algorithm
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test/yosupo/pow_of_formal_power_series_sparse.cpp
- View this file on GitHub
- Last update: 2026-03-12 07:30:09+09:00
- Link: https://atcoder.jp/contests/ABC265/editorial/4577
前提
- $p - 1$ が十分大きい $2$ べきで割り切れる場合に $\mod p$ で積を計算する
- modint ライブラリと併用する
例題
2D NTT
-
https://yukicoder.me/problems/no/1241
-
https://atcoder.jp/contests/jag2013spring/tasks/icpc2013spring_f (まだ)
Code
#pragma region satashun
#include <algorithm>
#include <array>
#include <atomic>
#include <bitset>
#include <cassert>
#include <chrono>
#include <complex>
#include <condition_variable>
#include <deque>
#include <exception>
#include <forward_list>
#include <fstream>
#include <functional>
#include <future>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iosfwd>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <locale>
#include <map>
#include <memory>
#include <mutex>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <ratio>
#include <regex>
#include <scoped_allocator>
#include <set>
#include <sstream>
#include <stack>
#include <stdexcept>
#include <streambuf>
#include <string>
#include <system_error>
#include <thread>
#include <tuple>
#include <type_traits>
#include <typeindex>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <valarray>
#include <vector>
// #pragma GCC optimize("Ofast")
// #pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>
using namespace std;
using uint = unsigned int;
using ll = long long;
using ull = unsigned long long;
using pii = pair<int, int>;
template <class T>
using V = vector<T>;
template <class T>
using VV = V<V<T>>;
template <class T>
V<T> make_vec(size_t a) {
return V<T>(a);
}
template <class T, class... Ts>
auto make_vec(size_t a, Ts... ts) {
return V<decltype(make_vec<T>(ts...))>(a, make_vec<T>(ts...));
}
template <typename T, typename V>
void fill_vec(T& v, const V& val) {
v = val;
}
template <typename T, typename V>
void fill_vec(vector<T>& vec, const V& val) {
for (auto& v : vec) fill_vec(v, val);
}
#define pb push_back
#define eb emplace_back
#define mp make_pair
#define fi first
#define se second
#define rep(i, n) rep2(i, 0, n)
#define rep2(i, m, n) for (int i = m; i < (n); i++)
#define per(i, b) per2(i, 0, b)
#define per2(i, a, b) for (int i = int(b) - 1; i >= int(a); i--)
#define ALL(c) (c).begin(), (c).end()
#define SZ(x) ((int)(x).size())
constexpr ll TEN(int n) { return (n == 0) ? 1 : 10 * TEN(n - 1); }
template <class T, class U>
void chmin(T& t, const U& u) {
if (t > u) t = u;
}
template <class T, class U>
void chmax(T& t, const U& u) {
if (t < u) t = u;
}
template <typename T>
int arglb(const V<T>& v, const T& x) {
return distance(v.begin(), lower_bound(ALL(v), x));
}
template <typename T>
int argub(const V<T>& v, const T& x) {
return distance(v.begin(), upper_bound(ALL(v), x));
}
template <class T>
void mkuni(vector<T>& v) {
sort(ALL(v));
v.erase(unique(ALL(v)), end(v));
}
template <class T>
vector<int> sort_by(const vector<T>& v, bool increasing = true) {
vector<int> res(v.size());
iota(res.begin(), res.end(), 0);
if (increasing) {
stable_sort(res.begin(), res.end(),
[&](int i, int j) { return v[i] < v[j]; });
} else {
stable_sort(res.begin(), res.end(),
[&](int i, int j) { return v[i] > v[j]; });
}
return res;
}
// prototype
template <class T, class U>
ostream& operator<<(ostream& os, const pair<T, U>& p);
template <class T>
ostream& operator<<(ostream& os, const vector<T>& v);
template <class T>
istream& operator>>(istream& is, vector<T>& v);
template <class T, size_t sz>
ostream& operator<<(ostream& os, const array<T, sz>& arr);
template <class T>
ostream& operator<<(ostream& os, const set<T>& ST);
template <class T>
ostream& operator<<(ostream& os, const unordered_set<T>& ST);
template <class T>
ostream& operator<<(ostream& os, const multiset<T>& ST);
template <class T, class U>
ostream& operator<<(ostream& os, const map<T, U>& MP);
template <class T, class U>
ostream& operator<<(ostream& os, const unordered_map<T, U>& MP);
ostream& operator<<(ostream& o, __int128_t x);
// io functions
template <class T, class U>
istream& operator>>(istream& is, pair<T, U>& p) {
is >> p.first >> p.second;
return is;
}
template <class T, class U>
ostream& operator<<(ostream& os, const pair<T, U>& p) {
os << "(" << p.first << "," << p.second << ")";
return os;
}
template <class T>
istream& operator>>(istream& is, vector<T>& v) {
for (auto& x : v) {
is >> x;
}
return is;
}
template <class T>
ostream& operator<<(ostream& os, const vector<T>& v) {
os << "{";
rep(i, v.size()) {
if (i) os << ",";
os << v[i];
}
os << "}";
return os;
}
template <class T, size_t sz>
ostream& operator<<(ostream& os, const array<T, sz>& arr) {
os << '[';
for (auto v : arr) os << v << ',';
os << ']';
return os;
}
template <class T>
ostream& operator<<(ostream& os, const set<T>& ST) {
os << "{";
for (auto it = ST.begin(); it != ST.end(); ++it) {
if (it != ST.begin()) os << ",";
os << *it;
}
os << "}";
return os;
}
template <class T>
ostream& operator<<(ostream& os, const unordered_set<T>& ST) {
os << "{";
for (auto it = ST.begin(); it != ST.end(); ++it) {
if (it != ST.begin()) os << ",";
os << *it;
}
os << "}";
return os;
}
template <class T>
ostream& operator<<(ostream& os, const multiset<T>& ST) {
os << "{";
for (auto it = ST.begin(); it != ST.end(); ++it) {
if (it != ST.begin()) os << ",";
os << *it;
}
os << "}";
return os;
}
template <class T, class U>
ostream& operator<<(ostream& os, const map<T, U>& MP) {
for (auto it = MP.begin(); it != MP.end(); ++it) {
os << "(" << it->first << ": " << it->second << ")";
}
return os;
}
template <class T, class U>
ostream& operator<<(ostream& os, const unordered_map<T, U>& MP) {
for (auto it = MP.begin(); it != MP.end(); ++it) {
os << "(" << it->first << ": " << it->second << ")";
}
return os;
}
string to_string(__int128_t x) {
if (x == 0) return "0";
string result;
if (x < 0) {
result += "-";
x *= -1;
}
string t;
while (x) {
t.push_back('0' + x % 10);
x /= 10;
}
reverse(t.begin(), t.end());
return result + t;
}
ostream& operator<<(ostream& o, __int128_t x) { return o << to_string(x); }
#ifdef LOCAL
void debug_out() { cerr << endl; }
template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
cerr << " " << H;
debug_out(T...);
}
#define debug(...) \
cerr << __LINE__ << " [" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#define dump(x) cerr << __LINE__ << " " << #x << " = " << (x) << endl
#else
#define debug(...) (void(0))
#define dump(x) (void(0))
#endif
template <class T>
V<T>& operator+=(V<T>& vec, const T& v) {
for (auto& x : vec) x += v;
return vec;
}
template <class T>
V<T>& operator-=(V<T>& vec, const T& v) {
for (auto& x : vec) x -= v;
return vec;
}
// suc : 1 = newline, 2 = space
template <class T>
void print(T x, int suc = 1) {
cout << x;
if (suc == 1)
cout << "\n";
else if (suc == 2)
cout << " ";
}
template <class T>
void print(const vector<T>& v, int suc = 1) {
for (int i = 0; i < v.size(); ++i)
print(v[i], i == int(v.size()) - 1 ? suc : 2);
}
template <class T>
void show(T x) {
print(x, 1);
}
template <typename Head, typename... Tail>
void show(Head H, Tail... T) {
print(H, 2);
show(T...);
}
int topbit(int t) { return t == 0 ? -1 : 31 - __builtin_clz(t); }
int topbit(ll t) { return t == 0 ? -1 : 63 - __builtin_clzll(t); }
int botbit(int a) { return a == 0 ? 32 : __builtin_ctz(a); }
int botbit(ll a) { return a == 0 ? 64 : __builtin_ctzll(a); }
int popcount(int t) { return __builtin_popcount(t); }
int popcount(ll t) { return __builtin_popcountll(t); }
int bit_parity(int t) { return __builtin_parity(t); }
int bit_parity(ll t) { return __builtin_parityll(t); }
struct prepare_io {
prepare_io() {
cin.tie(nullptr);
ios::sync_with_stdio(false);
cout << fixed << setprecision(10);
}
} prep_io;
#pragma endregion satashun
template <unsigned int MOD>
struct ModInt {
using uint = unsigned int;
using ull = unsigned long long;
using M = ModInt;
uint v;
ModInt(ll _v = 0) { set_norm(_v % MOD + MOD); }
M& set_norm(uint _v) { //[0, MOD * 2)->[0, MOD)
v = (_v < MOD) ? _v : _v - MOD;
return *this;
}
explicit operator bool() const { return v != 0; }
M operator+(const M& a) const { return M().set_norm(v + a.v); }
M operator-(const M& a) const { return M().set_norm(v + MOD - a.v); }
M operator*(const M& a) const { return M().set_norm(ull(v) * a.v % MOD); }
M operator/(const M& a) const { return *this * a.inv(); }
M& operator+=(const M& a) { return *this = *this + a; }
M& operator-=(const M& a) { return *this = *this - a; }
M& operator*=(const M& a) { return *this = *this * a; }
M& operator/=(const M& a) { return *this = *this / a; }
M operator-() const { return M() - *this; }
M& operator++(int) { return *this = *this + 1; }
M& operator--(int) { return *this = *this - 1; }
M pow(ll n) const {
if (n < 0) return inv().pow(-n);
M x = *this, res = 1;
while (n) {
if (n & 1) res *= x;
x *= x;
n >>= 1;
}
return res;
}
M inv() const {
ll a = v, b = MOD, p = 1, q = 0, t;
while (b != 0) {
t = a / b;
swap(a -= t * b, b);
swap(p -= t * q, q);
}
return M(p);
}
bool operator==(const M& a) const { return v == a.v; }
bool operator!=(const M& a) const { return v != a.v; }
friend ostream& operator<<(ostream& os, const M& a) { return os << a.v; }
static uint get_mod() { return MOD; }
};
using Mint = ModInt<998244353>;
V<Mint> fact, ifact, inv;
VV<Mint> small_comb;
void mod_init() {
const int maxv = 1000010;
const int maxvv = 5000;
fact.resize(maxv);
ifact.resize(maxv);
inv.resize(maxv);
small_comb = make_vec<Mint>(maxvv, maxvv);
fact[0] = 1;
for (int i = 1; i < maxv; ++i) {
fact[i] = fact[i - 1] * i;
}
ifact[maxv - 1] = fact[maxv - 1].inv();
for (int i = maxv - 2; i >= 0; --i) {
ifact[i] = ifact[i + 1] * (i + 1);
}
for (int i = 1; i < maxv; ++i) {
inv[i] = ifact[i] * fact[i - 1];
}
for (int i = 0; i < maxvv; ++i) {
small_comb[i][0] = small_comb[i][i] = 1;
for (int j = 1; j < i; ++j) {
small_comb[i][j] = small_comb[i - 1][j] + small_comb[i - 1][j - 1];
}
}
}
Mint comb(int n, int r) {
if (n < 0 || r < 0 || r > n) return Mint(0);
if (n < small_comb.size()) return small_comb[n][r];
return fact[n] * ifact[r] * ifact[n - r];
}
Mint inv_comb(int n, int r) {
if (n < 0 || r < 0 || r > n) return Mint(0);
return ifact[n] * fact[r] * fact[n - r];
}
// O(k)
Mint comb_slow(ll n, ll k) {
if (n < 0 || k < 0 || k > n) return Mint(0);
Mint res = ifact[k];
for (int i = 0; i < k; ++i) {
res = res * (n - i);
}
return res;
}
// line up
// a 'o' + b 'x'
Mint comb2(int a, int b) {
if (a < 0 || b < 0) return 0;
return comb(a + b, a);
}
// divide a into b groups
Mint nhr(int a, int b) {
if (b == 0) return Mint(a == 0);
return comb(a + b - 1, a);
}
// O(p + log_p n)
Mint lucas(ll n, ll k, int p) {
if (n < 0 || k < 0 || k > n) return Mint(0);
Mint res = 1;
while (n > 0) {
res *= comb(n % p, k % p);
n /= p;
k /= p;
}
return res;
}
struct ModPrepare {
ModPrepare() { mod_init(); }
} prep_mod;
/**
* @docs docs/ntt.md
*/
// 解説: https://atcoder.jp/contests/ABC265/editorial/4577
template <class D>
struct NumberTheoreticTransform {
D root;
V<D> roots = {0, 1};
V<int> rev = {0, 1};
int base = 1, max_base = -1;
void init() {
int mod = D::get_mod();
int tmp = mod - 1;
max_base = 0;
while (tmp % 2 == 0) {
tmp /= 2;
max_base++;
}
root = 2;
while (true) {
if (root.pow(1 << max_base).v == 1) {
if (root.pow(1 << (max_base - 1)).v != 1) {
break;
}
}
root++;
}
}
void ensure_base(int nbase) {
if (max_base == -1) init();
if (nbase <= base) return;
assert(nbase <= max_base);
rev.resize(1 << nbase);
for (int i = 0; i < (1 << nbase); ++i) {
rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
}
roots.resize(1 << nbase);
while (base < nbase) {
D z = root.pow(1 << (max_base - 1 - base));
for (int i = 1 << (base - 1); i < (1 << base); ++i) {
roots[i << 1] = roots[i];
roots[(i << 1) + 1] = roots[i] * z;
}
++base;
}
}
// FPS24 - R
void ntt(V<D>& a, bool inv = false) {
int n = a.size();
// assert((n & (n - 1)) == 0);
int zeros = __builtin_ctz(n);
ensure_base(zeros);
int shift = base - zeros;
for (int i = 0; i < n; i++) {
if (i < (rev[i] >> shift)) {
swap(a[i], a[rev[i] >> shift]);
}
}
for (int k = 1; k < n; k <<= 1) {
for (int i = 0; i < n; i += 2 * k) {
for (int j = 0; j < k; j++) {
D x = a[i + j];
D y = a[i + j + k] * roots[j + k];
a[i + j] = x + y;
a[i + j + k] = x - y;
}
}
}
if (inv) {
int v = D(n).inv().v;
reverse(a.begin() + 1, a.end());
for (int i = 0; i < n; i++) {
a[i] *= v;
}
}
}
V<D> mul(V<D> a, V<D> b) {
if (a.size() == 0 && b.size() == 0) return {};
int s = a.size() + b.size() - 1;
int nbase = 1;
while ((1 << nbase) < s) nbase++;
int sz = 1 << nbase;
if (sz <= 16) {
V<D> ret(s);
for (int i = 0; i < a.size(); i++) {
for (int j = 0; j < b.size(); j++) ret[i + j] += a[i] * b[j];
}
return ret;
}
a.resize(sz);
b.resize(sz);
ntt(a);
ntt(b);
for (int i = 0; i < sz; i++) {
a[i] *= b[i];
}
ntt(a, true);
a.resize(s);
return a;
}
};
// T : modint
template <class T>
void ntt_2d(VV<T>& a, bool rev) {
if (a.size() == 0 || a[0].size() == 0) return;
int h = a.size(), w = a[0].size();
NumberTheoreticTransform<T> fft;
fft.init();
for (auto& v : a) {
fft.ntt(v, rev);
}
rep(j, w) {
V<T> vh(h);
rep(i, h) { vh[i] = a[i][j]; }
fft.ntt(vh, rev);
rep(i, h) { a[i][j] = vh[i]; }
}
}
NumberTheoreticTransform<Mint> ntt;
struct prepare_NTT {
prepare_NTT() { ntt.init(); }
} prep_NTT;
template <class D>
struct Poly : public V<D> {
template <class... Args>
Poly(Args... args) : V<D>(args...) {}
Poly(initializer_list<D> init) : V<D>(init.begin(), init.end()) {}
int size() const { return V<D>::size(); }
D at(int p) const { return (p < this->size() ? (*this)[p] : D(0)); }
// first len terms
Poly pref(int len) const {
return Poly(this->begin(), this->begin() + min(this->size(), len));
}
// for polynomial division
Poly rev() const {
Poly res = *this;
reverse(res.begin(), res.end());
return res;
}
Poly operator+(const Poly& r) const {
auto n = max(size(), r.size());
V<D> tmp(n);
for (int i = 0; i < n; ++i) {
tmp[i] = at(i) + r.at(i);
}
return tmp;
}
Poly operator-(const Poly& r) const {
auto n = max(size(), r.size());
V<D> tmp(n);
for (int i = 0; i < n; ++i) {
tmp[i] = at(i) - r.at(i);
}
return tmp;
}
// scalar
Poly operator*(const D& k) const {
int n = size();
V<D> tmp(n);
for (int i = 0; i < n; ++i) {
tmp[i] = at(i) * k;
}
return tmp;
}
Poly operator*(const Poly& r) const {
Poly a = *this;
Poly b = r;
auto v = ntt.mul(a, b);
return v;
}
// scalar
Poly operator/(const D& k) const { return *this * k.inv(); }
Poly operator/(const Poly& r) const {
if (size() < r.size()) {
return {{}};
}
int d = size() - r.size() + 1;
return (rev().pref(d) * r.rev().inv(d)).pref(d).rev();
}
Poly operator%(const Poly& r) const {
auto res = *this - *this / r * r;
while (res.size() && !res.back()) {
res.pop_back();
}
return res;
}
Poly diff() const {
V<D> res(max(0, size() - 1));
for (int i = 1; i < size(); ++i) {
res[i - 1] = at(i) * i;
}
return res;
}
Poly inte() const {
V<D> res(size() + 1);
for (int i = 0; i < size(); ++i) {
res[i + 1] = at(i) / (D)(i + 1);
}
return res;
}
// f * f.inv(m) === 1 mod (x^m)
// f_0 ^ -1 must exist
Poly inv(int m) const {
Poly res = Poly({D(1) / at(0)});
for (int i = 1; i < m; i *= 2) {
res = (res * D(2) - res * res * pref(i * 2)).pref(i * 2);
}
return res.pref(m);
}
// f_0 = 1 must hold
Poly log(int n) const {
auto f = pref(n);
return (f.diff() * f.inv(n - 1)).pref(n - 1).inte();
}
// f_0 = 0 must hold
Poly exp(int n) const {
auto h = diff();
Poly f({1}), g({1});
for (int m = 1; m < n; m *= 2) {
g = (g * D(2) - f * g * g).pref(m);
auto q = h.pref(m - 1);
auto w = (q + g * (f.diff() - f * q)).pref(m * 2 - 1);
f = (f + f * (*this - w.inte()).pref(m * 2)).pref(m * 2);
}
return f.pref(n);
}
// be careful when k = 0
Poly pow(int n, ll k) const { return (log(n) * (D)k).exp(n); }
// f_0 = 1 must hold (use it with modular sqrt)
// CF250E
Poly sqrt(int n) const {
Poly f = pref(n);
Poly g({1});
for (int i = 1; i < n; i *= 2) {
g = (g + f.pref(i * 2) * g.inv(i * 2)) * D(2).inv();
}
return g.pref(n);
}
D eval(D x) const {
D res = 0, c = 1;
for (auto a : *this) {
res += a * c;
c *= x;
}
return res;
}
Poly& operator+=(const Poly& r) { return *this = *this + r; }
Poly& operator-=(const Poly& r) { return *this = *this - r; }
Poly& operator*=(const D& r) { return *this = *this * r; }
Poly& operator*=(const Poly& r) { return *this = *this * r; }
Poly& operator/=(const Poly& r) { return *this = *this / r; }
Poly& operator/=(const D& r) { return *this = *this / r; }
Poly& operator%=(const Poly& r) { return *this = *this % r; }
friend ostream& operator<<(ostream& os, const Poly& pl) {
if (pl.size() == 0) return os << "0";
for (int i = 0; i < pl.size(); ++i) {
if (pl[i]) {
os << pl[i] << "x^" << i;
if (i + 1 != pl.size()) os << ",";
}
}
return os;
}
};
// first d terms of f^k, where f is sparse.
// O(d * |i | f_i != 0|)
// FPS24 - A
template <class T>
V<T> pow_sparse(const Poly<T>& f, ll k, int d) {
Poly<T> g(d);
if (k == 0) {
if (d > 0) g[0] = 1;
return g;
}
V<pair<int, T>> fv;
rep(i, f.size()) {
if (f[i] != 0) fv.emplace_back(i, f[i]);
}
if (SZ(fv) == 0) {
return g;
}
auto [head_pos, head_val] = fv[0];
fv.erase(fv.begin());
for (auto& [j, val] : fv) {
j -= head_pos;
}
Poly<T> gg(d);
gg[0] = head_val.pow(k);
Mint rev = head_val.inv();
for (int i = 0; i < d - 1; i++) {
for (auto [j, val] : fv) {
if (i + 1 - j < 0) break;
gg[i + 1] += gg[i + 1 - j] * val * (T(k) * j - (i - j + 1));
}
gg[i + 1] *= inv[i + 1] * rev;
}
rep(i, d) {
// k * head_pos < d-i
if (head_pos > 0 && k >= (d - i + head_pos - 1) / head_pos) break;
ll p = k * head_pos + i;
g[p] = gg[i];
}
return g;
}
int main() {
int N, K;
ll M;
cin >> N >> K >> M;
Poly<Mint> pl(N);
rep(i, K) {
int j, a;
cin >> j >> a;
pl[j] = a;
}
pl = pow_sparse(pl, M, N);
rep(i, N) { cout << pl[i] << (i != N - 1 ? ' ' : '\n'); }
return 0;
}#line 1 "test/yosupo/pow_of_formal_power_series_sparse.cpp"
#pragma region satashun
#include <algorithm>
#include <array>
#include <atomic>
#include <bitset>
#include <cassert>
#include <chrono>
#include <complex>
#include <condition_variable>
#include <deque>
#include <exception>
#include <forward_list>
#include <fstream>
#include <functional>
#include <future>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iosfwd>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <locale>
#include <map>
#include <memory>
#include <mutex>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <ratio>
#include <regex>
#include <scoped_allocator>
#include <set>
#include <sstream>
#include <stack>
#include <stdexcept>
#include <streambuf>
#include <string>
#include <system_error>
#include <thread>
#include <tuple>
#include <type_traits>
#include <typeindex>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <valarray>
#include <vector>
// #pragma GCC optimize("Ofast")
// #pragma GCC optimize("unroll-loops")
#include <bits/stdc++.h>
using namespace std;
using uint = unsigned int;
using ll = long long;
using ull = unsigned long long;
using pii = pair<int, int>;
template <class T>
using V = vector<T>;
template <class T>
using VV = V<V<T>>;
template <class T>
V<T> make_vec(size_t a) {
return V<T>(a);
}
template <class T, class... Ts>
auto make_vec(size_t a, Ts... ts) {
return V<decltype(make_vec<T>(ts...))>(a, make_vec<T>(ts...));
}
template <typename T, typename V>
void fill_vec(T& v, const V& val) {
v = val;
}
template <typename T, typename V>
void fill_vec(vector<T>& vec, const V& val) {
for (auto& v : vec) fill_vec(v, val);
}
#define pb push_back
#define eb emplace_back
#define mp make_pair
#define fi first
#define se second
#define rep(i, n) rep2(i, 0, n)
#define rep2(i, m, n) for (int i = m; i < (n); i++)
#define per(i, b) per2(i, 0, b)
#define per2(i, a, b) for (int i = int(b) - 1; i >= int(a); i--)
#define ALL(c) (c).begin(), (c).end()
#define SZ(x) ((int)(x).size())
constexpr ll TEN(int n) { return (n == 0) ? 1 : 10 * TEN(n - 1); }
template <class T, class U>
void chmin(T& t, const U& u) {
if (t > u) t = u;
}
template <class T, class U>
void chmax(T& t, const U& u) {
if (t < u) t = u;
}
template <typename T>
int arglb(const V<T>& v, const T& x) {
return distance(v.begin(), lower_bound(ALL(v), x));
}
template <typename T>
int argub(const V<T>& v, const T& x) {
return distance(v.begin(), upper_bound(ALL(v), x));
}
template <class T>
void mkuni(vector<T>& v) {
sort(ALL(v));
v.erase(unique(ALL(v)), end(v));
}
template <class T>
vector<int> sort_by(const vector<T>& v, bool increasing = true) {
vector<int> res(v.size());
iota(res.begin(), res.end(), 0);
if (increasing) {
stable_sort(res.begin(), res.end(),
[&](int i, int j) { return v[i] < v[j]; });
} else {
stable_sort(res.begin(), res.end(),
[&](int i, int j) { return v[i] > v[j]; });
}
return res;
}
// prototype
template <class T, class U>
ostream& operator<<(ostream& os, const pair<T, U>& p);
template <class T>
ostream& operator<<(ostream& os, const vector<T>& v);
template <class T>
istream& operator>>(istream& is, vector<T>& v);
template <class T, size_t sz>
ostream& operator<<(ostream& os, const array<T, sz>& arr);
template <class T>
ostream& operator<<(ostream& os, const set<T>& ST);
template <class T>
ostream& operator<<(ostream& os, const unordered_set<T>& ST);
template <class T>
ostream& operator<<(ostream& os, const multiset<T>& ST);
template <class T, class U>
ostream& operator<<(ostream& os, const map<T, U>& MP);
template <class T, class U>
ostream& operator<<(ostream& os, const unordered_map<T, U>& MP);
ostream& operator<<(ostream& o, __int128_t x);
// io functions
template <class T, class U>
istream& operator>>(istream& is, pair<T, U>& p) {
is >> p.first >> p.second;
return is;
}
template <class T, class U>
ostream& operator<<(ostream& os, const pair<T, U>& p) {
os << "(" << p.first << "," << p.second << ")";
return os;
}
template <class T>
istream& operator>>(istream& is, vector<T>& v) {
for (auto& x : v) {
is >> x;
}
return is;
}
template <class T>
ostream& operator<<(ostream& os, const vector<T>& v) {
os << "{";
rep(i, v.size()) {
if (i) os << ",";
os << v[i];
}
os << "}";
return os;
}
template <class T, size_t sz>
ostream& operator<<(ostream& os, const array<T, sz>& arr) {
os << '[';
for (auto v : arr) os << v << ',';
os << ']';
return os;
}
template <class T>
ostream& operator<<(ostream& os, const set<T>& ST) {
os << "{";
for (auto it = ST.begin(); it != ST.end(); ++it) {
if (it != ST.begin()) os << ",";
os << *it;
}
os << "}";
return os;
}
template <class T>
ostream& operator<<(ostream& os, const unordered_set<T>& ST) {
os << "{";
for (auto it = ST.begin(); it != ST.end(); ++it) {
if (it != ST.begin()) os << ",";
os << *it;
}
os << "}";
return os;
}
template <class T>
ostream& operator<<(ostream& os, const multiset<T>& ST) {
os << "{";
for (auto it = ST.begin(); it != ST.end(); ++it) {
if (it != ST.begin()) os << ",";
os << *it;
}
os << "}";
return os;
}
template <class T, class U>
ostream& operator<<(ostream& os, const map<T, U>& MP) {
for (auto it = MP.begin(); it != MP.end(); ++it) {
os << "(" << it->first << ": " << it->second << ")";
}
return os;
}
template <class T, class U>
ostream& operator<<(ostream& os, const unordered_map<T, U>& MP) {
for (auto it = MP.begin(); it != MP.end(); ++it) {
os << "(" << it->first << ": " << it->second << ")";
}
return os;
}
string to_string(__int128_t x) {
if (x == 0) return "0";
string result;
if (x < 0) {
result += "-";
x *= -1;
}
string t;
while (x) {
t.push_back('0' + x % 10);
x /= 10;
}
reverse(t.begin(), t.end());
return result + t;
}
ostream& operator<<(ostream& o, __int128_t x) { return o << to_string(x); }
#ifdef LOCAL
void debug_out() { cerr << endl; }
template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
cerr << " " << H;
debug_out(T...);
}
#define debug(...) \
cerr << __LINE__ << " [" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#define dump(x) cerr << __LINE__ << " " << #x << " = " << (x) << endl
#else
#define debug(...) (void(0))
#define dump(x) (void(0))
#endif
template <class T>
V<T>& operator+=(V<T>& vec, const T& v) {
for (auto& x : vec) x += v;
return vec;
}
template <class T>
V<T>& operator-=(V<T>& vec, const T& v) {
for (auto& x : vec) x -= v;
return vec;
}
// suc : 1 = newline, 2 = space
template <class T>
void print(T x, int suc = 1) {
cout << x;
if (suc == 1)
cout << "\n";
else if (suc == 2)
cout << " ";
}
template <class T>
void print(const vector<T>& v, int suc = 1) {
for (int i = 0; i < v.size(); ++i)
print(v[i], i == int(v.size()) - 1 ? suc : 2);
}
template <class T>
void show(T x) {
print(x, 1);
}
template <typename Head, typename... Tail>
void show(Head H, Tail... T) {
print(H, 2);
show(T...);
}
int topbit(int t) { return t == 0 ? -1 : 31 - __builtin_clz(t); }
int topbit(ll t) { return t == 0 ? -1 : 63 - __builtin_clzll(t); }
int botbit(int a) { return a == 0 ? 32 : __builtin_ctz(a); }
int botbit(ll a) { return a == 0 ? 64 : __builtin_ctzll(a); }
int popcount(int t) { return __builtin_popcount(t); }
int popcount(ll t) { return __builtin_popcountll(t); }
int bit_parity(int t) { return __builtin_parity(t); }
int bit_parity(ll t) { return __builtin_parityll(t); }
struct prepare_io {
prepare_io() {
cin.tie(nullptr);
ios::sync_with_stdio(false);
cout << fixed << setprecision(10);
}
} prep_io;
#pragma endregion satashun
template <unsigned int MOD>
struct ModInt {
using uint = unsigned int;
using ull = unsigned long long;
using M = ModInt;
uint v;
ModInt(ll _v = 0) { set_norm(_v % MOD + MOD); }
M& set_norm(uint _v) { //[0, MOD * 2)->[0, MOD)
v = (_v < MOD) ? _v : _v - MOD;
return *this;
}
explicit operator bool() const { return v != 0; }
M operator+(const M& a) const { return M().set_norm(v + a.v); }
M operator-(const M& a) const { return M().set_norm(v + MOD - a.v); }
M operator*(const M& a) const { return M().set_norm(ull(v) * a.v % MOD); }
M operator/(const M& a) const { return *this * a.inv(); }
M& operator+=(const M& a) { return *this = *this + a; }
M& operator-=(const M& a) { return *this = *this - a; }
M& operator*=(const M& a) { return *this = *this * a; }
M& operator/=(const M& a) { return *this = *this / a; }
M operator-() const { return M() - *this; }
M& operator++(int) { return *this = *this + 1; }
M& operator--(int) { return *this = *this - 1; }
M pow(ll n) const {
if (n < 0) return inv().pow(-n);
M x = *this, res = 1;
while (n) {
if (n & 1) res *= x;
x *= x;
n >>= 1;
}
return res;
}
M inv() const {
ll a = v, b = MOD, p = 1, q = 0, t;
while (b != 0) {
t = a / b;
swap(a -= t * b, b);
swap(p -= t * q, q);
}
return M(p);
}
bool operator==(const M& a) const { return v == a.v; }
bool operator!=(const M& a) const { return v != a.v; }
friend ostream& operator<<(ostream& os, const M& a) { return os << a.v; }
static uint get_mod() { return MOD; }
};
using Mint = ModInt<998244353>;
V<Mint> fact, ifact, inv;
VV<Mint> small_comb;
void mod_init() {
const int maxv = 1000010;
const int maxvv = 5000;
fact.resize(maxv);
ifact.resize(maxv);
inv.resize(maxv);
small_comb = make_vec<Mint>(maxvv, maxvv);
fact[0] = 1;
for (int i = 1; i < maxv; ++i) {
fact[i] = fact[i - 1] * i;
}
ifact[maxv - 1] = fact[maxv - 1].inv();
for (int i = maxv - 2; i >= 0; --i) {
ifact[i] = ifact[i + 1] * (i + 1);
}
for (int i = 1; i < maxv; ++i) {
inv[i] = ifact[i] * fact[i - 1];
}
for (int i = 0; i < maxvv; ++i) {
small_comb[i][0] = small_comb[i][i] = 1;
for (int j = 1; j < i; ++j) {
small_comb[i][j] = small_comb[i - 1][j] + small_comb[i - 1][j - 1];
}
}
}
Mint comb(int n, int r) {
if (n < 0 || r < 0 || r > n) return Mint(0);
if (n < small_comb.size()) return small_comb[n][r];
return fact[n] * ifact[r] * ifact[n - r];
}
Mint inv_comb(int n, int r) {
if (n < 0 || r < 0 || r > n) return Mint(0);
return ifact[n] * fact[r] * fact[n - r];
}
// O(k)
Mint comb_slow(ll n, ll k) {
if (n < 0 || k < 0 || k > n) return Mint(0);
Mint res = ifact[k];
for (int i = 0; i < k; ++i) {
res = res * (n - i);
}
return res;
}
// line up
// a 'o' + b 'x'
Mint comb2(int a, int b) {
if (a < 0 || b < 0) return 0;
return comb(a + b, a);
}
// divide a into b groups
Mint nhr(int a, int b) {
if (b == 0) return Mint(a == 0);
return comb(a + b - 1, a);
}
// O(p + log_p n)
Mint lucas(ll n, ll k, int p) {
if (n < 0 || k < 0 || k > n) return Mint(0);
Mint res = 1;
while (n > 0) {
res *= comb(n % p, k % p);
n /= p;
k /= p;
}
return res;
}
struct ModPrepare {
ModPrepare() { mod_init(); }
} prep_mod;
/**
* @docs docs/ntt.md
*/
// 解説: https://atcoder.jp/contests/ABC265/editorial/4577
template <class D>
struct NumberTheoreticTransform {
D root;
V<D> roots = {0, 1};
V<int> rev = {0, 1};
int base = 1, max_base = -1;
void init() {
int mod = D::get_mod();
int tmp = mod - 1;
max_base = 0;
while (tmp % 2 == 0) {
tmp /= 2;
max_base++;
}
root = 2;
while (true) {
if (root.pow(1 << max_base).v == 1) {
if (root.pow(1 << (max_base - 1)).v != 1) {
break;
}
}
root++;
}
}
void ensure_base(int nbase) {
if (max_base == -1) init();
if (nbase <= base) return;
assert(nbase <= max_base);
rev.resize(1 << nbase);
for (int i = 0; i < (1 << nbase); ++i) {
rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
}
roots.resize(1 << nbase);
while (base < nbase) {
D z = root.pow(1 << (max_base - 1 - base));
for (int i = 1 << (base - 1); i < (1 << base); ++i) {
roots[i << 1] = roots[i];
roots[(i << 1) + 1] = roots[i] * z;
}
++base;
}
}
// FPS24 - R
void ntt(V<D>& a, bool inv = false) {
int n = a.size();
// assert((n & (n - 1)) == 0);
int zeros = __builtin_ctz(n);
ensure_base(zeros);
int shift = base - zeros;
for (int i = 0; i < n; i++) {
if (i < (rev[i] >> shift)) {
swap(a[i], a[rev[i] >> shift]);
}
}
for (int k = 1; k < n; k <<= 1) {
for (int i = 0; i < n; i += 2 * k) {
for (int j = 0; j < k; j++) {
D x = a[i + j];
D y = a[i + j + k] * roots[j + k];
a[i + j] = x + y;
a[i + j + k] = x - y;
}
}
}
if (inv) {
int v = D(n).inv().v;
reverse(a.begin() + 1, a.end());
for (int i = 0; i < n; i++) {
a[i] *= v;
}
}
}
V<D> mul(V<D> a, V<D> b) {
if (a.size() == 0 && b.size() == 0) return {};
int s = a.size() + b.size() - 1;
int nbase = 1;
while ((1 << nbase) < s) nbase++;
int sz = 1 << nbase;
if (sz <= 16) {
V<D> ret(s);
for (int i = 0; i < a.size(); i++) {
for (int j = 0; j < b.size(); j++) ret[i + j] += a[i] * b[j];
}
return ret;
}
a.resize(sz);
b.resize(sz);
ntt(a);
ntt(b);
for (int i = 0; i < sz; i++) {
a[i] *= b[i];
}
ntt(a, true);
a.resize(s);
return a;
}
};
// T : modint
template <class T>
void ntt_2d(VV<T>& a, bool rev) {
if (a.size() == 0 || a[0].size() == 0) return;
int h = a.size(), w = a[0].size();
NumberTheoreticTransform<T> fft;
fft.init();
for (auto& v : a) {
fft.ntt(v, rev);
}
rep(j, w) {
V<T> vh(h);
rep(i, h) { vh[i] = a[i][j]; }
fft.ntt(vh, rev);
rep(i, h) { a[i][j] = vh[i]; }
}
}
NumberTheoreticTransform<Mint> ntt;
struct prepare_NTT {
prepare_NTT() { ntt.init(); }
} prep_NTT;
template <class D>
struct Poly : public V<D> {
template <class... Args>
Poly(Args... args) : V<D>(args...) {}
Poly(initializer_list<D> init) : V<D>(init.begin(), init.end()) {}
int size() const { return V<D>::size(); }
D at(int p) const { return (p < this->size() ? (*this)[p] : D(0)); }
// first len terms
Poly pref(int len) const {
return Poly(this->begin(), this->begin() + min(this->size(), len));
}
// for polynomial division
Poly rev() const {
Poly res = *this;
reverse(res.begin(), res.end());
return res;
}
Poly operator+(const Poly& r) const {
auto n = max(size(), r.size());
V<D> tmp(n);
for (int i = 0; i < n; ++i) {
tmp[i] = at(i) + r.at(i);
}
return tmp;
}
Poly operator-(const Poly& r) const {
auto n = max(size(), r.size());
V<D> tmp(n);
for (int i = 0; i < n; ++i) {
tmp[i] = at(i) - r.at(i);
}
return tmp;
}
// scalar
Poly operator*(const D& k) const {
int n = size();
V<D> tmp(n);
for (int i = 0; i < n; ++i) {
tmp[i] = at(i) * k;
}
return tmp;
}
Poly operator*(const Poly& r) const {
Poly a = *this;
Poly b = r;
auto v = ntt.mul(a, b);
return v;
}
// scalar
Poly operator/(const D& k) const { return *this * k.inv(); }
Poly operator/(const Poly& r) const {
if (size() < r.size()) {
return {{}};
}
int d = size() - r.size() + 1;
return (rev().pref(d) * r.rev().inv(d)).pref(d).rev();
}
Poly operator%(const Poly& r) const {
auto res = *this - *this / r * r;
while (res.size() && !res.back()) {
res.pop_back();
}
return res;
}
Poly diff() const {
V<D> res(max(0, size() - 1));
for (int i = 1; i < size(); ++i) {
res[i - 1] = at(i) * i;
}
return res;
}
Poly inte() const {
V<D> res(size() + 1);
for (int i = 0; i < size(); ++i) {
res[i + 1] = at(i) / (D)(i + 1);
}
return res;
}
// f * f.inv(m) === 1 mod (x^m)
// f_0 ^ -1 must exist
Poly inv(int m) const {
Poly res = Poly({D(1) / at(0)});
for (int i = 1; i < m; i *= 2) {
res = (res * D(2) - res * res * pref(i * 2)).pref(i * 2);
}
return res.pref(m);
}
// f_0 = 1 must hold
Poly log(int n) const {
auto f = pref(n);
return (f.diff() * f.inv(n - 1)).pref(n - 1).inte();
}
// f_0 = 0 must hold
Poly exp(int n) const {
auto h = diff();
Poly f({1}), g({1});
for (int m = 1; m < n; m *= 2) {
g = (g * D(2) - f * g * g).pref(m);
auto q = h.pref(m - 1);
auto w = (q + g * (f.diff() - f * q)).pref(m * 2 - 1);
f = (f + f * (*this - w.inte()).pref(m * 2)).pref(m * 2);
}
return f.pref(n);
}
// be careful when k = 0
Poly pow(int n, ll k) const { return (log(n) * (D)k).exp(n); }
// f_0 = 1 must hold (use it with modular sqrt)
// CF250E
Poly sqrt(int n) const {
Poly f = pref(n);
Poly g({1});
for (int i = 1; i < n; i *= 2) {
g = (g + f.pref(i * 2) * g.inv(i * 2)) * D(2).inv();
}
return g.pref(n);
}
D eval(D x) const {
D res = 0, c = 1;
for (auto a : *this) {
res += a * c;
c *= x;
}
return res;
}
Poly& operator+=(const Poly& r) { return *this = *this + r; }
Poly& operator-=(const Poly& r) { return *this = *this - r; }
Poly& operator*=(const D& r) { return *this = *this * r; }
Poly& operator*=(const Poly& r) { return *this = *this * r; }
Poly& operator/=(const Poly& r) { return *this = *this / r; }
Poly& operator/=(const D& r) { return *this = *this / r; }
Poly& operator%=(const Poly& r) { return *this = *this % r; }
friend ostream& operator<<(ostream& os, const Poly& pl) {
if (pl.size() == 0) return os << "0";
for (int i = 0; i < pl.size(); ++i) {
if (pl[i]) {
os << pl[i] << "x^" << i;
if (i + 1 != pl.size()) os << ",";
}
}
return os;
}
};
// first d terms of f^k, where f is sparse.
// O(d * |i | f_i != 0|)
// FPS24 - A
template <class T>
V<T> pow_sparse(const Poly<T>& f, ll k, int d) {
Poly<T> g(d);
if (k == 0) {
if (d > 0) g[0] = 1;
return g;
}
V<pair<int, T>> fv;
rep(i, f.size()) {
if (f[i] != 0) fv.emplace_back(i, f[i]);
}
if (SZ(fv) == 0) {
return g;
}
auto [head_pos, head_val] = fv[0];
fv.erase(fv.begin());
for (auto& [j, val] : fv) {
j -= head_pos;
}
Poly<T> gg(d);
gg[0] = head_val.pow(k);
Mint rev = head_val.inv();
for (int i = 0; i < d - 1; i++) {
for (auto [j, val] : fv) {
if (i + 1 - j < 0) break;
gg[i + 1] += gg[i + 1 - j] * val * (T(k) * j - (i - j + 1));
}
gg[i + 1] *= inv[i + 1] * rev;
}
rep(i, d) {
// k * head_pos < d-i
if (head_pos > 0 && k >= (d - i + head_pos - 1) / head_pos) break;
ll p = k * head_pos + i;
g[p] = gg[i];
}
return g;
}
int main() {
int N, K;
ll M;
cin >> N >> K >> M;
Poly<Mint> pl(N);
rep(i, K) {
int j, a;
cin >> j >> a;
pl[j] = a;
}
pl = pow_sparse(pl, M, N);
rep(i, N) { cout << pl[i] << (i != N - 1 ? ' ' : '\n'); }
return 0;
}