geo/dynamicupperhull.hpp

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• Last update: 2023-11-07 01:10:45+09:00
• Include: #include "geo/dynamicupperhull.hpp"
• Link: https://codeforces.com/blog/entry/75929

Depends on

• geo/base.hpp

Verified with

• test/library_checker/geometry/convex_layers.test.cpp

Code

#include "geo/base.hpp"

/*
https://codeforces.com/blog/entry/75929
動的凸包。
x 座標でソートして完全二分木のセグ木の形にしておく。
セグ木のマージ部分（次の bridge を求める）で二分探索する。
bridge 同士の 4 点での上側凸包を見れば、次に探索するべき区間対が分かる。

構築 O(NlogN)、更新 O(Nlog^2N)
座標 10^9 以下の整数を仮定
*/
template <typename Point>
struct DynamicUpperHull {
  struct node {
    int l, r;   // 範囲 (-1 if no vertex)

    int bl, br; // bridge idx

  };
  int N, sz;
  vc<Point> P;
  vc<node> seg;
  // 受け取ったインデックスとの対応

  vc<int> to_original_idx, to_seg_idx;

  DynamicUpperHull(vc<Point> P) : DynamicUpperHull(P, 0) {}
  DynamicUpperHull(vc<Point> P, bool b)
      : DynamicUpperHull(P, vc<bool>(len(P), b)) {}

  DynamicUpperHull(vc<Point> _P, vc<bool> isin) : N(len(_P)), P(_P) {
    to_original_idx = argsort(P);
    sort(all(P));
    sz = 1;
    while (sz < N) sz *= 2;
    to_seg_idx.resize(N);
    seg.assign(sz + sz, {-1, -1, -1, -1});
    for (int i = 0; i < N; ++i) to_seg_idx[to_original_idx[i]] = i;
    for (int i = 0; i < N; ++i)
      if (isin[to_original_idx[i]]) { seg[sz + i] = {i, i + 1, i, i}; }
    FOR3_R(i, 1, sz) update(i);
  }

  void insert(int i) {
    i = to_seg_idx[i];
    seg[sz + i] = {i, i + 1, i, i};
    i = (sz + i) / 2;
    while (i) {
      update(i);
      i /= 2;
    }
  }
  void add(int i) { insert(i); }

  void erase(int i) {
    i = to_seg_idx[i];
    seg[sz + i] = {-1, -1, -1, -1};
    i = (sz + i) / 2;
    while (i) {
      update(i);
      i /= 2;
    }
  }
  void remove(int i) { insert(i); }

  inline bool exist(int i) { return seg[i].r != -1; }

  void update(int i) {
    if (!exist(2 * i + 0) && !exist(2 * i + 1)) {
      seg[i].r = -1;
      return;
    }
    if (!exist(2 * i + 0)) {
      seg[i] = seg[2 * i + 1];
      return;
    }
    if (!exist(2 * i + 1)) {
      seg[i] = seg[2 * i + 0];
      return;
    }
    int p = 2 * i, q = 2 * i + 1;
    ll X = P[seg[q].l].x;
    while (p < sz || q < sz) {
      if (p < sz && !exist(2 * p + 0)) {
        p = 2 * p + 1;
        continue;
      }
      if (p < sz && !exist(2 * p + 1)) {
        p = 2 * p + 0;
        continue;
      }
      if (q < sz && !exist(2 * q + 0)) {
        q = 2 * q + 1;
        continue;
      }
      if (q < sz && !exist(2 * q + 1)) {
        q = 2 * q + 0;
        continue;
      }
      int a = seg[p].bl, b = seg[p].br, c = seg[q].bl, d = seg[q].br;
      if (a != b && (P[b] - P[a]).det(P[c] - P[a]) > 0) p = p * 2 + 0;
      elif (c != d && (P[c] - P[b]).det(P[d] - P[b]) > 0) q = 2 * q + 1;
      elif (a == b) q = 2 * q + 0;
      elif (c == d) p = 2 * p + 1;
      else {
        i128 c1 = (P[b] - P[a]).det(P[c] - P[a]);
        i128 c2 = (P[a] - P[b]).det(P[d] - P[b]);
        if (c1 + c2 == 0 || c1 * P[d].x + c2 * P[c].x < X * (c1 + c2)) {
          p = 2 * p + 1;
        } else {
          q = 2 * q + 0;
        }
      }
    }
    seg[i].l = seg[2 * i].l, seg[i].r = seg[2 * i + 1].r;
    seg[i].bl = seg[p].l, seg[i].br = seg[q].l;
  }

  vc<int> get() {
    // output sensitive complexity

    vc<int> res;
    auto dfs = [&](auto self, int k, int l, int r) -> void {
      if (!exist(k) || l >= r) return;
      if (k >= sz) {
        res.eb(seg[k].l);
        return;
      }
      if (!exist(2 * k + 0)) return self(self, 2 * k + 1, l, r);
      if (!exist(2 * k + 1)) return self(self, 2 * k + 0, l, r);
      if (r <= seg[k].bl) return self(self, 2 * k + 0, l, r);
      if (seg[k].br <= l) return self(self, 2 * k + 1, l, r);
      self(self, 2 * k + 0, l, seg[k].bl + 1);
      self(self, 2 * k + 1, seg[k].br, r);
    };
    dfs(dfs, 1, 0, N);
    for (auto&& i: res) i = to_original_idx[i];
    return res;
  }

  void debug() {
    print("points");
    FOR(i, len(P)) print(i, P[i].x, P[i].y);
    print("seg");
    FOR(i, len(seg)) print(i, seg[i].l, seg[i].r, seg[i].bl, seg[i].br);
    print("get");
    print(get());
  }
};

#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 2 "geo/dynamicupperhull.hpp"

/*
https://codeforces.com/blog/entry/75929
動的凸包。
x 座標でソートして完全二分木のセグ木の形にしておく。
セグ木のマージ部分（次の bridge を求める）で二分探索する。
bridge 同士の 4 点での上側凸包を見れば、次に探索するべき区間対が分かる。

構築 O(NlogN)、更新 O(Nlog^2N)
座標 10^9 以下の整数を仮定
*/
template <typename Point>
struct DynamicUpperHull {
  struct node {
    int l, r;   // 範囲 (-1 if no vertex)

    int bl, br; // bridge idx

  };
  int N, sz;
  vc<Point> P;
  vc<node> seg;
  // 受け取ったインデックスとの対応

  vc<int> to_original_idx, to_seg_idx;

  DynamicUpperHull(vc<Point> P) : DynamicUpperHull(P, 0) {}
  DynamicUpperHull(vc<Point> P, bool b)
      : DynamicUpperHull(P, vc<bool>(len(P), b)) {}

  DynamicUpperHull(vc<Point> _P, vc<bool> isin) : N(len(_P)), P(_P) {
    to_original_idx = argsort(P);
    sort(all(P));
    sz = 1;
    while (sz < N) sz *= 2;
    to_seg_idx.resize(N);
    seg.assign(sz + sz, {-1, -1, -1, -1});
    for (int i = 0; i < N; ++i) to_seg_idx[to_original_idx[i]] = i;
    for (int i = 0; i < N; ++i)
      if (isin[to_original_idx[i]]) { seg[sz + i] = {i, i + 1, i, i}; }
    FOR3_R(i, 1, sz) update(i);
  }

  void insert(int i) {
    i = to_seg_idx[i];
    seg[sz + i] = {i, i + 1, i, i};
    i = (sz + i) / 2;
    while (i) {
      update(i);
      i /= 2;
    }
  }
  void add(int i) { insert(i); }

  void erase(int i) {
    i = to_seg_idx[i];
    seg[sz + i] = {-1, -1, -1, -1};
    i = (sz + i) / 2;
    while (i) {
      update(i);
      i /= 2;
    }
  }
  void remove(int i) { insert(i); }

  inline bool exist(int i) { return seg[i].r != -1; }

  void update(int i) {
    if (!exist(2 * i + 0) && !exist(2 * i + 1)) {
      seg[i].r = -1;
      return;
    }
    if (!exist(2 * i + 0)) {
      seg[i] = seg[2 * i + 1];
      return;
    }
    if (!exist(2 * i + 1)) {
      seg[i] = seg[2 * i + 0];
      return;
    }
    int p = 2 * i, q = 2 * i + 1;
    ll X = P[seg[q].l].x;
    while (p < sz || q < sz) {
      if (p < sz && !exist(2 * p + 0)) {
        p = 2 * p + 1;
        continue;
      }
      if (p < sz && !exist(2 * p + 1)) {
        p = 2 * p + 0;
        continue;
      }
      if (q < sz && !exist(2 * q + 0)) {
        q = 2 * q + 1;
        continue;
      }
      if (q < sz && !exist(2 * q + 1)) {
        q = 2 * q + 0;
        continue;
      }
      int a = seg[p].bl, b = seg[p].br, c = seg[q].bl, d = seg[q].br;
      if (a != b && (P[b] - P[a]).det(P[c] - P[a]) > 0) p = p * 2 + 0;
      elif (c != d && (P[c] - P[b]).det(P[d] - P[b]) > 0) q = 2 * q + 1;
      elif (a == b) q = 2 * q + 0;
      elif (c == d) p = 2 * p + 1;
      else {
        i128 c1 = (P[b] - P[a]).det(P[c] - P[a]);
        i128 c2 = (P[a] - P[b]).det(P[d] - P[b]);
        if (c1 + c2 == 0 || c1 * P[d].x + c2 * P[c].x < X * (c1 + c2)) {
          p = 2 * p + 1;
        } else {
          q = 2 * q + 0;
        }
      }
    }
    seg[i].l = seg[2 * i].l, seg[i].r = seg[2 * i + 1].r;
    seg[i].bl = seg[p].l, seg[i].br = seg[q].l;
  }

  vc<int> get() {
    // output sensitive complexity

    vc<int> res;
    auto dfs = [&](auto self, int k, int l, int r) -> void {
      if (!exist(k) || l >= r) return;
      if (k >= sz) {
        res.eb(seg[k].l);
        return;
      }
      if (!exist(2 * k + 0)) return self(self, 2 * k + 1, l, r);
      if (!exist(2 * k + 1)) return self(self, 2 * k + 0, l, r);
      if (r <= seg[k].bl) return self(self, 2 * k + 0, l, r);
      if (seg[k].br <= l) return self(self, 2 * k + 1, l, r);
      self(self, 2 * k + 0, l, seg[k].bl + 1);
      self(self, 2 * k + 1, seg[k].br, r);
    };
    dfs(dfs, 1, 0, N);
    for (auto&& i: res) i = to_original_idx[i];
    return res;
  }

  void debug() {
    print("points");
    FOR(i, len(P)) print(i, P[i].x, P[i].y);
    print("seg");
    FOR(i, len(seg)) print(i, seg[i].l, seg[i].r, seg[i].bl, seg[i].br);
    print("get");
    print(get());
  }
};

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