#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; using lint = long long; using pint = pair; using plint = pair; struct fast_ios { fast_ios(){ cin.tie(nullptr), ios::sync_with_stdio(false), cout << fixed << setprecision(20); }; } fast_ios_; #define ALL(x) (x).begin(), (x).end() #define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i=i##_begin_;i--) #define REP(i, n) FOR(i,0,n) #define IREP(i, n) IFOR(i,0,n) template void ndarray(vector& vec, const V& val, int len) { vec.assign(len, val); } template void ndarray(vector& vec, const V& val, int len, Args... args) { vec.resize(len), for_each(begin(vec), end(vec), [&](T& v) { ndarray(v, val, args...); }); } template bool chmax(T &m, const T q) { return m < q ? (m = q, true) : false; } template bool chmin(T &m, const T q) { return m > q ? (m = q, true) : false; } int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); } template pair operator+(const pair &l, const pair &r) { return make_pair(l.first + r.first, l.second + r.second); } template pair operator-(const pair &l, const pair &r) { return make_pair(l.first - r.first, l.second - r.second); } template vector sort_unique(vector vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; } template int arglb(const std::vector &v, const T &x) { return std::distance(v.begin(), std::lower_bound(v.begin(), v.end(), x)); } template int argub(const std::vector &v, const T &x) { return std::distance(v.begin(), std::upper_bound(v.begin(), v.end(), x)); } template istream &operator>>(istream &is, vector &vec) { for (auto &v : vec) is >> v; return is; } template ostream &operator<<(ostream &os, const vector &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; } template ostream &operator<<(ostream &os, const array &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; } #if __cplusplus >= 201703L template istream &operator>>(istream &is, tuple &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; } template ostream &operator<<(ostream &os, const tuple &tpl) { os << '('; std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os << ')'; } #endif template ostream &operator<<(ostream &os, const deque &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; } template ostream &operator<<(ostream &os, const set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const unordered_set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const unordered_multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const pair &pa) { os << '(' << pa.first << ',' << pa.second << ')'; return os; } template ostream &operator<<(ostream &os, const map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const unordered_map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } #ifdef HITONANODE_LOCAL const string COLOR_RESET = "\033[0m", BRIGHT_GREEN = "\033[1;32m", BRIGHT_RED = "\033[1;31m", BRIGHT_CYAN = "\033[1;36m", NORMAL_CROSSED = "\033[0;9;37m", RED_BACKGROUND = "\033[1;41m", NORMAL_FAINT = "\033[0;2m"; #define dbg(x) cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << endl #define dbgif(cond, x) ((cond) ? cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << endl : cerr) #else #define dbg(x) (x) #define dbgif(cond, x) 0 #endif template struct ModInt { #if __cplusplus >= 201402L #define MDCONST constexpr #else #define MDCONST #endif using lint = long long; MDCONST static int mod() { return md; } static int get_primitive_root() { static int primitive_root = 0; if (!primitive_root) { primitive_root = [&]() { std::set fac; int v = md - 1; for (lint i = 2; i * i <= v; i++) while (v % i == 0) fac.insert(i), v /= i; if (v > 1) fac.insert(v); for (int g = 1; g < md; g++) { bool ok = true; for (auto i : fac) if (ModInt(g).pow((md - 1) / i) == 1) { ok = false; break; } if (ok) return g; } return -1; }(); } return primitive_root; } int val; MDCONST ModInt() : val(0) {} MDCONST ModInt &_setval(lint v) { return val = (v >= md ? v - md : v), *this; } MDCONST ModInt(lint v) { _setval(v % md + md); } MDCONST explicit operator bool() const { return val != 0; } MDCONST ModInt operator+(const ModInt &x) const { return ModInt()._setval((lint)val + x.val); } MDCONST ModInt operator-(const ModInt &x) const { return ModInt()._setval((lint)val - x.val + md); } MDCONST ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val * x.val % md); } MDCONST ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val * x.inv() % md); } MDCONST ModInt operator-() const { return ModInt()._setval(md - val); } MDCONST ModInt &operator+=(const ModInt &x) { return *this = *this + x; } MDCONST ModInt &operator-=(const ModInt &x) { return *this = *this - x; } MDCONST ModInt &operator*=(const ModInt &x) { return *this = *this * x; } MDCONST ModInt &operator/=(const ModInt &x) { return *this = *this / x; } friend MDCONST ModInt operator+(lint a, const ModInt &x) { return ModInt()._setval(a % md + x.val); } friend MDCONST ModInt operator-(lint a, const ModInt &x) { return ModInt()._setval(a % md - x.val + md); } friend MDCONST ModInt operator*(lint a, const ModInt &x) { return ModInt()._setval(a % md * x.val % md); } friend MDCONST ModInt operator/(lint a, const ModInt &x) { return ModInt()._setval(a % md * x.inv() % md); } MDCONST bool operator==(const ModInt &x) const { return val == x.val; } MDCONST bool operator!=(const ModInt &x) const { return val != x.val; } MDCONST bool operator<(const ModInt &x) const { return val < x.val; } // To use std::map friend std::istream &operator>>(std::istream &is, ModInt &x) { lint t; return is >> t, x = ModInt(t), is; } MDCONST friend std::ostream &operator<<(std::ostream &os, const ModInt &x) { return os << x.val; } MDCONST ModInt pow(lint n) const { ModInt ans = 1, tmp = *this; while (n) { if (n & 1) ans *= tmp; tmp *= tmp, n >>= 1; } return ans; } static std::vector facs, facinvs, invs; MDCONST static void _precalculation(int N) { int l0 = facs.size(); if (N > md) N = md; if (N <= l0) return; facs.resize(N), facinvs.resize(N), invs.resize(N); for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i; facinvs[N - 1] = facs.back().pow(md - 2); for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1); for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1]; } MDCONST lint inv() const { if (this->val < std::min(md >> 1, 1 << 21)) { while (this->val >= int(facs.size())) _precalculation(facs.size() * 2); return invs[this->val].val; } else { return this->pow(md - 2).val; } } MDCONST ModInt fac() const { while (this->val >= int(facs.size())) _precalculation(facs.size() * 2); return facs[this->val]; } MDCONST ModInt facinv() const { while (this->val >= int(facs.size())) _precalculation(facs.size() * 2); return facinvs[this->val]; } MDCONST ModInt doublefac() const { lint k = (this->val + 1) / 2; return (this->val & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac()) : ModInt(k).fac() * ModInt(2).pow(k); } MDCONST ModInt nCr(const ModInt &r) const { return (this->val < r.val) ? 0 : this->fac() * (*this - r).facinv() * r.facinv(); } MDCONST ModInt nPr(const ModInt &r) const { return (this->val < r.val) ? 0 : this->fac() * (*this - r).facinv(); } ModInt sqrt() const { if (val == 0) return 0; if (md == 2) return val; if (pow((md - 1) / 2) != 1) return 0; ModInt b = 1; while (b.pow((md - 1) / 2) == 1) b += 1; int e = 0, m = md - 1; while (m % 2 == 0) m >>= 1, e++; ModInt x = pow((m - 1) / 2), y = (*this) * x * x; x *= (*this); ModInt z = b.pow(m); while (y != 1) { int j = 0; ModInt t = y; while (t != 1) j++, t *= t; z = z.pow(1LL << (e - j - 1)); x *= z, z *= z, y *= z; e = j; } return ModInt(std::min(x.val, md - x.val)); } }; template std::vector> ModInt::facs = {1}; template std::vector> ModInt::facinvs = {1}; template std::vector> ModInt::invs = {0}; using mint = ModInt<998244353>; // Integer convolution for arbitrary mod // with NTT (and Garner's algorithm) for ModInt / ModIntRuntime class. // We skip Garner's algorithm if `skip_garner` is true or mod is in `nttprimes`. // input: a (size: n), b (size: m) // return: vector (size: n + m - 1) template std::vector nttconv(std::vector a, std::vector b, bool skip_garner); constexpr int nttprimes[3] = {998244353, 167772161, 469762049}; // Integer FFT (Fast Fourier Transform) for ModInt class // (Also known as Number Theoretic Transform, NTT) // is_inverse: inverse transform // ** Input size must be 2^n ** template void ntt(std::vector &a, bool is_inverse = false) { int n = a.size(); if (n == 1) return; static const int mod = MODINT::mod(); static const MODINT root = MODINT::get_primitive_root(); assert(__builtin_popcount(n) == 1 and (mod - 1) % n == 0); static std::vector w{1}, iw{1}; for (int m = w.size(); m < n / 2; m *= 2) { MODINT dw = root.pow((mod - 1) / (4 * m)), dwinv = 1 / dw; w.resize(m * 2), iw.resize(m * 2); for (int i = 0; i < m; i++) w[m + i] = w[i] * dw, iw[m + i] = iw[i] * dwinv; } if (!is_inverse) { for (int m = n; m >>= 1;) { for (int s = 0, k = 0; s < n; s += 2 * m, k++) { for (int i = s; i < s + m; i++) { MODINT x = a[i], y = a[i + m] * w[k]; a[i] = x + y, a[i + m] = x - y; } } } } else { for (int m = 1; m < n; m *= 2) { for (int s = 0, k = 0; s < n; s += 2 * m, k++) { for (int i = s; i < s + m; i++) { MODINT x = a[i], y = a[i + m]; a[i] = x + y, a[i + m] = (x - y) * iw[k]; } } } int n_inv = MODINT(n).inv(); for (auto &v : a) v *= n_inv; } } template std::vector> nttconv_(const std::vector &a, const std::vector &b) { int sz = a.size(); assert(a.size() == b.size() and __builtin_popcount(sz) == 1); std::vector> ap(sz), bp(sz); for (int i = 0; i < sz; i++) ap[i] = a[i], bp[i] = b[i]; ntt(ap, false); if (a == b) bp = ap; else ntt(bp, false); for (int i = 0; i < sz; i++) ap[i] *= bp[i]; ntt(ap, true); return ap; } long long garner_ntt_(int r0, int r1, int r2, int mod) { using mint2 = ModInt; static const long long m01 = 1LL * nttprimes[0] * nttprimes[1]; static const long long m0_inv_m1 = ModInt(nttprimes[0]).inv(); static const long long m01_inv_m2 = mint2(m01).inv(); int v1 = (m0_inv_m1 * (r1 + nttprimes[1] - r0)) % nttprimes[1]; auto v2 = (mint2(r2) - r0 - mint2(nttprimes[0]) * v1) * m01_inv_m2; return (r0 + 1LL * nttprimes[0] * v1 + m01 % mod * v2.val) % mod; } template std::vector nttconv(std::vector a, std::vector b, bool skip_garner) { if (a.empty() or b.empty()) return {}; int sz = 1, n = a.size(), m = b.size(); while (sz < n + m) sz <<= 1; if (sz <= 16) { std::vector ret(n + m - 1); for (int i = 0; i < n; i++) { for (int j = 0; j < m; j++) ret[i + j] += a[i] * b[j]; } return ret; } int mod = MODINT::mod(); if (skip_garner or std::find(std::begin(nttprimes), std::end(nttprimes), mod) != std::end(nttprimes)) { a.resize(sz), b.resize(sz); if (a == b) { ntt(a, false); b = a; } else { ntt(a, false), ntt(b, false); } for (int i = 0; i < sz; i++) a[i] *= b[i]; ntt(a, true); a.resize(n + m - 1); } else { std::vector ai(sz), bi(sz); for (int i = 0; i < n; i++) ai[i] = a[i].val; for (int i = 0; i < m; i++) bi[i] = b[i].val; auto ntt0 = nttconv_(ai, bi); auto ntt1 = nttconv_(ai, bi); auto ntt2 = nttconv_(ai, bi); a.resize(n + m - 1); for (int i = 0; i < n + m - 1; i++) a[i] = garner_ntt_(ntt0[i].val, ntt1[i].val, ntt2[i].val, mod); } return a; } template std::vector nttconv(const std::vector &a, const std::vector &b) { return nttconv(a, b, false); } vector Q; vector ret; /* (Recursive) Centroid Decomposition Verification: Codeforces #190 Div.1 C https://codeforces.com/contest/321/submission/59093583 fix_root(int r): Build information of the tree which `r` belongs to. detect_centroid(int r): Enumerate centroid(s) of the tree which `r` belongs to. */ struct CentroidDecomposition { int NO_PARENT = -1; int V; int E; std::vector>> to; // (node_id, edge_id) std::vector par; // parent node_id par[root] = -1 std::vector> chi; // children id's std::vector subtree_size; // size of each subtree std::vector available_edge; // If 0, ignore the corresponding edge. CentroidDecomposition(int v = 0) : V(v), E(0), to(v), par(v, NO_PARENT), chi(v), subtree_size(v) {} CentroidDecomposition(const std::vector> &to_) : CentroidDecomposition(to_.size()) { for (int i = 0; i < V; i++) { for (auto j : to_[i]) { if (i < j) { add_edge(i, j); } } } } void add_edge(int v1, int v2) { to[v1].emplace_back(v2, E), to[v2].emplace_back(v1, E), E++; available_edge.emplace_back(1); } int _dfs_fixroot(int now, int prv) { chi[now].clear(), subtree_size[now] = 1; for (auto nxt : to[now]) { if (nxt.first != prv and available_edge[nxt.second]) { par[nxt.first] = now, chi[now].push_back(nxt.first); subtree_size[now] += _dfs_fixroot(nxt.first, now); } } return subtree_size[now]; } void fix_root(int root) { par[root] = NO_PARENT; _dfs_fixroot(root, -1); } //// Centroid Decpmposition //// std::vector centroid_cand_tmp; void _dfs_detect_centroids(int now, int prv, int n) { bool is_centroid = true; for (auto nxt : to[now]) { if (nxt.first != prv and available_edge[nxt.second]) { _dfs_detect_centroids(nxt.first, now, n); if (subtree_size[nxt.first] > n / 2) is_centroid = false; } } if (n - subtree_size[now] > n / 2) is_centroid = false; if (is_centroid) centroid_cand_tmp.push_back(now); } std::pair detect_centroids(int r) { // ([centroid_node_id1], ([centroid_node_id2]|-1)) centroid_cand_tmp.clear(); while (par[r] != NO_PARENT) r = par[r]; int n = subtree_size[r]; _dfs_detect_centroids(r, -1, n); if (centroid_cand_tmp.size() == 1) return std::make_pair(centroid_cand_tmp[0], -1); else return std::make_pair(centroid_cand_tmp[0], centroid_cand_tmp[1]); } std::vector _cd_vertices; void _centroid_decomposition(int now) { fix_root(now); now = detect_centroids(now).first; _cd_vertices.emplace_back(now); auto gen_affect = [&](vector v) { reverse(v.begin(), v.end()); int n = v.size(); vector trans(v.size() + 4); REP(i, trans.size()) { trans[i] = mint(i + 1).inv() * mint(i + 1).inv(); } v = nttconv(v, trans); v.erase(v.begin(), v.begin() + n - 1); return v; }; dbg(now); auto q0 = Q[now]; vector coeffs{0}; for (auto [nxt, eid] : to[now]) { if (!available_edge[eid]) continue; vector tmp; auto rec1 = [&](auto &&self, int now, int prv, int depth) -> void { if (depth >= int(tmp.size())) tmp.push_back(0); tmp[depth] += Q[now]; for (auto [nxt, eid] : to[now]) { if (!available_edge[eid]) continue; if (nxt == prv) continue; self(self, nxt, now, depth + 1); } ret[now] += q0 * mint(depth + 2).inv() * mint(depth + 2).inv(); }; rec1(rec1, nxt, now, 0); while (coeffs.size() < tmp.size()) coeffs.push_back(0); REP(j, tmp.size()) coeffs[j] += tmp[j]; auto v = gen_affect(tmp); auto rec2 = [&](auto &&self, int now, int prv, int depth) -> void { ret[now] -= v[depth]; for (auto [nxt, eid] : to[now]) { if (!available_edge[eid]) continue; if (nxt == prv) continue; self(self, nxt, now, depth + 1); } }; rec2(rec2, nxt, now, 2); // ret[now] -= v[1]; } coeffs = gen_affect(coeffs); dbg(coeffs); auto rec3 = [&](auto &&self, int now, int prv, int depth) -> void { ret[now] += coeffs[depth]; for (auto [nxt, eid] : to[now]) { if (!available_edge[eid]) continue; if (nxt == prv) continue; self(self, nxt, now, depth + 1); } }; rec3(rec3, now, -1, 1); for (auto p : to[now]) { int nxt, eid; std::tie(nxt, eid) = p; if (available_edge[eid] == 0) continue; available_edge[eid] = 0; _centroid_decomposition(nxt); } } std::vector centroid_decomposition(int x) { _cd_vertices.clear(); _centroid_decomposition(x); return _cd_vertices; } }; int main() { int N; cin >> N; Q.resize(N); cin >> Q; CentroidDecomposition cd(N); REP(i, N - 1) { int u, v; cin >> u >> v; u--, v--; cd.add_edge(u, v); } ret = Q; dbg(Q); dbg(ret); cd.centroid_decomposition(0); auto k0 = mint(N).fac() * mint(N).fac(); dbg(k0); for (auto &x : ret) cout << x * k0 << '\n'; }