多項式版 floor sum
(math/polynomial-floor-sum.hpp)

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Last update: 2025-10-17 21:43:09+09:00
Include: #include "math/polynomial-floor-sum.hpp"

使い方

以下の問題を $O(pq(p+q)\log m)$ 時間で解く．

整数 $l,r,m,a,b\ (m\neq 0)$ および非負整数 $p,q$ に対し以下の値を求めよ： $f_{p,q}(l,r,m,a,b)=\sum_{i=l}^{r-1}i^p\left\lfloor\frac{ai+b}{m}\right\rfloor^q$

型 U は $an+b$ が正しく計算できる必要がある．

Depends on

Code

#pragma once
#include "math/util.hpp"
#include "math/floor-monoid-product.hpp"

template <class T, int D1, int D2>
struct PolynomialFloorSumMonoid {
  static constexpr int D = max(D1, D2);
  using M = PolynomialFloorSumMonoid;
  using P = array<array<T, D2 + 1>, D1 + 1>;
  T x, y;
  P f;
  static M e() { return {0, 0, P{}}; }
  static M op(M a, M b) {
    static T binom[D + 1][D + 1];
    if (binom[0][0] != T(1)) {
      binom[0][0] = T(1);
      for (int i = 0; i < D; i++)
        for (int j = 0; j <= i; j++) {
          binom[i + 1][j] += binom[i][j];
          binom[i + 1][j + 1] += binom[i][j];
        }
    }
    T pow_x[D1 + 1], pow_y[D2 + 1];
    pow_x[0] = 1, pow_y[0] = 1;
    for (int i = 0; i < D1; i++) pow_x[i + 1] = pow_x[i] * a.x;
    for (int i = 0; i < D2; i++) pow_y[i + 1] = pow_y[i] * a.y;
    for (int i = 0; i <= D1; i++)
      for (int j = 0; j <= D2; j++) {
        T v = b.f[i][j];
        for (int k = i + 1; k <= D1; k++)
          b.f[k][j] += v * pow_x[k - i] * binom[k][i];
      }
    for (int i = 0; i <= D1; i++)
      for (int j = 0; j <= D2; j++) {
        T v = b.f[i][j];
        for (int k = j; k <= D2; k++)
          a.f[i][k] += v * pow_y[k - j] * binom[k][j];
      }
    return {a.x + b.x, a.y + b.y, a.f};
  }
  static M elm_x() {
    M t = e();
    t.x = 1, t.f[0][0] = 1;
    return t;
  }
  static M elm_y() {
    M t = e();
    t.y = 1;
    return t;
  }
};

// enumerate sum{k=l}^{r-1}k^i*floor((a*k+b)/m))^j
template <class T, int D1, int D2, std::signed_integral I = int64_t>
array<array<T, D2 + 1>, D1 + 1> PolynomialFloorSum(I l, I r, I m, I a, I b) {
  assert(l <= r && m != 0);
  using U = conditional_t<is_same_v<I, __int128_t>, __uint128_t, uint64_t>;
  using M = PolynomialFloorSumMonoid<T, D1, D2>;
  using P = array<array<T, D2 + 1>, D1 + 1>;
  if (m < 0) m = -m, a = -a, b = -b;
  if (a < 0) {
    P c = PolynomialFloorSum<T, D1, D2, I>(-r + 1, -l + 1, m, -a, b);
    for (int i = 1; i <= D1; i += 2)
      for (int j = 0; j <= D2; j++)
        c[i][j] = -c[i][j];
    return c;
  }
  b += a * l;
  I q = Math::floor(b, m);
  b -= q * m;
  M t = M::e();
  t.x = l, t.y = q;
  M z = FloorMonoidProduct<M, M::op, M::e, U>(r - l, m, a, b, M::elm_x(), M::elm_y());
  return M::op(t, z).f;
}

// P(x,y): 2 variables polynomial, deg_x(P)<=D1, deg_y(Q)<=D2
// find sum{k=0}^{n-1}P(k,floor((a*k+b)/m))
template <class T, int D1, int D2, std::signed_integral I = int64_t>
T PolynomialFloorSum(array<array<T, D2 + 1>, D1 + 1> poly, I l, I r, I m, I a, I b) {
  using M = PolynomialFloorSumMonoid<T, D1, D2>;
  auto c = PolynomialFloorSum<T, D1, D2, I>(l, r, m, a, b);
  T res = 0;
  for (int i = 0; i <= D1; i++)
    for (int j = 0; j <= D2; j++)
      res += poly[i][j] * c[i][j];
  return res;
}

/**
 * @brief 多項式版 floor sum
 * @docs docs/math/polynomial-floor-sum.md
 */

#line 2 "math/util.hpp"

namespace Math {
template <class T>
T safe_mod(T a, T b) {
  assert(b != 0);
  if (b < 0) a = -a, b = -b;
  a %= b;
  return a >= 0 ? a : a + b;
}
template <class T>
T floor(T a, T b) {
  assert(b != 0);
  if (b < 0) a = -a, b = -b;
  return a >= 0 ? a / b : (a + 1) / b - 1;
}
template <class T>
T ceil(T a, T b) {
  assert(b != 0);
  if (b < 0) a = -a, b = -b;
  return a > 0 ? (a - 1) / b + 1 : a / b;
}
long long isqrt(long long n) {
  if (n <= 0) return 0;
  long long x = sqrt(n);
  while ((x + 1) * (x + 1) <= n) x++;
  while (x * x > n) x--;
  return x;
}
// return g=gcd(a,b)
// a*x+b*y=g
// - b!=0 -> 0<=x<|b|/g
// - b=0  -> ax=g
template <class T>
T ext_gcd(T a, T b, T& x, T& y) {
  T a0 = a, b0 = b;
  bool sgn_a = a < 0, sgn_b = b < 0;
  if (sgn_a) a = -a;
  if (sgn_b) b = -b;
  if (b == 0) {
    x = sgn_a ? -1 : 1;
    y = 0;
    return a;
  }
  T x00 = 1, x01 = 0, x10 = 0, x11 = 1;
  while (b != 0) {
    T q = a / b, r = a - b * q;
    x00 -= q * x01;
    x10 -= q * x11;
    swap(x00, x01);
    swap(x10, x11);
    a = b, b = r;
  }
  x = x00, y = x10;
  if (sgn_a) x = -x;
  if (sgn_b) y = -y;
  if (b0 != 0) {
    a0 /= a, b0 /= a;
    if (b0 < 0) a0 = -a0, b0 = -b0;
    T q = x >= 0 ? x / b0 : (x + 1) / b0 - 1;
    x -= b0 * q;
    y += a0 * q;
  }
  return a;
}
template <class T>
T inv_mod(T x, T m) {
  x %= m;
  if (x < 0) x += m;
  T a = m, b = x;
  T y0 = 0, y1 = 1;
  while (b > 0) {
    T q = a / b;
    swap(a -= q * b, b);
    swap(y0 -= q * y1, y1);
  }
  if (y0 < 0) y0 += m / a;
  return y0;
}
template <class T>
T pow_mod(T x, T n, T m) {
  x = (x % m + m) % m;
  T y = 1;
  while (n) {
    if (n & 1) y = y * x % m;
    x = x * x % m;
    n >>= 1;
  }
  return y;
}
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
  if (m == 1) return 0;
  unsigned int _m = (unsigned int)(m);
  unsigned long long r = 1;
  unsigned long long y = x % m;
  if (y >= m) y += m;
  while (n) {
    if (n & 1) r = (r * y) % _m;
    y = (y * y) % _m;
    n >>= 1;
  }
  return r;
}
};  // namespace Math
#line 2 "math/floor-monoid-product.hpp"

template <class T, T (*op)(T, T), T (*e)(), std::unsigned_integral U = uint64_t>
T FloorMonoidProduct(U n, U m, U a, U b, T x, T y, function<T(T, U)> pow = nullptr) {
  if (!pow) {
    pow = [](T t, U n) {
      T p = e();
      while (n) {
        if (n & 1) p = op(p, t);
        t = op(t, t);
        n >>= 1;
      }
      return p;
    };
  }
  assert(m != 0);
  T pl = e(), pr = e();
  while (true) {
    if (a >= m) {
      U q = a / m;
      x = op(x, pow(y, q));
      a -= m * q;
    }
    if (b >= m) {
      U q = b / m;
      pl = op(pl, pow(y, q));
      b -= m * q;
    }
    U c = a * n + b;
    if (c < m) {
      pl = op(pl, pow(x, n));
      break;
    }
    pr = op(op(y, pow(x, c % m / a)), pr);
    n = c / m - 1;
    b = m + a - b - 1;
    swap(a, m);
    swap(x, y);
  }
  return op(pl, pr);
}

/**
 * @brief モノイド版 Floor Sum
 * @docs docs/math/floor-monoid-product.md
 */
#line 4 "math/polynomial-floor-sum.hpp"

template <class T, int D1, int D2>
struct PolynomialFloorSumMonoid {
  static constexpr int D = max(D1, D2);
  using M = PolynomialFloorSumMonoid;
  using P = array<array<T, D2 + 1>, D1 + 1>;
  T x, y;
  P f;
  static M e() { return {0, 0, P{}}; }
  static M op(M a, M b) {
    static T binom[D + 1][D + 1];
    if (binom[0][0] != T(1)) {
      binom[0][0] = T(1);
      for (int i = 0; i < D; i++)
        for (int j = 0; j <= i; j++) {
          binom[i + 1][j] += binom[i][j];
          binom[i + 1][j + 1] += binom[i][j];
        }
    }
    T pow_x[D1 + 1], pow_y[D2 + 1];
    pow_x[0] = 1, pow_y[0] = 1;
    for (int i = 0; i < D1; i++) pow_x[i + 1] = pow_x[i] * a.x;
    for (int i = 0; i < D2; i++) pow_y[i + 1] = pow_y[i] * a.y;
    for (int i = 0; i <= D1; i++)
      for (int j = 0; j <= D2; j++) {
        T v = b.f[i][j];
        for (int k = i + 1; k <= D1; k++)
          b.f[k][j] += v * pow_x[k - i] * binom[k][i];
      }
    for (int i = 0; i <= D1; i++)
      for (int j = 0; j <= D2; j++) {
        T v = b.f[i][j];
        for (int k = j; k <= D2; k++)
          a.f[i][k] += v * pow_y[k - j] * binom[k][j];
      }
    return {a.x + b.x, a.y + b.y, a.f};
  }
  static M elm_x() {
    M t = e();
    t.x = 1, t.f[0][0] = 1;
    return t;
  }
  static M elm_y() {
    M t = e();
    t.y = 1;
    return t;
  }
};

// enumerate sum{k=l}^{r-1}k^i*floor((a*k+b)/m))^j
template <class T, int D1, int D2, std::signed_integral I = int64_t>
array<array<T, D2 + 1>, D1 + 1> PolynomialFloorSum(I l, I r, I m, I a, I b) {
  assert(l <= r && m != 0);
  using U = conditional_t<is_same_v<I, __int128_t>, __uint128_t, uint64_t>;
  using M = PolynomialFloorSumMonoid<T, D1, D2>;
  using P = array<array<T, D2 + 1>, D1 + 1>;
  if (m < 0) m = -m, a = -a, b = -b;
  if (a < 0) {
    P c = PolynomialFloorSum<T, D1, D2, I>(-r + 1, -l + 1, m, -a, b);
    for (int i = 1; i <= D1; i += 2)
      for (int j = 0; j <= D2; j++)
        c[i][j] = -c[i][j];
    return c;
  }
  b += a * l;
  I q = Math::floor(b, m);
  b -= q * m;
  M t = M::e();
  t.x = l, t.y = q;
  M z = FloorMonoidProduct<M, M::op, M::e, U>(r - l, m, a, b, M::elm_x(), M::elm_y());
  return M::op(t, z).f;
}

// P(x,y): 2 variables polynomial, deg_x(P)<=D1, deg_y(Q)<=D2
// find sum{k=0}^{n-1}P(k,floor((a*k+b)/m))
template <class T, int D1, int D2, std::signed_integral I = int64_t>
T PolynomialFloorSum(array<array<T, D2 + 1>, D1 + 1> poly, I l, I r, I m, I a, I b) {
  using M = PolynomialFloorSumMonoid<T, D1, D2>;
  auto c = PolynomialFloorSum<T, D1, D2, I>(l, r, m, a, b);
  T res = 0;
  for (int i = 0; i <= D1; i++)
    for (int j = 0; j <= D2; j++)
      res += poly[i][j] * c[i][j];
  return res;
}

/**
 * @brief 多項式版 floor sum
 * @docs docs/math/polynomial-floor-sum.md
 */

多項式版 floor sum (math/polynomial-floor-sum.hpp)

使い方

Depends on

Code

多項式版 floor sum
(math/polynomial-floor-sum.hpp)