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formal_power_series/test/sparse_fps_exp.test.cpp
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- Last update: 2026-03-01 22:00:22+09:00
- Problem: https://judge.yosupo.jp/problem/exp_of_formal_power_series_sparse
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Code
#define PROBLEM "https://judge.yosupo.jp/problem/exp_of_formal_power_series_sparse"
#include "../sparse_fps.hpp"
#include "../../modint.hpp"
#include <iostream>
#include <vector>
using namespace std;
using mint = ModInt<998244353>;
int main() {
cin.tie(nullptr), ios::sync_with_stdio(false);
int N, K;
cin >> N >> K;
vector<mint> f(N);
while (K--) {
int i, a;
cin >> i >> a;
f.at(i) = a;
}
const auto ret = sparse_fps::exp(f, N - 1);
for (auto e : ret) cout << e << ' ';
cout << '\n';
}#line 1 "formal_power_series/test/sparse_fps_exp.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/exp_of_formal_power_series_sparse"
#line 2 "formal_power_series/sparse_fps.hpp"
#include <algorithm>
#include <cassert>
#include <concepts>
#include <optional>
#include <utility>
#include <vector>
namespace sparse_fps {
// https://github.com/yosupo06/library-checker-problems/issues/767#issuecomment-1166414020
// Calculate f(x)^k up to x^max_deg
template <typename Vec>
requires std::derived_from<Vec, std::vector<typename Vec::value_type>>
Vec pow(const Vec &f, int max_deg, long long k) {
using T = typename Vec::value_type;
assert(k >= 0);
Vec ret(max_deg + 1);
if (k == 0) {
ret[0] = T{1};
return ret;
}
std::vector<std::pair<int, T>> terms;
int d0 = 0;
while (d0 < int(f.size()) and d0 <= max_deg and f[d0] == T()) ++d0;
if (d0 == int(f.size()) or d0 > max_deg) return ret;
if (d0 and max_deg / d0 < k) return ret;
for (int d = d0 + 1; d < std::min<int>(max_deg + 1, f.size()); ++d) {
if (f[d] != T{}) terms.emplace_back(d - d0, f[d]);
}
const int bias = d0 * k;
ret[bias] = f[d0].pow(k);
const T fd0inv = 1 / f[d0];
for (int d = 0; bias + d + 1 < int(ret.size()); ++d) {
// Compare [x^d](k f'g - fg')
T tmp{0};
for (auto [i, fi] : terms) {
int j = d - i;
if (0 <= j) tmp -= fi * ret[bias + j + 1] * (j + 1);
j = d - (i - 1);
if (0 <= j) tmp += fi * i * ret[bias + j] * T(k);
}
ret[bias + d + 1] = tmp * fd0inv / (d + 1);
}
return ret;
}
template <typename Vec>
requires std::derived_from<Vec, std::vector<typename Vec::value_type>>
Vec inv(const Vec &f, int max_deg) {
using T = typename Vec::value_type;
assert(!f.empty() and f[0] != T{0});
Vec ret(max_deg + 1);
std::vector<std::pair<int, T>> terms;
for (int d = 1; d < (int)f.size() and d <= max_deg; ++d) {
if (f[d] != T{}) terms.emplace_back(d, f[d]);
}
const T f0inv = f[0].inv();
ret[0] = f0inv;
for (int d = 1; d <= max_deg; ++d) {
T tmp{0};
for (auto [i, fi] : terms) {
if (i > d) break;
tmp -= fi * ret[d - i];
}
ret[d] = tmp * f0inv;
}
return ret;
}
template <typename Vec>
requires std::derived_from<Vec, std::vector<typename Vec::value_type>>
Vec log(const Vec &f, int max_deg) {
using T = typename Vec::value_type;
assert(!f.empty() and f[0] != T{0});
const auto inv = sparse_fps::inv(f, max_deg);
std::vector<std::pair<int, T>> df_terms;
for (int d = 1; d < (int)f.size() and d <= max_deg; ++d) {
if (f[d] != T{}) df_terms.emplace_back(d - 1, f[d] * T{d});
}
Vec ret(max_deg + 1);
for (int d = 0; d < max_deg; ++d) {
for (auto [i, fi] : df_terms) {
const int j = d + i + 1;
if (j > max_deg) break;
ret[j] += inv[d] * fi * T{j}.inv();
}
}
return ret;
}
template <typename Vec>
requires std::derived_from<Vec, std::vector<typename Vec::value_type>>
Vec exp(const Vec &f, int max_deg) {
using T = typename Vec::value_type;
assert(f.empty() or f[0] == T{0});
std::vector<std::pair<int, T>> df_terms;
for (int d = 1; d < (int)f.size() and d <= max_deg; ++d) {
if (f[d] != T{}) df_terms.emplace_back(d - 1, f[d] * T{d});
}
Vec ret(max_deg + 1);
ret[0] = T{1};
// F' = F * f'
for (int d = 1; d <= max_deg; ++d) {
T tmp{0};
for (auto [i, dfi] : df_terms) {
if (i > d - 1) break;
tmp += dfi * ret[d - 1 - i];
}
ret[d] = tmp * T{d}.inv();
}
return ret;
}
template <typename Vec>
requires std::derived_from<Vec, std::vector<typename Vec::value_type>>
std::optional<Vec> sqrt(const Vec &f, int max_deg) {
using T = typename Vec::value_type;
Vec ret(max_deg + 1);
int d0 = 0;
while (d0 < int(f.size()) and d0 <= max_deg and f[d0] == T{}) ++d0;
if (d0 == int(f.size()) or d0 > max_deg) return ret;
if (d0 & 1) return std::nullopt;
const T sqrtf0 = f[d0].sqrt();
if (sqrtf0 == T{}) return std::nullopt;
std::vector<std::pair<int, T>> terms;
const T fd0inv = 1 / f[d0];
for (int d = d0 + 1; d < std::min<int>(max_deg + 1, f.size()); ++d) {
if (f[d] != T{}) terms.emplace_back(d - d0, f[d] * fd0inv);
}
const int bias = d0 / 2;
const T inv2 = T{2}.inv();
ret[bias] = sqrtf0;
for (int d = 0; bias + d + 1 < int(ret.size()); ++d) {
T tmp{0};
for (auto [i, fi] : terms) {
if (i > d + 1) break;
int j = d - i;
if (0 <= j) tmp -= fi * ret[bias + j + 1] * (j + 1);
j = d - (i - 1);
if (0 <= j) tmp += fi * i * ret[bias + j] * inv2;
}
ret[bias + d + 1] = tmp / (d + 1);
}
return ret;
}
} // namespace sparse_fps
#line 3 "modint.hpp"
#include <iostream>
#include <set>
#line 6 "modint.hpp"
template <int md> struct ModInt {
static_assert(md > 1);
using lint = long long;
constexpr static int mod() { return md; }
static int get_primitive_root() {
static int primitive_root = 0;
if (!primitive_root) {
primitive_root = [&]() {
std::set<int> fac;
int v = md - 1;
for (lint i = 2; i * i <= v; i++)
while (v % i == 0) fac.insert(i), v /= i;
if (v > 1) fac.insert(v);
for (int g = 1; g < md; g++) {
bool ok = true;
for (auto i : fac)
if (ModInt(g).pow((md - 1) / i) == 1) {
ok = false;
break;
}
if (ok) return g;
}
return -1;
}();
}
return primitive_root;
}
int val_;
int val() const noexcept { return val_; }
constexpr ModInt() : val_(0) {}
constexpr ModInt &_setval(lint v) { return val_ = (v >= md ? v - md : v), *this; }
constexpr ModInt(lint v) { _setval(v % md + md); }
constexpr explicit operator bool() const { return val_ != 0; }
constexpr ModInt operator+(const ModInt &x) const {
return ModInt()._setval((lint)val_ + x.val_);
}
constexpr ModInt operator-(const ModInt &x) const {
return ModInt()._setval((lint)val_ - x.val_ + md);
}
constexpr ModInt operator*(const ModInt &x) const {
return ModInt()._setval((lint)val_ * x.val_ % md);
}
constexpr ModInt operator/(const ModInt &x) const {
return ModInt()._setval((lint)val_ * x.inv().val() % md);
}
constexpr ModInt operator-() const { return ModInt()._setval(md - val_); }
constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; }
constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; }
constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; }
constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; }
friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt(a) + x; }
friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt(a) - x; }
friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt(a) * x; }
friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt(a) / x; }
constexpr bool operator==(const ModInt &x) const { return val_ == x.val_; }
constexpr bool operator!=(const ModInt &x) const { return val_ != x.val_; }
constexpr bool operator<(const ModInt &x) const {
return val_ < x.val_;
} // To use std::map<ModInt, T>
friend std::istream &operator>>(std::istream &is, ModInt &x) {
lint t;
return is >> t, x = ModInt(t), is;
}
constexpr friend std::ostream &operator<<(std::ostream &os, const ModInt &x) {
return os << x.val_;
}
constexpr ModInt pow(lint n) const {
ModInt ans = 1, tmp = *this;
while (n) {
if (n & 1) ans *= tmp;
tmp *= tmp, n >>= 1;
}
return ans;
}
static constexpr int cache_limit = std::min(md, 1 << 21);
static std::vector<ModInt> facs, facinvs, invs;
constexpr static void _precalculation(int N) {
const int l0 = facs.size();
if (N > md) N = md;
if (N <= l0) return;
facs.resize(N), facinvs.resize(N), invs.resize(N);
for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i;
facinvs[N - 1] = facs.back().pow(md - 2);
for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1);
for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1];
}
constexpr ModInt inv() const {
if (this->val_ < cache_limit) {
if (facs.empty()) facs = {1}, facinvs = {1}, invs = {0};
while (this->val_ >= int(facs.size())) _precalculation(facs.size() * 2);
return invs[this->val_];
} else {
return this->pow(md - 2);
}
}
constexpr static ModInt fac(int n) {
assert(n >= 0);
if (n >= md) return ModInt(0);
while (n >= int(facs.size())) _precalculation(facs.size() * 2);
return facs[n];
}
constexpr static ModInt facinv(int n) {
assert(n >= 0);
if (n >= md) return ModInt(0);
while (n >= int(facs.size())) _precalculation(facs.size() * 2);
return facinvs[n];
}
constexpr static ModInt doublefac(int n) {
assert(n >= 0);
if (n >= md) return ModInt(0);
long long k = (n + 1) / 2;
return (n & 1) ? ModInt::fac(k * 2) / (ModInt(2).pow(k) * ModInt::fac(k))
: ModInt::fac(k) * ModInt(2).pow(k);
}
constexpr static ModInt nCr(int n, int r) {
assert(n >= 0);
if (r < 0 or n < r) return ModInt(0);
return ModInt::fac(n) * ModInt::facinv(r) * ModInt::facinv(n - r);
}
constexpr static ModInt nPr(int n, int r) {
assert(n >= 0);
if (r < 0 or n < r) return ModInt(0);
return ModInt::fac(n) * ModInt::facinv(n - r);
}
static ModInt binom(long long n, long long r) {
static long long bruteforce_times = 0;
if (r < 0 or n < r) return ModInt(0);
if (n <= bruteforce_times or n < (int)facs.size()) return ModInt::nCr(n, r);
r = std::min(r, n - r);
assert((int)r == r);
ModInt ret = ModInt::facinv(r);
for (int i = 0; i < r; ++i) ret *= n - i;
bruteforce_times += r;
return ret;
}
// Multinomial coefficient, (k_1 + k_2 + ... + k_m)! / (k_1! k_2! ... k_m!)
// Complexity: O(sum(ks))
// Verify: https://yukicoder.me/problems/no/3178
template <class Vec> static ModInt multinomial(const Vec &ks) {
ModInt ret{1};
int sum = 0;
for (int k : ks) {
assert(k >= 0);
ret *= ModInt::facinv(k), sum += k;
}
return ret * ModInt::fac(sum);
}
template <class... Args> static ModInt multinomial(Args... args) {
int sum = (0 + ... + args);
ModInt result = (1 * ... * ModInt::facinv(args));
return ModInt::fac(sum) * result;
}
// Catalan number, C_n = binom(2n, n) / (n + 1) = # of Dyck words of length 2n
// C_0 = 1, C_1 = 1, C_2 = 2, C_3 = 5, C_4 = 14, ...
// https://oeis.org/A000108
// Complexity: O(n)
static ModInt catalan(int n) {
if (n < 0) return ModInt(0);
return ModInt::fac(n * 2) * ModInt::facinv(n + 1) * ModInt::facinv(n);
}
ModInt sqrt() const {
if (val_ == 0) return 0;
if (md == 2) return val_;
if (pow((md - 1) / 2) != 1) return 0;
ModInt b = 1;
while (b.pow((md - 1) / 2) == 1) b += 1;
int e = 0, m = md - 1;
while (m % 2 == 0) m >>= 1, e++;
ModInt x = pow((m - 1) / 2), y = (*this) * x * x;
x *= (*this);
ModInt z = b.pow(m);
while (y != 1) {
int j = 0;
ModInt t = y;
while (t != 1) j++, t *= t;
z = z.pow(1LL << (e - j - 1));
x *= z, z *= z, y *= z;
e = j;
}
return ModInt(std::min(x.val_, md - x.val_));
}
};
template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0};
using ModInt998244353 = ModInt<998244353>;
// using mint = ModInt<998244353>;
// using mint = ModInt<1000000007>;
#line 6 "formal_power_series/test/sparse_fps_exp.test.cpp"
using namespace std;
using mint = ModInt<998244353>;
int main() {
cin.tie(nullptr), ios::sync_with_stdio(false);
int N, K;
cin >> N >> K;
vector<mint> f(N);
while (K--) {
int i, a;
cin >> i >> a;
f.at(i) = a;
}
const auto ret = sparse_fps::exp(f, N - 1);
for (auto e : ret) cout << e << ' ';
cout << '\n';
}