Black Box Linear Algebra
(matrix/black_box_linear_algebra.hpp)

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• Last update: 2024-08-06 16:15:06+09:00
• Include: #include "matrix/black_box_linear_algebra.hpp"

説明

Black Box Linear Algebraは、行列を陽に与えずに、行列 $A$ とベクトル $v$ の積の演算を行う関数（black box） $Ax$ を与えることで各種の計算を行うものである。以下、 $A$ と ベクトル $v$ の積の計算量を $T(N)$ とする。

matrix_minimum_poly(int n, F Ax)

$N$ 次正方行列 $A$ の最小多項式を求める。 $O(N^2)$

pow(int n, F Ax, std::vector b, long long k)

$A^k b$ を求める。多項式乗算の計算量を $M(N)$ とすると、 $O(N^2 + M(N)\log{k} + N T(N))$

det(int n, F Ax)

$N$ 次正方行列 $A$ の行列式を求める。 $O(N^2 + N T(N))$

Depends on

• Berlekamp Massey (fps/berlekamp_massey.hpp)
• Formal Power Series (fps/fps.hpp)
• fps/poly_mod_pow.hpp
• modint/base.hpp
• Random Number Generator (utility/random_number_generator.hpp)

Verified with

• test/matrix/Determinant_of_Sparse_Matrix.test.cpp
• test/yuki/yuki_1112.test.cpp

Code

#pragma once

#include <cassert>
#include <vector>

#include "../fps/berlekamp_massey.hpp"
#include "../fps/poly_mod_pow.hpp"
#include "../modint/base.hpp"
#include "../utility/random_number_generator.hpp"

namespace ebi {

template <Modint mint, class F>
std::vector<mint> matrix_minimum_poly(int n, F Ax) {
    static random_number_generator rng;
    std::vector<mint> s(2 * n + 10, 0), u(n), b(n);
    for (int i = 0; i < n; i++) {
        u[i] = rng.get(0, mint::mod());
        b[i] = rng.get(0, mint::mod());
    }
    for (int i = 0; i < 2 * n + 10; i++) {
        for (int j = 0; j < n; j++) {
            s[i] += u[j] * b[j];
        }
        b = Ax(b);
    }
    auto c = berlekamp_massey(s);
    std::reverse(c.begin(), c.end());
    return c;
}

template <Modint mint, class F>
std::vector<mint> pow(int n, F Ax, const std::vector<mint> &b, long long k) {
    assert(n == (int)b.size());
    using FPS = FormalPowerSeries<mint>;
    auto g = matrix_minimum_poly<mint>(n, Ax);
    auto c = poly_mod_pow<mint>({0, 1}, k, g);
    FPS res(n, 0), Ab = b;
    for (int i = 0; i < (int)c.size(); i++) {
        res += Ab * c[i];
        Ab = FPS(Ax(Ab));
    }
    return res;
}

template <Modint mint, class F> mint det(int n, F Ax) {
    static random_number_generator rng;
    std::vector<mint> d(n);
    mint r = 1;
    for (int i = 0; i < n; i++) {
        d[i] = rng.get(1, mint::mod());
        r *= d[i];
    }
    auto ADx = [&](std::vector<mint> v) -> std::vector<mint> {
        assert(n == (int)v.size());
        for (int i = 0; i < n; i++) {
            v[i] *= d[i];
        }
        return Ax(v);
    };
    auto f = matrix_minimum_poly<mint>(n, ADx);
    mint res = ((int)f.size() == n + 1 ? f[0] : 0);
    if (n % 2 == 1) res = -res;
    return res / r;
}

}  // namespace ebi

#line 2 "matrix/black_box_linear_algebra.hpp"

#include <cassert>
#include <vector>

#line 2 "fps/berlekamp_massey.hpp"

#include <algorithm>
#line 5 "fps/berlekamp_massey.hpp"

#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 7 "fps/berlekamp_massey.hpp"

namespace ebi {

template <Modint mint>
std::vector<mint> berlekamp_massey(const std::vector<mint> &s) {
    std::vector<mint> C = {1}, B = {1};
    int L = 0, m = 1;
    mint b = 1;
    for (int n = 0; n < (int)s.size(); n++) {
        mint d = s[n];
        for (int i = 1; i <= L; i++) {
            d += s[n - i] * C[i];
        }
        if (d == 0) {
            m++;
        } else if (2 * L <= n) {
            auto T = C;
            mint f = d / b;
            C.resize((int)B.size() + m);
            for (int i = 0; i < (int)B.size(); i++) {
                C[i + m] -= f * B[i];
            }
            L = n + 1 - L;
            B = T;
            b = d;
            m = 1;
        } else {
            mint f = d / b;
            for (int i = 0; i < (int)B.size(); i++) {
                C[i + m] -= f * B[i];
            }
            m++;
        }
    }
    return C;
}

}  // namespace ebi
#line 2 "fps/poly_mod_pow.hpp"

#line 2 "fps/fps.hpp"

#line 5 "fps/fps.hpp"
#include <optional>
#line 7 "fps/fps.hpp"

#line 9 "fps/fps.hpp"

namespace ebi {

template <Modint mint> struct FormalPowerSeries : std::vector<mint> {
  private:
    using std::vector<mint>::vector;
    using std::vector<mint>::vector::operator=;
    using FPS = FormalPowerSeries;

  public:
    FormalPowerSeries(const std::vector<mint> &a) {
        *this = a;
    }

    FPS operator+(const FPS &rhs) const noexcept {
        return FPS(*this) += rhs;
    }
    FPS operator-(const FPS &rhs) const noexcept {
        return FPS(*this) -= rhs;
    }
    FPS operator*(const FPS &rhs) const noexcept {
        return FPS(*this) *= rhs;
    }
    FPS operator/(const FPS &rhs) const noexcept {
        return FPS(*this) /= rhs;
    }
    FPS operator%(const FPS &rhs) const noexcept {
        return FPS(*this) %= rhs;
    }

    FPS operator+(const mint &rhs) const noexcept {
        return FPS(*this) += rhs;
    }
    FPS operator-(const mint &rhs) const noexcept {
        return FPS(*this) -= rhs;
    }
    FPS operator*(const mint &rhs) const noexcept {
        return FPS(*this) *= rhs;
    }
    FPS operator/(const mint &rhs) const noexcept {
        return FPS(*this) /= rhs;
    }

    FPS &operator+=(const FPS &rhs) noexcept {
        if (this->size() < rhs.size()) this->resize(rhs.size());
        for (int i = 0; i < (int)rhs.size(); ++i) {
            (*this)[i] += rhs[i];
        }
        return *this;
    }

    FPS &operator-=(const FPS &rhs) noexcept {
        if (this->size() < rhs.size()) this->resize(rhs.size());
        for (int i = 0; i < (int)rhs.size(); ++i) {
            (*this)[i] -= rhs[i];
        }
        return *this;
    }

    FPS &operator*=(const FPS &);

    FPS &operator/=(const FPS &rhs) noexcept {
        int n = deg() - 1;
        int m = rhs.deg() - 1;
        if (n < m) {
            *this = {};
            return *this;
        }
        *this = (*this).rev() * rhs.rev().inv(n - m + 1);
        (*this).resize(n - m + 1);
        std::reverse((*this).begin(), (*this).end());
        return *this;
    }

    FPS &operator%=(const FPS &rhs) noexcept {
        *this -= *this / rhs * rhs;
        shrink();
        return *this;
    }

    FPS &operator+=(const mint &rhs) noexcept {
        if (this->empty()) this->resize(1);
        (*this)[0] += rhs;
        return *this;
    }

    FPS &operator-=(const mint &rhs) noexcept {
        if (this->empty()) this->resize(1);
        (*this)[0] -= rhs;
        return *this;
    }

    FPS &operator*=(const mint &rhs) noexcept {
        for (int i = 0; i < deg(); ++i) {
            (*this)[i] *= rhs;
        }
        return *this;
    }
    FPS &operator/=(const mint &rhs) noexcept {
        mint inv_rhs = rhs.inv();
        for (int i = 0; i < deg(); ++i) {
            (*this)[i] *= inv_rhs;
        }
        return *this;
    }

    FPS operator>>(int d) const {
        if (deg() <= d) return {};
        FPS f = *this;
        f.erase(f.begin(), f.begin() + d);
        return f;
    }

    FPS operator<<(int d) const {
        FPS f = *this;
        f.insert(f.begin(), d, 0);
        return f;
    }

    FPS operator-() const {
        FPS g(this->size());
        for (int i = 0; i < (int)this->size(); i++) g[i] = -(*this)[i];
        return g;
    }

    FPS pre(int sz) const {
        return FPS(this->begin(), this->begin() + std::min(deg(), sz));
    }

    FPS rev() const {
        auto f = *this;
        std::reverse(f.begin(), f.end());
        return f;
    }

    FPS differential() const {
        int n = deg();
        FPS g(std::max(0, n - 1));
        for (int i = 0; i < n - 1; i++) {
            g[i] = (*this)[i + 1] * (i + 1);
        }
        return g;
    }

    FPS integral() const {
        int n = deg();
        FPS g(n + 1);
        g[0] = 0;
        if (n > 0) g[1] = 1;
        auto mod = mint::mod();
        for (int i = 2; i <= n; i++) g[i] = (-g[mod % i]) * (mod / i);
        for (int i = 0; i < n; i++) g[i + 1] *= (*this)[i];
        return g;
    }

    FPS inv(int d = -1) const {
        int n = 1;
        if (d < 0) d = deg();
        FPS g(n);
        g[0] = (*this)[0].inv();
        while (n < d) {
            n <<= 1;
            g = (g * 2 - g * g * this->pre(n)).pre(n);
        }
        g.resize(d);
        return g;
    }

    FPS log(int d = -1) const {
        assert((*this)[0].val() == 1);
        if (d < 0) d = deg();
        return ((*this).differential() * (*this).inv(d)).pre(d - 1).integral();
    }

    FPS exp(int d = -1) const {
        assert((*this)[0].val() == 0);
        int n = 1;
        if (d < 0) d = deg();
        FPS g(n);
        g[0] = 1;
        while (n < d) {
            n <<= 1;
            g = (g * (this->pre(n) - g.log(n) + 1)).pre(n);
        }
        g.resize(d);
        return g;
    }

    FPS pow(long long k, int d = -1) const {
        assert(k >= 0);
        int n = deg();
        if (d < 0) d = n;
        if (k == 0) {
            FPS f(d);
            if (d > 0) f[0] = 1;
            return f;
        }
        int low = d;
        for (int i = n - 1; i >= 0; i--)
            if ((*this)[i] != 0) low = i;
        if (low >= (d + k - 1) / k) return FPS(d, 0);
        int offset = k * low;
        mint c = (*this)[low];
        FPS g(d - offset);
        for (int i = 0; i < std::min(n - low, d - offset); i++) {
            g[i] = (*this)[i + low];
        }
        g /= c;
        g = g.pow_1(k);
        return (g << offset) * c.pow(k);
    }

    FPS pow_1(mint k, int d = -1) const {
        assert((*this)[0] == 1);
        return ((*this).log(d) * k).exp(d);
    }

    FPS pow_newton(long long k, int d = -1) const {
        assert(k >= 0);
        const int n = deg();
        if (d < 0) d = n;
        if (k == 0) {
            FPS f(d);
            if (d > 0) f[0] = 1;
            return f;
        }
        for (int i = 0; i < n; i++) {
            if ((*this)[i] != 0) {
                mint rev = (*this)[i].inv();
                FPS f = (((*this * rev) >> i).log(d) * k).exp(d);
                f *= (*this)[i].pow(k);
                f = (f << (i * k)).pre(d);
                if (f.deg() < d) f.resize(d);
                return f;
            }
            if (i + 1 >= (d + k - 1) / k) break;
        }
        return FPS(d);
    }

    int deg() const {
        return (*this).size();
    }

    void shrink() {
        while ((!this->empty()) && this->back() == 0) this->pop_back();
    }

    int count_terms() const {
        int c = 0;
        for (int i = 0; i < deg(); i++) {
            if ((*this)[i] != 0) c++;
        }
        return c;
    }

    std::optional<FPS> sqrt(int d = -1) const;

    static FPS exp_x(int n) {
        FPS f(n);
        mint fact = 1;
        for (int i = 1; i < n; i++) fact *= i;
        f[n - 1] = fact.inv();
        for (int i = n - 1; i >= 0; i--) f[i - 1] = f[i] * i;
        return f;
    }

    void fft();
    void ifft();
};

}  // namespace ebi
#line 5 "fps/poly_mod_pow.hpp"

namespace ebi {

template <Modint mint>
FormalPowerSeries<mint> poly_mod_pow(FormalPowerSeries<mint> f, long long k,
                                     const FormalPowerSeries<mint> &g) {
    FormalPowerSeries<mint> res = {1};
    while (k > 0) {
        if (k & 1) {
            res *= f;
            res %= g;
            res.shrink();
        }
        f *= f;
        f %= g;
        f.shrink();
        k >>= 1;
    }
    return res;
}

}  // namespace ebi
#line 2 "utility/random_number_generator.hpp"

#line 4 "utility/random_number_generator.hpp"
#include <cstdint>

#include <numeric>

#include <random>

#line 8 "utility/random_number_generator.hpp"

namespace ebi {

struct random_number_generator {
    random_number_generator(int seed = -1) {
        if (seed < 0) seed = rnd();
        mt.seed(seed);
    }

    void set_seed(int seed) {
        mt.seed(seed);
    }

    template <class T> T get(T a, T b) {
        std::uniform_int_distribution<T> dist(a, b - 1);
        return dist(mt);
    }

    std::vector<int> get_permutation(int n) {
        std::vector<int> p(n);
        std::iota(p.begin(), p.end(), 0);
        std::shuffle(p.begin(), p.end(), mt);
        return p;
    }

  private:
    std::mt19937_64 mt;
    std::random_device rnd;
};

}  // namespace ebi
#line 10 "matrix/black_box_linear_algebra.hpp"

namespace ebi {

template <Modint mint, class F>
std::vector<mint> matrix_minimum_poly(int n, F Ax) {
    static random_number_generator rng;
    std::vector<mint> s(2 * n + 10, 0), u(n), b(n);
    for (int i = 0; i < n; i++) {
        u[i] = rng.get(0, mint::mod());
        b[i] = rng.get(0, mint::mod());
    }
    for (int i = 0; i < 2 * n + 10; i++) {
        for (int j = 0; j < n; j++) {
            s[i] += u[j] * b[j];
        }
        b = Ax(b);
    }
    auto c = berlekamp_massey(s);
    std::reverse(c.begin(), c.end());
    return c;
}

template <Modint mint, class F>
std::vector<mint> pow(int n, F Ax, const std::vector<mint> &b, long long k) {
    assert(n == (int)b.size());
    using FPS = FormalPowerSeries<mint>;
    auto g = matrix_minimum_poly<mint>(n, Ax);
    auto c = poly_mod_pow<mint>({0, 1}, k, g);
    FPS res(n, 0), Ab = b;
    for (int i = 0; i < (int)c.size(); i++) {
        res += Ab * c[i];
        Ab = FPS(Ax(Ab));
    }
    return res;
}

template <Modint mint, class F> mint det(int n, F Ax) {
    static random_number_generator rng;
    std::vector<mint> d(n);
    mint r = 1;
    for (int i = 0; i < n; i++) {
        d[i] = rng.get(1, mint::mod());
        r *= d[i];
    }
    auto ADx = [&](std::vector<mint> v) -> std::vector<mint> {
        assert(n == (int)v.size());
        for (int i = 0; i < n; i++) {
            v[i] *= d[i];
        }
        return Ax(v);
    };
    auto f = matrix_minimum_poly<mint>(n, ADx);
    mint res = ((int)f.size() == n + 1 ? f[0] : 0);
    if (n % 2 == 1) res = -res;
    return res / r;
}

}  // namespace ebi

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