#define PROBLEM "https://judge.yosupo.jp/problem/convolution_mod" #include "../../template/template.hpp" // #include "../../misc/fastio.hpp" #include "../../modint/montgomery-modint.hpp" #include "../../ntt/cooley-tukey-ntt.hpp" using namespace Nyaan; void Nyaan::solve() { using mint = LazyMontgomeryModInt<998244353>; ArbitraryLengthNTT<mint> ntt(7 * 17 * (1 << 14)); int N, M, x; rd(N, M); V<mint> a(N), b(M); for (int i = 0; i < N; i++) rd(x), a[i] = x; for (int i = 0; i < M; i++) rd(x), b[i] = x; auto c = ntt.multiply(a, b); for (int i = 0; i < (int)N + M - 1; i++) { wt(c[i].get(), (i == (int)N + M - 1 ? '\n' : ' ')); } }
#line 1 "verify/verify-yosupo-ntt/yosupo-convolution-arbitrarylengthntt.test.cpp" #define PROBLEM "https://judge.yosupo.jp/problem/convolution_mod" #line 2 "template/template.hpp" using namespace std; // intrinstic #include <immintrin.h> #include <algorithm> #include <array> #include <bitset> #include <cassert> #include <cctype> #include <cfenv> #include <cfloat> #include <chrono> #include <cinttypes> #include <climits> #include <cmath> #include <complex> #include <cstdarg> #include <cstddef> #include <cstdint> #include <cstdio> #include <cstdlib> #include <cstring> #include <deque> #include <fstream> #include <functional> #include <initializer_list> #include <iomanip> #include <ios> #include <iostream> #include <istream> #include <iterator> #include <limits> #include <list> #include <map> #include <memory> #include <new> #include <numeric> #include <ostream> #include <queue> #include <random> #include <set> #include <sstream> #include <stack> #include <streambuf> #include <string> #include <tuple> #include <type_traits> #include <typeinfo> #include <unordered_map> #include <unordered_set> #include <utility> #include <vector> // utility #line 1 "template/util.hpp" namespace Nyaan { using ll = long long; using i64 = long long; using u64 = unsigned long long; using i128 = __int128_t; using u128 = __uint128_t; template <typename T> using V = vector<T>; template <typename T> using VV = vector<vector<T>>; using vi = vector<int>; using vl = vector<long long>; using vd = V<double>; using vs = V<string>; using vvi = vector<vector<int>>; using vvl = vector<vector<long long>>; template <typename T> using minpq = priority_queue<T, vector<T>, greater<T>>; template <typename T, typename U> struct P : pair<T, U> { template <typename... Args> P(Args... args) : pair<T, U>(args...) {} using pair<T, U>::first; using pair<T, U>::second; P &operator+=(const P &r) { first += r.first; second += r.second; return *this; } P &operator-=(const P &r) { first -= r.first; second -= r.second; return *this; } P &operator*=(const P &r) { first *= r.first; second *= r.second; return *this; } template <typename S> P &operator*=(const S &r) { first *= r, second *= r; return *this; } P operator+(const P &r) const { return P(*this) += r; } P operator-(const P &r) const { return P(*this) -= r; } P operator*(const P &r) const { return P(*this) *= r; } template <typename S> P operator*(const S &r) const { return P(*this) *= r; } P operator-() const { return P{-first, -second}; } }; using pl = P<ll, ll>; using pi = P<int, int>; using vp = V<pl>; constexpr int inf = 1001001001; constexpr long long infLL = 4004004004004004004LL; template <typename T> int sz(const T &t) { return t.size(); } template <typename T, typename U> inline bool amin(T &x, U y) { return (y < x) ? (x = y, true) : false; } template <typename T, typename U> inline bool amax(T &x, U y) { return (x < y) ? (x = y, true) : false; } template <typename T> inline T Max(const vector<T> &v) { return *max_element(begin(v), end(v)); } template <typename T> inline T Min(const vector<T> &v) { return *min_element(begin(v), end(v)); } template <typename T> inline long long Sum(const vector<T> &v) { return accumulate(begin(v), end(v), 0LL); } template <typename T> int lb(const vector<T> &v, const T &a) { return lower_bound(begin(v), end(v), a) - begin(v); } template <typename T> int ub(const vector<T> &v, const T &a) { return upper_bound(begin(v), end(v), a) - begin(v); } constexpr long long TEN(int n) { long long ret = 1, x = 10; for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1); return ret; } template <typename T, typename U> pair<T, U> mkp(const T &t, const U &u) { return make_pair(t, u); } template <typename T> vector<T> mkrui(const vector<T> &v, bool rev = false) { vector<T> ret(v.size() + 1); if (rev) { for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1]; } else { for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i]; } return ret; }; template <typename T> vector<T> mkuni(const vector<T> &v) { vector<T> ret(v); sort(ret.begin(), ret.end()); ret.erase(unique(ret.begin(), ret.end()), ret.end()); return ret; } template <typename F> vector<int> mkord(int N, F f) { vector<int> ord(N); iota(begin(ord), end(ord), 0); sort(begin(ord), end(ord), f); return ord; } template <typename T> vector<int> mkinv(vector<T> &v) { int max_val = *max_element(begin(v), end(v)); vector<int> inv(max_val + 1, -1); for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i; return inv; } vector<int> mkiota(int n) { vector<int> ret(n); iota(begin(ret), end(ret), 0); return ret; } template <typename T> T mkrev(const T &v) { T w{v}; reverse(begin(w), end(w)); return w; } template <typename T> bool nxp(vector<T> &v) { return next_permutation(begin(v), end(v)); } // 返り値の型は入力の T に依存 // i 要素目 : [0, a[i]) template <typename T> vector<vector<T>> product(const vector<T> &a) { vector<vector<T>> ret; vector<T> v; auto dfs = [&](auto rc, int i) -> void { if (i == (int)a.size()) { ret.push_back(v); return; } for (int j = 0; j < a[i]; j++) v.push_back(j), rc(rc, i + 1), v.pop_back(); }; dfs(dfs, 0); return ret; } // F : function(void(T&)), mod を取る操作 // T : 整数型のときはオーバーフローに注意する template <typename T> T Power(T a, long long n, const T &I, const function<void(T &)> &f) { T res = I; for (; n; f(a = a * a), n >>= 1) { if (n & 1) f(res = res * a); } return res; } // T : 整数型のときはオーバーフローに注意する template <typename T> T Power(T a, long long n, const T &I) { return Power(a, n, I, function<void(T &)>{[](T &) -> void {}}); } } // namespace Nyaan #line 58 "template/template.hpp" // bit operation #line 1 "template/bitop.hpp" namespace Nyaan { __attribute__((target("popcnt"))) inline int popcnt(const u64 &a) { return _mm_popcnt_u64(a); } inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; } inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; } inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; } template <typename T> inline int gbit(const T &a, int i) { return (a >> i) & 1; } template <typename T> inline void sbit(T &a, int i, bool b) { if (gbit(a, i) != b) a ^= T(1) << i; } constexpr long long PW(int n) { return 1LL << n; } constexpr long long MSK(int n) { return (1LL << n) - 1; } } // namespace Nyaan #line 61 "template/template.hpp" // inout #line 1 "template/inout.hpp" namespace Nyaan { template <typename T, typename U> ostream &operator<<(ostream &os, const pair<T, U> &p) { os << p.first << " " << p.second; return os; } template <typename T, typename U> istream &operator>>(istream &is, pair<T, U> &p) { is >> p.first >> p.second; return is; } template <typename T> ostream &operator<<(ostream &os, const vector<T> &v) { int s = (int)v.size(); for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i]; return os; } template <typename T> istream &operator>>(istream &is, vector<T> &v) { for (auto &x : v) is >> x; return is; } istream &operator>>(istream &is, __int128_t &x) { string S; is >> S; x = 0; int flag = 0; for (auto &c : S) { if (c == '-') { flag = true; continue; } x *= 10; x += c - '0'; } if (flag) x = -x; return is; } istream &operator>>(istream &is, __uint128_t &x) { string S; is >> S; x = 0; for (auto &c : S) { x *= 10; x += c - '0'; } return is; } ostream &operator<<(ostream &os, __int128_t x) { if (x == 0) return os << 0; if (x < 0) os << '-', x = -x; string S; while (x) S.push_back('0' + x % 10), x /= 10; reverse(begin(S), end(S)); return os << S; } ostream &operator<<(ostream &os, __uint128_t x) { if (x == 0) return os << 0; string S; while (x) S.push_back('0' + x % 10), x /= 10; reverse(begin(S), end(S)); return os << S; } void in() {} template <typename T, class... U> void in(T &t, U &...u) { cin >> t; in(u...); } void out() { cout << "\n"; } template <typename T, class... U, char sep = ' '> void out(const T &t, const U &...u) { cout << t; if (sizeof...(u)) cout << sep; out(u...); } struct IoSetupNya { IoSetupNya() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(15); cerr << fixed << setprecision(7); } } iosetupnya; } // namespace Nyaan #line 64 "template/template.hpp" // debug #line 1 "template/debug.hpp" namespace DebugImpl { template <typename U, typename = void> struct is_specialize : false_type {}; template <typename U> struct is_specialize< U, typename conditional<false, typename U::iterator, void>::type> : true_type {}; template <typename U> struct is_specialize< U, typename conditional<false, decltype(U::first), void>::type> : true_type {}; template <typename U> struct is_specialize<U, enable_if_t<is_integral<U>::value, void>> : true_type { }; void dump(const char& t) { cerr << t; } void dump(const string& t) { cerr << t; } void dump(const bool& t) { cerr << (t ? "true" : "false"); } void dump(__int128_t t) { if (t == 0) cerr << 0; if (t < 0) cerr << '-', t = -t; string S; while (t) S.push_back('0' + t % 10), t /= 10; reverse(begin(S), end(S)); cerr << S; } void dump(__uint128_t t) { if (t == 0) cerr << 0; string S; while (t) S.push_back('0' + t % 10), t /= 10; reverse(begin(S), end(S)); cerr << S; } template <typename U, enable_if_t<!is_specialize<U>::value, nullptr_t> = nullptr> void dump(const U& t) { cerr << t; } template <typename T> void dump(const T& t, enable_if_t<is_integral<T>::value>* = nullptr) { string res; if (t == Nyaan::inf) res = "inf"; if constexpr (is_signed<T>::value) { if (t == -Nyaan::inf) res = "-inf"; } if constexpr (sizeof(T) == 8) { if (t == Nyaan::infLL) res = "inf"; if constexpr (is_signed<T>::value) { if (t == -Nyaan::infLL) res = "-inf"; } } if (res.empty()) res = to_string(t); cerr << res; } template <typename T, typename U> void dump(const pair<T, U>&); template <typename T> void dump(const pair<T*, int>&); template <typename T> void dump(const T& t, enable_if_t<!is_void<typename T::iterator>::value>* = nullptr) { cerr << "[ "; for (auto it = t.begin(); it != t.end();) { dump(*it); cerr << (++it == t.end() ? "" : ", "); } cerr << " ]"; } template <typename T, typename U> void dump(const pair<T, U>& t) { cerr << "( "; dump(t.first); cerr << ", "; dump(t.second); cerr << " )"; } template <typename T> void dump(const pair<T*, int>& t) { cerr << "[ "; for (int i = 0; i < t.second; i++) { dump(t.first[i]); cerr << (i == t.second - 1 ? "" : ", "); } cerr << " ]"; } void trace() { cerr << endl; } template <typename Head, typename... Tail> void trace(Head&& head, Tail&&... tail) { cerr << " "; dump(head); if (sizeof...(tail) != 0) cerr << ","; trace(forward<Tail>(tail)...); } } // namespace DebugImpl #ifdef NyaanDebug #define trc(...) \ do { \ cerr << "## " << #__VA_ARGS__ << " = "; \ DebugImpl::trace(__VA_ARGS__); \ } while (0) #else #define trc(...) (void(0)) #endif #ifdef NyaanLocal #define trc2(...) \ do { \ cerr << "## " << #__VA_ARGS__ << " = "; \ DebugImpl::trace(__VA_ARGS__); \ } while (0) #else #define trc2(...) (void(0)) #endif #line 67 "template/template.hpp" // macro #line 1 "template/macro.hpp" #define each(x, v) for (auto&& x : v) #define each2(x, y, v) for (auto&& [x, y] : v) #define all(v) (v).begin(), (v).end() #define rep(i, N) for (long long i = 0; i < (long long)(N); i++) #define repr(i, N) for (long long i = (long long)(N)-1; i >= 0; i--) #define rep1(i, N) for (long long i = 1; i <= (long long)(N); i++) #define repr1(i, N) for (long long i = (N); (long long)(i) > 0; i--) #define reg(i, a, b) for (long long i = (a); i < (b); i++) #define regr(i, a, b) for (long long i = (b)-1; i >= (a); i--) #define fi first #define se second #define ini(...) \ int __VA_ARGS__; \ in(__VA_ARGS__) #define inl(...) \ long long __VA_ARGS__; \ in(__VA_ARGS__) #define ins(...) \ string __VA_ARGS__; \ in(__VA_ARGS__) #define in2(s, t) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i]); \ } #define in3(s, t, u) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i], u[i]); \ } #define in4(s, t, u, v) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i], u[i], v[i]); \ } #define die(...) \ do { \ Nyaan::out(__VA_ARGS__); \ return; \ } while (0) #line 70 "template/template.hpp" namespace Nyaan { void solve(); } int main() { Nyaan::solve(); } #line 4 "verify/verify-yosupo-ntt/yosupo-convolution-arbitrarylengthntt.test.cpp" // #line 2 "misc/fastio.hpp" #line 8 "misc/fastio.hpp" using namespace std; #line 2 "internal/internal-type-traits.hpp" #line 4 "internal/internal-type-traits.hpp" using namespace std; namespace internal { template <typename T> using is_broadly_integral = typename conditional_t<is_integral_v<T> || is_same_v<T, __int128_t> || is_same_v<T, __uint128_t>, true_type, false_type>::type; template <typename T> using is_broadly_signed = typename conditional_t<is_signed_v<T> || is_same_v<T, __int128_t>, true_type, false_type>::type; template <typename T> using is_broadly_unsigned = typename conditional_t<is_unsigned_v<T> || is_same_v<T, __uint128_t>, true_type, false_type>::type; #define ENABLE_VALUE(x) \ template <typename T> \ constexpr bool x##_v = x<T>::value; ENABLE_VALUE(is_broadly_integral); ENABLE_VALUE(is_broadly_signed); ENABLE_VALUE(is_broadly_unsigned); #undef ENABLE_VALUE #define ENABLE_HAS_TYPE(var) \ template <class, class = void> \ struct has_##var : false_type {}; \ template <class T> \ struct has_##var<T, void_t<typename T::var>> : true_type {}; \ template <class T> \ constexpr auto has_##var##_v = has_##var<T>::value; #define ENABLE_HAS_VAR(var) \ template <class, class = void> \ struct has_##var : false_type {}; \ template <class T> \ struct has_##var<T, void_t<decltype(T::var)>> : true_type {}; \ template <class T> \ constexpr auto has_##var##_v = has_##var<T>::value; } // namespace internal #line 12 "misc/fastio.hpp" namespace fastio { static constexpr int SZ = 1 << 17; static constexpr int offset = 64; char inbuf[SZ], outbuf[SZ]; int in_left = 0, in_right = 0, out_right = 0; struct Pre { char num[40000]; constexpr Pre() : num() { for (int i = 0; i < 10000; i++) { int n = i; for (int j = 3; j >= 0; j--) { num[i * 4 + j] = n % 10 + '0'; n /= 10; } } } } constexpr pre; void load() { int len = in_right - in_left; memmove(inbuf, inbuf + in_left, len); in_right = len + fread(inbuf + len, 1, SZ - len, stdin); in_left = 0; } void flush() { fwrite(outbuf, 1, out_right, stdout); out_right = 0; } void skip_space() { if (in_left + offset > in_right) load(); while (inbuf[in_left] <= ' ') in_left++; } void single_read(char& c) { if (in_left + offset > in_right) load(); skip_space(); c = inbuf[in_left++]; } void single_read(string& S) { skip_space(); while (true) { if (in_left == in_right) load(); int i = in_left; for (; i != in_right; i++) { if (inbuf[i] <= ' ') break; } copy(inbuf + in_left, inbuf + i, back_inserter(S)); in_left = i; if (i != in_right) break; } } template <typename T, enable_if_t<internal::is_broadly_integral_v<T>>* = nullptr> void single_read(T& x) { if (in_left + offset > in_right) load(); skip_space(); char c = inbuf[in_left++]; [[maybe_unused]] bool minus = false; if constexpr (internal::is_broadly_signed_v<T>) { if (c == '-') minus = true, c = inbuf[in_left++]; } x = 0; while (c >= '0') { x = x * 10 + (c & 15); c = inbuf[in_left++]; } if constexpr (internal::is_broadly_signed_v<T>) { if (minus) x = -x; } } void rd() {} template <typename Head, typename... Tail> void rd(Head& head, Tail&... tail) { single_read(head); rd(tail...); } void single_write(const char& c) { if (out_right > SZ - offset) flush(); outbuf[out_right++] = c; } void single_write(const bool& b) { if (out_right > SZ - offset) flush(); outbuf[out_right++] = b ? '1' : '0'; } void single_write(const string& S) { flush(), fwrite(S.data(), 1, S.size(), stdout); } void single_write(const char* p) { flush(), fwrite(p, 1, strlen(p), stdout); } template <typename T, enable_if_t<internal::is_broadly_integral_v<T>>* = nullptr> void single_write(const T& _x) { if (out_right > SZ - offset) flush(); if (_x == 0) { outbuf[out_right++] = '0'; return; } T x = _x; if constexpr (internal::is_broadly_signed_v<T>) { if (x < 0) outbuf[out_right++] = '-', x = -x; } constexpr int buffer_size = sizeof(T) * 10 / 4; char buf[buffer_size]; int i = buffer_size; while (x >= 10000) { i -= 4; memcpy(buf + i, pre.num + (x % 10000) * 4, 4); x /= 10000; } if (x < 100) { if (x < 10) { outbuf[out_right] = '0' + x; ++out_right; } else { uint32_t q = (uint32_t(x) * 205) >> 11; uint32_t r = uint32_t(x) - q * 10; outbuf[out_right] = '0' + q; outbuf[out_right + 1] = '0' + r; out_right += 2; } } else { if (x < 1000) { memcpy(outbuf + out_right, pre.num + (x << 2) + 1, 3); out_right += 3; } else { memcpy(outbuf + out_right, pre.num + (x << 2), 4); out_right += 4; } } memcpy(outbuf + out_right, buf + i, buffer_size - i); out_right += buffer_size - i; } void wt() {} template <typename Head, typename... Tail> void wt(const Head& head, const Tail&... tail) { single_write(head); wt(forward<const Tail>(tail)...); } template <typename... Args> void wtn(const Args&... x) { wt(forward<const Args>(x)...); wt('\n'); } struct Dummy { Dummy() { atexit(flush); } } dummy; } // namespace fastio using fastio::rd; using fastio::skip_space; using fastio::wt; using fastio::wtn; #line 2 "modint/montgomery-modint.hpp" template <uint32_t mod> struct LazyMontgomeryModInt { using mint = LazyMontgomeryModInt; using i32 = int32_t; using u32 = uint32_t; using u64 = uint64_t; static constexpr u32 get_r() { u32 ret = mod; for (i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret; return ret; } static constexpr u32 r = get_r(); static constexpr u32 n2 = -u64(mod) % mod; static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30"); static_assert((mod & 1) == 1, "invalid, mod % 2 == 0"); static_assert(r * mod == 1, "this code has bugs."); u32 a; constexpr LazyMontgomeryModInt() : a(0) {} constexpr LazyMontgomeryModInt(const int64_t &b) : a(reduce(u64(b % mod + mod) * n2)){}; static constexpr u32 reduce(const u64 &b) { return (b + u64(u32(b) * u32(-r)) * mod) >> 32; } constexpr mint &operator+=(const mint &b) { if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod; return *this; } constexpr mint &operator-=(const mint &b) { if (i32(a -= b.a) < 0) a += 2 * mod; return *this; } constexpr mint &operator*=(const mint &b) { a = reduce(u64(a) * b.a); return *this; } constexpr mint &operator/=(const mint &b) { *this *= b.inverse(); return *this; } constexpr mint operator+(const mint &b) const { return mint(*this) += b; } constexpr mint operator-(const mint &b) const { return mint(*this) -= b; } constexpr mint operator*(const mint &b) const { return mint(*this) *= b; } constexpr mint operator/(const mint &b) const { return mint(*this) /= b; } constexpr bool operator==(const mint &b) const { return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a); } constexpr bool operator!=(const mint &b) const { return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a); } constexpr mint operator-() const { return mint() - mint(*this); } constexpr mint operator+() const { return mint(*this); } constexpr mint pow(u64 n) const { mint ret(1), mul(*this); while (n > 0) { if (n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } constexpr mint inverse() const { int x = get(), y = mod, u = 1, v = 0, t = 0, tmp = 0; while (y > 0) { t = x / y; x -= t * y, u -= t * v; tmp = x, x = y, y = tmp; tmp = u, u = v, v = tmp; } return mint{u}; } friend ostream &operator<<(ostream &os, const mint &b) { return os << b.get(); } friend istream &operator>>(istream &is, mint &b) { int64_t t; is >> t; b = LazyMontgomeryModInt<mod>(t); return (is); } constexpr u32 get() const { u32 ret = reduce(a); return ret >= mod ? ret - mod : ret; } static constexpr u32 get_mod() { return mod; } }; #line 2 "ntt/cooley-tukey-ntt.hpp" #line 2 "ntt/rader-ntt.hpp" #line 2 "math/constexpr-primitive-root.hpp" constexpr unsigned int constexpr_primitive_root(unsigned int mod) { using u32 = unsigned int; using u64 = unsigned long long; if(mod == 2) return 1; u64 m = mod - 1, ds[32] = {}, idx = 0; for (u64 i = 2; i * i <= m; ++i) { if (m % i == 0) { ds[idx++] = i; while (m % i == 0) m /= i; } } if (m != 1) ds[idx++] = m; for (u32 _pr = 2, flg = true;; _pr++, flg = true) { for (u32 i = 0; i < idx && flg; ++i) { u64 a = _pr, b = (mod - 1) / ds[i], r = 1; for (; b; a = a * a % mod, b >>= 1) if (b & 1) r = r * a % mod; if (r == 1) flg = false; } if (flg == true) return _pr; } } #line 2 "ntt/arbitrary-ntt.hpp" #line 2 "ntt/ntt.hpp" template <typename mint> struct NTT { static constexpr uint32_t get_pr() { uint32_t _mod = mint::get_mod(); using u64 = uint64_t; u64 ds[32] = {}; int idx = 0; u64 m = _mod - 1; for (u64 i = 2; i * i <= m; ++i) { if (m % i == 0) { ds[idx++] = i; while (m % i == 0) m /= i; } } if (m != 1) ds[idx++] = m; uint32_t _pr = 2; while (1) { int flg = 1; for (int i = 0; i < idx; ++i) { u64 a = _pr, b = (_mod - 1) / ds[i], r = 1; while (b) { if (b & 1) r = r * a % _mod; a = a * a % _mod; b >>= 1; } if (r == 1) { flg = 0; break; } } if (flg == 1) break; ++_pr; } return _pr; }; static constexpr uint32_t mod = mint::get_mod(); static constexpr uint32_t pr = get_pr(); static constexpr int level = __builtin_ctzll(mod - 1); mint dw[level], dy[level]; void setwy(int k) { mint w[level], y[level]; w[k - 1] = mint(pr).pow((mod - 1) / (1 << k)); y[k - 1] = w[k - 1].inverse(); for (int i = k - 2; i > 0; --i) w[i] = w[i + 1] * w[i + 1], y[i] = y[i + 1] * y[i + 1]; dw[1] = w[1], dy[1] = y[1], dw[2] = w[2], dy[2] = y[2]; for (int i = 3; i < k; ++i) { dw[i] = dw[i - 1] * y[i - 2] * w[i]; dy[i] = dy[i - 1] * w[i - 2] * y[i]; } } NTT() { setwy(level); } void fft4(vector<mint> &a, int k) { if ((int)a.size() <= 1) return; if (k == 1) { mint a1 = a[1]; a[1] = a[0] - a[1]; a[0] = a[0] + a1; return; } if (k & 1) { int v = 1 << (k - 1); for (int j = 0; j < v; ++j) { mint ajv = a[j + v]; a[j + v] = a[j] - ajv; a[j] += ajv; } } int u = 1 << (2 + (k & 1)); int v = 1 << (k - 2 - (k & 1)); mint one = mint(1); mint imag = dw[1]; while (v) { // jh = 0 { int j0 = 0; int j1 = v; int j2 = j1 + v; int j3 = j2 + v; for (; j0 < v; ++j0, ++j1, ++j2, ++j3) { mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3]; mint t0p2 = t0 + t2, t1p3 = t1 + t3; mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag; a[j0] = t0p2 + t1p3, a[j1] = t0p2 - t1p3; a[j2] = t0m2 + t1m3, a[j3] = t0m2 - t1m3; } } // jh >= 1 mint ww = one, xx = one * dw[2], wx = one; for (int jh = 4; jh < u;) { ww = xx * xx, wx = ww * xx; int j0 = jh * v; int je = j0 + v; int j2 = je + v; for (; j0 < je; ++j0, ++j2) { mint t0 = a[j0], t1 = a[j0 + v] * xx, t2 = a[j2] * ww, t3 = a[j2 + v] * wx; mint t0p2 = t0 + t2, t1p3 = t1 + t3; mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag; a[j0] = t0p2 + t1p3, a[j0 + v] = t0p2 - t1p3; a[j2] = t0m2 + t1m3, a[j2 + v] = t0m2 - t1m3; } xx *= dw[__builtin_ctzll((jh += 4))]; } u <<= 2; v >>= 2; } } void ifft4(vector<mint> &a, int k) { if ((int)a.size() <= 1) return; if (k == 1) { mint a1 = a[1]; a[1] = a[0] - a[1]; a[0] = a[0] + a1; return; } int u = 1 << (k - 2); int v = 1; mint one = mint(1); mint imag = dy[1]; while (u) { // jh = 0 { int j0 = 0; int j1 = v; int j2 = v + v; int j3 = j2 + v; for (; j0 < v; ++j0, ++j1, ++j2, ++j3) { mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3]; mint t0p1 = t0 + t1, t2p3 = t2 + t3; mint t0m1 = t0 - t1, t2m3 = (t2 - t3) * imag; a[j0] = t0p1 + t2p3, a[j2] = t0p1 - t2p3; a[j1] = t0m1 + t2m3, a[j3] = t0m1 - t2m3; } } // jh >= 1 mint ww = one, xx = one * dy[2], yy = one; u <<= 2; for (int jh = 4; jh < u;) { ww = xx * xx, yy = xx * imag; int j0 = jh * v; int je = j0 + v; int j2 = je + v; for (; j0 < je; ++j0, ++j2) { mint t0 = a[j0], t1 = a[j0 + v], t2 = a[j2], t3 = a[j2 + v]; mint t0p1 = t0 + t1, t2p3 = t2 + t3; mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy; a[j0] = t0p1 + t2p3, a[j2] = (t0p1 - t2p3) * ww; a[j0 + v] = t0m1 + t2m3, a[j2 + v] = (t0m1 - t2m3) * ww; } xx *= dy[__builtin_ctzll(jh += 4)]; } u >>= 4; v <<= 2; } if (k & 1) { u = 1 << (k - 1); for (int j = 0; j < u; ++j) { mint ajv = a[j] - a[j + u]; a[j] += a[j + u]; a[j + u] = ajv; } } } void ntt(vector<mint> &a) { if ((int)a.size() <= 1) return; fft4(a, __builtin_ctz(a.size())); } void intt(vector<mint> &a) { if ((int)a.size() <= 1) return; ifft4(a, __builtin_ctz(a.size())); mint iv = mint(a.size()).inverse(); for (auto &x : a) x *= iv; } vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) { int l = a.size() + b.size() - 1; if (min<int>(a.size(), b.size()) <= 40) { vector<mint> s(l); for (int i = 0; i < (int)a.size(); ++i) for (int j = 0; j < (int)b.size(); ++j) s[i + j] += a[i] * b[j]; return s; } int k = 2, M = 4; while (M < l) M <<= 1, ++k; setwy(k); vector<mint> s(M); for (int i = 0; i < (int)a.size(); ++i) s[i] = a[i]; fft4(s, k); if (a.size() == b.size() && a == b) { for (int i = 0; i < M; ++i) s[i] *= s[i]; } else { vector<mint> t(M); for (int i = 0; i < (int)b.size(); ++i) t[i] = b[i]; fft4(t, k); for (int i = 0; i < M; ++i) s[i] *= t[i]; } ifft4(s, k); s.resize(l); mint invm = mint(M).inverse(); for (int i = 0; i < l; ++i) s[i] *= invm; return s; } void ntt_doubling(vector<mint> &a) { int M = (int)a.size(); auto b = a; intt(b); mint r = 1, zeta = mint(pr).pow((mint::get_mod() - 1) / (M << 1)); for (int i = 0; i < M; i++) b[i] *= r, r *= zeta; ntt(b); copy(begin(b), end(b), back_inserter(a)); } }; #line 5 "ntt/arbitrary-ntt.hpp" namespace ArbitraryNTT { using i64 = int64_t; using u128 = __uint128_t; constexpr int32_t m0 = 167772161; constexpr int32_t m1 = 469762049; constexpr int32_t m2 = 754974721; using mint0 = LazyMontgomeryModInt<m0>; using mint1 = LazyMontgomeryModInt<m1>; using mint2 = LazyMontgomeryModInt<m2>; constexpr int r01 = mint1(m0).inverse().get(); constexpr int r02 = mint2(m0).inverse().get(); constexpr int r12 = mint2(m1).inverse().get(); constexpr int r02r12 = i64(r02) * r12 % m2; constexpr i64 w1 = m0; constexpr i64 w2 = i64(m0) * m1; template <typename T, typename submint> vector<submint> mul(const vector<T> &a, const vector<T> &b) { static NTT<submint> ntt; vector<submint> s(a.size()), t(b.size()); for (int i = 0; i < (int)a.size(); ++i) s[i] = i64(a[i] % submint::get_mod()); for (int i = 0; i < (int)b.size(); ++i) t[i] = i64(b[i] % submint::get_mod()); return ntt.multiply(s, t); } template <typename T> vector<int> multiply(const vector<T> &s, const vector<T> &t, int mod) { auto d0 = mul<T, mint0>(s, t); auto d1 = mul<T, mint1>(s, t); auto d2 = mul<T, mint2>(s, t); int n = d0.size(); vector<int> ret(n); const int W1 = w1 % mod; const int W2 = w2 % mod; for (int i = 0; i < n; i++) { int n1 = d1[i].get(), n2 = d2[i].get(), a = d0[i].get(); int b = i64(n1 + m1 - a) * r01 % m1; int c = (i64(n2 + m2 - a) * r02r12 + i64(m2 - b) * r12) % m2; ret[i] = (i64(a) + i64(b) * W1 + i64(c) * W2) % mod; } return ret; } template <typename mint> vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) { if (a.size() == 0 && b.size() == 0) return {}; if (min<int>(a.size(), b.size()) < 128) { vector<mint> ret(a.size() + b.size() - 1); for (int i = 0; i < (int)a.size(); ++i) for (int j = 0; j < (int)b.size(); ++j) ret[i + j] += a[i] * b[j]; return ret; } vector<int> s(a.size()), t(b.size()); for (int i = 0; i < (int)a.size(); ++i) s[i] = a[i].get(); for (int i = 0; i < (int)b.size(); ++i) t[i] = b[i].get(); vector<int> u = multiply<int>(s, t, mint::get_mod()); vector<mint> ret(u.size()); for (int i = 0; i < (int)u.size(); ++i) ret[i] = mint(u[i]); return ret; } template <typename T> vector<u128> multiply_u128(const vector<T> &s, const vector<T> &t) { if (s.size() == 0 && t.size() == 0) return {}; if (min<int>(s.size(), t.size()) < 128) { vector<u128> ret(s.size() + t.size() - 1); for (int i = 0; i < (int)s.size(); ++i) for (int j = 0; j < (int)t.size(); ++j) ret[i + j] += i64(s[i]) * t[j]; return ret; } auto d0 = mul<T, mint0>(s, t); auto d1 = mul<T, mint1>(s, t); auto d2 = mul<T, mint2>(s, t); int n = d0.size(); vector<u128> ret(n); for (int i = 0; i < n; i++) { i64 n1 = d1[i].get(), n2 = d2[i].get(); i64 a = d0[i].get(); i64 b = (n1 + m1 - a) * r01 % m1; i64 c = ((n2 + m2 - a) * r02r12 + (m2 - b) * r12) % m2; ret[i] = a + b * w1 + u128(c) * w2; } return ret; } } // namespace ArbitraryNTT #line 5 "ntt/rader-ntt.hpp" template <typename mint> struct RaderNTT { int p, pr, len; const vector<mint>& w; vector<int> prs, iprs; RaderNTT() {} RaderNTT(int _p, int _len, const vector<mint>& _w) : p(_p), pr(constexpr_primitive_root(p)), len(_len), w(_w) { prs.resize(p - 1); iprs.resize(p, -1); for (int i = 0; i < p - 1; i++) prs[i] = i ? prs[i - 1] * pr % p : 1; for (int i = 0; i < p - 1; i++) iprs[prs[i]] = i; } void ntt(vector<mint>& a) { vector<mint> s(p - 1), t(p - 1); for (int i = 0; i < p - 1; i++) s[i] = a[prs[i]]; for (int i = 0, ldp = len / p; i < p - 1; i++) t[i] = w[ldp * prs[i ? p - 1 - i : 0]]; vector<mint> u = ArbitraryNTT::multiply(s, t); s.resize(p); fill(begin(s), end(s), a[0]); for (int i = 1; i < p; i++) s[0] += a[i]; for (int i = 0, y = 0; i < (int)u.size(); i++) { s[prs[y]] += u[i]; if (--y < 0) y += p - 1; } swap(a, s); } }; /** * @brief Rader's FFT Algorithm */ #line 4 "ntt/cooley-tukey-ntt.hpp" template <typename mint> struct ArbitraryLengthNTT { using i64 = long long; int factor(int n) { for (int i = 2; i * i <= n; i++) if (n % i == 0) return i; return n; } vector<int> divisor(int n) { vector<int> ret; for (int i = 2; i * i <= n; i++) { if (n % i == 0) { ret.push_back(i); ret.push_back(n / i); } } ret.push_back(n); sort(begin(ret), end(ret)); ret.erase(unique(begin(ret), end(ret)), end(ret)); return ret; } int len; vector<mint> w; vector<int> divisors; map<int, RaderNTT<mint> *> rader; ArbitraryLengthNTT(int len_ = -1) { set_len(len_); } void set_len(int len_ = -1) { int mod = mint::get_mod(); if ((len = len_) == -1) len = mod - 1; if (mod >= 3 && len <= 1) len = 2; while ((mod - 1) % len != 0) ++len; mint pr = mint(constexpr_primitive_root(mod)).pow((mod - 1) / len); w.resize(len + 1); for (int i = 0; i <= len; i++) w[i] = i ? w[i - 1] * pr : mint(1); divisors = divisor(len); } void dft(vector<mint> &a) { int N = a.size(); if (N == 2) { mint a01 = a[0] + a[1]; a[1] = a[0] - a[1]; a[0] = a01; return; } int d = len / N; vector<mint> b(N); for (int i = 0, dk = 0; i < N; i++, dk += d) { for (int j = 0, k = 0; j < N; j++) { b[j] += a[i] * w[k]; if ((k += dk) >= len) k -= len; } } swap(a, b); } void ntt_base2(vector<mint> &a) { static vector<int> btr; int N = a.size(); assert(N % 2 == 0); if (btr.size() != a.size()) { btr.resize(N); int b = __builtin_ctz(N); assert(N == (1 << b)); for (int i = 0; i < N; i++) { btr[i] = (btr[i >> 1] >> 1) + ((i & 1) << (b - 1)); } } static vector<mint> basis; if (basis.size() < a.size()) { basis.resize(N); mint b = w[len / N]; for (int i = N >> 1; i > 0; i >>= 1) { mint c = 1; for (int j = 0; j < i; ++j) { basis[i + j] = c; c *= b; } b *= b; } } for (int i = 0; i < N; i++) if (i < btr[i]) swap(a[i], a[btr[i]]); for (int k = 1; k < N; k <<= 1) { for (int i = 0; i < N; i += 2 * k) { for (int j = 0; j < k; j++) { mint z = a[i + j + k] * basis[j + k]; a[i + j + k] = a[i + j] - z; a[i + j] = a[i + j] + z; } } } } void pntt(vector<mint> &a) { int P = a.size(); if (P <= 64) { dft(a); return; } if (rader.find(P) == end(rader)) rader[P] = new RaderNTT<mint>(P, len, w); rader[P]->ntt(a); } void ntt(vector<mint> &a) { assert(len % a.size() == 0); int N = (int)a.size(); if (N <= 1) return; if (N <= 64) { dft(a); return; } int P = factor(N); if (P == N) { pntt(a); return; } if (P == 2) { P = 1 << __builtin_ctz(N); if (N == P) { ntt_base2(a); return; } } int Q = N / P; vector<mint> t(N), u(P); { vector<mint> s(Q); for (int p = 0, lN = len / N, d = 0; p < P; p++, d += lN) { for (int q = 0, qP = 0; q < Q; q++, qP += P) s[q] = a[qP + p]; ntt(s); for (int r = 0, n = 0, pQ = p * Q; r < Q; ++r, n += d) { t[pQ + r] = w[n] * s[r]; } } } for (int r = 0; r < Q; r++) { for (int p = 0, pQ = 0; p < P; p++, pQ += Q) u[p] = t[pQ + r]; if (P <= 64) dft(u); else if (P & 1) pntt(u); else ntt_base2(u); for (int s = 0, sQ = 0; s < P; s++, sQ += Q) a[sQ + r] = u[s]; } } void intt(vector<mint> &a) { reverse(begin(a) + 1, end(a)); ntt(a); mint invn = mint(a.size()).inverse(); for (auto &x : a) x *= invn; } vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) { int N = (int)a.size() + (int)b.size() - 1; assert(N <= len); vector<mint> s(a), t(b); int l = *lower_bound(begin(divisors), end(divisors), N); s.resize(l, mint(0)); t.resize(l, mint(0)); ntt(s); ntt(t); for (int i = 0; i < l; i++) s[i] *= t[i]; intt(s); s.resize(N); return s; } }; /** * @brief Cooley-Tukey FFT Algorithm */ #line 8 "verify/verify-yosupo-ntt/yosupo-convolution-arbitrarylengthntt.test.cpp" using namespace Nyaan; void Nyaan::solve() { using mint = LazyMontgomeryModInt<998244353>; ArbitraryLengthNTT<mint> ntt(7 * 17 * (1 << 14)); int N, M, x; rd(N, M); V<mint> a(N), b(M); for (int i = 0; i < N; i++) rd(x), a[i] = x; for (int i = 0; i < M; i++) rd(x), b[i] = x; auto c = ntt.multiply(a, b); for (int i = 0; i < (int)N + M - 1; i++) { wt(c[i].get(), (i == (int)N + M - 1 ? '\n' : ' ')); } }