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Verify/LC_polynomial_interpolation.test.cpp
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- Last update: 2024-04-30 06:38:42+09:00
- Problem: https://judge.yosupo.jp/problem/polynomial_interpolation
Depends on
- Number Theoretic Transform
(Convolution/ntt.hpp)
- Formal Power Series (NTT-friendly mod)
(FPS/fps.hpp)
- Multipoint Evaluation
(FPS/multieval.hpp)
- Modint
(Math/modint.hpp)
- Template/template.hpp
- Fast IO
(Utility/fastio.hpp)
Code
#define PROBLEM "https://judge.yosupo.jp/problem/polynomial_interpolation"
#include "Template/template.hpp"
#include "Utility/fastio.hpp"
#include "Math/modint.hpp"
#include "Convolution/ntt.hpp"
#include "FPS/fps.hpp"
#include "FPS/multieval.hpp"
using Fp = fp<998244353>;
NTT<Fp> ntt;
template <> void Poly<Fp>::NTT(bool inv) {
ntt.ntt(*this, inv);
}
int main() {
int n;
cin >> n;
vector<Fp> a(n), b(n);
for (auto &x : a)
cin >> x;
for (auto &x : b)
cin >> x;
MultiEval<Fp> multi(a);
auto ret = multi.build(b);
rep(i, 0, n) cout << ret[i] << '\n';
return 0;
}#line 1 "Verify/LC_polynomial_interpolation.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/polynomial_interpolation"
#line 1 "Template/template.hpp"
#include <bits/stdc++.h>
using namespace std;
#define rep(i, a, b) for (int i = (int)(a); i < (int)(b); i++)
#define rrep(i, a, b) for (int i = (int)(b-1); i >= (int)(a); i--)
#define ALL(v) (v).begin(), (v).end()
#define UNIQUE(v) sort(ALL(v)), (v).erase(unique(ALL(v)), (v).end())
#define SZ(v) (int)v.size()
#define MIN(v) *min_element(ALL(v))
#define MAX(v) *max_element(ALL(v))
#define LB(v, x) int(lower_bound(ALL(v), (x)) - (v).begin())
#define UB(v, x) int(upper_bound(ALL(v), (x)) - (v).begin())
using uint = unsigned int;
using ll = long long int;
using ull = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
const int inf = 0x3fffffff;
const ll INF = 0x1fffffffffffffff;
template <typename T> inline bool chmax(T &a, T b) {
if (a < b) {
a = b;
return 1;
}
return 0;
}
template <typename T> inline bool chmin(T &a, T b) {
if (a > b) {
a = b;
return 1;
}
return 0;
}
template <typename T, typename U> T ceil(T x, U y) {
assert(y != 0);
if (y < 0)
x = -x, y = -y;
return (x > 0 ? (x + y - 1) / y : x / y);
}
template <typename T, typename U> T floor(T x, U y) {
assert(y != 0);
if (y < 0)
x = -x, y = -y;
return (x > 0 ? x / y : (x - y + 1) / y);
}
template <typename T> int popcnt(T x) {
return __builtin_popcountll(x);
}
template <typename T> int topbit(T x) {
return (x == 0 ? -1 : 63 - __builtin_clzll(x));
}
template <typename T> int lowbit(T x) {
return (x == 0 ? -1 : __builtin_ctzll(x));
}
#ifdef LOCAL
#define show(...) _show(0, #__VA_ARGS__, __VA_ARGS__)
#else
#define show(...) true
#endif
template <typename T> void _show(int i, T name) {
cerr << '\n';
}
template <typename T1, typename T2, typename... T3>
void _show(int i, const T1 &a, const T2 &b, const T3 &...c) {
for (; a[i] != ',' && a[i] != '\0'; i++)
cerr << a[i];
cerr << ":" << b << " ";
_show(i + 1, a, c...);
}
template <class T, class U>
ostream &operator<<(ostream &os, const pair<T, U> &p) {
os << "P(" << p.first << ", " << p.second << ")";
return os;
}
template <typename T, template <class> class C>
ostream &operator<<(ostream &os, const C<T> &v) {
os << "[";
for (auto d : v)
os << d << ", ";
os << "]";
return os;
}
#line 2 "Utility/fastio.hpp"
#include <unistd.h>
namespace fastio {
static constexpr uint32_t SZ = 1 << 17;
char ibuf[SZ];
char obuf[SZ];
char out[100];
// pointer of ibuf, obuf
uint32_t pil = 0, pir = 0, por = 0;
struct Pre {
char num[10000][4];
constexpr Pre() : num() {
for (int i = 0; i < 10000; i++) {
int n = i;
for (int j = 3; j >= 0; j--) {
num[i][j] = n % 10 | '0';
n /= 10;
}
}
}
} constexpr pre;
inline void load() {
memmove(ibuf, ibuf + pil, pir - pil);
pir = pir - pil + fread(ibuf + pir - pil, 1, SZ - pir + pil, stdin);
pil = 0;
if (pir < SZ)
ibuf[pir++] = '\n';
}
inline void flush() {
fwrite(obuf, 1, por, stdout);
por = 0;
}
void rd(char &c) {
do {
if (pil + 1 > pir)
load();
c = ibuf[pil++];
} while (isspace(c));
}
void rd(string &x) {
x.clear();
char c;
do {
if (pil + 1 > pir)
load();
c = ibuf[pil++];
} while (isspace(c));
do {
x += c;
if (pil == pir)
load();
c = ibuf[pil++];
} while (!isspace(c));
}
template <typename T> void rd_real(T &x) {
string s;
rd(s);
x = stod(s);
}
template <typename T> void rd_integer(T &x) {
if (pil + 100 > pir)
load();
char c;
do
c = ibuf[pil++];
while (c < '-');
bool minus = 0;
if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
if (c == '-') {
minus = 1, c = ibuf[pil++];
}
}
x = 0;
while ('0' <= c) {
x = x * 10 + (c & 15), c = ibuf[pil++];
}
if constexpr (is_signed<T>::value || is_same_v<T, i128>) {
if (minus)
x = -x;
}
}
void rd(int &x) {
rd_integer(x);
}
void rd(ll &x) {
rd_integer(x);
}
void rd(i128 &x) {
rd_integer(x);
}
void rd(uint &x) {
rd_integer(x);
}
void rd(ull &x) {
rd_integer(x);
}
void rd(u128 &x) {
rd_integer(x);
}
void rd(double &x) {
rd_real(x);
}
void rd(long double &x) {
rd_real(x);
}
template <class T, class U> void rd(pair<T, U> &p) {
return rd(p.first), rd(p.second);
}
template <size_t N = 0, typename T> void rd_tuple(T &t) {
if constexpr (N < std::tuple_size<T>::value) {
auto &x = std::get<N>(t);
rd(x);
rd_tuple<N + 1>(t);
}
}
template <class... T> void rd(tuple<T...> &tpl) {
rd_tuple(tpl);
}
template <size_t N = 0, typename T> void rd(array<T, N> &x) {
for (auto &d : x)
rd(d);
}
template <class T> void rd(vector<T> &x) {
for (auto &d : x)
rd(d);
}
void read() {}
template <class H, class... T> void read(H &h, T &...t) {
rd(h), read(t...);
}
void wt(const char c) {
if (por == SZ)
flush();
obuf[por++] = c;
}
void wt(const string s) {
for (char c : s)
wt(c);
}
void wt(const char *s) {
size_t len = strlen(s);
for (size_t i = 0; i < len; i++)
wt(s[i]);
}
template <typename T> void wt_integer(T x) {
if (por > SZ - 100)
flush();
if (x < 0) {
obuf[por++] = '-', x = -x;
}
int outi;
for (outi = 96; x >= 10000; outi -= 4) {
memcpy(out + outi, pre.num[x % 10000], 4);
x /= 10000;
}
if (x >= 1000) {
memcpy(obuf + por, pre.num[x], 4);
por += 4;
} else if (x >= 100) {
memcpy(obuf + por, pre.num[x] + 1, 3);
por += 3;
} else if (x >= 10) {
int q = (x * 103) >> 10;
obuf[por] = q | '0';
obuf[por + 1] = (x - q * 10) | '0';
por += 2;
} else
obuf[por++] = x | '0';
memcpy(obuf + por, out + outi + 4, 96 - outi);
por += 96 - outi;
}
template <typename T> void wt_real(T x) {
ostringstream oss;
oss << fixed << setprecision(15) << double(x);
string s = oss.str();
wt(s);
}
void wt(int x) {
wt_integer(x);
}
void wt(ll x) {
wt_integer(x);
}
void wt(i128 x) {
wt_integer(x);
}
void wt(uint x) {
wt_integer(x);
}
void wt(ull x) {
wt_integer(x);
}
void wt(u128 x) {
wt_integer(x);
}
void wt(double x) {
wt_real(x);
}
void wt(long double x) {
wt_real(x);
}
template <class T, class U> void wt(const pair<T, U> val) {
wt(val.first);
wt(' ');
wt(val.second);
}
template <size_t N = 0, typename T> void wt_tuple(const T t) {
if constexpr (N < std::tuple_size<T>::value) {
if constexpr (N > 0) {
wt(' ');
}
const auto x = std::get<N>(t);
wt(x);
wt_tuple<N + 1>(t);
}
}
template <class... T> void wt(tuple<T...> tpl) {
wt_tuple(tpl);
}
template <class T, size_t S> void wt(const array<T, S> val) {
auto n = val.size();
for (size_t i = 0; i < n; i++) {
if (i)
wt(' ');
wt(val[i]);
}
}
template <class T> void wt(const vector<T> val) {
auto n = val.size();
for (size_t i = 0; i < n; i++) {
if (i)
wt(' ');
wt(val[i]);
}
}
void print() {
wt('\n');
}
template <class Head, class... Tail> void print(Head &&head, Tail &&...tail) {
wt(head);
if (sizeof...(Tail))
wt(' ');
print(forward<Tail>(tail)...);
}
void __attribute__((destructor)) _d() {
flush();
}
} // namespace fastio
using fastio::flush;
using fastio::print;
using fastio::read;
inline void first(bool i = true) {
print(i ? "first" : "second");
}
inline void Alice(bool i = true) {
print(i ? "Alice" : "Bob");
}
inline void yes(bool i = true) {
print(i ? "yes" : "no");
}
inline void Yes(bool i = true) {
print(i ? "Yes" : "No");
}
inline void No() {
print("No");
}
inline void YES(bool i = true) {
print(i ? "YES" : "NO");
}
inline void NO() {
print("NO");
}
inline void Yay(bool i = true) {
print(i ? "Yay!" : ":(");
}
inline void Possible(bool i = true) {
print(i ? "Possible" : "Impossible");
}
inline void POSSIBLE(bool i = true) {
print(i ? "POSSIBLE" : "IMPOSSIBLE");
}
/**
* @brief Fast IO
*/
#line 2 "Math/modint.hpp"
template <unsigned mod = 1000000007> struct fp {
unsigned v;
static constexpr int get_mod() {
return mod;
}
constexpr unsigned inv() const {
assert(v != 0);
int x = v, y = mod, p = 1, q = 0, t = 0, tmp = 0;
while (y > 0) {
t = x / y;
x -= t * y, p -= t * q;
tmp = x, x = y, y = tmp;
tmp = p, p = q, q = tmp;
}
if (p < 0)
p += mod;
return p;
}
constexpr fp(ll x = 0) : v(x >= 0 ? x % mod : (mod - (-x) % mod) % mod) {}
fp operator-() const {
return fp() - *this;
}
fp pow(ull t) {
fp res = 1, b = *this;
while (t) {
if (t & 1)
res *= b;
b *= b;
t >>= 1;
}
return res;
}
fp &operator+=(const fp &x) {
if ((v += x.v) >= mod)
v -= mod;
return *this;
}
fp &operator-=(const fp &x) {
if ((v += mod - x.v) >= mod)
v -= mod;
return *this;
}
fp &operator*=(const fp &x) {
v = ull(v) * x.v % mod;
return *this;
}
fp &operator/=(const fp &x) {
v = ull(v) * x.inv() % mod;
return *this;
}
fp operator+(const fp &x) const {
return fp(*this) += x;
}
fp operator-(const fp &x) const {
return fp(*this) -= x;
}
fp operator*(const fp &x) const {
return fp(*this) *= x;
}
fp operator/(const fp &x) const {
return fp(*this) /= x;
}
bool operator==(const fp &x) const {
return v == x.v;
}
bool operator!=(const fp &x) const {
return v != x.v;
}
friend istream &operator>>(istream &is, fp &x) {
return is >> x.v;
}
friend ostream &operator<<(ostream &os, const fp &x) {
return os << x.v;
}
};
template <unsigned mod> void rd(fp<mod> &x) {
fastio::rd(x.v);
}
template <unsigned mod> void wt(fp<mod> x) {
fastio::wt(x.v);
}
template <typename T> T Inv(ll n) {
static const int md = T::get_mod();
static vector<T> buf({0, 1});
assert(n > 0);
n %= md;
while (SZ(buf) <= n) {
int k = SZ(buf), q = (md + k - 1) / k;
buf.push_back(buf[k * q - md] * q);
}
return buf[n];
}
template <typename T> T Fact(ll n, bool inv = 0) {
static const int md = T::get_mod();
static vector<T> buf({1, 1}), ibuf({1, 1});
assert(n >= 0 and n < md);
while (SZ(buf) <= n) {
buf.push_back(buf.back() * SZ(buf));
ibuf.push_back(ibuf.back() * Inv<T>(SZ(ibuf)));
}
return inv ? ibuf[n] : buf[n];
}
template <typename T> T nPr(int n, int r, bool inv = 0) {
if (n < 0 || n < r || r < 0)
return 0;
return Fact<T>(n, inv) * Fact<T>(n - r, inv ^ 1);
}
template <typename T> T nCr(int n, int r, bool inv = 0) {
if (n < 0 || n < r || r < 0)
return 0;
return Fact<T>(n, inv) * Fact<T>(r, inv ^ 1) * Fact<T>(n - r, inv ^ 1);
}
template <typename T> T nHr(int n, int r, bool inv = 0) {
return nCr<T>(n + r - 1, r, inv);
}
/**
* @brief Modint
*/
#line 2 "Convolution/ntt.hpp"
template <typename T> struct NTT {
static constexpr int rank2 = __builtin_ctzll(T::get_mod() - 1);
std::array<T, rank2 + 1> root; // root[i]^(2^i) == 1
std::array<T, rank2 + 1> iroot; // root[i] * iroot[i] == 1
std::array<T, std::max(0, rank2 - 2 + 1)> rate2;
std::array<T, std::max(0, rank2 - 2 + 1)> irate2;
std::array<T, std::max(0, rank2 - 3 + 1)> rate3;
std::array<T, std::max(0, rank2 - 3 + 1)> irate3;
NTT() {
T g = 2;
while (g.pow((T::get_mod() - 1) >> 1) == 1) {
g += 1;
}
root[rank2] = g.pow((T::get_mod() - 1) >> rank2);
iroot[rank2] = root[rank2].inv();
for (int i = rank2 - 1; i >= 0; i--) {
root[i] = root[i + 1] * root[i + 1];
iroot[i] = iroot[i + 1] * iroot[i + 1];
}
{
T prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 2; i++) {
rate2[i] = root[i + 2] * prod;
irate2[i] = iroot[i + 2] * iprod;
prod *= iroot[i + 2];
iprod *= root[i + 2];
}
}
{
T prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 3; i++) {
rate3[i] = root[i + 3] * prod;
irate3[i] = iroot[i + 3] * iprod;
prod *= iroot[i + 3];
iprod *= root[i + 3];
}
}
}
void ntt(std::vector<T> &a, bool type = 0) {
int n = int(a.size());
int h = __builtin_ctzll((unsigned int)n);
a.resize(1 << h);
if (type) {
int len = h; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
while (len) {
if (len == 1) {
int p = 1 << (h - len);
T irot = 1;
for (int s = 0; s < (1 << (len - 1)); s++) {
int offset = s << (h - len + 1);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p];
a[i + offset] = l + r;
a[i + offset + p] =
(unsigned long long)(T::get_mod() + l.v - r.v) *
irot.v;
;
}
if (s + 1 != (1 << (len - 1)))
irot *= irate2[__builtin_ctzll(~(unsigned int)(s))];
}
len--;
} else {
// 4-base
int p = 1 << (h - len);
T irot = 1, iimag = iroot[2];
for (int s = 0; s < (1 << (len - 2)); s++) {
T irot2 = irot * irot;
T irot3 = irot2 * irot;
int offset = s << (h - len + 2);
for (int i = 0; i < p; i++) {
auto a0 = 1ULL * a[i + offset + 0 * p].v;
auto a1 = 1ULL * a[i + offset + 1 * p].v;
auto a2 = 1ULL * a[i + offset + 2 * p].v;
auto a3 = 1ULL * a[i + offset + 3 * p].v;
auto a2na3iimag =
1ULL * T((T::get_mod() + a2 - a3) * iimag.v).v;
a[i + offset] = a0 + a1 + a2 + a3;
a[i + offset + 1 * p] =
(a0 + (T::get_mod() - a1) + a2na3iimag) *
irot.v;
a[i + offset + 2 * p] =
(a0 + a1 + (T::get_mod() - a2) +
(T::get_mod() - a3)) *
irot2.v;
a[i + offset + 3 * p] =
(a0 + (T::get_mod() - a1) +
(T::get_mod() - a2na3iimag)) *
irot3.v;
}
if (s + 1 != (1 << (len - 2)))
irot *= irate3[__builtin_ctzll(~(unsigned int)(s))];
}
len -= 2;
}
}
T e = T(n).inv();
for (auto &x : a)
x *= e;
} else {
int len = 0; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
while (len < h) {
if (h - len == 1) {
int p = 1 << (h - len - 1);
T rot = 1;
for (int s = 0; s < (1 << len); s++) {
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p] * rot;
a[i + offset] = l + r;
a[i + offset + p] = l - r;
}
if (s + 1 != (1 << len))
rot *= rate2[__builtin_ctzll(~(unsigned int)(s))];
}
len++;
} else {
// 4-base
int p = 1 << (h - len - 2);
T rot = 1, imag = root[2];
for (int s = 0; s < (1 << len); s++) {
T rot2 = rot * rot;
T rot3 = rot2 * rot;
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto mod2 = 1ULL * T::get_mod() * T::get_mod();
auto a0 = 1ULL * a[i + offset].v;
auto a1 = 1ULL * a[i + offset + p].v * rot.v;
auto a2 = 1ULL * a[i + offset + 2 * p].v * rot2.v;
auto a3 = 1ULL * a[i + offset + 3 * p].v * rot3.v;
auto a1na3imag =
1ULL * T(a1 + mod2 - a3).v * imag.v;
auto na2 = mod2 - a2;
a[i + offset] = a0 + a2 + a1 + a3;
a[i + offset + 1 * p] =
a0 + a2 + (2 * mod2 - (a1 + a3));
a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
a[i + offset + 3 * p] =
a0 + na2 + (mod2 - a1na3imag);
}
if (s + 1 != (1 << len))
rot *= rate3[__builtin_ctzll(~(unsigned int)(s))];
}
len += 2;
}
}
}
}
vector<T> mult(const vector<T> &a, const vector<T> &b) {
if (a.empty() or b.empty())
return vector<T>();
int as = a.size(), bs = b.size();
int n = as + bs - 1;
if (as <= 30 or bs <= 30) {
if (as > 30)
return mult(b, a);
vector<T> res(n);
rep(i, 0, as) rep(j, 0, bs) res[i + j] += a[i] * b[j];
return res;
}
int m = 1;
while (m < n)
m <<= 1;
vector<T> res(m);
rep(i, 0, as) res[i] = a[i];
ntt(res);
if (a == b)
rep(i, 0, m) res[i] *= res[i];
else {
vector<T> c(m);
rep(i, 0, bs) c[i] = b[i];
ntt(c);
rep(i, 0, m) res[i] *= c[i];
}
ntt(res, 1);
res.resize(n);
return res;
}
};
/**
* @brief Number Theoretic Transform
*/
#line 2 "FPS/fps.hpp"
template <typename T> struct Poly : vector<T> {
Poly(int n = 0) {
this->assign(n, T());
}
Poly(const initializer_list<T> f) : vector<T>::vector(f) {}
Poly(const vector<T> &f) {
this->assign(ALL(f));
}
int deg() const {
return this->size() - 1;
}
T eval(const T &x) {
T res;
for (int i = this->size() - 1; i >= 0; i--)
res *= x, res += this->at(i);
return res;
}
Poly rev() const {
Poly res = *this;
reverse(ALL(res));
return res;
}
void shrink() {
while (!this->empty() and this->back() == 0)
this->pop_back();
}
Poly operator>>(ll sz) const {
if ((int)this->size() <= sz)
return {};
Poly ret(*this);
ret.erase(ret.begin(), ret.begin() + sz);
return ret;
}
Poly operator<<(ll sz) const {
Poly ret(*this);
ret.insert(ret.begin(), sz, T(0));
return ret;
}
Poly<T> mult(const Poly<T> &a, const Poly<T> &b) {
if (a.empty() or b.empty())
return {};
int as = a.size(), bs = b.size();
int n = as + bs - 1;
if (as <= 30 or bs <= 30) {
if (as > 30)
return mult(b, a);
Poly<T> res(n);
rep(i, 0, as) rep(j, 0, bs) res[i + j] += a[i] * b[j];
return res;
}
int m = 1;
while (m < n)
m <<= 1;
Poly<T> res(m);
rep(i, 0, as) res[i] = a[i];
res.NTT(0);
if (a == b)
rep(i, 0, m) res[i] *= res[i];
else {
Poly<T> c(m);
rep(i, 0, bs) c[i] = b[i];
c.NTT(0);
rep(i, 0, m) res[i] *= c[i];
}
res.NTT(1);
res.resize(n);
return res;
}
Poly square() const {
return Poly(mult(*this, *this));
}
Poly operator-() const {
return Poly() - *this;
}
Poly operator+(const Poly &g) const {
return Poly(*this) += g;
}
Poly operator+(const T &g) const {
return Poly(*this) += g;
}
Poly operator-(const Poly &g) const {
return Poly(*this) -= g;
}
Poly operator-(const T &g) const {
return Poly(*this) -= g;
}
Poly operator*(const Poly &g) const {
return Poly(*this) *= g;
}
Poly operator*(const T &g) const {
return Poly(*this) *= g;
}
Poly operator/(const Poly &g) const {
return Poly(*this) /= g;
}
Poly operator/(const T &g) const {
return Poly(*this) /= g;
}
Poly operator%(const Poly &g) const {
return Poly(*this) %= g;
}
pair<Poly, Poly> divmod(const Poly &g) const {
Poly q = *this / g, r = *this - g * q;
r.shrink();
return {q, r};
}
Poly &operator+=(const Poly &g) {
if (g.size() > this->size())
this->resize(g.size());
rep(i, 0, g.size()) {
(*this)[i] += g[i];
}
return *this;
}
Poly &operator+=(const T &g) {
if (this->empty())
this->push_back(0);
(*this)[0] += g;
return *this;
}
Poly &operator-=(const Poly &g) {
if (g.size() > this->size())
this->resize(g.size());
rep(i, 0, g.size()) {
(*this)[i] -= g[i];
}
return *this;
}
Poly &operator-=(const T &g) {
if (this->empty())
this->push_back(0);
(*this)[0] -= g;
return *this;
}
Poly &operator*=(const Poly &g) {
*this = mult(*this, g);
return *this;
}
Poly &operator*=(const T &g) {
rep(i, 0, this->size())(*this)[i] *= g;
return *this;
}
Poly &operator/=(const Poly &g) {
if (g.size() > this->size()) {
this->clear();
return *this;
}
Poly g2 = g;
reverse(ALL(*this));
reverse(ALL(g2));
int n = this->size() - g2.size() + 1;
this->resize(n);
g2.resize(n);
*this *= g2.inv();
this->resize(n);
reverse(ALL(*this));
shrink();
return *this;
}
Poly &operator/=(const T &g) {
rep(i, 0, this->size())(*this)[i] /= g;
return *this;
}
Poly &operator%=(const Poly &g) {
*this -= *this / g * g;
shrink();
return *this;
}
Poly diff() const {
Poly res(this->size() - 1);
rep(i, 0, res.size()) res[i] = (*this)[i + 1] * (i + 1);
return res;
}
Poly inte() const {
Poly res(this->size() + 1);
for (int i = res.size() - 1; i; i--)
res[i] = (*this)[i - 1] / i;
return res;
}
Poly log() const {
assert(this->front() == 1);
const int n = this->size();
Poly res = diff() * inv();
res = res.inte();
res.resize(n);
return res;
}
Poly shift(const int &c) const {
const int n = this->size();
Poly res = *this, g(n);
g[0] = 1;
rep(i, 1, n) g[i] = g[i - 1] * c / i;
vector<T> fact(n, 1);
rep(i, 0, n) {
if (i)
fact[i] = fact[i - 1] * i;
res[i] *= fact[i];
}
res = res.rev();
res *= g;
res.resize(n);
res = res.rev();
rep(i, 0, n) res[i] /= fact[i];
return res;
}
Poly inv() const {
const int n = this->size();
Poly res(1);
res.front() = T(1) / this->front();
for (int k = 1; k < n; k <<= 1) {
Poly f(k * 2), g(k * 2);
rep(i, 0, min(n, k * 2)) f[i] = (*this)[i];
rep(i, 0, k) g[i] = res[i];
f.NTT(0);
g.NTT(0);
rep(i, 0, k * 2) f[i] *= g[i];
f.NTT(1);
rep(i, 0, k) {
f[i] = 0;
f[i + k] = -f[i + k];
}
f.NTT(0);
rep(i, 0, k * 2) f[i] *= g[i];
f.NTT(1);
rep(i, 0, k) f[i] = res[i];
swap(res, f);
}
res.resize(n);
return res;
}
Poly exp() const {
const int n = this->size();
if (n == 1)
return Poly({T(1)});
Poly b(2), c(1), z1, z2(2);
b[0] = c[0] = z2[0] = z2[1] = 1;
b[1] = (*this)[1];
for (int k = 2; k < n; k <<= 1) {
Poly y = b;
y.resize(k * 2);
y.NTT(0);
z1 = z2;
Poly z(k);
rep(i, 0, k) z[i] = y[i] * z1[i];
z.NTT(1);
rep(i, 0, k >> 1) z[i] = 0;
z.NTT(0);
rep(i, 0, k) z[i] *= -z1[i];
z.NTT(1);
c.insert(c.end(), z.begin() + (k >> 1), z.end());
z2 = c;
z2.resize(k * 2);
z2.NTT(0);
Poly x = *this;
x.resize(k);
x = x.diff();
x.resize(k);
x.NTT(0);
rep(i, 0, k) x[i] *= y[i];
x.NTT(1);
Poly bb = b.diff();
rep(i, 0, k - 1) x[i] -= bb[i];
x.resize(k * 2);
rep(i, 0, k - 1) {
x[k + i] = x[i];
x[i] = 0;
}
x.NTT(0);
rep(i, 0, k * 2) x[i] *= z2[i];
x.NTT(1);
x.pop_back();
x = x.inte();
rep(i, k, min(n, k * 2)) x[i] += (*this)[i];
rep(i, 0, k) x[i] = 0;
x.NTT(0);
rep(i, 0, k * 2) x[i] *= y[i];
x.NTT(1);
b.insert(b.end(), x.begin() + k, x.end());
}
b.resize(n);
return b;
}
Poly pow(ll t) {
if (t == 0) {
Poly res(this->size());
res[0] = 1;
return res;
}
int n = this->size(), k = 0;
while (k < n and (*this)[k] == 0)
k++;
Poly res(n);
if (__int128_t(t) * k >= n)
return res;
n -= t * k;
Poly g(n);
T c = (*this)[k], ic = c.inv();
rep(i, 0, n) g[i] = (*this)[i + k] * ic;
g = g.log();
for (auto &x : g)
x *= t;
g = g.exp();
c = c.pow(t);
rep(i, 0, n) res[i + t * k] = g[i] * c;
return res;
}
void NTT(bool inv);
};
/**
* @brief Formal Power Series (NTT-friendly mod)
*/
#line 2 "FPS/multieval.hpp"
template<typename T>struct MultiEval{
int m,n; vector<Poly<T>> t;
MultiEval(vector<T>& v){
m=v.size(),n=1; while(n<m)n<<=1;
t.resize(n<<1);
rep(i,0,n){
T w=(i<m?v[i]:0);
t[n+i]=Poly<T>({-w,T(1)});
}
for(int i=n-1;i;i--)t[i]=t[i*2]*t[i*2+1];
}
vector<T> run(const vector<T>& f){
if(f.empty())return vector<T>(m);
vector<Poly<T>> c(n*2);
auto v=t[1].rev();
v.resize(f.size());
v=v.inv().rev()*Poly<T>(f);
v.erase(v.begin(),v.begin()+f.size()-1);
v.resize(n); reverse(ALL(v)); c[1]=v;
rep(i,1,n){
int d=c[i].size();
rep(k,0,2){
auto add=t[i*2+(k^1)];
add.resize(d/2+1);
add=add.rev();
add*=c[i];
add.resize(d);
c[i*2+k]=vector<T>(add.begin()+d/2,add.end());
}
}
vector<T> res(m); rep(i,0,m)res[i]=c[n+i][0];
return res;
}
vector<T> build(vector<T>& ys){
auto w=t[1].rev();
w.resize(m+1);
auto vs=run(w.rev().diff());
rep(i,0,m)ys[i]/=vs[i];
vector<Poly<T>> c(n*2);
rep(i,0,n){
if(i<m)c[n+i]=Poly<T>({ys[i]});
else c[n+i]=Poly<T>({T()});
}
for(int i=n-1;i;i--)c[i]=c[i*2]*t[i*2+1]+c[i*2+1]*t[i*2];
c[1]=vector<T>(c[1].begin()+(n-m),c[1].end());
c[1].resize(m);
return c[1];
}
};
/**
* @brief Multipoint Evaluation
*/
#line 9 "Verify/LC_polynomial_interpolation.test.cpp"
using Fp = fp<998244353>;
NTT<Fp> ntt;
template <> void Poly<Fp>::NTT(bool inv) {
ntt.ntt(*this, inv);
}
int main() {
int n;
cin >> n;
vector<Fp> a(n), b(n);
for (auto &x : a)
cin >> x;
for (auto &x : b)
cin >> x;
MultiEval<Fp> multi(a);
auto ret = multi.build(b);
rep(i, 0, n) cout << ret[i] << '\n';
return 0;
}