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#define PROBLEM \ "https://judge.yosupo.jp/problem/sqrt_of_formal_power_series_sparse" #include "../../convolution/ntt.hpp" #include "../../fps/fps.hpp" #include "../../fps/fps_sqrt.hpp" #include "../../modint/modint.hpp" using namespace ebi; using mint = modint998244353; using FPS = FormalPowerSeries<mint, convolution>; int main() { int n, k; std::cin >> n >> k; FPS f(n); for (int i = 0; i < k; i++) { int idx, a; std::cin >> idx >> a; f[idx] = a; } auto opt = f.sqrt(n); if (!opt) { std::cout << "-1\n"; return 0; } auto g = opt.value(); for (int i = 0; i < n; i++) { std::cout << g[i].val() << " \n"[i == n - 1]; } }
#line 1 "test/polynomial/Sqrt_of_Formal_Power_Series_Sparse.test.cpp" #define PROBLEM \ "https://judge.yosupo.jp/problem/sqrt_of_formal_power_series_sparse" #line 2 "convolution/ntt.hpp" #include <algorithm> #include <array> #include <bit> #include <cassert> #include <vector> #line 2 "math/internal_math.hpp" #line 4 "math/internal_math.hpp" namespace ebi { namespace internal { constexpr int primitive_root_constexpr(int m) { if (m == 2) return 1; if (m == 167772161) return 3; if (m == 469762049) return 3; if (m == 754974721) return 11; if (m == 998244353) return 3; if (m == 880803841) return 26; if (m == 924844033) return 5; return -1; } template <int m> constexpr int primitive_root = primitive_root_constexpr(m); } // namespace internal } // namespace ebi #line 2 "modint/base.hpp" #include <concepts> #include <iostream> #include <utility> namespace ebi { template <class T> concept Modint = requires(T a, T b) { a + b; a - b; a * b; a / b; a.inv(); a.val(); a.pow(std::declval<long long>()); T::mod(); }; template <Modint mint> std::istream &operator>>(std::istream &os, mint &a) { long long x; os >> x; a = x; return os; } template <Modint mint> std::ostream &operator<<(std::ostream &os, const mint &a) { return os << a.val(); } } // namespace ebi #line 11 "convolution/ntt.hpp" namespace ebi { namespace internal { template <Modint mint, int g = internal::primitive_root<mint::mod()>> struct ntt_info { static constexpr int rank2 = std::countr_zero((unsigned int)(mint::mod() - 1)); std::array<mint, rank2 + 1> root, inv_root; ntt_info() { root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2); inv_root[rank2] = root[rank2].inv(); for (int i = rank2 - 1; i >= 0; i--) { root[i] = root[i + 1] * root[i + 1]; inv_root[i] = inv_root[i + 1] * inv_root[i + 1]; } } }; template <Modint mint> void butterfly(std::vector<mint>& a) { static const ntt_info<mint> info; int n = int(a.size()); int bit_size = std::countr_zero(a.size()); assert(n == (int)std::bit_ceil(a.size())); // bit reverse for (int i = 0, j = 1; j < n - 1; j++) { for (int k = n >> 1; k > (i ^= k); k >>= 1) ; if (j < i) { std::swap(a[i], a[j]); } } for (int bit = 0; bit < bit_size; bit++) { for (int i = 0; i < n / (1 << (bit + 1)); i++) { mint zeta1 = 1; mint zeta2 = info.root[1]; for (int j = 0; j < (1 << bit); j++) { int idx = i * (1 << (bit + 1)) + j; int jdx = idx + (1 << bit); mint p1 = a[idx]; mint p2 = a[jdx]; a[idx] = p1 + zeta1 * p2; a[jdx] = p1 + zeta2 * p2; zeta1 *= info.root[bit + 1]; zeta2 *= info.root[bit + 1]; } } } } template <Modint mint> void butterfly_inv(std::vector<mint>& a) { static const ntt_info<mint> info; int n = int(a.size()); int bit_size = std::countr_zero(a.size()); assert(n == (int)std::bit_ceil(a.size())); // bit reverse for (int i = 0, j = 1; j < n - 1; j++) { for (int k = n >> 1; k > (i ^= k); k >>= 1) ; if (j < i) { std::swap(a[i], a[j]); } } for (int bit = 0; bit < bit_size; bit++) { for (int i = 0; i < n / (1 << (bit + 1)); i++) { mint zeta1 = 1; mint zeta2 = info.inv_root[1]; for (int j = 0; j < (1 << bit); j++) { int idx = i * (1 << (bit + 1)) + j; int jdx = idx + (1 << bit); mint p1 = a[idx]; mint p2 = a[jdx]; a[idx] = p1 + zeta1 * p2; a[jdx] = p1 + zeta2 * p2; zeta1 *= info.inv_root[bit + 1]; zeta2 *= info.inv_root[bit + 1]; } } } mint inv_n = mint(n).inv(); for (int i = 0; i < n; i++) { a[i] *= inv_n; } } } // namespace internal template <Modint mint> std::vector<mint> convolution_naive(const std::vector<mint>& f, const std::vector<mint>& g) { if (f.empty() || g.empty()) return {}; int n = int(f.size()), m = int(g.size()); std::vector<mint> c(n + m - 1); if (n < m) { for (int j = 0; j < m; j++) { for (int i = 0; i < n; i++) { c[i + j] += f[i] * g[j]; } } } else { for (int i = 0; i < n; i++) { for (int j = 0; j < m; j++) { c[i + j] += f[i] * g[j]; } } } return c; } template <Modint mint> std::vector<mint> convolution(const std::vector<mint>& f, const std::vector<mint>& g) { if (f.empty() || g.empty()) return {}; if (std::min(f.size(), g.size()) < 60) return convolution_naive(f, g); int n = std::bit_ceil(f.size() + g.size() - 1); std::vector<mint> a(n), b(n); std::copy(f.begin(), f.end(), a.begin()); std::copy(g.begin(), g.end(), b.begin()); internal::butterfly(a); internal::butterfly(b); for (int i = 0; i < n; i++) { a[i] *= b[i]; } internal::butterfly_inv(a); a.resize(f.size() + g.size() - 1); return a; } } // namespace ebi #line 2 "fps/fps.hpp" #line 5 "fps/fps.hpp" #include <optional> #line 7 "fps/fps.hpp" #line 9 "fps/fps.hpp" namespace ebi { template <Modint mint, std::vector<mint> (*convolution)(const std::vector<mint> &, const std::vector<mint> &)> struct FormalPowerSeries : std::vector<mint> { private: using std::vector<mint>::vector; using std::vector<mint>::vector::operator=; using FPS = FormalPowerSeries; public: FormalPowerSeries(const std::vector<mint> &a) { *this = a; } FPS operator+(const FPS &rhs) const noexcept { return FPS(*this) += rhs; } FPS operator-(const FPS &rhs) const noexcept { return FPS(*this) -= rhs; } FPS operator*(const FPS &rhs) const noexcept { return FPS(*this) *= rhs; } FPS operator/(const FPS &rhs) const noexcept { return FPS(*this) /= rhs; } FPS operator%(const FPS &rhs) const noexcept { return FPS(*this) %= rhs; } FPS operator+(const mint &rhs) const noexcept { return FPS(*this) += rhs; } FPS operator-(const mint &rhs) const noexcept { return FPS(*this) -= rhs; } FPS operator*(const mint &rhs) const noexcept { return FPS(*this) *= rhs; } FPS operator/(const mint &rhs) const noexcept { return FPS(*this) /= rhs; } FPS &operator+=(const FPS &rhs) noexcept { if (this->size() < rhs.size()) this->resize(rhs.size()); for (int i = 0; i < (int)rhs.size(); ++i) { (*this)[i] += rhs[i]; } return *this; } FPS &operator-=(const FPS &rhs) noexcept { if (this->size() < rhs.size()) this->resize(rhs.size()); for (int i = 0; i < (int)rhs.size(); ++i) { (*this)[i] -= rhs[i]; } return *this; } FPS &operator*=(const FPS &rhs) noexcept { *this = convolution(*this, rhs); return *this; } FPS &operator/=(const FPS &rhs) noexcept { int n = deg() - 1; int m = rhs.deg() - 1; if (n < m) { *this = {}; return *this; } *this = (*this).rev() * rhs.rev().inv(n - m + 1); (*this).resize(n - m + 1); std::reverse((*this).begin(), (*this).end()); return *this; } FPS &operator%=(const FPS &rhs) noexcept { *this -= *this / rhs * rhs; shrink(); return *this; } FPS &operator+=(const mint &rhs) noexcept { if (this->empty()) this->resize(1); (*this)[0] += rhs; return *this; } FPS &operator-=(const mint &rhs) noexcept { if (this->empty()) this->resize(1); (*this)[0] -= rhs; return *this; } FPS &operator*=(const mint &rhs) noexcept { for (int i = 0; i < deg(); ++i) { (*this)[i] *= rhs; } return *this; } FPS &operator/=(const mint &rhs) noexcept { mint inv_rhs = rhs.inv(); for (int i = 0; i < deg(); ++i) { (*this)[i] *= inv_rhs; } return *this; } FPS operator>>(int d) const { if (deg() <= d) return {}; FPS f = *this; f.erase(f.begin(), f.begin() + d); return f; } FPS operator<<(int d) const { FPS f = *this; f.insert(f.begin(), d, 0); return f; } FPS operator-() const { FPS g(this->size()); for (int i = 0; i < (int)this->size(); i++) g[i] = -(*this)[i]; return g; } FPS pre(int sz) const { return FPS(this->begin(), this->begin() + std::min(deg(), sz)); } FPS rev() const { auto f = *this; std::reverse(f.begin(), f.end()); return f; } FPS differential() const { int n = deg(); FPS g(std::max(0, n - 1)); for (int i = 0; i < n - 1; i++) { g[i] = (*this)[i + 1] * (i + 1); } return g; } FPS integral() const { int n = deg(); FPS g(n + 1); g[0] = 0; if (n > 0) g[1] = 1; auto mod = mint::mod(); for (int i = 2; i <= n; i++) g[i] = (-g[mod % i]) * (mod / i); for (int i = 0; i < n; i++) g[i + 1] *= (*this)[i]; return g; } FPS inv(int d = -1) const { int n = 1; if (d < 0) d = deg(); FPS g(n); g[0] = (*this)[0].inv(); while (n < d) { n <<= 1; g = (g * 2 - g * g * this->pre(n)).pre(n); } g.resize(d); return g; } FPS log(int d = -1) const { assert((*this)[0].val() == 1); if (d < 0) d = deg(); return ((*this).differential() * (*this).inv(d)).pre(d - 1).integral(); } FPS exp(int d = -1) const { assert((*this)[0].val() == 0); int n = 1; if (d < 0) d = deg(); FPS g(n); g[0] = 1; while (n < d) { n <<= 1; g = (g * (this->pre(n) - g.log(n) + 1)).pre(n); } g.resize(d); return g; } FPS pow(int64_t k, int d = -1) const { const int n = deg(); if (d < 0) d = n; if (k == 0) { FPS f(d); if (d > 0) f[0] = 1; return f; } for (int i = 0; i < n; i++) { if ((*this)[i] != 0) { mint rev = (*this)[i].inv(); FPS f = (((*this * rev) >> i).log(d) * k).exp(d); f *= (*this)[i].pow(k); f = (f << (i * k)).pre(d); if (f.deg() < d) f.resize(d); return f; } if (i + 1 >= (d + k - 1) / k) break; } return FPS(d); } int deg() const { return (*this).size(); } void shrink() { while ((!this->empty()) && this->back() == 0) this->pop_back(); } int count_terms() const { int c = 0; for (int i = 0; i < deg(); i++) { if ((*this)[i] != 0) c++; } return c; } std::optional<FPS> sqrt(int d = -1) const; static FPS exp_x(int n) { FPS f(n); mint fact = 1; for (int i = 1; i < n; i++) fact *= i; f[n - 1] = fact.inv(); for (int i = n - 1; i >= 0; i--) f[i - 1] = f[i] * i; return f; } }; } // namespace ebi #line 2 "fps/fps_sqrt.hpp" #line 2 "fps/fps_sparse.hpp" #line 5 "fps/fps_sparse.hpp" #line 2 "math/mod_inv.hpp" #line 5 "math/mod_inv.hpp" #line 7 "math/mod_inv.hpp" namespace ebi { template <Modint mint> mint inv(int n) { static const int mod = mint::mod(); static std::vector<mint> dat = {0, 1}; assert(0 <= n); if (n >= mod) n -= mod; while (int(dat.size()) <= n) { int num = dat.size(); int q = (mod + num - 1) / num; dat.emplace_back(dat[num * q - mod] * mint(q)); } return dat[n]; } } // namespace ebi #line 8 "fps/fps_sparse.hpp" namespace ebi { template <Modint 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 <Modint 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 <Modint 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; 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] *= inv<mint>(i + 1); } return g; } template <Modint 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); for (int i = 0; i < d - 1; i++) { g[i + 1] = dg[i] * inv<mint>(i + 1); } return g; } template <Modint 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; 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] *= inv<mint>(i + 1); } return g; } template <Modint 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 ebi #line 2 "math/mod_sqrt.hpp" #include <cstdint> #line 5 "math/mod_sqrt.hpp" #line 2 "modint/dynamic_modint.hpp" #line 4 "modint/dynamic_modint.hpp" #line 6 "modint/dynamic_modint.hpp" namespace ebi { template <int id> struct dynamic_modint { private: using modint = dynamic_modint; public: static void set_mod(int p) { assert(1 <= p); m = p; } static int mod() { return m; } modint raw(int v) { modint x; x._v = v; return x; } dynamic_modint() : _v(0) {} dynamic_modint(long long v) { v %= (long long)umod(); if (v < 0) v += (long long)umod(); _v = (unsigned int)v; } unsigned int val() const { return _v; } unsigned int value() const { return val(); } modint &operator++() { _v++; if (_v == umod()) _v = 0; return *this; } modint &operator--() { if (_v == 0) _v = umod(); _v--; return *this; } modint &operator+=(const modint &rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } modint &operator-=(const modint &rhs) { _v -= rhs._v; if (_v >= umod()) _v += umod(); return *this; } modint &operator*=(const modint &rhs) { unsigned long long x = _v; x *= rhs._v; _v = (unsigned int)(x % (unsigned long long)umod()); return *this; } modint &operator/=(const modint &rhs) { return *this = *this * rhs.inv(); } modint operator+() const { return *this; } modint operator-() const { return modint() - *this; } modint pow(long long n) const { assert(0 <= n); modint x = *this, res = 1; while (n) { if (n & 1) res *= x; x *= x; n >>= 1; } return res; } modint inv() const { assert(_v); return pow(umod() - 2); } friend modint operator+(const modint &lhs, const modint &rhs) { return modint(lhs) += rhs; } friend modint operator-(const modint &lhs, const modint &rhs) { return modint(lhs) -= rhs; } friend modint operator*(const modint &lhs, const modint &rhs) { return modint(lhs) *= rhs; } friend modint operator/(const modint &lhs, const modint &rhs) { return modint(lhs) /= rhs; } friend bool operator==(const modint &lhs, const modint &rhs) { return lhs.val() == rhs.val(); } friend bool operator!=(const modint &lhs, const modint &rhs) { return !(lhs == rhs); } private: unsigned int _v = 0; static int m; static unsigned int umod() { return m; } }; template <int id> int dynamic_modint<id>::m = 998244353; } // namespace ebi #line 7 "math/mod_sqrt.hpp" namespace ebi { std::optional<std::int64_t> mod_sqrt(const std::int64_t &a, const std::int64_t &p) { if (a == 0 || a == 1) return a; using mint = dynamic_modint<100>; mint::set_mod(p); if (mint(a).pow((p - 1) >> 1) != 1) return std::nullopt; mint b = 1; while (b.pow((p - 1) >> 1) == 1) b += 1; std::int64_t m = p - 1, e = 0; while (m % 2 == 0) m >>= 1, e++; mint x = mint(a).pow((m - 1) >> 1); mint y = mint(a) * x * x; x *= a; mint z = b.pow(m); while (y != 1) { std::int64_t j = 0; mint t = y; while (t != 1) { j++; t *= t; } z = z.pow(1ll << (e - j - 1)); x *= z; z *= z; y *= z; e = j; } return x.val(); } } // namespace ebi #line 7 "fps/fps_sqrt.hpp" namespace ebi { template <Modint mint, std::vector<mint> (*convolution)(const std::vector<mint> &, const std::vector<mint> &)> std::optional<FormalPowerSeries<mint, convolution>> FormalPowerSeries<mint, convolution>::sqrt(int d) const { using FPS = FormalPowerSeries<mint, convolution>; if (d < 0) d = deg(); if ((*this)[0] == 0) { for (int i = 1; i < this->deg(); i++) { if ((*this)[i] != 0) { if (i & 1) return std::nullopt; if (d - i / 2 <= 0) break; auto opt = ((*this) >> i).sqrt(d - i / 2); if (!opt) return std::nullopt; auto ret = opt.value() << (i / 2); if ((int)ret.deg() < d) ret.resize(d); return ret; } } return FPS(d, 0); } auto s = mod_sqrt((*this)[0].val(), mint::mod()); if (!s) { return std::nullopt; } if (this->count_terms() <= 200) { mint y = s.value(); std::vector<mint> sqrt_f = pow_sparse_1(*this / (*this)[0], mint(2).inv().val(), d); FPS g(d); for (int i = 0; i < d; i++) g[i] = sqrt_f[i] * y; return g; } int n = 1; FPS g(n); g[0] = s.value(); mint inv_two = mint(2).inv(); while (n < d) { n <<= 1; g = (g + this->pre(n) * g.inv(n)).pre(n) * inv_two; } g.resize(d); return g; } } // namespace ebi #line 2 "modint/modint.hpp" #line 5 "modint/modint.hpp" #line 7 "modint/modint.hpp" namespace ebi { template <int m> struct static_modint { private: using modint = static_modint; public: static constexpr int mod() { return m; } static constexpr modint raw(int v) { modint x; x._v = v; return x; } constexpr static_modint() : _v(0) {} constexpr static_modint(long long v) { v %= (long long)umod(); if (v < 0) v += (long long)umod(); _v = (unsigned int)v; } constexpr unsigned int val() const { return _v; } constexpr unsigned int value() const { return val(); } constexpr modint &operator++() { _v++; if (_v == umod()) _v = 0; return *this; } constexpr modint &operator--() { if (_v == 0) _v = umod(); _v--; return *this; } constexpr modint operator++(int) { modint res = *this; ++*this; return res; } constexpr modint operator--(int) { modint res = *this; --*this; return res; } constexpr modint &operator+=(const modint &rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } constexpr modint &operator-=(const modint &rhs) { _v -= rhs._v; if (_v >= umod()) _v += umod(); return *this; } constexpr modint &operator*=(const modint &rhs) { unsigned long long x = _v; x *= rhs._v; _v = (unsigned int)(x % (unsigned long long)umod()); return *this; } constexpr modint &operator/=(const modint &rhs) { return *this = *this * rhs.inv(); } constexpr modint operator+() const { return *this; } constexpr modint operator-() const { return modint() - *this; } constexpr modint pow(long long n) const { assert(0 <= n); modint x = *this, res = 1; while (n) { if (n & 1) res *= x; x *= x; n >>= 1; } return res; } constexpr modint inv() const { assert(_v); return pow(umod() - 2); } friend modint operator+(const modint &lhs, const modint &rhs) { return modint(lhs) += rhs; } friend modint operator-(const modint &lhs, const modint &rhs) { return modint(lhs) -= rhs; } friend modint operator*(const modint &lhs, const modint &rhs) { return modint(lhs) *= rhs; } friend modint operator/(const modint &lhs, const modint &rhs) { return modint(lhs) /= rhs; } friend bool operator==(const modint &lhs, const modint &rhs) { return lhs.val() == rhs.val(); } friend bool operator!=(const modint &lhs, const modint &rhs) { return !(lhs == rhs); } private: unsigned int _v = 0; static constexpr unsigned int umod() { return m; } }; using modint998244353 = static_modint<998244353>; using modint1000000007 = static_modint<1000000007>; } // namespace ebi #line 8 "test/polynomial/Sqrt_of_Formal_Power_Series_Sparse.test.cpp" using namespace ebi; using mint = modint998244353; using FPS = FormalPowerSeries<mint, convolution>; int main() { int n, k; std::cin >> n >> k; FPS f(n); for (int i = 0; i < k; i++) { int idx, a; std::cin >> idx >> a; f[idx] = a; } auto opt = f.sqrt(n); if (!opt) { std::cout << "-1\n"; return 0; } auto g = opt.value(); for (int i = 0; i < n; i++) { std::cout << g[i].val() << " \n"[i == n - 1]; } }