#908069 (C++23) No.2445 奇行列式

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結果

問題	No.2445 奇行列式
コンテスト
ユーザー	Aeren
提出日時	2023-08-25 23:17:00
言語	C++23 (gcc 13.3.0 + boost 1.87.0)
結果	AC
実行時間	1,046 ms / 3,000 ms
コード長	16,619 bytes
コンパイル時間	5,016 ms
コンパイル使用メモリ	370,976 KB
実行使用メモリ	5,248 KB
最終ジャッジ日時	2024-12-24 11:07:37
合計ジャッジ時間	12,686 ms
ジャッジサーバーID （参考情報）	judge1 / judge3
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ファイルパターン	結果
other	AC * 20
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ソースコード

raw source code
#include <x86intrin.h>
#include <bits/stdc++.h>
using namespace std;
#if __cplusplus > 201703L
#include <ranges>
using namespace numbers;
#endif

template<int id>
struct modular_unfixed_base{
	static unsigned int _mod;
	static unsigned long long _inverse_mod;
	static unsigned int &mod(){
		return _mod;
	}
	static void precalc_barrett(){
		_inverse_mod = (unsigned long long)-1 / _mod + 1;
	}
	static void setup(unsigned int mod = 0){
		if(!mod) cin >> mod;
		_mod = mod;
		assert(_mod >= 1);
		precalc_barrett();
	}
	template<class T>
	static vector<modular_unfixed_base> precalc_power(T base, int SZ){
		vector<modular_unfixed_base> res(SZ + 1, 1);
		for(auto i = 1; i <= SZ; ++ i) res[i] = res[i - 1] * base;
		return res;
	}
	static vector<modular_unfixed_base> _INV;
	static void precalc_inverse(int SZ){
		if(_INV.empty()) _INV.assign(2, 1);
		for(auto x = _INV.size(); x <= SZ; ++ x) _INV.push_back(_mod / x * -_INV[_mod % x]);
	}
	// _mod must be a prime
	static modular_unfixed_base _primitive_root;
	static modular_unfixed_base primitive_root(){
		if(_primitive_root) return _primitive_root;
		if(_mod == 2) return _primitive_root = 1;
		if(_mod == 998244353) return _primitive_root = 3;
		unsigned int divs[20] = {};
		divs[0] = 2;
		int cnt = 1;
		unsigned int x = (_mod - 1) / 2;
		while(x % 2 == 0) x /= 2;
		for(auto i = 3; 1LL * i * i <= x; i += 2){
			if(x % i == 0){
				divs[cnt ++] = i;
				while(x % i == 0) x /= i;
			}
		}
		if(x > 1) divs[cnt ++] = x;
		for(auto g = 2; ; ++ g){
			bool ok = true;
			for(auto i = 0; i < cnt; ++ i){
				if((modular_unfixed_base(g).power((_mod - 1) / divs[i])) == 1){
					ok = false;
					break;
				}
			}
			if(ok) return _primitive_root = g;
		}
	}
	constexpr modular_unfixed_base(): data(){ }
	modular_unfixed_base(const double &x){ data = normalize(llround(x)); }
	modular_unfixed_base(const long double &x){ data = normalize(llround(x)); }
	template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_unfixed_base(const T &x){ data = normalize(x); }
	template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr> static unsigned int normalize(const T &x){
		if(_mod == 1) return 0;
		assert(_inverse_mod);
		int sign = x >= 0 ? 1 : -1;
		unsigned int v = _mod <= sign * x ? sign * x - ((__uint128_t)(sign * x) * _inverse_mod >> 64) * _mod : sign * x;
		if(v >= _mod) v += _mod;
		if(sign == -1 && v) v = _mod - v;
		return v;
	}
	const unsigned int &operator()() const{ return data; }
	template<class T> operator T() const{ return data; }
	modular_unfixed_base &operator+=(const modular_unfixed_base &otr){ if((data += otr.data) >= _mod) data -= _mod; return *this; }
	modular_unfixed_base &operator-=(const modular_unfixed_base &otr){ if((data += _mod - otr.data) >= _mod) data -= _mod; return *this; }
	template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_unfixed_base &operator+=(const T &otr){ return *this += modular_unfixed_base(otr); }
	template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_unfixed_base &operator-=(const T &otr){ return *this -= modular_unfixed_base(otr); }
	modular_unfixed_base &operator++(){ return *this += 1; }
	modular_unfixed_base &operator--(){ return *this += _mod - 1; }
	modular_unfixed_base operator++(int){ modular_unfixed_base result(*this); *this += 1; return result; }
	modular_unfixed_base operator--(int){ modular_unfixed_base result(*this); *this += _mod - 1; return result; }
	modular_unfixed_base operator-() const{ return modular_unfixed_base(_mod - data); }
	modular_unfixed_base &operator*=(const modular_unfixed_base &rhs){
		data = normalize((unsigned long long)data * rhs.data);
		return *this;
	}
	template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr>
	modular_unfixed_base &inplace_power(T e){
		if(e < 0) *this = 1 / *this, e = -e;
		modular_unfixed_base res = 1;
		for(; e; *this *= *this, e >>= 1) if(e & 1) res *= *this;
		return *this = res;
	}
	template<class T, typename enable_if<is_integral<T>::value>::type* = nullptr>
	modular_unfixed_base power(T e) const{
		return modular_unfixed_base(*this).inplace_power(e);
	}
	modular_unfixed_base &operator/=(const modular_unfixed_base &otr){
		int a = otr.data, m = _mod, u = 0, v = 1;
		if(a < _INV.size()) return *this *= _INV[a];
		while(a){
			int t = m / a;
			m -= t * a; swap(a, m);
			u -= t * v; swap(u, v);
		}
		assert(m == 1);
		return *this *= u;
	}
	unsigned int data;
};
template<int id> unsigned int modular_unfixed_base<id>::_mod;
template<int id> unsigned long long modular_unfixed_base<id>::_inverse_mod;
template<int id> vector<modular_unfixed_base<id>> modular_unfixed_base<id>::_INV;
template<int id> modular_unfixed_base<id> modular_unfixed_base<id>::_primitive_root;
template<int id> bool operator==(const modular_unfixed_base<id> &lhs, const modular_unfixed_base<id> &rhs){ return lhs.data == rhs.data; }
template<int id, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> bool operator==(const modular_unfixed_base<id> &lhs, T rhs){ return lhs == modular_unfixed_base<id>(rhs); }
template<int id, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> bool operator==(T lhs, const modular_unfixed_base<id> &rhs){ return modular_unfixed_base<id>(lhs) == rhs; }
template<int id> bool operator!=(const modular_unfixed_base<id> &lhs, const modular_unfixed_base<id> &rhs){ return !(lhs == rhs); }
template<int id, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> bool operator!=(const modular_unfixed_base<id> &lhs, T rhs){ return !(lhs == rhs); }
template<int id, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> bool operator!=(T lhs, const modular_unfixed_base<id> &rhs){ return !(lhs == rhs); }
template<int id> bool operator<(const modular_unfixed_base<id> &lhs, const modular_unfixed_base<id> &rhs){ return lhs.data < rhs.data; }
template<int id> bool operator>(const modular_unfixed_base<id> &lhs, const modular_unfixed_base<id> &rhs){ return lhs.data > rhs.data; }
template<int id> bool operator<=(const modular_unfixed_base<id> &lhs, const modular_unfixed_base<id> &rhs){ return lhs.data <= rhs.data; }
template<int id> bool operator>=(const modular_unfixed_base<id> &lhs, const modular_unfixed_base<id> &rhs){ return lhs.data >= rhs.data; }
template<int id> modular_unfixed_base<id> operator+(const modular_unfixed_base<id> &lhs, const modular_unfixed_base<id> &rhs){ return modular_unfixed_base<id>(lhs) += rhs; }
template<int id, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_unfixed_base<id> operator+(const modular_unfixed_base<id> &lhs, T rhs){ return modular_unfixed_base<id>(lhs) += rhs; }
template<int id, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_unfixed_base<id> operator+(T lhs, const modular_unfixed_base<id> &rhs){ return modular_unfixed_base<id>(lhs) += rhs; }
template<int id> modular_unfixed_base<id> operator-(const modular_unfixed_base<id> &lhs, const modular_unfixed_base<id> &rhs){ return modular_unfixed_base<id>(lhs) -= rhs; }
template<int id, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_unfixed_base<id> operator-(const modular_unfixed_base<id> &lhs, T rhs){ return modular_unfixed_base<id>(lhs) -= rhs; }
template<int id, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_unfixed_base<id> operator-(T lhs, const modular_unfixed_base<id> &rhs){ return modular_unfixed_base<id>(lhs) -= rhs; }
template<int id> modular_unfixed_base<id> operator*(const modular_unfixed_base<id> &lhs, const modular_unfixed_base<id> &rhs){ return modular_unfixed_base<id>(lhs) *= rhs; }
template<int id, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_unfixed_base<id> operator*(const modular_unfixed_base<id> &lhs, T rhs){ return modular_unfixed_base<id>(lhs) *= rhs; }
template<int id, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_unfixed_base<id> operator*(T lhs, const modular_unfixed_base<id> &rhs){ return modular_unfixed_base<id>(lhs) *= rhs; }
template<int id> modular_unfixed_base<id> operator/(const modular_unfixed_base<id> &lhs, const modular_unfixed_base<id> &rhs) { return modular_unfixed_base<id>(lhs) /= rhs; }
template<int id, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_unfixed_base<id> operator/(const modular_unfixed_base<id> &lhs, T rhs) { return modular_unfixed_base<id>(lhs) /= rhs; }
template<int id, class T, typename enable_if<is_integral<T>::value>::type* = nullptr> modular_unfixed_base<id> operator/(T lhs, const modular_unfixed_base<id> &rhs) { return modular_unfixed_base<id>(lhs) /= rhs; }
template<int id> istream &operator>>(istream &in, modular_unfixed_base<id> &number){
	long long x;
	in >> x;
	number.data = modular_unfixed_base<id>::normalize(x);
	return in;
}
// #define _PRINT_AS_FRACTION
template<int id> ostream &operator<<(ostream &out, const modular_unfixed_base<id> &number){
#ifdef LOCAL
#ifdef _PRINT_AS_FRACTION
	out << number();
	cerr << "(";
	for(auto d = 1; ; ++ d){
		if((number * d).data <= 1000000){
			cerr << (number * d).data << "/" << d;
			break;
		}
		else if((-number * d).data <= 1000000){
			cerr << "-" << (-number * d).data << "/" << d;
			break;
		}
	}
	cerr << ")";
	return out;
#else
	return out << number();
#endif
#else
	return out << number();
#endif
}
#undef _PRINT_AS_FRACTION

using modular = modular_unfixed_base<0>;

// T must support +=, -=, *, *=, ==, and !=
template<class T>
struct matrix{
	using ring_t = T;
	using domain_t = vector<T>;
	using range_t = vector<T>;
	int n, m;
	vector<vector<T>> data;
	vector<T> &operator()(int i){
		assert(0 <= i && i < n);
		return data[i];
	}
	const vector<T> &operator()(int i) const{
		assert(0 <= i && i < n);
		return data[i];
	}
	T &operator()(int i, int j){
		assert(0 <= i && i < n && 0 <= j && j < m);
		return data[i][j];
	}
	const T &operator()(int i, int j) const{
		assert(0 <= i && i < n && 0 <= j && j < m);
		return data[i][j];
	}
	bool operator==(const matrix &a) const{
		assert(n == a.n && m == a.m);
		return data == a.data;
	}
	bool operator!=(const matrix &a) const{
		assert(n == a.n && m == a.m);
		return data != a.data;
	}
	matrix &operator+=(const matrix &a){
		assert(n == a.n && m == a.m);
		for(auto i = 0; i < n; ++ i) for(auto j = 0; j < m; ++ j) data[i][j] += a(i, j);
		return *this;
	}
	matrix operator+(const matrix &a) const{
		return matrix(*this) += a;
	}
	matrix &operator-=(const matrix &a){
		assert(n == a.n && m == a.m);
		for(auto i = 0; i < n; ++ i) for(auto j = 0; j < m; ++ j) data[i][j] -= a(i, j);
		return *this;
	}
	matrix operator-(const matrix &a) const{
		return matrix(*this) -= a;
	}
	matrix operator*=(const matrix &a){
		assert(m == a.n);
		int l = a.m;
		matrix res(n, l);
		for(auto i = 0; i < n; ++ i) for(auto j = 0; j < m; ++ j) for(auto k = 0; k < l; ++ k) res(i, k) += data[i][j] * a(j, k);
		return *this = res;
	}
	matrix operator*(const matrix &a) const{
		return matrix(*this) *= a;
	}
	matrix &operator*=(T c){
		for(auto i = 0; i < n; ++ i) for(auto j = 0; j < m; ++ j) data[i][j] *= c;
		return *this;
	}
	matrix operator*(T c) const{
		return matrix(*this) *= c;
	}
	template<class U, typename enable_if<is_integral<U>::value>::type* = nullptr>
	matrix &inplace_power(U e){
		assert(n == m && e >= 0);
		matrix res(n, n, T(1));
		for(; e; *this *= *this, e >>= 1) if(e & 1) res *= *this;
		return *this = res;
	}
	template<class U>
	matrix power(U e) const{
		return matrix(*this).inplace_power(e);
	}
	matrix transposed() const{
		matrix res(m, n);
		for(auto i = 0; i < n; ++ i) for(auto j = 0; j < m; ++ j) res(j, i) = data[i][j];
		return res;
	}
	matrix &transpose(){
		return *this = transposed();
	}
	range_t operator*(const domain_t &v) const{
		assert(m == (int)v.size());
		vector<T> res(n, T(0));
		for(auto i = 0; i < n; ++ i) for(auto j = 0; j < m; ++ j) res[i] += data[i][j] * v[j];
		return res;
	}
	// Assumes T is a field
	// find_inverse() must return optional<T>
	// O(n) find_inverse() calls along with O(n^3) operations on T
	T determinant(auto find_inverse) const{
		assert(n == m);
		if(n == 0) return T(1);
		auto a = data;
		T res = T(1);
		for(auto j = 0; j < n; ++ j){
			int pivot = -1;
			for(auto i = j; i < n; ++ i) if(a[i][j] != T(0)){
				pivot = i;
				break;
			}
			if(!~pivot) return T(0);
			swap(a[j], a[pivot]);
			res *= a[j][j] * (j != pivot ? -1 : 1);
			auto invp = find_inverse(a[j][j]);
			assert(invp);
			T inv = *invp;
			for(auto i = j + 1; i < n; ++ i) if(i != j && a[i][j] != T(0)){
				T d = a[i][j] * inv;
				for(auto jj = j; jj < n; ++ jj) a[i][jj] -= d * a[j][jj];
			}
		}
		return res;
	}
	// Assumes T is a field
	// find_inverse() must return optional<T>
	// O(n) find_inverse() calls along with O(n^3) operations on T
	optional<matrix> inverse(auto find_inverse) const{
		assert(n == m);
		if(n == 0) return *this;
		auto a = data;
		vector<vector<T>> res(n, vector<T>(n, T(0)));
		for(auto i = 0; i < n; ++ i) res[i][i] = T(1);
		for(auto j = 0; j < n; ++ j){
			int pivot = -1;
			for(auto i = j; i < n; ++ i) if(a[i][j] != T(0)){
				pivot = i;
				break;
			}
			if(!~pivot) return {};
			swap(a[j], a[pivot]), swap(res[j], res[pivot]);
			auto invp = find_inverse(a[j][j]);
			assert(invp);
			T inv = *invp;
			for(auto jj = 0; jj < n; ++ jj) a[j][jj] *= inv, res[j][jj] *= inv;
			for(auto i = 0; i < n; ++ i) if(i != j && a[i][j] != T(0)){
				T d = a[i][j];
				for(auto jj = 0; jj < n; ++ jj) a[i][jj] -= d * a[j][jj], res[i][jj] -= d * res[j][jj];
			}
		}
		return res;
	}
	template<class output_stream>
	friend output_stream &operator<<(output_stream &out, const matrix &a){
		out << "\n";
		for(auto i = 0; i < a.n; ++ i){
			for(auto j = 0; j < a.m; ++ j){
				out << a(i, j) << " ";
			}
			out << "\n";
		}
		return out;
	}
	matrix(int n, int m, T init_diagonal = T(0), T init_off_diagonal = T(0)): n(n), m(m){
		data.assign(n, vector<T>(m, T(0)));
		for(auto i = 0; i < n; ++ i) for(auto j = 0; j < m; ++ j) data[i][j] = i == j ? init_diagonal : init_off_diagonal;
	}
	matrix(const vector<vector<T>> &arr, int _n = -1, int _m = -1): n(~_n ? _n : (int)arr.size()), m(~_m ? _m : (int)arr[0].size()), data(arr){ }
};
template<class T>
matrix<T> operator*(T c, matrix<T> a){
	for(auto i = 0; i < a.n; ++ i) for(auto j = 0; j < a.m; ++ j) a(i, j) = c * a(i, j);
	return a;
}
// Multiply a row vector v on the left
template<class T>
typename matrix<T>::domain_t operator*(const typename matrix<T>::range_t &v, const matrix<T> &a){
	assert(a.n == (int)size(v));
	typename matrix<T>::domain_t res(a.m, T(0));
	for(auto i = 0; i < a.n; ++ i) for(auto j = 0; j < a.m; ++ j) res[j] += v[i] * a(i, j);
	return res;
}

// T must be integral or modular type
// O(n^3 + n^2 * log(MAXVAL))
template<class T>
T determinant_integral(vector<vector<T>> M){
	if(M.empty()) return 1;
	int n = (int)M.size();
	assert((int)M[0].size() == n);
	T res = 1;
	for(auto i = 0; i < n; ++ i){
		for(auto j = i + 1; j < n; ++ j) while(M[j][i]){
			T t;
			if constexpr(is_integral<T>::value) t = M[i][i] / M[j][i];
			else t = M[i][i].data / M[j][i].data;
			if(t) for(auto k = i; k < n; ++ k) M[i][k] -= M[j][k] * t;
			swap(M[i], M[j]);
			res *= -1;
		}
		res *= M[i][i];
		if(!res) return 0;
	}
	return res;
}

int main(){
	cin.tie(0)->sync_with_stdio(0);
	cin.exceptions(ios::badbit | ios::failbit);
	int n, mod;
	cin >> n >> mod, mod <<= 1;
	modular::setup(mod);
	matrix<modular> a(n, n);
	vector<vector<modular>> mat(n, vector<modular>(n));
	for(auto i = 0; i < n; ++ i){
		for(auto j = 0; j < n; ++ j){
			cin >> a(i, j);
			mat[i][j] = a(i, j);
		}
	}
	auto det = determinant_integral(mat);
	modular perm = 0;
	for(auto mask = 0; mask < 1 << n; ++ mask){
		modular cur = 1;
		for(auto i = 0; i < n; ++ i){
			modular sum = 0;
			for(auto j = 0; j < n; ++ j){
				if(mask & 1 << j){
					sum += a(i, j);
				}
			}
			cur *= sum;
		}
		if(__builtin_popcount(mask) & 1){
			cur = -cur;
		}
		perm += cur;
	}
	if(n & 1){
		perm = -perm;
	}
	cout << (perm - det).data / 2 << "\n";
	return 0;
}

/*

*/

////////////////////////////////////////////////////////////////////////////////////////
//                                                                                    //
//                                   Coded by Aeren                                   //
//                                                                                    //
////////////////////////////////////////////////////////////////////////////////////////
yukicoder

結果

ソースコード
