結果
| 問題 | No.1907 DETERMINATION |
| ユーザー |
 松杰方
|
| 提出日時 | 2023-10-28 20:02:39 |
| 言語 | C++23 (gcc 12.3.0 + boost 1.83.0) |
| 結果 |
WA
|
| 実行時間 | - |
| コード長 | 7,165 bytes |
| コンパイル時間 | 3,673 ms |
| コンパイル使用メモリ | 259,112 KB |
| 実行使用メモリ | 7,168 KB |
| 最終ジャッジ日時 | 2024-09-25 16:33:58 |
| 合計ジャッジ時間 | 32,677 ms |
|
ジャッジサーバーID （参考情報） |
judge2 / judge5 |
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テストケース
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ソースコード
#include <bits/stdc++.h>
using namespace std;
// https://github.com/hos-lyric/libra/blob/master/algebra/modint.h
template <unsigned M_> struct ModInt {
static constexpr unsigned M = M_;
unsigned x;
constexpr ModInt() : x(0U) {}
constexpr ModInt(unsigned x_) : x(x_ % M) {}
constexpr ModInt(unsigned long long x_) : x(x_ % M) {}
constexpr ModInt(int x_) : x(((x_ %= static_cast<int>(M)) < 0) ? (x_ + static_cast<int>(M)) : x_) {}
constexpr ModInt(long long x_) : x(((x_ %= static_cast<long long>(M)) < 0) ? (x_ + static_cast<long long>(M)) : x_) {}
ModInt &operator+=(const ModInt &a) { x = ((x += a.x) >= M) ? (x - M) : x; return *this; }
ModInt &operator-=(const ModInt &a) { x = ((x -= a.x) >= M) ? (x + M) : x; return *this; }
ModInt &operator*=(const ModInt &a) { x = (static_cast<unsigned long long>(x) * a.x) % M; return *this; }
ModInt &operator/=(const ModInt &a) { return (*this *= a.inv()); }
ModInt pow(long long e) const {
if (e < 0) return inv().pow(-e);
ModInt a = *this, b = 1U; for (; e; e >>= 1) { if (e & 1) b *= a; a *= a; } return b;
}
ModInt inv() const {
unsigned a = M, b = x; int y = 0, z = 1;
for (; b; ) { const unsigned q = a / b; const unsigned c = a - q * b; a = b; b = c; const int w = y - static_cast<int>(q) * z; y = z; z = w; }
assert(a == 1U); return ModInt(y);
}
ModInt operator+() const { return *this; }
ModInt operator-() const { ModInt a; a.x = x ? (M - x) : 0U; return a; }
ModInt operator+(const ModInt &a) const { return (ModInt(*this) += a); }
ModInt operator-(const ModInt &a) const { return (ModInt(*this) -= a); }
ModInt operator*(const ModInt &a) const { return (ModInt(*this) *= a); }
ModInt operator/(const ModInt &a) const { return (ModInt(*this) /= a); }
template <class T> friend ModInt operator+(T a, const ModInt &b) { return (ModInt(a) += b); }
template <class T> friend ModInt operator-(T a, const ModInt &b) { return (ModInt(a) -= b); }
template <class T> friend ModInt operator*(T a, const ModInt &b) { return (ModInt(a) *= b); }
template <class T> friend ModInt operator/(T a, const ModInt &b) { return (ModInt(a) /= b); }
explicit operator bool() const { return x; }
bool operator==(const ModInt &a) const { return (x == a.x); }
bool operator!=(const ModInt &a) const { return (x != a.x); }
friend std::ostream &operator<<(std::ostream &os, const ModInt &a) { return os << a.x; }
};
// https://yukicoder.me/submissions/737488
// Editorial 解
#include <cassert>
#include <iostream>
#include <utility>
#include <vector>
using namespace std;
// Library Checker Characteristic Polynomial https://judge.yosupo.jp/problem/characteristic_polynomial
// 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];
}
// Library Checker ここまで
template <class T>
std::vector<T> det_of_first_degree_mat(std::vector<std::vector<T>> M0, std::vector<std::vector<T>> M1) {
const int N = M0.size();
int multiply_by_x = 0; // 「特定の列に x をかける」操作を行った回数
T detAdetBinv = 1; // 解説中の 1 / (det A det B) の値
for (int p = 0; p < N; ++p) {
// M1[p][p] に nonzero を持ってきて、M1 の第 p 列を掃き出す
int pivot = -1;
for (int row = p; row < N; ++row) {
if (M1[row][p] != T()) {
pivot = row;
break;
}
}
if (pivot < 0) {
++multiply_by_x;
if (multiply_by_x > N) return std::vector<T>(N + 1);
// M1 の第 p 列で pivot が見つからなかった場合、M0 + x M1 の第 p 列に x をかけたい
// かける前に M1 の第 p 列を第 1 ~ (p - 1) 列を使って掃き出して、
// x をかけた後で x の二次の項が出てこないようにする
for (int row = 0; row < p; ++row) {
T v = M1[row][p];
M1[row][p] = 0;
for (int i = 0; i < N; ++i) M0[i][p] -= v * M0[i][row];
}
for (int i = 0; i < N; ++i) swap(M0[i][p], M1[i][p]);
--p; // 第 p 列をもう一度やり直す この処理は高々 N 回しか走らないので全体の計算量は O(n^3) が保たれる
continue;
}
if (pivot != p) {
M1[pivot].swap(M1[p]);
M0[pivot].swap(M0[p]);
detAdetBinv *= -1;
}
// p 行目を定数倍して M1[p][p] == 1 にする
T v = M1[p][p], vinv = v.inv();
detAdetBinv *= v;
for (int col = 0; col < N; ++col) {
M0[p][col] *= vinv;
M1[p][col] *= vinv;
}
// p 行目を使用して M1 の p 列目を p 行目以外ゼロにする
for (int row = 0; row < N; ++row) {
if (row == p) continue;
T v = M1[row][p];
for (int col = 0; col < N; ++col) {
M0[row][col] -= M0[p][col] * v;
M1[row][col] -= M1[p][col] * v;
}
}
}
// この時点で M1 = I なので det(xI + M0) を求める
for (auto &vec : M0) {
for (auto &x : vec) x = -x;
}
auto poly = characteristic_poly(M0);
for (auto &x : poly) x *= detAdetBinv;
poly.erase(poly.begin(), poly.begin() + multiply_by_x);
poly.resize(N + 1);
return poly;
}
using mint = ModInt<998244353>;
int main() {
int n;
cin >> n;
vector<vector<mint>> M0(n, vector<mint>(n)), M1(n, vector<mint>(n));
for (auto& v : M0) for (auto& x : v) {
int x1;
cin >> x1;
x = x1;
}
for (auto& v : M1) for (auto& x : v) {
int x1;
cin >> x1;
x = x1;
}
vector<mint> res = det_of_first_degree_mat(M0, M1);
for (int i = 0; i <= n; i++) {
cout << res[i] << " \n"[i == n];
}
}