#include #include #include #include #include #include #include #include using namespace std; #include using mint = atcoder::static_modint<1000000009>; // using mint = atcoder::modint1000000007; template std::vector mat_vec_mul(const std::vector> &mat, const std::vector &vec) { int H = mat.size(), W = mat[0].size(); assert(W == int(vec.size())); std::vector ret(H); for (int i = 0; i < H; ++i) { for (int j = 0; j < W; ++j) ret[i] += mat[i][j] * vec[j]; } return ret; } // Try to calculate inverse of M and return rank of M (destructive) template int inverse_matrix(std::vector> &M) { const int N = M.size(); assert(N and M[0].size() == M.size()); std::vector> ret(N, std::vector(N)); for (int i = 0; i < N; ++i) ret[i][i] = 1; int rank = 0; for (int i = 0; i < N; ++i) { int ti = i; while (ti < N and M[ti][i] == 0) ti++; if (ti == N) { continue; } ++rank; ret[i].swap(ret[ti]), M[i].swap(M[ti]); T inv = T(1) / M[i][i]; for (int j = 0; j < N; ++j) ret[i][j] *= inv; for (int j = i + 1; j < N; ++j) M[i][j] *= inv; for (int h = 0; h < N; ++h) { if (i == h) continue; const T c = -M[h][i]; for (int j = 0; j < N; ++j) ret[h][j] += ret[i][j] * c; for (int j = i + 1; j < N; ++j) M[h][j] += M[i][j] * c; } } M = ret; return rank; } template std::vector linear_matroid_parity(const std::vector, std::vector>> &bcs) { if (bcs.empty()) return {}; const int r = bcs[0].first.size(), m = bcs.size(), r2 = (r + 1) / 2; std::mt19937 mt(std::chrono::steady_clock::now().time_since_epoch().count()); std::uniform_int_distribution d(0, ModInt::mod() - 1); auto gen_random_vector = [&]() -> std::vector { std::vector v(r2 * 2); for (int i = 0; i < r2 * 2; i++) v[i] = d(mt); return v; }; std::vector x(m); std::vector, vector>> bcadd(r2); std::vector> Yinv; // r2 * r2 matrices int rankY = -1; while (rankY < r2 * 2) { Yinv.assign(r2 * 2, std::vector(r2 * 2, 0)); for (auto &[b, c] : bcadd) { b = gen_random_vector(), c = gen_random_vector(); for (int j = 0; j < r2 * 2; j++) { for (int k = 0; k < r2 * 2; k++) Yinv[j][k] += b[j] * c[k] - c[j] * b[k]; } } rankY = inverse_matrix(Yinv); } std::vector> tmpmat(r2 * 2, std::vector(r2 * 2)); std::vector ret(m, -1); int additional_dim = bcadd.size(); for (int i = 0; i < m; i++) { { x[i] = d(mt); auto b = bcs[i].first, c = bcs[i].second; b.resize(r2 * 2, 0), c.resize(r2 * 2, 0); std::vector Yib = mat_vec_mul(Yinv, b), Yic = mat_vec_mul(Yinv, c); ModInt bYic = std::inner_product(b.begin(), b.end(), Yic.begin(), ModInt(0)); ModInt v = 1 + x[i] * bYic; if (v == 0) break; // failed const auto coeff = x[i] / v; for (int j = 0; j < r2 * 2; j++) { for (int k = 0; k < r2 * 2; k++) { tmpmat[j][k] = Yib[j] * Yic[k] - Yic[j] * Yib[k]; } } for (int j = 0; j < r2 * 2; j++) { for (int k = 0; k < r2 * 2; k++) Yinv[j][k] -= tmpmat[j][k] * coeff; } } if (additional_dim) { const auto &[b, c] = bcadd[additional_dim - 1]; std::vector Yib = mat_vec_mul(Yinv, b), Yic = mat_vec_mul(Yinv, c); ModInt bYic = std::inner_product(b.begin(), b.end(), Yic.begin(), ModInt(0)); const ModInt v = 1 + bYic; if (v != 0) { // 消しても正則 additional_dim--; const auto coeff = 1 / v; for (int j = 0; j < r2 * 2; j++) { for (int k = 0; k < r2 * 2; k++) tmpmat[j][k] = Yib[j] * Yic[k] - Yic[j] * Yib[k]; } for (int j = 0; j < r2 * 2; j++) { for (int k = 0; k < r2 * 2; k++) Yinv[j][k] -= tmpmat[j][k] * coeff; } } } ret[i] = r2 - additional_dim; } return ret; } vector solve(int N, vector, pair>> bcs) { vector, vector>> vs; for (auto [ab, cd] : bcs) { auto [a, b] = ab; auto [c, d] = cd; vector B(N), C(N); B.at(a) += 1; B.at(b) -= 1; C.at(c) += 1; C.at(d) -= 1; vs.emplace_back(B, C); } return linear_matroid_parity(vs); } int main() { int N, M; cin >> N >> M; vector, pair>> edges; while (M--) { int u, v, w; cin >> u >> v >> w; u--, v--, w--; edges.push_back({{u, w}, {v, w}}); } auto ret = solve(N, edges); for (auto x : ret) cout << x << '\n'; }