#753797 (C++23) No.1907 DETERMINATION

提出ソース

結果

問題	No.1907 DETERMINATION
ユーザー	hitonanode
提出日時	2022-04-14 23:24:57
言語	C++23 (gcc 12.3.0 + boost 1.83.0)
結果	AC
実行時間	700 ms / 4,000 ms
コード長	13,770 bytes
コンパイル時間	2,384 ms
コンパイル使用メモリ	155,372 KB
実行使用メモリ	17,124 KB
最終ジャッジ日時	2024-06-06 19:07:10
合計ジャッジ時間	30,156 ms
ジャッジサーバーID （参考情報）	judge1 / judge2

このコードへのチャレンジ
（要ログイン）

テストケース

テストケース表示

入力	結果	実行時間実行使用メモリ
testcase_00	AC	15 ms 11,008 KB
testcase_01	AC	16 ms 11,008 KB
testcase_02	AC	15 ms 11,008 KB
testcase_03	AC	15 ms 10,880 KB
testcase_04	AC	16 ms 11,136 KB
testcase_05	AC	15 ms 11,136 KB
testcase_06	AC	16 ms 11,136 KB
testcase_07	AC	285 ms 14,596 KB
testcase_08	AC	121 ms 12,712 KB
testcase_09	AC	198 ms 13,988 KB
testcase_10	AC	595 ms 16,932 KB
testcase_11	AC	140 ms 16,668 KB
testcase_12	AC	632 ms 16,840 KB
testcase_13	AC	622 ms 16,764 KB
testcase_14	AC	570 ms 17,092 KB
testcase_15	AC	138 ms 13,116 KB
testcase_16	AC	51 ms 11,776 KB
testcase_17	AC	549 ms 16,636 KB
testcase_18	AC	406 ms 15,036 KB
testcase_19	AC	26 ms 11,264 KB
testcase_20	AC	589 ms 16,800 KB
testcase_21	AC	70 ms 12,288 KB
testcase_22	AC	226 ms 16,804 KB
testcase_23	AC	599 ms 16,792 KB
testcase_24	AC	200 ms 13,856 KB
testcase_25	AC	16 ms 11,136 KB
testcase_26	AC	667 ms 17,036 KB
testcase_27	AC	700 ms 16,996 KB
testcase_28	AC	656 ms 16,996 KB
testcase_29	AC	677 ms 16,996 KB
testcase_30	AC	16 ms 11,136 KB
testcase_31	AC	680 ms 16,868 KB
testcase_32	AC	684 ms 16,868 KB
testcase_33	AC	658 ms 16,992 KB
testcase_34	AC	652 ms 16,996 KB
testcase_35	AC	17 ms 11,264 KB
testcase_36	AC	16 ms 11,136 KB
testcase_37	AC	15 ms 11,008 KB
testcase_38	AC	654 ms 17,124 KB
testcase_39	AC	659 ms 16,996 KB
testcase_40	AC	663 ms 17,124 KB
testcase_41	AC	681 ms 16,996 KB
testcase_42	AC	685 ms 16,996 KB
testcase_43	AC	666 ms 16,996 KB
testcase_44	AC	663 ms 16,996 KB
testcase_45	AC	666 ms 16,996 KB
testcase_46	AC	648 ms 16,844 KB
testcase_47	AC	660 ms 17,100 KB
testcase_48	AC	665 ms 16,864 KB
testcase_49	AC	655 ms 16,868 KB
testcase_50	AC	663 ms 16,996 KB
testcase_51	AC	671 ms 16,996 KB
testcase_52	AC	16 ms 11,008 KB
testcase_53	AC	244 ms 16,740 KB
testcase_54	AC	252 ms 16,872 KB
testcase_55	AC	17 ms 11,008 KB
testcase_56	AC	245 ms 16,864 KB
testcase_57	AC	243 ms 16,868 KB
testcase_58	AC	482 ms 16,968 KB
testcase_59	AC	489 ms 16,992 KB
testcase_60	AC	493 ms 17,000 KB
testcase_61	AC	536 ms 16,892 KB
testcase_62	AC	488 ms 17,124 KB
testcase_63	AC	650 ms 17,124 KB
testcase_64	AC	17 ms 11,008 KB
testcase_65	AC	17 ms 11,008 KB
testcase_66	AC	16 ms 10,880 KB

権限があれば一括ダウンロードができます

ソースコード

raw source code

#include <cassert>
#include <iostream>
#include <utility>
#include <vector>
using namespace std;

#include <chrono>
#include <random>
struct rand_int_ {
    using lint = long long;
    mt19937 mt;
    rand_int_() : mt(chrono::steady_clock::now().time_since_epoch().count()) {}
    lint operator()(lint x) { return this->operator()(0, x); } // [0, x)
    lint operator()(lint l, lint r) {
        uniform_int_distribution<lint> d(l, r - 1);
        return d(mt);
    }
} rnd;



// Upper Hessenberg reduction of square matrices
// Complexity: O(n^3)
// Reference:
// http://www.phys.uri.edu/nigh/NumRec/bookfpdf/f11-5.pdf
template <class Tp> void hessenberg_reduction(std::vector<std::vector<Tp>> &M) {
    assert(M.size() == M[0].size());
    const int N = M.size();
    for (int r = 0; r < N - 2; r++) {
        int piv = -1;
        for (int h = r + 1; h < N; ++h) {
            if (M[h][r] != 0) {
                piv = h;
                break;
            }
        }
        if (piv < 0) continue;
        for (int i = 0; i < N; i++) std::swap(M[r + 1][i], M[piv][i]);
        for (int i = 0; i < N; i++) std::swap(M[i][r + 1], M[i][piv]);

        const auto rinv = Tp(1) / M[r + 1][r];
        for (int i = r + 2; i < N; i++) {
            const auto n = M[i][r] * rinv;
            for (int j = 0; j < N; j++) M[i][j] -= M[r + 1][j] * n;
            for (int j = 0; j < N; j++) M[j][r + 1] += M[j][i] * n;
        }
    }
}

// Characteristic polynomial of matrix M (|xI - M|)
// Complexity: O(n^3)
// R. Rehman, I. C. Ipsen, "La Budde's Method for Computing Characteristic Polynomials," 2011.
template <class Tp> std::vector<Tp> characteristic_poly(std::vector<std::vector<Tp>> M) {
    hessenberg_reduction(M);
    const int N = M.size();
    // p[i + 1] = (Characteristic polynomial of i-th leading principal minor)
    std::vector<std::vector<Tp>> p(N + 1);
    p[0] = {1};
    for (int i = 0; i < N; i++) {
        p[i + 1].assign(i + 2, 0);
        for (int j = 0; j < i + 1; j++) p[i + 1][j + 1] += p[i][j];
        for (int j = 0; j < i + 1; j++) p[i + 1][j] -= p[i][j] * M[i][i];
        Tp betas = 1;
        for (int j = i - 1; j >= 0; j--) {
            betas *= M[j + 1][j];
            Tp hb = -M[j][i] * betas;
            for (int k = 0; k < j + 1; k++) p[i + 1][k] += hb * p[j][k];
        }
    }
    return p[N];
}


#include <algorithm>
#include <cassert>
#include <cmath>
#include <iterator>
#include <type_traits>
#include <utility>
#include <vector>

namespace matrix_ {
struct has_id_method_impl {
    template <class T_> static auto check(T_ *) -> decltype(T_::id(), std::true_type());
    template <class T_> static auto check(...) -> std::false_type;
};
template <class T_> struct has_id : decltype(has_id_method_impl::check<T_>(nullptr)) {};
} // namespace matrix_

template <typename T> struct matrix {
    int H, W;
    std::vector<T> elem;
    typename std::vector<T>::iterator operator[](int i) { return elem.begin() + i * W; }
    inline T &at(int i, int j) { return elem[i * W + j]; }
    inline T get(int i, int j) const { return elem[i * W + j]; }
    int height() const { return H; }
    int width() const { return W; }
    std::vector<std::vector<T>> vecvec() const {
        std::vector<std::vector<T>> ret(H);
        for (int i = 0; i < H; i++) {
            std::copy(elem.begin() + i * W, elem.begin() + (i + 1) * W, std::back_inserter(ret[i]));
        }
        return ret;
    }
    operator std::vector<std::vector<T>>() const { return vecvec(); }
    matrix() = default;
    matrix(int H, int W) : H(H), W(W), elem(H * W) {}
    matrix(const std::vector<std::vector<T>> &d) : H(d.size()), W(d.size() ? d[0].size() : 0) {
        for (auto &raw : d) std::copy(raw.begin(), raw.end(), std::back_inserter(elem));
    }

    template <typename T2, typename std::enable_if<matrix_::has_id<T2>::value>::type * = nullptr>
    static T2 _T_id() {
        return T2::id();
    }
    template <typename T2, typename std::enable_if<!matrix_::has_id<T2>::value>::type * = nullptr>
    static T2 _T_id() {
        return T2(1);
    }

    static matrix Identity(int N) {
        matrix ret(N, N);
        for (int i = 0; i < N; i++) ret.at(i, i) = _T_id<T>();
        return ret;
    }

    matrix operator-() const {
        matrix ret(H, W);
        for (int i = 0; i < H * W; i++) ret.elem[i] = -elem[i];
        return ret;
    }
    matrix operator*(const T &v) const {
        matrix ret = *this;
        for (auto &x : ret.elem) x *= v;
        return ret;
    }
    matrix operator/(const T &v) const {
        matrix ret = *this;
        const T vinv = _T_id<T>() / v;
        for (auto &x : ret.elem) x *= vinv;
        return ret;
    }
    matrix operator+(const matrix &r) const {
        matrix ret = *this;
        for (int i = 0; i < H * W; i++) ret.elem[i] += r.elem[i];
        return ret;
    }
    matrix operator-(const matrix &r) const {
        matrix ret = *this;
        for (int i = 0; i < H * W; i++) ret.elem[i] -= r.elem[i];
        return ret;
    }
    matrix operator*(const matrix &r) const {
        matrix ret(H, r.W);
        for (int i = 0; i < H; i++) {
            for (int k = 0; k < W; k++) {
                for (int j = 0; j < r.W; j++) ret.at(i, j) += this->get(i, k) * r.get(k, j);
            }
        }
        return ret;
    }
    matrix &operator*=(const T &v) { return *this = *this * v; }
    matrix &operator/=(const T &v) { return *this = *this / v; }
    matrix &operator+=(const matrix &r) { return *this = *this + r; }
    matrix &operator-=(const matrix &r) { return *this = *this - r; }
    matrix &operator*=(const matrix &r) { return *this = *this * r; }
    bool operator==(const matrix &r) const { return H == r.H and W == r.W and elem == r.elem; }
    bool operator!=(const matrix &r) const { return H != r.H or W != r.W or elem != r.elem; }
    bool operator<(const matrix &r) const { return elem < r.elem; }
    matrix pow(int64_t n) const {
        matrix ret = Identity(H);
        bool ret_is_id = true;
        if (n == 0) return ret;
        for (int i = 63 - __builtin_clzll(n); i >= 0; i--) {
            if (!ret_is_id) ret *= ret;
            if ((n >> i) & 1) ret *= (*this), ret_is_id = false;
        }
        return ret;
    }
    std::vector<T> pow_vec(int64_t n, std::vector<T> vec) const {
        matrix x = *this;
        while (n) {
            if (n & 1) vec = x * vec;
            x *= x;
            n >>= 1;
        }
        return vec;
    };
    matrix transpose() const {
        matrix ret(W, H);
        for (int i = 0; i < H; i++) {
            for (int j = 0; j < W; j++) ret.at(j, i) = this->get(i, j);
        }
        return ret;
    }
    // Gauss-Jordan elimination
    // - Require inverse for every non-zero element
    // - Complexity: O(H^2 W)
    template <typename T2, typename std::enable_if<std::is_floating_point<T2>::value>::type * = nullptr>
    static int choose_pivot(const matrix<T2> &mtr, int h, int c) noexcept {
        int piv = -1;
        for (int j = h; j < mtr.H; j++) {
            if (mtr.get(j, c) and (piv < 0 or std::abs(mtr.get(j, c)) > std::abs(mtr.get(piv, c))))
                piv = j;
        }
        return piv;
    }
    template <typename T2, typename std::enable_if<!std::is_floating_point<T2>::value>::type * = nullptr>
    static int choose_pivot(const matrix<T2> &mtr, int h, int c) noexcept {
        for (int j = h; j < mtr.H; j++) {
            if (mtr.get(j, c) != T2()) return j;
        }
        return -1;
    }
    matrix gauss_jordan() const {
        int c = 0;
        matrix mtr(*this);
        std::vector<int> ws;
        ws.reserve(W);
        for (int h = 0; h < H; h++) {
            if (c == W) break;
            int piv = choose_pivot(mtr, h, c);
            if (piv == -1) {
                c++;
                h--;
                continue;
            }
            if (h != piv) {
                for (int w = 0; w < W; w++) {
                    std::swap(mtr[piv][w], mtr[h][w]);
                    mtr.at(piv, w) *= -_T_id<T>(); // To preserve sign of determinant
                }
            }
            ws.clear();
            for (int w = c; w < W; w++) {
                if (mtr.at(h, w) != T()) ws.emplace_back(w);
            }
            const T hcinv = _T_id<T>() / mtr.at(h, c);
            for (int hh = 0; hh < H; hh++)
                if (hh != h) {
                    const T coeff = mtr.at(hh, c) * hcinv;
                    for (auto w : ws) mtr.at(hh, w) -= mtr.at(h, w) * coeff;
                    mtr.at(hh, c) = T();
                }
            c++;
        }
        return mtr;
    }
    int rank_of_gauss_jordan() const {
        for (int i = H * W - 1; i >= 0; i--) {
            if (elem[i] != 0) return i / W + 1;
        }
        return 0;
    }
    T determinant_of_upper_triangle() const {
        T ret = _T_id<T>();
        for (int i = 0; i < H; i++) ret *= get(i, i);
        return ret;
    }
    int inverse() {
        assert(H == W);
        std::vector<std::vector<T>> ret = Identity(H), tmp = *this;
        int rank = 0;
        for (int i = 0; i < H; i++) {
            int ti = i;
            while (ti < H and tmp[ti][i] == 0) ti++;
            if (ti == H) {
                continue;
            } else {
                rank++;
            }
            ret[i].swap(ret[ti]), tmp[i].swap(tmp[ti]);
            T inv = _T_id<T>() / tmp[i][i];
            for (int j = 0; j < W; j++) ret[i][j] *= inv;
            for (int j = i + 1; j < W; j++) tmp[i][j] *= inv;
            for (int h = 0; h < H; h++) {
                if (i == h) continue;
                const T c = -tmp[h][i];
                for (int j = 0; j < W; j++) ret[h][j] += ret[i][j] * c;
                for (int j = i + 1; j < W; j++) tmp[h][j] += tmp[i][j] * c;
            }
        }
        *this = ret;
        return rank;
    }
    friend std::vector<T> operator*(const matrix &m, const std::vector<T> &v) {
        assert(m.W == int(v.size()));
        std::vector<T> ret(m.H);
        for (int i = 0; i < m.H; i++) {
            for (int j = 0; j < m.W; j++) ret[i] += m.get(i, j) * v[j];
        }
        return ret;
    }
    friend std::vector<T> operator*(const std::vector<T> &v, const matrix &m) {
        assert(int(v.size()) == m.H);
        std::vector<T> ret(m.W);
        for (int i = 0; i < m.H; i++) {
            for (int j = 0; j < m.W; j++) ret[j] += v[i] * m.get(i, j);
        }
        return ret;
    }
    std::vector<T> prod(const std::vector<T> &v) const { return (*this) * v; }
    std::vector<T> prod_left(const std::vector<T> &v) const { return v * (*this); }
    template <class OStream> friend OStream &operator<<(OStream &os, const matrix &x) {
        os << "[(" << x.H << " * " << x.W << " matrix)";
        os << "\n[column sums: ";
        for (int j = 0; j < x.W; j++) {
            T s = 0;
            for (int i = 0; i < x.H; i++) s += x.get(i, j);
            os << s << ",";
        }
        os << "]";
        for (int i = 0; i < x.H; i++) {
            os << "\n[";
            for (int j = 0; j < x.W; j++) os << x.get(i, j) << ",";
            os << "]";
        }
        os << "]\n";
        return os;
    }
    template <class IStream> friend IStream &operator>>(IStream &is, matrix &x) {
        for (auto &v : x.elem) is >> v;
        return is;
    }
};


#include <atcoder/modint>
using mint = atcoder::modint998244353;

#include <cassert>
#include <iostream>
#include <vector>

// Utility functions for atcoder::static_modint<md>
template <int md> std::istream &operator>>(std::istream &is, atcoder::static_modint<md> &x) {
    long long t;
    return is >> t, x = t, is;
}
template <int md> std::ostream &operator<<(std::ostream &os, const atcoder::static_modint<md> &x) {
    return os << x.val();
}
// atcoder::static_modint<P>, P: prime number
template <typename modint> struct acl_fac {
    std::vector<modint> facs, facinvs;
    acl_fac(int N) {
        assert(-1 <= N and N < modint::mod());
        facs.resize(N + 1, 1);
        for (int i = 1; i <= N; i++) facs[i] = facs[i - 1] * i;
        facinvs.assign(N + 1, facs.back().inv());
        for (int i = N; i > 0; i--) facinvs[i - 1] = facinvs[i] * i;
    }
    modint ncr(int n, int r) const {
        if (n < 0 or r < 0 or n < r) return 0;
        return facs[n] * facinvs[r] * facinvs[n - r];
    }
    modint operator[](int i) const { return facs[i]; }
    modint finv(int i) const { return facinvs[i]; }
};
acl_fac<mint> fac(1000000);




int main() {
    cin.tie(nullptr), ios::sync_with_stdio(false);
    int N;
    cin >> N;

    vector M0(N, vector<mint>(N)), M1(N, vector<mint>(N));

    for (auto &vec : M0) {
        for (auto &x : vec) {
            int v;
            cin >> v;
            x = v;
        }
    }

    for (auto &vec : M1) {
        for (auto &x : vec) {
            int v;
            cin >> v;
            x = v;
        }
    }

    const mint a = rnd(mint::mod());

    // M0 + M1(x + a) の行列式を求める

    matrix<mint> M0_plus_aM1 = matrix(M0) + matrix(M1) * a;
    auto M_inv = M0_plus_aM1;
    int rank = M_inv.inverse();


    if (rank < N) {
        for (int i = 0; i <= N; ++i) cout << 0 << '\n';
        return 0;
    }

    mint det = M0_plus_aM1.gauss_jordan().determinant_of_upper_triangle();

    cerr << det.val() << '\n';

    auto poly = characteristic_poly((-matrix(M1) * M_inv).vecvec());
    reverse(poly.begin(), poly.end());

    for (auto e : poly) cerr << e << ' ';
    cerr << endl;

    vector<mint> ret(N + 1);
    for (int d = 0; d <= N; ++d) {
        // poly[d] * (x + a)^d
        for (int e = 0; e <= d; ++e) ret[e] += poly[d] * fac.ncr(d, e) * mint(-a).pow(d - e) * det;
    }

    for (auto x : ret) cout << x.val() << '\n';
}

yukicoder

結果

テストケース

ソースコード