C++ Library for Competitive Programming
/*
* @title 数学/多項式
*
* verification-helper: IGNORE
* verification-helper: PROBLEM https://judge.yosupo.jp/problem/division_of_polynomials
*/
#include <iostream>
#include "emthrm/math/modint.hpp"
#include "emthrm/math/polynomial.hpp"
int main() {
using ModInt = emthrm::MInt<998244353>;
int n, m;
std::cin >> n >> m;
emthrm::Polynomial<ModInt> f(n - 1), g(m - 1);
for (int i = 0; i < n; ++i) {
std::cin >> f[i];
}
for (int i = 0; i < m; ++i) {
std::cin >> g[i];
}
auto [q, r] = f.divide(g);
q.shrink();
r.shrink();
const int u = (q == emthrm::Polynomial<ModInt>{0} ? 0 : q.degree() + 1);
const int v = (r == emthrm::Polynomial<ModInt>{0} ? 0 : r.degree() + 1);
std::cout << u << ' ' << v << '\n';
for (int i = 0; i < u; ++i) {
std::cout << q[i];
if (i + 1 < u) std::cout << ' ';
}
std::cout << '\n';
for (int i = 0; i < v; ++i) {
std::cout << r[i];
if (i + 1 < v) std::cout << ' ';
}
std::cout << '\n';
return 0;
}
#line 1 "test/math/polynomial.test.cpp"
/*
* @title 数学/多項式
*
* verification-helper: IGNORE
* verification-helper: PROBLEM https://judge.yosupo.jp/problem/division_of_polynomials
*/
#include <iostream>
#line 1 "include/emthrm/math/modint.hpp"
#ifndef ARBITRARY_MODINT
# include <cassert>
#endif
#include <compare>
#line 9 "include/emthrm/math/modint.hpp"
// #include <numeric>
#include <utility>
#include <vector>
namespace emthrm {
#ifndef ARBITRARY_MODINT
template <unsigned int M>
struct MInt {
unsigned int v;
constexpr MInt() : v(0) {}
constexpr MInt(const long long x) : v(x >= 0 ? x % M : x % M + M) {}
static constexpr MInt raw(const int x) {
MInt x_;
x_.v = x;
return x_;
}
static constexpr int get_mod() { return M; }
static constexpr void set_mod(const int divisor) {
assert(std::cmp_equal(divisor, M));
}
static void init(const int x) {
inv<true>(x);
fact(x);
fact_inv(x);
}
template <bool MEMOIZES = false>
static MInt inv(const int n) {
// assert(0 <= n && n < M && std::gcd(n, M) == 1);
static std::vector<MInt> inverse{0, 1};
const int prev = inverse.size();
if (n < prev) return inverse[n];
if constexpr (MEMOIZES) {
// "n!" and "M" must be disjoint.
inverse.resize(n + 1);
for (int i = prev; i <= n; ++i) {
inverse[i] = -inverse[M % i] * raw(M / i);
}
return inverse[n];
}
int u = 1, v = 0;
for (unsigned int a = n, b = M; b;) {
const unsigned int q = a / b;
std::swap(a -= q * b, b);
std::swap(u -= q * v, v);
}
return u;
}
static MInt fact(const int n) {
static std::vector<MInt> factorial{1};
if (const int prev = factorial.size(); n >= prev) {
factorial.resize(n + 1);
for (int i = prev; i <= n; ++i) {
factorial[i] = factorial[i - 1] * i;
}
}
return factorial[n];
}
static MInt fact_inv(const int n) {
static std::vector<MInt> f_inv{1};
if (const int prev = f_inv.size(); n >= prev) {
f_inv.resize(n + 1);
f_inv[n] = inv(fact(n).v);
for (int i = n; i > prev; --i) {
f_inv[i - 1] = f_inv[i] * i;
}
}
return f_inv[n];
}
static MInt nCk(const int n, const int k) {
if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) :
fact_inv(n - k) * fact_inv(k));
}
static MInt nPk(const int n, const int k) {
return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k);
}
static MInt nHk(const int n, const int k) {
return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k));
}
static MInt large_nCk(long long n, const int k) {
if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
inv<true>(k);
MInt res = 1;
for (int i = 1; i <= k; ++i) {
res *= inv(i) * n--;
}
return res;
}
constexpr MInt pow(long long exponent) const {
MInt res = 1, tmp = *this;
for (; exponent > 0; exponent >>= 1) {
if (exponent & 1) res *= tmp;
tmp *= tmp;
}
return res;
}
constexpr MInt& operator+=(const MInt& x) {
if ((v += x.v) >= M) v -= M;
return *this;
}
constexpr MInt& operator-=(const MInt& x) {
if ((v += M - x.v) >= M) v -= M;
return *this;
}
constexpr MInt& operator*=(const MInt& x) {
v = (unsigned long long){v} * x.v % M;
return *this;
}
MInt& operator/=(const MInt& x) { return *this *= inv(x.v); }
constexpr auto operator<=>(const MInt& x) const = default;
constexpr MInt& operator++() {
if (++v == M) [[unlikely]] v = 0;
return *this;
}
constexpr MInt operator++(int) {
const MInt res = *this;
++*this;
return res;
}
constexpr MInt& operator--() {
v = (v == 0 ? M - 1 : v - 1);
return *this;
}
constexpr MInt operator--(int) {
const MInt res = *this;
--*this;
return res;
}
constexpr MInt operator+() const { return *this; }
constexpr MInt operator-() const { return raw(v ? M - v : 0); }
constexpr MInt operator+(const MInt& x) const { return MInt(*this) += x; }
constexpr MInt operator-(const MInt& x) const { return MInt(*this) -= x; }
constexpr MInt operator*(const MInt& x) const { return MInt(*this) *= x; }
MInt operator/(const MInt& x) const { return MInt(*this) /= x; }
friend std::ostream& operator<<(std::ostream& os, const MInt& x) {
return os << x.v;
}
friend std::istream& operator>>(std::istream& is, MInt& x) {
long long v;
is >> v;
x = MInt(v);
return is;
}
};
#else // ARBITRARY_MODINT
template <int ID>
struct MInt {
unsigned int v;
constexpr MInt() : v(0) {}
MInt(const long long x) : v(x >= 0 ? x % mod() : x % mod() + mod()) {}
static constexpr MInt raw(const int x) {
MInt x_;
x_.v = x;
return x_;
}
static int get_mod() { return mod(); }
static void set_mod(const unsigned int divisor) { mod() = divisor; }
static void init(const int x) {
inv<true>(x);
fact(x);
fact_inv(x);
}
template <bool MEMOIZES = false>
static MInt inv(const int n) {
// assert(0 <= n && n < mod() && std::gcd(x, mod()) == 1);
static std::vector<MInt> inverse{0, 1};
const int prev = inverse.size();
if (n < prev) return inverse[n];
if constexpr (MEMOIZES) {
// "n!" and "M" must be disjoint.
inverse.resize(n + 1);
for (int i = prev; i <= n; ++i) {
inverse[i] = -inverse[mod() % i] * raw(mod() / i);
}
return inverse[n];
}
int u = 1, v = 0;
for (unsigned int a = n, b = mod(); b;) {
const unsigned int q = a / b;
std::swap(a -= q * b, b);
std::swap(u -= q * v, v);
}
return u;
}
static MInt fact(const int n) {
static std::vector<MInt> factorial{1};
if (const int prev = factorial.size(); n >= prev) {
factorial.resize(n + 1);
for (int i = prev; i <= n; ++i) {
factorial[i] = factorial[i - 1] * i;
}
}
return factorial[n];
}
static MInt fact_inv(const int n) {
static std::vector<MInt> f_inv{1};
if (const int prev = f_inv.size(); n >= prev) {
f_inv.resize(n + 1);
f_inv[n] = inv(fact(n).v);
for (int i = n; i > prev; --i) {
f_inv[i - 1] = f_inv[i] * i;
}
}
return f_inv[n];
}
static MInt nCk(const int n, const int k) {
if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) :
fact_inv(n - k) * fact_inv(k));
}
static MInt nPk(const int n, const int k) {
return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k);
}
static MInt nHk(const int n, const int k) {
return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k));
}
static MInt large_nCk(long long n, const int k) {
if (n < 0 || n < k || k < 0) [[unlikely]] return MInt();
inv<true>(k);
MInt res = 1;
for (int i = 1; i <= k; ++i) {
res *= inv(i) * n--;
}
return res;
}
MInt pow(long long exponent) const {
MInt res = 1, tmp = *this;
for (; exponent > 0; exponent >>= 1) {
if (exponent & 1) res *= tmp;
tmp *= tmp;
}
return res;
}
MInt& operator+=(const MInt& x) {
if ((v += x.v) >= mod()) v -= mod();
return *this;
}
MInt& operator-=(const MInt& x) {
if ((v += mod() - x.v) >= mod()) v -= mod();
return *this;
}
MInt& operator*=(const MInt& x) {
v = (unsigned long long){v} * x.v % mod();
return *this;
}
MInt& operator/=(const MInt& x) { return *this *= inv(x.v); }
auto operator<=>(const MInt& x) const = default;
MInt& operator++() {
if (++v == mod()) [[unlikely]] v = 0;
return *this;
}
MInt operator++(int) {
const MInt res = *this;
++*this;
return res;
}
MInt& operator--() {
v = (v == 0 ? mod() - 1 : v - 1);
return *this;
}
MInt operator--(int) {
const MInt res = *this;
--*this;
return res;
}
MInt operator+() const { return *this; }
MInt operator-() const { return raw(v ? mod() - v : 0); }
MInt operator+(const MInt& x) const { return MInt(*this) += x; }
MInt operator-(const MInt& x) const { return MInt(*this) -= x; }
MInt operator*(const MInt& x) const { return MInt(*this) *= x; }
MInt operator/(const MInt& x) const { return MInt(*this) /= x; }
friend std::ostream& operator<<(std::ostream& os, const MInt& x) {
return os << x.v;
}
friend std::istream& operator>>(std::istream& is, MInt& x) {
long long v;
is >> v;
x = MInt(v);
return is;
}
private:
static unsigned int& mod() {
static unsigned int divisor = 0;
return divisor;
}
};
#endif // ARBITRARY_MODINT
} // namespace emthrm
#line 1 "include/emthrm/math/polynomial.hpp"
#include <algorithm>
#include <cassert>
#include <functional>
#include <initializer_list>
#include <iterator>
#include <numeric>
#line 12 "include/emthrm/math/polynomial.hpp"
namespace emthrm {
template <typename T>
struct Polynomial {
std::vector<T> coef;
explicit Polynomial(const int deg = 0) : coef(deg + 1, 0) {}
explicit Polynomial(const std::vector<T>& coef) : coef(coef) {}
Polynomial(const std::initializer_list<T> init)
: coef(init.begin(), init.end()) {}
template <typename InputIter>
explicit Polynomial(const InputIter first, const InputIter last)
: coef(first, last) {}
inline const T& operator[](const int term) const { return coef[term]; }
inline T& operator[](const int term) { return coef[term]; }
using Mult = std::function<std::vector<T>(const std::vector<T>&,
const std::vector<T>&)>;
static void set_mult(const Mult mult) { get_mult() = mult; }
void resize(const int deg) { coef.resize(deg + 1, 0); }
void shrink() {
while (coef.size() > 1 && coef.back() == 0) coef.pop_back();
}
int degree() const { return std::ssize(coef) - 1; }
Polynomial& operator=(const std::vector<T>& coef_) {
coef = coef_;
return *this;
}
Polynomial& operator=(const Polynomial& x) = default;
Polynomial& operator+=(const Polynomial& x) {
const int deg_x = x.degree();
if (deg_x > degree()) resize(deg_x);
for (int i = 0; i <= deg_x; ++i) {
coef[i] += x[i];
}
return *this;
}
Polynomial& operator-=(const Polynomial& x) {
const int deg_x = x.degree();
if (deg_x > degree()) resize(deg_x);
for (int i = 0; i <= deg_x; ++i) {
coef[i] -= x[i];
}
return *this;
}
Polynomial& operator*=(const T x) {
for (T& e : coef) e *= x;
return *this;
}
Polynomial& operator*=(const Polynomial& x) {
return *this = get_mult()(coef, x.coef);
}
Polynomial& operator/=(const T x) {
assert(x != 0);
return *this *= static_cast<T>(1) / x;
}
std::pair<Polynomial, Polynomial> divide(Polynomial x) const {
x.shrink();
Polynomial rem = *this;
const int n = rem.degree(), m = x.degree(), deg = n - m;
if (deg < 0) return {Polynomial{0}, rem};
Polynomial quo(deg);
for (int i = 0; i <= deg; ++i) {
quo[deg - i] = rem[n - i] / x[m];
for (int j = 0; j <= m; ++j) {
rem[n - i - j] -= x[m - j] * quo[deg - i];
}
}
rem.resize(deg);
return {quo, rem};
}
Polynomial& operator/=(const Polynomial& x) {
return *this = divide(x).first;
}
Polynomial& operator%=(const Polynomial& x) {
return *this = divide(x).second;
}
Polynomial& operator<<=(const int n) {
coef.insert(coef.begin(), n, 0);
return *this;
}
bool operator==(Polynomial x) const {
x.shrink();
Polynomial y = *this;
y.shrink();
return x.coef == y.coef;
}
Polynomial operator+() const { return *this; }
Polynomial operator-() const {
Polynomial res = *this;
for (T& e : res.coef) e = -e;
return res;
}
Polynomial operator+(const Polynomial& x) const {
return Polynomial(*this) += x;
}
Polynomial operator-(const Polynomial& x) const {
return Polynomial(*this) -= x;
}
Polynomial operator*(const T x) const {
return Polynomial(*this) *= x;
}
Polynomial operator*(const Polynomial& x) const {
return Polynomial(*this) *= x;
}
Polynomial operator/(const T x) const { return Polynomial(*this) /= x; }
Polynomial operator/(const Polynomial& x) const {
return Polynomial(*this) /= x;
}
Polynomial operator%(const Polynomial& x) const {
return Polynomial(*this) %= x;
}
Polynomial operator<<(const int n) const { return Polynomial(*this) <<= n; }
T horner(const T x) const {
return std::accumulate(
coef.rbegin(), coef.rend(), static_cast<T>(0),
[x](const T l, const T r) -> T { return l * x + r; });
}
Polynomial differential() const {
const int deg = degree();
assert(deg >= 0);
Polynomial res(std::max(deg - 1, 0));
for (int i = 1; i <= deg; ++i) {
res[i - 1] = coef[i] * i;
}
return res;
}
Polynomial pow(int exponent) const {
Polynomial res{1}, base = *this;
for (; exponent > 0; exponent >>= 1) {
if (exponent & 1) res *= base;
base *= base;
}
return res;
}
Polynomial translate(const T c) const {
const int n = coef.size();
std::vector<T> fact(n, 1), inv_fact(n, 1);
for (int i = 1; i < n; ++i) {
fact[i] = fact[i - 1] * i;
}
inv_fact[n - 1] = static_cast<T>(1) / fact[n - 1];
for (int i = n - 1; i > 0; --i) {
inv_fact[i - 1] = inv_fact[i] * i;
}
std::vector<T> g(n), ex(n);
for (int i = 0; i < n; ++i) {
g[i] = coef[i] * fact[i];
}
std::reverse(g.begin(), g.end());
T pow_c = 1;
for (int i = 0; i < n; ++i) {
ex[i] = pow_c * inv_fact[i];
pow_c *= c;
}
const std::vector<T> conv = get_mult()(g, ex);
Polynomial res(n - 1);
for (int i = 0; i < n; ++i) {
res[i] = conv[n - 1 - i] * inv_fact[i];
}
return res;
}
private:
static Mult& get_mult() {
static Mult mult = [](const std::vector<T>& a, const std::vector<T>& b)
-> std::vector<T> {
const int n = a.size(), m = b.size();
std::vector<T> res(n + m - 1, 0);
for (int i = 0; i < n; ++i) {
for (int j = 0; j < m; ++j) {
res[i + j] += a[i] * b[j];
}
}
return res;
};
return mult;
}
};
} // namespace emthrm
#line 12 "test/math/polynomial.test.cpp"
int main() {
using ModInt = emthrm::MInt<998244353>;
int n, m;
std::cin >> n >> m;
emthrm::Polynomial<ModInt> f(n - 1), g(m - 1);
for (int i = 0; i < n; ++i) {
std::cin >> f[i];
}
for (int i = 0; i < m; ++i) {
std::cin >> g[i];
}
auto [q, r] = f.divide(g);
q.shrink();
r.shrink();
const int u = (q == emthrm::Polynomial<ModInt>{0} ? 0 : q.degree() + 1);
const int v = (r == emthrm::Polynomial<ModInt>{0} ? 0 : r.degree() + 1);
std::cout << u << ' ' << v << '\n';
for (int i = 0; i < u; ++i) {
std::cout << q[i];
if (i + 1 < u) std::cout << ' ';
}
std::cout << '\n';
for (int i = 0; i < v; ++i) {
std::cout << r[i];
if (i + 1 < v) std::cout << ' ';
}
std::cout << '\n';
return 0;
}