#907697 (C++23) No.2443 特殊線形群の標準表現

提出ソース
結果

問題	No.2443 特殊線形群の標準表現
コンテスト
ユーザー	Aeren
提出日時	2023-08-25 21:42:24
言語	C++23 (gcc 13.3.0 + boost 1.87.0)
結果	AC
実行時間	217 ms / 3,000 ms
コード長	19,755 bytes
コンパイル時間	4,716 ms
コンパイル使用メモリ	367,088 KB
実行使用メモリ	8,936 KB
最終ジャッジ日時	2024-12-24 08:18:38
合計ジャッジ時間	7,439 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 * 21
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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, size_t N, size_t M>
struct matrix_fixed{
	using ring_t = T;
	using domain_t = array<T, M>;
	using range_t = array<T, N>;
	static constexpr int n = N, m = M;
	array<array<T, M>, N> data;
	array<T, M> &operator()(int i){
		assert(0 <= i && i < n);
		return data[i];
	}
	const array<T, M> &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_fixed &a) const{
		assert(n == a.n && m == a.m);
		return data == a.data;
	}
	bool operator!=(const matrix_fixed &a) const{
		assert(n == a.n && m == a.m);
		return data != a.data;
	}
	matrix_fixed &operator+=(const matrix_fixed &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_fixed operator+(const matrix_fixed &a) const{
		assert(n == a.n && m == a.m);
		return matrix_fixed(*this) += a;
	}
	matrix_fixed &operator-=(const matrix_fixed &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_fixed operator-(const matrix_fixed &a) const{
		assert(n == a.n && m == a.m);
		return matrix_fixed(*this) += a;
	}
	template<size_t N2, size_t M2>
	matrix_fixed<T, N, M2> operator*(const matrix_fixed<T, N2, M2> &a) const{
		assert(m == a.n);
		int l = M2;
		matrix_fixed<T, N, M2> res;
		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 res;
	}
	template<size_t N2, size_t M2>
	matrix_fixed &operator*=(const matrix_fixed<T, N2, M2> &a){
		return *this = *this * a;
	}
	matrix_fixed &operator*=(T c){
		for(auto i = 0; i < n; ++ i) for(auto j = 0; j < m; ++ j) data[i][j] *= c;
		return *this;
	}
	matrix_fixed operator*(T c) const{
		return matrix_fixed(*this) *= c;
	}
	template<class U, typename enable_if<is_integral<U>::value>::type* = nullptr>
	matrix_fixed &inplace_power(U e){
		assert(n == m && e >= 0);
		matrix_fixed res(1, 0);
		for(; e; *this *= *this, e >>= 1) if(e & 1) res *= *this;
		return *this = res;
	}
	template<class U>
	matrix_fixed power(U e) const{
		return matrix_fixed(*this).inplace_power(e);
	}
	matrix_fixed<T, M, N> transposed() const{
		matrix_fixed<T, M, N> res;
		for(auto i = 0; i < n; ++ i) for(auto j = 0; j < m; ++ j) res(j, i) = data[i][j];
		return res;
	}
	matrix_fixed &transpose(){
		return *this = transposed();
	}
	// Multiply a column vector v on the right
	range_t operator*(const domain_t &v) const{
		range_t res;
		res.fill(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_fixed> inverse(auto find_inverse) const{
		assert(n == m);
		if(n == 0) return *this;
		auto a = data;
		array<array<T, M>, N> res;
		for(auto i = 0; i < n; ++ i) res[i].fill(T(0)), 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 matrix_fixed(n, n, res);
	}
	template<class output_stream>
	friend output_stream &operator<<(output_stream &out, const matrix_fixed &a){
		out << "{";
		for(auto i = 0; i < a.n; ++ i){
			out << "{";
			for(auto j = 0; j < a.m; ++ j){
				out << a(i, j);
				if(j != a.m - 1) out << ", ";
			}
			out << "}";
			if(i != a.n - 1) out << ", ";
		}
		return out << "}";
	}
	matrix_fixed(): matrix_fixed(T(0), T(0)){ }
	matrix_fixed(const T &init_diagonal, const T &init_off_diagonal){
		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_fixed(const array<array<T, M>, N> &arr): data(arr){ }
	static matrix_fixed additive_identity(){
		return matrix_fixed(T(1), T(0));
	}
	static matrix_fixed multiplicative_identity(){
		return matrix_fixed(T(0), T(0));
	}
};
template<class T, size_t N, size_t M>
matrix_fixed<T, N, M> operator*(T c, matrix_fixed<T, N, M> 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, size_t N, size_t M>
matrix_fixed<T, N, M>::domain_t operator*(const typename matrix_fixed<T, N, M>::range_t &v, const matrix_fixed<T, N, M> &a){
	typename matrix_fixed<T, N, M>::domain_t res;
	res.fill(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;
}

template<class T, class F>
struct segment_tree{
	int n, size, log;
	vector<T> data;
	F TT; // monoid operation (always adjacent)
	T T_id; // monoid identity
	// O(n)
	segment_tree(int n, F TT, T T_id): TT(TT), T_id(T_id){
		init(n);
	}
	// O(n)
	segment_tree(int n, T x, F TT, T T_id): TT(TT), T_id(T_id){
		init(n, x);
	}
	// O(n)
	segment_tree(const vector<T> &a, F TT, T T_id): TT(TT), T_id(T_id){
		init(a);
	}
	segment_tree &operator=(const segment_tree &otr){
		n = otr.n;
		size = otr.size;
		log = otr.log;
		data = otr.data;
		return *this;
	}
	// O(n)
	void init(int n){
		init(n, T_id);
	}
	// O(n)
	void init(int n, T x){
		this->n = n;
		size = 1;
		while(size < n) size <<= 1;
		log = __lg(size);
		data.assign(size << 1, T_id);
		fill(data.begin() + size, data.begin() + size + n, x);
		for(auto i = size - 1; i >= 1; -- i) refresh(i);
	}
	// O(n)
	void init(const vector<T> &a){
		n = (int)a.size();
		size = 1;
		while(size < n) size <<= 1;
		log = __lg(size);
		data.assign(size << 1, T_id);
		copy(a.begin(), a.end(), data.begin() + size);
		for(auto i = size - 1; i >= 1; -- i) refresh(i);
	}
	// O(1)
	void refresh(int i){
		data[i] = TT(data[i << 1], data[i << 1 | 1]);
	}
	// O(log(n))
	void set(int p, T x){
		assert(0 <= p && p < n);
		data[p += size] = x;
		for(auto i = 1; i <= log; ++ i) refresh(p >> i);
	}
	// O(log(n))
	void update(int p, T x){
		assert(0 <= p && p < n);
		p += size;
		data[p] = TT(data[p], x);
		for(auto i = 1; i <= log; ++ i) refresh(p >> i);
	}
	// O(1)
	T query(int p) const{
		assert(0 <= p && p < n);
		return data[p + size];
	}
	// O(log(n))
	T query(int l, int r) const{
		assert(0 <= l && l <= r && r <= n);
		T res_left = T_id, res_right = T_id;
		for(l += size, r += size; l < r; l >>= 1, r >>= 1){
			if(l & 1) res_left = TT(res_left, data[l ++]);
			if(r & 1) res_right = TT(data[-- r], res_right);
		}
		return TT(res_left, res_right);
	}
	// O(1)
	T query_all() const{
		return data[1];
	}
	// pred(sum[l, r)) is T, T, ..., T, F, F, ..., F
	// Returns max r with T
	// O(log(n))
	int max_pref(int l, auto pred) const{
		assert(0 <= l && l <= n && pred(T_id));
		if(l == n) return n;
		l += size;
		T sm = T_id;
		do{
			while(~l & 1) l >>= 1;
			if(!pred(TT(sm, data[l]))){
				while(l < size){
					l = l << 1;
					if(pred(TT(sm, data[l]))) sm = TT(sm, data[l ++]);
				}
				return l - size;
			}
			sm = TT(sm, data[l ++]);
		}while((l & -l) != l);
		return n;
	}
	// pred(sum[l, r)) is F, F, ..., F, T, T, ..., T
	// Returns min l with T
	// O(log(n))
	int min_suff(int r, auto pred) const{
		assert(0 <= r && r <= n && pred(T_id));
		if(r == 0) return 0;
		r += size;
		T sm = T_id;
		do{
			-- r;
			while(r > 1 && r & 1) r >>= 1;
			if(!pred(TT(data[r], sm))){
				while(r < size){
					r = r << 1 | 1;
					if(pred(TT(data[r], sm))) sm = TT(data[r --], sm);
				}
				return r + 1 - size;
			}
			sm = TT(data[r], sm);
		}while((r & -r) != r);
		return 0;
	}
	template<class output_stream>
	friend output_stream &operator<<(output_stream &out, const segment_tree<T, F> &seg){
		out << "[";
		for(auto i = 0; i < seg.n; ++ i){
			out << seg.query(i);
			if(i != seg.n - 1) out << ", ";
		}
		return out << ']';
	}
};

int main(){
	cin.tie(0)->sync_with_stdio(0);
	cin.exceptions(ios::badbit | ios::failbit);
	int n, qn;
	cin >> n;
	modular::setup();
	cin >> qn;
	vector<matrix_fixed<modular, 2, 2>> init(n);
	for(auto i = 0; i < n; ++ i){
		for(auto x = 0; x < 2; ++ x){
			for(auto y = 0; y < 2; ++ y){
				cin >> init[i](x, y);
			}
		}
	}
	segment_tree seg(init, [&](const auto &x, const auto &y){ return y * x; }, matrix_fixed<modular, 2, 2>(1, 0));
	for(auto qi = 0; qi < qn; ++ qi){
		int ql, qr;
		modular x, y;
		cin >> ql >> qr >> x >> y;
		auto res = seg.query(ql, qr) * array{x, y};
		cout << res[0] << " " << res[1] << "\n";
	}
	return 0;
}

/*

*/

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

結果

ソースコード
