(n);
  FOR(i, n) I[i] = i;
  shuffle(I);

  points = rearrange(points, I);

  tuple<int, int, int> c = {0, -1, -1};
  auto contain = [&](Point<T> p) -> bool {
    auto [i, j, k] = c;
    if (j == -1) { return p == points[i]; }
    if (k == -1) { return (points[i] - p).dot(points[j] - p) <= 0; }
    return outcircle_side(points[i], points[j], points[k], p) >= 0;
  };

  FOR(i, 1, n) {
    if (contain(points[i])) continue;
    c = {0, i, -1};
    FOR(j, 1, i) {
      if (contain(points[j])) continue;
      c = {i, j, -1};
      FOR(k, j) {
        if (contain(points[k])) continue;
        c = {i, j, k};
      }
    }
  }
  auto [i, j, k] = c;
  if (k == -1) {
    REAL x1 = points[i].x;
    REAL y1 = points[i].y;
    REAL x2 = points[j].x;
    REAL y2 = points[j].y;
    Point<REAL> O = {0.5 * (x1 + x2), 0.5 * (y1 + y2)};
    REAL r = sqrtl((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2)) / 2;
    Circle<REAL> C(O, r);
    return {C, I[i], I[j], -1};
  }
  Circle<REAL> C = outcircle<REAL>(points[i], points[j], points[k]);
  return {C, I[i], I[j], I[k]};
}

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