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#define PROBLEM \ "https://judge.yosupo.jp/problem/exp_of_formal_power_series_sparse" #include "../../template/template.hpp" #include "../../fps/fps_sparse.hpp" #include "../../utility/modint.hpp" using namespace lib; using mint = modint998244353; int main() { int n, k; std::cin >> n >> k; std::vector<mint> f(n); for (int i = 0; i < k; i++) { int idx, a; std::cin >> idx >> a; f[idx] = a; } auto g = exp_sparse(f, n); for (int i = 0; i < n; i++) { std::cout << g[i].val() << " \n"[i == n - 1]; } }
#line 1 "test/polynomial/Exp_of_FPS_Sparse.test.cpp" #define PROBLEM \ "https://judge.yosupo.jp/problem/exp_of_formal_power_series_sparse" #line 2 "template/template.hpp" #include <bits/stdc++.h> #define rep(i, s, n) for (int i = (int)(s); i < (int)(n); i++) #define rrep(i, s, n) for (int i = (int)(n)-1; i >= (int)(s); i--) #define all(v) v.begin(), v.end() using ll = long long; using ld = long double; using ull = unsigned long long; template <typename T> bool chmin(T &a, const T &b) { if (a <= b) return false; a = b; return true; } template <typename T> bool chmax(T &a, const T &b) { if (a >= b) return false; a = b; return true; } namespace lib { using namespace std; } // namespace lib // using namespace lib; #line 2 "fps/fps_sparse.hpp" #line 2 "math/factorial.hpp" #line 4 "math/factorial.hpp" namespace lib { template<typename T> struct Binom{ Binom(int lim = 300000){ if (kaijo.empty()){ kaijo = {1,1}; kainv = {1,1}; } extend(lim); } static T fact(int x) { if (x < 0) return T(0); return kaijo[x]; } static T ifact(int x){ if (x < 0) return T(0); return kainv[x]; } static T C(int n, int r){ if (n < 0 || n < r || r < 0) return T(0); return kaijo[n] * kainv[r] * kainv[n-r]; } static T P(int n, int r){ if (n < 0 || n < r || r < 0) return T(0); return kaijo[n] * kainv[n-r]; } static T Inv(int n){ assert(0 < n); return ifact(n) * fact(n-1); } T operator()(int n, int r){ return C(n,r); } private: static vector<T> kaijo, kainv; static void extend(int lim){ if ((int)kaijo.size() > lim) return ; int pre = kaijo.size(); kaijo.resize(lim+1); kainv.resize(lim+1); for (int i = pre; i <= lim; i++) kaijo[i] = kaijo[i-1] * T(i); kainv[lim] = kaijo[lim].inv(); for (int i = lim-1; i >= pre; i--) kainv[i] = kainv[i+1] * T(i+1); } }; template<typename T> vector<T>Binom<T>::kaijo = vector<T>(2,T(1)); template<typename T> vector<T>Binom<T>::kainv = vector<T>(2,T(1)); } // namespace lib #line 5 "fps/fps_sparse.hpp" namespace lib { template <class mint> std::vector<mint> mul_sparse(const std::vector<mint> &f, const std::vector<mint> &g) { int n = f.size(); int m = g.size(); std::vector<std::pair<int, mint>> cf, cg; for (int i = 0; i < n; i++) { if (f[i] != 0) cf.emplace_back(i, f[i]); } for (int i = 0; i < m; i++) { if (g[i] != 0) cg.emplace_back(i, g[i]); } std::vector<mint> h(n + m - 1); for (auto [i, p] : cf) { for (auto [j, q] : cg) { h[i + j] += p * q; } } return h; } template <class mint> std::vector<mint> inv_sparse(const std::vector<mint> &f, int d = -1) { assert(f[0] != 0); if (d < 0) { d = f.size(); } std::vector<std::pair<int, mint>> ret; for (int i = 1; i < int(f.size()); i++) { if (f[i] != 0) { ret.emplace_back(i, f[i]); } } std::vector<mint> g(d); g[0] = f[0].inv(); for (int i = 1; i < d; i++) { for (auto [k, p] : ret) { if (i - k < 0) break; g[i] -= g[i - k] * p; } g[i] *= g[0]; } return g; } template <class mint> std::vector<mint> exp_sparse(const std::vector<mint> &f, int d = -1) { int n = f.size(); if (d < 0) d = n; std::vector<std::pair<int, mint>> ret; for (int i = 1; i < n; i++) { if (f[i] != 0) { ret.emplace_back(i - 1, f[i] * i); } } std::vector<mint> g(d); g[0] = 1; Binom<mint> binom(d); for (int i = 0; i < d - 1; i++) { for (auto [k, p] : ret) { if (i - k < 0) break; g[i + 1] += g[i - k] * p; } g[i + 1] *= binom.Inv(i + 1); } return g; } template <class mint> std::vector<mint> log_sparse(const std::vector<mint> &f, int d = -1) { int n = f.size(); if (d < 0) d = n; std::vector<mint> df(d); for (int i = 0; i < std::min(d, n - 1); i++) { df[i] = f[i + 1] * (i + 1); } auto dg = mul_sparse(df, inv_sparse(f)); dg.resize(d); std::vector<mint> g(d); Binom<mint> binom(d); for (int i = 0; i < d - 1; i++) { g[i + 1] = dg[i] * binom.Inv(i + 1); } return g; } template <class mint> std::vector<mint> pow_sparse_1(const std::vector<mint> &f, long long k, int d = -1) { int n = f.size(); assert(n == 0 || f[0] == 1); std::vector<std::pair<int, mint>> ret; for (int i = 1; i < n; i++) { if (f[i] != 0) ret.emplace_back(i, f[i]); } std::vector<mint> g(d); g[0] = 1; Binom<mint> binom(d); for (int i = 0; i < d - 1; i++) { for (const auto &[j, cf] : ret) { if (i + 1 - j < 0) break; g[i + 1] += (mint(k) * mint(j) - mint(i - j + 1)) * cf * g[i + 1 - j]; } g[i + 1] *= binom.Inv(i + 1); } return g; } template <class mint> std::vector<mint> pow_sparse(const std::vector<mint> &f, long long k, int d = -1) { int n = f.size(); if (d < 0) d = n; assert(k >= 0); if (k == 0) { std::vector<mint> g(d); if (d > 0) g[0] = 1; return g; } for (int i = 0; i < n; i++) { if (f[i] != 0) { mint rev = f[i].inv(); std::vector<mint> f2(n - i); for (int j = i; j < n; j++) { f2[j - i] = f[j] * rev; } f2 = pow_sparse_1(f2, k, d); mint fk = f[i].pow(k); std::vector<mint> g(d); for (int j = 0; j < int(f2.size()); j++) { if (j + i * k >= d) break; g[j + i * k] = f2[j] * fk; } return g; } if (i >= (d + k - 1) / k) break; } return std::vector<mint>(d); } } // namespace lib #line 2 "utility/modint.hpp" #line 4 "utility/modint.hpp" namespace lib { template <ll m> struct modint { using mint = modint; ll a; modint(ll x = 0) : a((x % m + m) % m) {} static constexpr ll mod() { return m; } ll val() const { return a; } ll& val() { return a; } mint pow(ll n) const { mint res = 1; mint x = a; while (n) { if (n & 1) res *= x; x *= x; n >>= 1; } return res; } mint inv() const { return pow(m - 2); } mint& operator+=(const mint rhs) { a += rhs.a; if (a >= m) a -= m; return *this; } mint& operator-=(const mint rhs) { if (a < rhs.a) a += m; a -= rhs.a; return *this; } mint& operator*=(const mint rhs) { a = a * rhs.a % m; return *this; } mint& operator/=(mint rhs) { *this *= rhs.inv(); return *this; } friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; } friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; } friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; } friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; } friend bool operator==(const modint &lhs, const modint &rhs) { return lhs.a == rhs.a; } friend bool operator!=(const modint &lhs, const modint &rhs) { return !(lhs == rhs); } mint operator+() const { return *this; } mint operator-() const { return mint() - *this; } }; using modint998244353 = modint<998244353>; using modint1000000007 = modint<1'000'000'007>; } // namespace lib #line 7 "test/polynomial/Exp_of_FPS_Sparse.test.cpp" using namespace lib; using mint = modint998244353; int main() { int n, k; std::cin >> n >> k; std::vector<mint> f(n); for (int i = 0; i < k; i++) { int idx, a; std::cin >> idx >> a; f[idx] = a; } auto g = exp_sparse(f, n); for (int i = 0; i < n; i++) { std::cout << g[i].val() << " \n"[i == n - 1]; } }