#include #include #include #include template* = nullptr> struct modulo_matrix{ private: using m32 = Elem; public: int h; int w; std::vector elems; modulo_matrix(int new_h, int new_w){ h = new_h; w = new_w; elems.assign(h * w, 0); } modulo_matrix(const modulo_matrix&) = default; typename std::vector::iterator operator[](int y){ return elems.begin() + (y * w); } typename std::vector::const_iterator operator[](int y) const { return elems.begin() + (y * w); } static modulo_matrix identity(int idx){ auto res = modulo_matrix(idx, idx); for(int i = 0; i < idx; i++) res[i][i] = 1; return res; } modulo_matrix operator*(const modulo_matrix& r) const { assert(w == r.h); auto res = modulo_matrix(h, r.w); for (int i=0; i=i; k--) g[j][k] -= g[j][i] * g[i][k]; } return ans; } int rank() const { modulo_matrix g = *this; int y = 0; for (int d=0; d=d; j--) g[i][j] -= g[i][d] * g[y][j]; y++; } return y; } std::pair, modulo_matrix> linear_equation() const { modulo_matrix g = *this; int y = 0; std::vector> det_var; std::vector rank_var; for (int d=0; d=d; j--) g[i][j] -= g[i][d] * g[y][j]; det_var.push_back(std::make_pair(d,y)); y++; } for (int i=y; i(), modulo_matrix(0,0)); std::vector solution(w-1, 0); for (auto [x,i] : det_var) { solution[x] = g[i][w-1]; } modulo_matrix solution_rank(rank_var.size(), w-1); for (int d=0; d #include using namespace std; using i32 = int32_t; using u32 = uint32_t; using i64 = int64_t; using u64 = uint64_t; #define rep(i,n) for(int i=0; i<(n); i++) using m32 = atcoder::static_modint<998244353>; using Matrix = modulo_matrix; struct EquationTerm{ int idx_1; int idx_2; m32 coeff; }; // ... = 0 struct Equation{ vector terms; m32 constant_term; m32 get_coeff(int idx_1, int idx_2){ for(auto& t : terms){ if(t.idx_1 == idx_1 && t.idx_2 == idx_2) return t.coeff; } return 0; } void add(EquationTerm term){ for(auto& t : terms){ if(t.idx_1 == term.idx_1 && t.idx_2 == term.idx_2){ t.coeff += term.coeff; if(t.coeff.val() == 0){ swap(t, terms.back()); terms.pop_back(); return; } return; } } terms.push_back(term); } void output(){ rep(i,terms.size()){ cout << terms[i].coeff.val() << " f( " << terms[i].idx_1 << ", " << terms[i].idx_2 << " ) + "; } cout << constant_term.val() << " = 0" << endl; } }; int N; int num_edge1; int num_edge2; vector> initial_state; m32 invN; m32 invWeight; m32 invNumOpenSlot; vector potential; Equation get_neighbor_equation(int idx_1, int idx_2){ Equation res; res.constant_term = invN; res.terms.push_back(EquationTerm{ idx_1, idx_2, m32(-1) }); m32 num_open_idx = ((N-1) - idx_1 - idx_2); // move an edge of type 1 if(idx_1 != 0){ // in -> out res.add(EquationTerm{ idx_1-1, idx_2, m32(1) * invWeight * idx_1 * (m32(1) - invNumOpenSlot * (num_open_idx + 1)) }); // in -> in res.add(EquationTerm{ idx_1, idx_2, m32(1) * invWeight * idx_1 * invNumOpenSlot * (num_open_idx + 1) }); } if(idx_1 != num_edge1){ // out -> out res.add(EquationTerm{ idx_1, idx_2, m32(1) * invWeight * (num_edge1-idx_1) * (m32(1) - invNumOpenSlot * num_open_idx) }); // out -> in res.add(EquationTerm{ idx_1+1, idx_2, m32(1) * invWeight * (num_edge1-idx_1) * invNumOpenSlot * num_open_idx }); } // move an edge of type 2 if(idx_2 != 0){ // in -> out res.terms.push_back(EquationTerm{ idx_1, idx_2-1, m32(1) * invWeight * idx_2 * 2 * (m32(1) - invNumOpenSlot * (num_open_idx + 1)) }); // in -> in res.add(EquationTerm{ idx_1, idx_2, m32(1) * invWeight * idx_2 * 2 * invNumOpenSlot * (num_open_idx + 1) }); } if(idx_2 != num_edge2){ // out -> out res.add(EquationTerm{ idx_1, idx_2, m32(1) * invWeight * (num_edge2-idx_2) * 2 * (m32(1) - invNumOpenSlot * num_open_idx) }); // out -> in res.add(EquationTerm{ idx_1, idx_2+1, m32(1) * invWeight * (num_edge2-idx_2) * 2 * invNumOpenSlot * num_open_idx }); } for(auto& term : res.terms) term.coeff /= (invWeight * invNumOpenSlot); res.constant_term /= (invWeight * invNumOpenSlot); //res.output(); return res; } int main() { cin >> N; invN = m32(N).inv(); num_edge1 = num_edge2 = 0; initial_state.assign(N, make_pair(0,0)); rep(i,N-1){ int u,v,c; cin >> u >> v >> c; u--; v--; if(c == 1){ num_edge1++; initial_state[u].first++; initial_state[v].first++; } else{ num_edge2++; initial_state[u].second++; initial_state[v].second++; } } if(num_edge1 == 0){ swap(num_edge1, num_edge2); for(auto& a : initial_state) swap(a.first, a.second); } invWeight = m32(num_edge1 + 2 * num_edge2).inv(); invNumOpenSlot = m32(N * (N-1) / 2 - (N-2)).inv(); potential.assign(num_edge1+1, Matrix(num_edge2+1, num_edge1+2)); const int constant_term_idx = num_edge1 + 1; const int equation_size = constant_term_idx + 1; for(int idx_1 = 0; idx_1 <= num_edge1; idx_1++) potential[idx_1][0][idx_1] = 1; for(int idx_2 = 1; idx_2 <= num_edge2; idx_2++){ for(int idx_1 = 0; idx_1 <= num_edge1; idx_1++){ auto eq = get_neighbor_equation(idx_1, idx_2-1); auto pcoeff = -eq.get_coeff(idx_1, idx_2); eq.add(EquationTerm{ idx_1, idx_2, pcoeff }); pcoeff = pcoeff.inv(); for(auto eqterm : eq.terms){ rep(i,equation_size) potential[idx_1][idx_2][i] += potential[eqterm.idx_1][eqterm.idx_2][i] * eqterm.coeff; } potential[idx_1][idx_2][constant_term_idx] += eq.constant_term; rep(i,equation_size) potential[idx_1][idx_2][i] *= pcoeff; } } Matrix final_equation = Matrix(num_edge1+1, equation_size); for(int idx_1 = 0; idx_1 < num_edge1; idx_1++){ auto eq = get_neighbor_equation(idx_1, num_edge2); for(auto eqterm : eq.terms){ rep(i,equation_size) final_equation[idx_1][i] += eqterm.coeff * potential[eqterm.idx_1][eqterm.idx_2][i]; } final_equation[idx_1][constant_term_idx] += eq.constant_term; } rep(i,equation_size) final_equation[num_edge1][i] += potential[num_edge1][num_edge2][i]; rep(i,equation_size) final_equation[num_edge1][i] += potential[1][0][i] * num_edge1; rep(i,equation_size) final_equation[num_edge1][i] += potential[0][1][i] * num_edge2; for(int idx_1 = 0; idx_1 <= num_edge1; idx_1++) final_equation[idx_1][constant_term_idx] *= -1; auto potential_solution = final_equation.linear_equation().first; potential_solution.push_back(1); m32 ans = 0; for(auto [idx_1, idx_2] : initial_state){ //cout << idx_1 << ", " << idx_2 << endl; rep(i,equation_size) ans += potential_solution[i] * potential[idx_1][idx_2][i]; } cout << ans.val() << endl; return 0; } struct ios_do_not_sync{ ios_do_not_sync(){ ios::sync_with_stdio(false); cin.tie(nullptr); } } ios_do_not_sync_instance;