結果

問題 No.1907 DETERMINATION
ユーザー flselfflself
提出日時 2023-10-09 00:08:11
言語 C++23
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 1,829 ms / 4,000 ms
コード長 18,169 bytes
コンパイル時間 3,640 ms
コンパイル使用メモリ 272,116 KB
実行使用メモリ 7,168 KB
最終ジャッジ日時 2024-07-26 18:16:36
合計ジャッジ時間 58,777 ms
ジャッジサーバーID
（参考情報） 		judge1 / judge4
このコードへのチャレンジ
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ファイルパターン 結果
sample AC * 4
other AC * 63
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

diff # 

#include<bits/stdc++.h>

#ifdef LOCAL
#include "debugger.hpp"
#else
#define dbg(...)
#endif

using i64 = long long;
template<class T>
constexpr T power(T a, i64 b) {
    T res = 1;
    for (; b; b /= 2, a *= a) {
        if (b % 2) {
            res *= a;
        }
    }
    return res;
}
 
constexpr i64 mul(i64 a, i64 b, i64 p) {
    i64 res = a * b - i64(1.L * a * b / p) * p;
    res %= p;
    if (res < 0) {
        res += p;
    }
    return res;
}
template<i64 P>
struct MLong {
    i64 x;
    constexpr MLong() : x{} {}
    constexpr MLong(i64 x) : x{norm(x % getMod())} {}
    
    static i64 Mod;
    constexpr static i64 getMod() {
        if (P > 0) {
            return P;
        } else {
            return Mod;
        }
    }
    constexpr static int getRoot() {
        if (getMod() == 1231453023109121) return 3;
        assert(false);
    }
    constexpr static void setMod(i64 Mod_) {
        Mod = Mod_;
    }
    constexpr i64 norm(i64 x) const {
        if (x < 0) {
            x += getMod();
        }
        if (x >= getMod()) {
            x -= getMod();
        }
        return x;
    }
    constexpr i64 val() const { return x; }
    explicit constexpr operator i64() const { return x; }
    explicit constexpr operator bool() const { return x != 0;}
    constexpr MLong operator-() const { MLong res; res.x = norm(getMod() - x); return res; }
    constexpr MLong inv() const { 
        i64 a = getMod(), b = x;
        i64 y = 0, z = 1;
        for (; b; ) {
            i64 k = a / b;
            std::swap(a -= k * b, b);
            std::swap(y -= k * z, z);
        }
        assert(a == 1);
        return MLong(y);
    }
    constexpr MLong &operator*=(MLong rhs) & { x = mul(x, rhs.x, getMod()); return *this; }
    constexpr MLong &operator+=(MLong rhs) & { x = norm(x + rhs.x); return *this; }
    constexpr MLong &operator-=(MLong rhs) & { x = norm(x - rhs.x); return *this; }
    constexpr MLong &operator/=(MLong rhs) & { return *this *= rhs.inv(); }
    friend constexpr MLong operator*(MLong lhs, MLong rhs) { MLong res = lhs; res *= rhs; return res; }
    friend constexpr MLong operator+(MLong lhs, MLong rhs) { MLong res = lhs; res += rhs; return res; }
    friend constexpr MLong operator-(MLong lhs, MLong rhs) { MLong res = lhs; res -= rhs; return res; }
    friend constexpr MLong operator/(MLong lhs, MLong rhs) { MLong res = lhs; res /= rhs; return res; }
    friend constexpr std::istream &operator>>(std::istream &is, MLong &a) { i64 v; is >> v; a = MLong(v); return is; }
    friend constexpr std::ostream &operator<<(std::ostream &os, const MLong &a) { return os << a.val(); }
    friend constexpr bool operator==(MLong lhs, MLong rhs) { return lhs.val() == rhs.val(); }
    friend constexpr bool operator!=(MLong lhs, MLong rhs) { return lhs.val() != rhs.val(); }
};
 
template<>
i64 MLong<0LL>::Mod = i64(1E18) + 9;
 
template<int P>
struct MInt {
    int x;
    constexpr MInt() : x{} {}
    constexpr MInt(i64 x) : x{norm(x % getMod())} {}
    
    static int Mod;
    constexpr static int getMod() {
        if (P > 0) {
            return P;
        } else {
            return Mod;
        }
    }
    constexpr static int getRoot() {
        if (getMod() == 998244353) return 3;
        assert(false);
    }
    constexpr static void setMod(int Mod_) { Mod = Mod_; }
    constexpr int norm(int x) const {
        if (x < 0) {
            x += getMod();
        }
        if (x >= getMod()) {
            x -= getMod();
        }
        return x;
    }
    constexpr int val() const { return x; }
    explicit operator MLong<P>() const { return MLong<P>(x); }
    explicit constexpr operator int() const { return x; }
    explicit constexpr operator bool() const { return x != 0;}
    constexpr MInt operator-() const { MInt res; res.x = norm(getMod() - x); return res; }
    constexpr MInt inv() const {
        unsigned a = getMod(), b = x;
        int y = 0, z = 1;
        for (; b; ) {
            int k = a / b;
            std::swap(a -= k * b, b);
            std::swap(y -= k * z, z);
        }
        assert(a == 1U);
        return MInt(y);
    }
    constexpr MInt &operator*=(MInt rhs) & { x = 1LL * x * rhs.x % getMod(); return *this; }
    constexpr MInt &operator+=(MInt rhs) & { x = norm(x + rhs.x); return *this; }
    constexpr MInt &operator-=(MInt rhs) & { x = norm(x - rhs.x); return *this; }
    constexpr MInt &operator/=(MInt rhs) & { return *this *= rhs.inv(); }
    friend constexpr MInt operator*(MInt lhs, MInt rhs) { MInt res = lhs; res *= rhs; return res; }
    friend constexpr MInt operator+(MInt lhs, MInt rhs) { MInt res = lhs; res += rhs; return res; }
    friend constexpr MInt operator-(MInt lhs, MInt rhs) { MInt res = lhs; res -= rhs; return res; }
    friend constexpr MInt operator/(MInt lhs, MInt rhs) { MInt res = lhs; res /= rhs; return res; }
    friend constexpr std::istream &operator>>(std::istream &is, MInt &a) { i64 v; is >> v; a = MInt(v); return is; }
    friend constexpr std::ostream &operator<<(std::ostream &os, const MInt &a) { return os << a.val(); }
    friend constexpr bool operator==(MInt lhs, MInt rhs) { return lhs.val() == rhs.val(); }
    friend constexpr bool operator!=(MInt lhs, MInt rhs) { return lhs.val() != rhs.val(); }
};
 
template<>
int MInt<0>::Mod = 998244353;
 
template<int V, int P>
constexpr MInt<P> CInv = MInt<P>(V).inv();
 
constexpr int P = 998244353;
using Z = MInt<P>;



std::vector<int> rev;
std::vector<Z> roots{0, 1};
void dft(std::vector<Z> &a) {
    int n = a.size();
    
    if ((int)(rev.size()) != n) {
        int k = __builtin_ctz(n) - 1;
        rev.resize(n);
        for (int i = 0; i < n; i++) {
            rev[i] = rev[i >> 1] >> 1 | (i & 1) << k;
        }
    }
    
    for (int i = 0; i < n; i++) {
        if (rev[i] < i) {
            std::swap(a[i], a[rev[i]]);
        }
    }
    if ((int)(roots.size()) < n) {
        int k = __builtin_ctz(roots.size());
        roots.resize(n);
        while ((1 << k) < n) {
            Z e = power(Z(Z::getRoot()), (Z::getMod() - 1) >> (k + 1));
            for (int i = 1 << (k - 1); i < (1 << k); i++) {
                roots[2 * i] = roots[i];
                roots[2 * i + 1] = roots[i] * e;
            }
            k++;
        }
    }
    for (int k = 1; k < n; k *= 2) {
        for (int i = 0; i < n; i += 2 * k) {
            for (int j = 0; j < k; j++) {
                Z u = a[i + j];
                Z v = a[i + j + k] * roots[k + j];
                a[i + j] = u + v;
                a[i + j + k] = u - v;
            }
        }
    }
}
void idft(std::vector<Z> &a) {
    int n = a.size();
    std::reverse(a.begin() + 1, a.end());
    dft(a);
    Z inv = (1 - Z::getMod()) / n;
    for (int i = 0; i < n; i++) {
        a[i] *= inv;
    }
}
template<typename T>
struct Poly {
    std::vector<T> a;
    Poly() {}
    Poly(const std::vector<T> &a) : a(a) {}
    Poly(const std::initializer_list<T> &a) : a(a) {}
    int size() const {
        return a.size();
    }
    void resize(int n) {
        a.resize(n);
    }
    T operator[](int idx) const {
        if (idx < size()) {
            return a[idx];
        } else {
            return 0;
        }
    }
    T &operator[](int idx) {
        return a[idx];
    }
    Poly mulxk(int k) const {
        auto b = a;
        b.insert(b.begin(), k, 0);
        return Poly(b);
    }
    Poly modxk(int k) const {
        k = std::min(k, size());
        return Poly(std::vector<T>(a.begin(), a.begin() + k));
    }
    Poly divxk(int k) const {
        if (size() <= k) {
            return Poly();
        }
        return Poly(std::vector<T>(a.begin() + k, a.end()));
    }
    friend Poly operator+(const Poly &a, const Poly &b) {
        std::vector<T> res(std::max(a.size(), b.size()));
        for (int i = 0; i < (int)(res.size()); i++) {
            res[i] = a[i] + b[i];
        }
        return Poly(res);
    }
    friend Poly operator-(const Poly &a, const Poly &b) {
        std::vector<T> res(std::max(a.size(), b.size()));
        for (int i = 0; i < (int)(res.size()); i++) {
            res[i] = a[i] - b[i];
        }
        return Poly(res);
    }
    friend Poly operator*(Poly a, Poly b) {
        if (a.size() == 0 || b.size() == 0) {
            return Poly();
        }
        int sz = 1, tot = a.size() + b.size() - 1;
        while (sz < tot) {
            sz *= 2;
        }
        a.a.resize(sz);
        b.a.resize(sz);
        dft(a.a);
        dft(b.a);
        for (int i = 0; i < sz; ++i) {
            a.a[i] = a[i] * b[i];
        }
        idft(a.a);
        a.resize(tot);
        return a;
    }
    friend Poly operator*(T a, Poly b) {
        for (int i = 0; i < (int)(b.size()); i++) {
            b[i] *= a;
        }
        return b;
    }
    friend Poly operator*(Poly a, T b) {
        for (int i = 0; i < (int)(a.size()); i++) {
            a[i] *= b;
        }
        return a;
    }
    Poly &operator+=(Poly b) {
        return (*this) = (*this) + b;
    }
    Poly &operator-=(Poly b) {
        return (*this) = (*this) - b;
    }
    Poly &operator*=(Poly b) {
        return (*this) = (*this) * b;
    }
    Poly deriv() const {
        if (a.empty()) {
            return Poly();
        }
        std::vector<T> res(size() - 1);
        for (int i = 0; i < size() - 1; ++i) {
            res[i] = (i + 1) * a[i + 1];
        }
        return Poly(res);
    }
    Poly integr() const {
        std::vector<T> res(size() + 1);
        for (int i = 0; i < size(); ++i) {
            res[i + 1] = a[i] / (i + 1);
        }
        return Poly(res);
    }
    Poly inv(int m) const {
        Poly x{a[0].inv()};
        int k = 1;
        while (k < m) {
            k *= 2;
            x = (x * (Poly{2} - modxk(k) * x)).modxk(k);
        }
        return x.modxk(m);
    }
    Poly log(int m) const {
        return (deriv() * inv(m)).integr().modxk(m);
    }
    Poly exp(int m) const {
        Poly x{1};
        int k = 1;
        while (k < m) {
            k *= 2;
            x = (x * (Poly{1} - x.log(k) + modxk(k))).modxk(k);
        }
        return x.modxk(m);
    }
    Poly pow(int k, int m) const {
        int i = 0;
        while (i < size() && a[i].val() == 0) {
            i++;
        }
        if (i == size() || 1LL * i * k >= m) {
            return Poly(std::vector<T>(m));
        }
        T v = a[i];
        auto f = divxk(i) * v.inv();
        return (f.log(m - i * k) * k).exp(m - i * k).mulxk(i * k) * power(v, k);
    }
    Poly sqrt(int m) const {
        Poly x{1};
        int k = 1;
        while (k < m) {
            k *= 2;
            x = (x + (modxk(k) * x.inv(k)).modxk(k)) * ((T::getMod() + 1) / 2);
        }
        return x.modxk(m);
    }
    Poly mulT(Poly b) const {
        if (b.size() == 0) {
            return Poly();
        }
        int n = b.size();
        std::reverse(b.a.begin(), b.a.end());
        return ((*this) * b).divxk(n - 1);
    }
    std::vector<T> eval(std::vector<T> x) const {
        if (size() == 0) {
            return std::vector<T>(x.size(), 0);
        }
        const int n = std::max((int)(x.size()), size());
        std::vector<Poly> q(4 * n);
        std::vector<T> ans(x.size());
        x.resize(n);
        std::function<void(int, int, int)> build = [&](int p, int l, int r) {
            if (r - l == 1) {
                q[p] = Poly{1, -x[l]};
            } else {
                int m = (l + r) / 2;
                build(2 * p, l, m);
                build(2 * p + 1, m, r);
                q[p] = q[2 * p] * q[2 * p + 1];
            }
        };
        build(1, 0, n);
        std::function<void(int, int, int, const Poly &)> work = [&](int p, int l, int r, const Poly &num) {
            if (r - l == 1) {
                if (l < (int)(ans.size())) {
                    ans[l] = num[0];
                }
            } else {
                int m = (l + r) / 2;
                work(2 * p, l, m, num.mulT(q[2 * p + 1]).modxk(m - l));
                work(2 * p + 1, m, r, num.mulT(q[2 * p]).modxk(r - m));
            }
        };
        work(1, 0, n, mulT(q[1].inv(n)));
        return ans;
    }
};

template<typename T>
T count_minor(const std::vector<std::vector<int>> matrix, int idx = 0, int idy = 0) {
    assert(matrix.size() == matrix[0].size());
    int n = matrix.size();
    assert(idx < n && idy < n);
    if (n == 1) {
        return 0;
    }
    std::vector minor(n-1, std::vector<T>(n-1));
    for (int i = 0; i < n; ++i) {
        if (i == idx) continue;
        for (int j = 0; j < n; ++j) {
            if (j == idy) continue;
            minor[i - (i > idx)][j - (j > idy)] = matrix[i][j];
        }
    }
    n--;
    auto gauss=[&]()->T {
        for (int i = 0; i < n; ++i) {
            if (minor[i][i] == 0) {
                for (int j = i + 1; j < n; ++j) {
                    if (minor[j][i]) {
                        // for (int k = 0; k < n; ++k) {
                        //     std::swap(minor[i][k], minor[j][k]);
                        // }
                        std::swap(minor[i], minor[j]);
                        break;
                    }
                }
            }
            if (minor[i][i] == 0) {
                return 0;
            }
            for (int j = i + 1; j < n; ++j) {
                if (i == j) continue;
                T mul = minor[j][i] / minor[i][i];
                for (int k = i; k < n; ++k) {
                    minor[j][k] -= mul * minor[i][k];
                }
            }
        }
        T res = 1;
        for (int i = 0; i < n; ++i) {
            res *= minor[i][i];
        }
        return res;
    };
    return gauss();
}

// Kirchhoff 
Z Kirchhoff(const std::vector<std::vector<int>> &G) {
    int n = G.size();
    std::vector<std::vector<int>> L(n, std::vector<int>(n));
    for (int i = 0; i < n; ++i) {
        L[i][i] = G[i].size();
        for (auto &j : G[i]) {
            L[i][j]--;
        }
    }
    return count_minor<Z>(L);
}


/// @param A square matrix of size n
/// @return characteristic polynomial of A, which is `det(xI - M)`
template <typename T>
std::vector<T> char_poly(std::vector<std::vector<T>> M) {
    int N = M.size();
    assert(N == M[0].size());

    // Hessenberg Reduction
    for (int i = 0; i < N - 2; i++) {
        int p = -1;
        for (int j = i + 1; j < N; j++) {
            if (M[j][i] != T(0)) {
                p = j;
                break;
            }
        }
        if (p == -1) {
            continue;
        }
        M[i + 1].swap(M[p]);
        for (int j = 0; j < N; j++) {
            std::swap(M[j][i + 1], M[j][p]);
        }

        T r = T(1) / M[i + 1][i];
        for (int j = i + 2; j < N; j++) {
            T c = M[j][i] * r;
            for (int k = 0; k < N; k++) M[j][k] -= M[i + 1][k] * c;
            for (int k = 0; k < N; k++) M[k][i + 1] += M[k][j] * c;
        }
    }

    // La Budde's Method
    std::vector<std::vector<T>> P = {{T(1)}};
    for (int i = 0; i < N; i++) {
        std::vector<T> f(i + 2, 0);
        for (int j = 0; j <= i; j++) f[j + 1] += P[i][j];
        for (int j = 0; j <= i; j++) f[j] -= P[i][j] * M[i][i];

        T b = 1;
        for (int j = i - 1; j >= 0; j--) {
            b *= M[j + 1][j];
            T h = -M[j][i] * b;
            for (int k = 0; k <= j; k++) f[k] += h * P[j][k];
        }
        P.push_back(f);
    }
    return P.back();
}

/// @brief calculate `det(Ax + B)`, where A and B are square matrices of size n
/// @tparam T usually Z
/// @tparam Matrix usually std::vector<std::vector<T>>, or custom matrix class
/// @param A 
/// @param B 
/// @return `det(Ax + B)`
template <typename T>
std::vector<T> det_linear(std::vector<std::vector<T>> A, std::vector<std::vector<T>> B) {
    int N = A.size(), nu = 0;
    T det = 1;
    for (int i = 0; i < N; i++) {
        // do normal gaussian elimination
        int p = -1;
        for (int j = i; j < N; j++) {
            if (A[j][i] != T(0)) {
                p = j;
                break;
            }
        }
        // replace B with A
        if (p == -1) {
            // Increase nullity by 1
            if (++nu > N) {
                return std::vector<T>(N + 1, 0);
            }
            // i-th column is empty
            for (int j = 0; j < i; j++) {
                for (int k = 0; k < N; k++) {
                    B[k][i] -= B[k][j] * A[j][i];
                }
                A[j][i] = 0;
            }
            for (int j = 0; j < N; j++) {
                std::swap(A[j][i], B[j][i]);
            }
            // retry
            --i;
            continue;
        }

        if (p != i) {
            A[i].swap(A[p]);
            B[i].swap(B[p]);
            det = -det;
        }

        det *= A[i][i];
        T c = T(1) / A[i][i];
        for (int j = 0; j < N; j++) {
            A[i][j] *= c, B[i][j] *= c;
        }

        for (int j = 0; j < N; j++) {
            if (j != i) {
                T c = A[j][i];
                for (int k = 0; k < N; k++) {
                    A[j][k] -= A[i][k] * c, B[j][k] -= B[i][k] * c;
                }
            }
        }
    }

    for (auto &y : B) {
        for (T &x : y) {
            x = -x;
        }
    }

    auto f = char_poly(B);
    for (T &x : f) {
        x *= det;
    }
    f.erase(f.begin(), f.begin() + nu);
    f.resize(N + 1);
    return f;
}

void solv() {
    int n;
    std::cin >> n;
    
    std::vector<std::vector<Z>> D1(n, std::vector<Z>(n));
    auto D2 = D1;

    for (int i = 0; i < n; ++i) {
        for (int j = 0; j < n; ++j) {
            std::cin >> D2[i][j];
        }
    }
    for (int i = 0; i < n; ++i) {
        for (int j = 0; j < n; ++j) {
            std::cin >> D1[i][j];
        }
    }

    auto poly = det_linear(D1, D2);
    for (int i = 0; i <= n; ++i) {
        if (i >= poly.size()) {
            std::cout << "0\n";
        } else {
            std::cout << poly[i] << '\n';
        }
    }
}

signed main() {
    std::ios::sync_with_stdio(false), std::cin.tie(nullptr), std::cout.tie(nullptr);
    
    int tt = 1;
    // std::cin >> tt;
    while (tt--) {
        solv();
    }

    return 0;
}
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