lib/internal/internal_fft.hpp

• View this file on GitHub
• Last update: 2025-05-22 15:25:19+09:00
• Include: #include "lib/internal/internal_fft.hpp"

Depends on

• Montgomery modular multiplication (lib/internal/internal_math.hpp)
• lib/internal/internal_type_traits.hpp
• lib/math/modint.hpp

Required by

• 多倍長整数 (lib/data_structure/bigint.hpp)
• lib/fft/formal_power_series.hpp
• 畳み込み (lib/fft/ntt.hpp)
• 任意mod畳み込み (lib/fft/ntt_mod.hpp)
• lib/math/bell.hpp
• Bostan-Mori法 (lib/math/bostan_mori.hpp)

Verified with

• test/aoj/ntl/addition.test.cpp
• test/aoj/ntl/difference.test.cpp
• test/aoj/ntl/division.test.cpp
• test/aoj/ntl/multiplication.test.cpp
• test/aoj/ntl/multiplication2.test.cpp
• test/aoj/ntl/remainder.test.cpp
• test/yosupo/convolution/convolution.test.cpp
• test/yosupo/convolution/convolution_mod.test.cpp
• test/yosupo/data_structure/addition_of_big_integers.test.cpp
• test/yosupo/enumerative_conbinatorics/bell_number.test.cpp
• test/yosupo/math/bostan_mori.test.cpp
• test/yosupo/polynomial/division_of_polynomials.test.cpp
• test/yosupo/polynomial/exp_of_formal_power_series.test.cpp
• test/yosupo/polynomial/inv_of_formal_power_series.test.cpp
• test/yosupo/polynomial/log_of_formal_power_series.test.cpp
• test/yosupo/polynomial/multipoint_evaluation.test.cpp
• test/yosupo/polynomial/polynomial_interpolation.test.cpp
• test/yosupo/polynomial/polynomial_taylor_shift.test.cpp
• test/yosupo/polynomial/pow_of_formal_power_series.test.cpp
• test/yosupo/polynomial/product_of_polynomial_sequence.test.cpp

Code

#pragma once
#include <algorithm>
#include <array>
#include <bit>
#include <cassert>
#include <cstdint>
#include <type_traits>
#include <vector>
#include "internal/internal_math.hpp"
#include "internal/internal_type_traits.hpp"
#include "math/modint.hpp"

namespace internal {

template <class mint, int g = internal::primitive_root<mint::mod()>, internal::is_static_modint_t<mint> * = nullptr>
struct fft_info {
    static constexpr int rank2 = std::countr_zero<unsigned>(mint::mod() - 1);
    std::array<mint, rank2 + 1> root, iroot;
    std::array<mint, std::max(0, rank2 - 2 + 1)> rate2, irate2;
    std::array<mint, std::max(0, rank2 - 3 + 1)> rate3, irate3;

    fft_info() {
        root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2);
        iroot[rank2] = root[rank2].inv();
        for (int i = rank2 - 1; i >= 0; i--) {
            root[i] = root[i + 1] * root[i + 1];
            iroot[i] = iroot[i + 1] * iroot[i + 1];
        }

        {
            mint prod = 1, iprod = 1;
            for (int i = 0; i <= rank2 - 2; i++) {
                rate2[i] = root[i + 2] * prod;
                irate2[i] = iroot[i + 2] * iprod;
                prod *= iroot[i + 2];
                iprod *= root[i + 2];
            }
        }
        {
            mint prod = 1, iprod = 1;
            for (int i = 0; i <= rank2 - 3; i++) {
                rate3[i] = root[i + 3] * prod;
                irate3[i] = iroot[i + 3] * iprod;
                prod *= iroot[i + 3];
                iprod *= root[i + 3];
            }
        }
    }
};

template <class mint, internal::is_static_modint_t<mint> * = nullptr>
void butterfly(std::vector<mint> &a) {
    int n = int(a.size());
    int h = std::countr_zero<unsigned>(n);
    static const fft_info<mint> info;
    int len = 0;
    while (len < h) {
        if (h - len == 1) {
            int p = 1 << (h - len - 1);
            mint rot = 1;
            for (int s = 0; s < (1 << len); s++) {
                int offset = s << (h - len);
                for (int i = 0; i < p; i++) {
                    auto l = a[i + offset], r = a[i + offset + p] * rot;
                    a[i + offset] = l + r, a[i + offset + p] = l - r;
                }
                if (s + 1 != (1 << len)) rot *= info.rate2[std::countr_zero(~(unsigned int)(s))];
            }
            len++;
        } else {
            int p = 1 << (h - len - 2);
            mint rot = 1, imag = info.root[2];
            for (int s = 0; s < (1 << len); s++) {
                mint rot2 = rot * rot;
                mint rot3 = rot2 * rot;
                int offset = s << (h - len);
                for (int i = 0; i < p; i++) {
                    auto mod2 = 1ull * mint::mod() * mint::mod();
                    auto a0 = 1ull * a[i + offset].val();
                    auto a1 = 1ull * a[i + offset + p].val() * rot.val();
                    auto a2 = 1ull * a[i + offset + 2 * p].val() * rot2.val();
                    auto a3 = 1ull * a[i + offset + 3 * p].val() * rot3.val();
                    auto a1na3imag = 1ull * mint(a1 + mod2 - a3).val() * imag.val();
                    auto na2 = mod2 - a2;
                    a[i + offset] = a0 + a2 + a1 + a3;
                    a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
                    a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
                    a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
                }
                if (s + 1 != (1 << len)) rot *= info.rate3[std::countr_zero(~(unsigned int)(s))];
            }
            len += 2;
        }
    }
}

template <class mint, internal::is_static_modint_t<mint> * = nullptr>
void butterfly_inv(std::vector<mint> &a) {
    int n = int(a.size());
    int h = std::countr_zero<unsigned>(n);
    static const fft_info<mint> info;
    int len = h;
    while (len) {
        if (len == 1) {
            int p = 1 << (h - len);
            mint irot = 1;
            for (int s = 0; s < (1 << (len - 1)); s++) {
                int offset = s << (h - len + 1);
                for (int i = 0; i < p; i++) {
                    auto l = a[i + offset], r = a[i + offset + p];
                    a[i + offset] = l + r;
                    a[i + offset + p] = (std::uint64_t)(mint::mod() + l.val() - r.val()) * irot.val();
                }
                if (s + 1 != (1 << (len - 1))) irot *= info.irate2[std::countr_zero(~(unsigned int)(s))];
            }
            len--;
        } else {
            int p = 1 << (h - len);
            mint irot = 1, iimag = info.iroot[2];
            for (int s = 0; s < (1 << (len - 2)); s++) {
                mint irot2 = irot * irot;
                mint irot3 = irot2 * irot;
                int offset = s << (h - len + 2);
                for (int i = 0; i < p; i++) {
                    auto a0 = 1ull * a[i + offset + 0 * p].val();
                    auto a1 = 1ull * a[i + offset + 1 * p].val();
                    auto a2 = 1ull * a[i + offset + 2 * p].val();
                    auto a3 = 1ull * a[i + offset + 3 * p].val();
                    auto a2na3iimag = 1ull * mint((mint::mod() + a2 - a3) * iimag.val()).val();
                    a[i + offset] = a0 + a1 + a2 + a3;
                    a[i + offset + 1 * p] = (a0 + (mint::mod() - a1) + a2na3iimag) * irot.val();
                    a[i + offset + 2 * p] = (a0 + a1 + (mint::mod() - a2) + (mint::mod() - a3)) * irot2.val();
                    a[i + offset + 3 * p] = (a0 + (mint::mod() - a1) + (mint::mod() - a2na3iimag)) * irot3.val();
                }
                if (s + 1 != (1 << (len - 2))) irot *= info.irate3[std::countr_zero(~(unsigned int)(s))];
            }
            len -= 2;
        }
    }
}

template <class mint, internal::is_static_modint_t<mint> * = nullptr>
std::vector<mint> convolution_naive(const std::vector<mint> &a, const std::vector<mint> &b) {
    int n = int(a.size()), m = int(b.size());
    std::vector<mint> ans(n + m - 1);
    if (n < m) {
        for (int j = 0; j < m; j++) {
            for (int i = 0; i < n; i++) ans[i + j] += a[i] * b[j];
        }
    } else {
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) ans[i + j] += a[i] * b[j];
        }
    }
    return ans;
}

template <class mint, internal::is_static_modint_t<mint> * = nullptr>
std::vector<mint> convolution_fft(std::vector<mint> a, std::vector<mint> b) {
    int n = int(a.size()), m = int(b.size());
    int z = std::bit_ceil<unsigned>(n + m - 1);
    a.resize(z);
    internal::butterfly(a);
    b.resize(z);
    internal::butterfly(b);
    for (int i = 0; i < z; i++) { a[i] *= b[i]; }
    internal::butterfly_inv(a);
    a.resize(n + m - 1);
    mint iz = mint(z).inv();
    for (int i = 0; i < n + m - 1; i++) a[i] *= iz;
    return a;
}

}  // namespace internal

Traceback (most recent call last):
  File "/home/runner/.local/lib/python3.12/site-packages/competitive_verifier/oj/resolver.py", line 291, in resolve
    bundled_code = language.bundle(path, basedir=basedir)
                   ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
  File "/home/runner/.local/lib/python3.12/site-packages/competitive_verifier/oj/verify/languages/cplusplus.py", line 242, in bundle
    bundler.update(path)
  File "/home/runner/.local/lib/python3.12/site-packages/competitive_verifier/oj/verify/languages/cplusplus_bundle.py", line 479, in update
    self._resolve(pathlib.Path(included), included_from=path)
  File "/home/runner/.local/lib/python3.12/site-packages/competitive_verifier/oj/verify/languages/cplusplus_bundle.py", line 286, in _resolve
    raise BundleErrorAt(path, -1, "no such header")
competitive_verifier.oj.verify.languages.cplusplus_bundle.BundleErrorAt: internal/internal_math.hpp: line -1: no such header

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