Library

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:heavy_check_mark: Convolution
(convolution/convolution.hpp)

説明

$a$ と $b$ を畳み込み、その配列を返す。

convolution_naive(std::vector a, std::vector b)

愚直に畳み込む。 $O(N^2)$

convolution(std::vector a, std::vector b)

NTT friendlyな素数における $O(N\log N)$ 時間での畳み込み。 $O(N\log N)$

Depends on

Required by

Verified with

Code

#pragma once

#include <algorithm>
#include <bit>
#include <vector>

#include "../convolution/ntt.hpp"
#include "../modint/base.hpp"

namespace ebi {

template <Modint mint>
std::vector<mint> convolution_naive(const std::vector<mint>& f,
                                    const std::vector<mint>& g) {
    if (f.empty() || g.empty()) return {};
    int n = int(f.size()), m = int(g.size());
    std::vector<mint> c(n + m - 1);
    if (n < m) {
        for (int j = 0; j < m; j++) {
            for (int i = 0; i < n; i++) {
                c[i + j] += f[i] * g[j];
            }
        }
    } else {
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                c[i + j] += f[i] * g[j];
            }
        }
    }
    return c;
}

template <Modint mint>
std::vector<mint> convolution(const std::vector<mint>& f,
                              const std::vector<mint>& g) {
    if (f.empty() || g.empty()) return {};
    if (std::min(f.size(), g.size()) < 60) return convolution_naive(f, g);
    int n = (int)std::bit_ceil(f.size() + g.size() - 1);
    std::vector<mint> a(n), b(n);
    std::copy(f.begin(), f.end(), a.begin());
    std::copy(g.begin(), g.end(), b.begin());
    internal::fft4(a);
    internal::fft4(b);
    for (int i = 0; i < n; i++) {
        a[i] *= b[i];
    }
    internal::ifft4(a);
    a.resize(f.size() + g.size() - 1);
    mint inv_n = mint(n).inv();
    for (auto& x : a) x *= inv_n;
    return a;
}

}  // namespace ebi
#line 2 "convolution/convolution.hpp"

#include <algorithm>
#include <bit>
#include <vector>

#line 2 "convolution/ntt.hpp"

#line 4 "convolution/ntt.hpp"
#include <array>
#line 6 "convolution/ntt.hpp"
#include <cassert>
#line 8 "convolution/ntt.hpp"

#line 2 "math/internal_math.hpp"

#line 4 "math/internal_math.hpp"

namespace ebi {

namespace internal {

constexpr int primitive_root_constexpr(int m) {
    if (m == 2) return 1;
    if (m == 167772161) return 3;
    if (m == 469762049) return 3;
    if (m == 754974721) return 11;
    if (m == 998244353) return 3;
    if (m == 880803841) return 26;
    if (m == 924844033) return 5;
    return -1;
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

}  // namespace internal

}  // namespace ebi
#line 2 "modint/base.hpp"

#include <concepts>
#include <iostream>
#include <utility>

namespace ebi {

template <class T>
concept Modint = requires(T a, T b) {
    a + b;
    a - b;
    a * b;
    a / b;
    a.inv();
    a.val();
    a.pow(std::declval<long long>());
    T::mod();
};

template <Modint mint> std::istream &operator>>(std::istream &os, mint &a) {
    long long x;
    os >> x;
    a = x;
    return os;
}

template <Modint mint>
std::ostream &operator<<(std::ostream &os, const mint &a) {
    return os << a.val();
}

}  // namespace ebi
#line 2 "template/int_alias.hpp"

#include <cstdint>

namespace ebi {

using ld = long double;
using std::size_t;
using i8 = std::int8_t;
using u8 = std::uint8_t;
using i16 = std::int16_t;
using u16 = std::uint16_t;
using i32 = std::int32_t;
using u32 = std::uint32_t;
using i64 = std::int64_t;
using u64 = std::uint64_t;
using i128 = __int128_t;
using u128 = __uint128_t;

}  // namespace ebi
#line 12 "convolution/ntt.hpp"

namespace ebi {

namespace internal {

template <Modint mint, int g = internal::primitive_root<mint::mod()>>
struct ntt_info {
    static constexpr int rank2 =
        std::countr_zero((unsigned int)(mint::mod() - 1));

    std::array<mint, rank2 + 1> root, inv_root;

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

template <Modint mint> void fft2(std::vector<mint>& a) {
    static const ntt_info<mint> info;
    int n = int(a.size());
    int bit_size = std::countr_zero(a.size());
    assert(n == 1 << bit_size);
    for (int bit = bit_size - 1; bit >= 0; bit--) {
        int m = 1 << bit;
        for (int i = 0; i < n; i += 2 * m) {
            mint w = 1;
            for (int j = 0; j < m; j++) {
                mint p1 = a[i + j];
                mint p2 = a[i + j + m];
                a[i + j] = p1 + p2;
                a[i + j + m] = (p1 - p2) * w;
                w *= info.root[bit + 1];
            }
        }
    }
}

template <Modint mint> void ifft2(std::vector<mint>& a) {
    static const ntt_info<mint> info;
    int n = int(a.size());
    int bit_size = std::countr_zero(a.size());
    assert(n == 1 << bit_size);

    for (int bit = 0; bit < bit_size; bit++) {
        for (int i = 0; i < n / (1 << (bit + 1)); i++) {
            mint w = 1;
            for (int j = 0; j < (1 << bit); j++) {
                int idx = i * (1 << (bit + 1)) + j;
                int jdx = idx + (1 << bit);
                mint p1 = a[idx];
                mint p2 = w * a[jdx];
                a[idx] = p1 + p2;
                a[jdx] = p1 - p2;
                w *= info.inv_root[bit + 1];
            }
        }
    }
}

template <Modint mint> void fft4(std::vector<mint>& a) {
    static const ntt_info<mint> info;
    const u32 mod = mint::mod();
    const u64 iw = info.root[2].val();
    int n = int(a.size());
    int bit_size = std::countr_zero(a.size());
    assert(n == 1 << bit_size);
    int len = bit_size;
    while (len > 0) {
        if (len == 1) {
            for (int i = 0; i < n; i += 2) {
                mint p0 = a[i];
                mint p1 = a[i + 1];
                a[i] = p0 + p1;
                a[i + 1] = p0 - p1;
            }
            len--;
        } else {
            int m = 1 << (len - 2);
            u64 w1 = 1, w2 = 1, w3 = 1, iw1 = iw, iw3 = iw;
            for (int i = 0; i < m; i++) {
                for (int j = 0; j < n; j += 4 * m) {
                    int i0 = i + j, i1 = i0 + m, i2 = i1 + m, i3 = i2 + m;
                    u32 a0 = a[i0].val();
                    u32 a1 = a[i1].val();
                    u32 a2 = a[i2].val();
                    u32 a3 = a[i3].val();
                    u32 a0_plus_a2 = a0 + a2;
                    u32 a1_plus_a3 = a1 + a3;
                    u32 a0_minus_a2 = a0 + mod - a2;
                    u32 a1_minus_a3 = a1 + mod - a3;
                    a[i0] = a0_plus_a2 + a1_plus_a3;
                    a[i1] = a0_minus_a2 * w1 + a1_minus_a3 * iw1;
                    a[i2] = (a0_plus_a2 + 2 * mod - a1_plus_a3) * w2;
                    a[i3] = a0_minus_a2 * w3 + (2 * mod - a1_minus_a3) * iw3;
                }
                w1 = w1 * info.root[len].val() % mod;
                w2 = w1 * w1 % mod;
                w3 = w2 * w1 % mod;
                iw1 = iw * w1 % mod;
                iw3 = iw * w3 % mod;
            }
            len -= 2;
        }
    }
}

template <Modint mint> void ifft4(std::vector<mint>& a) {
    static const ntt_info<mint> info;
    const u32 mod = mint::mod();
    const u64 mod2 = u64(mod) * mod;
    const u64 iw = info.inv_root[2].val();
    int n = int(a.size());
    int bit_size = std::countr_zero(a.size());
    assert(n == 1 << bit_size);
    int len = (bit_size & 1 ? 1 : 2);
    while (len <= bit_size) {
        if (len == 1) {
            for (int i = 0; i < n; i += 2) {
                mint a0 = a[i];
                mint a1 = a[i + 1];
                a[i] = a0 + a1;
                a[i + 1] = a0 - a1;
            }
        } else {
            int m = 1 << (len - 2);
            u64 w1 = 1, w2 = 1, w3 = 1, iw1 = iw, iw3 = iw;
            for (int i = 0; i < m; i++) {
                for (int j = 0; j < n; j += 4 * m) {
                    int i0 = i + j, i1 = i0 + m, i2 = i1 + m, i3 = i2 + m;
                    u64 a0 = a[i0].val();
                    u64 a1 = w1 * a[i1].val();
                    u64 a2 = w2 * a[i2].val();
                    u64 a3 = w3 * a[i3].val();
                    u64 b1 = iw1 * a[i1].val();
                    u64 b3 = iw3 * a[i3].val();
                    u64 a0_plus_a2 = a0 + a2;
                    u64 a1_plus_a3 = a1 + a3;
                    u64 a0_minus_a2 = a0 + mod2 - a2;
                    u64 b1_minus_b3 = b1 + mod2 - b3;
                    a[i0] = a0_plus_a2 + a1_plus_a3;
                    a[i1] = a0_minus_a2 + b1_minus_b3;
                    a[i2] = a0_plus_a2 + mod2 * 2 - a1_plus_a3;
                    a[i3] = a0_minus_a2 + mod2 * 2 - b1_minus_b3;
                }
                w1 = w1 * info.inv_root[len].val() % mod;
                w2 = w1 * w1 % mod;
                w3 = w2 * w1 % mod;
                iw1 = iw * w1 % mod;
                iw3 = iw * w3 % mod;
            }
        }
        len += 2;
    }
}

}  // namespace internal

}  // namespace ebi
#line 9 "convolution/convolution.hpp"

namespace ebi {

template <Modint mint>
std::vector<mint> convolution_naive(const std::vector<mint>& f,
                                    const std::vector<mint>& g) {
    if (f.empty() || g.empty()) return {};
    int n = int(f.size()), m = int(g.size());
    std::vector<mint> c(n + m - 1);
    if (n < m) {
        for (int j = 0; j < m; j++) {
            for (int i = 0; i < n; i++) {
                c[i + j] += f[i] * g[j];
            }
        }
    } else {
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                c[i + j] += f[i] * g[j];
            }
        }
    }
    return c;
}

template <Modint mint>
std::vector<mint> convolution(const std::vector<mint>& f,
                              const std::vector<mint>& g) {
    if (f.empty() || g.empty()) return {};
    if (std::min(f.size(), g.size()) < 60) return convolution_naive(f, g);
    int n = (int)std::bit_ceil(f.size() + g.size() - 1);
    std::vector<mint> a(n), b(n);
    std::copy(f.begin(), f.end(), a.begin());
    std::copy(g.begin(), g.end(), b.begin());
    internal::fft4(a);
    internal::fft4(b);
    for (int i = 0; i < n; i++) {
        a[i] *= b[i];
    }
    internal::ifft4(a);
    a.resize(f.size() + g.size() - 1);
    mint inv_n = mint(n).inv();
    for (auto& x : a) x *= inv_n;
    return a;
}

}  // namespace ebi
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