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#define PROBLEM "https://judge.yosupo.jp/problem/subset_convolution" #include "../subset_convolution.hpp" #include "modint.hpp" #include <iostream> using namespace std; int main() { cin.tie(nullptr), ios::sync_with_stdio(false); int N; cin >> N; vector<ModInt<998244353>> A(1 << N), B(1 << N); for (auto &x : A) cin >> x; for (auto &x : B) cin >> x; for (auto x : subset_convolution(A, B)) cout << x << ' '; }
#line 1 "set_power_series/test/subset_conv.test.cpp" #define PROBLEM "https://judge.yosupo.jp/problem/subset_convolution" #line 2 "set_power_series/subset_convolution.hpp" #include <algorithm> #include <cassert> #include <vector> // CUT begin // Subset sum (fast zeta transform) // Complexity: O(N 2^N) for array of size 2^N template <typename T> void subset_sum(std::vector<T> &f) { const int sz = f.size(), n = __builtin_ctz(sz); assert(__builtin_popcount(sz) == 1); for (int d = 0; d < n; d++) { for (int S = 0; S < 1 << n; S++) if (S & (1 << d)) f[S] += f[S ^ (1 << d)]; } } // Inverse of subset sum (fast moebius transform) // Complexity: O(N 2^N) for array of size 2^N template <typename T> void subset_sum_inv(std::vector<T> &g) { const int sz = g.size(), n = __builtin_ctz(sz); assert(__builtin_popcount(sz) == 1); for (int d = 0; d < n; d++) { for (int S = 0; S < 1 << n; S++) if (S & (1 << d)) g[S] -= g[S ^ (1 << d)]; } } // Superset sum / its inverse (fast zeta/moebius transform) // Complexity: O(N 2^N) for array of size 2^N template <typename T> void superset_sum(std::vector<T> &f) { const int sz = f.size(), n = __builtin_ctz(sz); assert(__builtin_popcount(sz) == 1); for (int d = 0; d < n; d++) { for (int S = 0; S < 1 << n; S++) if (!(S & (1 << d))) f[S] += f[S | (1 << d)]; } } template <typename T> void superset_sum_inv(std::vector<T> &g) { const int sz = g.size(), n = __builtin_ctz(sz); assert(__builtin_popcount(sz) == 1); for (int d = 0; d < n; d++) { for (int S = 0; S < 1 << n; S++) if (!(S & (1 << d))) g[S] -= g[S | (1 << d)]; } } template <typename T> std::vector<std::vector<T>> build_zeta_(int D, const std::vector<T> &f) { int n = f.size(); std::vector<std::vector<T>> ret(D, std::vector<T>(n)); for (int i = 0; i < n; i++) ret[__builtin_popcount(i)][i] += f[i]; for (auto &vec : ret) subset_sum(vec); return ret; } template <typename T> std::vector<T> get_moebius_of_prod_(const std::vector<std::vector<T>> &mat1, const std::vector<std::vector<T>> &mat2) { int D = mat1.size(), n = mat1[0].size(); std::vector<std::vector<int>> pc2i(D); for (int i = 0; i < n; i++) pc2i[__builtin_popcount(i)].push_back(i); std::vector<T> tmp, ret(mat1[0].size()); for (int d = 0; d < D; d++) { tmp.assign(mat1[d].size(), 0); for (int e = 0; e <= d; e++) { for (int i = 0; i < int(tmp.size()); i++) tmp[i] += mat1[e][i] * mat2[d - e][i]; } subset_sum_inv(tmp); for (auto i : pc2i[d]) ret[i] = tmp[i]; } return ret; }; // Subset convolution // h[S] = \sum_T f[T] * g[S - T] // Complexity: O(N^2 2^N) for arrays of size 2^N template <typename T> std::vector<T> subset_convolution(std::vector<T> f, std::vector<T> g) { const int sz = f.size(), m = __builtin_ctz(sz) + 1; assert(__builtin_popcount(sz) == 1 and f.size() == g.size()); auto ff = build_zeta_(m, f), fg = build_zeta_(m, g); return get_moebius_of_prod_(ff, fg); } // https://hos-lyric.hatenablog.com/entry/2021/01/14/201231 template <class T, class Function> void subset_func(std::vector<T> &f, const Function &func) { const int sz = f.size(), m = __builtin_ctz(sz) + 1; assert(__builtin_popcount(sz) == 1); auto ff = build_zeta_(m, f); std::vector<T> p(m); for (int i = 0; i < sz; i++) { for (int d = 0; d < m; d++) p[d] = ff[d][i]; func(p); for (int d = 0; d < m; d++) ff[d][i] = p[d]; } for (auto &vec : ff) subset_sum_inv(vec); for (int i = 0; i < sz; i++) f[i] = ff[__builtin_popcount(i)][i]; } // log(f(x)) for f(x), f(0) == 1 // Requires inv() template <class T> void poly_log(std::vector<T> &f) { assert(f.at(0) == T(1)); static std::vector<T> invs{0}; const int m = f.size(); std::vector<T> finv(m); for (int d = 0; d < m; d++) { finv[d] = (d == 0); if (int(invs.size()) <= d) invs.push_back(T(d).inv()); for (int e = 0; e < d; e++) finv[d] -= finv[e] * f[d - e]; } std::vector<T> ret(m); for (int d = 1; d < m; d++) { for (int e = 0; d + e < m; e++) ret[d + e] += f[d] * d * finv[e] * invs[d + e]; } f = ret; } // log(f(S)) for set function f(S), f(0) == 1 // Requires inv() // Complexity: O(n^2 2^n) // https://atcoder.jp/contests/abc213/tasks/abc213_g template <class T> void subset_log(std::vector<T> &f) { subset_func(f, poly_log<T>); } // exp(f(S)) for set function f(S), f(0) == 0 // Complexity: O(n^2 2^n) // https://codeforces.com/blog/entry/92183 template <class T> void subset_exp(std::vector<T> &f) { const int sz = f.size(), m = __builtin_ctz(sz); assert(sz == 1 << m); assert(f.at(0) == 0); std::vector<T> ret{T(1)}; ret.reserve(sz); for (int d = 0; d < m; d++) { auto c = subset_convolution({f.begin() + (1 << d), f.begin() + (1 << (d + 1))}, ret); ret.insert(ret.end(), c.begin(), c.end()); } f = ret; } // sqrt(f(x)), f(x) == 1 // Requires inv of 2 // Compelxity: O(n^2) template <class T> void poly_sqrt(std::vector<T> &f) { assert(f.at(0) == T(1)); const int m = f.size(); static const auto inv2 = T(2).inv(); for (int d = 1; d < m; d++) { if (~(d & 1)) f[d] -= f[d / 2] * f[d / 2]; f[d] *= inv2; for (int e = 1; e < d - e; e++) f[d] -= f[e] * f[d - e]; } } // sqrt(f(S)) for set function f(S), f(0) == 1 // Requires inv() // https://atcoder.jp/contests/xmascon20/tasks/xmascon20_h template <class T> void subset_sqrt(std::vector<T> &f) { subset_func(f, poly_sqrt<T>); } // exp(f(S)) for set function f(S), f(0) == 0 template <class T> void poly_exp(std::vector<T> &P) { const int m = P.size(); assert(m and P[0] == 0); std::vector<T> Q(m), logQ(m), Qinv(m); Q[0] = Qinv[0] = T(1); static std::vector<T> invs{0}; auto set_invlog = [&](int d) { Qinv[d] = 0; for (int e = 0; e < d; e++) Qinv[d] -= Qinv[e] * Q[d - e]; while (d >= int(invs.size())) { int sz = invs.size(); invs.push_back(T(sz).inv()); } logQ[d] = 0; for (int e = 1; e <= d; e++) logQ[d] += Q[e] * e * Qinv[d - e]; logQ[d] *= invs[d]; }; for (int d = 1; d < m; d++) { Q[d] += P[d] - logQ[d]; set_invlog(d); assert(logQ[d] == P[d]); if (d + 1 < m) set_invlog(d + 1); } P = Q; } // f(S)^k for set function f(S) // Requires inv() template <class T> void subset_pow(std::vector<T> &f, long long k) { auto poly_pow = [&](std::vector<T> &f) { const int m = f.size(); if (k == 0) f[0] = 1, std::fill(f.begin() + 1, f.end(), T(0)); if (k <= 1) return; int nzero = 0; while (nzero < int(f.size()) and f[nzero] == T(0)) nzero++; int rem = std::max<long long>((long long)f.size() - nzero * k, 0LL); if (rem == 0) { std::fill(f.begin(), f.end(), 0); return; } f.erase(f.begin(), f.begin() + nzero); f.resize(rem); const T f0 = f.at(0), f0inv = f0.inv(), f0pow = f0.pow(k); for (auto &x : f) x *= f0inv; poly_log(f); for (auto &x : f) x *= k; poly_exp(f); for (auto &x : f) x *= f0pow; f.resize(rem, 0); f.insert(f.begin(), m - int(f.size()), T(0)); }; subset_func(f, poly_pow); } #line 3 "modint.hpp" #include <iostream> #include <set> #line 6 "modint.hpp" template <int md> struct ModInt { using lint = long long; constexpr static int mod() { return md; } static int get_primitive_root() { static int primitive_root = 0; if (!primitive_root) { primitive_root = [&]() { std::set<int> 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_; int val() const noexcept { return val_; } constexpr ModInt() : val_(0) {} constexpr ModInt &_setval(lint v) { return val_ = (v >= md ? v - md : v), *this; } constexpr ModInt(lint v) { _setval(v % md + md); } constexpr explicit operator bool() const { return val_ != 0; } constexpr ModInt operator+(const ModInt &x) const { return ModInt()._setval((lint)val_ + x.val_); } constexpr ModInt operator-(const ModInt &x) const { return ModInt()._setval((lint)val_ - x.val_ + md); } constexpr ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val_ * x.val_ % md); } constexpr ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val_ * x.inv().val() % md); } constexpr ModInt operator-() const { return ModInt()._setval(md - val_); } constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; } constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; } constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; } constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; } friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt(a) + x; } friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt(a) - x; } friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt(a) * x; } friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt(a) / x; } constexpr bool operator==(const ModInt &x) const { return val_ == x.val_; } constexpr bool operator!=(const ModInt &x) const { return val_ != x.val_; } constexpr bool operator<(const ModInt &x) const { return val_ < x.val_; } // To use std::map<ModInt, T> friend std::istream &operator>>(std::istream &is, ModInt &x) { lint t; return is >> t, x = ModInt(t), is; } constexpr friend std::ostream &operator<<(std::ostream &os, const ModInt &x) { return os << x.val_; } constexpr ModInt pow(lint n) const { ModInt ans = 1, tmp = *this; while (n) { if (n & 1) ans *= tmp; tmp *= tmp, n >>= 1; } return ans; } static constexpr int cache_limit = std::min(md, 1 << 21); static std::vector<ModInt> facs, facinvs, invs; constexpr static void _precalculation(int N) { const 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]; } constexpr ModInt inv() const { if (this->val_ < cache_limit) { if (facs.empty()) facs = {1}, facinvs = {1}, invs = {0}; while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2); return invs[this->val_]; } else { return this->pow(md - 2); } } constexpr ModInt fac() const { while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2); return facs[this->val_]; } constexpr ModInt facinv() const { while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2); return facinvs[this->val_]; } constexpr 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); } constexpr ModInt nCr(int r) const { if (r < 0 or this->val_ < r) return ModInt(0); return this->fac() * (*this - r).facinv() * ModInt(r).facinv(); } constexpr ModInt nPr(int r) const { if (r < 0 or this->val_ < r) return ModInt(0); return this->fac() * (*this - r).facinv(); } static ModInt binom(int n, int r) { static long long bruteforce_times = 0; if (r < 0 or n < r) return ModInt(0); if (n <= bruteforce_times or n < (int)facs.size()) return ModInt(n).nCr(r); r = std::min(r, n - r); ModInt ret = ModInt(r).facinv(); for (int i = 0; i < r; ++i) ret *= n - i; bruteforce_times += r; return ret; } // Multinomial coefficient, (k_1 + k_2 + ... + k_m)! / (k_1! k_2! ... k_m!) // Complexity: O(sum(ks)) template <class Vec> static ModInt multinomial(const Vec &ks) { ModInt ret{1}; int sum = 0; for (int k : ks) { assert(k >= 0); ret *= ModInt(k).facinv(), sum += k; } return ret * ModInt(sum).fac(); } // Catalan number, C_n = binom(2n, n) / (n + 1) // C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, ... // https://oeis.org/A000108 // Complexity: O(n) static ModInt catalan(int n) { if (n < 0) return ModInt(0); return ModInt(n * 2).fac() * ModInt(n + 1).facinv() * ModInt(n).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 <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1}; template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1}; template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0}; using ModInt998244353 = ModInt<998244353>; // using mint = ModInt<998244353>; // using mint = ModInt<1000000007>; #line 5 "set_power_series/test/subset_conv.test.cpp" using namespace std; int main() { cin.tie(nullptr), ios::sync_with_stdio(false); int N; cin >> N; vector<ModInt<998244353>> A(1 << N), B(1 << N); for (auto &x : A) cin >> x; for (auto &x : B) cin >> x; for (auto x : subset_convolution(A, B)) cout << x << ' '; }