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#define PROBLEM "https://judge.yosupo.jp/problem/bitwise_and_convolution" #include "../../modint.hpp" #include "../hadamard.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 : andconv(A, B)) cout << x << ' '; }
#line 1 "convolution/test/bitwise_and_conv.test.cpp" #define PROBLEM "https://judge.yosupo.jp/problem/bitwise_and_convolution" #line 2 "modint.hpp" #include <cassert> #include <iostream> #include <set> #include <vector> 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 4 "convolution/hadamard.hpp" // CUT begin // Fast Walsh-Hadamard transform and its abstraction // Tutorials: <https://codeforces.com/blog/entry/71899> // <https://csacademy.com/blog/fast-fourier-transform-and-variations-of-it> template <typename T, typename F> void abstract_fwht(std::vector<T> &seq, F f) { const int n = seq.size(); assert(__builtin_popcount(n) == 1); for (int w = 1; w < n; w *= 2) { for (int i = 0; i < n; i += w * 2) { for (int j = 0; j < w; j++) { f(seq[i + j], seq[i + j + w]); } } } } template <typename T, typename F1, typename F2> std::vector<T> bitwise_conv(std::vector<T> x, std::vector<T> y, F1 f, F2 finv) { const int n = x.size(); assert(__builtin_popcount(n) == 1); assert(x.size() == y.size()); if (x == y) { abstract_fwht(x, f), y = x; } else { abstract_fwht(x, f), abstract_fwht(y, f); } for (size_t i = 0; i < x.size(); i++) { x[i] *= y[i]; } abstract_fwht(x, finv); return x; } // bitwise xor convolution (FWHT-based) // ret[i] = \sum_j x[j] * y[i ^ j] // if T is integer, ||x||_1 * ||y||_1 * 2 < numeric_limits<T>::max() template <typename T> std::vector<T> xorconv(std::vector<T> x, std::vector<T> y) { auto f = [](T &lo, T &hi) { T c = lo + hi; hi = lo - hi, lo = c; }; auto finv = [](T &lo, T &hi) { T c = lo + hi; hi = (lo - hi) / 2, lo = c / 2; // Reconsider HEAVY complexity of division by 2 when T is ModInt }; return bitwise_conv(x, y, f, finv); } // bitwise AND conolution // ret[i] = \sum_{(j & k) == i} x[j] * y[k] template <typename T> std::vector<T> andconv(std::vector<T> x, std::vector<T> y) { return bitwise_conv( x, y, [](T &lo, T &hi) { lo += hi; }, [](T &lo, T &hi) { lo -= hi; }); } // bitwise OR convolution // ret[i] = \sum_{(j | k) == i} x[j] * y[k] template <typename T> std::vector<T> orconv(std::vector<T> x, std::vector<T> y) { return bitwise_conv( x, y, [](T &lo, T &hi) { hi += lo; }, [](T &lo, T &hi) { hi -= lo; }); } #line 5 "convolution/test/bitwise_and_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 : andconv(A, B)) cout << x << ' '; }