library
This documentation is automatically generated by online-judge-tools/verification-helper
MultipointEvaluation(多点評価)
(math/poly/MultipointEvaluation.hpp)
- View this file on GitHub
- Last update: 2024-12-16 12:15:38+09:00
- Include:
#include "math/poly/MultipointEvaluation.hpp"
概要
多点評価をする。
-
vector<T> multipoint_evaluation(FormalPowerSeries<T> f, vector<T> p): 各 $n = \deg(f), m = \lvert p \rvert$ として、 $0 \leq i < m$ について $f(p_i)$ を計算する。 $\Theta(n \log n + m \log^2 m)$ 。
特に評価する点が等比数列 $p_i = ar^i$ になっているときに Chirp-Z Transform を用いるとより高速に計算できる。 $t_i = \frac{i(i-1)}{2}$ とおくと $ki = t_{k+i}-t_{k}-t_{i}$ であることより、 $b_i = [x^i] f$ とすると、
$f(ar^i) = \sum_{k=0}^{n-1} b_k(ar^i)^k = \sum_{k=0}^{n-1} b_ka^kr^{ki} = r^{-t_i} \sum_{k=0}^{n-1} b_ka^kr^{-t_k}r^{t_{k+i}}$
より畳み込める。長さ $n, n+m-1$ の列を畳み込んだ $2n+m-2$ 個の値のうち、 $[n-1, n+m-1)$ のみ用いるので $n+m-1$ の巡回畳み込みで良い。
-
vector<T> multipoint_evaluation_geometric(FormalPowerSeries<T> f, T a, T r, ll m): 各 $n = \deg(f)$ として、 $0 \leq i < m$ について $f(ar^i)$ を計算する。 $\Theta((n+m)\log(n+m))$ 。
Depends on
- Combinatorics
(math/Combinatorics.hpp)
- ModInt
(math/ModInt.hpp)
- MontgomeryModInt(モンゴメリ乗算)
(math/MontgomeryModInt.hpp)
- SqrtMod(平方剰余)
(math/SqrtMod.hpp)
- Convolution(畳み込み)
(math/convolution/Convolution.hpp)
- FormalPowerSeries(形式的冪級数)
(math/poly/FormalPowerSeries.hpp)
- other/template.hpp
- template/alias.hpp
- template/bitop.hpp
- template/func.hpp
- template/in.hpp
- template/macros.hpp
- template/out.hpp
- template/type_traits.hpp
- template/util.hpp
Required by
- Factorial(階乗)
(math/Factorial.hpp)
- PolynomialInterpolation(多項式補間)
(math/poly/PolynomialInterpolation.hpp)
Verified with
- test/yosupo/enumerative_combinatorics/factorial.test.cpp
- test/yosupo/polynomial/multipoint_evaluation.test.cpp
- test/yosupo/polynomial/multipoint_evaluation_on_geometric_sequence.test.cpp
- test/yosupo/polynomial/polynomial_interpolation.test.cpp
- test/yosupo/polynomial/polynomial_interpolation_on_geometric_sequence.test.cpp
Code
#pragma once
#include "../../other/template.hpp"
#include "FormalPowerSeries.hpp"
namespace internal {
template<class T> class ProductTree {
private:
int n;
std::vector<FormalPowerSeries<T>> dat;
public:
ProductTree(const std::vector<T>& xs) {
n = xs.size();
dat.resize(n << 1);
rep (i, n) dat[i + n] = FormalPowerSeries<T>{-xs[i], 1};
rrep (i, n, 1) dat[i] = dat[i << 1] * dat[i << 1 | 1];
}
const FormalPowerSeries<T>& operator[](int k) const& { return dat[k]; }
FormalPowerSeries<T> operator[](int k) && { return std::move(dat[k]); }
};
template<class T>
std::vector<T> multipoint_evaluation(const FormalPowerSeries<T>& a,
const std::vector<T>& b,
const ProductTree<T>& c) {
int m = b.size();
std::vector<FormalPowerSeries<T>> d(m << 1);
d[1] = a % c[1];
rep (i, 2, m << 1) d[i] = d[i >> 1] % c[i];
std::vector<T> e(m);
rep (i, m) e[i] = d[m + i].empty() ? T{0} : d[m + i][0];
return e;
}
} // namespace internal
template<class T>
std::vector<T> multipoint_evaluation(const FormalPowerSeries<T>& a,
const std::vector<T>& b) {
if (a.empty() || b.empty()) return std::vector<T>(b.size(), T{0});
if (a.size() <= 32 || b.size() <= 32) {
std::vector<T> res(b.size());
rep (i, b.size()) res[i] = a.eval(b[i]);
return res;
}
return internal::multipoint_evaluation(a, b, internal::ProductTree<T>(b));
}
template<class T>
std::vector<T> multipoint_evaluation_geometric(const FormalPowerSeries<T>& f,
T a, T r, int m) {
if (f.empty() || m == 0) return std::vector<T>(m, T{0});
if (a == 0 || r == 1) return std::vector<T>(m, f.eval(a));
if (f.size() <= 32 || m <= 32) {
std::vector<T> res(m);
rep (i, m) {
res[i] = f.eval(a);
a *= r;
}
return res;
}
if (r == 0) {
std::vector<T> res(m, f.eval(0));
res[0] = f.eval(a);
return res;
}
int n = f.size();
int l = 1 << bitop::ceil_log2(n + m - 1);
std::vector<T> p(l), q(l);
T ir = T{1} / r, t = 1, t2 = 1;
rep (i, n) {
p[n - i - 1] = f[i] * t;
t *= a * t2;
t2 *= ir;
}
t = t2 = 1;
rep (i, n + m - 1) {
q[i] = t;
t *= t2;
t2 *= r;
}
number_theoretic_transform(p);
number_theoretic_transform(q);
rep (i, l) p[i] *= q[i];
inverse_number_theoretic_transform(p);
std::vector<T> ans(p.begin() + (n - 1), p.begin() + (n + m - 1));
t = t2 = 1;
rep (i, m) {
ans[i] *= t;
t *= t2;
t2 *= ir;
}
return ans;
}
/**
* @brief MultipointEvaluation(多点評価)
* @docs docs/math/poly/MultipointEvaluation.md
*/#line 2 "math/poly/MultipointEvaluation.hpp"
#line 2 "other/template.hpp"
#include <bits/stdc++.h>
#line 2 "template/macros.hpp"
#line 4 "template/macros.hpp"
#ifndef __COUNTER__
#define __COUNTER__ __LINE__
#endif
#define OVERLOAD5(a, b, c, d, e, ...) e
#define REP1_0(b, c) REP1_1(b, c)
#define REP1_1(b, c) \
for (ll REP_COUNTER_##c = 0; REP_COUNTER_##c < (ll)(b); ++REP_COUNTER_##c)
#define REP1(b) REP1_0(b, __COUNTER__)
#define REP2(i, b) for (ll i = 0; i < (ll)(b); ++i)
#define REP3(i, a, b) for (ll i = (ll)(a); i < (ll)(b); ++i)
#define REP4(i, a, b, c) for (ll i = (ll)(a); i < (ll)(b); i += (ll)(c))
#define rep(...) OVERLOAD5(__VA_ARGS__, REP4, REP3, REP2, REP1)(__VA_ARGS__)
#define RREP2(i, a) for (ll i = (ll)(a)-1; i >= 0; --i)
#define RREP3(i, a, b) for (ll i = (ll)(a)-1; i >= (ll)(b); --i)
#define RREP4(i, a, b, c) for (ll i = (ll)(a)-1; i >= (ll)(b); i -= (ll)(c))
#define rrep(...) OVERLOAD5(__VA_ARGS__, RREP4, RREP3, RREP2)(__VA_ARGS__)
#define REPS2(i, b) for (ll i = 1; i <= (ll)(b); ++i)
#define REPS3(i, a, b) for (ll i = (ll)(a) + 1; i <= (ll)(b); ++i)
#define REPS4(i, a, b, c) for (ll i = (ll)(a) + 1; i <= (ll)(b); i += (ll)(c))
#define reps(...) OVERLOAD5(__VA_ARGS__, REPS4, REPS3, REPS2)(__VA_ARGS__)
#define RREPS2(i, a) for (ll i = (ll)(a); i > 0; --i)
#define RREPS3(i, a, b) for (ll i = (ll)(a); i > (ll)(b); --i)
#define RREPS4(i, a, b, c) for (ll i = (ll)(a); i > (ll)(b); i -= (ll)(c))
#define rreps(...) OVERLOAD5(__VA_ARGS__, RREPS4, RREPS3, RREPS2)(__VA_ARGS__)
#define each_for(...) for (auto&& __VA_ARGS__)
#define each_const(...) for (const auto& __VA_ARGS__)
#define all(v) std::begin(v), std::end(v)
#define rall(v) std::rbegin(v), std::rend(v)
#if __cpp_if_constexpr >= 201606L
#define IF_CONSTEXPR constexpr
#else
#define IF_CONSTEXPR
#endif
#define IO_BUFFER_SIZE (1 << 17)
#line 2 "template/alias.hpp"
#line 4 "template/alias.hpp"
using ll = long long;
using uint = unsigned int;
using ull = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
using ld = long double;
using PLL = std::pair<ll, ll>;
template<class T>
using prique = std::priority_queue<T, std::vector<T>, std::greater<T>>;
template<class T> struct infinity {
static constexpr T value = std::numeric_limits<T>::max() / 2;
static constexpr T mvalue = std::numeric_limits<T>::lowest() / 2;
static constexpr T max = std::numeric_limits<T>::max();
static constexpr T min = std::numeric_limits<T>::lowest();
};
#if __cplusplus <= 201402L
template<class T> constexpr T infinity<T>::value;
template<class T> constexpr T infinity<T>::mvalue;
template<class T> constexpr T infinity<T>::max;
template<class T> constexpr T infinity<T>::min;
#endif
#if __cpp_variable_templates >= 201304L
template<class T> constexpr T INF = infinity<T>::value;
#endif
constexpr ll inf = infinity<ll>::value;
constexpr ld EPS = 1e-8;
constexpr ld PI = 3.1415926535897932384626;
#line 2 "template/type_traits.hpp"
#line 5 "template/type_traits.hpp"
template<class T, class... Args> struct function_traits_impl {
using result_type = T;
template<std::size_t idx>
using argument_type =
typename std::tuple_element<idx, std::tuple<Args...>>::type;
using argument_tuple = std::tuple<Args...>;
static constexpr std::size_t arg_size() { return sizeof...(Args); }
};
template<class> struct function_traits_helper;
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...)> {
using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...)&> {
using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) const> {
using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) const&> {
using type = function_traits_impl<Res, Args...>;
};
#if __cpp_noexcept_function_type >= 201510L
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) noexcept> {
using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...)& noexcept> {
using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) const noexcept> {
using type = function_traits_impl<Res, Args...>;
};
template<class Res, class Tp, class... Args>
struct function_traits_helper<Res (Tp::*)(Args...) const& noexcept> {
using type = function_traits_impl<Res, Args...>;
};
#endif
template<class F>
using function_traits = typename function_traits_helper<
decltype(&std::remove_reference<F>::type::operator())>::type;
template<class F>
using function_result_type = typename function_traits<F>::result_type;
template<class F, std::size_t idx>
using function_argument_type =
typename function_traits<F>::template argument_type<idx>;
template<class F>
using function_argument_tuple = typename function_traits<F>::argument_tuple;
template<class T>
using is_signed_int =
std::integral_constant<bool, (std::is_integral<T>::value &&
std::is_signed<T>::value) ||
std::is_same<T, i128>::value>;
template<class T>
using is_unsigned_int =
std::integral_constant<bool, (std::is_integral<T>::value &&
std::is_unsigned<T>::value) ||
std::is_same<T, u128>::value>;
template<class T>
using is_int = std::integral_constant<bool, is_signed_int<T>::value ||
is_unsigned_int<T>::value>;
template<class T>
using make_signed_int = typename std::conditional<
std::is_same<T, i128>::value || std::is_same<T, u128>::value,
std::common_type<i128>, std::make_signed<T>>::type;
template<class T>
using make_unsigned_int = typename std::conditional<
std::is_same<T, i128>::value || std::is_same<T, u128>::value,
std::common_type<u128>, std::make_unsigned<T>>::type;
template<class T, class = void> struct is_range : std::false_type {};
template<class T>
struct is_range<
T,
decltype(all(std::declval<typename std::add_lvalue_reference<T>::type>()),
(void)0)> : std::true_type {};
template<class T, bool = is_range<T>::value>
struct range_rank : std::integral_constant<std::size_t, 0> {};
template<class T>
struct range_rank<T, true>
: std::integral_constant<std::size_t,
range_rank<typename T::value_type>::value + 1> {};
template<std::size_t size> struct int_least {
static_assert(size <= 128, "size must be less than or equal to 128");
using type = typename std::conditional<
size <= 8, std::int_least8_t,
typename std::conditional<
size <= 16, std::int_least16_t,
typename std::conditional<
size <= 32, std::int_least32_t,
typename std::conditional<size <= 64, std::int_least64_t,
i128>::type>::type>::type>::type;
};
template<std::size_t size> using int_least_t = typename int_least<size>::type;
template<std::size_t size> struct uint_least {
static_assert(size <= 128, "size must be less than or equal to 128");
using type = typename std::conditional<
size <= 8, std::uint_least8_t,
typename std::conditional<
size <= 16, std::uint_least16_t,
typename std::conditional<
size <= 32, std::uint_least32_t,
typename std::conditional<size <= 64, std::uint_least64_t,
u128>::type>::type>::type>::type;
};
template<std::size_t size> using uint_least_t = typename uint_least<size>::type;
template<class T>
using double_size_int = int_least<std::numeric_limits<T>::digits * 2 + 1>;
template<class T> using double_size_int_t = typename double_size_int<T>::type;
template<class T>
using double_size_uint = uint_least<std::numeric_limits<T>::digits * 2>;
template<class T> using double_size_uint_t = typename double_size_uint<T>::type;
template<class T>
using double_size =
typename std::conditional<is_signed_int<T>::value, double_size_int<T>,
double_size_uint<T>>::type;
template<class T> using double_size_t = typename double_size<T>::type;
#line 2 "template/in.hpp"
#line 4 "template/in.hpp"
#include <unistd.h>
#line 8 "template/in.hpp"
template<std::size_t buf_size = IO_BUFFER_SIZE,
std::size_t decimal_precision = 16>
class Scanner {
private:
template<class, class = void> struct has_scan : std::false_type {};
template<class T>
struct has_scan<
T, decltype(std::declval<T>().scan(std::declval<Scanner&>()), (void)0)>
: std::true_type {};
int fd;
int idx, sz;
bool state;
std::array<char, IO_BUFFER_SIZE + 1> buffer;
inline char cur() {
if (idx == sz) load();
if (idx == sz) {
state = false;
return '\0';
}
return buffer[idx];
}
inline void next() {
if (idx == sz) load();
if (idx == sz) return;
++idx;
}
public:
inline void load() {
int len = sz - idx;
if (idx < len) return;
std::memcpy(buffer.begin(), buffer.begin() + idx, len);
sz = len + read(fd, buffer.data() + len, buf_size - len);
buffer[sz] = 0;
idx = 0;
}
Scanner(int fd) : fd(fd), idx(0), sz(0), state(true) {}
Scanner(FILE* fp) : fd(fileno(fp)), idx(0), sz(0), state(true) {}
inline char scan_char() {
if (idx == sz) load();
return idx == sz ? '\0' : buffer[idx++];
}
Scanner ignore(int n = 1) {
if (idx + n > sz) load();
idx += n;
return *this;
}
inline void discard_space() {
if (idx == sz) load();
while (('\t' <= buffer[idx] && buffer[idx] <= '\r') ||
buffer[idx] == ' ') {
if (++idx == sz) load();
}
}
void scan(char& a) {
discard_space();
a = scan_char();
}
void scan(bool& a) {
discard_space();
a = scan_char() != '0';
}
void scan(std::string& a) {
discard_space();
a.clear();
while (cur() != '\0' && (buffer[idx] < '\t' || '\r' < buffer[idx]) &&
buffer[idx] != ' ') {
a += scan_char();
}
}
template<std::size_t len> void scan(std::bitset<len>& a) {
discard_space();
if (idx + len > sz) load();
rrep (i, len) a[i] = buffer[idx++] != '0';
}
template<class T,
typename std::enable_if<is_signed_int<T>::value &&
!has_scan<T>::value>::type* = nullptr>
void scan(T& a) {
discard_space();
if (buffer[idx] == '-') {
++idx;
if (idx + 40 > sz &&
(idx == sz || ('0' <= buffer[sz - 1] && buffer[sz - 1] <= '9')))
load();
a = 0;
while ('0' <= buffer[idx] && buffer[idx] <= '9') {
a = a * 10 - (buffer[idx++] - '0');
}
}
else {
if (idx + 40 > sz && '0' <= buffer[sz - 1] && buffer[sz - 1] <= '9')
load();
a = 0;
while ('0' <= buffer[idx] && buffer[idx] <= '9') {
a = a * 10 + (buffer[idx++] - '0');
}
}
}
template<class T,
typename std::enable_if<is_unsigned_int<T>::value &&
!has_scan<T>::value>::type* = nullptr>
void scan(T& a) {
discard_space();
if (idx + 40 > sz && '0' <= buffer[sz - 1] && buffer[sz - 1] <= '9')
load();
a = 0;
while ('0' <= buffer[idx] && buffer[idx] <= '9') {
a = a * 10 + (buffer[idx++] - '0');
}
}
template<class T,
typename std::enable_if<std::is_floating_point<T>::value &&
!has_scan<T>::value>::type* = nullptr>
void scan(T& a) {
discard_space();
bool sgn = false;
if (cur() == '-') {
sgn = true;
next();
}
a = 0;
while ('0' <= cur() && cur() <= '9') {
a = a * 10 + cur() - '0';
next();
}
if (cur() == '.') {
next();
T n = 0, d = 1;
for (int i = 0;
'0' <= cur() && cur() <= '9' && i < (int)decimal_precision;
++i) {
n = n * 10 + cur() - '0';
d *= 10;
next();
}
while ('0' <= cur() && cur() <= '9') next();
a += n / d;
}
if (sgn) a = -a;
}
private:
template<std::size_t i, class... Args> void scan(std::tuple<Args...>& a) {
if IF_CONSTEXPR (i < sizeof...(Args)) {
scan(std::get<i>(a));
scan<i + 1, Args...>(a);
}
}
public:
template<class... Args> void scan(std::tuple<Args...>& a) {
scan<0, Args...>(a);
}
template<class T, class U> void scan(std::pair<T, U>& a) {
scan(a.first);
scan(a.second);
}
template<class T,
typename std::enable_if<is_range<T>::value &&
!has_scan<T>::value>::type* = nullptr>
void scan(T& a) {
for (auto&& i : a) scan(i);
}
template<class T,
typename std::enable_if<has_scan<T>::value>::type* = nullptr>
void scan(T& a) {
a.scan(*this);
}
void operator()() {}
template<class Head, class... Args>
void operator()(Head& head, Args&... args) {
scan(head);
operator()(args...);
}
template<class T> Scanner& operator>>(T& a) {
scan(a);
return *this;
}
explicit operator bool() const { return state; }
friend Scanner& getline(Scanner& scan, std::string& a) {
a.erase();
char c;
if ((c = scan.scan_char()) == '\n' || c == '\0') return scan;
a += c;
while ((c = scan.scan_char()) != '\n' && c != '\0') a += c;
scan.state = true;
return scan;
}
};
Scanner<> scan(0);
#line 2 "template/out.hpp"
#line 8 "template/out.hpp"
struct NumberToString {
char buf[10000][4];
constexpr NumberToString() : buf{} {
rep (i, 10000) {
int n = i;
rrep (j, 4) {
buf[i][j] = (char)('0' + n % 10);
n /= 10;
}
}
}
} constexpr precalc_number_to_string;
template<std::size_t buf_size = IO_BUFFER_SIZE, bool debug = false>
class Printer {
private:
template<class, bool = debug, class = void>
struct has_print : std::false_type {};
template<class T>
struct has_print<T, false,
decltype(std::declval<T>().print(std::declval<Printer&>()),
(void)0)> : std::true_type {};
template<class T>
struct has_print<T, true,
decltype(std::declval<T>().debug(std::declval<Printer&>()),
(void)0)> : std::true_type {};
int fd;
std::size_t idx;
std::array<char, buf_size> buffer;
std::size_t decimal_precision;
public:
inline void print_char(char c) {
#if SHIO_LOCAL
buffer[idx++] = c;
if (idx == buf_size) flush();
#else
if IF_CONSTEXPR (!debug) {
buffer[idx++] = c;
if (idx == buf_size) flush();
}
#endif
}
inline void flush() {
idx = write(fd, buffer.begin(), idx);
idx = 0;
}
Printer(int fd) : fd(fd), idx(0), decimal_precision(16) {}
Printer(FILE* fp) : fd(fileno(fp)), idx(0), decimal_precision(16) {}
~Printer() { flush(); }
void set_decimal_precision(std::size_t decimal_precision) {
this->decimal_precision = decimal_precision;
}
void print(char c) {
if IF_CONSTEXPR (debug) print_char('\'');
print_char(c);
if IF_CONSTEXPR (debug) print_char('\'');
}
void print(bool b) { print_char((char)(b + '0')); }
void print(const char* a) {
if IF_CONSTEXPR (debug) print_char('"');
for (; *a != '\0'; ++a) print_char(*a);
if IF_CONSTEXPR (debug) print_char('"');
}
template<std::size_t len> void print(const char (&a)[len]) {
if IF_CONSTEXPR (debug) print_char('"');
for (auto i : a) print_char(i);
if IF_CONSTEXPR (debug) print_char('"');
}
void print(const std::string& a) {
if IF_CONSTEXPR (debug) print_char('"');
for (auto i : a) print_char(i);
if IF_CONSTEXPR (debug) print_char('"');
}
template<std::size_t len> void print(const std::bitset<len>& a) {
rrep (i, len) print_char((char)(a[i] + '0'));
}
template<class T,
typename std::enable_if<is_int<T>::value &&
!has_print<T>::value>::type* = nullptr>
void print(T a) {
if (!a) {
print_char('0');
return;
}
if IF_CONSTEXPR (is_signed_int<T>::value) {
if (a < 0) {
print_char('-');
using U = typename make_unsigned_int<T>::type;
print(static_cast<U>(-static_cast<U>(a)));
return;
}
}
if (idx + 40 >= buf_size) flush();
static char s[40];
int t = 40;
while (a >= 10000) {
int i = a % 10000;
a /= 10000;
t -= 4;
std::memcpy(s + t, precalc_number_to_string.buf[i], 4);
}
if (a >= 1000) {
std::memcpy(buffer.begin() + idx, precalc_number_to_string.buf[a],
4);
idx += 4;
}
else if (a >= 100) {
std::memcpy(buffer.begin() + idx,
precalc_number_to_string.buf[a] + 1, 3);
idx += 3;
}
else if (a >= 10) {
std::memcpy(buffer.begin() + idx,
precalc_number_to_string.buf[a] + 2, 2);
idx += 2;
}
else {
buffer[idx++] = '0' | a;
}
std::memcpy(buffer.begin() + idx, s + t, 40 - t);
idx += 40 - t;
}
template<class T,
typename std::enable_if<std::is_floating_point<T>::value &&
!has_print<T>::value>::type* = nullptr>
void print(T a) {
if (a == std::numeric_limits<T>::infinity()) {
print("inf");
return;
}
if (a == -std::numeric_limits<T>::infinity()) {
print("-inf");
return;
}
if (std::isnan(a)) {
print("nan");
return;
}
if (a < 0) {
print_char('-');
a = -a;
}
T b = a;
if (b < 1) {
print_char('0');
}
else {
std::string s;
while (b >= 1) {
s += (char)('0' + (int)std::fmod(b, 10.0));
b /= 10;
}
for (auto i = s.rbegin(); i != s.rend(); ++i) print_char(*i);
}
print_char('.');
rep (decimal_precision) {
a *= 10;
print_char((char)('0' + (int)std::fmod(a, 10.0)));
}
}
private:
template<std::size_t i, class... Args>
void print(const std::tuple<Args...>& a) {
if IF_CONSTEXPR (i < sizeof...(Args)) {
if IF_CONSTEXPR (debug) print_char(',');
print_char(' ');
print(std::get<i>(a));
print<i + 1, Args...>(a);
}
}
public:
template<class... Args> void print(const std::tuple<Args...>& a) {
if IF_CONSTEXPR (debug) print_char('(');
if IF_CONSTEXPR (sizeof...(Args) != 0) print(std::get<0>(a));
print<1, Args...>(a);
if IF_CONSTEXPR (debug) print_char(')');
}
template<class T, class U> void print(const std::pair<T, U>& a) {
if IF_CONSTEXPR (debug) print_char('(');
print(a.first);
if IF_CONSTEXPR (debug) print_char(',');
print_char(' ');
print(a.second);
if IF_CONSTEXPR (debug) print_char(')');
}
template<class T,
typename std::enable_if<is_range<T>::value &&
!has_print<T>::value>::type* = nullptr>
void print(const T& a) {
if IF_CONSTEXPR (debug) print_char('{');
for (auto i = std::begin(a); i != std::end(a); ++i) {
if (i != std::begin(a)) {
if IF_CONSTEXPR (debug) print_char(',');
print_char(' ');
}
print(*i);
}
if IF_CONSTEXPR (debug) print_char('}');
}
template<class T, typename std::enable_if<has_print<T>::value &&
!debug>::type* = nullptr>
void print(const T& a) {
a.print(*this);
}
template<class T, typename std::enable_if<has_print<T>::value &&
debug>::type* = nullptr>
void print(const T& a) {
a.debug(*this);
}
void operator()() {}
template<class Head, class... Args>
void operator()(const Head& head, const Args&... args) {
print(head);
operator()(args...);
}
template<class T> Printer& operator<<(const T& a) {
print(a);
return *this;
}
Printer& operator<<(Printer& (*pf)(Printer&)) { return pf(*this); }
};
template<std::size_t buf_size, bool debug>
Printer<buf_size, debug>& endl(Printer<buf_size, debug>& pr) {
pr.print_char('\n');
pr.flush();
return pr;
}
template<std::size_t buf_size, bool debug>
Printer<buf_size, debug>& flush(Printer<buf_size, debug>& pr) {
pr.flush();
return pr;
}
struct SetPrec {
int n;
template<class Pr> void print(Pr& pr) const { pr.set_decimal_precision(n); }
template<class Pr> void debug(Pr& pr) const { pr.set_decimal_precision(n); }
};
SetPrec setprec(int n) { return SetPrec{n}; };
Printer<> print(1), eprint(2);
void prints() { print.print_char('\n'); }
template<class T> auto prints(const T& v) -> decltype(print << v, (void)0) {
print << v;
print.print_char('\n');
}
template<class Head, class... Tail>
auto prints(const Head& head, const Tail&... tail)
-> decltype(print << head, (void)0) {
print << head;
print.print_char(' ');
prints(tail...);
}
Printer<IO_BUFFER_SIZE, true> debug(1), edebug(2);
void debugs() { debug.print_char('\n'); }
template<class T> auto debugs(const T& v) -> decltype(debug << v, (void)0) {
debug << v;
debug.print_char('\n');
}
template<class Head, class... Tail>
auto debugs(const Head& head, const Tail&... tail)
-> decltype(debug << head, (void)0) {
debug << head;
debug.print_char(' ');
debugs(tail...);
}
#line 2 "template/bitop.hpp"
#line 6 "template/bitop.hpp"
namespace bitop {
#define KTH_BIT(b, k) (((b) >> (k)) & 1)
#define POW2(k) (1ull << (k))
inline ull next_combination(int n, ull x) {
if (n == 0) return 1;
ull a = x & -x;
ull b = x + a;
return (x & ~b) / a >> 1 | b;
}
#define rep_comb(i, n, k) \
for (ull i = (1ull << (k)) - 1; i < (1ull << (n)); \
i = bitop::next_combination((n), i))
inline constexpr int msb(ull x) {
int res = x ? 0 : -1;
if (x & 0xFFFFFFFF00000000) x &= 0xFFFFFFFF00000000, res += 32;
if (x & 0xFFFF0000FFFF0000) x &= 0xFFFF0000FFFF0000, res += 16;
if (x & 0xFF00FF00FF00FF00) x &= 0xFF00FF00FF00FF00, res += 8;
if (x & 0xF0F0F0F0F0F0F0F0) x &= 0xF0F0F0F0F0F0F0F0, res += 4;
if (x & 0xCCCCCCCCCCCCCCCC) x &= 0xCCCCCCCCCCCCCCCC, res += 2;
return res + ((x & 0xAAAAAAAAAAAAAAAA) ? 1 : 0);
}
inline constexpr int ceil_log2(ull x) { return x ? msb(x - 1) + 1 : 0; }
inline constexpr ull reverse(ull x) {
x = ((x & 0xAAAAAAAAAAAAAAAA) >> 1) | ((x & 0x5555555555555555) << 1);
x = ((x & 0xCCCCCCCCCCCCCCCC) >> 2) | ((x & 0x3333333333333333) << 2);
x = ((x & 0xF0F0F0F0F0F0F0F0) >> 4) | ((x & 0x0F0F0F0F0F0F0F0F) << 4);
x = ((x & 0xFF00FF00FF00FF00) >> 8) | ((x & 0x00FF00FF00FF00FF) << 8);
x = ((x & 0xFFFF0000FFFF0000) >> 16) | ((x & 0x0000FFFF0000FFFF) << 16);
return (x >> 32) | (x << 32);
}
inline constexpr ull reverse(ull x, int n) { return reverse(x) >> (64 - n); }
} // namespace bitop
inline constexpr int popcnt(ull x) noexcept {
#if __cplusplus >= 202002L
return std::popcount(x);
#endif
x = (x & 0x5555555555555555) + ((x >> 1) & 0x5555555555555555);
x = (x & 0x3333333333333333) + ((x >> 2) & 0x3333333333333333);
x = (x & 0x0f0f0f0f0f0f0f0f) + ((x >> 4) & 0x0f0f0f0f0f0f0f0f);
x = (x & 0x00ff00ff00ff00ff) + ((x >> 8) & 0x00ff00ff00ff00ff);
x = (x & 0x0000ffff0000ffff) + ((x >> 16) & 0x0000ffff0000ffff);
return (x & 0x00000000ffffffff) + ((x >> 32) & 0x00000000ffffffff);
}
#line 2 "template/func.hpp"
#line 6 "template/func.hpp"
template<class T, class U, class Comp = std::less<>>
inline constexpr bool chmin(T& a, const U& b,
Comp cmp = Comp()) noexcept(noexcept(cmp(b, a))) {
return cmp(b, a) ? a = b, true : false;
}
template<class T, class U, class Comp = std::less<>>
inline constexpr bool chmax(T& a, const U& b,
Comp cmp = Comp()) noexcept(noexcept(cmp(a, b))) {
return cmp(a, b) ? a = b, true : false;
}
inline constexpr ll gcd(ll a, ll b) {
if (a < 0) a = -a;
if (b < 0) b = -b;
while (b) {
const ll c = a;
a = b;
b = c % b;
}
return a;
}
inline constexpr ll lcm(ll a, ll b) { return a / gcd(a, b) * b; }
inline constexpr bool is_prime(ll N) {
if (N <= 1) return false;
for (ll i = 2; i * i <= N; ++i) {
if (N % i == 0) return false;
}
return true;
}
inline std::vector<ll> prime_factor(ll N) {
std::vector<ll> res;
for (ll i = 2; i * i <= N; ++i) {
while (N % i == 0) {
res.push_back(i);
N /= i;
}
}
if (N != 1) res.push_back(N);
return res;
}
inline constexpr ll my_pow(ll a, ll b) {
ll res = 1;
while (b) {
if (b & 1) res *= a;
b >>= 1;
a *= a;
}
return res;
}
inline constexpr ll mod_pow(ll a, ll b, ll mod) {
assert(mod > 0);
if (mod == 1) return 0;
a %= mod;
ll res = 1;
while (b) {
if (b & 1) (res *= a) %= mod;
b >>= 1;
(a *= a) %= mod;
}
return res;
}
inline PLL extGCD(ll a, ll b) {
const ll n = a, m = b;
ll x = 1, y = 0, u = 0, v = 1;
ll t;
while (b) {
t = a / b;
std::swap(a -= t * b, b);
std::swap(x -= t * u, u);
std::swap(y -= t * v, v);
}
if (x < 0) {
x += m;
y -= n;
}
return {x, y};
}
inline ll mod_inv(ll a, ll mod) {
ll b = mod;
ll x = 1, u = 0;
ll t;
while (b) {
t = a / b;
std::swap(a -= t * b, b);
std::swap(x -= t * u, u);
}
if (x < 0) x += mod;
assert(a == 1);
return x;
}
#line 2 "template/util.hpp"
#line 6 "template/util.hpp"
template<class F> class RecLambda {
private:
F f;
public:
explicit constexpr RecLambda(F&& f_) : f(std::forward<F>(f_)) {}
template<class... Args>
constexpr auto operator()(Args&&... args)
-> decltype(f(*this, std::forward<Args>(args)...)) {
return f(*this, std::forward<Args>(args)...);
}
};
template<class F> inline constexpr RecLambda<F> rec_lambda(F&& f) {
return RecLambda<F>(std::forward<F>(f));
}
template<class Head, class... Tail> struct multi_dim_vector {
using type = std::vector<typename multi_dim_vector<Tail...>::type>;
};
template<class T> struct multi_dim_vector<T> { using type = T; };
template<class T, class Arg>
constexpr std::vector<T> make_vec(int n, Arg&& arg) {
return std::vector<T>(n, std::forward<Arg>(arg));
}
template<class T, class... Args>
constexpr typename multi_dim_vector<Args..., T>::type make_vec(int n,
Args&&... args) {
return typename multi_dim_vector<Args..., T>::type(
n, make_vec<T>(std::forward<Args>(args)...));
}
template<class T, class Comp = std::less<T>> class compressor {
private:
std::vector<T> dat;
Comp cmp;
bool sorted = false;
public:
compressor() : compressor(Comp()) {}
compressor(const Comp& cmp) : cmp(cmp) {}
compressor(const std::vector<T>& vec, bool f = false,
const Comp& cmp = Comp())
: dat(vec), cmp(cmp) {
if (f) build();
}
compressor(std::vector<T>&& vec, bool f = false, const Comp& cmp = Comp())
: dat(std::move(vec)), cmp(cmp) {
if (f) build();
}
compressor(std::initializer_list<T> il, bool f = false,
const Comp& cmp = Comp())
: dat(all(il)), cmp(cmp) {
if (f) build();
}
void reserve(int n) {
assert(!sorted);
dat.reserve(n);
}
void push_back(const T& v) {
assert(!sorted);
dat.push_back(v);
}
void push_back(T&& v) {
assert(!sorted);
dat.push_back(std::move(v));
}
template<class... Args> void emplace_back(Args&&... args) {
assert(!sorted);
dat.emplace_back(std::forward<Args>(args)...);
}
void push(const std::vector<T>& vec) {
assert(!sorted);
const int n = dat.size();
dat.resize(n + vec.size());
rep (i, vec.size()) dat[n + i] = vec[i];
}
int build() {
assert(!sorted);
sorted = true;
std::sort(all(dat), cmp);
dat.erase(std::unique(all(dat),
[&](const T& a, const T& b) -> bool {
return !cmp(a, b) && !cmp(b, a);
}),
dat.end());
return dat.size();
}
const T& operator[](int k) const& {
assert(sorted);
assert(0 <= k && k < (int)dat.size());
return dat[k];
}
int get(const T& val) const {
assert(sorted);
auto itr = std::lower_bound(all(dat), val, cmp);
assert(itr != dat.end() && !cmp(val, *itr));
return itr - dat.begin();
}
int lower_bound(const T& val) const {
assert(sorted);
auto itr = std::lower_bound(all(dat), val, cmp);
return itr - dat.begin();
}
int upper_bound(const T& val) const {
assert(sorted);
auto itr = std::upper_bound(all(dat), val, cmp);
return itr - dat.begin();
}
bool contains(const T& val) const {
assert(sorted);
return std::binary_search(all(dat), val, cmp);
}
std::vector<int> pressed(const std::vector<T>& vec) const {
assert(sorted);
std::vector<int> res(vec.size());
rep (i, vec.size()) res[i] = get(vec[i]);
return res;
}
void press(std::vector<T>& vec) const {
assert(sorted);
for (auto&& i : vec) i = get(i);
}
int size() const {
assert(sorted);
return dat.size();
}
};
#line 2 "math/poly/FormalPowerSeries.hpp"
#line 2 "math/convolution/Convolution.hpp"
#line 2 "math/ModInt.hpp"
#line 4 "math/ModInt.hpp"
template<class T, T mod> class StaticModInt {
static_assert(std::is_integral<T>::value, "T must be integral");
static_assert(std::is_unsigned<T>::value, "T must be unsigned");
static_assert(mod > 0, "mod must be positive");
static_assert(mod <= std::numeric_limits<T>::max() / 2,
"mod * 2 must be less than or equal to T::max()");
private:
using large_t = typename double_size_uint<T>::type;
using signed_t = typename std::make_signed<T>::type;
T val;
static constexpr unsigned int inv1000000007[] = {
0, 1, 500000004, 333333336, 250000002, 400000003,
166666668, 142857144, 125000001, 111111112, 700000005};
static constexpr unsigned int inv998244353[] = {
0, 1, 499122177, 332748118, 748683265, 598946612,
166374059, 855638017, 873463809, 443664157, 299473306};
static constexpr ll mod_inv(ll a) {
ll b = mod;
ll x = 1, u = 0;
ll t = 0, tmp = 0;
while (b) {
t = a / b;
tmp = (a - t * b);
a = b;
b = tmp;
tmp = (x - t * u);
x = u;
u = tmp;
}
if (x < 0) x += mod;
return x;
}
public:
constexpr StaticModInt() : val(0) {}
template<class U,
typename std::enable_if<std::is_integral<U>::value &&
std::is_signed<U>::value>::type* = nullptr>
constexpr StaticModInt(U v) : val{} {
v %= static_cast<signed_t>(mod);
if (v < 0) v += static_cast<signed_t>(mod);
val = static_cast<T>(v);
}
template<class U, typename std::enable_if<
std::is_integral<U>::value &&
std::is_unsigned<U>::value>::type* = nullptr>
constexpr StaticModInt(U v) : val(v % mod) {}
constexpr T get() const { return val; }
static constexpr T get_mod() { return mod; }
static constexpr StaticModInt raw(T v) {
StaticModInt res;
res.val = v;
return res;
}
constexpr StaticModInt inv() const {
if IF_CONSTEXPR (mod == 1000000007) {
if (val <= 10) return inv1000000007[val];
}
else if IF_CONSTEXPR (mod == 998244353) {
if (val <= 10) return inv998244353[val];
}
return mod_inv(val);
}
constexpr StaticModInt& operator++() {
++val;
if (val == mod) val = 0;
return *this;
}
constexpr StaticModInt operator++(int) {
StaticModInt res = *this;
++*this;
return res;
}
constexpr StaticModInt& operator--() {
if (val == 0) val = mod;
--val;
return *this;
}
constexpr StaticModInt operator--(int) {
StaticModInt res = *this;
--*this;
return res;
}
constexpr StaticModInt& operator+=(const StaticModInt& other) {
val += other.val;
if (val >= mod) val -= mod;
return *this;
}
constexpr StaticModInt& operator-=(const StaticModInt& other) {
if (val < other.val) val += mod;
val -= other.val;
return *this;
}
constexpr StaticModInt& operator*=(const StaticModInt& other) {
large_t a = val;
a *= other.val;
a %= mod;
val = a;
return *this;
}
constexpr StaticModInt& operator/=(const StaticModInt& other) {
*this *= other.inv();
return *this;
}
friend constexpr StaticModInt operator+(const StaticModInt& lhs,
const StaticModInt& rhs) {
return StaticModInt(lhs) += rhs;
}
friend constexpr StaticModInt operator-(const StaticModInt& lhs,
const StaticModInt& rhs) {
return StaticModInt(lhs) -= rhs;
}
friend constexpr StaticModInt operator*(const StaticModInt& lhs,
const StaticModInt& rhs) {
return StaticModInt(lhs) *= rhs;
}
friend constexpr StaticModInt operator/(const StaticModInt& lhs,
const StaticModInt& rhs) {
return StaticModInt(lhs) /= rhs;
}
constexpr StaticModInt operator+() const { return StaticModInt(*this); }
constexpr StaticModInt operator-() const { return StaticModInt() - *this; }
friend constexpr bool operator==(const StaticModInt& lhs,
const StaticModInt& rhs) {
return lhs.val == rhs.val;
}
friend constexpr bool operator!=(const StaticModInt& lhs,
const StaticModInt& rhs) {
return lhs.val != rhs.val;
}
constexpr StaticModInt pow(ll a) const {
StaticModInt v = *this, res = 1;
while (a) {
if (a & 1) res *= v;
a >>= 1;
v *= v;
}
return res;
}
template<class Pr> void print(Pr& a) const { a.print(val); }
template<class Pr> void debug(Pr& a) const { a.print(val); }
template<class Sc> void scan(Sc& a) {
ll v;
a.scan(v);
*this = v;
}
};
#if __cplusplus < 201703L
template<class T, T mod>
constexpr unsigned int StaticModInt<T, mod>::inv1000000007[];
template<class T, T mod>
constexpr unsigned int StaticModInt<T, mod>::inv998244353[];
#endif
template<unsigned int p> using static_modint = StaticModInt<unsigned int, p>;
using modint1000000007 = static_modint<1000000007>;
using modint998244353 = static_modint<998244353>;
template<class T, int id> class DynamicModInt {
static_assert(std::is_integral<T>::value, "T must be integral");
static_assert(std::is_unsigned<T>::value, "T must be unsigned");
private:
using large_t = typename double_size_uint<T>::type;
using signed_t = typename std::make_signed<T>::type;
T val;
static T mod;
public:
constexpr DynamicModInt() : val(0) {}
template<class U,
typename std::enable_if<std::is_integral<U>::value &&
std::is_signed<U>::value>::type* = nullptr>
constexpr DynamicModInt(U v) : val{} {
v %= static_cast<signed_t>(mod);
if (v < 0) v += static_cast<signed_t>(mod);
val = static_cast<T>(v);
}
template<class U, typename std::enable_if<
std::is_integral<U>::value &&
std::is_unsigned<U>::value>::type* = nullptr>
constexpr DynamicModInt(U v) : val(v % mod) {}
T get() const { return val; }
static T get_mod() { return mod; }
static void set_mod(T v) {
assert(v > 0);
assert(v <= std::numeric_limits<T>::max() / 2);
mod = v;
}
static DynamicModInt raw(T v) {
DynamicModInt res;
res.val = v;
return res;
}
DynamicModInt inv() const { return mod_inv(val, mod); }
DynamicModInt& operator++() {
++val;
if (val == mod) val = 0;
return *this;
}
DynamicModInt operator++(int) {
DynamicModInt res = *this;
++*this;
return res;
}
DynamicModInt& operator--() {
if (val == 0) val = mod;
--val;
return *this;
}
DynamicModInt operator--(int) {
DynamicModInt res = *this;
--*this;
return res;
}
DynamicModInt& operator+=(const DynamicModInt& other) {
val += other.val;
if (val >= mod) val -= mod;
return *this;
}
DynamicModInt& operator-=(const DynamicModInt& other) {
if (val < other.val) val += mod;
val -= other.val;
return *this;
}
DynamicModInt& operator*=(const DynamicModInt& other) {
large_t a = val;
a *= other.val;
a %= mod;
val = a;
return *this;
}
DynamicModInt& operator/=(const DynamicModInt& other) {
*this *= other.inv();
return *this;
}
friend DynamicModInt operator+(const DynamicModInt& lhs,
const DynamicModInt& rhs) {
return DynamicModInt(lhs) += rhs;
}
friend DynamicModInt operator-(const DynamicModInt& lhs,
const DynamicModInt& rhs) {
return DynamicModInt(lhs) -= rhs;
}
friend DynamicModInt operator*(const DynamicModInt& lhs,
const DynamicModInt& rhs) {
return DynamicModInt(lhs) *= rhs;
}
friend DynamicModInt operator/(const DynamicModInt& lhs,
const DynamicModInt& rhs) {
return DynamicModInt(lhs) /= rhs;
}
DynamicModInt operator+() const { return DynamicModInt(*this); }
DynamicModInt operator-() const { return DynamicModInt() - *this; }
friend bool operator==(const DynamicModInt& lhs, const DynamicModInt& rhs) {
return lhs.val == rhs.val;
}
friend bool operator!=(const DynamicModInt& lhs, const DynamicModInt& rhs) {
return lhs.val != rhs.val;
}
DynamicModInt pow(ll a) const {
DynamicModInt v = *this, res = 1;
while (a) {
if (a & 1) res *= v;
a >>= 1;
v *= v;
}
return res;
}
template<class Pr> void print(Pr& a) const { a.print(val); }
template<class Pr> void debug(Pr& a) const { a.print(val); }
template<class Sc> void scan(Sc& a) {
ll v;
a.scan(v);
*this = v;
}
};
template<class T, int id> T DynamicModInt<T, id>::mod = 998244353;
template<int id> using dynamic_modint = DynamicModInt<unsigned int, id>;
using modint = dynamic_modint<-1>;
/**
* @brief ModInt
* @docs docs/math/ModInt.md
*/
#line 5 "math/convolution/Convolution.hpp"
constexpr ull primitive_root_for_convolution(ull p) {
if (p == 2) return 1;
if (p == 998244353) return 3;
if (p == 469762049) return 3;
if (p == 1811939329) return 11;
if (p == 2013265921) return 11;
rep (g, 2, p) {
if (mod_pow(g, (p - 1) >> 1, p) != 1) return g;
}
return -1;
}
namespace internal {
template<class T> class NthRoot {
private:
static constexpr unsigned int lg =
bitop::msb((T::get_mod() - 1) & (1 - T::get_mod()));
T root[lg + 1];
T inv_root[lg + 1];
T rate[lg + 1];
T inv_rate[lg + 1];
public:
constexpr NthRoot() : root{}, inv_root{}, rate{}, inv_rate{} {
root[lg] = T{primitive_root_for_convolution(T::get_mod())}.pow(
(T::get_mod() - 1) >> lg);
inv_root[lg] = root[lg].inv();
rrep (i, lg) {
root[i] = root[i + 1] * root[i + 1];
inv_root[i] = inv_root[i + 1] * inv_root[i + 1];
}
T r = 1;
rep (i, 2, lg + 1) {
rate[i - 2] = r * root[i];
r = r * inv_root[i];
}
r = 1;
rep (i, 2, lg + 1) {
inv_rate[i - 2] = r * inv_root[i];
r = r * root[i];
}
}
static constexpr unsigned int get_lg() { return lg; }
constexpr T get(int n) const { return root[n]; }
constexpr T inv(int n) const { return inv_root[n]; }
constexpr T get_rate(int n) const { return rate[n]; }
constexpr T get_inv_rate(int n) const { return inv_rate[n]; }
};
template<class T> void number_theoretic_transform(std::vector<T>& a) {
static constexpr NthRoot<T> nth_root;
int n = a.size();
for (int i = n >> 1; i > 0; i >>= 1) {
T z = T::raw(1);
rep (j, 0, n, i << 1) {
rep (k, i) {
const T x = a[j + k];
const T y = a[j + i + k] * z;
a[j + k] = x + y;
a[j + i + k] = x - y;
}
z *= nth_root.get_rate(popcnt(j & ~(j + (i << 1))));
}
}
}
template<class T> void inverse_number_theoretic_transform(std::vector<T>& a) {
static constexpr NthRoot<T> nth_root;
int n = a.size();
for (int i = 1; i < n; i <<= 1) {
T z = T::raw(1);
rep (j, 0, n, i << 1) {
rep (k, i) {
const T x = a[j + k];
const T y = a[j + i + k];
a[j + k] = x + y;
a[j + i + k] = (x - y) * z;
}
z *= nth_root.get_inv_rate(popcnt(j & ~(j + (i << 1))));
}
}
T inv_n = T(1) / n;
for (auto&& x : a) x *= inv_n;
}
template<class T>
std::vector<T> convolution_naive(const std::vector<T>& a,
const std::vector<T>& b) {
int n = a.size(), m = b.size();
std::vector<T> c(n + m - 1);
rep (i, n)
rep (j, m) c[i + j] += a[i] * b[j];
return c;
}
template<class T> std::vector<T> convolution_pow2(std::vector<T> a) {
int n = a.size() * 2 - 1;
int lg = bitop::msb(n - 1) + 1;
if (n - (1 << (lg - 1)) <= 5) {
--lg;
int m = a.size() - (1 << (lg - 1));
std::vector<T> a1(a.begin(), a.begin() + m), a2(a.begin() + m, a.end());
std::vector<T> c(n);
std::vector<T> c1 = convolution_naive(a1, a1);
std::vector<T> c2 = convolution_naive(a1, a2);
std::vector<T> c3 = convolution_pow2(a2);
rep (i, c1.size()) c[i] += c1[i];
rep (i, c2.size()) c[i + m] += c2[i] * 2;
rep (i, c3.size()) c[i + m * 2] += c3[i];
return c;
}
int m = 1 << lg;
a.resize(m);
number_theoretic_transform(a);
rep (i, m) a[i] *= a[i];
inverse_number_theoretic_transform(a);
a.resize(n);
return a;
}
template<class T>
std::vector<T> convolution(std::vector<T> a, std::vector<T> b) {
int n = a.size() + b.size() - 1;
int lg = bitop::ceil_log2(n);
int m = 1 << lg;
if (n - (1 << (lg - 1)) <= 5) {
--lg;
if (a.size() < b.size()) std::swap(a, b);
int m = n - (1 << lg);
std::vector<T> a1(a.begin(), a.begin() + m), a2(a.begin() + m, a.end());
std::vector<T> c(n);
std::vector<T> c1 = convolution_naive(a1, b);
std::vector<T> c2 = convolution(a2, b);
rep (i, c1.size()) c[i] += c1[i];
rep (i, c2.size()) c[i + m] += c2[i];
return c;
}
a.resize(m);
b.resize(m);
number_theoretic_transform(a);
number_theoretic_transform(b);
rep (i, m) a[i] *= b[i];
inverse_number_theoretic_transform(a);
a.resize(n);
return a;
}
} // namespace internal
using internal::inverse_number_theoretic_transform;
using internal::number_theoretic_transform;
template<unsigned int p>
std::vector<static_modint<p>>
convolution_for_any_mod(const std::vector<static_modint<p>>& a,
const std::vector<static_modint<p>>& b);
template<unsigned int p>
std::vector<static_modint<p>>
convolution(const std::vector<static_modint<p>>& a,
const std::vector<static_modint<p>>& b) {
unsigned int n = a.size(), m = b.size();
if (n == 0 || m == 0) return {};
if (n <= 60 || m <= 60) return internal::convolution_naive(a, b);
if (n + m - 1 <= ((1 - p) & (p - 1))) {
if (n == m && a == b) return internal::convolution_pow2(a);
return internal::convolution(a, b);
}
return convolution_for_any_mod(a, b);
}
template<unsigned int p>
std::vector<ll> convolution(const std::vector<ll>& a,
const std::vector<ll>& b) {
int n = a.size(), m = b.size();
std::vector<static_modint<p>> a2(n), b2(m);
rep (i, n) a2[i] = a[i];
rep (i, m) b2[i] = b[i];
auto c2 = convolution(a2, b2);
std::vector<ll> c(c2.size());
rep (i, c2.size()) c[i] = c2[i].get();
return c;
}
template<unsigned int p>
std::vector<static_modint<p>>
convolution_for_any_mod(const std::vector<static_modint<p>>& a,
const std::vector<static_modint<p>>& b) {
int n = a.size(), m = b.size();
assert(n + m - 1 <= (1 << 26));
std::vector<ll> a2(n), b2(m);
rep (i, n) a2[i] = a[i].get();
rep (i, m) b2[i] = b[i].get();
static constexpr ll MOD1 = 469762049;
static constexpr ll MOD2 = 1811939329;
static constexpr ll MOD3 = 2013265921;
static constexpr ll INV1_2 = mod_pow(MOD1, MOD2 - 2, MOD2);
static constexpr ll INV1_3 = mod_pow(MOD1, MOD3 - 2, MOD3);
static constexpr ll INV2_3 = mod_pow(MOD2, MOD3 - 2, MOD3);
auto c1 = convolution<MOD1>(a2, b2);
auto c2 = convolution<MOD2>(a2, b2);
auto c3 = convolution<MOD3>(a2, b2);
std::vector<static_modint<p>> res(n + m - 1);
rep (i, n + m - 1) {
ll t1 = c1[i];
ll t2 = (c2[i] - t1 + MOD2) * INV1_2 % MOD2;
if (t2 < 0) t2 += MOD2;
ll t3 =
((c3[i] - t1 + MOD3) * INV1_3 % MOD3 - t2 + MOD3) * INV2_3 % MOD3;
if (t3 < 0) t3 += MOD3;
res[i] = static_modint<p>(t1 + (t2 + t3 * MOD2) % p * MOD1);
}
return res;
}
template<class T> void ntt_doubling_(std::vector<T>& a, std::vector<T> b) {
static constexpr internal::NthRoot<T> nth_root;
int n = a.size();
const T z = nth_root.get(bitop::msb(n) + 1);
T r = 1;
rep (i, n) {
b[i] *= r;
r *= z;
}
number_theoretic_transform(b);
a.reserve(2 * n);
a.insert(a.end(), all(b));
}
template<class T> void ntt_doubling_(std::vector<T>& a) {
static constexpr internal::NthRoot<T> nth_root;
int n = a.size();
auto b = a;
inverse_number_theoretic_transform(b);
const T z = nth_root.get(bitop::msb(n) + 1);
T r = 1;
rep (i, n) {
b[i] *= r;
r *= z;
}
number_theoretic_transform(b);
a.reserve(2 * n);
a.insert(a.end(), all(b));
}
template<unsigned int p> struct is_ntt_friendly : std::false_type {};
template<> struct is_ntt_friendly<998244353> : std::true_type {};
/**
* @brief Convolution(畳み込み)
* @docs docs/math/convolution/Convolution.md
*/
#line 2 "math/Combinatorics.hpp"
#line 5 "math/Combinatorics.hpp"
template<class T> class Combinatorics {
private:
static std::vector<T> factorial;
static std::vector<T> factinv;
public:
static void init(ll n) {
const int b = factorial.size();
if (n < b) return;
factorial.resize(n + 1);
rep (i, b, n + 1) factorial[i] = factorial[i - 1] * i;
factinv.resize(n + 1);
factinv[n] = T(1) / factorial[n];
rreps (i, n, b) factinv[i - 1] = factinv[i] * i;
}
static T fact(ll x) {
if (x < 0) return 0;
init(x);
return factorial[x];
}
static T finv(ll x) {
if (x < 0) return 0;
init(x);
return factinv[x];
}
static T inv(ll x) {
if (x <= 0) return 0;
init(x);
return factorial[x - 1] * factinv[x];
}
static T perm(ll n, ll r) {
if (r < 0 || r > n) return 0;
init(n);
return factorial[n] * factinv[n - r];
}
static T comb(ll n, ll r) {
if (n < 0) return 0;
if (r < 0 || r > n) return 0;
init(n);
return factorial[n] * factinv[n - r] * factinv[r];
}
static T homo(ll n, ll r) { return comb(n + r - 1, r); }
static T small_perm(ll n, ll r) {
if (r < 0 || r > n) return 0;
T res = 1;
reps (i, r) res *= n - r + i;
return res;
}
static T small_comb(ll n, ll r) {
if (r < 0 || r > n) return 0;
chmin(r, n - r);
init(r);
T res = factinv[r];
reps (i, r) res *= n - r + i;
return res;
}
static T small_homo(ll n, ll r) { return small_comb(n + r - 1, r); }
};
template<class T>
std::vector<T> Combinatorics<T>::factorial = std::vector<T>(1, 1);
template<class T>
std::vector<T> Combinatorics<T>::factinv = std::vector<T>(1, 1);
/**
* @brief Combinatorics
* @docs docs/math/Combinatorics.md
*/
#line 2 "math/SqrtMod.hpp"
#line 2 "math/MontgomeryModInt.hpp"
#line 4 "math/MontgomeryModInt.hpp"
template<class T> class MontgomeryReduction {
static_assert(std::is_integral<T>::value, "T must be integral");
static_assert(std::is_unsigned<T>::value, "T must be unsigned");
private:
using large_t = typename double_size_uint<T>::type;
static constexpr int lg = std::numeric_limits<T>::digits;
T mod;
T r;
T r2; // r^2 mod m
T calc_minv() {
T t = 0, res = 0;
rep (i, lg) {
if (~t & 1) {
t += mod;
res += static_cast<T>(1) << i;
}
t >>= 1;
}
return res;
}
T minv;
public:
MontgomeryReduction(T v) { set_mod(v); }
static constexpr int get_lg() { return lg; }
void set_mod(T v) {
assert(v > 0);
assert(v & 1);
assert(v <= std::numeric_limits<T>::max() / 2);
mod = v;
r = (-static_cast<T>(mod)) % mod;
r2 = (-static_cast<large_t>(mod)) % mod;
minv = calc_minv();
}
inline T get_mod() const { return mod; }
inline T get_r() const { return r; }
T reduce(large_t x) const {
large_t tmp =
(x + static_cast<large_t>(static_cast<T>(x) * minv) * mod) >> lg;
return tmp >= mod ? tmp - mod : tmp;
}
T transform(large_t x) const { return reduce(x * r2); }
};
template<class T, int id> class MontgomeryModInt {
private:
using large_t = typename double_size_uint<T>::type;
using signed_t = typename std::make_signed<T>::type;
T val;
static MontgomeryReduction<T> mont;
public:
MontgomeryModInt() : val(0) {}
template<class U, typename std::enable_if<
std::is_integral<U>::value &&
std::is_unsigned<U>::value>::type* = nullptr>
MontgomeryModInt(U x)
: val(mont.transform(
x < (static_cast<large_t>(mont.get_mod()) << mont.get_lg())
? x
: x % mont.get_mod())) {}
template<class U,
typename std::enable_if<std::is_integral<U>::value &&
std::is_signed<U>::value>::type* = nullptr>
MontgomeryModInt(U x)
: MontgomeryModInt(static_cast<typename std::make_unsigned<U>::type>(
x < 0 ? -x : x)) {
if (x < 0 && val) val = mont.get_mod() - val;
}
T get() const { return mont.reduce(val); }
static T get_mod() { return mont.get_mod(); }
static void set_mod(T v) { mont.set_mod(v); }
MontgomeryModInt operator+() const { return *this; }
MontgomeryModInt operator-() const {
MontgomeryModInt res;
if (val) res.val = mont.get_mod() - val;
return res;
}
MontgomeryModInt& operator++() {
val += mont.get_r();
if (val >= mont.get_mod()) val -= mont.get_mod();
return *this;
}
MontgomeryModInt& operator--() {
if (val < mont.get_r()) val += mont.get_mod();
val -= mont.get_r();
return *this;
}
MontgomeryModInt operator++(int) {
MontgomeryModInt res = *this;
++*this;
return res;
}
MontgomeryModInt operator--(int) {
MontgomeryModInt res = *this;
--*this;
return res;
}
MontgomeryModInt& operator+=(const MontgomeryModInt& rhs) {
val += rhs.val;
if (val >= mont.get_mod()) val -= mont.get_mod();
return *this;
}
MontgomeryModInt& operator-=(const MontgomeryModInt& rhs) {
if (val < rhs.val) val += mont.get_mod();
val -= rhs.val;
return *this;
}
MontgomeryModInt& operator*=(const MontgomeryModInt& rhs) {
val = mont.reduce(static_cast<large_t>(val) * rhs.val);
return *this;
}
MontgomeryModInt pow(ull n) const {
MontgomeryModInt res = 1, x = *this;
while (n) {
if (n & 1) res *= x;
x *= x;
n >>= 1;
}
return res;
}
MontgomeryModInt inv() const { return pow(mont.get_mod() - 2); }
MontgomeryModInt& operator/=(const MontgomeryModInt& rhs) {
return *this *= rhs.inv();
}
friend MontgomeryModInt operator+(const MontgomeryModInt& lhs,
const MontgomeryModInt& rhs) {
return MontgomeryModInt(lhs) += rhs;
}
friend MontgomeryModInt operator-(const MontgomeryModInt& lhs,
const MontgomeryModInt& rhs) {
return MontgomeryModInt(lhs) -= rhs;
}
friend MontgomeryModInt operator*(const MontgomeryModInt& lhs,
const MontgomeryModInt& rhs) {
return MontgomeryModInt(lhs) *= rhs;
}
friend MontgomeryModInt operator/(const MontgomeryModInt& lhs,
const MontgomeryModInt& rhs) {
return MontgomeryModInt(lhs) /= rhs;
}
friend bool operator==(const MontgomeryModInt& lhs,
const MontgomeryModInt& rhs) {
return lhs.val == rhs.val;
}
friend bool operator!=(const MontgomeryModInt& lhs,
const MontgomeryModInt& rhs) {
return lhs.val != rhs.val;
}
template<class Pr> void print(Pr& a) const { a.print(mont.reduce(val)); }
template<class Pr> void debug(Pr& a) const { a.print(mont.reduce(val)); }
template<class Sc> void scan(Sc& a) {
ll v;
a.scan(v);
*this = v;
}
};
template<class T, int id>
MontgomeryReduction<T>
MontgomeryModInt<T, id>::mont = MontgomeryReduction<T>(998244353);
using mmodint = MontgomeryModInt<unsigned int, -1>;
/**
* @brief MontgomeryModInt(モンゴメリ乗算)
* @docs docs/math/MontgomeryModInt.md
*/
#line 5 "math/SqrtMod.hpp"
template<class T> ll sqrt_mod(ll a) {
const ll p = T::get_mod();
if (p == 2) return a;
if (a == 0) return 0;
if (T{a}.pow((p - 1) >> 1) != 1) return -1;
T b = 2;
while (T{b}.pow((p - 1) >> 1) == 1) ++b;
ll s = 0, t = p - 1;
while ((t & 1) == 0) t >>= 1, ++s;
T x = T{a}.pow((t + 1) >> 1);
T w = T{a}.pow(t);
T v = T{b}.pow(t);
while (w != 1) {
ll k = 0;
T y = w;
while (y != 1) {
y *= y;
++k;
}
T z = v;
rep (s - k - 1) z *= z;
x *= z;
w *= z * z;
}
return std::min<ll>(x.get(), p - x.get());
}
ll sqrt_mod(ll a, ll p) {
if (p == 2) return a;
using mint = MontgomeryModInt<unsigned int, 493174342>;
mint::set_mod(p);
return sqrt_mod<mint>(a);
}
/**
* @brief SqrtMod(平方剰余)
* @docs docs/math/SqrtMod.md
* @see https://37zigen.com/tonelli-shanks-algorithm/
*/
#line 7 "math/poly/FormalPowerSeries.hpp"
template<class T> class FormalPowerSeries : public std::vector<T> {
private:
using Base = std::vector<T>;
using Comb = Combinatorics<T>;
public:
using Base::Base;
FormalPowerSeries(const Base& v) : Base(v) {}
FormalPowerSeries(Base&& v) : Base(std::move(v)) {}
FormalPowerSeries& shrink() {
while (!this->empty() && this->back() == T{0}) this->pop_back();
return *this;
}
T eval(T x) const {
T res = 0;
rrep (i, this->size()) {
res *= x;
res += (*this)[i];
}
return res;
}
FormalPowerSeries prefix(int deg) const {
assert(0 <= deg);
if (deg < (int)this->size()) {
return FormalPowerSeries(this->begin(), this->begin() + deg);
}
FormalPowerSeries res(*this);
res.resize(deg);
return res;
}
FormalPowerSeries operator+() const { return *this; }
FormalPowerSeries operator-() const {
FormalPowerSeries res(this->size());
rep (i, this->size()) res[i] = -(*this)[i];
return res;
}
FormalPowerSeries& operator<<=(int n) {
this->insert(this->begin(), n, T{0});
return *this;
}
FormalPowerSeries& operator>>=(int n) {
this->erase(this->begin(),
this->begin() + std::min(n, (int)this->size()));
return *this;
}
friend FormalPowerSeries operator<<(const FormalPowerSeries& lhs, int rhs) {
return FormalPowerSeries(lhs) <<= rhs;
}
friend FormalPowerSeries operator>>(const FormalPowerSeries& lhs, int rhs) {
return FormalPowerSeries(lhs) >>= rhs;
}
FormalPowerSeries& operator+=(const FormalPowerSeries& rhs) {
if (this->size() < rhs.size()) this->resize(rhs.size());
rep (i, rhs.size()) (*this)[i] += rhs[i];
return *this;
}
FormalPowerSeries& operator-=(const FormalPowerSeries& rhs) {
if (this->size() < rhs.size()) this->resize(rhs.size());
rep (i, rhs.size()) (*this)[i] -= rhs[i];
return *this;
}
friend FormalPowerSeries operator+(const FormalPowerSeries& lhs,
const FormalPowerSeries& rhs) {
return FormalPowerSeries(lhs) += rhs;
}
friend FormalPowerSeries operator-(const FormalPowerSeries& lhs,
const FormalPowerSeries& rhs) {
return FormalPowerSeries(lhs) -= rhs;
}
friend FormalPowerSeries operator*(const FormalPowerSeries& lhs,
const FormalPowerSeries& rhs) {
return FormalPowerSeries(convolution(lhs, rhs));
}
FormalPowerSeries& operator*=(const FormalPowerSeries& rhs) {
return *this = *this * rhs;
}
FormalPowerSeries& operator*=(const T& rhs) {
rep (i, this->size()) (*this)[i] *= rhs;
return *this;
}
friend FormalPowerSeries operator*(const FormalPowerSeries& lhs,
const T& rhs) {
return FormalPowerSeries(lhs) *= rhs;
}
friend FormalPowerSeries operator*(const T& lhs,
const FormalPowerSeries& rhs) {
return FormalPowerSeries(rhs) *= lhs;
}
FormalPowerSeries& operator/=(const T& rhs) {
rep (i, this->size()) (*this)[i] /= rhs;
return *this;
}
friend FormalPowerSeries operator/(const FormalPowerSeries& lhs,
const T& rhs) {
return FormalPowerSeries(lhs) /= rhs;
}
FormalPowerSeries rev() const {
FormalPowerSeries res(*this);
std::reverse(all(res));
return res;
}
friend FormalPowerSeries div(FormalPowerSeries lhs, FormalPowerSeries rhs) {
lhs.shrink();
rhs.shrink();
if (lhs.size() < rhs.size()) {
return FormalPowerSeries{};
}
int n = lhs.size() - rhs.size() + 1;
if (rhs.size() <= 32) {
FormalPowerSeries res(n);
T iv = rhs.back().inv();
rrep (i, n) {
T d = lhs[i + rhs.size() - 1] * iv;
res[i] = d;
rep (j, rhs.size()) lhs[i + j] -= d * rhs[j];
}
return res;
}
return (lhs.rev().prefix(n) * rhs.rev().inv(n)).prefix(n).rev();
}
friend FormalPowerSeries operator%(FormalPowerSeries lhs,
FormalPowerSeries rhs) {
lhs.shrink();
rhs.shrink();
if (lhs.size() < rhs.size()) {
return lhs;
}
int n = lhs.size() - rhs.size() + 1;
if (rhs.size() <= 32) {
T iv = rhs.back().inv();
rrep (i, n) {
T d = lhs[i + rhs.size() - 1] * iv;
rep (j, rhs.size()) lhs[i + j] -= d * rhs[j];
}
return lhs.shrink();
}
return (lhs - div(lhs, rhs) * rhs).shrink();
}
friend std::pair<FormalPowerSeries, FormalPowerSeries>
divmod(FormalPowerSeries lhs, FormalPowerSeries rhs) {
lhs.shrink();
rhs.shrink();
if (lhs.size() < rhs.size()) {
return {FormalPowerSeries{}, lhs};
}
int n = lhs.size() - rhs.size() + 1;
if (rhs.size() <= 32) {
FormalPowerSeries res(n);
T iv = rhs.back().inv();
rrep (i, n) {
T d = lhs[i + rhs.size() - 1] * iv;
res[i] = d;
rep (j, rhs.size()) lhs[i + j] -= d * rhs[j];
}
return {res, lhs.shrink()};
}
FormalPowerSeries q = div(lhs, rhs);
return {q, (lhs - q * rhs).shrink()};
}
FormalPowerSeries& operator%=(const FormalPowerSeries& rhs) {
return *this = *this % rhs;
}
FormalPowerSeries diff() const {
if (this->empty()) return {};
FormalPowerSeries res(this->size() - 1);
rep (i, res.size()) res[i] = (*this)[i + 1] * (i + 1);
return res;
}
FormalPowerSeries integral() const {
FormalPowerSeries res(this->size() + 1);
res[0] = 0;
Comb::init(this->size());
rep (i, this->size()) res[i + 1] = (*this)[i] * Comb::inv(i + 1);
return res;
}
template<bool AlwaysTrue = true,
typename std::enable_if<
AlwaysTrue && is_ntt_friendly<T::get_mod()>::value>::type* =
nullptr>
FormalPowerSeries inv(int deg = -1) const {
assert(this->size() > 0 && (*this)[0] != 0);
if (deg == -1) deg = this->size();
FormalPowerSeries f(1, (*this)[0].inv());
for (int m = 1; m < deg; m <<= 1) {
FormalPowerSeries t = this->prefix(2 * m);
f.resize(2 * m);
FormalPowerSeries dft_f = f;
number_theoretic_transform(t);
number_theoretic_transform(dft_f);
rep (i, 2 * m) t[i] *= dft_f[i];
inverse_number_theoretic_transform(t);
std::fill(t.begin(), t.begin() + m, T{0});
number_theoretic_transform(t);
rep (i, 2 * m) dft_f[i] *= t[i];
inverse_number_theoretic_transform(dft_f);
rep (i, m, 2 * m) f[i] = -dft_f[i];
}
return f.prefix(deg);
}
template<bool AlwaysTrue = true,
typename std::enable_if<
AlwaysTrue && !is_ntt_friendly<T::get_mod()>::value>::type* =
nullptr>
FormalPowerSeries inv(int deg = -1) const {
assert(this->size() > 0 && (*this)[0] != 0);
if (deg == -1) deg = this->size();
FormalPowerSeries res(1, (*this)[0].inv());
for (int m = 1; m < deg; m <<= 1) {
res = res * 2 - (res * res * this->prefix(2 * m)).prefix(2 * m);
}
return res.prefix(deg);
}
FormalPowerSeries log(int deg = -1) const {
assert(this->size() > 0 && (*this)[0] == 1);
if (deg == -1) deg = this->size();
return (diff().prefix(deg - 1) * inv(deg - 1)).prefix(deg - 1).integral();
}
template<bool AlwaysTrue = true,
typename std::enable_if<
AlwaysTrue && is_ntt_friendly<T::get_mod()>::value>::type* =
nullptr>
FormalPowerSeries exp(int deg = -1) const {
assert(this->size() > 0 && (*this)[0] == 0);
if (deg == -1) deg = this->size();
FormalPowerSeries df = this->diff();
FormalPowerSeries f(1, 1);
FormalPowerSeries g(1, 1);
FormalPowerSeries dft_f = f;
number_theoretic_transform(dft_f);
for (int m = 1; m < deg; m <<= 1) {
dft_f.ntt_doubling(f);
f.resize(2 * m);
g.resize(2 * m);
FormalPowerSeries dft_g = g;
number_theoretic_transform(dft_g);
FormalPowerSeries t = df.prefix(2 * m);
number_theoretic_transform(t);
rep (i, 2 * m) t[i] *= dft_f[i];
inverse_number_theoretic_transform(t);
std::fill(t.begin(), t.begin() + m - 1, T{0});
number_theoretic_transform(t);
rep (i, 2 * m) t[i] *= dft_g[i];
inverse_number_theoretic_transform(t);
std::fill(t.begin(), t.begin() + m - 1, T{0});
t = t.prefix(2 * m - 1).integral();
number_theoretic_transform(t);
rep (i, 2 * m) t[i] *= dft_f[i];
inverse_number_theoretic_transform(t);
rep (i, m, 2 * m) f[i] = t[i];
if (2 * m < deg) {
dft_f = f;
number_theoretic_transform(dft_f);
FormalPowerSeries t = dft_f;
rep (i, 2 * m) t[i] *= dft_g[i];
inverse_number_theoretic_transform(t);
std::fill(t.begin(), t.begin() + m, T{0});
number_theoretic_transform(t);
rep (i, 2 * m) t[i] *= dft_g[i];
inverse_number_theoretic_transform(t);
rep (i, m, 2 * m) g[i] = -t[i];
}
}
return f.prefix(deg);
}
template<bool AlwaysTrue = true,
typename std::enable_if<
AlwaysTrue && !is_ntt_friendly<T::get_mod()>::value>::type* =
nullptr>
FormalPowerSeries exp(int deg = -1) const {
assert(this->size() > 0 && (*this)[0] == 0);
if (deg == -1) deg = this->size();
FormalPowerSeries res(1, 1);
for (int m = 1; m < deg; m <<= 1) {
res = (res * (prefix(2 * m) - res.log(2 * m)) + res).prefix(2 * m);
}
return res.prefix(deg);
}
FormalPowerSeries pow(ll k, int deg = -1) const {
if (deg == -1) deg = this->size();
if (deg == 0) return {};
if (k == 0) {
FormalPowerSeries res(deg);
res[0] = 1;
return res;
}
if (k == 1) return prefix(deg);
if (k == 2) return (*this * *this).prefix(deg);
T a;
int d = -1;
rep (i, this->size()) {
if ((*this)[i] != 0) {
a = (*this)[i];
d = i;
break;
}
}
if (d == -1) {
FormalPowerSeries res(deg);
return res;
}
if ((i128)d * k >= deg) {
FormalPowerSeries res(deg);
return res;
}
deg -= d * k;
FormalPowerSeries res = (((*this >> d) / a).log(deg) * k).exp(deg);
res *= a.pow(k);
res <<= d * k;
return res;
}
template<bool AlwaysTrue = true,
typename std::enable_if<
AlwaysTrue && is_ntt_friendly<T::get_mod()>::value>::type* =
nullptr>
FormalPowerSeries sqrt(int deg = -1) const {
if (deg == -1) deg = this->size();
T a;
int d = -1;
rep (i, this->size()) {
if ((*this)[i] != 0) {
a = (*this)[i];
d = i;
break;
}
}
if (d == -1) {
FormalPowerSeries res(deg);
return res;
}
if (d & 1) return {};
deg -= (d >> 1);
if (deg <= 0) {
FormalPowerSeries res(deg);
return res;
}
FormalPowerSeries t = (*this >> d);
T sq = sqrt_mod<T>(a.get());
if (sq == -1) return {};
FormalPowerSeries f(1, sq), g(1, 1 / sq), dft_f = f;
number_theoretic_transform(dft_f);
for (int m = 1; m < deg; m <<= 1) {
dft_f.ntt_doubling(f);
f.resize(2 * m);
g.resize(2 * m);
FormalPowerSeries dft_g = g;
number_theoretic_transform(dft_g);
FormalPowerSeries u = dft_f;
rep (i, 2 * m) u[i] *= dft_f[i];
FormalPowerSeries tx = t.prefix(2 * m);
number_theoretic_transform(tx);
rep (i, 2 * m) u[i] = (tx[i] - u[i]) * dft_g[i];
inverse_number_theoretic_transform(u);
rep (i, m, 2 * m) f[i] = u[i] / 2;
if (2 * m < deg) {
dft_f = f;
number_theoretic_transform(dft_f);
FormalPowerSeries u = dft_g;
rep (i, 2 * m) u[i] *= dft_f[i];
inverse_number_theoretic_transform(u);
std::fill(u.begin(), u.begin() + m, T{0});
number_theoretic_transform(u);
rep (i, 2 * m) u[i] *= dft_g[i];
inverse_number_theoretic_transform(u);
rep (i, m, 2 * m) g[i] = -u[i];
}
}
return f.prefix(deg) << (d >> 1);
}
template<bool AlwaysTrue = true,
typename std::enable_if<
AlwaysTrue && !is_ntt_friendly<T::get_mod()>::value>::type* =
nullptr>
FormalPowerSeries sqrt(int deg = -1) const {
if (deg == -1) deg = this->size();
T a;
int d = -1;
rep (i, this->size()) {
if ((*this)[i] != 0) {
a = (*this)[i];
d = i;
break;
}
}
if (d == -1) {
FormalPowerSeries res(deg);
return res;
}
if (d & 1) return {};
deg -= (d >> 1);
if (deg <= 0) {
FormalPowerSeries res(deg);
return res;
}
FormalPowerSeries t = (*this >> d);
T sq = sqrt_mod<T>(a.get());
if (sq == -1) return {};
FormalPowerSeries f(1, sq);
for (int m = 1; m < deg; m <<= 1) {
f = (f + t * f.inv(2 * m)).prefix(2 * m) / 2;
}
return f.prefix(deg) << (d >> 1);
}
FormalPowerSeries compose(FormalPowerSeries g, int deg = -1) const {
if (this->empty()) return {};
if (g.empty()) return {(*this)[0]};
assert(g[0] == 0);
int n = deg == -1 ? this->size() : deg;
int m = 1 << (bitop::ceil_log2(
std::max<int>(1, std::sqrt(n / std::log2(n)))) +
1);
FormalPowerSeries p = g.prefix(m), q = g >> m;
p.shrink();
q.shrink();
int l = (n + m - 1) / m;
std::vector<FormalPowerSeries> fs(this->size());
rep (i, this->size()) fs[i] = FormalPowerSeries{(*this)[i]};
FormalPowerSeries pd = p.diff();
int z = 0;
while (z < (int)pd.size() && pd[z] == T{0}) z++;
if (z == (int)pd.size()) {
FormalPowerSeries ans;
rrep (i, l) {
ans = ((ans * q) << m).prefix(n - i * m) +
FormalPowerSeries{(*this)[i]};
}
return ans;
}
pd = (pd >> z).inv(n);
FormalPowerSeries t = p;
for (int k = 1; fs.size() > 1; k <<= 1) {
std::vector<FormalPowerSeries> nfs((fs.size() + 1) / 2);
t.resize(1 << (bitop::ceil_log2(t.size()) + 1));
number_theoretic_transform(t);
rep (i, fs.size() / 2) {
nfs[i] = std::move(fs[2 * i]);
fs[2 * i + 1].resize(t.size());
number_theoretic_transform(fs[2 * i + 1]);
rep (j, t.size()) fs[2 * i + 1][j] *= t[j];
inverse_number_theoretic_transform(fs[2 * i + 1]);
if ((int)fs[2 * i + 1].size() > n) fs[2 * i + 1].resize(n);
nfs[i] += fs[2 * i + 1];
}
if (fs.size() & 1) nfs.back() = std::move(fs.back());
fs = std::move(nfs);
if (fs.size() > 1) {
rep (i, t.size()) t[i] *= t[i];
inverse_number_theoretic_transform(t);
if ((int)t.size() > n) t.resize(n);
}
}
FormalPowerSeries fp = fs[0].prefix(n);
FormalPowerSeries res = fp;
int n2 = 1 << (bitop::ceil_log2(n) + 1);
FormalPowerSeries qpow(n2);
qpow[0] = 1;
q.resize(n2);
number_theoretic_transform(q);
pd.resize(n2);
number_theoretic_transform(pd);
rep (i, 1, l) {
if ((n - i * m) * 4 <= n2) {
while ((n - i * m) * 4 <= n2) {
n2 /= 2;
}
inverse_number_theoretic_transform(q);
q.resize(n - i * m);
q.resize(n2);
number_theoretic_transform(q);
inverse_number_theoretic_transform(pd);
pd.resize(n - i * m);
pd.resize(n2);
number_theoretic_transform(pd);
}
qpow.resize(n - i * m);
qpow.resize(n2);
number_theoretic_transform(qpow);
rep (j, n2) qpow[j] *= q[j];
inverse_number_theoretic_transform(qpow);
qpow.resize(n - i * m);
fp = fp.diff() >> z;
fp.resize(n - i * m);
fp.resize(n2);
number_theoretic_transform(fp);
rep (j, n2) fp[j] *= pd[j];
inverse_number_theoretic_transform(fp);
fp.resize(n - i * m);
res += ((qpow * fp).prefix(n - i * m) * Comb::finv(i)) << (i * m);
}
return res;
}
FormalPowerSeries compinv(int deg = -1) const {
assert(this->size() >= 2 && (*this)[0] == 0 && (*this)[1] != 0);
if (deg == -1) deg = this->size();
FormalPowerSeries fd = diff();
FormalPowerSeries x{0, 1};
FormalPowerSeries res{0, (*this)[1].inv()};
for (int m = 2; m < deg; m <<= 1) {
auto tmp = prefix(2 * m).compose(res);
auto d = tmp.diff();
auto gd = res.diff();
res -=
((tmp - x) * (d.inv(2 * m) * gd).prefix(2 * m)).prefix(2 * m);
}
return res.prefix(deg);
}
template<bool AlwaysTrue = true,
typename std::enable_if<
AlwaysTrue && is_ntt_friendly<T::get_mod()>::value>::type* =
nullptr>
FormalPowerSeries& ntt_doubling() {
ntt_doubling_(*this);
return *this;
}
template<bool AlwaysTrue = true,
typename std::enable_if<
AlwaysTrue && is_ntt_friendly<T::get_mod()>::value>::type* =
nullptr>
FormalPowerSeries& ntt_doubling(const std::vector<T>& b) {
ntt_doubling_(*this, b);
return *this;
}
};
/**
* @brief FormalPowerSeries(形式的冪級数)
* @docs docs/math/poly/FormalPowerSeries.md
* @see https://nyaannyaan.github.io/library/fps/formal-power-series.hpp
*/
#line 5 "math/poly/MultipointEvaluation.hpp"
namespace internal {
template<class T> class ProductTree {
private:
int n;
std::vector<FormalPowerSeries<T>> dat;
public:
ProductTree(const std::vector<T>& xs) {
n = xs.size();
dat.resize(n << 1);
rep (i, n) dat[i + n] = FormalPowerSeries<T>{-xs[i], 1};
rrep (i, n, 1) dat[i] = dat[i << 1] * dat[i << 1 | 1];
}
const FormalPowerSeries<T>& operator[](int k) const& { return dat[k]; }
FormalPowerSeries<T> operator[](int k) && { return std::move(dat[k]); }
};
template<class T>
std::vector<T> multipoint_evaluation(const FormalPowerSeries<T>& a,
const std::vector<T>& b,
const ProductTree<T>& c) {
int m = b.size();
std::vector<FormalPowerSeries<T>> d(m << 1);
d[1] = a % c[1];
rep (i, 2, m << 1) d[i] = d[i >> 1] % c[i];
std::vector<T> e(m);
rep (i, m) e[i] = d[m + i].empty() ? T{0} : d[m + i][0];
return e;
}
} // namespace internal
template<class T>
std::vector<T> multipoint_evaluation(const FormalPowerSeries<T>& a,
const std::vector<T>& b) {
if (a.empty() || b.empty()) return std::vector<T>(b.size(), T{0});
if (a.size() <= 32 || b.size() <= 32) {
std::vector<T> res(b.size());
rep (i, b.size()) res[i] = a.eval(b[i]);
return res;
}
return internal::multipoint_evaluation(a, b, internal::ProductTree<T>(b));
}
template<class T>
std::vector<T> multipoint_evaluation_geometric(const FormalPowerSeries<T>& f,
T a, T r, int m) {
if (f.empty() || m == 0) return std::vector<T>(m, T{0});
if (a == 0 || r == 1) return std::vector<T>(m, f.eval(a));
if (f.size() <= 32 || m <= 32) {
std::vector<T> res(m);
rep (i, m) {
res[i] = f.eval(a);
a *= r;
}
return res;
}
if (r == 0) {
std::vector<T> res(m, f.eval(0));
res[0] = f.eval(a);
return res;
}
int n = f.size();
int l = 1 << bitop::ceil_log2(n + m - 1);
std::vector<T> p(l), q(l);
T ir = T{1} / r, t = 1, t2 = 1;
rep (i, n) {
p[n - i - 1] = f[i] * t;
t *= a * t2;
t2 *= ir;
}
t = t2 = 1;
rep (i, n + m - 1) {
q[i] = t;
t *= t2;
t2 *= r;
}
number_theoretic_transform(p);
number_theoretic_transform(q);
rep (i, l) p[i] *= q[i];
inverse_number_theoretic_transform(p);
std::vector<T> ans(p.begin() + (n - 1), p.begin() + (n + m - 1));
t = t2 = 1;
rep (i, m) {
ans[i] *= t;
t *= t2;
t2 *= ir;
}
return ans;
}
/**
* @brief MultipointEvaluation(多点評価)
* @docs docs/math/poly/MultipointEvaluation.md
*/