Tifa's CP Library

:heavy_check_mark: lagrange_interp0 (src/code/math/lagrange_interp0.hpp)

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#ifndef TIFALIBS_MATH_LAGRANGE_INTERP0
#define TIFALIBS_MATH_LAGRANGE_INTERP0

#include "../comb/gen_ifact.hpp"
#include "mul_mod_s.hpp"
#include "mul_mod_u.hpp"

namespace tifa_libs::math {

constexpr i64 lagrange_interp0(vec<i64> const &v, u64 x, u64 mod, vecu64 const &ifact) {
  u32 n = (u32)v.size();
  assert(n);
  if (n == 1) return v[0];
  if (x < n) return v[x];
  vecu64 pre(n);
  for (u32 i = 0; i < n; ++i) pre[i] = x - i;
  for (u32 i = 1; i < n; ++i) pre[i] = mul_mod_u(pre[i], pre[i - 1], mod);
  vecu64 suc(n);
  for (u32 i = 0; i < n; ++i) suc[i] = x - i;
  for (u32 i = n - 2; ~i; --i) suc[i] = mul_mod_u(suc[i], suc[i + 1], mod);
  i64 ans = 0;
  for (u32 i = 0; i < n; ++i) {
    i64 _ = v[i];
    if (i) _ = mul_mod_s(_, (i64)pre[i - 1], mod);
    if (i + 1 < n) _ = mul_mod_s(_, (i64)suc[i + 1], mod);
    _ = mul_mod_s(mul_mod_s(_, (i64)ifact[i], mod), (i64)ifact[n - i - 1], mod);
    ans = (ans + ((n - i) % 2 ? _ : (i64)mod - _)) % (i64)mod;
  }
  return ans;
}
constexpr i64 lagrange_interp0(vec<i64> const &v, u64 x, u64 mod) { return lagrange_interp0(v, x, mod, gen_ifact((u32)v.size(), mod)); }
template <class mint>
constexpr mint lagrange_interp0(vec<mint> const &v, u64 x, vec<mint> const &ifact) {
  vec<i64> _(v.size());
  for (u32 i = 0; i < (u32)v.size(); ++i) _[i] = v[i].val();
  vecu64 ifa(ifact.size());
  for (u32 i = 0; i < (u32)ifact.size(); ++i) ifa[i] = ifact[i].val();
  return mint(lagrange_interp0(_, x, mint::mod(), ifa));
}
template <class mint>
constexpr mint lagrange_interp0(vec<mint> const &v, u64 x) { return lagrange_interp0(v, x, mint::mod(), gen_ifact<mint>(v.size())); }

}  // namespace tifa_libs::math

#endif
#line 1 "src/code/math/lagrange_interp0.hpp"



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



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



#line 5 "src/code/comb/gen_inv.hpp"

namespace tifa_libs::math {

// i^{-1} from i=0..n-1
constexpr vecu64 gen_inv(u32 n, u64 mod) {
  if (n == 0) return {};
  if (n == 1) return {1};
  vecu64 ans(n);
  ans[0] = ans[1] = 1;
  for (u32 i = 2; i < n; ++i) ans[i] = mul_mod_u(mod - mod / i, ans[mod % i], mod);
  return ans;
}
// i^{-1} from i=0..n-1
template <class mint>
constexpr vec<mint> gen_inv(u32 n) {
  vec<mint> ans(n);
  auto _ = gen_inv(n, mint::mod());
  for (u32 i = 0; i < n; ++i) ans[i] = _[i];
  return ans;
}

}  // namespace tifa_libs::math


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

namespace tifa_libs::math {

// (i!)^{-1} from i=0..n-1
constexpr vecu64 gen_ifact(u32 n, u64 mod, vecu64 inv) {
  for (u32 i = 2; i < n; ++i) inv[i] = mul_mod_u(inv[i], inv[i - 1], mod);
  return inv;
}
// (i!)^{-1} from i=0..n-1
constexpr vecu64 gen_ifact(u32 n, u64 mod) { return gen_ifact(n, mod, gen_inv(n, mod)); }
// (i!)^{-1} from i=0..n-1
template <class mint>
constexpr vec<mint> gen_ifact(u32 n, vec<mint> inv) {
  for (u32 i = 2; i < n; ++i) inv[i] *= inv[i - 1];
  return inv;
}
// (i!)^{-1} from i=0..n-1
template <class mint>
constexpr vec<mint> gen_ifact(u32 n) { return gen_ifact(n, gen_inv<mint>(n)); }

}  // namespace tifa_libs::math


#line 1 "src/code/math/mul_mod_s.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 5 "src/code/math/mul_mod_s.hpp"

namespace tifa_libs::math {

constexpr i64 mul_mod_s(i64 a, i64 b, u64 mod) {
  if (std::bit_width((u64)abs(a)) + std::bit_width((u64)abs(b)) < 64) return safe_mod(a * b % (i64)mod, mod);
  else return safe_mod((i64)((i128)a * b % mod), mod);
}

}  // namespace tifa_libs::math


#line 7 "src/code/math/lagrange_interp0.hpp"

namespace tifa_libs::math {

constexpr i64 lagrange_interp0(vec<i64> const &v, u64 x, u64 mod, vecu64 const &ifact) {
  u32 n = (u32)v.size();
  assert(n);
  if (n == 1) return v[0];
  if (x < n) return v[x];
  vecu64 pre(n);
  for (u32 i = 0; i < n; ++i) pre[i] = x - i;
  for (u32 i = 1; i < n; ++i) pre[i] = mul_mod_u(pre[i], pre[i - 1], mod);
  vecu64 suc(n);
  for (u32 i = 0; i < n; ++i) suc[i] = x - i;
  for (u32 i = n - 2; ~i; --i) suc[i] = mul_mod_u(suc[i], suc[i + 1], mod);
  i64 ans = 0;
  for (u32 i = 0; i < n; ++i) {
    i64 _ = v[i];
    if (i) _ = mul_mod_s(_, (i64)pre[i - 1], mod);
    if (i + 1 < n) _ = mul_mod_s(_, (i64)suc[i + 1], mod);
    _ = mul_mod_s(mul_mod_s(_, (i64)ifact[i], mod), (i64)ifact[n - i - 1], mod);
    ans = (ans + ((n - i) % 2 ? _ : (i64)mod - _)) % (i64)mod;
  }
  return ans;
}
constexpr i64 lagrange_interp0(vec<i64> const &v, u64 x, u64 mod) { return lagrange_interp0(v, x, mod, gen_ifact((u32)v.size(), mod)); }
template <class mint>
constexpr mint lagrange_interp0(vec<mint> const &v, u64 x, vec<mint> const &ifact) {
  vec<i64> _(v.size());
  for (u32 i = 0; i < (u32)v.size(); ++i) _[i] = v[i].val();
  vecu64 ifa(ifact.size());
  for (u32 i = 0; i < (u32)ifact.size(); ++i) ifa[i] = ifact[i].val();
  return mint(lagrange_interp0(_, x, mint::mod(), ifa));
}
template <class mint>
constexpr mint lagrange_interp0(vec<mint> const &v, u64 x) { return lagrange_interp0(v, x, mint::mod(), gen_ifact<mint>(v.size())); }

}  // namespace tifa_libs::math


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