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Composition of Formal Power Series (math/fps/composition-of-formal-power-series.hpp)
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- Last update: 2026-08-02 17:17:16+00:00
- Include:
#include "math/fps/composition-of-formal-power-series.hpp"
2 つの形式的冪級数の合成を計算する。
composition_of_formal_power_series
template <typename Mint>
FormalPowerSeriesFriendlyNTT<Mint> composition_of_formal_power_series(
const FormalPowerSeriesFriendlyNTT<Mint>& outer,
const FormalPowerSeriesFriendlyNTT<Mint>& inner, int deg = -1)
形式的冪級数 $f(x)=\mathrm{outer}(x)$ と $g(x)=\mathrm{inner}(x)$ に対して、
\[f(g(x)) \bmod x^{\mathrm{deg}}\]を返す。返り値の要素数は常に deg であり、outer と inner の不足する係数は $0$ とみなす。
deg を省略した場合は outer.size() 項を返す。
制約
- $g(0)=0$（
innerが空の場合も含む） deg >= 0-
Mintは NTT-friendly な素数 mod 上の ModInt -
deg >= 2のとき、2 * bit_ceil(deg)点の NTT を実行できる
計算量
deg を $N$ とすると、
- 時間計算量: $O(N\log^2 N)$
- 空間計算量: $O(N)$
Depends on
- Number Theoretic Transform Friendly ModInt (math/fft/number-theoretic-transform-friendly-mod-int.hpp)
- Formal Power Series Friendly NTT (NTTmod用形式的冪級数) (math/fps/formal-power-series-friendly-ntt.hpp)
Verified with
Code
#pragma once
#include "formal-power-series-friendly-ntt.hpp"
/**
* @brief Composition of Formal Power Series
*
* Returns outer(inner(x)) modulo x^deg. Missing coefficients are treated as
* zero, and the returned formal power series has exactly deg coefficients.
*
* This is the transposed power-projection algorithm. The two-dimensional
* transforms are kept in NTT representation between recursive levels.
*
* @note This implementation requires an NTT-friendly modulus.
* @complexity O(deg log^2 deg)
*/
template <typename Mint>
FormalPowerSeriesFriendlyNTT<Mint> composition_of_formal_power_series(
const FormalPowerSeriesFriendlyNTT<Mint>& outer,
const FormalPowerSeriesFriendlyNTT<Mint>& inner, int deg = -1) {
using Poly = FormalPowerSeriesFriendlyNTT<Mint>;
using NTT = NumberTheoreticTransformFriendlyModInt<Mint>;
if (deg == -1) deg = (int)outer.size();
assert(deg >= 0);
if (deg == 0) return {};
assert(inner.empty() || inner[0] == Mint(0));
if (deg == 1) {
return Poly{outer.empty() ? Mint(0) : outer[0]};
}
const int result_size = deg;
int size = 1;
while (size < result_size) size <<= 1;
NTT::init();
assert(__builtin_ctz(size) + 1 <= NTT::max_base);
vector<Mint> f(size), g(size);
copy_n(outer.begin(), min((int)outer.size(), result_size), f.begin());
copy_n(inner.begin(), min((int)inner.size(), result_size), g.begin());
// W is the table of powers of a primitive 2*size-th inverse root, stored in
// the bit-reversed order used by the radix-4 NTT. inverse_w stores W^{-1}.
vector<Mint> w(size), inverse_w(size);
vector<int> bit_reverse(size);
const int lg = __builtin_ctz(size);
for (int i = 1; i < size; i++) {
bit_reverse[i] = (bit_reverse[i >> 1] >> 1) | ((i & 1) << (lg - 1));
}
const Mint dw = NTT::iroots[lg + 1];
const Mint inverse_dw = NTT::roots[lg + 1];
Mint power = 1, inverse_power = 1;
for (int index : bit_reverse) {
w[index] = power;
inverse_w[index] = inverse_power;
power *= dw;
inverse_power *= inverse_dw;
}
auto rec = [&](auto& self, int n, int k,
vector<Mint>& denominator) -> vector<Mint> {
// n * k is invariant and is equal to size.
assert((int)denominator.size() == 4 * n * k);
if (n == 1) {
reverse(f.begin(), f.end());
NTT::transposed_intt(f);
const Mint coefficient = Mint(1) / Mint(k);
for (auto& value : f) value *= coefficient;
vector<Mint> result(4 * k);
for (int i = 0; i < k; i++) result[2 * i] = f[i];
return result;
}
auto double_y = [&](vector<Mint>& values, int left, int right,
bool transpose) -> void {
const Mint z = inverse_w[k / 2];
vector<Mint> buffer(k);
if (!transpose) {
for (int x = left; x < right; x++) {
for (int y = 0; y < k; y++) {
buffer[y] = values[2 * n * y + x];
}
NTT::intt(buffer);
Mint multiplier = 1;
for (int y = 1; y < k; y++) {
multiplier *= z;
buffer[y] *= multiplier;
}
NTT::ntt(buffer);
for (int y = 0; y < k; y++) {
values[2 * n * (k + y) + x] = buffer[y];
}
}
} else {
for (int x = left; x < right; x++) {
for (int y = 0; y < k; y++) {
buffer[y] = values[2 * n * (k + y) + x];
}
NTT::transposed_ntt(buffer);
Mint multiplier = 1;
for (int y = 1; y < k; y++) {
multiplier *= z;
buffer[y] *= multiplier;
}
NTT::transposed_intt(buffer);
for (int y = 0; y < k; y++) {
values[2 * n * y + x] += buffer[y];
}
}
}
};
auto transform_x = [&](vector<Mint>& values, int left, int right,
bool transpose) -> void {
vector<Mint> buffer(2 * n);
for (int y = left; y < right; y++) {
copy_n(values.begin() + (size_t)2 * n * y, 2 * n, buffer.begin());
if (transpose) {
NTT::transposed_ntt(buffer);
} else {
NTT::ntt(buffer);
}
copy(buffer.begin(), buffer.end(), values.begin() + (size_t)2 * n * y);
}
};
// Transform the denominator in both variables. The order is selected so
// that the smaller number of transforms is performed before doubling.
if (n <= k) {
double_y(denominator, 1, n, false);
transform_x(denominator, 0, 2 * k, false);
} else {
transform_x(denominator, 0, k, false);
double_y(denominator, 0, 2 * n, false);
}
for (int i = 0; i < 2 * n * k; i++) denominator[i] += Mint(1);
for (int i = 2 * n * k; i < 4 * n * k; i++) {
denominator[i] -= Mint(1);
}
vector<Mint> next_denominator(4 * n * k);
vector<Mint> reduced_row(n);
for (int y = 0; y < 2 * k; y++) {
const auto row = denominator.begin() + (size_t)2 * n * y;
for (int x = 0; x < n; x++) {
reduced_row[x] = row[2 * x] * row[2 * x + 1];
}
NTT::intt(reduced_row);
copy_n(reduced_row.begin(), n / 2,
next_denominator.begin() + (size_t)n * y);
}
for (int y = 0; y < 4 * k; y++) {
next_denominator[(size_t)n * y] = Mint(0);
}
auto result = self(self, n / 2, k * 2, next_denominator);
vector<Mint> buffer(n), reconstructed(2 * n);
for (int y = 2 * k; y-- > 0;) {
copy_n(result.begin() + (size_t)n * y, n / 2, buffer.begin());
fill(buffer.begin() + n / 2, buffer.end(), Mint(0));
NTT::transposed_intt(buffer);
const auto row = denominator.begin() + (size_t)2 * n * y;
for (int x = 0; x < n; x++) {
buffer[x] *= w[x];
reconstructed[2 * x] = row[2 * x + 1] * buffer[x];
reconstructed[2 * x + 1] = -row[2 * x] * buffer[x];
}
copy(reconstructed.begin(), reconstructed.end(),
result.begin() + (size_t)2 * n * y);
}
// Apply the transposes in reverse order.
if (n <= k) {
transform_x(result, 0, 2 * k, true);
double_y(result, 0, n, true);
} else {
double_y(result, 0, 2 * n, true);
transform_x(result, 0, k, true);
}
return result;
};
vector<Mint> denominator(4 * size);
for (int i = 0; i < size; i++) denominator[i] = -g[i];
auto result = rec(rec, size, 1, denominator);
result.resize(size);
reverse(result.begin(), result.end());
result.resize(result_size);
return Poly(result.begin(), result.end());
}
#line 2 "math/fps/composition-of-formal-power-series.hpp"
#line 2 "math/fps/formal-power-series-friendly-ntt.hpp"
#line 1 "math/fft/number-theoretic-transform-friendly-mod-int.hpp"
/**
* @brief Number Theoretic Transform Friendly ModInt
*/
template <typename Mint>
struct NumberTheoreticTransformFriendlyModInt {
static vector<Mint> roots, iroots, rate2, irate2, rate3, irate3;
static int max_base;
NumberTheoreticTransformFriendlyModInt() = default;
static void init() {
if (roots.empty()) {
const unsigned mod = Mint::mod();
assert(mod >= 3 && mod % 2 == 1);
auto tmp = mod - 1;
max_base = 0;
while (tmp % 2 == 0) tmp >>= 1, max_base++;
Mint root = 2;
while (root.pow((mod - 1) >> 1) == 1) {
root += 1;
}
assert(root.pow(mod - 1) == 1);
roots.resize(max_base + 1);
iroots.resize(max_base + 1);
rate2.resize(max_base + 1);
irate2.resize(max_base + 1);
rate3.resize(max_base + 1);
irate3.resize(max_base + 1);
roots[max_base] = root.pow((mod - 1) >> max_base);
iroots[max_base] = Mint(1) / roots[max_base];
for (int i = max_base - 1; i >= 0; i--) {
roots[i] = roots[i + 1] * roots[i + 1];
iroots[i] = iroots[i + 1] * iroots[i + 1];
}
{
Mint prod = 1, iprod = 1;
for (int i = 0; i <= max_base - 2; i++) {
rate2[i] = roots[i + 2] * prod;
irate2[i] = iroots[i + 2] * iprod;
prod *= iroots[i + 2];
iprod *= roots[i + 2];
}
}
{
Mint prod = 1, iprod = 1;
for (int i = 0; i <= max_base - 3; i++) {
rate3[i] = roots[i + 3] * prod;
irate3[i] = iroots[i + 3] * iprod;
prod *= iroots[i + 3];
iprod *= roots[i + 3];
}
}
}
}
static void ntt(vector<Mint>& a) {
init();
const int n = (int)a.size();
assert((n & (n - 1)) == 0);
int h = __builtin_ctz(n);
assert(h <= max_base);
int len = 0;
Mint imag = roots[2];
if (h & 1) {
int p = 1 << (h - 1);
for (int i = 0; i < p; i++) {
auto r = a[i + p];
a[i + p] = a[i] - r;
a[i] += r;
}
len++;
}
for (; len + 1 < h; len += 2) {
int p = 1 << (h - len - 2);
{ // s = 0
for (int i = 0; i < p; i++) {
auto a0 = a[i];
auto a1 = a[i + p];
auto a2 = a[i + 2 * p];
auto a3 = a[i + 3 * p];
auto a1na3imag = (a1 - a3) * imag;
auto a0a2 = a0 + a2;
auto a1a3 = a1 + a3;
auto a0na2 = a0 - a2;
a[i] = a0a2 + a1a3;
a[i + 1 * p] = a0a2 - a1a3;
a[i + 2 * p] = a0na2 + a1na3imag;
a[i + 3 * p] = a0na2 - a1na3imag;
}
}
Mint rot = rate3[0];
for (int s = 1; s < (1 << len); s++) {
int offset = s << (h - len);
Mint rot2 = rot * rot;
Mint rot3 = rot2 * rot;
for (int i = 0; i < p; i++) {
auto a0 = a[i + offset];
auto a1 = a[i + offset + p] * rot;
auto a2 = a[i + offset + 2 * p] * rot2;
auto a3 = a[i + offset + 3 * p] * rot3;
auto a1na3imag = (a1 - a3) * imag;
auto a0a2 = a0 + a2;
auto a1a3 = a1 + a3;
auto a0na2 = a0 - a2;
a[i + offset] = a0a2 + a1a3;
a[i + offset + 1 * p] = a0a2 - a1a3;
a[i + offset + 2 * p] = a0na2 + a1na3imag;
a[i + offset + 3 * p] = a0na2 - a1na3imag;
}
rot *= rate3[__builtin_ctz(~s)];
}
}
}
static void intt(vector<Mint>& a, bool f = true) {
init();
const int n = (int)a.size();
assert((n & (n - 1)) == 0);
int h = __builtin_ctz(n);
assert(h <= max_base);
int len = h;
Mint iimag = iroots[2];
for (; len > 1; len -= 2) {
int p = 1 << (h - len);
{ // s = 0
for (int i = 0; i < p; i++) {
auto a0 = a[i];
auto a1 = a[i + 1 * p];
auto a2 = a[i + 2 * p];
auto a3 = a[i + 3 * p];
auto a2na3iimag = (a2 - a3) * iimag;
auto a0na1 = a0 - a1;
auto a0a1 = a0 + a1;
auto a2a3 = a2 + a3;
a[i] = a0a1 + a2a3;
a[i + 1 * p] = (a0na1 + a2na3iimag);
a[i + 2 * p] = (a0a1 - a2a3);
a[i + 3 * p] = (a0na1 - a2na3iimag);
}
}
Mint irot = irate3[0];
for (int s = 1; s < (1 << (len - 2)); s++) {
int offset = s << (h - len + 2);
Mint irot2 = irot * irot;
Mint irot3 = irot2 * irot;
for (int i = 0; i < p; i++) {
auto a0 = a[i + offset];
auto a1 = a[i + offset + 1 * p];
auto a2 = a[i + offset + 2 * p];
auto a3 = a[i + offset + 3 * p];
auto a2na3iimag = (a2 - a3) * iimag;
auto a0na1 = a0 - a1;
auto a0a1 = a0 + a1;
auto a2a3 = a2 + a3;
a[i + offset] = a0a1 + a2a3;
a[i + offset + 1 * p] = (a0na1 + a2na3iimag) * irot;
a[i + offset + 2 * p] = (a0a1 - a2a3) * irot2;
a[i + offset + 3 * p] = (a0na1 - a2na3iimag) * irot3;
}
irot *= irate3[__builtin_ctz(~s)];
}
}
if (len >= 1) {
int p = 1 << (h - 1);
for (int i = 0; i < p; i++) {
auto ajp = a[i] - a[i + p];
a[i] += a[i + p];
a[i + p] = ajp;
}
}
if (f) {
Mint inv_sz = Mint(1) / n;
for (int i = 0; i < n; i++) a[i] *= inv_sz;
}
}
/**
* @brief Transpose of ntt()
*/
static void transposed_ntt(vector<Mint>& a) {
init();
const int n = (int)a.size();
assert((n & (n - 1)) == 0);
const int h = __builtin_ctz(n);
assert(h <= max_base);
int len = h;
const Mint imag = roots[2];
while (len > 0) {
if (len == 1) {
const int p = 1 << (h - len);
Mint rot = 1;
for (int s = 0; s < (1 << (len - 1)); s++) {
const int offset = s << (h - len + 1);
for (int i = 0; i < p; i++) {
const auto lhs = a[i + offset];
const auto rhs = a[i + offset + p];
a[i + offset] = lhs + rhs;
a[i + offset + p] = (lhs - rhs) * rot;
}
rot *= rate2[__builtin_ctz(~s)];
}
len--;
} else {
const int p = 1 << (h - len);
Mint rot = 1;
for (int s = 0; s < (1 << (len - 2)); s++) {
const int offset = s << (h - len + 2);
const Mint rot2 = rot * rot;
const Mint rot3 = rot2 * rot;
for (int i = 0; i < p; i++) {
const auto a0 = a[i + offset];
const auto a1 = a[i + offset + p];
const auto a2 = a[i + offset + 2 * p];
const auto a3 = a[i + offset + 3 * p];
const auto x = (a2 - a3) * imag;
a[i + offset] = a0 + a1 + a2 + a3;
a[i + offset + p] = (a0 - a1 + x) * rot;
a[i + offset + 2 * p] = (a0 + a1 - a2 - a3) * rot2;
a[i + offset + 3 * p] = (a0 - a1 - x) * rot3;
}
rot *= rate3[__builtin_ctz(~s)];
}
len -= 2;
}
}
}
/**
* @brief Transpose of intt()
*/
static void transposed_intt(vector<Mint>& a, bool f = true) {
init();
const int n = (int)a.size();
assert((n & (n - 1)) == 0);
const int h = __builtin_ctz(n);
assert(h <= max_base);
if (f) {
const Mint inv_sz = Mint(1) / n;
for (auto& value : a) value *= inv_sz;
}
int len = 0;
const Mint iimag = iroots[2];
while (len < h) {
if (len == h - 1) {
const int p = 1 << (h - len - 1);
Mint irot = 1;
for (int s = 0; s < (1 << len); s++) {
const int offset = s << (h - len);
for (int i = 0; i < p; i++) {
const auto lhs = a[i + offset];
const auto rhs = a[i + offset + p] * irot;
a[i + offset] = lhs + rhs;
a[i + offset + p] = lhs - rhs;
}
irot *= irate2[__builtin_ctz(~s)];
}
len++;
} else {
const int p = 1 << (h - len - 2);
Mint irot = 1;
for (int s = 0; s < (1 << len); s++) {
const Mint irot2 = irot * irot;
const Mint irot3 = irot2 * irot;
const int offset = s << (h - len);
for (int i = 0; i < p; i++) {
const auto a0 = a[i + offset];
const auto a1 = a[i + offset + p] * irot;
const auto a2 = a[i + offset + 2 * p] * irot2;
const auto a3 = a[i + offset + 3 * p] * irot3;
const auto x = (a1 - a3) * iimag;
a[i + offset] = a0 + a2 + a1 + a3;
a[i + offset + p] = a0 + a2 - a1 - a3;
a[i + offset + 2 * p] = a0 - a2 + x;
a[i + offset + 3 * p] = a0 - a2 - x;
}
irot *= irate3[__builtin_ctz(~s)];
}
len += 2;
}
}
}
static vector<Mint> multiply(vector<Mint> a, vector<Mint> b) {
int need = a.size() + b.size() - 1;
int nbase = 1;
while ((1 << nbase) < need) nbase++;
int sz = 1 << nbase;
a.resize(sz, 0);
b.resize(sz, 0);
ntt(a);
ntt(b);
Mint inv_sz = Mint(1) / sz;
for (int i = 0; i < sz; i++) a[i] *= b[i] * inv_sz;
intt(a, false);
a.resize(need);
return a;
}
};
template <typename Mint>
vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::roots =
vector<Mint>();
template <typename Mint>
vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::iroots =
vector<Mint>();
template <typename Mint>
vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::rate2 =
vector<Mint>();
template <typename Mint>
vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::irate2 =
vector<Mint>();
template <typename Mint>
vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::rate3 =
vector<Mint>();
template <typename Mint>
vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::irate3 =
vector<Mint>();
template <typename Mint>
int NumberTheoreticTransformFriendlyModInt<Mint>::max_base = 0;
#line 4 "math/fps/formal-power-series-friendly-ntt.hpp"
template <typename T>
struct FormalPowerSeriesFriendlyNTT : vector<T> {
using vector<T>::vector;
using P = FormalPowerSeriesFriendlyNTT;
using NTT = NumberTheoreticTransformFriendlyModInt<T>;
P pre(int deg) const {
return P(begin(*this), begin(*this) + min((int)this->size(), deg));
}
P rev(int deg = -1) const {
P ret(*this);
if (deg != -1) ret.resize(deg, T(0));
reverse(begin(ret), end(ret));
return ret;
}
void shrink() {
while (this->size() && this->back() == T(0)) this->pop_back();
}
P operator+(const P& r) const { return P(*this) += r; }
P operator+(const T& v) const { return P(*this) += v; }
P operator-(const P& r) const { return P(*this) -= r; }
P operator-(const T& v) const { return P(*this) -= v; }
P operator*(const P& r) const { return P(*this) *= r; }
P operator*(const T& v) const { return P(*this) *= v; }
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) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i];
return *this;
}
P& operator-=(const P& r) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
return *this;
}
// https://judge.yosupo.jp/problem/convolution_mod
P& operator*=(const P& r) {
if (this->empty() || r.empty()) {
this->clear();
return *this;
}
auto ret = NTT::multiply(*this, r);
return *this = {begin(ret), end(ret)};
}
P& operator/=(const P& r) {
if (this->size() < r.size()) {
this->clear();
return *this;
}
int n = this->size() - r.size() + 1;
return *this = (rev().pre(n) * r.rev().inv(n)).pre(n).rev(n);
}
P& operator%=(const P& r) {
*this -= *this / r * r;
shrink();
return *this;
}
// https://judge.yosupo.jp/problem/division_of_polynomials
pair<P, P> div_mod(const P& r) {
P q = *this / r;
P x = *this - q * r;
x.shrink();
return make_pair(q, x);
}
P operator-() const {
P ret(this->size());
for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
return ret;
}
P& operator+=(const T& r) {
if (this->empty()) this->resize(1);
(*this)[0] += r;
return *this;
}
P& operator-=(const T& r) {
if (this->empty()) this->resize(1);
(*this)[0] -= r;
return *this;
}
P& operator*=(const T& v) {
for (int i = 0; i < (int)this->size(); i++) (*this)[i] *= v;
return *this;
}
P dot(P r) const {
P ret(min(this->size(), r.size()));
for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
return ret;
}
P operator>>(int sz) const {
if ((int)this->size() <= sz) return {};
P ret(*this);
ret.erase(ret.begin(), ret.begin() + sz);
return ret;
}
P operator<<(int sz) const {
P ret(*this);
ret.insert(ret.begin(), sz, T(0));
return ret;
}
T operator()(T x) const {
T r = 0, w = 1;
for (auto& v : *this) {
r += w * v;
w *= x;
}
return r;
}
P diff() const {
const int n = (int)this->size();
P ret(max(0, n - 1));
for (int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * T(i);
return ret;
}
P integral() const {
const int n = (int)this->size();
P ret(n + 1);
ret[0] = T(0);
for (int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / T(i + 1);
return ret;
}
// https://judge.yosupo.jp/problem/inv_of_formal_power_series
// F(0) must not be 0
P inv(int deg = -1) const {
assert(((*this)[0]) != T(0));
const int n = (int)this->size();
if (deg == -1) deg = n;
P res(deg);
res[0] = {T(1) / (*this)[0]};
for (int d = 1; d < deg; d <<= 1) {
P f(2 * d), g(2 * d);
for (int j = 0; j < min(n, 2 * d); j++) f[j] = (*this)[j];
for (int j = 0; j < d; j++) g[j] = res[j];
NTT::ntt(f);
NTT::ntt(g);
f = f.dot(g);
NTT::intt(f);
for (int j = 0; j < d; j++) f[j] = 0;
NTT::ntt(f);
for (int j = 0; j < 2 * d; j++) f[j] *= g[j];
NTT::intt(f);
for (int j = d; j < min(2 * d, deg); j++) res[j] = -f[j];
}
return res;
}
// https://judge.yosupo.jp/problem/log_of_formal_power_series
// F(0) must be 1
P log(int deg = -1) const {
assert((*this)[0] == T(1));
const int n = (int)this->size();
if (deg == -1) deg = n;
return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
}
// https://judge.yosupo.jp/problem/sqrt_of_formal_power_series
P sqrt(
int deg = -1,
const function<T(T)>& get_sqrt = [](T) { return T(1); }) const {
const int n = (int)this->size();
if (deg == -1) deg = n;
if ((*this)[0] == T(0)) {
for (int i = 1; i < n; i++) {
if ((*this)[i] != T(0)) {
if (i & 1) return {};
if (deg - i / 2 <= 0) break;
auto ret = (*this >> i).sqrt(deg - i / 2, get_sqrt);
if (ret.empty()) return {};
ret = ret << (i / 2);
if ((int)ret.size() < deg) ret.resize(deg, T(0));
return ret;
}
}
return P(deg, 0);
}
auto sqr = T(get_sqrt((*this)[0]));
if (sqr * sqr != (*this)[0]) return {};
P ret{sqr};
T inv2 = T(1) / T(2);
for (int i = 1; i < deg; i <<= 1) {
ret = (ret + pre(i << 1) * ret.inv(i << 1)) * inv2;
}
return ret.pre(deg);
}
P sqrt(const function<T(T)>& get_sqrt, int deg = -1) const {
return sqrt(deg, get_sqrt);
}
// https://judge.yosupo.jp/problem/exp_of_formal_power_series
// F(0) must be 0
P exp(int deg = -1) const {
if (deg == -1) deg = this->size();
assert((*this)[0] == T(0));
P inv;
inv.reserve(deg + 1);
inv.push_back(T(0));
inv.push_back(T(1));
auto inplace_integral = [&](P& F) -> void {
const int n = (int)F.size();
auto mod = T::mod();
while ((int)inv.size() <= n) {
int i = inv.size();
inv.push_back((-inv[mod % i]) * (mod / i));
}
F.insert(begin(F), T(0));
for (int i = 1; i <= n; i++) F[i] *= inv[i];
};
auto inplace_diff = [](P& F) -> void {
if (F.empty()) return;
F.erase(begin(F));
T coeff = 1, one = 1;
for (int i = 0; i < (int)F.size(); i++) {
F[i] *= coeff;
coeff += one;
}
};
P b{1, 1 < (int)this->size() ? (*this)[1] : 0}, c{1}, z1, z2{1, 1};
for (int m = 2; m < deg; m *= 2) {
auto y = b;
y.resize(2 * m);
NTT::ntt(y);
z1 = z2;
P z(m);
for (int i = 0; i < m; ++i) z[i] = y[i] * z1[i];
NTT::intt(z);
fill(begin(z), begin(z) + m / 2, T(0));
NTT::ntt(z);
for (int i = 0; i < m; ++i) z[i] *= -z1[i];
NTT::intt(z);
c.insert(end(c), begin(z) + m / 2, end(z));
z2 = c;
z2.resize(2 * m);
NTT::ntt(z2);
P x(begin(*this), begin(*this) + min<int>(this->size(), m));
inplace_diff(x);
x.push_back(T(0));
NTT::ntt(x);
for (int i = 0; i < m; ++i) x[i] *= y[i];
NTT::intt(x);
x -= b.diff();
x.resize(2 * m);
for (int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = T(0);
NTT::ntt(x);
for (int i = 0; i < 2 * m; ++i) x[i] *= z2[i];
NTT::intt(x);
x.pop_back();
inplace_integral(x);
for (int i = m; i < min<int>(this->size(), 2 * m); ++i)
x[i] += (*this)[i];
fill(begin(x), begin(x) + m, T(0));
NTT::ntt(x);
for (int i = 0; i < 2 * m; ++i) x[i] *= y[i];
NTT::intt(x);
b.insert(end(b), begin(x) + m, end(x));
}
return P{begin(b), begin(b) + deg};
}
// https://judge.yosupo.jp/problem/pow_of_formal_power_series
P pow(int64_t k, int deg = -1) const {
const int n = (int)this->size();
if (deg == -1) deg = n;
if (k == 0) {
P ret(deg, T(0));
ret[0] = T(1);
return ret;
}
for (int i = 0; i < n; i++) {
if (i * k > deg) return P(deg, T(0));
if ((*this)[i] != T(0)) {
T rev = T(1) / (*this)[i];
P ret = (((*this * rev) >> i).log() * k).exp() * ((*this)[i].pow(k));
ret = (ret << (i * k)).pre(deg);
if ((int)ret.size() < deg) ret.resize(deg, T(0));
return ret;
}
}
return *this;
}
P mod_pow(int64_t k, P g) const {
P modinv = g.rev().inv();
auto get_div = [&](P base) {
if (base.size() < g.size()) {
base.clear();
return base;
}
int n = base.size() - g.size() + 1;
return (base.rev().pre(n) * modinv.pre(n)).pre(n).rev(n);
};
P x(*this), ret{1};
while (k > 0) {
if (k & 1) {
ret *= x;
ret -= get_div(ret) * g;
ret.shrink();
}
x *= x;
x -= get_div(x) * g;
x.shrink();
k >>= 1;
}
return ret;
}
// https://judge.yosupo.jp/problem/polynomial_taylor_shift
P taylor_shift(T c) const {
int n = (int)this->size();
vector<T> fact(n), rfact(n);
fact[0] = rfact[0] = T(1);
for (int i = 1; i < n; i++) fact[i] = fact[i - 1] * T(i);
rfact[n - 1] = T(1) / fact[n - 1];
for (int i = n - 1; i > 1; i--) rfact[i - 1] = rfact[i] * T(i);
P p(*this);
for (int i = 0; i < n; i++) p[i] *= fact[i];
p = p.rev();
P bs(n, T(1));
for (int i = 1; i < n; i++) bs[i] = bs[i - 1] * c * rfact[i] * fact[i - 1];
p = (p * bs).pre(n);
p = p.rev();
for (int i = 0; i < n; i++) p[i] *= rfact[i];
return p;
}
};
template <typename Mint>
using FPS = FormalPowerSeriesFriendlyNTT<Mint>;
#line 4 "math/fps/composition-of-formal-power-series.hpp"
/**
* @brief Composition of Formal Power Series
*
* Returns outer(inner(x)) modulo x^deg. Missing coefficients are treated as
* zero, and the returned formal power series has exactly deg coefficients.
*
* This is the transposed power-projection algorithm. The two-dimensional
* transforms are kept in NTT representation between recursive levels.
*
* @note This implementation requires an NTT-friendly modulus.
* @complexity O(deg log^2 deg)
*/
template <typename Mint>
FormalPowerSeriesFriendlyNTT<Mint> composition_of_formal_power_series(
const FormalPowerSeriesFriendlyNTT<Mint>& outer,
const FormalPowerSeriesFriendlyNTT<Mint>& inner, int deg = -1) {
using Poly = FormalPowerSeriesFriendlyNTT<Mint>;
using NTT = NumberTheoreticTransformFriendlyModInt<Mint>;
if (deg == -1) deg = (int)outer.size();
assert(deg >= 0);
if (deg == 0) return {};
assert(inner.empty() || inner[0] == Mint(0));
if (deg == 1) {
return Poly{outer.empty() ? Mint(0) : outer[0]};
}
const int result_size = deg;
int size = 1;
while (size < result_size) size <<= 1;
NTT::init();
assert(__builtin_ctz(size) + 1 <= NTT::max_base);
vector<Mint> f(size), g(size);
copy_n(outer.begin(), min((int)outer.size(), result_size), f.begin());
copy_n(inner.begin(), min((int)inner.size(), result_size), g.begin());
// W is the table of powers of a primitive 2*size-th inverse root, stored in
// the bit-reversed order used by the radix-4 NTT. inverse_w stores W^{-1}.
vector<Mint> w(size), inverse_w(size);
vector<int> bit_reverse(size);
const int lg = __builtin_ctz(size);
for (int i = 1; i < size; i++) {
bit_reverse[i] = (bit_reverse[i >> 1] >> 1) | ((i & 1) << (lg - 1));
}
const Mint dw = NTT::iroots[lg + 1];
const Mint inverse_dw = NTT::roots[lg + 1];
Mint power = 1, inverse_power = 1;
for (int index : bit_reverse) {
w[index] = power;
inverse_w[index] = inverse_power;
power *= dw;
inverse_power *= inverse_dw;
}
auto rec = [&](auto& self, int n, int k,
vector<Mint>& denominator) -> vector<Mint> {
// n * k is invariant and is equal to size.
assert((int)denominator.size() == 4 * n * k);
if (n == 1) {
reverse(f.begin(), f.end());
NTT::transposed_intt(f);
const Mint coefficient = Mint(1) / Mint(k);
for (auto& value : f) value *= coefficient;
vector<Mint> result(4 * k);
for (int i = 0; i < k; i++) result[2 * i] = f[i];
return result;
}
auto double_y = [&](vector<Mint>& values, int left, int right,
bool transpose) -> void {
const Mint z = inverse_w[k / 2];
vector<Mint> buffer(k);
if (!transpose) {
for (int x = left; x < right; x++) {
for (int y = 0; y < k; y++) {
buffer[y] = values[2 * n * y + x];
}
NTT::intt(buffer);
Mint multiplier = 1;
for (int y = 1; y < k; y++) {
multiplier *= z;
buffer[y] *= multiplier;
}
NTT::ntt(buffer);
for (int y = 0; y < k; y++) {
values[2 * n * (k + y) + x] = buffer[y];
}
}
} else {
for (int x = left; x < right; x++) {
for (int y = 0; y < k; y++) {
buffer[y] = values[2 * n * (k + y) + x];
}
NTT::transposed_ntt(buffer);
Mint multiplier = 1;
for (int y = 1; y < k; y++) {
multiplier *= z;
buffer[y] *= multiplier;
}
NTT::transposed_intt(buffer);
for (int y = 0; y < k; y++) {
values[2 * n * y + x] += buffer[y];
}
}
}
};
auto transform_x = [&](vector<Mint>& values, int left, int right,
bool transpose) -> void {
vector<Mint> buffer(2 * n);
for (int y = left; y < right; y++) {
copy_n(values.begin() + (size_t)2 * n * y, 2 * n, buffer.begin());
if (transpose) {
NTT::transposed_ntt(buffer);
} else {
NTT::ntt(buffer);
}
copy(buffer.begin(), buffer.end(), values.begin() + (size_t)2 * n * y);
}
};
// Transform the denominator in both variables. The order is selected so
// that the smaller number of transforms is performed before doubling.
if (n <= k) {
double_y(denominator, 1, n, false);
transform_x(denominator, 0, 2 * k, false);
} else {
transform_x(denominator, 0, k, false);
double_y(denominator, 0, 2 * n, false);
}
for (int i = 0; i < 2 * n * k; i++) denominator[i] += Mint(1);
for (int i = 2 * n * k; i < 4 * n * k; i++) {
denominator[i] -= Mint(1);
}
vector<Mint> next_denominator(4 * n * k);
vector<Mint> reduced_row(n);
for (int y = 0; y < 2 * k; y++) {
const auto row = denominator.begin() + (size_t)2 * n * y;
for (int x = 0; x < n; x++) {
reduced_row[x] = row[2 * x] * row[2 * x + 1];
}
NTT::intt(reduced_row);
copy_n(reduced_row.begin(), n / 2,
next_denominator.begin() + (size_t)n * y);
}
for (int y = 0; y < 4 * k; y++) {
next_denominator[(size_t)n * y] = Mint(0);
}
auto result = self(self, n / 2, k * 2, next_denominator);
vector<Mint> buffer(n), reconstructed(2 * n);
for (int y = 2 * k; y-- > 0;) {
copy_n(result.begin() + (size_t)n * y, n / 2, buffer.begin());
fill(buffer.begin() + n / 2, buffer.end(), Mint(0));
NTT::transposed_intt(buffer);
const auto row = denominator.begin() + (size_t)2 * n * y;
for (int x = 0; x < n; x++) {
buffer[x] *= w[x];
reconstructed[2 * x] = row[2 * x + 1] * buffer[x];
reconstructed[2 * x + 1] = -row[2 * x] * buffer[x];
}
copy(reconstructed.begin(), reconstructed.end(),
result.begin() + (size_t)2 * n * y);
}
// Apply the transposes in reverse order.
if (n <= k) {
transform_x(result, 0, 2 * k, true);
double_y(result, 0, n, true);
} else {
double_y(result, 0, 2 * n, true);
transform_x(result, 0, k, true);
}
return result;
};
vector<Mint> denominator(4 * size);
for (int i = 0; i < size; i++) denominator[i] = -g[i];
auto result = rec(rec, size, 1, denominator);
result.resize(size);
reverse(result.begin(), result.end());
result.resize(result_size);
return Poly(result.begin(), result.end());
}