#include #include #include #include #include #include // Gauss-Jordan elimination of n * m matrix M // Complexity: O(nm + nm rank(M) / 64) // Verified: abc276_h (2000 x 8000) template std::vector> f2_gauss_jordan(int W, std::vector> M) { assert(W <= Wmax); int H = M.size(), c = 0; for (int h = 0; h < H and c < W; ++h, ++c) { int piv = -1; for (int j = h; j < H; ++j) { if (M[j][c]) { piv = j; break; } } if (piv == -1) { --h; continue; } std::swap(M[piv], M[h]); for (int hh = 0; hh < H; ++hh) { if (hh != h and M[hh][c]) M[hh] ^= M[h]; } } return M; } // Rank of Gauss-Jordan eliminated matrix template int f2_rank_gauss_jordan(int W, const std::vector> &M) { assert(W <= Wmax); for (int h = (int)M.size() - 1; h >= 0; h--) { int j = 0; while (j < W and !M[h][j]) ++j; if (j < W) return h + 1; } return 0; } // determinant of F2 matrix. // Return 0 if the matrix is singular, otherwise return 1. // Complexity: O(W^3 / 64) template int f2_determinant(const std::vector> &M) { const int H = M.size(); if (H > Wmax) return 0; auto tmp = M; for (int h = 0; h < H; ++h) { int piv = -1; for (int j = h; j < H; ++j) { if (tmp.at(j).test(h)) { piv = j; break; } } if (piv == -1) return 0; // singular if (piv != h) std::swap(tmp.at(piv), tmp.at(h)); for (int hh = h + 1; hh < H; ++hh) { if (tmp.at(hh).test(h)) tmp.at(hh) ^= tmp.at(h); } } return 1; // nonsingular } template std::vector> f2_matmul(const std::vector> &A, const std::vector> &B) { int H = A.size(), K = B.size(); std::vector> C(H); for (int i = 0; i < H; i++) { for (int j = 0; j < K; j++) { if (A.at(i).test(j)) C.at(i) ^= B.at(j); } } return C; } template std::vector> f2_matpower(std::vector> X, long long n) { int D = X.size(); std::vector> ret(D); for (int i = 0; i < D; i++) ret[i][i] = 1; while (n) { if (n & 1) ret = f2_matmul(ret, X); X = f2_matmul(X, X), n >>= 1; } return ret; } // Solve Ax = b on F_2 // - retval: {true, one of the solutions, {freedoms}} (if solution exists) // {false, {}, {}} (otherwise) // Complexity: O(HW + HW rank(A) / 64 + W^2 len(freedoms)) template std::tuple, std::vector>> f2_system_of_linear_equations(std::vector> A, Vec b, int W) { int H = A.size(); assert(W <= Wmax); assert(A.size() == b.size()); std::vector> M(H); for (int i = 0; i < H; ++i) { for (int j = 0; j < W; ++j) M[i][j] = A[i][j]; M[i][W] = b[i]; } M = f2_gauss_jordan(W + 1, M); std::vector ss(W, -1); std::vector ss_nonneg_js; for (int i = 0; i < H; i++) { int j = 0; while (j <= W and !M[i][j]) ++j; if (j == W) return {false, 0, {}}; if (j < W) { ss_nonneg_js.push_back(j); ss[j] = i; } } std::bitset x; std::vector> D; for (int j = 0; j < W; ++j) { if (ss[j] == -1) { // This part may require W^2 space complexity in output std::bitset d; d[j] = 1; for (int jj : ss_nonneg_js) d[jj] = M[ss[jj]][j]; D.emplace_back(d); } else { x[j] = M[ss[j]][W]; } } return std::make_tuple(true, x, D); } #include #include #include using namespace std; template bool chmin(T &m, const T q) { if (m > q) { m = q; return true; } else return false; } void bad() { puts("-1"); exit(0); } int main() { cin.tie(nullptr), ios::sync_with_stdio(false); int N, M; cin >> N >> M; using ull = unsigned long long; vector> basis; while (M--) { int a; cin >> a; ull mask = 0; while (a--) { int b; cin >> b; b--; mask += 1ULL << b; } int y; cin >> y; for (auto [v, w] : basis) { if (chmin(mask, mask ^ v)) y ^= w; } if (!mask and y) bad(); if (mask) basis.emplace_back(mask, y); } vector ret(N); for (int d = 0; d < 30; ++d) { constexpr int Wmax = 320; vector> A; vector b; for (int i = 0; i < int(basis.size()); ++i) { b.push_back((basis[i].second >> d) & 1); bitset a; a.reset(); for (int j = 0; j < N; ++j) { if ((basis[i].first >> j) & 1) a[j] = 1; } A.emplace_back(a); } auto [ok, solution, freedoms] = f2_system_of_linear_equations(A, b, N); if (!ok) bad(); for (int i = 0; i < N; ++i) ret[i] += int(solution[i]) << d; } for (auto x : ret) cout << x << '\n'; }