結果
| 問題 | No.3619 Compositional Power with Schröder Coordinate |
| コンテスト | |
| ユーザー |
 shingo0909
|
| 提出日時 | 2026-08-10 22:22:39 |
| 言語 | C++23 (gcc 15.2.0 + boost 1.90.0) |
| 結果 |
AC
|
| 実行時間 | 1,163 ms / 10,000 ms |
| + 435µs | |
| コード長 | 35,959 bytes |
| 記録 | |
| コンパイル時間 | 4,572 ms |
| コンパイル使用メモリ | 389,128 KB |
| 実行使用メモリ | 149,420 KB |
| 最終ジャッジ日時 | 2026-08-10 22:22:52 |
| 合計ジャッジ時間 | 12,815 ms |
|
ジャッジサーバーID （参考情報） |
judge3_0 / judge2_0 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 2 |
| other | AC * 6 |
ソースコード
// drkenさんのgithubより引用
// https://github.com/drken1215/algorithm/blob/master/MathAlgebra/composition_formal_power_series.cpp
//
// FPS の合成 (Kinoshita-Li 法, O(N (log N)^2))
//
// verified:
// Yosupo Library Checker - Composition of Formal Power Series (Large)
// https://judge.yosupo.jp/problem/composition_of_formal_power_series_large
//
#pragma GCC optimize("Ofast")
#pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>
using namespace std;
//------------------------------//
// mod algorithms
//------------------------------//
// modint
template <int MOD = 998244353, bool PRIME = true>
struct Fp {
// inner value
unsigned int val;
// constructor
constexpr Fp() : val(0) {}
template <std::signed_integral T>
constexpr Fp(T v) {
long long tmp = (long long)(v % (long long)(get_umod()));
if (tmp < 0)
tmp += get_umod();
val = (unsigned int)(tmp);
}
template <std::unsigned_integral T>
constexpr Fp(T v) {
val = (unsigned int)(v % get_umod());
}
constexpr long long get() const { return val; }
constexpr static int get_mod() { return MOD; }
constexpr static unsigned int get_umod() { return MOD; }
// arithmetic operators
constexpr Fp operator+() const { return Fp(*this); }
constexpr Fp operator-() const { return Fp() - Fp(*this); }
constexpr Fp operator+(const Fp &r) const { return Fp(*this) += r; }
constexpr Fp operator-(const Fp &r) const { return Fp(*this) -= r; }
constexpr Fp operator*(const Fp &r) const { return Fp(*this) *= r; }
constexpr Fp operator/(const Fp &r) const { return Fp(*this) /= r; }
constexpr Fp &operator+=(const Fp &r) {
val += r.val;
if (val >= get_umod())
val -= get_umod();
return *this;
}
constexpr Fp &operator-=(const Fp &r) {
val -= r.val;
if (val >= get_umod())
val += get_umod();
return *this;
}
constexpr Fp &operator*=(const Fp &r) {
unsigned long long tmp = val;
tmp *= r.val;
val = (unsigned int)(tmp % get_umod());
return *this;
}
constexpr Fp &operator/=(const Fp &r) {
return *this = *this * r.inv();
}
constexpr Fp pow(long long n) const {
assert(n >= 0);
Fp res(1), mul(*this);
while (n) {
if (n & 1)
res *= mul;
mul *= mul;
n >>= 1;
}
return res;
}
constexpr Fp inv() const {
assert(val);
if (PRIME) {
return pow(get_umod() - 2);
} else {
assert(gcd(val, get_umod()) == 1);
long long m = get_umod(), a = val, b = m, u = 1, v = 0;
while (b > 0) {
auto t = a / b;
a -= t * b, swap(a, b);
u -= t * v, swap(u, v);
}
return Fp(u);
}
}
// other operators
constexpr bool operator==(const Fp &r) const {
return this->val == r.val;
}
constexpr bool operator!=(const Fp &r) const {
return this->val != r.val;
}
constexpr bool operator<(const Fp &r) const {
return this->val < r.val;
}
constexpr bool operator>(const Fp &r) const {
return this->val > r.val;
}
constexpr bool operator<=(const Fp &r) const {
return this->val <= r.val;
}
constexpr bool operator>=(const Fp &r) const {
return this->val >= r.val;
}
constexpr Fp &operator++() {
++val;
if (val == get_umod())
val = 0;
return *this;
}
constexpr Fp &operator--() {
if (val == 0)
val = get_umod();
--val;
return *this;
}
constexpr Fp operator++(int) {
Fp res = *this;
++*this;
return res;
}
constexpr Fp operator--(int) {
Fp res = *this;
--*this;
return res;
}
friend constexpr istream &operator>>(istream &is, Fp &x) {
long long tmp = 1;
is >> tmp;
tmp = tmp % (long long)(get_umod());
if (tmp < 0)
tmp += get_umod();
x.val = (unsigned int)(tmp);
return is;
}
friend constexpr ostream &operator<<(ostream &os, const Fp &x) {
return os << x.val;
}
friend constexpr Fp pow(const Fp &r, long long n) {
return r.pow(n);
}
friend constexpr Fp inv(const Fp &r) {
return r.inv();
}
};
// Binomial coefficient
template <class mint>
struct BiCoef {
vector<mint> fact_, inv_, finv_;
constexpr BiCoef() {}
constexpr BiCoef(int n) : fact_(n, 1), inv_(n, 1), finv_(n, 1) {
init(n);
}
constexpr void init(int n) {
fact_.assign(n, 1), inv_.assign(n, 1), finv_.assign(n, 1);
int MOD = fact_[0].get_mod();
for (int i = 2; i < n; i++) {
fact_[i] = fact_[i - 1] * i;
inv_[i] = -inv_[MOD % i] * (MOD / i);
finv_[i] = finv_[i - 1] * inv_[i];
}
}
constexpr mint com(int n, int k) const {
if (n < k || n < 0 || k < 0)
return 0;
return fact_[n] * finv_[k] * finv_[n - k];
}
constexpr mint fact(int n) const {
if (n < 0)
return 0;
return fact_[n];
}
constexpr mint inv(int n) const {
if (n < 0)
return 0;
return inv_[n];
}
constexpr mint finv(int n) const {
if (n < 0)
return 0;
return finv_[n];
}
};
//------------------------------//
// NTT
//------------------------------//
// calc primitive root
constexpr int calc_primitive_root(long long m) {
if (m == 1)
return -1;
if (m == 2)
return 1;
if (m == 998244353)
return 3;
if (m == 167772161)
return 3;
if (m == 469762049)
return 3;
if (m == 754974721)
return 11;
if (m == 645922817)
return 3;
if (m == 897581057)
return 3;
auto mod_pow = [&](long long a, long long n, long long m) {
long long res = 1;
while (n > 0) {
if (n % 2 == 1)
res = res * a % m;
a = a * a % m;
n >>= 1;
}
return res;
};
long long divs[20] = {};
divs[0] = 2;
long long cnt = 1;
long long x = (m - 1) / 2;
while (x % 2 == 0)
x /= 2;
for (long long i = 3; i * i <= x; i += 2) {
if (x % i == 0) {
divs[cnt++] = i;
while (x % i == 0)
x /= i;
}
}
if (x > 1)
divs[cnt++] = x;
for (long long g = 2;; g++) {
bool ok = true;
for (int i = 0; i < cnt; i++) {
if (mod_pow(g, (m - 1) / divs[i], m) == 1) {
ok = false;
break;
}
}
if (ok)
return g;
}
}
// NTT setup
template <class mint, int MOD = mint::get_mod(), int g = calc_primitive_root(mint::get_mod())>
struct ntt_setup {
static constexpr int bsf_constexpr(unsigned int x) {
int i = 0;
while (!(x & (1 << i)))
i++;
return i;
};
static constexpr int rank = bsf_constexpr(MOD - 1);
array<mint, rank + 1> root, iroot; // root[i]^(2^i) = 1, root[i] * iroot[i] = 1
array<mint, max(0, rank - 1)> rate2, irate2;
array<mint, max(0, rank - 2)> rate3, irate3;
ntt_setup() {
root[rank] = mint(g).pow((MOD - 1) >> rank);
iroot[rank] = root[rank].inv();
for (int i = rank - 1; i >= 0; i--) {
root[i] = root[i + 1] * root[i + 1];
iroot[i] = iroot[i + 1] * iroot[i + 1];
}
mint prod = 1, iprod = 1;
for (int i = 0; i < rank - 1; i++) {
rate2[i] = root[i + 2] * prod;
irate2[i] = iroot[i + 2] * iprod;
prod *= iroot[i + 2];
iprod *= root[i + 2];
}
prod = 1, iprod = 1;
for (int i = 0; i < rank - 2; i++) {
rate3[i] = root[i + 3] * prod;
irate3[i] = iroot[i + 3] * iprod;
prod *= iroot[i + 3];
iprod *= root[i + 3];
}
}
};
// NTT transformation
template <class mint, int MOD = mint::get_mod()>
void ntt_trans(vector<mint> &v) {
int n = (int)v.size();
int h = 0;
while ((1U << h) < (unsigned int)(n))
h++;
static const ntt_setup<mint> setup;
int len = 0;
while (len < h) {
if (h - len == 1) {
int p = 1 << (h - len - 1);
mint rot = 1;
for (int s = 0; s < (1 << len); s++) {
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto l = v[i + offset];
auto r = v[i + offset + p] * rot;
v[i + offset] = l + r;
v[i + offset + p] = l - r;
}
if (s + 1 != (1 << len)) {
rot *= setup.rate2[setup.bsf_constexpr(~(unsigned int)(s))];
}
}
len++;
} else {
int p = 1 << (h - len - 2);
mint rot = 1, imag = setup.root[2];
for (int s = 0; s < (1 << len); s++) {
mint rot2 = rot * rot, rot3 = rot2 * rot;
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto mod2 = 1ULL * MOD * MOD;
auto a0 = 1ULL * v[i + offset].val;
auto a1 = 1ULL * v[i + offset + p].val * rot.val;
auto a2 = 1ULL * v[i + offset + p * 2].val * rot2.val;
auto a3 = 1ULL * v[i + offset + p * 3].val * rot3.val;
auto tmp = 1ULL * mint(a1 + mod2 - a3).val * imag.val;
auto na2 = mod2 - a2;
v[i + offset] = a0 + a2 + a1 + a3;
v[i + offset + p] = a0 + a2 + (mod2 * 2 - (a1 + a3));
v[i + offset + p * 2] = a0 + na2 + tmp;
v[i + offset + p * 3] = a0 + na2 + (mod2 - tmp);
}
if (s + 1 != (1 << len)) {
rot *= setup.rate3[setup.bsf_constexpr(~(unsigned int)(s))];
}
}
len += 2;
}
}
}
// NTT inv-transformation
template <class mint, int MOD = mint::get_mod()>
void ntt_trans_inv(vector<mint> &v) {
int n = (int)v.size();
int h = 0;
while ((1U << h) < (unsigned int)(n))
h++;
static const ntt_setup<mint> setup;
int len = h;
while (len) {
if (len == 1) {
int p = 1 << (h - len);
mint 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 = v[i + offset];
auto r = v[i + offset + p];
v[i + offset] = l + r;
v[i + offset + p] = (unsigned long long)((long long)(MOD) + l.val - r.val) * irot.val;
}
if (s + 1 != (1 << (len - 1))) {
irot *= setup.irate2[setup.bsf_constexpr(~(unsigned int)(s))];
}
}
len--;
} else {
int p = 1 << (h - len);
mint irot = 1, iimag = setup.iroot[2];
for (int s = 0; s < (1 << (len - 2)); s++) {
mint irot2 = irot * irot, irot3 = irot2 * irot;
int offset = s << (h - len + 2);
for (int i = 0; i < p; i++) {
auto a0 = 1ULL * v[i + offset].val;
auto a1 = 1ULL * v[i + offset + p].val;
auto a2 = 1ULL * v[i + offset + p * 2].val;
auto a3 = 1ULL * v[i + offset + p * 3].val;
auto tmp = 1ULL * mint((MOD + a2 - a3) * iimag.val).val;
v[i + offset] = a0 + a1 + a2 + a3;
v[i + offset + p] = (a0 + (MOD - a1) + tmp) * irot.val;
v[i + offset + p * 2] = (a0 + a1 + (MOD - a2) + (MOD - a3)) * irot2.val;
v[i + offset + p * 3] = (a0 + (MOD - a1) + (MOD - tmp)) * irot3.val;
}
if (s + 1 != (1 << (len - 2))) {
irot *= setup.irate3[setup.bsf_constexpr(~(unsigned int)(s))];
}
}
len -= 2;
}
}
mint in = mint(n).inv();
for (int i = 0; i < n; i++)
v[i] *= in;
}
// naive convolution
template <class VEC>
VEC convolution_naive(const VEC &a, const VEC &b) {
int n = (int)a.size(), m = (int)b.size();
if (!n || !m)
return {};
VEC res(n + m - 1);
if (n < m) {
for (int j = 0; j < m; j++)
for (int i = 0; i < n; i++)
res[i + j] += a[i] * b[j];
} else {
for (int i = 0; i < n; i++)
for (int j = 0; j < m; j++)
res[i + j] += a[i] * b[j];
}
return res;
}
// ntt convolution
template <class mint>
vector<mint> convolution_ntt(vector<mint> a, vector<mint> b) {
int MOD = mint::get_mod();
int n = (int)a.size(), m = (int)b.size();
if (!n || !m)
return {};
int z = (int)bit_ceil((unsigned int)(n + m - 1));
assert((MOD - 1) % z == 0);
a.resize(z), b.resize(z);
ntt_trans(a), ntt_trans(b);
for (int i = 0; i < z; i++)
a[i] *= b[i];
ntt_trans_inv(a);
a.resize(n + m - 1);
return a;
}
// convolution long long (if u64 is necessary, use convolution_ull)
template <class VEC>
VEC convolution_ll(const VEC &a, const VEC &b) {
int n = (int)a.size(), m = (int)b.size();
if (!n || !m)
return VEC();
if (min(n, m) <= 60)
return convolution_naive(a, b);
static constexpr int MOD0 = 754974721; // 2^24
static constexpr int MOD1 = 167772161; // 2^25
static constexpr int MOD2 = 469762049; // 2^26
using mint0 = Fp<MOD0>;
using mint1 = Fp<MOD1>;
using mint2 = Fp<MOD2>;
static const mint1 imod0 = 95869806; // modinv(MOD0, MOD1);
static const mint2 imod1 = 104391568; // modinv(MOD1, MOD2);
static const mint2 imod01 = 187290749; // imod1 / MOD0;
vector<mint0> a0(n, 0), b0(m, 0);
vector<mint1> a1(n, 0), b1(m, 0);
vector<mint2> a2(n, 0), b2(m, 0);
for (int i = 0; i < n; ++i)
a0[i] = a[i], a1[i] = a[i], a2[i] = a[i];
for (int i = 0; i < m; ++i)
b0[i] = b[i], b1[i] = b[i], b2[i] = b[i];
auto c0 = convolution_ntt(std::move(a0), std::move(b0));
auto c1 = convolution_ntt(std::move(a1), std::move(b1));
auto c2 = convolution_ntt(std::move(a2), std::move(b2));
VEC res(n + m - 1);
long long mod0 = MOD0, mod01 = mod0 * MOD1;
for (int i = 0; i < n + m - 1; ++i) {
unsigned int y0 = c0[i].val;
unsigned int y1 = (imod0 * (c1[i] - mint1(y0))).val;
unsigned int y2 = (imod01 * (c2[i] - mint2(y0)) - imod1 * y1).val;
res[i] = mod01 * y2 + mod0 * y1 + y0;
}
return res;
}
// convolution in general mod
template <class mint>
vector<mint> convolution_general_mod(const vector<mint> &a, const vector<mint> &b) {
int n = (int)a.size(), m = (int)b.size();
if (!n || !m)
return {};
if (min(n, m) <= 60)
return convolution_naive(a, b);
if constexpr (std::is_same_v<mint, Fp<998244353>>)
return convolution_ntt(a, b);
static constexpr int MOD0 = 754974721; // 2^24
static constexpr int MOD1 = 167772161; // 2^25
static constexpr int MOD2 = 469762049; // 2^26
using mint0 = Fp<MOD0>;
using mint1 = Fp<MOD1>;
using mint2 = Fp<MOD2>;
static const mint1 imod0 = 95869806; // modinv(MOD0, MOD1);
static const mint2 imod1 = 104391568; // modinv(MOD1, MOD2);
static const mint2 imod01 = 187290749; // imod1 / MOD0;
vector<mint0> a0(n, 0), b0(m, 0);
vector<mint1> a1(n, 0), b1(m, 0);
vector<mint2> a2(n, 0), b2(m, 0);
for (int i = 0; i < n; ++i)
a0[i] = a[i].val, a1[i] = a[i].val, a2[i] = a[i].val;
for (int i = 0; i < m; ++i)
b0[i] = b[i].val, b1[i] = b[i].val, b2[i] = b[i].val;
auto c0 = convolution_ntt(std::move(a0), std::move(b0));
auto c1 = convolution_ntt(std::move(a1), std::move(b1));
auto c2 = convolution_ntt(std::move(a2), std::move(b2));
vector<mint> res(n + m - 1);
mint mod0 = MOD0, mod01 = mod0 * MOD1;
for (int i = 0; i < n + m - 1; ++i) {
unsigned int y0 = c0[i].val;
unsigned int y1 = (imod0 * (c1[i] - mint1(y0))).val;
unsigned int y2 = (imod01 * (c2[i] - mint2(y0)) - imod1 * y1).val;
res[i] = mod01 * y2 + mod0 * y1 + y0;
}
return res;
}
// convolution overall
template <class T>
vector<T> convolution(const vector<T> &a, const vector<T> &b) {
int n = (int)a.size(), m = (int)b.size();
if (!n || !m)
return {};
if (min(n, m) <= 60)
return convolution_naive(a, b);
if constexpr (std::is_same_v<T, Fp<998244353>>)
return convolution_ntt(a, b);
else if constexpr (std::is_integral_v<T>)
return convolution_ll(a, b);
else
return convolution_general_mod(a, b);
}
//------------------------------//
// FPS
//------------------------------//
// Formal Power Series
template <class mint>
struct FPS : vector<mint> {
static const int SPARSE_BOARDER = 60;
using vector<mint>::vector;
// constructor
constexpr FPS(const vector<mint> &r) : vector<mint>(r) {}
// core operator
constexpr FPS pre(int siz) const {
return FPS(begin(*this), begin(*this) + min((int)this->size(), siz));
}
constexpr FPS rev() const {
FPS res = *this;
reverse(begin(res), end(res));
return res;
}
constexpr FPS &normalize() {
while (!this->empty() && this->back() == 0)
this->pop_back();
return *this;
}
constexpr mint eval(const mint &v) const {
mint res = 0;
for (int i = (int)this->size() - 1; i >= 0; --i) {
res *= v;
res += (*this)[i];
}
return res;
}
constexpr int count_terms() const {
int res = 0;
for (int i = 0; i < (int)this->size(); i++)
if ((*this)[i] != mint(0))
res++;
return res;
}
// basic operator
constexpr FPS operator-() const noexcept {
FPS res = (*this);
for (int i = 0; i < (int)res.size(); ++i)
res[i] = -res[i];
return res;
}
constexpr FPS operator+(const mint &v) const { return FPS(*this) += v; }
constexpr FPS operator+(const FPS &r) const { return FPS(*this) += r; }
constexpr FPS operator-(const mint &v) const { return FPS(*this) -= v; }
constexpr FPS operator-(const FPS &r) const { return FPS(*this) -= r; }
constexpr FPS operator*(const mint &v) const { return FPS(*this) *= v; }
constexpr FPS operator*(const FPS &r) const { return FPS(*this) *= r; }
constexpr FPS operator/(const mint &v) const { return FPS(*this) /= v; }
constexpr FPS operator/(const FPS &r) const { return FPS(*this) /= r; }
constexpr FPS operator%(const FPS &r) const { return FPS(*this) %= r; }
constexpr FPS operator<<(int x) const { return FPS(*this) <<= x; }
constexpr FPS operator>>(int x) const { return FPS(*this) >>= x; }
constexpr FPS &operator+=(const mint &v) {
if (this->empty())
this->reserve(1), this->resize(1);
(*this)[0] += v;
return *this;
}
constexpr FPS &operator+=(const FPS &r) {
if (r.size() > this->size())
this->reserve(r.size()), this->resize(r.size());
for (int i = 0; i < (int)r.size(); ++i)
(*this)[i] += r[i];
return this->normalize();
}
constexpr FPS &operator-=(const mint &v) {
if (this->empty())
this->reserve(1), this->resize(1);
(*this)[0] -= v;
return *this;
}
constexpr FPS &operator-=(const FPS &r) {
if (r.size() > this->size())
this->reserve(r.size()), this->resize(r.size());
for (int i = 0; i < (int)r.size(); ++i)
(*this)[i] -= r[i];
return this->normalize();
}
constexpr FPS &operator*=(const mint &v) {
for (int i = 0; i < (int)this->size(); ++i)
(*this)[i] *= v;
return *this;
}
constexpr FPS &operator*=(const FPS &r) {
return *this = convolution((*this), r);
}
constexpr FPS ÷_by_modint(const mint &v) {
assert(v != 0);
mint iv = v.inv();
for (int i = 0; i < (int)this->size(); ++i)
(*this)[i] *= iv;
return *this;
}
constexpr FPS ÷_by_integer(const mint &v) {
assert(v != 0);
for (int i = 0; i < (int)this->size(); ++i)
(*this)[i] /= v;
return *this;
}
constexpr FPS &operator/=(const mint &v) {
assert(v != 0);
if constexpr (std::is_integral_v<mint>)
return divide_by_integer(v);
else
return divide_by_modint(v);
}
// division, r must be normalized (r.back() must not be 0)
constexpr FPS &operator/=(const FPS &r) {
assert(!r.empty());
assert(r.back() != 0);
this->normalize();
if (this->size() < r.size()) {
this->clear();
return *this;
}
int need = (int)this->size() - (int)r.size() + 1;
*this = (rev().pre(need) * r.rev().inv(need)).pre(need).rev();
return *this;
}
constexpr FPS &operator%=(const FPS &r) {
assert(!r.empty());
assert(r.back() != 0);
this->normalize();
FPS q = (*this) / r;
return *this -= q * r;
}
constexpr FPS &operator<<=(int x) {
FPS res(x, 0);
res.insert(res.end(), begin(*this), end(*this));
return *this = res;
}
constexpr FPS &operator>>=(int x) {
FPS res;
res.insert(res.end(), begin(*this) + x, end(*this));
return *this = res;
}
// advanced operation
// df/dx
constexpr FPS diff() const {
int n = (int)this->size();
if (n <= 0)
return FPS();
FPS res(n - 1);
for (int i = 1; i < n; ++i)
res[i - 1] = (*this)[i] * i;
return res;
}
// \int f dx
constexpr FPS integral() const {
int n = (int)this->size();
FPS res(n + 1, 0);
for (int i = 0; i < n; ++i)
res[i + 1] = (*this)[i] / (i + 1);
return res;
}
// inv(f), f[0] must not be 0
constexpr FPS inv(int deg = -1) const {
if (count_terms() <= SPARSE_BOARDER)
return inv_sparse(deg);
if constexpr (std::is_same_v<mint, Fp<998244353>>)
return inv_ntt_friendly(deg);
assert(this->size() >= 1 && (*this)[0] != 0);
if (deg < 0)
deg = (int)this->size();
FPS res({mint(1) / (*this)[0]});
for (int d = 1; d < deg; d <<= 1) {
res = (res + res - res * res * pre(d << 1)).pre(d << 1);
}
res.resize(deg);
return res;
}
constexpr FPS inv_ntt_friendly(int deg = -1) const {
assert(this->size() >= 1 && (*this)[0] != 0);
if (deg < 0)
deg = (int)this->size();
FPS res(deg);
res[0] = mint(1) / (*this)[0];
for (int d = 1; d < deg; d <<= 1) {
FPS g(d * 2), h(d * 2);
mint iv = mint(d * 2).inv();
for (int i = 0; i < min((int)this->size(), d * 2); i++)
g[i] = (*this)[i];
for (int i = 0; i < d; i++)
h[i] = res[i];
ntt_trans(g), ntt_trans(h);
for (int i = 0; i < d * 2; i++)
g[i] *= h[i];
ntt_trans_inv(g);
for (int i = 0; i < d; i++)
g[i] = 0;
ntt_trans(g);
for (int i = 0; i < d * 2; i++)
g[i] *= h[i];
ntt_trans_inv(g);
for (int i = d; i < min(deg, d * 2); i++)
res[i] = -g[i];
}
return res.pre(deg);
}
constexpr FPS inv_sparse(int deg = -1) const {
assert(this->size() >= 1 && (*this)[0] != 0);
if (deg < 0)
deg = (int)this->size();
vector<pair<int, mint>> dat;
for (int i = 1; i < (int)this->size(); i++)
if ((*this)[i] != mint(0)) {
dat.emplace_back(i, (*this)[i]);
}
vector<mint> res(deg);
res[0] = (*this)[0].inv();
for (int i = 1; i < deg; i++) {
mint r = 0;
for (auto &&[k, val] : dat) {
if (k > i)
break;
r -= val * res[i - k];
}
res[i] = r * res[0];
}
return res;
}
// log(f) = \int f'/f dx, f[0] must be 1
constexpr FPS log(int deg = -1) const {
assert(this->size() >= 1 && (*this)[0] == 1);
if (count_terms() <= SPARSE_BOARDER)
return log_sparse(deg);
if (deg < 0)
deg = (int)this->size();
return ((diff() * inv(deg)).pre(deg - 1)).integral();
}
constexpr FPS log_sparse(int deg = -1) const {
assert(this->size() >= 1 && (*this)[0] == 1);
if (deg < 0)
deg = (int)this->size();
vector<pair<int, mint>> dat;
for (int i = 1; i < (int)this->size(); i++)
if ((*this)[i] != mint(0)) {
dat.emplace_back(i, (*this)[i]);
}
BiCoef<mint> bc(deg);
vector<mint> res(deg), tmp(deg);
for (int i = 0; i < deg - 1; i++) {
mint r = mint(i + 1) * (*this)[i + 1];
for (auto &&[k, val] : dat) {
if (k > i)
break;
r -= val * tmp[i - k];
}
tmp[i] = r;
res[i + 1] = r * bc.inv(i + 1);
}
return res;
}
// exp(f), f[0] must be 0
constexpr FPS exp(int deg = -1) const {
if ((int)this->size() == 0)
return {mint(1)};
if (count_terms() <= SPARSE_BOARDER)
return exp_sparse(deg);
if constexpr (std::is_same_v<mint, Fp<998244353>>)
return exp_ntt_friendly(deg);
assert((*this)[0] == 0);
if (deg < 0)
deg = (int)this->size();
FPS res(1, 1);
for (int d = 1; d < deg; d <<= 1) {
res = res * (pre(d << 1) - res.log(d << 1) + 1).pre(d << 1);
}
res.resize(deg);
return res;
}
constexpr FPS exp_ntt_friendly(int deg = -1) const {
if ((int)this->size() == 0)
return {mint(1)};
assert((*this)[0] == 0);
if (deg < 0)
deg = (int)this->size();
FPS fiv;
fiv.reserve(deg + 1);
fiv.emplace_back(mint(0));
fiv.emplace_back(mint(1));
auto inplace_integral = [&](FPS &F) -> void {
const int n = (int)F.size();
auto mod = mint::get_mod();
while ((int)fiv.size() <= n) {
int i = fiv.size();
fiv.emplace_back((-fiv[mod % i]) * (mod / i));
}
F.insert(begin(F), mint(0));
for (int i = 1; i <= n; i++)
F[i] *= fiv[i];
};
auto inplace_diff = [](FPS &F) -> void {
if (F.empty())
return;
F.erase(begin(F));
mint coef = 1;
for (int i = 0; i < (int)F.size(); i++) {
F[i] *= coef;
coef++;
}
};
FPS b{1, (1 < (int)this->size() ? (*this)[1] : 0)}, c{1}, z1, z2{1, 1};
for (int m = 2; m < deg; m <<= 1) {
auto y = b;
y.resize(m * 2);
ntt_trans(y);
z1 = z2;
FPS z(m);
for (int i = 0; i < m; i++)
z[i] = y[i] * z1[i];
ntt_trans_inv(z);
fill(begin(z), begin(z) + m / 2, mint(0));
ntt_trans(z);
for (int i = 0; i < m; i++)
z[i] *= -z1[i];
ntt_trans_inv(z);
c.insert(end(c), begin(z) + m / 2, end(z));
z2 = c;
z2.resize(m * 2);
ntt_trans(z2);
FPS x(begin(*this), begin(*this) + min((int)this->size(), m));
inplace_diff(x);
x.emplace_back(mint(0));
ntt_trans(x);
for (int i = 0; i < m; i++)
x[i] *= y[i];
ntt_trans_inv(x);
x -= b.diff();
x.resize(m * 2);
for (int i = 0; i < m - 1; i++)
x[m + i] = x[i], x[i] = mint(0);
ntt_trans(x);
for (int i = 0; i < m * 2; i++)
x[i] *= z2[i];
ntt_trans_inv(x);
x.pop_back();
inplace_integral(x);
for (int i = m; i < min((int)this->size(), m * 2); i++)
x[i] += (*this)[i];
fill(begin(x), begin(x) + m, mint(0));
ntt_trans(x);
for (int i = 0; i < m * 2; i++)
x[i] *= y[i];
ntt_trans_inv(x);
b.insert(end(b), begin(x) + m, end(x));
}
return FPS(begin(b), begin(b) + deg);
}
constexpr FPS exp_sparse(int deg = -1) const {
if ((int)this->size() == 0)
return {mint(1)};
assert((*this)[0] == 0);
if (deg < 0)
deg = (int)this->size();
vector<pair<int, mint>> dat;
for (int i = 1; i < (int)this->size(); i++)
if ((*this)[i] != mint(0)) {
dat.emplace_back(i - 1, (*this)[i] * i);
}
BiCoef<mint> bc(deg);
vector<mint> res(deg);
res[0] = 1;
for (int i = 1; i < deg; i++) {
mint r = 0;
for (auto &&[k, val] : dat) {
if (k > i - 1)
break;
r += val * res[i - k - 1];
}
res[i] = r * bc.inv(i);
}
return res;
}
// pow(f) = exp(e * log f)
constexpr FPS pow(long long e, int deg = -1) const {
if (count_terms() <= SPARSE_BOARDER)
return pow_sparse(e, deg);
assert(e >= 0);
if (deg < 0)
deg = (int)this->size();
if (deg == 0)
return FPS();
if (e == 0) {
FPS res(deg, 0);
res[0] = 1;
return res;
}
long long ord = 0;
while (ord < (int)this->size() && (*this)[ord] == 0)
ord++;
if (ord == (int)this->size() || ord > (deg - 1) / e)
return FPS(deg, 0);
mint k = (*this)[ord];
FPS res = ((((*this) >> ord) / k).log(deg) * e).exp(deg) * mint(k).pow(e) << (e * ord);
res.resize(deg);
return res;
}
constexpr FPS pow_sparse(long long e, int deg = -1) const {
assert(e >= 0);
if (deg < 0)
deg = (int)this->size();
if (deg == 0)
return FPS();
if (e == 0) {
FPS res(deg, 0);
res[0] = 1;
return res;
}
long long ord = 0;
while (ord < (int)this->size() && (*this)[ord] == 0)
ord++;
if (ord == (int)this->size() || ord > (deg - 1) / e)
return FPS(deg, 0);
if ((*this)[0] == 1)
return pow_sparse_constant1(e, deg);
auto f = (*this);
rotate(f.begin(), f.begin() + ord, f.end());
mint con = f[0], icon = f[0].inv();
for (int i = 0; i < deg; i++)
f[i] *= icon;
auto res = f.pow_sparse_constant1(e, deg);
int ord2 = e * ord;
rotate(res.begin(), res.begin() + (deg - ord2), res.end());
fill(res.begin(), res.begin() + ord2, mint(0));
mint pw = con.pow(e);
for (int i = ord2; i < deg; i++)
res[i] *= pw;
return res;
}
constexpr FPS pow_sparse_constant1(mint e, int deg = -1) const {
assert((int)this->size() > 0 && (*this)[0] == 1);
if (deg < 0)
deg = (int)this->size();
vector<pair<int, mint>> dat;
for (int i = 1; i < (int)this->size(); i++)
if ((*this)[i] != mint(0)) {
dat.emplace_back(i, (*this)[i]);
}
BiCoef<mint> bc(deg);
vector<mint> res(deg);
res[0] = 1;
for (int i = 0; i < deg - 1; i++) {
mint &r = res[i + 1];
for (auto &&[k, val] : dat) {
if (k > i + 1)
break;
mint t = val * res[i - k + 1];
r += t * (mint(k) * e - mint(i - k + 1));
}
r *= bc.inv(i + 1);
}
return res;
}
// friend operators
friend constexpr FPS diff(const FPS &f) { return f.diff(); }
friend constexpr FPS integral(const FPS &f) { return f.integral(); }
friend constexpr FPS inv(const FPS &f, int deg = -1) { return f.inv(deg); }
friend constexpr FPS log(const FPS &f, int deg = -1) { return f.log(deg); }
friend constexpr FPS exp(const FPS &f, int deg = -1) { return f.exp(deg); }
friend constexpr FPS pow(const FPS &f, long long e, int deg = -1) { return f.pow(e, deg); }
};
// composition of FPS, calc g(f(x)), O(N (log N)^2)
template <class mint>
FPS<mint> composition(FPS<mint> g, FPS<mint> f, int deg = -1) {
auto rec = [&](auto &&rec, FPS<mint> Q, int n, int h, int k) -> FPS<mint> {
if (n == 0) {
FPS<mint> T{begin(Q), begin(Q) + k};
T.emplace_back(mint(1));
FPS<mint> u = g * T.rev().inv().rev();
FPS<mint> P(h * k);
for (int i = 0; i < (int)g.size(); i++)
P[k - i - 1] = u[i + k];
return P;
}
FPS<mint> nQ(h * k * 4), nR(h * k * 2);
for (int i = 0; i < k; i++) {
copy(begin(Q) + i * h, begin(Q) + i * h + n + 1, begin(nQ) + i * h * 2);
}
nQ[h * k * 2] += 1;
ntt_trans(nQ);
for (int i = 0; i < h * k * 4; i += 2)
swap(nQ[i], nQ[i + 1]);
for (int i = 0; i < h * k * 2; i++)
nR[i] = nQ[i * 2] * nQ[i * 2 + 1];
ntt_trans_inv(nR);
nR[0] -= 1;
Q.assign(h * k, 0);
for (int i = 0; i < k * 2; i++)
for (int j = 0; j <= n / 2; j++) {
Q[i * h / 2 + j] = nR[i * h + j];
}
auto P = rec(rec, Q, n / 2, h / 2, k * 2);
FPS<mint> nP(h * k * 4);
for (int i = 0; i < k * 2; i++)
for (int j = 0; j <= n / 2; j++) {
nP[i * h * 2 + j * 2 + n % 2] = P[i * h / 2 + j];
}
ntt_trans(nP);
for (int i = 1; i < h * k * 4; i <<= 1)
reverse(begin(nQ) + i, begin(nQ) + i * 2);
for (int i = 0; i < h * k * 4; i++)
nP[i] *= nQ[i];
ntt_trans_inv(nP);
P.assign(h * k, 0);
for (int i = 0; i < k; i++) {
copy(begin(nP) + i * h * 2, begin(nP) + i * h * 2 + n + 1, begin(P) + i * h);
}
return P;
};
if (deg == -1)
deg = max((int)f.size(), (int)g.size());
f.resize(deg), g.resize(deg);
int n = (int)f.size() - 1, h = 1, k = 1;
while (h < n + 1)
h *= 2;
FPS<mint> Q(h * k);
for (int i = 0; i <= n; i++)
Q[i] = -f[i];
FPS<mint> P = rec(rec, Q, n, h, k);
return P.pre(n + 1).rev();
}
#define rep(i, n) for (int i = 0; i < (int)(n); i++)
int main() {
cin.tie(nullptr);
ios_base::sync_with_stdio(false);
int n, m;
cin >> n >> m;
using mint = Fp<998244353>;
vector<mint> a(n), b(n), c(n);
rep(i, n) cin >> a[i];
rep(i, n) cin >> b[i];
rep(i, n) cin >> c[i];
rep(i, n) b[i] *= a[1].pow(m);
auto d = composition<mint>(c, b);
rep(i, n) cout << d[i] << " ";
cout << endl;
return 0;
}