Tifa's CP Library

:heavy_check_mark: exlucas (src/code/comb/exlucas.hpp)

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#ifndef TIFALIBS_COMB_EXLUCAS
#define TIFALIBS_COMB_EXLUCAS

#include "../math/isqrt.hpp"
#include "../nt/crt.hpp"
#include "lucas_pp.hpp"

namespace tifa_libs::math {

// Calculate binom(m, n) mod p, p can be ANY INTEGER
class ExLucas {
  u32 m_;
  vecu64 ms;
  vec<LucasPP> cs;

 public:
  explicit constexpr ExLucas(u32 md) : m_(md) {
    assert(md < 100'000'000);
    for (u32 i = 2, ed_ = isqrt(md); i <= ed_; ++i)
      if (md % i == 0) {
        u32 j = 0, k = 1;
        while (md % i == 0) {
          md /= i;
          ++j;
          k *= i;
        }
        ms.push_back(k);
        cs.emplace_back(i, j);
      }
    if (md > 1) {
      ms.push_back(md);
      cs.emplace_back(md, 1);
    }
  }

  constexpr u64 operator()(i64 m, i64 n) const {
    if (m_ == 1 || m < n || n < 0) return 0;
    vec<i64> b;
    b.reserve(cs.size());
    for (auto const &i : cs) b.push_back((i64)i(m, n));
    return crt(b, ms)->first;
  }
};

}  // namespace tifa_libs::math

#endif
#line 1 "src/code/comb/exlucas.hpp"



#line 1 "src/code/math/isqrt.hpp"



#line 1 "src/code/util/util.hpp"



#include <bits/stdc++.h>

template <class T>
constexpr T abs(T x) { return x < 0 ? -x : x; }

using i8 = int8_t;
using i16 = int16_t;
using i32 = int32_t;
using i64 = int64_t;
using i128 = __int128_t;
using isz = ptrdiff_t;

using u8 = uint8_t;
using u16 = uint16_t;
using u32 = uint32_t;
using u64 = uint64_t;
using u128 = __uint128_t;
using usz = size_t;

using f32 = float;
using f64 = double;
using f128 = long double;

template <class T>
using ptt = std::pair<T, T>;
template <class T>
using pt3 = std::tuple<T, T, T>;
template <class T>
using pt4 = std::tuple<T, T, T, T>;

template <class T, usz N>
using arr = std::array<T, N>;
template <class T>
using vec = std::vector<T>;
template <class T>
using vvec = vec<vec<T>>;
template <class T>
using v3ec = vec<vvec<T>>;
template <class U, class T>
using vecp = vec<std::pair<U, T>>;
template <class U, class T>
using vvecp = vvec<std::pair<U, T>>;
template <class T>
using vecpt = vec<ptt<T>>;
template <class T>
using vvecpt = vvec<ptt<T>>;

template <class T, class C = std::less<T>>
using pq = std::priority_queue<T, vec<T>, C>;
template <class T>
using pqg = std::priority_queue<T, vec<T>, std::greater<T>>;

using strn = std::string;
using strnv = std::string_view;

using vecu = vec<u32>;
using vvecu = vvec<u32>;
using v3ecu = v3ec<u32>;
using vecu64 = vec<u64>;
using vecb = vec<bool>;
using vvecb = vvec<bool>;

#ifdef ONLINE_JUDGE
#undef assert
#define assert(x) 42
#endif

using namespace std::literals;

constexpr i8 operator""_i8(unsigned long long x) { return (i8)x; }
constexpr i16 operator""_i16(unsigned long long x) { return (i16)x; }
constexpr i32 operator""_i32(unsigned long long x) { return (i32)x; }
constexpr i64 operator""_i64(unsigned long long x) { return (i64)x; }
constexpr isz operator""_iz(unsigned long long x) { return (isz)x; }

constexpr u8 operator""_u8(unsigned long long x) { return (u8)x; }
constexpr u16 operator""_u16(unsigned long long x) { return (u16)x; }
constexpr u32 operator""_u32(unsigned long long x) { return (u32)x; }
constexpr u64 operator""_u64(unsigned long long x) { return (u64)x; }
constexpr usz operator""_uz(unsigned long long x) { return (usz)x; }

inline const auto fn_0 = [](auto&&...) {};


#line 5 "src/code/math/isqrt.hpp"

namespace tifa_libs::math {

constexpr u32 isqrt(u64 x) {
  if (!x) return 0;
  int c = i32(std::bit_width(x) - 1) / 2, sh = 31 - c;
  u32 u = [](u64 x) {
    constexpr u8 TAB[192] = {128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 144, 145, 146, 147, 148, 149, 150, 151, 151, 152, 153, 154, 155, 156, 156, 157, 158, 159, 160, 160, 161, 162, 163, 164, 164, 165, 166, 167, 167, 168, 169, 170, 170, 171, 172, 173, 173, 174, 175, 176, 176, 177, 178, 179, 179, 180, 181, 181, 182, 183, 183, 184, 185, 186, 186, 187, 188, 188, 189, 190, 190, 191, 192, 192, 193, 194, 194, 195, 196, 196, 197, 198, 198, 199, 200, 200, 201, 201, 202, 203, 203, 204, 205, 205, 206, 206, 207, 208, 208, 209, 210, 210, 211, 211, 212, 213, 213, 214, 214, 215, 216, 216, 217, 217, 218, 219, 219, 220, 220, 221, 221, 222, 223, 223, 224, 224, 225, 225, 226, 227, 227, 228, 228, 229, 229, 230, 230, 231, 232, 232, 233, 233, 234, 234, 235, 235, 236, 237, 237, 238, 238, 239, 239, 240, 240, 241, 241, 242, 242, 243, 243, 244, 244, 245, 246, 246, 247, 247, 248, 248, 249, 249, 250, 250, 251, 251, 252, 252, 253, 253, 254, 254, 255, 255, 255};
    u32 u = TAB[(x >> 56) - 64];
    u = (u << 7) + (u32)(x >> 41) / u;
    return (u << 15) + (u32)((x >> 17) / u);
  }(x << 2 * sh);
  u >>= sh;
  u -= (u64)u * u > x;
  return u;
}

}  // namespace tifa_libs::math


#line 1 "src/code/nt/crt.hpp"



#line 1 "src/code/math/safe_mod.hpp"



#line 1 "src/code/util/traits.hpp"



#line 5 "src/code/util/traits.hpp"

namespace tifa_libs {

template <class T>
concept iterable_c = requires(T v) {
  { v.begin() } -> std::same_as<typename T::iterator>;
  { v.end() } -> std::same_as<typename T::iterator>;
};

template <class T>
concept container_c = iterable_c<T> && !std::derived_from<T, std::basic_string<typename T::value_type>>;

template <class T>
constexpr bool is_char_v = std::is_same_v<T, char> || std::is_same_v<T, signed char> || std::is_same_v<T, unsigned char>;
template <class T>
concept char_c = is_char_v<T>;

template <class T>
constexpr bool is_s128_v = std::is_same_v<T, __int128_t> || std::is_same_v<T, __int128>;
template <class T>
concept s128_c = is_s128_v<T>;

template <class T>
constexpr bool is_u128_v = std::is_same_v<T, __uint128_t> || std::is_same_v<T, unsigned __int128>;
template <class T>
concept u128_c = is_u128_v<T>;

template <class T>
constexpr bool is_i128_v = is_s128_v<T> || is_u128_v<T>;
template <class T>
concept i128_c = is_u128_v<T>;

template <class T>
constexpr bool is_int_v = std::is_integral_v<T> || is_i128_v<T>;
template <class T>
concept int_c = is_int_v<T>;

template <class T>
constexpr bool is_sint_v = is_s128_v<T> || (is_int_v<T> && std::is_signed_v<T>);
template <class T>
concept sint_c = is_sint_v<T>;

template <class T>
constexpr bool is_uint_v = is_u128_v<T> || (is_int_v<T> && std::is_unsigned_v<T>);
template <class T>
concept uint_c = is_uint_v<T>;

template <class T>
concept mint_c = requires(T x) {
  { x.mod() } -> uint_c;
  { x.val() } -> uint_c;
};

template <class T>
constexpr bool is_arithm_v = std::is_arithmetic_v<T> || is_int_v<T>;
template <class T>
concept arithm_c = is_arithm_v<T>;

template <class T>
struct to_sint : std::make_signed<T> {};
template <>
struct to_sint<u128> {
  using type = u128;
};
template <>
struct to_sint<i128> {
  using type = u128;
};
template <class T>
using to_sint_t = typename to_sint<T>::type;

template <class T>
struct to_uint : std::make_unsigned<T> {};
template <>
struct to_uint<u128> {
  using type = u128;
};
template <>
struct to_uint<i128> {
  using type = u128;
};
template <class T>
using to_uint_t = typename to_uint<T>::type;

}  // namespace tifa_libs


#line 5 "src/code/math/safe_mod.hpp"

namespace tifa_libs::math {

template <sint_c T>
constexpr T safe_mod(T x, to_uint_t<T> mod) { return ((x %= (T)mod) < 0 ? x + (T)mod : x); }

}  // namespace tifa_libs::math


#line 1 "src/code/nt/inv_gcd.hpp"



#line 1 "src/code/nt/exgcd.hpp"



#line 5 "src/code/nt/exgcd.hpp"

namespace tifa_libs::math {

// @return tuple(g, x, y) s.t. g = gcd(a, b), xa + yb = g, |x| + |y| is the minimal (primary) and x <= y (secondarily)
template <sint_c T>
constexpr auto exgcd(T a, T b) {
  T x1 = 1, x2 = 0, x3 = 0, x4 = 1;
  while (b) {
    T c = a / b;
    std::tie(x1, x2, x3, x4, a, b) = std::make_tuple(x3, x4, x1 - x3 * c, x2 - x4 * c, b, a - b * c);
  }
  return std::make_tuple(to_uint_t<T>(a), x1, x2);
}

}  // namespace tifa_libs::math


#line 6 "src/code/nt/inv_gcd.hpp"

namespace tifa_libs::math {

template <uint_c T>
constexpr ptt<T> inv_gcd(T n, T mod) {
  using U = to_sint_t<T>;
  auto [g, x, y] = exgcd(U(n % mod), (U)mod);
  return {g, safe_mod(x, mod)};
}

}  // namespace tifa_libs::math


#line 6 "src/code/nt/crt.hpp"

namespace tifa_libs::math {
namespace crt_impl_ {
constexpr std::optional<ptt<i64>> crt2(i64 a0, u64 m0, i64 a1, u64 m1) {
  if (m0 < m1) return crt2(a1, m1, a0, m0);
  auto [d, x] = inv_gcd(m0, m1);
  i64 a1_a0 = a1 - a0, a1_a0_d = a1_a0 / (i64)d;
  if (a1_a0 != a1_a0_d * (i64)d) return {};
  i64 m1_d = (i64)(m1 / d), k0 = (i64)x % m1_d * (a1_a0_d % m1_d) % m1_d;
  if (k0 < 0) k0 += m1_d;
  return std::make_pair(a0 + k0 * (i64)m0, (i64)m0 * m1_d);
}
}  // namespace crt_impl_

// Returns (remainder, modular)
constexpr std::optional<ptt<u64>> crt(vec<i64> const &a, vecu64 const &m) {
  if (a.size() != m.size()) return {};
  i64 A = 0;
  u64 M = 1;
  u32 n = (u32)a.size();
  for (u32 i = 0; i < n; ++i) {
    auto res = crt_impl_::crt2(safe_mod(a[i], m[i]), m[i], A, M);
    if (!res) return {};
    std::tie(A, M) = res.value();
  }
  return std::make_pair(A, M);
}

}  // namespace tifa_libs::math


#line 1 "src/code/comb/lucas_pp.hpp"



#line 1 "src/code/math/qpow_mod.hpp"



#line 1 "src/code/math/mul_mod_u.hpp"



#line 5 "src/code/math/mul_mod_u.hpp"

namespace tifa_libs::math {

constexpr u64 mul_mod_u(u64 a, u64 b, u64 mod) {
  if (std::bit_width(a) + std::bit_width(b) <= 64) return a * b % mod;
  else return (u64)((u128)a * b % mod);
}

}  // namespace tifa_libs::math


#line 5 "src/code/math/qpow_mod.hpp"

namespace tifa_libs::math {

constexpr u64 qpow_mod(u64 a, u64 b, u64 mod) {
  u64 res(1);
  for (a %= mod; b; b >>= 1, a = mul_mod_u(a, a, mod))
    if (b & 1) res = mul_mod_u(res, a, mod);
  return res;
}

}  // namespace tifa_libs::math


#line 1 "src/code/comb/gen_invseq.hpp"



#line 1 "src/code/nt/inverse.hpp"



#line 5 "src/code/nt/inverse.hpp"

namespace tifa_libs::math {

template <uint_c T, uint_c U>
constexpr U inverse(T n, U mod) {
  auto [g, x] = inv_gcd(U(n % mod), mod);
  assert(g == 1);
  return x;
}

}  // namespace tifa_libs::math


#line 6 "src/code/comb/gen_invseq.hpp"

namespace tifa_libs::math {

// i^{-1} for i in v
constexpr vecu64 gen_invseq(vecu64 const &v, u64 mod) {
  u32 n = (u32)v.size();
  if (n == 0) return {};
  if (n == 1) return {inverse(v[0], mod)};
  vecu64 ans(n);
  ans[0] = v[1];
  for (u32 i = 1; i < n; ++i) ans[i] = mul_mod_u(ans[i - 1], v[i], mod);
  u64 _ = inverse(ans.back(), mod);
  for (u32 i = n - 1; i; --i) {
    ans[i] = mul_mod_u(_, ans[i - 1], mod);
    _ = mul_mod_u(_, v[i], mod);
  }
  ans[0] = _;
  return ans;
}
// i^{-1} for i in v
template <class mint>
constexpr vec<mint> gen_invseq(vec<mint> const &v) {
  u32 n = (u32)v.size();
  vec<mint> ans(n);
  auto _ = gen_invseq(v, mint::mod());
  for (u32 i = 0; i < n; ++i) ans[i] = _[i];
  return ans;
}

}  // namespace tifa_libs::math


#line 6 "src/code/comb/lucas_pp.hpp"

namespace tifa_libs::math {

class LucasPP {
  u32 p, q;
  u64 m_;
  bool no_proot;
  vecu64 facp, ifacp;

 public:
  // mod = p ** q
  // @param p MUSU be prime
  constexpr LucasPP(u32 p, u32 q) : p(p), q(q) {
    assert(p <= 100'000'000 && q > 0);
    m_ = 1;
    while (q--) {
      m_ *= p;
      assert(m_ <= 100'000'000);
    }
    no_proot = (p == 2 && q >= 3);
    facp.resize(m_);
    facp[0] = facp[1] = 1;
    for (u32 i = 2; i < m_; ++i)
      if (i % p == 0) {
        facp[i] = facp[i - 1];
        facp[i + 1] = facp[i - 1] * (i + 1) % m_;
        ++i;
      } else facp[i] = facp[i - 1] * i % m_;
    ifacp = gen_invseq(facp, m_);
  }

  constexpr u64 mod() const { return m_; }

  constexpr u64 operator()(i64 m, i64 n) const {
    if (m < n || n < 0) return 0;
    i64 r = m - n;
    i32 e0 = 0, eq = 0;
    u32 i = 0;
    u64 res = 1;
    while (m) {
      res = res * facp[(u64)m % m_] % m_ * ifacp[(u64)n % m_] % m_ * ifacp[(u64)r % m_] % m_;
      m /= p;
      n /= p;
      r /= p;
      i32 eps = (i32)(m - n - r);
      if ((e0 += eps) >= (i32)q) return 0;
      if (++i >= q) eq += eps;
    }
    return (!no_proot && (eq & 1) ? m_ - res : res) * qpow_mod(p, (u32)e0, m_) % m_;
  }
};

}  // namespace tifa_libs::math


#line 7 "src/code/comb/exlucas.hpp"

namespace tifa_libs::math {

// Calculate binom(m, n) mod p, p can be ANY INTEGER
class ExLucas {
  u32 m_;
  vecu64 ms;
  vec<LucasPP> cs;

 public:
  explicit constexpr ExLucas(u32 md) : m_(md) {
    assert(md < 100'000'000);
    for (u32 i = 2, ed_ = isqrt(md); i <= ed_; ++i)
      if (md % i == 0) {
        u32 j = 0, k = 1;
        while (md % i == 0) {
          md /= i;
          ++j;
          k *= i;
        }
        ms.push_back(k);
        cs.emplace_back(i, j);
      }
    if (md > 1) {
      ms.push_back(md);
      cs.emplace_back(md, 1);
    }
  }

  constexpr u64 operator()(i64 m, i64 n) const {
    if (m_ == 1 || m < n || n < 0) return 0;
    vec<i64> b;
    b.reserve(cs.size());
    for (auto const &i : cs) b.push_back((i64)i(m, n));
    return crt(b, ms)->first;
  }
};

}  // namespace tifa_libs::math


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