cp-library-cpp
This documentation is automatically generated by online-judge-tools/verification-helper
test/src/convolution/polynomial_eval_multipoint_eval/nim_counting.test.cpp
- View this file on GitHub
- Last update: 2023-09-15 20:02:25+09:00
- Problem: https://atcoder.jp/contests/abc212/tasks/abc212_h
Depends on
- 列を変数として持つ多項式の評価 (多点評価版)
(library/convolution/polynomial_eval_multipoint_eval.hpp)
- 逆元テーブル
(library/math/inv_mods.hpp)
- Modint Extension
(library/math/modint_extension.hpp)
- 形式的冪級数
(library/polynomial/fps.hpp)
- FFT-free な形式的べき級数
(library/polynomial/fps_naive.hpp)
- Multi Point Evaluation
(library/polynomial/multi_point_eval.hpp)
- クロネッカー冪による線形変換 (仮称)
(library/transform/kronecker_power.hpp)
- Walsh Hadamard 変換
(library/transform/walsh_hadamard.hpp)
- Type Traits
(library/type_traits/type_traits.hpp)
- Default Operator
(library/util/default_operator.hpp)
Code
#define PROBLEM "https://atcoder.jp/contests/abc212/tasks/abc212_h"
#include <iostream>
#include <atcoder/convolution>
#include <atcoder/modint>
#include "library/polynomial/fps.hpp"
#include "library/transform/walsh_hadamard.hpp"
#include "library/convolution/polynomial_eval_multipoint_eval.hpp"
using namespace suisen;
using mint = atcoder::modint998244353;
constexpr int M = 1 << 16;
int main() {
FPS<mint>::set_multiplication([](const auto& f, const auto& g) { return atcoder::convolution(f, g); });
std::ios::sync_with_stdio(false);
std::cin.tie(nullptr);
int n, k;
std::cin >> n >> k;
std::vector<mint> c(M, 0);
for (int i = 0; i < k; ++i) {
int v;
std::cin >> v;
++c[v];
}
FPS<mint> f(n + 1, 1);
f[0] = 0;
using namespace walsh_hadamard_transform;
auto res = polynomial_eval<mint, walsh_hadamard<mint>, walsh_hadamard_inv<mint>>(c, f);
std::cout << std::accumulate(res.begin() + 1, res.end(), mint(0)).val() << std::endl;
return 0;
}#line 1 "test/src/convolution/polynomial_eval_multipoint_eval/nim_counting.test.cpp"
#define PROBLEM "https://atcoder.jp/contests/abc212/tasks/abc212_h"
#include <iostream>
#include <atcoder/convolution>
#include <atcoder/modint>
#line 1 "library/polynomial/fps.hpp"
#include <algorithm>
#include <cassert>
#line 7 "library/polynomial/fps.hpp"
#include <queue>
#line 1 "library/polynomial/fps_naive.hpp"
#line 5 "library/polynomial/fps_naive.hpp"
#include <cmath>
#include <limits>
#include <type_traits>
#include <vector>
#line 1 "library/type_traits/type_traits.hpp"
#line 7 "library/type_traits/type_traits.hpp"
namespace suisen {
template <typename ...Constraints> using constraints_t = std::enable_if_t<std::conjunction_v<Constraints...>, std::nullptr_t>;
template <typename T, typename = std::nullptr_t> struct bitnum { static constexpr int value = 0; };
template <typename T> struct bitnum<T, constraints_t<std::is_integral<T>>> { static constexpr int value = std::numeric_limits<std::make_unsigned_t<T>>::digits; };
template <typename T> static constexpr int bitnum_v = bitnum<T>::value;
template <typename T, size_t n> struct is_nbit { static constexpr bool value = bitnum_v<T> == n; };
template <typename T, size_t n> static constexpr bool is_nbit_v = is_nbit<T, n>::value;
template <typename T, typename = std::nullptr_t> struct safely_multipliable { using type = T; };
template <typename T> struct safely_multipliable<T, constraints_t<std::is_signed<T>, is_nbit<T, 32>>> { using type = long long; };
template <typename T> struct safely_multipliable<T, constraints_t<std::is_signed<T>, is_nbit<T, 64>>> { using type = __int128_t; };
template <typename T> struct safely_multipliable<T, constraints_t<std::is_unsigned<T>, is_nbit<T, 32>>> { using type = unsigned long long; };
template <typename T> struct safely_multipliable<T, constraints_t<std::is_unsigned<T>, is_nbit<T, 64>>> { using type = __uint128_t; };
template <typename T> using safely_multipliable_t = typename safely_multipliable<T>::type;
template <typename T, typename = void> struct rec_value_type { using type = T; };
template <typename T> struct rec_value_type<T, std::void_t<typename T::value_type>> {
using type = typename rec_value_type<typename T::value_type>::type;
};
template <typename T> using rec_value_type_t = typename rec_value_type<T>::type;
template <typename T> class is_iterable {
template <typename T_> static auto test(T_ e) -> decltype(e.begin(), e.end(), std::true_type{});
static std::false_type test(...);
public:
static constexpr bool value = decltype(test(std::declval<T>()))::value;
};
template <typename T> static constexpr bool is_iterable_v = is_iterable<T>::value;
template <typename T> class is_writable {
template <typename T_> static auto test(T_ e) -> decltype(std::declval<std::ostream&>() << e, std::true_type{});
static std::false_type test(...);
public:
static constexpr bool value = decltype(test(std::declval<T>()))::value;
};
template <typename T> static constexpr bool is_writable_v = is_writable<T>::value;
template <typename T> class is_readable {
template <typename T_> static auto test(T_ e) -> decltype(std::declval<std::istream&>() >> e, std::true_type{});
static std::false_type test(...);
public:
static constexpr bool value = decltype(test(std::declval<T>()))::value;
};
template <typename T> static constexpr bool is_readable_v = is_readable<T>::value;
} // namespace suisen
#line 11 "library/polynomial/fps_naive.hpp"
#line 1 "library/math/modint_extension.hpp"
#line 5 "library/math/modint_extension.hpp"
#include <optional>
/**
* refernce: https://37zigen.com/tonelli-shanks-algorithm/
* calculates x s.t. x^2 = a mod p in O((log p)^2).
*/
template <typename mint>
std::optional<mint> safe_sqrt(mint a) {
static int p = mint::mod();
if (a == 0) return std::make_optional(0);
if (p == 2) return std::make_optional(a);
if (a.pow((p - 1) / 2) != 1) return std::nullopt;
mint b = 1;
while (b.pow((p - 1) / 2) == 1) ++b;
static int tlz = __builtin_ctz(p - 1), q = (p - 1) >> tlz;
mint x = a.pow((q + 1) / 2);
b = b.pow(q);
for (int shift = 2; x * x != a; ++shift) {
mint e = a.inv() * x * x;
if (e.pow(1 << (tlz - shift)) != 1) x *= b;
b *= b;
}
return std::make_optional(x);
}
/**
* calculates x s.t. x^2 = a mod p in O((log p)^2).
* if not exists, raises runtime error.
*/
template <typename mint>
auto sqrt(mint a) -> decltype(mint::mod(), mint()) {
return *safe_sqrt(a);
}
template <typename mint>
auto log(mint a) -> decltype(mint::mod(), mint()) {
assert(a == 1);
return 0;
}
template <typename mint>
auto exp(mint a) -> decltype(mint::mod(), mint()) {
assert(a == 0);
return 1;
}
template <typename mint, typename T>
auto pow(mint a, T b) -> decltype(mint::mod(), mint()) {
return a.pow(b);
}
template <typename mint>
auto inv(mint a) -> decltype(mint::mod(), mint()) {
return a.inv();
}
#line 1 "library/math/inv_mods.hpp"
#line 5 "library/math/inv_mods.hpp"
namespace suisen {
template <typename mint>
class inv_mods {
public:
inv_mods() = default;
inv_mods(int n) { ensure(n); }
const mint& operator[](int i) const {
ensure(i);
return invs[i];
}
static void ensure(int n) {
int sz = invs.size();
if (sz < 2) invs = { 0, 1 }, sz = 2;
if (sz < n + 1) {
invs.resize(n + 1);
for (int i = sz; i <= n; ++i) invs[i] = mint(mod - mod / i) * invs[mod % i];
}
}
private:
static std::vector<mint> invs;
static constexpr int mod = mint::mod();
};
template <typename mint>
std::vector<mint> inv_mods<mint>::invs{};
template <typename mint>
std::vector<mint> get_invs(const std::vector<mint>& vs) {
const int n = vs.size();
mint p = 1;
for (auto& e : vs) {
p *= e;
assert(e != 0);
}
mint ip = p.inv();
std::vector<mint> rp(n + 1);
rp[n] = 1;
for (int i = n - 1; i >= 0; --i) {
rp[i] = rp[i + 1] * vs[i];
}
std::vector<mint> res(n);
for (int i = 0; i < n; ++i) {
res[i] = ip * rp[i + 1];
ip *= vs[i];
}
return res;
}
}
#line 14 "library/polynomial/fps_naive.hpp"
namespace suisen {
template <typename T>
struct FPSNaive : std::vector<T> {
static inline int MAX_SIZE = std::numeric_limits<int>::max() / 2;
using value_type = T;
using element_type = rec_value_type_t<T>;
using std::vector<value_type>::vector;
FPSNaive(const std::initializer_list<value_type> l) : std::vector<value_type>::vector(l) {}
FPSNaive(const std::vector<value_type>& v) : std::vector<value_type>::vector(v) {}
static void set_max_size(int n) {
FPSNaive<T>::MAX_SIZE = n;
}
const value_type operator[](int n) const {
return n <= deg() ? unsafe_get(n) : value_type{ 0 };
}
value_type& operator[](int n) {
return ensure_deg(n), unsafe_get(n);
}
int size() const {
return std::vector<value_type>::size();
}
int deg() const {
return size() - 1;
}
int normalize() {
while (size() and this->back() == value_type{ 0 }) this->pop_back();
return deg();
}
FPSNaive& cut_inplace(int n) {
if (size() > n) this->resize(std::max(0, n));
return *this;
}
FPSNaive cut(int n) const {
FPSNaive f = FPSNaive(*this).cut_inplace(n);
return f;
}
FPSNaive operator+() const {
return FPSNaive(*this);
}
FPSNaive operator-() const {
FPSNaive f(*this);
for (auto& e : f) e = -e;
return f;
}
FPSNaive& operator++() { return ++(*this)[0], * this; }
FPSNaive& operator--() { return --(*this)[0], * this; }
FPSNaive& operator+=(const value_type x) { return (*this)[0] += x, *this; }
FPSNaive& operator-=(const value_type x) { return (*this)[0] -= x, *this; }
FPSNaive& operator+=(const FPSNaive& g) {
ensure_deg(g.deg());
for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) += g.unsafe_get(i);
return *this;
}
FPSNaive& operator-=(const FPSNaive& g) {
ensure_deg(g.deg());
for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) -= g.unsafe_get(i);
return *this;
}
FPSNaive& operator*=(const FPSNaive& g) { return *this = *this * g; }
FPSNaive& operator*=(const value_type x) {
for (auto& e : *this) e *= x;
return *this;
}
FPSNaive& operator/=(const FPSNaive& g) { return *this = *this / g; }
FPSNaive& operator%=(const FPSNaive& g) { return *this = *this % g; }
FPSNaive& operator<<=(const int shamt) {
this->insert(this->begin(), shamt, value_type{ 0 });
return *this;
}
FPSNaive& operator>>=(const int shamt) {
if (shamt > size()) this->clear();
else this->erase(this->begin(), this->begin() + shamt);
return *this;
}
friend FPSNaive operator+(FPSNaive f, const FPSNaive& g) { f += g; return f; }
friend FPSNaive operator+(FPSNaive f, const value_type& x) { f += x; return f; }
friend FPSNaive operator-(FPSNaive f, const FPSNaive& g) { f -= g; return f; }
friend FPSNaive operator-(FPSNaive f, const value_type& x) { f -= x; return f; }
friend FPSNaive operator*(const FPSNaive& f, const FPSNaive& g) {
if (f.empty() or g.empty()) return FPSNaive{};
const int n = f.size(), m = g.size();
FPSNaive h(std::min(MAX_SIZE, n + m - 1));
for (int i = 0; i < n; ++i) for (int j = 0; j < m; ++j) {
if (i + j >= MAX_SIZE) break;
h.unsafe_get(i + j) += f.unsafe_get(i) * g.unsafe_get(j);
}
return h;
}
friend FPSNaive operator*(FPSNaive f, const value_type& x) { f *= x; return f; }
friend FPSNaive operator/(FPSNaive f, const FPSNaive& g) { return std::move(f.div_mod(g).first); }
friend FPSNaive operator%(FPSNaive f, const FPSNaive& g) { return std::move(f.div_mod(g).second); }
friend FPSNaive operator*(const value_type x, FPSNaive f) { f *= x; return f; }
friend FPSNaive operator<<(FPSNaive f, const int shamt) { f <<= shamt; return f; }
friend FPSNaive operator>>(FPSNaive f, const int shamt) { f >>= shamt; return f; }
std::pair<FPSNaive, FPSNaive> div_mod(FPSNaive g) const {
FPSNaive f = *this;
const int fd = f.normalize(), gd = g.normalize();
assert(gd >= 0);
if (fd < gd) return { FPSNaive{}, f };
if (gd == 0) return { f *= g.unsafe_get(0).inv(), FPSNaive{} };
const int k = f.deg() - gd;
value_type head_inv = g.unsafe_get(gd).inv();
FPSNaive q(k + 1);
for (int i = k; i >= 0; --i) {
value_type div = f.unsafe_get(i + gd) * head_inv;
q.unsafe_get(i) = div;
for (int j = 0; j <= gd; ++j) f.unsafe_get(i + j) -= div * g.unsafe_get(j);
}
f.cut_inplace(gd);
f.normalize();
return { q, f };
}
friend bool operator==(const FPSNaive& f, const FPSNaive& g) {
const int n = f.size(), m = g.size();
if (n < m) return g == f;
for (int i = 0; i < m; ++i) if (f.unsafe_get(i) != g.unsafe_get(i)) return false;
for (int i = m; i < n; ++i) if (f.unsafe_get(i) != 0) return false;
return true;
}
friend bool operator!=(const FPSNaive& f, const FPSNaive& g) {
return not (f == g);
}
FPSNaive mul(const FPSNaive& g, int n = -1) const {
if (n < 0) n = size();
if (this->empty() or g.empty()) return FPSNaive{};
const int m = size(), k = g.size();
FPSNaive h(std::min(n, m + k - 1));
for (int i = 0; i < m; ++i) {
for (int j = 0, jr = std::min(k, n - i); j < jr; ++j) {
h.unsafe_get(i + j) += unsafe_get(i) * g.unsafe_get(j);
}
}
return h;
}
FPSNaive diff() const {
if (this->empty()) return {};
FPSNaive g(size() - 1);
for (int i = 1; i <= deg(); ++i) g.unsafe_get(i - 1) = unsafe_get(i) * i;
return g;
}
FPSNaive intg() const {
const int n = size();
FPSNaive g(n + 1);
for (int i = 0; i < n; ++i) g.unsafe_get(i + 1) = unsafe_get(i) * invs[i + 1];
if (g.deg() > MAX_SIZE) g.cut_inplace(MAX_SIZE);
return g;
}
FPSNaive inv(int n = -1) const {
if (n < 0) n = size();
FPSNaive g(n);
const value_type inv_f0 = ::inv(unsafe_get(0));
g.unsafe_get(0) = inv_f0;
for (int i = 1; i < n; ++i) {
for (int j = 1; j <= i; ++j) g.unsafe_get(i) -= g.unsafe_get(i - j) * (*this)[j];
g.unsafe_get(i) *= inv_f0;
}
return g;
}
FPSNaive exp(int n = -1) const {
if (n < 0) n = size();
assert(unsafe_get(0) == value_type{ 0 });
FPSNaive g(n);
g.unsafe_get(0) = value_type{ 1 };
for (int i = 1; i < n; ++i) {
for (int j = 1; j <= i; ++j) g.unsafe_get(i) += j * g.unsafe_get(i - j) * (*this)[j];
g.unsafe_get(i) *= invs[i];
}
return g;
}
FPSNaive log(int n = -1) const {
if (n < 0) n = size();
assert(unsafe_get(0) == value_type{ 1 });
FPSNaive g(n);
g.unsafe_get(0) = value_type{ 0 };
for (int i = 1; i < n; ++i) {
g.unsafe_get(i) = i * (*this)[i];
for (int j = 1; j < i; ++j) g.unsafe_get(i) -= (i - j) * g.unsafe_get(i - j) * (*this)[j];
g.unsafe_get(i) *= invs[i];
}
return g;
}
FPSNaive pow(const long long k, int n = -1) const {
if (n < 0) n = size();
if (k == 0) {
FPSNaive res(n);
res[0] = 1;
return res;
}
int z = 0;
while (z < size() and unsafe_get(z) == value_type{ 0 }) ++z;
if (z == size() or z > (n - 1) / k) return FPSNaive(n, 0);
const int m = n - z * k;
FPSNaive g(m);
const value_type inv_f0 = ::inv(unsafe_get(z));
g.unsafe_get(0) = unsafe_get(z).pow(k);
for (int i = 1; i < m; ++i) {
for (int j = 1; j <= i; ++j) g.unsafe_get(i) += (element_type{ k } *j - (i - j)) * g.unsafe_get(i - j) * (*this)[z + j];
g.unsafe_get(i) *= inv_f0 * invs[i];
}
g <<= z * k;
return g;
}
std::optional<FPSNaive> safe_sqrt(int n = -1) const {
if (n < 0) n = size();
int dl = 0;
while (dl < size() and unsafe_get(dl) == value_type{ 0 }) ++dl;
if (dl == size()) return FPSNaive(n, 0);
if (dl & 1) return std::nullopt;
const int m = n - dl / 2;
FPSNaive g(m);
auto opt_g0 = ::safe_sqrt((*this)[dl]);
if (not opt_g0.has_value()) return std::nullopt;
g.unsafe_get(0) = *opt_g0;
value_type inv_2g0 = ::inv(2 * g.unsafe_get(0));
for (int i = 1; i < m; ++i) {
g.unsafe_get(i) = (*this)[dl + i];
for (int j = 1; j < i; ++j) g.unsafe_get(i) -= g.unsafe_get(j) * g.unsafe_get(i - j);
g.unsafe_get(i) *= inv_2g0;
}
g <<= dl / 2;
return g;
}
FPSNaive sqrt(int n = -1) const {
if (n < 0) n = size();
return *safe_sqrt(n);
}
value_type eval(value_type x) const {
value_type y = 0;
for (int i = size() - 1; i >= 0; --i) y = y * x + unsafe_get(i);
return y;
}
private:
static inline inv_mods<element_type> invs;
void ensure_deg(int d) {
if (deg() < d) this->resize(d + 1, value_type{ 0 });
}
const value_type& unsafe_get(int i) const {
return std::vector<value_type>::operator[](i);
}
value_type& unsafe_get(int i) {
return std::vector<value_type>::operator[](i);
}
};
} // namespace suisen
template <typename mint>
suisen::FPSNaive<mint> sqrt(suisen::FPSNaive<mint> a) {
return a.sqrt();
}
template <typename mint>
suisen::FPSNaive<mint> log(suisen::FPSNaive<mint> a) {
return a.log();
}
template <typename mint>
suisen::FPSNaive<mint> exp(suisen::FPSNaive<mint> a) {
return a.exp();
}
template <typename mint, typename T>
suisen::FPSNaive<mint> pow(suisen::FPSNaive<mint> a, T b) {
return a.pow(b);
}
template <typename mint>
suisen::FPSNaive<mint> inv(suisen::FPSNaive<mint> a) {
return a.inv();
}
#line 12 "library/polynomial/fps.hpp"
namespace suisen {
template <typename mint>
using convolution_t = std::vector<mint>(*)(const std::vector<mint>&, const std::vector<mint>&);
template <typename mint>
struct FPS : public std::vector<mint> {
using base_type = std::vector<mint>;
using value_type = typename base_type::value_type;
using base_type::vector;
FPS(const std::initializer_list<mint> l) : std::vector<mint>::vector(l) {}
FPS(const std::vector<mint>& v) : std::vector<mint>::vector(v) {}
FPS(std::vector<mint>&& v) : std::vector<mint>::vector(std::move(v)) {}
static void set_multiplication(convolution_t<mint> multiplication) {
FPS<mint>::mult = multiplication;
}
int size() const noexcept {
return base_type::size();
}
int deg() const noexcept {
return size() - 1;
}
void ensure(int n) {
if (size() < n) this->resize(n);
}
value_type safe_get(int d) const {
return d <= deg() ? (*this)[d] : 0;
}
value_type& safe_get(int d) {
ensure(d + 1);
return (*this)[d];
}
FPS& cut_trailing_zeros() {
while (this->size() and this->back() == 0) this->pop_back();
return *this;
}
FPS& cut(int n) {
if (size() > n) this->resize(std::max(0, n));
return *this;
}
FPS cut_copy(int n) const {
FPS res(this->begin(), this->begin() + std::min(size(), n));
res.ensure(n);
return res;
}
FPS cut_copy(int l, int r) const {
if (l >= size()) return FPS(r - l, 0);
FPS res(this->begin() + l, this->begin() + std::min(size(), r));
res.ensure(r - l);
return res;
}
/* Unary Operations */
FPS operator+() const { return *this; }
FPS operator-() const {
FPS res = *this;
for (auto& e : res) e = -e;
return res;
}
FPS& operator++() { return ++safe_get(0), * this; }
FPS& operator--() { return --safe_get(0), * this; }
FPS operator++(int) {
FPS res = *this;
++(*this);
return res;
}
FPS operator--(int) {
FPS res = *this;
--(*this);
return res;
}
/* Binary Operations With Constant */
FPS& operator+=(const value_type& x) { return safe_get(0) += x, *this; }
FPS& operator-=(const value_type& x) { return safe_get(0) -= x, *this; }
FPS& operator*=(const value_type& x) {
for (auto& e : *this) e *= x;
return *this;
}
FPS& operator/=(const value_type& x) { return *this *= x.inv(); }
friend FPS operator+(FPS f, const value_type& x) { f += x; return f; }
friend FPS operator+(const value_type& x, FPS f) { f += x; return f; }
friend FPS operator-(FPS f, const value_type& x) { f -= x; return f; }
friend FPS operator-(const value_type& x, FPS f) { f -= x; return -f; }
friend FPS operator*(FPS f, const value_type& x) { f *= x; return f; }
friend FPS operator*(const value_type& x, FPS f) { f *= x; return f; }
friend FPS operator/(FPS f, const value_type& x) { f /= x; return f; }
/* Binary Operations With Formal Power Series */
FPS& operator+=(const FPS& g) {
const int n = g.size();
ensure(n);
for (int i = 0; i < n; ++i) (*this)[i] += g[i];
return *this;
}
FPS& operator-=(const FPS& g) {
const int n = g.size();
ensure(n);
for (int i = 0; i < n; ++i) (*this)[i] -= g[i];
return *this;
}
FPS& operator*=(const FPS& g) { return *this = *this * g; }
FPS& operator/=(const FPS& g) { return *this = *this / g; }
FPS& operator%=(const FPS& g) { return *this = *this % g; }
friend FPS operator+(FPS f, const FPS& g) { f += g; return f; }
friend FPS operator-(FPS f, const FPS& g) { f -= g; return f; }
friend FPS operator*(const FPS& f, const FPS& g) { return mult(f, g); }
friend FPS operator/(FPS f, FPS g) {
if (f.size() < 60) return FPSNaive<mint>(f).div_mod(g).first;
f.cut_trailing_zeros(), g.cut_trailing_zeros();
const int fd = f.deg(), gd = g.deg();
assert(gd >= 0);
if (fd < gd) return {};
if (gd == 0) {
f /= g[0];
return f;
}
std::reverse(f.begin(), f.end()), std::reverse(g.begin(), g.end());
const int qd = fd - gd;
FPS q = f * g.inv(qd + 1);
q.cut(qd + 1);
std::reverse(q.begin(), q.end());
return q;
}
friend FPS operator%(const FPS& f, const FPS& g) { return f.div_mod(g).second; }
std::pair<FPS, FPS> div_mod(const FPS& g) const {
if (size() < 60) {
auto [q, r] = FPSNaive<mint>(*this).div_mod(g);
return { q, r };
}
FPS q = *this / g, r = *this - g * q;
r.cut_trailing_zeros();
return { q, r };
}
/* Shift Operations */
FPS& operator<<=(const int shamt) {
return this->insert(this->begin(), shamt, 0), * this;
}
FPS& operator>>=(const int shamt) {
return this->erase(this->begin(), this->begin() + std::min(shamt, size())), * this;
}
friend FPS operator<<(FPS f, const int shamt) { f <<= shamt; return f; }
friend FPS operator>>(FPS f, const int shamt) { f >>= shamt; return f; }
/* Compare */
friend bool operator==(const FPS& f, const FPS& g) {
const int n = f.size(), m = g.size();
if (n < m) return g == f;
for (int i = 0; i < m; ++i) if (f[i] != g[i]) return false;
for (int i = m; i < n; ++i) if (f[i] != 0) return false;
return true;
}
friend bool operator!=(const FPS& f, const FPS& g) { return not (f == g); }
/* Other Operations */
FPS& diff_inplace() {
const int n = size();
for (int i = 1; i < n; ++i) (*this)[i - 1] = (*this)[i] * i;
return (*this)[n - 1] = 0, *this;
}
FPS diff() const {
FPS res = *this;
res.diff_inplace();
return res;
}
FPS& intg_inplace() {
const int n = size();
inv_mods<value_type> invs(n);
this->resize(n + 1);
for (int i = n; i > 0; --i) (*this)[i] = (*this)[i - 1] * invs[i];
return (*this)[0] = 0, *this;
}
FPS intg() const {
FPS res = *this;
res.intg_inplace();
return res;
}
FPS& inv_inplace(const int n = -1) { return *this = inv(n); }
FPS inv(int n = -1) const {
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(*this).inv(n);
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return inv_sparse(std::move(*sp_f), n);
FPS g{ (*this)[0].inv() };
for (int k = 1; k < n; k *= 2) {
FPS f_lo = cut_copy(k), f_hi = cut_copy(k, 2 * k);
FPS h = (f_hi * g).cut(k) + ((f_lo * g) >>= k);
FPS g_hi = g * h;
g.resize(2 * k);
for (int i = 0; i < k; ++i) g[k + i] = -g_hi[i];
}
g.resize(n);
return g;
}
FPS& log_inplace(int n = -1) { return *this = log(n); }
FPS log(int n = -1) const {
assert(safe_get(0) == 1);
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).log(n);
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return log_sparse(std::move(*sp_f), n);
FPS res = inv(n) * diff();
res.resize(n - 1);
return res.intg();
}
FPS& exp_inplace(int n = -1) { return *this = exp(n); }
FPS exp(int n = -1) {
assert(safe_get(0) == 0);
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).exp(n);
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return exp_sparse(std::move(*sp_f), n);
FPS res{ 1 };
for (int k = 1; k < n; k *= 2) res *= ++(cut_copy(k * 2) - res.log(k * 2)), res.cut(k * 2);
res.resize(n);
return res;
}
FPS& pow_inplace(long long k, int n = -1) { return *this = pow(k, n); }
FPS pow(const long long k, int n = -1) const {
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).pow(k, n);
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return pow_sparse(std::move(*sp_f), k, n);
if (k == 0) {
FPS f{ 1 };
f.resize(n);
return f;
}
int tlz = 0;
while (tlz < size() and (*this)[tlz] == 0) ++tlz;
if (tlz == size() or tlz > (n - 1) / k) return FPS(n, 0);
const int m = n - tlz * k;
FPS f = *this >> tlz;
value_type base = f[0];
return ((((f /= base).log(m) *= k).exp(m) *= base.pow(k)) <<= (tlz * k));
}
std::optional<FPS> safe_sqrt(int n = -1) const {
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).safe_sqrt(n);
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return safe_sqrt_sparse(std::move(*sp_f), n);
int tlz = 0;
while (tlz < size() and (*this)[tlz] == 0) ++tlz;
if (tlz == size()) return FPS(n, 0);
if (tlz & 1) return std::nullopt;
const int m = n - tlz / 2;
FPS h(this->begin() + tlz, this->end());
auto q0 = ::safe_sqrt(h[0]);
if (not q0.has_value()) return std::nullopt;
FPS f{ *q0 }, g{ q0->inv() };
mint inv_2 = mint(2).inv();
for (int k = 1; k < m; k *= 2) {
FPS tmp = h.cut_copy(2 * k) * f.inv(2 * k);
tmp.cut(2 * k);
f += tmp, f *= inv_2;
}
f.resize(m);
f <<= tlz / 2;
return f;
}
FPS& sqrt_inplace(int n = -1) { return *this = sqrt(n); }
FPS sqrt(int n = -1) const {
return *safe_sqrt(n);
}
mint eval(mint x) const {
mint y = 0;
for (int i = size() - 1; i >= 0; --i) y = y * x + (*this)[i];
return y;
}
static FPS prod(const std::vector<FPS>& fs) {
auto comp = [](const FPS& f, const FPS& g) { return f.size() > g.size(); };
std::priority_queue<FPS, std::vector<FPS>, decltype(comp)> pq{ comp };
for (const auto& f : fs) pq.push(f);
while (pq.size() > 1) {
auto f = pq.top();
pq.pop();
auto g = pq.top();
pq.pop();
pq.push(f * g);
}
return pq.top();
}
std::optional<std::vector<std::pair<int, value_type>>> sparse_fps_format(int max_size) const {
std::vector<std::pair<int, value_type>> res;
for (int i = 0; i <= deg() and int(res.size()) <= max_size; ++i) if (value_type v = (*this)[i]; v != 0) res.emplace_back(i, v);
if (int(res.size()) > max_size) return std::nullopt;
return res;
}
protected:
static convolution_t<mint> mult;
static FPS div_fps_sparse(const FPS& f, const std::vector<std::pair<int, value_type>>& g, int n) {
const int siz = g.size();
assert(siz and g[0].first == 0);
const value_type inv_g0 = g[0].second.inv();
FPS h(n);
for (int i = 0; i < n; ++i) {
value_type v = f.safe_get(i);
for (int idx = 1; idx < siz; ++idx) {
const auto& [j, gj] = g[idx];
if (j > i) break;
v -= gj * h[i - j];
}
h[i] = v * inv_g0;
}
return h;
}
static FPS inv_sparse(const std::vector<std::pair<int, value_type>>& g, const int n) {
return div_fps_sparse(FPS{ 1 }, g, n);
}
static FPS exp_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
const int siz = f.size();
assert(not siz or f[0].first != 0);
FPS g(n);
g[0] = 1;
inv_mods<value_type> invs(n);
for (int i = 1; i < n; ++i) {
value_type v = 0;
for (const auto& [j, fj] : f) {
if (j > i) break;
v += j * fj * g[i - j];
}
v *= invs[i];
g[i] = v;
}
return g;
}
static FPS log_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
const int siz = f.size();
assert(siz and f[0].first == 0 and f[0].second == 1);
FPS g(n);
for (int idx = 1; idx < siz; ++idx) {
const auto& [j, fj] = f[idx];
if (j >= n) break;
g[j] = j * fj;
}
inv_mods<value_type> invs(n);
for (int i = 1; i < n; ++i) {
value_type v = g[i];
for (int idx = 1; idx < siz; ++idx) {
const auto& [j, fj] = f[idx];
if (j > i) break;
v -= fj * g[i - j] * (i - j);
}
v *= invs[i];
g[i] = v;
}
return g;
}
static FPS pow_sparse(const std::vector<std::pair<int, value_type>>& f, const long long k, const int n) {
if (k == 0) {
FPS res(n, 0);
res[0] = 1;
return res;
}
const int siz = f.size();
if (not siz) return FPS(n, 0);
const int p = f[0].first;
if (p > (n - 1) / k) return FPS(n, 0);
const value_type inv_f0 = f[0].second.inv();
const int lz = p * k;
FPS g(n);
g[lz] = f[0].second.pow(k);
inv_mods<value_type> invs(n);
for (int i = 1; lz + i < n; ++i) {
value_type v = 0;
for (int idx = 1; idx < siz; ++idx) {
auto [j, fj] = f[idx];
j -= p;
if (j > i) break;
v += fj * g[lz + i - j] * (value_type(k) * j - (i - j));
}
v *= invs[i] * inv_f0;
g[lz + i] = v;
}
return g;
}
static std::optional<FPS> safe_sqrt_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
const int siz = f.size();
if (not siz) return FPS(n, 0);
const int p = f[0].first;
if (p % 2 == 1) return std::nullopt;
if (p / 2 >= n) return FPS(n, 0);
const value_type inv_f0 = f[0].second.inv();
const int lz = p / 2;
FPS g(n);
auto opt_g0 = ::safe_sqrt(f[0].second);
if (not opt_g0.has_value()) return std::nullopt;
g[lz] = *opt_g0;
value_type k = mint(2).inv();
inv_mods<value_type> invs(n);
for (int i = 1; lz + i < n; ++i) {
value_type v = 0;
for (int idx = 1; idx < siz; ++idx) {
auto [j, fj] = f[idx];
j -= p;
if (j > i) break;
v += fj * g[lz + i - j] * (k * j - (i - j));
}
v *= invs[i] * inv_f0;
g[lz + i] = v;
}
return g;
}
static FPS sqrt_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
return *safe_sqrt(f, n);
}
};
template <typename mint>
convolution_t<mint> FPS<mint>::mult = [](const auto&, const auto&) {
std::cerr << "convolution function is not available." << std::endl;
assert(false);
return std::vector<mint>{};
};
} // namespace suisen
template <typename mint>
suisen::FPS<mint> sqrt(suisen::FPS<mint> a) {
return a.sqrt();
}
template <typename mint>
suisen::FPS<mint> log(suisen::FPS<mint> a) {
return a.log();
}
template <typename mint>
suisen::FPS<mint> exp(suisen::FPS<mint> a) {
return a.exp();
}
template <typename mint, typename T>
suisen::FPS<mint> pow(suisen::FPS<mint> a, T b) {
return a.pow(b);
}
template <typename mint>
suisen::FPS<mint> inv(suisen::FPS<mint> a) {
return a.inv();
}
#line 1 "library/transform/walsh_hadamard.hpp"
#line 1 "library/transform/kronecker_power.hpp"
#line 5 "library/transform/kronecker_power.hpp"
#include <utility>
#line 7 "library/transform/kronecker_power.hpp"
#line 1 "library/util/default_operator.hpp"
namespace suisen {
namespace default_operator {
template <typename T>
auto zero() -> decltype(T { 0 }) { return T { 0 }; }
template <typename T>
auto one() -> decltype(T { 1 }) { return T { 1 }; }
template <typename T>
auto add(const T &x, const T &y) -> decltype(x + y) { return x + y; }
template <typename T>
auto sub(const T &x, const T &y) -> decltype(x - y) { return x - y; }
template <typename T>
auto mul(const T &x, const T &y) -> decltype(x * y) { return x * y; }
template <typename T>
auto div(const T &x, const T &y) -> decltype(x / y) { return x / y; }
template <typename T>
auto mod(const T &x, const T &y) -> decltype(x % y) { return x % y; }
template <typename T>
auto neg(const T &x) -> decltype(-x) { return -x; }
template <typename T>
auto inv(const T &x) -> decltype(one<T>() / x) { return one<T>() / x; }
} // default_operator
namespace default_operator_noref {
template <typename T>
auto zero() -> decltype(T { 0 }) { return T { 0 }; }
template <typename T>
auto one() -> decltype(T { 1 }) { return T { 1 }; }
template <typename T>
auto add(T x, T y) -> decltype(x + y) { return x + y; }
template <typename T>
auto sub(T x, T y) -> decltype(x - y) { return x - y; }
template <typename T>
auto mul(T x, T y) -> decltype(x * y) { return x * y; }
template <typename T>
auto div(T x, T y) -> decltype(x / y) { return x / y; }
template <typename T>
auto mod(T x, T y) -> decltype(x % y) { return x % y; }
template <typename T>
auto neg(T x) -> decltype(-x) { return -x; }
template <typename T>
auto inv(T x) -> decltype(one<T>() / x) { return one<T>() / x; }
} // default_operator
} // namespace suisen
#line 9 "library/transform/kronecker_power.hpp"
namespace suisen {
namespace kronecker_power_transform {
namespace internal {
template <typename UnitTransform, typename ReferenceGetter, std::size_t... Seq>
void unit_transform(UnitTransform transform, ReferenceGetter ref_getter, std::index_sequence<Seq...>) {
transform(ref_getter(Seq)...);
}
}
template <typename T, std::size_t D, auto unit_transform>
void kronecker_power_transform(std::vector<T> &x) {
const std::size_t n = x.size();
for (std::size_t block = 1; block < n; block *= D) {
for (std::size_t l = 0; l < n; l += D * block) {
for (std::size_t offset = l; offset < l + block; ++offset) {
const auto ref_getter = [&](std::size_t i) -> T& { return x[offset + i * block]; };
internal::unit_transform(unit_transform, ref_getter, std::make_index_sequence<D>());
}
}
}
}
template <typename T, typename UnitTransform>
void kronecker_power_transform(std::vector<T> &x, const std::size_t D, UnitTransform unit_transform) {
const std::size_t n = x.size();
std::vector<T> work(D);
for (std::size_t block = 1; block < n; block *= D) {
for (std::size_t l = 0; l < n; l += D * block) {
for (std::size_t offset = l; offset < l + block; ++offset) {
for (std::size_t i = 0; i < D; ++i) work[i] = x[offset + i * block];
unit_transform(work);
for (std::size_t i = 0; i < D; ++i) x[offset + i * block] = work[i];
}
}
}
}
template <typename T, auto e = default_operator::zero<T>, auto add = default_operator::add<T>, auto mul = default_operator::mul<T>>
auto kronecker_power_transform(std::vector<T> &x, const std::vector<std::vector<T>> &A) -> decltype(e(), add(std::declval<T>(), std::declval<T>()), mul(std::declval<T>(), std::declval<T>()), void()) {
const std::size_t D = A.size();
assert(D == A[0].size());
auto unit_transform = [&](std::vector<T> &x) {
std::vector<T> y(D, e());
for (std::size_t i = 0; i < D; ++i) for (std::size_t j = 0; j < D; ++j) {
y[i] = add(y[i], mul(A[i][j], x[j]));
}
x.swap(y);
};
kronecker_power_transform<T>(x, D, unit_transform);
}
}
} // namespace suisen
#line 5 "library/transform/walsh_hadamard.hpp"
namespace suisen::walsh_hadamard_transform {
namespace internal {
template <typename T, auto add = default_operator::add<T>, auto sub = default_operator::sub<T>>
void unit_transform(T& x0, T& x1) {
T y0 = x0, y1 = x1;
x0 = add(y0, y1); // 1, 1
x1 = sub(y0, y1); // 1, -1
}
} // namespace internal
using kronecker_power_transform::kronecker_power_transform;
template <typename T, auto add = default_operator::add<T>, auto sub = default_operator::sub<T>>
void walsh_hadamard(std::vector<T>& a) {
kronecker_power_transform<T, 2, internal::unit_transform<T, add, sub>>(a);
}
template <typename T, auto add = default_operator::add<T>, auto sub = default_operator::sub<T>, auto div = default_operator::div<T>, std::enable_if_t<std::is_integral_v<T>, std::nullptr_t> = nullptr>
void walsh_hadamard_inv(std::vector<T>& a) {
walsh_hadamard<T, add, sub>(a);
const T n{ a.size() };
for (auto& val : a) val = div(val, n);
}
template <typename T, auto add = default_operator::add<T>, auto sub = default_operator::sub<T>, auto mul = default_operator::mul<T>, auto inv = default_operator::inv<T>, std::enable_if_t<std::negation_v<std::is_integral<T>>, std::nullptr_t> = nullptr>
void walsh_hadamard_inv(std::vector<T>& a) {
walsh_hadamard<T, add, sub>(a);
const T n{ a.size() };
const T inv_n = inv(n);
for (auto& val : a) val = mul(val, inv_n);
}
} // namespace suisen::walsh_hadamard_transform
#line 1 "library/convolution/polynomial_eval_multipoint_eval.hpp"
#line 1 "library/polynomial/multi_point_eval.hpp"
#line 5 "library/polynomial/multi_point_eval.hpp"
namespace suisen {
template <typename FPSType, typename T>
std::vector<typename FPSType::value_type> multi_point_eval(const FPSType& f, const std::vector<T>& xs) {
int n = xs.size();
if (n == 0) return {};
std::vector<FPSType> seg(2 * n);
for (int i = 0; i < n; ++i) seg[n + i] = FPSType{ -xs[i], 1 };
for (int i = n - 1; i > 0; --i) seg[i] = seg[i * 2] * seg[i * 2 + 1];
seg[1] = f % seg[1];
for (int i = 2; i < 2 * n; ++i) seg[i] = seg[i / 2] % seg[i];
std::vector<typename FPSType::value_type> ys(n);
for (int i = 0; i < n; ++i) ys[i] = seg[n + i].size() ? seg[n + i][0] : 0;
return ys;
}
} // namespace suisen
#line 6 "library/convolution/polynomial_eval_multipoint_eval.hpp"
namespace suisen {
template <typename mint, auto transform, auto transform_inv>
std::vector<mint> polynomial_eval(std::vector<mint> &&a, const FPS<mint> &f) {
transform(a);
a = multi_point_eval(f, a);
transform_inv(a);
return a;
}
template <typename mint, auto transform, auto transform_inv>
std::vector<mint> polynomial_eval(const std::vector<mint> &a, const FPS<mint> &f) {
return polynomial_eval<mint, transform, transform_inv>(std::vector<mint>(a), f);
}
} // namespace suisen
#line 10 "test/src/convolution/polynomial_eval_multipoint_eval/nim_counting.test.cpp"
using namespace suisen;
using mint = atcoder::modint998244353;
constexpr int M = 1 << 16;
int main() {
FPS<mint>::set_multiplication([](const auto& f, const auto& g) { return atcoder::convolution(f, g); });
std::ios::sync_with_stdio(false);
std::cin.tie(nullptr);
int n, k;
std::cin >> n >> k;
std::vector<mint> c(M, 0);
for (int i = 0; i < k; ++i) {
int v;
std::cin >> v;
++c[v];
}
FPS<mint> f(n + 1, 1);
f[0] = 0;
using namespace walsh_hadamard_transform;
auto res = polynomial_eval<mint, walsh_hadamard<mint>, walsh_hadamard_inv<mint>>(c, f);
std::cout << std::accumulate(res.begin() + 1, res.end(), mint(0)).val() << std::endl;
return 0;
}