convex/lattice_point_count.hpp

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• Last update: 2025-02-12 17:59:58+09:00
• Include: #include "convex/lattice_point_count.hpp"

Depends on

• convex/line_min_function.hpp
• geo/base.hpp
• geo/convex_hull.hpp
• mod/floor_sum_of_linear.hpp

Verified with

• test/1_mytest/count_lattice_point_in_convex_polygon.test.cpp

Code

#include "convex/line_min_function.hpp"
#include "mod/floor_sum_of_linear.hpp"

// ax+by<=c という半平面たち
// 有界でないときは 0 を返す
// （格子点が 0 個ということはあるので答えが inf とは限らない）
// 入力が pow(10,18) 以下とかであればオーバーフローしないつもり
i128 lattice_point_count(vc<tuple<ll, ll, ll>> LINE) {
  ll L = -infty<ll>, R = infty<ll>;
  vc<tuple<ll, ll, ll>> LINE1, LINE2;
  for (auto& [a, b, c]: LINE) {
    if (b == 0) {
      assert(a != 0);
      if (a > 0) { chmin(R, floor<ll>(c, a) + 1); }
      elif (a < 0) { chmax(L, ceil<ll>(-c, -a)); }
    } else {
      if (b > 0) { LINE2.eb(-a, c, b); }
      if (b < 0) { LINE1.eb(a, -c, -b); }
    }
  }
  if (L >= R) return 0;
  if (LINE1.empty() || LINE2.empty()) return 0;

  auto LOWER = line_max_function_rational(LINE1, L, R);
  auto UPPER = line_min_function_rational(LINE2, L, R);

  i128 ANS = 0;

  bool bad = 0;
  auto wk = [&](ll L, ll R, ll a1, ll b1, ll c1, ll a2, ll b2, ll c2) -> void {
    // 交点 t/s
    i128 s = i128(a2) * c1 - i128(a1) * c2;
    i128 t = i128(b1) * c2 - i128(b2) * c1;
    if (s == 0) {
      if (t > 0) return;
    }
    if (s > 0) { chmax(L, ceil<i128>(t, s)); }
    if (s < 0) { chmin(R, floor<i128>(-t, -s) + 1); }
    if (L >= R) return;
    if (L == -infty<ll> || R == infty<ll>) {
      bad = 1;
      return;
    }
    ANS += floor_sum_of_linear<i128, i128>(L, R, a2, b2, c2);
    ANS -= floor_sum_of_linear<i128, i128>(L, R, a1, b1 - 1, c1);
  };

  merge_58(LOWER, UPPER, wk);
  if (bad) return 0;
  return ANS;
}

#line 2 "convex/line_min_function.hpp"

#line 2 "geo/convex_hull.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+=(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 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 4 "convex/line_min_function.hpp"

// 1 次関数の max を [L,R,a,b] の列として出力
// https://qoj.ac/contest/1576/problem/8505
template <typename Re, typename T>
vc<tuple<Re, Re, Re, Re>> line_min_function_real(vc<pair<T, T>> LINE) {
  assert(!LINE.empty());
  using P = Point<T>;
  vc<P> point;
  for (auto& [x, y]: LINE) point.eb(P(x, y));
  auto I = ConvexHull(point, "lower");
  point = rearrange(point, I);
  int N = len(point);
  if (N >= 2 && point[N - 1].x == point[N - 2].x) { POP(point), --N; }
  reverse(all(point)); // 傾きは大きい方から
  Re l = -infty<Re>;
  vc<tuple<Re, Re, Re, Re>> ANS;
  FOR(i, N) {
    Re r = infty<Re>;
    auto [a, b] = point[i];
    if (i + 1 < N) {
      auto [c, d] = point[i + 1];
      if (a == c) continue;
      assert(a > c);
      r = Re(d - b) / (a - c);
      chmax(r, l), chmin(r, infty<Re>);
    }
    if (l < r) ANS.eb(l, r, a, b), l = r;
  }
  return ANS;
}

// 1 次関数の max を [L,R,a,b] の列として出力
template <typename Re, typename T>
vc<tuple<Re, Re, Re, Re>> line_max_function_real(vc<pair<T, T>> LINE) {
  assert(!LINE.empty());
  for (auto& [a, b]: LINE) a = -a, b = -b;
  auto ANS = line_min_function_real<Re, T>(LINE);
  for (auto& [l, r, a, b]: ANS) a = -a, b = -b;
  return ANS;
}

// LINE(a,b,c): y=(ax+b)/c, 評価点は整数
// 1 次関数の min を [L,R,a,b,c] の列として出力
// オーバーフロー安全
vc<tuple<ll, ll, ll, ll, ll>> line_min_function_rational(vc<tuple<ll, ll, ll>> LINE, ll L, ll R) {
  // 傾き降順
  sort(all(LINE), [&](auto& L, auto& R) -> bool {
    auto& [a1, b1, c1] = L;
    auto& [a2, b2, c2] = R;
    return i128(a1) * c2 > i128(a2) * c1;
  });
  vc<tuple<ll, ll, ll, ll, ll>> ANS;
  for (auto& [a2, b2, c2]: LINE) {
    while (1) {
      if (ANS.empty()) {
        ANS.eb(L, R, a2, b2, c2);
        break;
      }
      auto& [L1, R1, a1, b1, c1] = ANS.back();
      i128 s = i128(c2) * a1 - i128(a2) * c1; // >= 0
      i128 t = i128(b2) * c1 - i128(b1) * c2;
      if (s == 0) {
        // 平行なので小さい方だけを残す
        if (t >= 0) break;
        ANS.pop_back();
        if (len(ANS)) get<1>(ANS.back()) = R;
        continue;
      }
      i128 x = ceil<i128>(t, s);
      // x 以上で 2 の方が下に来る
      if (x <= L1) {
        ANS.pop_back();
        continue;
      }
      if (x < R) {
        R1 = x;
        ANS.eb(x, R, a2, b2, c2);
        break;
      } else {
        break;
      }
    }
  }
  return ANS;
}

// LINE(a,b,c): y=(ax+b)/c, 評価点は整数
// 1 次関数の max を [L,R,a,b,c] の列として出力
// オーバーフロー安全
vc<tuple<ll, ll, ll, ll, ll>> line_max_function_rational(vc<tuple<ll, ll, ll>> LINE, ll L, ll R) {
  for (auto& [a, b, c]: LINE) a = -a, b = -b;
  auto ANS = line_min_function_rational(LINE, L, R);
  for (auto& [L, R, a, b, c]: ANS) a = -a, b = -b;
  return ANS;
}

// LINE(a,b): y=ax+b, 評価点は整数
// 1 次関数の min を [L,R,a,b] の列として出力
// オーバーフロー安全
vc<tuple<ll, ll, ll, ll>> line_min_function_integer(vc<pair<ll, ll>> LINE, ll L, ll R) {
  // 傾き降順
  sort(all(LINE), [&](auto& L, auto& R) -> bool {
    auto& [a1, b1] = L;
    auto& [a2, b2] = R;
    return a1 > a2;
  });
  vc<tuple<ll, ll, ll, ll>> ANS;
  for (auto& [a2, b2]: LINE) {
    while (1) {
      if (ANS.empty()) {
        ANS.eb(L, R, a2, b2);
        break;
      }
      auto& [L1, R1, a1, b1] = ANS.back();
      if (a1 == a2) {
        if (b1 <= b2) break;
        ANS.pop_back();
        if (len(ANS)) get<1>(ANS.back()) = R;
        continue;
      }
      ll x = ceil<ll>(b2 - b1, a1 - a2);
      // x 以上で 2 の方が下に来る
      if (x <= L1) {
        ANS.pop_back();
        continue;
      }
      if (x < R) {
        R1 = x;
        ANS.eb(x, R, a2, b2);
        break;
      } else {
        break;
      }
    }
  }
  return ANS;
}

// LINE(a,b,c): y=(ax+b)/c, 評価点は整数
// 1 次関数の min を [L,R,a,b,c] の列として出力
// c>0, (ax+b)c がオーバーフローしない,
vc<tuple<ll, ll, ll, ll>> line_max_function_integer(vc<pair<ll, ll>> LINE, ll L, ll R) {
  for (auto& [a, b]: LINE) a = -a, b = -b;
  auto ANS = line_min_function_integer(LINE, L, R);
  for (auto& [L, R, a, b]: ANS) a = -a, b = -b;
  return ANS;
}

// (L,R,func) の下側と上側をマージするときなどに使う用
template <typename T>
vc<tuple<T, T, T, T, T, T>> merge_46(vc<tuple<T, T, T, T>> A, vc<tuple<T, T, T, T>> B) {
  vc<tuple<T, T, T, T, T, T>> ANS;
  reverse(all(A));
  reverse(all(B));
  while (len(A) && len(B)) {
    auto& [l1, r1, a1, b1] = A.back();
    auto& [l2, r2, a2, b2] = B.back();
    assert(l1 == l2);
    T r = min(r1, r2);
    ANS.eb(l1, r, a1, b1, a2, b2);
    l1 = r, l2 = r;
    if (r1 == r) POP(A);
    if (r2 == r) POP(B);
  };
  return ANS;
}

// (L,R,func) の下側と上側をマージするときなどに使う用
// f(L,R,a1,b1,a2,b2)
template <typename T, typename F>
void merge_46(const vc<tuple<T, T, T, T>>& A, const vc<tuple<T, T, T, T>>& B, F f) {
  int i = 0, j = 0;
  while (i < len(A) && j < len(B)) {
    auto& [l1, r1, a1, b1] = A[i];
    auto& [l2, r2, a2, b2] = B[j];
    T l = max(l1, l2), r = min(r1, r2);
    if (l < r) f(l, r, a1, b1, a2, b2);
    (r1 < r2 ? i : j)++;
  }
}

// (L,R,func) の下側と上側をマージするときなどに使う用
// f(L,R,a1,b1,a2,b2)
template <typename T, typename F>
void merge_58(const vc<tuple<T, T, T, T, T>>& A, const vc<tuple<T, T, T, T, T>>& B, F f) {
  int i = 0, j = 0;
  while (i < len(A) && j < len(B)) {
    auto& [l1, r1, a1, b1, c1] = A[i];
    auto& [l2, r2, a2, b2, c2] = B[j];
    T l = max(l1, l2), r = min(r1, r2);
    if (l < r) f(l, r, a1, b1, c1, a2, b2, c2);
    (r1 < r2 ? i : j)++;
  }
}
#line 2 "mod/floor_sum_of_linear.hpp"

// sum_{x in [L,R)} floor(ax + b, mod)
// I は範囲内で ax+b がオーバーフローしない程度
template <typename O = i128, typename I = long long>
O floor_sum_of_linear(I L, I R, I a, I b, I mod) {
  assert(L <= R);
  O res = 0;
  b += L * a;
  I N = R - L;

  if (b < 0) {
    I k = ceil(-b, mod);
    b += k * mod;
    res -= O(N) * O(k);
  }

  while (N) {
    I q;
    tie(q, a) = divmod(a, mod);
    res += (N & 1 ? O(N) * O((N - 1) / 2) * O(q) : O(N / 2) * O(N - 1) * O(q));
    if (b >= mod) {
      tie(q, b) = divmod(b, mod);
      res += O(N) * q;
    }
    tie(N, b) = divmod(a * N + b, mod);
    tie(a, mod) = mp(mod, a);
  }
  return res;
}
#line 3 "convex/lattice_point_count.hpp"

// ax+by<=c という半平面たち
// 有界でないときは 0 を返す
// （格子点が 0 個ということはあるので答えが inf とは限らない）
// 入力が pow(10,18) 以下とかであればオーバーフローしないつもり
i128 lattice_point_count(vc<tuple<ll, ll, ll>> LINE) {
  ll L = -infty<ll>, R = infty<ll>;
  vc<tuple<ll, ll, ll>> LINE1, LINE2;
  for (auto& [a, b, c]: LINE) {
    if (b == 0) {
      assert(a != 0);
      if (a > 0) { chmin(R, floor<ll>(c, a) + 1); }
      elif (a < 0) { chmax(L, ceil<ll>(-c, -a)); }
    } else {
      if (b > 0) { LINE2.eb(-a, c, b); }
      if (b < 0) { LINE1.eb(a, -c, -b); }
    }
  }
  if (L >= R) return 0;
  if (LINE1.empty() || LINE2.empty()) return 0;

  auto LOWER = line_max_function_rational(LINE1, L, R);
  auto UPPER = line_min_function_rational(LINE2, L, R);

  i128 ANS = 0;

  bool bad = 0;
  auto wk = [&](ll L, ll R, ll a1, ll b1, ll c1, ll a2, ll b2, ll c2) -> void {
    // 交点 t/s
    i128 s = i128(a2) * c1 - i128(a1) * c2;
    i128 t = i128(b1) * c2 - i128(b2) * c1;
    if (s == 0) {
      if (t > 0) return;
    }
    if (s > 0) { chmax(L, ceil<i128>(t, s)); }
    if (s < 0) { chmin(R, floor<i128>(-t, -s) + 1); }
    if (L >= R) return;
    if (L == -infty<ll> || R == infty<ll>) {
      bad = 1;
      return;
    }
    ANS += floor_sum_of_linear<i128, i128>(L, R, a2, b2, c2);
    ANS -= floor_sum_of_linear<i128, i128>(L, R, a1, b1 - 1, c1);
  };

  merge_58(LOWER, UPPER, wk);
  if (bad) return 0;
  return ANS;
}

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