convex/count_lattice_point_in_convex_polygon_polynomial.hpp

• View this file on GitHub
• Last update: 2025-01-06 23:56:37+09:00
• Include: #include "convex/count_lattice_point_in_convex_polygon_polynomial.hpp"

Depends on

• alg/monoid/monoid_for_floor_sum.hpp
• alg/monoid_pow.hpp
• convex/line_min_function.hpp
• geo/base.hpp
• geo/convex_hull.hpp
• mod/floor_monoid_product.hpp
• mod/floor_sum_of_linear_polynomial.hpp

Code

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

// 格子点 (x,y) に対して x^iy^j の sum. i<=K, j<=L
template <typename mint, int K1, int K2>
array<array<mint, K2 + 1>, K1 + 1> count_lattice_point_in_convex_polygon_polynomial(ll L, ll R, vc<tuple<ll, ll, ll>> LINE) {
  vc<tuple<ll, ll, ll>> LINE1, LINE2;
  for (auto &[a, b, c]: LINE) {
    if (b == 0) {
      // ax<=c
      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);
      } else {
        LINE1.eb(a, -c, -b);
      }
    }
  }
  if (L >= R) { return {}; }
  assert(!LINE1.empty());
  assert(!LINE2.empty());

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

  array<array<mint, K2 + 2>, K1 + 1> S;
  FOR(i, K1 + 1) FOR(j, K2 + 1) S[i][j] = 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;
    }
    elif (s > 0) {
      // 上側の方が傾きが大きい
      i128 x = ceil<i128>(t, s);
      chmax(L, x);
    }
    else {
      i128 x = floor<i128>(-t, -s);
      chmin(R, x + 1);
    }
    if (L >= R) return;
    auto ADD = floor_sum_of_linear_polynomial<mint, K1, K2 + 1, ll>(L, R, a2, b2, c2);
    auto SUB = floor_sum_of_linear_polynomial<mint, K1, K2 + 1, ll>(L, R, a1, b1 - 1, c1);
    FOR(i, K1 + 1) FOR(j, K2 + 2) S[i][j] += ADD[i][j] - SUB[i][j];
  };

  merge_58(LOWER, UPPER, wk);
  array<array<mint, K2 + 1>, K1 + 1> ANS;
  FOR(i, K1 + 1) FOR(j, K2 + 1) ANS[i][j] = 0;

  static vvc<mint> CF;
  if (CF.empty()) { CF = faulhaber_formula_2d<mint>(K2); }
  FOR(i, K1 + 1) {
    FOR(j, K2 + 1) {
      FOR(k, j + 2) { ANS[i][j] += CF[j][k] * S[i][k]; }
    }
  }
  return ANS;
}

#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 2 "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 1 "mod/floor_sum_of_linear_polynomial.hpp"

#line 2 "alg/monoid_pow.hpp"

// chat gpt
template <typename U, typename Arg1, typename Arg2>
struct has_power_method {
private:
  // ヘルパー関数の実装
  template <typename V, typename A1, typename A2>
  static auto check(int)
      -> decltype(std::declval<V>().power(std::declval<A1>(),
                                          std::declval<A2>()),
                  std::true_type{});
  template <typename, typename, typename>
  static auto check(...) -> std::false_type;

public:
  // メソッドの有無を表す型
  static constexpr bool value = decltype(check<U, Arg1, Arg2>(0))::value;
};

template <typename Monoid>
typename Monoid::X monoid_pow(typename Monoid::X x, ll exp) {
  using X = typename Monoid::X;
  if constexpr (has_power_method<Monoid, X, ll>::value) {
    return Monoid::power(x, exp);
  } else {
    assert(exp >= 0);
    X res = Monoid::unit();
    while (exp) {
      if (exp & 1) res = Monoid::op(res, x);
      x = Monoid::op(x, x);
      exp >>= 1;
    }
    return res;
  }
}
#line 2 "mod/floor_monoid_product.hpp"

// https://yukicoder.me/submissions/883884
// https://qoj.ac/contest/1411/problem/7620
// U は範囲内で ax+b がオーバーフローしない程度
// yyy x yyyy x ... yyy x yyy (x を N 個)
// k 個目の x までに floor(ak+b,m) 個の y がある
// my<=ax+b における lattice path における辺の列と見なせる
template <typename Monoid, typename X, typename U>
X floor_monoid_product(X x, X y, U N, U a, U b, U m) {
  U c = (a * N + b) / m;
  X pre = Monoid::unit(), suf = Monoid::unit();
  while (1) {
    const U p = a / m, q = b / m;
    a %= m, b %= m;
    x = Monoid::op(x, monoid_pow<Monoid>(y, p));
    pre = Monoid::op(pre, monoid_pow<Monoid>(y, q));
    c -= (p * N + q);
    if (c == 0) break;
    const U d = (m * c - b - 1) / a + 1;
    suf = Monoid::op(y, Monoid::op(monoid_pow<Monoid>(x, N - d), suf));
    b = m - b - 1 + a, N = c - 1, c = d;
    swap(m, a), swap(x, y);
  }
  x = monoid_pow<Monoid>(x, N);
  return Monoid::op(Monoid::op(pre, x), suf);
}
#line 1 "alg/monoid/monoid_for_floor_sum.hpp"
// sum i^k1floor^k2: floor path で (x,y) から x 方向に進むときに x^k1y^k2 を足す
template <typename T, int K1, int K2>
struct Monoid_for_floor_sum {
  using ARR = array<array<T, K2 + 1>, K1 + 1>;
  struct Data {
    ARR dp;
    T dx, dy;
  };

  using value_type = Data;
  using X = value_type;
  static X op(X a, X b) {
    static constexpr int n = max(K1, K2);
    static T comb[n + 1][n + 1];
    if (comb[0][0] != T(1)) {
      comb[0][0] = T(1);
      FOR(i, n) FOR(j, i + 1) { comb[i + 1][j] += comb[i][j], comb[i + 1][j + 1] += comb[i][j]; }
    }

    array<T, K1 + 1> pow_x;
    array<T, K2 + 1> pow_y;
    pow_x[0] = 1, pow_y[0] = 1;
    FOR(i, K1) pow_x[i + 1] = pow_x[i] * a.dx;
    FOR(i, K2) pow_y[i + 1] = pow_y[i] * a.dy;

    // +dy
    FOR(i, K1 + 1) {
      FOR_R(j, K2 + 1) {
        T x = b.dp[i][j];
        FOR(k, j + 1, K2 + 1) b.dp[i][k] += comb[k][j] * pow_y[k - j] * x;
      }
    }
    // +dx
    FOR(j, K2 + 1) {
      FOR_R(i, K1 + 1) { FOR(k, i, K1 + 1) a.dp[k][j] += comb[k][i] * pow_x[k - i] * b.dp[i][j]; }
    }

    a.dx += b.dx, a.dy += b.dy;
    return a;
  }

  static X to_x() {
    X x = unit();
    x.dp[0][0] = 1, x.dx = 1;
    return x;
  }
  static X to_y() {
    X x = unit();
    x.dy = 1;
    return x;
  }
  static constexpr X unit() { return {ARR{}, T(0), T(0)}; }
  static constexpr bool commute = 0;
};
#line 4 "mod/floor_sum_of_linear_polynomial.hpp"

// 全部非負, T は答, U は ax+b がオーバーフローしない
template <typename T, int K1, int K2, typename U>
array<array<T, K2 + 1>, K1 + 1> floor_sum_of_linear_polynomial_nonnegative(U N, U a, U b, U mod) {
  static_assert(is_same_v<U, u64> || is_same_v<U, u128>);
  assert(a == 0 || N < (U(-1) - b) / a);
  using Mono = Monoid_for_floor_sum<T, K1, K2>;
  auto x = floor_monoid_product<Mono>(Mono::to_x(), Mono::to_y(), N, a, b, mod);
  return x.dp;
};

// sum_{L<=x<R} x^i floor(ax+b,mod)^j
// a+bx が I, U でオーバーフローしない
template <typename T, int K1, int K2, typename I>
array<array<T, K2 + 1>, K1 + 1> floor_sum_of_linear_polynomial(I L, I R, I a, I b, I mod) {
  static_assert(is_same_v<I, ll> || is_same_v<I, i128>);
  assert(L <= R && mod > 0);
  if (a < 0) {
    auto ANS = floor_sum_of_linear_polynomial<T, K1, K2, I>(-R + 1, -L + 1, -a, b, mod);
    FOR(i, K1 + 1) {
      if (i % 2 == 1) { FOR(j, K2 + 1) ANS[i][j] = -ANS[i][j]; }
    }
    return ANS;
  }
  assert(a >= 0);
  I ADD_X = L;
  I N = R - L;
  b += a * L;
  I ADD_Y = floor<I>(b, mod);
  b -= ADD_Y * mod;
  assert(a >= 0 && b >= 0);

  using Mono = Monoid_for_floor_sum<T, K1, K2>;
  using Data = typename Mono::Data;
  using U = std::conditional_t<is_same_v<I, ll>, i128, u128>;
  Data A = floor_monoid_product<Mono, Data, U>(Mono::to_x(), Mono::to_y(), N, a, b, mod);
  Data offset = Mono::unit();
  offset.dx = T(ADD_X), offset.dy = T(ADD_Y);
  A = Mono::op(offset, A);
  return A.dp;
};
#line 3 "convex/count_lattice_point_in_convex_polygon_polynomial.hpp"

// 格子点 (x,y) に対して x^iy^j の sum. i<=K, j<=L
template <typename mint, int K1, int K2>
array<array<mint, K2 + 1>, K1 + 1> count_lattice_point_in_convex_polygon_polynomial(ll L, ll R, vc<tuple<ll, ll, ll>> LINE) {
  vc<tuple<ll, ll, ll>> LINE1, LINE2;
  for (auto &[a, b, c]: LINE) {
    if (b == 0) {
      // ax<=c
      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);
      } else {
        LINE1.eb(a, -c, -b);
      }
    }
  }
  if (L >= R) { return {}; }
  assert(!LINE1.empty());
  assert(!LINE2.empty());

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

  array<array<mint, K2 + 2>, K1 + 1> S;
  FOR(i, K1 + 1) FOR(j, K2 + 1) S[i][j] = 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;
    }
    elif (s > 0) {
      // 上側の方が傾きが大きい
      i128 x = ceil<i128>(t, s);
      chmax(L, x);
    }
    else {
      i128 x = floor<i128>(-t, -s);
      chmin(R, x + 1);
    }
    if (L >= R) return;
    auto ADD = floor_sum_of_linear_polynomial<mint, K1, K2 + 1, ll>(L, R, a2, b2, c2);
    auto SUB = floor_sum_of_linear_polynomial<mint, K1, K2 + 1, ll>(L, R, a1, b1 - 1, c1);
    FOR(i, K1 + 1) FOR(j, K2 + 2) S[i][j] += ADD[i][j] - SUB[i][j];
  };

  merge_58(LOWER, UPPER, wk);
  array<array<mint, K2 + 1>, K1 + 1> ANS;
  FOR(i, K1 + 1) FOR(j, K2 + 1) ANS[i][j] = 0;

  static vvc<mint> CF;
  if (CF.empty()) { CF = faulhaber_formula_2d<mint>(K2); }
  FOR(i, K1 + 1) {
    FOR(j, K2 + 1) {
      FOR(k, j + 2) { ANS[i][j] += CF[j][k] * S[i][k]; }
    }
  }
  return ANS;
}

Back to top page