library_for_cpp

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View the Project on GitHub Kazun1998/library_for_cpp

:heavy_check_mark: verify/yosupo_library_checker/linear_algebra/Power_Matrix.test.cpp

Depends on

Code

#define PROBLEM "https://judge.yosupo.jp/problem/pow_of_matrix"

#include"../../../template/template.hpp"
#include"../../../Algebra/modint.hpp"
#include"../../../Linear_Algebra/Field_Matrix.hpp"

int main(){
    int N; cin >> N;
    ll K; cin >> K;
    Field_Matrix<modint<998244353>> A(N);
    cin >> A;
    cout << power(A, K) << endl;
}
#line 1 "verify/yosupo_library_checker/linear_algebra/Power_Matrix.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/pow_of_matrix"

#line 2 "template/template.hpp"

using namespace std;

// intrinstic
#include <immintrin.h>

#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tuple>
#include <type_traits>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>

// utility
#line 2 "template/utility.hpp"

using ll = long long;

// a ← max(a, b) を実行する. a が更新されたら, 返り値が true.
template<typename T, typename U>
inline bool chmax(T &a, const U b){
    return (a < b ? a = b, 1: 0);
}

// a ← min(a, b) を実行する. a が更新されたら, 返り値が true.
template<typename T, typename U>
inline bool chmin(T &a, const U b){
    return (a > b ? a = b, 1: 0);
}
#line 59 "template/template.hpp"

// math
#line 2 "template/math.hpp"

// 除算に関する関数

// floor(x / y) を求める.
template<typename T, typename U>
T div_floor(T x, U y){ return (x > 0 ? x / y: (x - y + 1) / y); }

// ceil(x / y) を求める.
template<typename T, typename U>
T div_ceil(T x, U y){ return (x > 0 ? (x + y - 1) / y: x / y) ;}

// x を y で割った余りを求める.
template<typename T, typename U>
T mod(T x, U y){
    T q = div_floor(x, y);
    return x - q * y ;
}

// x を y で割った商と余りを求める.
template<typename T, typename U>
pair<T, T> divmod(T x, U y){
    T q = div_floor(x, y);
    return {q, x - q * y};
}

// 四捨五入を求める.
template<typename T, typename U>
T round(T x, U y){
    T q, r;
    tie (q, r) = divmod(x, y);
    return (r >= div_ceil(y, 2)) ? q + 1 : q;
}

// 指数に関する関数

// x の y 乗を求める.
ll intpow(ll x, ll y){
    ll a = 1;
    while (y){
        if (y & 1) { a *= x; }
        x *= x;
        y >>= 1;
    }
    return a;
}

// x の y 乗を z で割った余りを求める.
ll modpow(ll x, ll y, ll z){
    ll a = 1;
    while (y){
        if (y & 1) { (a *= x) %= z; }
        (x *= x) %= z;
        y >>= 1;
    }
    return a;
}

// x の y 乗を z で割った余りを求める.
template<typename T, typename U>
T modpow(T x, U y, T z) {
    T a = 1;
    while (y) {
        if (y & 1) { (a *= x) %= z; }

        (x *= x) %= z;
        y >>= 1;
    }

    return a;
}

// vector の要素の総和を求める.
ll sum(vector<ll> &X){
    ll y = 0;
    for (auto &&x: X) { y+=x; }
    return y;
}

// vector の要素の総和を求める.
template<typename T>
T sum(vector<T> &X){
    T y = T(0);
    for (auto &&x: X) { y += x; }
    return y;
}

// a x + b y = gcd(a, b) を満たす整数の組 (a, b) に対して, (x, y, gcd(a, b)) を求める.
tuple<ll, ll, ll> Extended_Euclid(ll a, ll b) {
    ll s = 1, t = 0, u = 0, v = 1;
    while (b) {
        ll q;
        tie(q, a, b) = make_tuple(div_floor(a, b), b, mod(a, b));
        tie(s, t) = make_pair(t, s - q * t);
        tie(u, v) = make_pair(v, u - q * v);
    }

    return make_tuple(s, u, a);
}

// floor(sqrt(N)) を求める (N < 0 のときは, 0 とする).
ll isqrt(const ll &N) { 
    if (N <= 0) { return 0; }

    ll x = sqrt(N);
    while ((x + 1) * (x + 1) <= N) { x++; }
    while (x * x > N) { x--; }

    return x;
}

// floor(sqrt(N)) を求める (N < 0 のときは, 0 とする).
ll floor_sqrt(const ll &N) { return isqrt(N); }

// ceil(sqrt(N)) を求める (N < 0 のときは, 0 とする).
ll ceil_sqrt(const ll &N) {
    ll x = isqrt(N);
    return x * x == N ? x : x + 1;
}
#line 62 "template/template.hpp"

// inout
#line 1 "template/inout.hpp"
// 入出力
template<class... T>
void input(T&... a){ (cin >> ... >> a); }

void print(){ cout << "\n"; }

template<class T, class... Ts>
void print(const T& a, const Ts&... b){
    cout << a;
    (cout << ... << (cout << " ", b));
    cout << "\n";
}

template<typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &P){
    is >> P.first >> P.second;
    return is;
}

template<typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &P){
    os << P.first << " " << P.second;
    return os;
}

template<typename T>
vector<T> vector_input(int N, int index){
    vector<T> X(N+index);
    for (int i=index; i<index+N; i++) cin >> X[i];
    return X;
}

template<typename T>
istream &operator>>(istream &is, vector<T> &X){
    for (auto &x: X) { is >> x; }
    return is;
}

template<typename T>
ostream &operator<<(ostream &os, const vector<T> &X){
    int s = (int)X.size();
    for (int i = 0; i < s; i++) { os << (i ? " " : "") << X[i]; }
    return os;
}

template<typename T>
ostream &operator<<(ostream &os, const unordered_set<T> &S){
    int i = 0;
    for (T a: S) {os << (i ? " ": "") << a; i++;}
    return os;
}

template<typename T>
ostream &operator<<(ostream &os, const set<T> &S){
    int i = 0;
    for (T a: S) { os << (i ? " ": "") << a; i++; }
    return os;
}

template<typename T>
ostream &operator<<(ostream &os, const unordered_multiset<T> &S){
    int i = 0;
    for (T a: S) { os << (i ? " ": "") << a; i++; }
    return os;
}

template<typename T>
ostream &operator<<(ostream &os, const multiset<T> &S){
    int i = 0;
    for (T a: S) { os << (i ? " ": "") << a; i++; }
    return os;
}
#line 65 "template/template.hpp"

// macro
#line 2 "template/macro.hpp"

// マクロの定義
#define all(x) x.begin(), x.end()
#define len(x) ll(x.size())
#define elif else if
#define unless(cond) if (!(cond))
#define until(cond) while (!(cond))
#define loop while (true)

// オーバーロードマクロ
#define overload2(_1, _2, name, ...) name
#define overload3(_1, _2, _3, name, ...) name
#define overload4(_1, _2, _3, _4, name, ...) name
#define overload5(_1, _2, _3, _4, _5, name, ...) name

// 繰り返し系
#define rep1(n) for (ll i = 0; i < n; i++)
#define rep2(i, n) for (ll i = 0; i < n; i++)
#define rep3(i, a, b) for (ll i = a; i < b; i++)
#define rep4(i, a, b, c) for (ll i = a; i < b; i += c)
#define rep(...) overload4(__VA_ARGS__, rep4, rep3, rep2, rep1)(__VA_ARGS__)

#define foreach1(x, a) for (auto &&x: a)
#define foreach2(x, y, a) for (auto &&[x, y]: a)
#define foreach3(x, y, z, a) for (auto &&[x, y, z]: a)
#define foreach4(x, y, z, w, a) for (auto &&[x, y, z, w]: a)
#define foreach(...) overload5(__VA_ARGS__, foreach4, foreach3, foreach2, foreach1)(__VA_ARGS__)
#line 68 "template/template.hpp"

// bitop
#line 2 "template/bitop.hpp"

// 非負整数 x の bit legnth を求める.
ll bit_length(ll x) {
    if (x == 0) { return 0; }
    return (sizeof(long) * CHAR_BIT) - __builtin_clzll(x);
}

// 非負整数 x の popcount を求める.
ll popcount(ll x) { return __builtin_popcountll(x); }

// 正の整数 x に対して, floor(log2(x)) を求める.
ll floor_log2(ll x) { return bit_length(x) - 1; }

// 正の整数 x に対して, ceil(log2(x)) を求める.
ll ceil_log2(ll x) { return bit_length(x - 1); }

// x の第 k ビットを取得する
int get_bit(ll x, int k) { return (x >> k) & 1; }

// x のビット列を取得する.
// k はビット列の長さとする.
vector<int> get_bits(ll x, int k) {
    vector<int> bits(k);
    rep(i, k) {
        bits[i] = x & 1;
        x >>= 1;
    }

    return bits;
}

// x のビット列を取得する.
vector<int> get_bits(ll x) { return get_bits(x, bit_length(x)); }
#line 2 "Algebra/modint.hpp"

#line 4 "Algebra/modint.hpp"

template<int M>
class modint {
    public:
    static constexpr int Mod = M; 
    int64_t x;

    public:
    // 初期化
    constexpr modint(): x(0) {}
    constexpr modint(int64_t a): x((a % Mod + Mod) % Mod) {}

    // マイナス元
    modint operator-() const { return modint(-x); }

    // 加法
    modint& operator+=(const modint &b){
        if ((x += b.x) >= Mod) x -= Mod;
        return *this;
    }

    friend modint operator+(const modint &x, const modint &y) { return modint(x) += y; }

    // 減法
    modint& operator-=(const modint &b){
        if ((x += Mod - b.x) >= Mod) x -= Mod;
        return *this;
    }

    friend modint operator-(const modint &x, const modint &y) { return modint(x) -= y; }

    // 乗法
    modint& operator*=(const modint &b){
        (x *= b.x) %= Mod;
        return *this;
    }

    friend modint operator*(const modint &x, const modint &y) { return modint(x) *= y; }
    friend modint operator*(const int &x, const modint &y) { return modint(x) *= y; }
    friend modint operator*(const ll &x, const modint &y) { return modint(x) *= y; }

    // 除法
    modint& operator/=(const modint &b){ return (*this) *= b.inverse(); }

    friend modint operator/(const modint &x, const modint &y) { return modint(x) /= y; }

    modint inverse() const {
        int64_t s = 1, t = 0;
        int64_t a = x, b = Mod;

        while (b > 0) {
            int64_t q = a / b;

            a -= q * b; swap(a, b);
            s -= q * t; swap(s, t);
        }

        assert (a == 1);

        return modint(s);
    }

    // 比較
    friend bool operator==(const modint &a, const modint &b) { return (a.x == b.x); }
    friend bool operator==(const modint &a, const int &b) { return a.x == mod(b, Mod); }
    friend bool operator!=(const modint &a, const modint &b) { return (a.x != b.x); }

    // 入力
    friend istream &operator>>(istream &is, modint &a) {
        is >> a.x;
        a.x = (a.x % Mod + Mod) % Mod;
        return is;
    }

    // 出力
    friend ostream &operator<<(ostream &os, const modint &a) { return os << a.x; }

    bool is_zero() const { return x == 0; }
    bool is_member(ll a) const { return x == (a % Mod + Mod) % Mod; }
};

template<int Mod>
modint<Mod> pow(modint<Mod> x, long long n) {
    if (n < 0) { return pow(x, -n).inverse(); }

    auto res = modint<Mod>(1);
    for (; n; n >>= 1) {
        if (n & 1) { res *= x; }
        x *= x;
    }

    return res;
}
#line 2 "Linear_Algebra/Field_Matrix.hpp"

#line 4 "Linear_Algebra/Field_Matrix.hpp"

class SingularMatrixError: private exception{
    const char* what() const throw() {
        return "非正則行列に関する操作を行いました.";
    }
};

template<typename F>
class Field_Matrix{
    public:
    vector<vector<F>> mat;
    int row, col;

    public:
    Field_Matrix(int row, int col): row(row), col(col){
        mat.assign(row, vector<F>(col, F()));
    }

    Field_Matrix(int row): Field_Matrix(row, row){}

    Field_Matrix(vector<vector<F>> &ele): Field_Matrix(ele.size(), ele[0].size()){
        for (int i = 0; i < row; i++){
            copy(ele[i].begin(), ele[i].end(), mat[i].begin());
        }
    }

    static Field_Matrix Zero_Matrix(int row, int col) { return Field_Matrix(row, col); }

    static Field_Matrix Identity_Matrix(int size) {
        Field_Matrix A(size);
        for (int i = 0; i < size; i++) { A[i][i] = 1; }
        return A;
    }

    // サイズ
    pair<int, int> size() { return make_pair(row, col); }

    // 正方行列?
    bool is_square() const { return row == col; }

    // 要素
    inline const vector<F> &operator[](int i) const { return mat[i]; }
    inline vector<F> &operator[](int i) { return mat[i]; }

    // 比較
    bool operator==(const Field_Matrix &B) const {
        if (row != B.row || col != B.col){ return false; }

        for (int i = 0; i < row; i++){
            for (int j = 0; j < col; j++){
                if ((*this)[i] != B[i]) return false;
            }
        }
        return true;
    }

    bool operator!=(const Field_Matrix &B) const { return !((*this) == B); }

    // マイナス元
    Field_Matrix operator-() const {
        Field_Matrix A(row, col);
        for (int i = 0; i < row; i++){
            for (int j = 0; j < col; j++) A[i][j] = -(*this)[i][j];
        }
        return A;
    }

    // 加法
    Field_Matrix& operator+=(const Field_Matrix &B){
        assert (row == B.row && col == B.col);
        for (int i = 0; i < row; i++){
            for (int j = 0; j < col; j++){
                (*this)[i][j] += B[i][j];
            }
        }
        return *this;
    }

    Field_Matrix operator+(const Field_Matrix &B) const { return Field_Matrix(*this) += B; }

    // 減法
    Field_Matrix& operator-=(const Field_Matrix &B){
        assert (row == B.row && col == B.col);
        for (int i = 0; i < row; i++){
            for (int j = 0; j < col; j++){
                (*this)[i][j] -= B[i][j];
            }
        }
        return *this;
    }

    Field_Matrix operator-(const Field_Matrix &B) const  {return Field_Matrix(*this) -= B; }

    // 乗法
    Field_Matrix& operator*=(const Field_Matrix &B){
        assert (col == B.row);
        vector<vector<F>> C(row, vector<F>(B.col, F()));

        for (int i = 0; i < row; i++){
            for (int k = 0; k < col; k++){
                for (int j = 0; j < B.col; j++){
                    C[i][j] += (*this)[i][k] * B[k][j];
                }
            }
        }

        mat.swap(C); col = B.col;
        return *this;
    }

    Field_Matrix operator*(const Field_Matrix &B) const { return Field_Matrix(*this)*=B; }

    // スカラー倍
    friend Field_Matrix operator*(const F &scaler, const Field_Matrix &rhs){
        Field_Matrix res(rhs);
        for (int i = 0; i < rhs.row; i++){
            for (int j = 0; j < rhs.col; j++) { res[i][j] *= scaler; }
        }

        return res;
    }

    // 逆行列
    Field_Matrix inverse(){
        assert (is_square());
        int N = col;
        Field_Matrix A(*this), B(N,N);
        for (int i = 0; i < N; i++) B[i][i] = F(1);

        for (int j = 0; j < N; j++){
            if (A[j][j] == 0){
                int i = j + 1;
                for (; i < N; i++){
                    if (A[i][j] != 0) break;
                }

                if (i == N) { throw SingularMatrixError(); }

                swap(A[i], A[j]); swap(B[i], B[j]);
            }

            F a_inv = A[j][j].inverse();

            for (int k = 0; k < N; k++){
                A[j][k] *= a_inv;
                B[j][k] *= a_inv;
            }

            for (int i = 0; i < N; i++){
                if (i == j) { continue; }

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

        return B;
    }

    bool is_regular(){
        assert (is_square());
        int N = row;

        vector<vector<F>> A(N, vector<F>(N));

        for (int i = 0; i < N; i++){
            copy(mat[i].begin(), mat[i].end(), A[i].begin());
        }

        for (int j = 0; j < N; j++){
            if (A[j][j] == 0){
                int i = j + 1;
                for (; i < N; i++){
                    if (A[i][j] != 0) break;
                }
                if (i == N) return false;
                swap(A[i], A[j]);
            }

            F a_inv = A[j][j].inverse();
            for (int i = j + 1; i < N; i++){
                F c = -a_inv * A[i][j];

               for (int k = 0; k < N; k++){ A[i][k] += c * A[j][k]; }
            }
        }

        return true;
    }

    // 転置
    Field_Matrix transpose(){
        Field_Matrix B(col, row);
        for (int i = 0; i < col; i++){
            for (int j = 0; j < row; j++) B[i][j] = (*this)[j][i];
        }
        return B;
    }

    //
    bool is_valid(){return (row > 0) && (col > 0);}

    // 入力
    friend istream &operator>>(istream &is, Field_Matrix &A) {
        for (int i = 0; i < A.row; i++){
            for (int j = 0; j < A.col; j++){
                cin >> A[i][j];
            }
        }
        return is;
    }

    // 出力
    friend ostream &operator<<(ostream &os, const Field_Matrix &A){
        for (int i = 0; i < A.row; i++){
            for (int j = 0; j < A.col; j++){
                cout << (j ? " ": "") << A[i][j];
            }
            if (i < A.row - 1) cout << "\n";
        }
        return os;
    }
};

template<typename F>
Field_Matrix<F> power(Field_Matrix<F> A, int64_t n){
    assert (A.is_square());

    if (n == 0) { return Field_Matrix<F>::Identity_Matrix(A.row); }
    if (n < 0) { return power(A.inverse(), -n); }

    if (n % 2 == 0){
        Field_Matrix<F> B = power(A, n / 2);
        return B * B;
    } else {
        Field_Matrix<F> B = power(A, (n - 1) / 2);
        return A * B * B;
    }
}

// 行列 A の行列式を求める
template<typename F>
F Determinant(const Field_Matrix<F> &A){
    assert (A.is_square());

    int n = A.row;
    F det(1);
    Field_Matrix<F> B(A);

    for (int j = 0; j < n; j ++){
        if (B[j][j] == 0){
            int i = j + 1;
            for (; i < n; i++) {
                if (B[i][j] != 0) { break; }
            } 

            if (i == n) { return F(0); }

            swap(B[i], B[j]);
            det = -det;
        }

        F a_inv = B[j][j].inverse();
        for (int i = j + 1; i < n; i++){
            F c = -a_inv * B[i][j];
            for (int k = 0; k < n; k++) { B[i][k] += c * B[j][k]; }
        }

        det *= B[j][j];
    }

    return det;
}

// 第 (i, i) 要素が a[i] である対角行列を生成する.
template<typename F>
Field_Matrix<F> Diagonal_Matrix(vector<F> a) {
    int n = a.size();
    vector<vector<F>> X(n, vector<F>(n));

    for (int i = 0; i < n; i++) { X[i][i] = a[i]; }

    return X;
}
#line 6 "verify/yosupo_library_checker/linear_algebra/Power_Matrix.test.cpp"

int main(){
    int N; cin >> N;
    ll K; cin >> K;
    Field_Matrix<modint<998244353>> A(N);
    cin >> A;
    cout << power(A, K) << endl;
}
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