#include #include #include #include #include #include #include #include #include using namespace std; using lint = long long; using pint = pair; #define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i // using mint = atcoder::modint1000000007; using mint = atcoder::static_modint<1000000009>; template struct matrix { int H, W; std::vector elem; typename std::vector::iterator operator[](int i) { return elem.begin() + i * W; } inline T &at(int i, int j) { return elem[i * W + j]; } inline T get(int i, int j) const { return elem[i * W + j]; } int height() const { return H; } int width() const { return W; } std::vector> vecvec() const { std::vector> ret(H); for (int i = 0; i < H; i++) { std::copy(elem.begin() + i * W, elem.begin() + (i + 1) * W, std::back_inserter(ret[i])); } return ret; } operator std::vector>() const { return vecvec(); } matrix() = default; matrix(int H, int W) : H(H), W(W), elem(H * W) {} matrix(const std::vector> &d) : H(d.size()), W(d.size() ? d[0].size() : 0) { for (auto &raw : d) std::copy(raw.begin(), raw.end(), std::back_inserter(elem)); } static matrix Identity(int N) { matrix ret(N, N); for (int i = 0; i < N; i++) ret.at(i, i) = 1; return ret; } matrix operator-() const { matrix ret(H, W); for (int i = 0; i < H * W; i++) ret.elem[i] = -elem[i]; return ret; } matrix operator*(const T &v) const { matrix ret = *this; for (auto &x : ret.elem) x *= v; return ret; } matrix operator/(const T &v) const { matrix ret = *this; const T vinv = T(1) / v; for (auto &x : ret.elem) x *= vinv; return ret; } matrix operator+(const matrix &r) const { matrix ret = *this; for (int i = 0; i < H * W; i++) ret.elem[i] += r.elem[i]; return ret; } matrix operator-(const matrix &r) const { matrix ret = *this; for (int i = 0; i < H * W; i++) ret.elem[i] -= r.elem[i]; return ret; } matrix operator*(const matrix &r) const { matrix ret(H, r.W); for (int i = 0; i < H; i++) { for (int k = 0; k < W; k++) { for (int j = 0; j < r.W; j++) ret.at(i, j) += this->get(i, k) * r.get(k, j); } } return ret; } matrix &operator*=(const T &v) { return *this = *this * v; } matrix &operator/=(const T &v) { return *this = *this / v; } matrix &operator+=(const matrix &r) { return *this = *this + r; } matrix &operator-=(const matrix &r) { return *this = *this - r; } matrix &operator*=(const matrix &r) { return *this = *this * r; } bool operator==(const matrix &r) const { return H == r.H and W == r.W and elem == r.elem; } bool operator!=(const matrix &r) const { return H != r.H or W != r.W or elem != r.elem; } bool operator<(const matrix &r) const { return elem < r.elem; } matrix pow(int64_t n) const { matrix ret = Identity(H); bool ret_is_id = true; if (n == 0) return ret; for (int i = 63 - __builtin_clzll(n); i >= 0; i--) { if (!ret_is_id) ret *= ret; if ((n >> i) & 1) ret *= (*this), ret_is_id = false; } return ret; } std::vector pow_vec(int64_t n, std::vector vec) const { matrix x = *this; while (n) { if (n & 1) vec = x * vec; x *= x; n >>= 1; } return vec; }; matrix transpose() const { matrix ret(W, H); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) ret.at(j, i) = this->get(i, j); } return ret; } // Gauss-Jordan elimination // - Require inverse for every non-zero element // - Complexity: O(H^2 W) template ::value>::type * = nullptr> static int choose_pivot(const matrix &mtr, int h, int c) noexcept { int piv = -1; for (int j = h; j < mtr.H; j++) { if (mtr.get(j, c) and (piv < 0 or std::abs(mtr.get(j, c)) > std::abs(mtr.get(piv, c)))) piv = j; } return piv; } template ::value>::type * = nullptr> static int choose_pivot(const matrix &mtr, int h, int c) noexcept { for (int j = h; j < mtr.H; j++) { if (mtr.get(j, c)) return j; } return -1; } matrix gauss_jordan() const { int c = 0; matrix mtr(*this); std::vector ws; ws.reserve(W); for (int h = 0; h < H; h++) { if (c == W) break; int piv = choose_pivot(mtr, h, c); if (piv == -1) { c++; h--; continue; } if (h != piv) { for (int w = 0; w < W; w++) { std::swap(mtr[piv][w], mtr[h][w]); mtr.at(piv, w) *= -1; // To preserve sign of determinant } } ws.clear(); for (int w = c; w < W; w++) { if (mtr.at(h, w) != 0) ws.emplace_back(w); } const T hcinv = T(1) / mtr.at(h, c); for (int hh = 0; hh < H; hh++) if (hh != h) { const T coeff = mtr.at(hh, c) * hcinv; for (auto w : ws) mtr.at(hh, w) -= mtr.at(h, w) * coeff; mtr.at(hh, c) = 0; } c++; } return mtr; } int rank_of_gauss_jordan() const { for (int i = H * W - 1; i >= 0; i--) { if (elem[i]) return i / W + 1; } return 0; } int inverse() { assert(H == W); std::vector> ret = Identity(H), tmp = *this; int rank = 0; for (int i = 0; i < H; i++) { int ti = i; while (ti < H and tmp[ti][i] == 0) ti++; if (ti == H) { continue; } else { rank++; } ret[i].swap(ret[ti]), tmp[i].swap(tmp[ti]); T inv = T(1) / tmp[i][i]; for (int j = 0; j < W; j++) ret[i][j] *= inv; for (int j = i + 1; j < W; j++) tmp[i][j] *= inv; for (int h = 0; h < H; h++) { if (i == h) continue; const T c = -tmp[h][i]; for (int j = 0; j < W; j++) ret[h][j] += ret[i][j] * c; for (int j = i + 1; j < W; j++) tmp[h][j] += tmp[i][j] * c; } } *this = ret; return rank; } friend std::vector operator*(const matrix &m, const std::vector &v) { assert(m.W == int(v.size())); std::vector ret(m.H); for (int i = 0; i < m.H; i++) { for (int j = 0; j < m.W; j++) ret[i] += m.get(i, j) * v[j]; } return ret; } friend std::vector operator*(const std::vector &v, const matrix &m) { assert(int(v.size()) == m.H); std::vector ret(m.W); for (int i = 0; i < m.H; i++) { for (int j = 0; j < m.W; j++) ret[j] += v[i] * m.get(i, j); } return ret; } }; 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); matrix Y, Yinv; // r2 * r2 matrices int rankY = -1; while (rankY < r2 * 2) { Y = matrix(r2 * 2, r2 * 2); 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++) Y[j][k] += b[j] * c[k] - c[j] * b[k]; } } Yinv = Y; rankY = Yinv.inverse(); } 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 = Yinv * b, Yic = Yinv * c; ModInt bYic = std::inner_product(b.begin(), b.end(), Yic.begin(), ModInt(0)); ModInt v = 1 + x[i] * bYic; 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 = Yinv * b, Yic = 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> 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); } auto ret1 = linear_matroid_parity(vs); // auto ret2 = linear_matroid_parity(vs); // for (int i = 0; i < int(ret1.size()); i++) ret1[i] = max(ret1[i], ret2[i]); return ret1; } int main() { int N, M; cin >> N >> M; vector> 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'; }