混合基数の高速フーリエ変換

fft.cpp

/*
Copyright 2024 OGAWA KenIchi

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The above copyright notice and this permission notice shall be included in
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THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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*/

#include <chrono>
#include <cmath>
#include <complex>
#include <iomanip>
#include <iostream>
#include <iterator>
#include <numbers>
#include <random>
#include <unordered_map>
#include <vector>

template<typename T>
using twiddle_factor_map =
    std::unordered_map<std::size_t, std::vector<std::complex<T>>>;

namespace detail
{

// nの約数を返す
inline
std::size_t find_divisor(std::size_t n, std::size_t start = 2)
{
    for (std::size_t i = start; i * i < n; ++i) {
        if (n % i == 0) return i;
    }
    return n;
}

// 範囲[first, last)の値を離散フーリエ変換した結果を範囲[first, last)に上書きする。
// work_firstは[first, last)と同じサイズの作業用範囲の先頭。
// radixには範囲[first, last)の要素数の約数を指定する。
template<typename T, typename Iter>
void fft(
    Iter first,
    Iter last,
    Iter work_first,
    std::size_t radix,
    const twiddle_factor_map<T>& tf_map)
{
    const auto n = static_cast<std::size_t>(std::distance(first, last));
    if (n <= 1) return;

    const auto& tfs_r = tf_map.at(radix);
    const auto& tfs_n = tf_map.at(n);
    const auto s = n / radix;

    for (std::size_t i = 0; i < s; ++i) {
        auto it = work_first + i;
        for (std::size_t j = 0; j < radix; ++j) {
            *it = 0;
            for (std::size_t k = 0; k < radix; ++k) {
                *it += *(first + i + s * k) * tfs_r[j * k % radix];
            }
            *it *= tfs_n[i * j];
            it += s;
        }
    }

    for (std::size_t i = 0; i < radix; ++i) {
        const auto it = work_first + s * i;
        fft(it, it + s, first, find_divisor(s), tf_map);
    }

    for (std::size_t i = 0; i < s; ++i) {
        auto it = first + radix * i;
        for (std::size_t j = 0; j < radix; ++j) {
            *it = *(work_first + i + s * j);
            ++it;
        }
    }
}

}

// 分母がnの回転因子列を作る
template<typename T>
std::vector<std::complex<T>> make_twiddle_factors(std::size_t n)
{
    std::vector<std::complex<T>> result;
    const auto theta = -2 * std::numbers::pi_v<T> / n;
    for (std::size_t i = 0; i < n; ++i) {
        result.push_back(std::polar<T>(1, theta * i));
    }
    return result;
}


// 要素数nのFFTに必要な回転因子列のマップを作る
template<typename T>
twiddle_factor_map<T> make_twiddle_factor_map(std::size_t n)
{
    twiddle_factor_map<T> result;
    result.try_emplace(n, make_twiddle_factors<T>(n));
    for (auto d = detail::find_divisor(n); d != n; d = detail::find_divisor(n)) {
        if (!result.contains(d)) result.try_emplace(d, make_twiddle_factors<T>(d));
        n /= d;
        if (!result.contains(n)) result.try_emplace(n, make_twiddle_factors<T>(n));
    }
    return result;
}

// inputの値を離散フーリエ変換した結果を返す
template<typename T>
std::vector<std::complex<T>> fft(
    std::vector<std::complex<T>> input,
    const twiddle_factor_map<T>& tf_map)
{
    std::vector<std::complex<double>> work(input.size());
    detail::fft(
        input.begin(),
        input.end(),
        work.begin(),
        detail::find_divisor(input.size()),
        tf_map);
    return input;
}

// inputの値を離散フーリエ変換した結果を返す。
// 素朴な実装により計算する。
template<typename T>
std::vector<std::complex<T>> dft(
    const std::vector<std::complex<T>>& input,
    const std::vector<std::complex<T>>& twiddle_factors)
{
    const auto n = input.size();
    std::vector<std::complex<T>> result(n);
    for (std::size_t i = 0; i < n; ++i) {
        for (std::size_t j = 0; j < n; ++j) {
            result[i] += input[j] * twiddle_factors[i * j % n];
        }
    }
    return result;
}

// fft()とdft()の両方で計算して結果を比較する
template<typename T>
void calc(const std::vector<std::complex<T>>& a)
{
    namespace ch = std::chrono;
    const auto t1 = ch::steady_clock::now();
    const auto b1 = dft(a, make_twiddle_factors<T>(a.size()));
    const auto t2 = ch::steady_clock::now();
    const auto b2 = fft(a, make_twiddle_factor_map<T>(a.size()));
    const auto t3 = ch::steady_clock::now();
    const auto d1 = ch::duration_cast<ch::duration<double>>(t2 - t1).count();
    const auto d2 = ch::duration_cast<ch::duration<double>>(t3 - t2).count();
    T max_delta = 0;
    for (std::size_t i = 0; i < a.size(); ++i) {
        max_delta = std::max(max_delta, std::abs(b1[i] - b2[i]));
    }
    std::cout << std::fixed << std::setprecision(3)
        << "size\t" << a.size()
        << "\tratio\t" << std::right << std::setw(8) << d1 / d2
        << "\tdft\t"<< std::right << std::setw(8) << d1
        << "\tfft\t" << std::right << std::setw(8) << d2
        << "\tdelta\t" << std::scientific << max_delta
        << std::endl;
}

int main()
{
    std::minstd_rand rand;
    std::uniform_real_distribution<double> urd{-10, 10};

    std::vector<std::complex<double>> input = { {urd(rand), urd(rand) }, { urd(rand), urd(rand) } };

    // 1回目の計算だけ時間がかかってしまうようなのでここでダミーの計算をしている
    dft(input, make_twiddle_factors<double>(2));

    std::cout << "--- 2000まで ---" << std::endl;
    for (int i = 2; i < 2000; ++i) {
        input.resize(i, { urd(rand), urd(rand) });
        calc(input);
    }
 
    std::cout << "--- 2のべき乗 ---" << std::endl;
    for (int i = 2; i < 100000; i *= 2) {
        input.resize(i, { urd(rand), urd(rand) });
        calc(input);
    }

    // 3のべき乗
    std::cout << "--- 3のべき乗 ---" << std::endl;
    for (int i = 3; i < 100000; i *= 3) {
        input.resize(i, { urd(rand), urd(rand) });
        calc(input);
    }

    // 素数の積
    std::cout << "--- 素数の積 ---" << std::endl;
    std::size_t s = 1;
    for (auto i : { 2, 3, 5, 7, 11, 13 }) {
        s *= i;
        input.resize(s, { urd(rand), urd(rand) });
        calc(input);
    }

    // 素数
    std::cout << "--- 素数 ---" << std::endl;
    for (auto i : { 2, 3, 5, 7, 11, 13, 1009, 10009, 100003 }) {
        input.resize(i, { urd(rand), urd(rand) });
        calc(input);
    }
}