結果

問題 No.194 フィボナッチ数列の理解(1)
ユーザー legosuke
提出日時 2021-04-26 13:27:05
言語 C++17
(gcc 13.3.0 + boost 1.87.0)
結果
WA  
実行時間 -
コード長 13,016 bytes
コンパイル時間 2,815 ms
コンパイル使用メモリ 208,616 KB
最終ジャッジ日時 2025-01-21 01:09:32
ジャッジサーバーID
（参考情報） 		judge5 / judge5
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ファイルパターン 結果
sample AC * 3
other AC * 28 WA * 9
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#line 1 "test/01_Math/03_Algebra/01.01.02.01_yukicoder-194.test.cpp"
#define PROBLEM "https://yukicoder.me/problems/no/194"
#line 1 "template/template.hpp"
#include <bits/stdc++.h>
#define int int64_t
using namespace std;
#line 3 "01_Math/02_Combinatorics/01.01_mod-operation.hpp"
/**
 * @brief mod 上の基本演算
 */
template <typename T, typename M>
inline M mod(T a, M m) {
    return (a % m + m) % m;
}
template <typename T, typename U, typename M>
inline M add(T a, U b, M m) {
    return mod(mod(a, m) + mod(b, m), m);
}
template <typename T, typename U, typename M>
inline M sub(T a, U b, M m) {
    return mod(mod(a, m) - mod(b, m), m);
}
template <typename T, typename U, typename M>
inline M mul(T a, U b, M m) {
    return mod((__uint128_t)a * b, m);
}
#line 3 "01_Math/02_Combinatorics/01.02.00_modint-base.hpp"
#include <type_traits>
/**
 * @brief modint 構造体 (base)
 */
class modint_base {};
template <class T>
using is_modint = std::is_base_of<modint_base, T>;
#line 3 "01_Math/02_Combinatorics/01.03.02_mod-pow.big-mod.hpp"
/**
 * @brief 累乗 : $a^n\bmod{m}$ ($m$ が大きい場合)
 * @note O(log(n))
 */
std::uint64_t mod_pow(std::int64_t a, std::uint64_t n, std::uint64_t m) {
    a = mod(a, m);
    std::uint64_t res = 1;
    while (n) {
        if (n & 1) res = mul(res, a, m);
        a = mul(a, a, m);
        n >>= 1;
    }
    return res;
}
#line 5 "01_Math/01_NumberTheory/01.04.01_ext-gcd.hpp"
/**
 * @brief 拡張ユークリッドの互除法
 * @note O(min(log(a),log(b)))
 */
template <typename Integer1, typename Integer2, typename Integer3>
Integer1 ext_gcd(Integer1 a, Integer2 b, Integer3& x, Integer3& y) {
    static_assert(std::is_integral<Integer1>::value);
    static_assert(std::is_integral<Integer2>::value);
    static_assert(std::is_integral<Integer3>::value || std::is_same<Integer3, __int128_t>::value);
    if (b == 0) { x = 1; y = 0; return a; }
    auto g = ext_gcd(b, a % b, y, x);
    y -= a / b * x;
    return g;
}
#line 4 "01_Math/02_Combinatorics/01.04.03_mod-inv.ext-gcd.hpp"
/**
 * @brief 逆元 : $a^{-1}\bmod{m}$ (拡張ユークリッドの互助法)
 * @note O(log(m))
 * @warning a と m は互いに素
 */
std::uint64_t mod_inv(std::int64_t a, std::uint64_t m) {
    __int128_t x, y;
    auto g = ext_gcd(a, m, x, y);
    assert(g == 1);
    return mod(x, m);
}
#line 7 "01_Math/02_Combinatorics/01.02.01_modint.static.hpp"
/**
 * @brief modint 構造体 (静的 MOD)
 */
template <std::uint32_t MOD>
class static_modint : public modint_base {
    using mint = static_modint;
public:
    static_modint() = default;
    template <typename Integer>
    static_modint(const Integer& v) : _v((v % MOD + MOD) % MOD) {}
    std::uint32_t get_mod() const {
        return MOD;
    }
    std::uint32_t get_val() const {
        return _v;
    }
    template <typename Integer>
    mint& operator += (const Integer& rhs) {
        _v = add(_v, mint(rhs)._v, MOD);
        return *this;
    }
    template <typename Integer>
    mint& operator -= (const Integer& rhs)  {
        _v = sub(_v, mint(rhs)._v, MOD);
        return *this;
    }
    template <typename Integer>
    mint& operator *= (const Integer& rhs)  {
        _v = mul(_v, mint(rhs)._v, MOD);
        return *this;
    }
    template <typename Integer>
    mint& operator /= (const Integer& rhs)  {
        return *this *= mint(rhs).inv();
    }
    template <typename Integer>
    mint& operator = (const Integer& v) {
        static_assert(std::is_integral<Integer>::value);
        _v = mod(v, MOD);
        return *this;
    }
    mint pow(std::uint32_t n) const {
        return mint(mod_pow(_v, n, MOD));
    }
    mint inv() const {
        return mint(mod_inv(_v, MOD));
    }
    mint operator - () const {
        return mint(_v ? MOD - _v : 0);
    }
    friend std::ostream& operator << (std::ostream& os, const static_modint<MOD>& rhs) {
        return os << rhs._v;
    };
    friend std::istream& operator >> (std::istream& is, static_modint<MOD>& rhs) {
        is >> rhs._v;
        rhs._v = mod(rhs._v, MOD);
        return is;
    }
protected:
    std::uint32_t _v;
};
using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
template <std::uint32_t MOD>
const static_modint<MOD> operator + (const static_modint<MOD>& lhs, const static_modint<MOD>& rhs) {
    return static_modint<MOD>(lhs) += rhs;
}
template <std::uint32_t MOD, typename Integer>
const static_modint<MOD> operator + (const static_modint<MOD>& lhs, const Integer& rhs) {
    return static_modint<MOD>(lhs) += rhs;
}
template <std::uint32_t MOD, typename Integer>
const static_modint<MOD> operator + (const Integer& lhs, const static_modint<MOD>& rhs) {
    return static_modint<MOD>(rhs) += lhs;
}
template <std::uint32_t MOD>
const static_modint<MOD> operator - (const static_modint<MOD>& lhs, const static_modint<MOD>& rhs) {
    return static_modint<MOD>(lhs) -= rhs;
}
template <std::uint32_t MOD, typename Integer>
const static_modint<MOD> operator - (const static_modint<MOD>& lhs, const Integer& rhs) {
    return static_modint<MOD>(lhs) -= rhs;
}
template <std::uint32_t MOD, typename Integer>
const static_modint<MOD> operator - (const Integer& lhs, const static_modint<MOD>& rhs) {
    return static_modint<MOD>(rhs) -= lhs;
}
template <std::uint32_t MOD>
const static_modint<MOD> operator * (const static_modint<MOD>& lhs, const static_modint<MOD>& rhs) {
    return static_modint<MOD>(lhs) *= rhs;
}
template <std::uint32_t MOD, typename Integer>
const static_modint<MOD> operator * (const static_modint<MOD>& lhs, const Integer& rhs) {
    return static_modint<MOD>(lhs) *= rhs;
}
template <std::uint32_t MOD, typename Integer>
const static_modint<MOD> operator * (const Integer& lhs, const static_modint<MOD>& rhs) {
    static_assert(std::is_same<Integer, static_modint<MOD>>::value == false);
    return static_modint<MOD>(rhs) *= lhs;
}
template <std::uint32_t MOD>
const static_modint<MOD> operator / (const static_modint<MOD>& lhs, const static_modint<MOD>& rhs) {
    return static_modint<MOD>(lhs) /= rhs;
}
template <std::uint32_t MOD, typename Integer>
const static_modint<MOD> operator / (const static_modint<MOD>& lhs, const Integer& rhs) {
    return static_modint<MOD>(lhs) /= rhs;
}
template <std::uint32_t MOD, typename Integer>
const static_modint<MOD> operator / (const Integer& lhs, const static_modint<MOD>& rhs) {
    return static_modint<MOD>(rhs) /= lhs;
}
#line 3 "01_Math/03_Algebra/01.01.00_matrix-base.hpp"
/**
 * @brief 行列 (base)
 */
class matrix_base {};
template <class T>
using is_matrix = std::is_base_of<matrix_base, T>;
#line 7 "01_Math/03_Algebra/01.01.01.01_matrix.vector.hpp"
/**
 * @brief 行列 (vector)
 */
template <class T>
class matrix_vector : matrix_base {
public:
    using value_type = T;
    matrix_vector() = default;
    explicit matrix_vector(std::uint32_t n, std::uint32_t m, T x = T(0)) { init(n, m, x); }
    std::uint32_t height() const {
        return _n;
    }
    std::uint32_t width() const {
        return _m;
    }
    void fill(T x = T(0)) {
        _v.clear(); _v.assign(_n, std::vector<T>(_m, x));
    }
    void init(std::uint32_t n, std::uint32_t m, T x = T(0)) {
        _n = n; _m = m;
        fill(x);
    }
    const std::vector<T>& operator [] (std::uint32_t i) const {
        return (_v.at(i));
    }
    std::vector<T>& operator [] (std::uint32_t i) {
        return (_v.at(i));
    }
    friend std::ostream& operator << (std::ostream& os, const matrix_vector<T>& A) {
        for (std::uint32_t i = 0; i < A.height(); ++i) for (std::uint32_t j = 0; j < A.width(); ++j) {
            os << A[i][j] << " \n"[j + 1 == A.width()];
        }
        return os;
    }
protected:
    std::uint32_t _n, _m;
    std::vector<std::vector<T>> _v;
};
template <class T>
matrix_vector<T> operator + (const matrix_vector<T>& A, const T& x) {
    matrix_vector<T> res(A.height(), A.width());
    for (std::uint32_t i = 0; i < A.height(); ++i) for (std::uint32_t j = 0; j < A.width(); ++j) {
        res[i][j] = A[i][j] + x;
    }
    return res;
}
template <class T>
matrix_vector<T> operator + (const T& x, const matrix_vector<T>& A) {
    matrix_vector<T> res(A.height(), A.width());
    for (std::uint32_t i = 0; i < A.height(); ++i) for (std::uint32_t j = 0; j < A.width(); ++j) {
        res[i][j] = x + A[i][j];
    }
    return res;
}
template <class T>
matrix_vector<T> operator + (const matrix_vector<T>& A, const matrix_vector<T>& B) {
    assert(A.height() == B.height() && A.width() == B.width());
    matrix_vector<T> res(A.height(), A.width());
    for (std::uint32_t i = 0; i < A.height(); ++i) for (std::uint32_t j = 0; j < A.width(); ++j) {
        res[i][j] = A[i][j] + B[i][j];
    }
    return res;
}
template <class T>
matrix_vector<T> operator - (const matrix_vector<T>& A, const T& x) {
    matrix_vector<T> res(A.height(), A.width());
    for (std::uint32_t i = 0; i < A.height(); ++i) for (std::uint32_t j = 0; j < A.width(); ++j) {
        res[i][j] = A[i][j] - x;
    }
    return res;
}
template <class T>
matrix_vector<T> operator - (const T& x, const matrix_vector<T>& A) {
    matrix_vector<T> res(A.height(), A.width());
    for (std::uint32_t i = 0; i < A.height(); ++i) for (std::uint32_t j = 0; j < A.width(); ++j) {
        res[i][j] = x - A[i][j];
    }
    return res;
}
template <class T>
matrix_vector<T> operator - (const matrix_vector<T>& A, const matrix_vector<T>& B) {
    assert(A.height() == B.height() && A.width() == B.width());
    matrix_vector<T> res(A.height(), A.width());
    for (std::uint32_t i = 0; i < A.height(); ++i) for (std::uint32_t j = 0; j < A.width(); ++j) {
        res[i][j] = A[i][j] - B[i][j];
    }
    return res;
}
template <class T>
matrix_vector<T> operator * (const matrix_vector<T>& A, const T& x) {
    matrix_vector<T> res(A.height(), A.width());
    for (std::uint32_t i = 0; i < A.height(); ++i) for (std::uint32_t j = 0; j < A.width(); ++j) {
        res[i][j] = A[i][j] * x;
    }
    return res;
}
template <class T>
matrix_vector<T> operator * (const T& x, const matrix_vector<T>& A) {
    matrix_vector<T> res(A.height(), A.width());
    for (std::uint32_t i = 0; i < A.height(); ++i) for (std::uint32_t j = 0; j < A.width(); ++j) {
        res[i][j] = x * A[i][j];
    }
    return res;
}
template <class T>
std::vector<T> operator * (const matrix_vector<T>& A, const std::vector<T>& v) {
    assert(A.width() == (std::uint32_t)v.size());
    std::vector<T> u(A.height(), T(0));
    for (std::uint32_t i = 0; i < A.height(); ++i) for (std::uint32_t j = 0; j < A.width(); ++j) {
        u[i] = u[i] + A[i][j] * v[j];
    }
    return u;
}
template <class T>
matrix_vector<T> operator * (const matrix_vector<T>& A, const matrix_vector<T>& B) {
    assert(A.width() == B.height());
    matrix_vector<T> res(A.height(), B.width(), T(0));
    for (std::uint32_t i = 0; i < A.height(); ++i) for (std::uint32_t j = 0; j < B.width(); ++j) for (std::uint32_t k = 0; k < A.width(); ++k) {
        res[i][j] = res[i][j] + A[i][k] * B[k][j];
    }
    return res;
}
template <class T>
matrix_vector<T> operator / (const matrix_vector<T>& A, const T& x) {
    matrix_vector<T> res(A.height(), A.width());
    for (std::uint32_t i = 0; i < A.height(); ++i) for (std::uint32_t j = 0; j < A.width(); ++j) {
        res[i][j] = A[i][j] / x;
    }
    return res;
}
template <class T>
matrix_vector<T> operator ^ (matrix_vector<T> A, std::uint64_t n) {
    assert(A.height() == A.width());
    matrix_vector<T> B(A.height(), A.width());
    for (int i = 0; i < A.height(); ++i) B[i][i] = T(1);
    while (n) {
        if (n & 1) B = B * A;
        A = A * A;
        n >>= 1;
    }
    return B;
}
#line 5 "test/01_Math/03_Algebra/01.01.02.01_yukicoder-194.test.cpp"
signed main() {
    int N, K; cin >> N >> K;
    vector<int> A(N);
    for (int i = 0; i < N; ++i) cin >> A[i];
    if (N <= 10000 && K <= 1000000) {
        vector<modint1000000007> fib(K);
        for (int i = 0; i < N; ++i) {
            fib[i] = A[i];
            fib[N] += A[i];
        }
        for (int i = N + 1; i < K; ++i) {
            fib[i] += fib[i - 1] * 2;
            fib[i] -= fib[i - N - 1];
        }
        modint1000000007 sum(0);
        for (int i = 0; i < K; ++i) {
            sum += fib[i];            
        }
        cout << fib[K - 1] << " " << sum << endl;
    } else {
        matrix_vector<modint1000000007> M(N + 1, N + 1);
        for (int i = 0; i < N - 1; ++i) {
            for (int j = 0; j < N + 1; ++j) {
                if (j == i + 1) M[i][j] = 1;
                else M[i][j] = 0;
            }
        }
        for (int j = 0; j < N; ++j) {
            M[N - 1][j] = M[N][j] = 1;
        }
        M[N][N] = 1;
        M = M ^ (K - N);
        vector<modint1000000007> v(N + 1);
        for (int i = 0; i < N; ++i) {
            v[i] = A[i];
            v[N] += A[i];
        }
        auto w = M * v;
        cout << w[N - 1] << " " << w[N] << endl;
    }
}
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