#define PROBLEM "https://judge.yosupo.jp/problem/aplusb" // #include "../../template/template.hpp" // #include "../../matrix/matrix.hpp" #include "../../matrix/gauss-elimination.hpp" // namespace Normal { #include "../../matrix/linear-equation.hpp" } // namespace Normal namespace Fast { #include "../../modulo/gauss-elimination-fast.hpp" } // namespace Fast #include "../../misc/rng.hpp" #include "../../modint/montgomery-modint.hpp" using namespace Nyaan; using mint = LazyMontgomeryModInt<998244353>; void test(int n, int m, bool sparse) { Matrix<mint> a(n, m); each(v, a.A) { each(x, v) { if (sparse) { if (randint(0, m) == 0) x = rng(); } else { x = rng(); } } } // trc(n, m, sparse); // trc(a); if (n == m) { mint det1, det2, det3; { auto b = a; det1 = b.determinant(); } { auto b = a; det2 = GaussElimination<mint>(b.A).second; } { auto b = a; det3 = Fast::determinant(b.A); } // trc(det1, det2, det3); assert(det1 == det2 and det2 == det3); } vector<mint> c(n); if (rng() & 1) each(x, c) x = rng(); vector<vector<mint>> sol1, sol2; { auto b = a; sol1 = Normal::LinearEquation<mint>(b.A, c); // trc(sol1); } { auto b = a; sol2 = Fast::LinearEquation<mint>(b.A, c); // trc(sol2); } assert(sol1 == sol2); } void Nyaan::solve() { rep(i, TEN(5)) test(randint(1, 17), randint(1, 17), rng() % 2 == 1); int a, b; cin >> a >> b; cout << a + b << endl; }
#line 1 "verify/verify-unit-test/gauss-elimination.test.cpp" #define PROBLEM "https://judge.yosupo.jp/problem/aplusb" // #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 1 "template/util.hpp" namespace Nyaan { using ll = long long; using i64 = long long; using u64 = unsigned long long; using i128 = __int128_t; using u128 = __uint128_t; template <typename T> using V = vector<T>; template <typename T> using VV = vector<vector<T>>; using vi = vector<int>; using vl = vector<long long>; using vd = V<double>; using vs = V<string>; using vvi = vector<vector<int>>; using vvl = vector<vector<long long>>; template <typename T> using minpq = priority_queue<T, vector<T>, greater<T>>; template <typename T, typename U> struct P : pair<T, U> { template <typename... Args> P(Args... args) : pair<T, U>(args...) {} using pair<T, U>::first; using pair<T, U>::second; P &operator+=(const P &r) { first += r.first; second += r.second; return *this; } P &operator-=(const P &r) { first -= r.first; second -= r.second; return *this; } P &operator*=(const P &r) { first *= r.first; second *= r.second; return *this; } template <typename S> P &operator*=(const S &r) { first *= r, second *= r; return *this; } P operator+(const P &r) const { return P(*this) += r; } P operator-(const P &r) const { return P(*this) -= r; } P operator*(const P &r) const { return P(*this) *= r; } template <typename S> P operator*(const S &r) const { return P(*this) *= r; } P operator-() const { return P{-first, -second}; } }; using pl = P<ll, ll>; using pi = P<int, int>; using vp = V<pl>; constexpr int inf = 1001001001; constexpr long long infLL = 4004004004004004004LL; template <typename T> int sz(const T &t) { return t.size(); } template <typename T, typename U> inline bool amin(T &x, U y) { return (y < x) ? (x = y, true) : false; } template <typename T, typename U> inline bool amax(T &x, U y) { return (x < y) ? (x = y, true) : false; } template <typename T> inline T Max(const vector<T> &v) { return *max_element(begin(v), end(v)); } template <typename T> inline T Min(const vector<T> &v) { return *min_element(begin(v), end(v)); } template <typename T> inline long long Sum(const vector<T> &v) { return accumulate(begin(v), end(v), 0LL); } template <typename T> int lb(const vector<T> &v, const T &a) { return lower_bound(begin(v), end(v), a) - begin(v); } template <typename T> int ub(const vector<T> &v, const T &a) { return upper_bound(begin(v), end(v), a) - begin(v); } constexpr long long TEN(int n) { long long ret = 1, x = 10; for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1); return ret; } template <typename T, typename U> pair<T, U> mkp(const T &t, const U &u) { return make_pair(t, u); } template <typename T> vector<T> mkrui(const vector<T> &v, bool rev = false) { vector<T> ret(v.size() + 1); if (rev) { for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1]; } else { for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i]; } return ret; }; template <typename T> vector<T> mkuni(const vector<T> &v) { vector<T> ret(v); sort(ret.begin(), ret.end()); ret.erase(unique(ret.begin(), ret.end()), ret.end()); return ret; } template <typename F> vector<int> mkord(int N, F f) { vector<int> ord(N); iota(begin(ord), end(ord), 0); sort(begin(ord), end(ord), f); return ord; } template <typename T> vector<int> mkinv(vector<T> &v) { int max_val = *max_element(begin(v), end(v)); vector<int> inv(max_val + 1, -1); for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i; return inv; } vector<int> mkiota(int n) { vector<int> ret(n); iota(begin(ret), end(ret), 0); return ret; } template <typename T> T mkrev(const T &v) { T w{v}; reverse(begin(w), end(w)); return w; } template <typename T> bool nxp(vector<T> &v) { return next_permutation(begin(v), end(v)); } // 返り値の型は入力の T に依存 // i 要素目 : [0, a[i]) template <typename T> vector<vector<T>> product(const vector<T> &a) { vector<vector<T>> ret; vector<T> v; auto dfs = [&](auto rc, int i) -> void { if (i == (int)a.size()) { ret.push_back(v); return; } for (int j = 0; j < a[i]; j++) v.push_back(j), rc(rc, i + 1), v.pop_back(); }; dfs(dfs, 0); return ret; } // F : function(void(T&)), mod を取る操作 // T : 整数型のときはオーバーフローに注意する template <typename T> T Power(T a, long long n, const T &I, const function<void(T &)> &f) { T res = I; for (; n; f(a = a * a), n >>= 1) { if (n & 1) f(res = res * a); } return res; } // T : 整数型のときはオーバーフローに注意する template <typename T> T Power(T a, long long n, const T &I) { return Power(a, n, I, function<void(T &)>{[](T &) -> void {}}); } } // namespace Nyaan #line 58 "template/template.hpp" // bit operation #line 1 "template/bitop.hpp" namespace Nyaan { __attribute__((target("popcnt"))) inline int popcnt(const u64 &a) { return _mm_popcnt_u64(a); } inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; } inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; } inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; } template <typename T> inline int gbit(const T &a, int i) { return (a >> i) & 1; } template <typename T> inline void sbit(T &a, int i, bool b) { if (gbit(a, i) != b) a ^= T(1) << i; } constexpr long long PW(int n) { return 1LL << n; } constexpr long long MSK(int n) { return (1LL << n) - 1; } } // namespace Nyaan #line 61 "template/template.hpp" // inout #line 1 "template/inout.hpp" namespace Nyaan { template <typename T, typename U> ostream &operator<<(ostream &os, const pair<T, U> &p) { os << p.first << " " << p.second; return os; } template <typename T, typename U> istream &operator>>(istream &is, pair<T, U> &p) { is >> p.first >> p.second; return is; } template <typename T> ostream &operator<<(ostream &os, const vector<T> &v) { int s = (int)v.size(); for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i]; return os; } template <typename T> istream &operator>>(istream &is, vector<T> &v) { for (auto &x : v) is >> x; return is; } istream &operator>>(istream &is, __int128_t &x) { string S; is >> S; x = 0; int flag = 0; for (auto &c : S) { if (c == '-') { flag = true; continue; } x *= 10; x += c - '0'; } if (flag) x = -x; return is; } istream &operator>>(istream &is, __uint128_t &x) { string S; is >> S; x = 0; for (auto &c : S) { x *= 10; x += c - '0'; } return is; } ostream &operator<<(ostream &os, __int128_t x) { if (x == 0) return os << 0; if (x < 0) os << '-', x = -x; string S; while (x) S.push_back('0' + x % 10), x /= 10; reverse(begin(S), end(S)); return os << S; } ostream &operator<<(ostream &os, __uint128_t x) { if (x == 0) return os << 0; string S; while (x) S.push_back('0' + x % 10), x /= 10; reverse(begin(S), end(S)); return os << S; } void in() {} template <typename T, class... U> void in(T &t, U &...u) { cin >> t; in(u...); } void out() { cout << "\n"; } template <typename T, class... U, char sep = ' '> void out(const T &t, const U &...u) { cout << t; if (sizeof...(u)) cout << sep; out(u...); } struct IoSetupNya { IoSetupNya() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(15); cerr << fixed << setprecision(7); } } iosetupnya; } // namespace Nyaan #line 64 "template/template.hpp" // debug #line 1 "template/debug.hpp" namespace DebugImpl { template <typename U, typename = void> struct is_specialize : false_type {}; template <typename U> struct is_specialize< U, typename conditional<false, typename U::iterator, void>::type> : true_type {}; template <typename U> struct is_specialize< U, typename conditional<false, decltype(U::first), void>::type> : true_type {}; template <typename U> struct is_specialize<U, enable_if_t<is_integral<U>::value, void>> : true_type { }; void dump(const char& t) { cerr << t; } void dump(const string& t) { cerr << t; } void dump(const bool& t) { cerr << (t ? "true" : "false"); } void dump(__int128_t t) { if (t == 0) cerr << 0; if (t < 0) cerr << '-', t = -t; string S; while (t) S.push_back('0' + t % 10), t /= 10; reverse(begin(S), end(S)); cerr << S; } void dump(__uint128_t t) { if (t == 0) cerr << 0; string S; while (t) S.push_back('0' + t % 10), t /= 10; reverse(begin(S), end(S)); cerr << S; } template <typename U, enable_if_t<!is_specialize<U>::value, nullptr_t> = nullptr> void dump(const U& t) { cerr << t; } template <typename T> void dump(const T& t, enable_if_t<is_integral<T>::value>* = nullptr) { string res; if (t == Nyaan::inf) res = "inf"; if constexpr (is_signed<T>::value) { if (t == -Nyaan::inf) res = "-inf"; } if constexpr (sizeof(T) == 8) { if (t == Nyaan::infLL) res = "inf"; if constexpr (is_signed<T>::value) { if (t == -Nyaan::infLL) res = "-inf"; } } if (res.empty()) res = to_string(t); cerr << res; } template <typename T, typename U> void dump(const pair<T, U>&); template <typename T> void dump(const pair<T*, int>&); template <typename T> void dump(const T& t, enable_if_t<!is_void<typename T::iterator>::value>* = nullptr) { cerr << "[ "; for (auto it = t.begin(); it != t.end();) { dump(*it); cerr << (++it == t.end() ? "" : ", "); } cerr << " ]"; } template <typename T, typename U> void dump(const pair<T, U>& t) { cerr << "( "; dump(t.first); cerr << ", "; dump(t.second); cerr << " )"; } template <typename T> void dump(const pair<T*, int>& t) { cerr << "[ "; for (int i = 0; i < t.second; i++) { dump(t.first[i]); cerr << (i == t.second - 1 ? "" : ", "); } cerr << " ]"; } void trace() { cerr << endl; } template <typename Head, typename... Tail> void trace(Head&& head, Tail&&... tail) { cerr << " "; dump(head); if (sizeof...(tail) != 0) cerr << ","; trace(forward<Tail>(tail)...); } } // namespace DebugImpl #ifdef NyaanDebug #define trc(...) \ do { \ cerr << "## " << #__VA_ARGS__ << " = "; \ DebugImpl::trace(__VA_ARGS__); \ } while (0) #else #define trc(...) (void(0)) #endif #ifdef NyaanLocal #define trc2(...) \ do { \ cerr << "## " << #__VA_ARGS__ << " = "; \ DebugImpl::trace(__VA_ARGS__); \ } while (0) #else #define trc2(...) (void(0)) #endif #line 67 "template/template.hpp" // macro #line 1 "template/macro.hpp" #define each(x, v) for (auto&& x : v) #define each2(x, y, v) for (auto&& [x, y] : v) #define all(v) (v).begin(), (v).end() #define rep(i, N) for (long long i = 0; i < (long long)(N); i++) #define repr(i, N) for (long long i = (long long)(N)-1; i >= 0; i--) #define rep1(i, N) for (long long i = 1; i <= (long long)(N); i++) #define repr1(i, N) for (long long i = (N); (long long)(i) > 0; i--) #define reg(i, a, b) for (long long i = (a); i < (b); i++) #define regr(i, a, b) for (long long i = (b)-1; i >= (a); i--) #define fi first #define se second #define ini(...) \ int __VA_ARGS__; \ in(__VA_ARGS__) #define inl(...) \ long long __VA_ARGS__; \ in(__VA_ARGS__) #define ins(...) \ string __VA_ARGS__; \ in(__VA_ARGS__) #define in2(s, t) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i]); \ } #define in3(s, t, u) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i], u[i]); \ } #define in4(s, t, u, v) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i], u[i], v[i]); \ } #define die(...) \ do { \ Nyaan::out(__VA_ARGS__); \ return; \ } while (0) #line 70 "template/template.hpp" namespace Nyaan { void solve(); } int main() { Nyaan::solve(); } #line 4 "verify/verify-unit-test/gauss-elimination.test.cpp" // #line 2 "matrix/matrix.hpp" #line 2 "matrix/inverse-matrix.hpp" #line 2 "matrix/gauss-elimination.hpp" #line 5 "matrix/gauss-elimination.hpp" using namespace std; // {rank, det(非正方行列の場合は未定義)} を返す // 型が double や Rational でも動くはず?(未検証) // // pivot 候補 : [0, pivot_end) template <typename T> std::pair<int, T> GaussElimination(vector<vector<T>> &a, int pivot_end = -1, bool diagonalize = false) { int H = a.size(), W = a[0].size(), rank = 0; if (pivot_end == -1) pivot_end = W; T det = 1; for (int j = 0; j < pivot_end; j++) { int idx = -1; for (int i = rank; i < H; i++) { if (a[i][j] != T(0)) { idx = i; break; } } if (idx == -1) { det = 0; continue; } if (rank != idx) det = -det, swap(a[rank], a[idx]); det *= a[rank][j]; if (diagonalize && a[rank][j] != T(1)) { T coeff = T(1) / a[rank][j]; for (int k = j; k < W; k++) a[rank][k] *= coeff; } int is = diagonalize ? 0 : rank + 1; for (int i = is; i < H; i++) { if (i == rank) continue; if (a[i][j] != T(0)) { T coeff = a[i][j] / a[rank][j]; for (int k = j; k < W; k++) a[i][k] -= a[rank][k] * coeff; } } rank++; } return make_pair(rank, det); } #line 4 "matrix/inverse-matrix.hpp" template <typename mint> vector<vector<mint>> inverse_matrix(const vector<vector<mint>>& a) { int N = a.size(); assert(N > 0); assert(N == (int)a[0].size()); vector<vector<mint>> m(N, vector<mint>(2 * N)); for (int i = 0; i < N; i++) { copy(begin(a[i]), end(a[i]), begin(m[i])); m[i][N + i] = 1; } auto [rank, det] = GaussElimination(m, N, true); if (rank != N) return {}; vector<vector<mint>> b(N); for (int i = 0; i < N; i++) { copy(begin(m[i]) + N, end(m[i]), back_inserter(b[i])); } return b; } #line 4 "matrix/matrix.hpp" template <class T> struct Matrix { vector<vector<T> > A; Matrix() = default; Matrix(int n, int m) : A(n, vector<T>(m, T())) {} Matrix(int n) : A(n, vector<T>(n, T())){}; int H() const { return A.size(); } int W() const { return A[0].size(); } int size() const { return A.size(); } inline const vector<T> &operator[](int k) const { return A[k]; } inline vector<T> &operator[](int k) { return A[k]; } static Matrix I(int n) { Matrix mat(n); for (int i = 0; i < n; i++) mat[i][i] = 1; return (mat); } Matrix &operator+=(const Matrix &B) { int n = H(), m = W(); assert(n == B.H() && m == B.W()); for (int i = 0; i < n; i++) for (int j = 0; j < m; j++) (*this)[i][j] += B[i][j]; return (*this); } Matrix &operator-=(const Matrix &B) { int n = H(), m = W(); assert(n == B.H() && m == B.W()); for (int i = 0; i < n; i++) for (int j = 0; j < m; j++) (*this)[i][j] -= B[i][j]; return (*this); } Matrix &operator*=(const Matrix &B) { int n = H(), m = B.W(), p = W(); assert(p == B.H()); vector<vector<T> > C(n, vector<T>(m, T{})); for (int i = 0; i < n; i++) for (int k = 0; k < p; k++) for (int j = 0; j < m; j++) C[i][j] += (*this)[i][k] * B[k][j]; A.swap(C); return (*this); } Matrix &operator^=(long long k) { Matrix B = Matrix::I(H()); while (k > 0) { if (k & 1) B *= *this; *this *= *this; k >>= 1LL; } A.swap(B.A); return (*this); } Matrix operator+(const Matrix &B) const { return (Matrix(*this) += B); } Matrix operator-(const Matrix &B) const { return (Matrix(*this) -= B); } Matrix operator*(const Matrix &B) const { return (Matrix(*this) *= B); } Matrix operator^(const long long k) const { return (Matrix(*this) ^= k); } bool operator==(const Matrix &B) const { assert(H() == B.H() && W() == B.W()); for (int i = 0; i < H(); i++) for (int j = 0; j < W(); j++) if (A[i][j] != B[i][j]) return false; return true; } bool operator!=(const Matrix &B) const { assert(H() == B.H() && W() == B.W()); for (int i = 0; i < H(); i++) for (int j = 0; j < W(); j++) if (A[i][j] != B[i][j]) return true; return false; } Matrix inverse() const { assert(H() == W()); Matrix B(H()); B.A = inverse_matrix(A); return B; } friend ostream &operator<<(ostream &os, const Matrix &p) { int n = p.H(), m = p.W(); for (int i = 0; i < n; i++) { os << (i ? " " : "") << "["; for (int j = 0; j < m; j++) { os << p[i][j] << (j + 1 == m ? "]\n" : ","); } } return (os); } T determinant() const { Matrix B(*this); assert(H() == W()); T ret = 1; for (int i = 0; i < H(); i++) { int idx = -1; for (int j = i; j < W(); j++) { if (B[j][i] != 0) { idx = j; break; } } if (idx == -1) return 0; if (i != idx) { ret *= T(-1); swap(B[i], B[idx]); } ret *= B[i][i]; T inv = T(1) / B[i][i]; for (int j = 0; j < W(); j++) { B[i][j] *= inv; } for (int j = i + 1; j < H(); j++) { T a = B[j][i]; if (a == 0) continue; for (int k = i; k < W(); k++) { B[j][k] -= B[i][k] * a; } } } return ret; } }; /** * @brief 行列ライブラリ */ #line 7 "verify/verify-unit-test/gauss-elimination.test.cpp" // namespace Normal { #line 2 "matrix/linear-equation.hpp" #line 4 "matrix/linear-equation.hpp" // 解が存在する場合は, 解が v + C_1 w_1 + ... + C_k w_k と表せるとして // (v, w_1, ..., w_k) を返す // 解が存在しない場合は空のベクトルを返す // // double や Rational でも動くはず?(未検証) template <typename T> vector<vector<T>> LinearEquation(vector<vector<T>> a, vector<T> b) { int H = a.size(), W = a[0].size(); for (int i = 0; i < H; i++) a[i].push_back(b[i]); auto p = GaussElimination(a, W, true); int rank = p.first; for (int i = rank; i < H; ++i) { if (a[i][W] != 0) return vector<vector<T>>{}; } vector<vector<T>> res(1, vector<T>(W)); vector<int> pivot(W, -1); for (int i = 0, j = 0; i < rank; ++i) { while (a[i][j] == 0) ++j; res[0][j] = a[i][W], pivot[j] = i; } for (int j = 0; j < W; ++j) { if (pivot[j] == -1) { vector<T> x(W); x[j] = 1; for (int k = 0; k < j; ++k) { if (pivot[k] != -1) x[k] = -a[pivot[k]][j]; } res.push_back(x); } } return res; } #line 10 "verify/verify-unit-test/gauss-elimination.test.cpp" } // namespace Normal namespace Fast { #line 2 "modulo/gauss-elimination-fast.hpp" #line 2 "modint/simd-montgomery.hpp" #line 4 "modint/simd-montgomery.hpp" __attribute__((target("sse4.2"))) inline __m128i my128_mullo_epu32( const __m128i &a, const __m128i &b) { return _mm_mullo_epi32(a, b); } __attribute__((target("sse4.2"))) inline __m128i my128_mulhi_epu32( const __m128i &a, const __m128i &b) { __m128i a13 = _mm_shuffle_epi32(a, 0xF5); __m128i b13 = _mm_shuffle_epi32(b, 0xF5); __m128i prod02 = _mm_mul_epu32(a, b); __m128i prod13 = _mm_mul_epu32(a13, b13); __m128i prod = _mm_unpackhi_epi64(_mm_unpacklo_epi32(prod02, prod13), _mm_unpackhi_epi32(prod02, prod13)); return prod; } __attribute__((target("sse4.2"))) inline __m128i montgomery_mul_128( const __m128i &a, const __m128i &b, const __m128i &r, const __m128i &m1) { return _mm_sub_epi32( _mm_add_epi32(my128_mulhi_epu32(a, b), m1), my128_mulhi_epu32(my128_mullo_epu32(my128_mullo_epu32(a, b), r), m1)); } __attribute__((target("sse4.2"))) inline __m128i montgomery_add_128( const __m128i &a, const __m128i &b, const __m128i &m2, const __m128i &m0) { __m128i ret = _mm_sub_epi32(_mm_add_epi32(a, b), m2); return _mm_add_epi32(_mm_and_si128(_mm_cmpgt_epi32(m0, ret), m2), ret); } __attribute__((target("sse4.2"))) inline __m128i montgomery_sub_128( const __m128i &a, const __m128i &b, const __m128i &m2, const __m128i &m0) { __m128i ret = _mm_sub_epi32(a, b); return _mm_add_epi32(_mm_and_si128(_mm_cmpgt_epi32(m0, ret), m2), ret); } __attribute__((target("avx2"))) inline __m256i my256_mullo_epu32( const __m256i &a, const __m256i &b) { return _mm256_mullo_epi32(a, b); } __attribute__((target("avx2"))) inline __m256i my256_mulhi_epu32( const __m256i &a, const __m256i &b) { __m256i a13 = _mm256_shuffle_epi32(a, 0xF5); __m256i b13 = _mm256_shuffle_epi32(b, 0xF5); __m256i prod02 = _mm256_mul_epu32(a, b); __m256i prod13 = _mm256_mul_epu32(a13, b13); __m256i prod = _mm256_unpackhi_epi64(_mm256_unpacklo_epi32(prod02, prod13), _mm256_unpackhi_epi32(prod02, prod13)); return prod; } __attribute__((target("avx2"))) inline __m256i montgomery_mul_256( const __m256i &a, const __m256i &b, const __m256i &r, const __m256i &m1) { return _mm256_sub_epi32( _mm256_add_epi32(my256_mulhi_epu32(a, b), m1), my256_mulhi_epu32(my256_mullo_epu32(my256_mullo_epu32(a, b), r), m1)); } __attribute__((target("avx2"))) inline __m256i montgomery_add_256( const __m256i &a, const __m256i &b, const __m256i &m2, const __m256i &m0) { __m256i ret = _mm256_sub_epi32(_mm256_add_epi32(a, b), m2); return _mm256_add_epi32(_mm256_and_si256(_mm256_cmpgt_epi32(m0, ret), m2), ret); } __attribute__((target("avx2"))) inline __m256i montgomery_sub_256( const __m256i &a, const __m256i &b, const __m256i &m2, const __m256i &m0) { __m256i ret = _mm256_sub_epi32(a, b); return _mm256_add_epi32(_mm256_and_si256(_mm256_cmpgt_epi32(m0, ret), m2), ret); } #line 4 "modulo/gauss-elimination-fast.hpp" namespace Gauss { constexpr int MAT_SIZE = 4096; uint32_t a_buf_[MAT_SIZE][MAT_SIZE] __attribute__((aligned(64))); // return value: (rank, (-1) ^ (number of swap time)) template <typename mint> __attribute__((target("avx2"))) pair<int, mint> GaussianElimination( const vector<vector<mint>> &m, int LinearEquation = false) { mint(&a)[MAT_SIZE][MAT_SIZE] = *reinterpret_cast<mint(*)[MAT_SIZE][MAT_SIZE]>(a_buf_); int H = m.size(), W = m[0].size(), rank = 0; mint det = 1; for (int i = 0; i < H; i++) for (int j = 0; j < W; j++) a[i][j].a = m[i][j].a; __m256i r = _mm256_set1_epi32(mint::r); __m256i m0 = _mm256_set1_epi32(0); __m256i m1 = _mm256_set1_epi32(mint::get_mod()); __m256i m2 = _mm256_set1_epi32(mint::get_mod() << 1); for (int j = 0; j < (LinearEquation ? (W - 1) : W); j++) { // find basis if (rank == H) break; int idx = -1; for (int i = rank; i < H; i++) { if (a[i][j].get() != 0) { idx = i; break; } } if (idx == -1) { det = 0; continue; } // swap if (rank != idx) { det = -det; for (int l = j; l < W; l++) swap(a[rank][l], a[idx][l]); } det *= a[rank][j]; // normalize if (LinearEquation) { if (a[rank][j].get() != 1) { mint coeff = a[rank][j].inverse(); __m256i COEFF = _mm256_set1_epi32(coeff.a); for (int i = j / 8 * 8; i < W; i += 8) { __m256i R = _mm256_load_si256((__m256i *)(a[rank] + i)); __m256i RmulC = montgomery_mul_256(R, COEFF, r, m1); _mm256_store_si256((__m256i *)(a[rank] + i), RmulC); } } } // elimination for (int k = (LinearEquation ? 0 : rank + 1); k < H; k++) { if (k == rank) continue; if (a[k][j].get() != 0) { mint coeff = a[k][j] / a[rank][j]; __m256i COEFF = _mm256_set1_epi32(coeff.a); for (int i = j / 8 * 8; i < W; i += 8) { __m256i R = _mm256_load_si256((__m256i *)(a[rank] + i)); __m256i K = _mm256_load_si256((__m256i *)(a[k] + i)); __m256i RmulC = montgomery_mul_256(R, COEFF, r, m1); __m256i KmnsR = montgomery_sub_256(K, RmulC, m2, m0); _mm256_store_si256((__m256i *)(a[k] + i), KmnsR); } } } rank++; } return {rank, det}; } // calculate determinant template <typename mint> mint determinant(const vector<vector<mint>> &mat) { return GaussianElimination(mat).second; } // return V<V<mint>> // 0 column ... one of solutions // 1 ~ (W - rank) column ... bases // if not exist, return empty vector template <typename mint> vector<vector<mint>> LinearEquation(vector<vector<mint>> A, vector<mint> B) { int H = A.size(), W = A[0].size(); for (int i = 0; i < H; i++) A[i].push_back(B[i]); auto p = GaussianElimination(A, true); mint(&a)[MAT_SIZE][MAT_SIZE] = *reinterpret_cast<mint(*)[MAT_SIZE][MAT_SIZE]>(a_buf_); int rank = p.first; // check if solutions exist for (int i = rank; i < H; ++i) if (a[i][W] != 0) return vector<vector<mint>>{}; vector<vector<mint>> res(1, vector<mint>(W)); vector<int> pivot(W, -1); for (int i = 0, j = 0; i < rank; ++i) { while (a[i][j] == 0) ++j; res[0][j] = a[i][W], pivot[j] = i; } for (int j = 0; j < W; ++j) { if (pivot[j] == -1) { vector<mint> x(W); x[j] = 1; for (int k = 0; k < j; ++k) if (pivot[k] != -1) x[k] = -a[pivot[k]][j]; res.push_back(x); } } return res; } } // namespace Gauss using Gauss::determinant; using Gauss::LinearEquation; #line 13 "verify/verify-unit-test/gauss-elimination.test.cpp" } // namespace Fast #line 2 "misc/rng.hpp" #line 2 "internal/internal-seed.hpp" #line 4 "internal/internal-seed.hpp" using namespace std; namespace internal { unsigned long long non_deterministic_seed() { unsigned long long m = chrono::duration_cast<chrono::nanoseconds>( chrono::high_resolution_clock::now().time_since_epoch()) .count(); m ^= 9845834732710364265uLL; m ^= m << 24, m ^= m >> 31, m ^= m << 35; return m; } unsigned long long deterministic_seed() { return 88172645463325252UL; } // 64 bit の seed 値を生成 (手元では seed 固定) // 連続で呼び出すと同じ値が何度も返ってくるので注意 // #define RANDOMIZED_SEED するとシードがランダムになる unsigned long long seed() { #if defined(NyaanLocal) && !defined(RANDOMIZED_SEED) return deterministic_seed(); #else return non_deterministic_seed(); #endif } } // namespace internal #line 4 "misc/rng.hpp" namespace my_rand { using i64 = long long; using u64 = unsigned long long; // [0, 2^64 - 1) u64 rng() { static u64 _x = internal::seed(); return _x ^= _x << 7, _x ^= _x >> 9; } // [l, r] i64 rng(i64 l, i64 r) { assert(l <= r); return l + rng() % u64(r - l + 1); } // [l, r) i64 randint(i64 l, i64 r) { assert(l < r); return l + rng() % u64(r - l); } // choose n numbers from [l, r) without overlapping vector<i64> randset(i64 l, i64 r, i64 n) { assert(l <= r && n <= r - l); unordered_set<i64> s; for (i64 i = n; i; --i) { i64 m = randint(l, r + 1 - i); if (s.find(m) != s.end()) m = r - i; s.insert(m); } vector<i64> ret; for (auto& x : s) ret.push_back(x); sort(begin(ret), end(ret)); return ret; } // [0.0, 1.0) double rnd() { return rng() * 5.42101086242752217004e-20; } // [l, r) double rnd(double l, double r) { assert(l < r); return l + rnd() * (r - l); } template <typename T> void randshf(vector<T>& v) { int n = v.size(); for (int i = 1; i < n; i++) swap(v[i], v[randint(0, i + 1)]); } } // namespace my_rand using my_rand::randint; using my_rand::randset; using my_rand::randshf; using my_rand::rnd; using my_rand::rng; #line 2 "modint/montgomery-modint.hpp" template <uint32_t mod> struct LazyMontgomeryModInt { using mint = LazyMontgomeryModInt; using i32 = int32_t; using u32 = uint32_t; using u64 = uint64_t; static constexpr u32 get_r() { u32 ret = mod; for (i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret; return ret; } static constexpr u32 r = get_r(); static constexpr u32 n2 = -u64(mod) % mod; static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30"); static_assert((mod & 1) == 1, "invalid, mod % 2 == 0"); static_assert(r * mod == 1, "this code has bugs."); u32 a; constexpr LazyMontgomeryModInt() : a(0) {} constexpr LazyMontgomeryModInt(const int64_t &b) : a(reduce(u64(b % mod + mod) * n2)){}; static constexpr u32 reduce(const u64 &b) { return (b + u64(u32(b) * u32(-r)) * mod) >> 32; } constexpr mint &operator+=(const mint &b) { if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod; return *this; } constexpr mint &operator-=(const mint &b) { if (i32(a -= b.a) < 0) a += 2 * mod; return *this; } constexpr mint &operator*=(const mint &b) { a = reduce(u64(a) * b.a); return *this; } constexpr mint &operator/=(const mint &b) { *this *= b.inverse(); return *this; } constexpr mint operator+(const mint &b) const { return mint(*this) += b; } constexpr mint operator-(const mint &b) const { return mint(*this) -= b; } constexpr mint operator*(const mint &b) const { return mint(*this) *= b; } constexpr mint operator/(const mint &b) const { return mint(*this) /= b; } constexpr bool operator==(const mint &b) const { return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a); } constexpr bool operator!=(const mint &b) const { return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a); } constexpr mint operator-() const { return mint() - mint(*this); } constexpr mint operator+() const { return mint(*this); } constexpr mint pow(u64 n) const { mint ret(1), mul(*this); while (n > 0) { if (n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } constexpr mint inverse() const { int x = get(), y = mod, u = 1, v = 0, t = 0, tmp = 0; while (y > 0) { t = x / y; x -= t * y, u -= t * v; tmp = x, x = y, y = tmp; tmp = u, u = v, v = tmp; } return mint{u}; } friend ostream &operator<<(ostream &os, const mint &b) { return os << b.get(); } friend istream &operator>>(istream &is, mint &b) { int64_t t; is >> t; b = LazyMontgomeryModInt<mod>(t); return (is); } constexpr u32 get() const { u32 ret = reduce(a); return ret >= mod ? ret - mod : ret; } static constexpr u32 get_mod() { return mod; } }; #line 17 "verify/verify-unit-test/gauss-elimination.test.cpp" using namespace Nyaan; using mint = LazyMontgomeryModInt<998244353>; void test(int n, int m, bool sparse) { Matrix<mint> a(n, m); each(v, a.A) { each(x, v) { if (sparse) { if (randint(0, m) == 0) x = rng(); } else { x = rng(); } } } // trc(n, m, sparse); // trc(a); if (n == m) { mint det1, det2, det3; { auto b = a; det1 = b.determinant(); } { auto b = a; det2 = GaussElimination<mint>(b.A).second; } { auto b = a; det3 = Fast::determinant(b.A); } // trc(det1, det2, det3); assert(det1 == det2 and det2 == det3); } vector<mint> c(n); if (rng() & 1) each(x, c) x = rng(); vector<vector<mint>> sol1, sol2; { auto b = a; sol1 = Normal::LinearEquation<mint>(b.A, c); // trc(sol1); } { auto b = a; sol2 = Fast::LinearEquation<mint>(b.A, c); // trc(sol2); } assert(sol1 == sol2); } void Nyaan::solve() { rep(i, TEN(5)) test(randint(1, 17), randint(1, 17), rng() % 2 == 1); int a, b; cin >> a >> b; cout << a + b << endl; }