結果
| 問題 | No.1938 Lagrange Sum |
| ユーザー |
 mtsd
|
| 提出日時 | 2022-05-13 22:44:57 |
| 言語 | C++17 (gcc 12.3.0 + boost 1.83.0) |
| 結果 |
TLE
|
| 実行時間 | - |
| コード長 | 51,438 bytes |
| コンパイル時間 | 5,247 ms |
| コンパイル使用メモリ | 336,596 KB |
| 実行使用メモリ | 22,532 KB |
| 最終ジャッジ日時 | 2024-07-22 02:56:30 |
| 合計ジャッジ時間 | 13,725 ms |
|
ジャッジサーバーID （参考情報） |
judge1 / judge4 |
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テストケース
テストケース表示| 入力 | 結果 | 実行時間 実行使用メモリ |
|---|---|---|
| testcase_00 | TLE | - |
| testcase_01 | -- | - |
| testcase_02 | -- | - |
| testcase_03 | -- | - |
| testcase_04 | -- | - |
| testcase_05 | -- | - |
| testcase_06 | -- | - |
| testcase_07 | -- | - |
| testcase_08 | -- | - |
| testcase_09 | -- | - |
| testcase_10 | -- | - |
| testcase_11 | -- | - |
| testcase_12 | -- | - |
| testcase_13 | -- | - |
| testcase_14 | -- | - |
| testcase_15 | -- | - |
| testcase_16 | -- | - |
| testcase_17 | -- | - |
| testcase_18 | -- | - |
| testcase_19 | -- | - |
| testcase_20 | -- | - |
| testcase_21 | -- | - |
| testcase_22 | -- | - |
| testcase_23 | -- | - |
| testcase_24 | -- | - |
| testcase_25 | -- | - |
| testcase_26 | -- | - |
| testcase_27 | -- | - |
ソースコード
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef long double ld;
#define inf 1000000007
#define MP make_pair
#define MT make_tuple
#define PB push_back
#define fi first
#define se second
#define rep(i,n) for(int i = 0; i < (int)(n); ++i)
#define rrep(i,n) for(int i = (int)n-1; i >= 0; --i)
#define srep(i,a,b) for(int i = (int)a; i < (int)(b); ++i)
#define all(x) (x).begin(),(x).end()
#define SUM(v) accumulate(all(v), 0LL)
#define MIN(v) *min_element(all(v))
#define MAX(v) *max_element(all(v))
#define lb(c, x) distance((c).begin(), lower_bound(all(c), (x)))
#define ub(c, x) distance((c).begin(), upper_bound(all(c), (x)))
#define UNIQUE(x) sort(all(x)), x.erase(unique(all(x)), x.end())
#define SZ(c) (int)(c).size()
template<typename T>
ostream& operator << (ostream& os, vector<T>& vec) {
os << "{";
for (int i = 0; i<(int)vec.size(); i++) {
os << vec[i] << (i + 1 == (int)vec.size() ? "" : ", ");
}
os << "}";
return os;
}
// pair出力
template<typename T, typename U>
ostream& operator << (ostream& os, pair<T, U> pair_var) {
os << "(" << pair_var.first << ", " << pair_var.second << ")";
return os;
}
// map出力
template<typename T, typename U>
ostream& operator << (ostream& os, map<T, U>& map_var) {
os << "{";
for(auto itr = map_var.begin(); itr != map_var.end(); itr++){
os << "(" << itr->first << ", " << itr->second << ")";
itr++;
if(itr != map_var.end()) os << ", ";
itr--;
}
os << "}";
return os;
}
// set 出力
template<typename T>
ostream& operator << (ostream& os, set<T>& set_var) {
os << "{";
for(auto itr = set_var.begin(); itr != set_var.end(); itr++){
os << (*itr);
++itr;
if(itr != set_var.end()) os << ", ";
itr--;
}
os << "}";
return os;
}
// tuple 出力
template<int N,class Tuple>
void out(ostream &os,const Tuple &t){}
template<int N,class Tuple,class H,class ...Ts>
void out(ostream &os,const Tuple &t){
if(N)os<<", ";
os<<get<N>(t);
out<N+1,Tuple,Ts...>(os,t);
}
template<class ...Ts>
ostream& operator<<(ostream &os, const tuple<Ts...> &t){
os<<"(";
out<0,tuple<Ts...>,Ts...>(os,t);
os<<")";
return os;
}
#define overload2(_1, _2, name, ...) name
#define vec(type, name, ...) vector<type> name(__VA_ARGS__)
#define VEC(type, name, size) \
vector<type> name(size); \
IN(name)
#define vv(type, name, h, ...) vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define VV(type, name, h, w) \
vector<vector<type>> name(h, vector<type>(w)); \
IN(name)
#define vvv(type, name, h, w, ...) vector<vector<vector<type>>> name(h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...) \
vector<vector<vector<vector<type>>>> name(a, vector<vector<vector<type>>>(b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))
#define INT(...) \
int __VA_ARGS__; \
IN(__VA_ARGS__)
#define LL(...) \
ll __VA_ARGS__; \
IN(__VA_ARGS__)
#define STR(...) \
string __VA_ARGS__; \
IN(__VA_ARGS__)
#define CHR(...) \
char __VA_ARGS__; \
IN(__VA_ARGS__)
#define DBL(...) \
double __VA_ARGS__; \
IN(__VA_ARGS__)
int scan() { return getchar(); }
void scan(int &a) { cin >> a; }
void scan(long long &a) { cin >> a; }
void scan(char &a) { cin >> a; }
void scan(double &a) { cin >> a; }
void scan(string &a) { cin >> a; }
template <class T, class S> void scan(pair<T, S> &p) { scan(p.first), scan(p.second); }
template <class T> void scan(vector<T> &);
template <class T> void scan(vector<T> &a) {
for(auto &i : a) scan(i);
}
template <class T> void scan(T &a) { cin >> a; }
void IN() {}
template <class Head, class... Tail> void IN(Head &head, Tail &...tail) {
scan(head);
IN(tail...);
}
const string YESNO[2] = {"NO", "YES"};
const string YesNo[2] = {"No", "Yes"};
const string yesno[2] = {"no", "yes"};
void YES(bool t = 1) { cout << YESNO[t] << endl; }
void NO(bool t = 1) { YES(!t); }
void Yes(bool t = 1) { cout << YesNo[t] << endl; }
void No(bool t = 1) { Yes(!t); }
void yes(bool t = 1) { cout << yesno[t] << endl; }
void no(bool t = 1) { yes(!t); }
#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 dbg(...) \
cerr << __LINE__ << " [" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#define dump(x) cerr << __LINE__ << " " << #x << " = " << (x) << endl
#else
#define dbg(...) (void(0))
#define dump(x) (void(0))
#endif
template<typename A, typename T>
std::enable_if_t<std::is_convertible<T, A>::value> fill(A& array, const T& val)
{
array = val;
}
template<typename A, typename T>
std::enable_if_t<!std::is_convertible<T, A>::value> fill(A& array, const T& val)
{
for (auto& a : array) {
fill(a, val);
}
}
template <typename T, typename S> T ceil(T x, S y) {
assert(y);
return (y < 0 ? ceil(-x, -y) : (x > 0 ? (x + y - 1) / y : x / y));
}
template <typename T, typename S> T floor(T x, S y) {
assert(y);
return (y < 0 ? floor(-x, -y) : (x > 0 ? x / y : x / y - (x % y == 0 ? 0 : 1)));
}
vector<int> iota(int n) {vector<int> a(n);iota(all(a), 0);return a;}
template <class T> T POW(T x, int n) {T res = 1;for(; n; n >>= 1, x *= x){if(n & 1) res *= x;}return res;}
ll pow2(int i) { return 1LL << i; }
int topbit(signed t) { return t == 0 ? -1 : 31 - __builtin_clz(t); }
int topbit(ll t) { return t == 0 ? -1 : 63 - __builtin_clzll(t); }
int lowbit(signed a) { return a == 0 ? 32 : __builtin_ctz(a); }
int lowbit(ll a) { return a == 0 ? 64 : __builtin_ctzll(a); }
// int allbit(int n) { return (1 << n) - 1; }
ll allbit(ll n) { return (1LL << n) - 1; }
int popcount(signed t) { return __builtin_popcount(t); }
int popcount(ll t) { return __builtin_popcountll(t); }
bool ispow2(int i) { return i && (i & -i) == i; }
template <class S> void fold_in(vector<S> &v) {}
template <typename Head, typename... Tail, class S> void fold_in(vector<S> &v, Head &&a, Tail &&...tail) {
for(auto e : a) v.emplace_back(e);
fold_in(v, tail...);
}
template <class S> void renumber(vector<S> &v) {}
template <typename Head, typename... Tail, class S> void renumber(vector<S> &v, Head &&a, Tail &&...tail) {
for(auto &&e : a) e = lb(v, e);
renumber(v, tail...);
}
template <class S, class... Args> void zip(vector<S> &head, Args &&...args) {
vector<S> v;
fold_in(v, head, args...);
sort(all(v)), v.erase(unique(all(v)), v.end());
renumber(v, head, args...);
}
template<class T> inline bool chmax(T &a, T b){
if(a<b){
a = b;
return true;
}
return false;
}
template<class T> inline bool chmin(T &a, T b){
if(a>b){
a = b;
return true;
}
return false;
}
#include <immintrin.h>
template <typename mint>
struct FormalPowerSeries : vector<mint> {
using vector<mint>::vector;
using FPS = FormalPowerSeries;
FPS &operator+=(const FPS &r) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i];
return *this;
}
FPS &operator+=(const mint &r) {
if (this->empty()) this->resize(1);
(*this)[0] += r;
return *this;
}
FPS &operator-=(const FPS &r) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
return *this;
}
FPS &operator-=(const mint &r) {
if (this->empty()) this->resize(1);
(*this)[0] -= r;
return *this;
}
FPS &operator*=(const mint &v) {
for (int k = 0; k < (int)this->size(); k++) (*this)[k] *= v;
return *this;
}
FPS &operator/=(const FPS &r) {
if (this->size() < r.size()) {
this->clear();
return *this;
}
int n = this->size() - r.size() + 1;
if ((int)r.size() <= 64) {
FPS f(*this), g(r);
g.shrink();
mint coeff = g.back().inverse();
for (auto &x : g) x *= coeff;
int deg = (int)f.size() - (int)g.size() + 1;
int gs = g.size();
FPS quo(deg);
for (int i = deg - 1; i >= 0; i--) {
quo[i] = f[i + gs - 1];
for (int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j];
}
*this = quo * coeff;
this->resize(n, mint(0));
return *this;
}
return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev();
}
FPS &operator%=(const FPS &r) {
*this -= *this / r * r;
shrink();
return *this;
}
FPS operator+(const FPS &r) const { return FPS(*this) += r; }
FPS operator+(const mint &v) const { return FPS(*this) += v; }
FPS operator-(const FPS &r) const { return FPS(*this) -= r; }
FPS operator-(const mint &v) const { return FPS(*this) -= v; }
FPS operator*(const FPS &r) const { return FPS(*this) *= r; }
FPS operator*(const mint &v) const { return FPS(*this) *= v; }
FPS operator/(const FPS &r) const { return FPS(*this) /= r; }
FPS operator%(const FPS &r) const { return FPS(*this) %= r; }
FPS operator-() const {
FPS ret(this->size());
for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
return ret;
}
void shrink() {
while (this->size() && this->back() == mint(0)) this->pop_back();
}
FPS rev() const {
FPS ret(*this);
reverse(begin(ret), end(ret));
return ret;
}
FPS dot(FPS r) const {
FPS ret(min(this->size(), r.size()));
for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
return ret;
}
FPS pre(int sz) const {
return FPS(begin(*this), begin(*this) + min((int)this->size(), sz));
}
FPS operator>>(int sz) const {
if ((int)this->size() <= sz) return {};
FPS ret(*this);
ret.erase(ret.begin(), ret.begin() + sz);
return ret;
}
FPS operator<<(int sz) const {
FPS ret(*this);
ret.insert(ret.begin(), sz, mint(0));
return ret;
}
FPS diff() const {
const int n = (int)this->size();
FPS ret(max(0, n - 1));
mint one(1), coeff(1);
for (int i = 1; i < n; i++) {
ret[i - 1] = (*this)[i] * coeff;
coeff += one;
}
return ret;
}
FPS integral() const {
const int n = (int)this->size();
FPS ret(n + 1);
ret[0] = mint(0);
if (n > 0) ret[1] = mint(1);
auto mod = mint::get_mod();
for (int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i);
for (int i = 0; i < n; i++) ret[i + 1] *= (*this)[i];
return ret;
}
mint eval(mint x) const {
mint r = 0, w = 1;
for (auto &v : *this) r += w * v, w *= x;
return r;
}
FPS log(int deg = -1) const {
assert((*this)[0] == mint(1));
if (deg == -1) deg = (int)this->size();
return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
}
FPS pow(int64_t k, int deg = -1) const {
const int n = (int)this->size();
if (deg == -1) deg = n;
for (int i = 0; i < n; i++) {
if ((*this)[i] != mint(0)) {
if (i * k > deg) return FPS(deg, mint(0));
mint rev = mint(1) / (*this)[i];
FPS ret =
(((*this * rev) >> i).log(deg) * k).exp(deg) * ((*this)[i].pow(k));
ret = (ret << (i * k)).pre(deg);
if ((int)ret.size() < deg) ret.resize(deg, mint(0));
return ret;
}
}
return FPS(deg, mint(0));
}
static void *ntt_ptr;
static void set_fft();
FPS &operator*=(const FPS &r);
void ntt();
void intt();
void ntt_doubling();
static int ntt_pr();
FPS inv(int deg = -1) const;
FPS exp(int deg = -1) const;
};
template <typename mint>
void *FormalPowerSeries<mint>::ntt_ptr = nullptr;
/**
* @brief 多項式/形式的冪級数ライブラリ
* @docs docs/fps/formal-power-series.md
*/
#line 4 "fps/multipoint-evaluation.hpp"
template <typename mint>
struct ProductTree {
using fps = FormalPowerSeries<mint>;
const vector<mint> &xs;
vector<fps> buf;
int N, xsz;
vector<int> l, r;
ProductTree(const vector<mint> &xs_) : xs(xs_), xsz(xs.size()) {
N = 1;
while (N < (int)xs.size()) N *= 2;
buf.resize(2 * N);
l.resize(2 * N, xs.size());
r.resize(2 * N, xs.size());
fps::set_fft();
if (fps::ntt_ptr == nullptr)
build();
else
build_ntt();
}
void build() {
for (int i = 0; i < xsz; i++) {
l[i + N] = i;
r[i + N] = i + 1;
buf[i + N] = {-xs[i], 1};
}
for (int i = N - 1; i > 0; i--) {
l[i] = l[(i << 1) | 0];
r[i] = r[(i << 1) | 1];
if (buf[(i << 1) | 0].empty())
continue;
else if (buf[(i << 1) | 1].empty())
buf[i] = buf[(i << 1) | 0];
else
buf[i] = buf[(i << 1) | 0] * buf[(i << 1) | 1];
}
}
void build_ntt() {
fps f;
f.reserve(N * 2);
for (int i = 0; i < xsz; i++) {
l[i + N] = i;
r[i + N] = i + 1;
buf[i + N] = {-xs[i] + 1, -xs[i] - 1};
}
for (int i = N - 1; i > 0; i--) {
l[i] = l[(i << 1) | 0];
r[i] = r[(i << 1) | 1];
if (buf[(i << 1) | 0].empty())
continue;
else if (buf[(i << 1) | 1].empty())
buf[i] = buf[(i << 1) | 0];
else if (buf[(i << 1) | 0].size() == buf[(i << 1) | 1].size()) {
buf[i] = buf[(i << 1) | 0];
f.clear();
copy(begin(buf[(i << 1) | 1]), end(buf[(i << 1) | 1]),
back_inserter(f));
buf[i].ntt_doubling();
f.ntt_doubling();
for (int j = 0; j < (int)buf[i].size(); j++) buf[i][j] *= f[j];
} else {
buf[i] = buf[(i << 1) | 0];
f.clear();
copy(begin(buf[(i << 1) | 1]), end(buf[(i << 1) | 1]),
back_inserter(f));
buf[i].ntt_doubling();
f.intt();
f.resize(buf[i].size(), mint(0));
f.ntt();
for (int j = 0; j < (int)buf[i].size(); j++) buf[i][j] *= f[j];
}
}
for (int i = 0; i < 2 * N; i++) {
buf[i].intt();
buf[i].shrink();
}
}
};
template <typename mint>
vector<mint> InnerMultipointEvaluation(const FormalPowerSeries<mint> &f,
const vector<mint> &xs,
const ProductTree<mint> &ptree) {
using fps = FormalPowerSeries<mint>;
vector<mint> ret;
ret.reserve(xs.size());
auto rec = [&](auto self, fps a, int idx) {
if (ptree.l[idx] == ptree.r[idx]) return;
a %= ptree.buf[idx];
if ((int)a.size() <= 64) {
for (int i = ptree.l[idx]; i < ptree.r[idx]; i++)
ret.push_back(a.eval(xs[i]));
return;
}
self(self, a, (idx << 1) | 0);
self(self, a, (idx << 1) | 1);
};
rec(rec, f, 1);
return ret;
}
template <typename mint>
vector<mint> MultipointEvaluation(const FormalPowerSeries<mint> &f,
const vector<mint> &xs) {
if(f.empty() || xs.empty()) return vector<mint>(xs.size(), mint(0));
return InnerMultipointEvaluation(f, xs, ProductTree<mint>(xs));
}
/**
* @brief Multipoint Evaluation
*/
#line 2 "fps/ntt-friendly-fps.hpp"
#line 2 "ntt/ntt-avx2.hpp"
#line 2 "modint/simd-montgomery.hpp"
#line 4 "modint/simd-montgomery.hpp"
__attribute__((target("sse4.2"))) inline __m128i my128_mullo_epu32(
const __m128i &a, const __m128i &b) {
return _mm_mullo_epi32(a, b);
}
__attribute__((target("sse4.2"))) inline __m128i my128_mulhi_epu32(
const __m128i &a, const __m128i &b) {
__m128i a13 = _mm_shuffle_epi32(a, 0xF5);
__m128i b13 = _mm_shuffle_epi32(b, 0xF5);
__m128i prod02 = _mm_mul_epu32(a, b);
__m128i prod13 = _mm_mul_epu32(a13, b13);
__m128i prod = _mm_unpackhi_epi64(_mm_unpacklo_epi32(prod02, prod13),
_mm_unpackhi_epi32(prod02, prod13));
return prod;
}
__attribute__((target("sse4.2"))) inline __m128i montgomery_mul_128(
const __m128i &a, const __m128i &b, const __m128i &r, const __m128i &m1) {
return _mm_sub_epi32(
_mm_add_epi32(my128_mulhi_epu32(a, b), m1),
my128_mulhi_epu32(my128_mullo_epu32(my128_mullo_epu32(a, b), r), m1));
}
__attribute__((target("sse4.2"))) inline __m128i montgomery_add_128(
const __m128i &a, const __m128i &b, const __m128i &m2, const __m128i &m0) {
__m128i ret = _mm_sub_epi32(_mm_add_epi32(a, b), m2);
return _mm_add_epi32(_mm_and_si128(_mm_cmpgt_epi32(m0, ret), m2), ret);
}
__attribute__((target("sse4.2"))) inline __m128i montgomery_sub_128(
const __m128i &a, const __m128i &b, const __m128i &m2, const __m128i &m0) {
__m128i ret = _mm_sub_epi32(a, b);
return _mm_add_epi32(_mm_and_si128(_mm_cmpgt_epi32(m0, ret), m2), ret);
}
__attribute__((target("avx2"))) inline __m256i my256_mullo_epu32(
const __m256i &a, const __m256i &b) {
return _mm256_mullo_epi32(a, b);
}
__attribute__((target("avx2"))) inline __m256i my256_mulhi_epu32(
const __m256i &a, const __m256i &b) {
__m256i a13 = _mm256_shuffle_epi32(a, 0xF5);
__m256i b13 = _mm256_shuffle_epi32(b, 0xF5);
__m256i prod02 = _mm256_mul_epu32(a, b);
__m256i prod13 = _mm256_mul_epu32(a13, b13);
__m256i prod = _mm256_unpackhi_epi64(_mm256_unpacklo_epi32(prod02, prod13),
_mm256_unpackhi_epi32(prod02, prod13));
return prod;
}
__attribute__((target("avx2"))) inline __m256i montgomery_mul_256(
const __m256i &a, const __m256i &b, const __m256i &r, const __m256i &m1) {
return _mm256_sub_epi32(
_mm256_add_epi32(my256_mulhi_epu32(a, b), m1),
my256_mulhi_epu32(my256_mullo_epu32(my256_mullo_epu32(a, b), r), m1));
}
__attribute__((target("avx2"))) inline __m256i montgomery_add_256(
const __m256i &a, const __m256i &b, const __m256i &m2, const __m256i &m0) {
__m256i ret = _mm256_sub_epi32(_mm256_add_epi32(a, b), m2);
return _mm256_add_epi32(_mm256_and_si256(_mm256_cmpgt_epi32(m0, ret), m2),
ret);
}
__attribute__((target("avx2"))) inline __m256i montgomery_sub_256(
const __m256i &a, const __m256i &b, const __m256i &m2, const __m256i &m0) {
__m256i ret = _mm256_sub_epi32(a, b);
return _mm256_add_epi32(_mm256_and_si256(_mm256_cmpgt_epi32(m0, ret), m2),
ret);
}
#line 4 "ntt/ntt-avx2.hpp"
namespace ntt_inner {
using u64 = uint64_t;
constexpr uint32_t get_pr(uint32_t mod) {
if (mod == 2) return 1;
u64 ds[32] = {};
int idx = 0;
u64 m = mod - 1;
for (u64 i = 2; i * i <= m; ++i) {
if (m % i == 0) {
ds[idx++] = i;
while (m % i == 0) m /= i;
}
}
if (m != 1) ds[idx++] = m;
uint32_t pr = 2;
while (1) {
int flg = 1;
for (int i = 0; i < idx; ++i) {
u64 a = pr, b = (mod - 1) / ds[i], r = 1;
while (b) {
if (b & 1) r = r * a % mod;
a = a * a % mod;
b >>= 1;
}
if (r == 1) {
flg = 0;
break;
}
}
if (flg == 1) break;
++pr;
}
return pr;
}
constexpr int SZ_FFT_BUF = 1 << 23;
uint32_t _buf1[SZ_FFT_BUF] __attribute__((aligned(64)));
uint32_t _buf2[SZ_FFT_BUF] __attribute__((aligned(64)));
} // namespace ntt_inner
template <typename mint>
struct NTT {
static constexpr uint32_t mod = mint::get_mod();
static constexpr uint32_t pr = ntt_inner::get_pr(mint::get_mod());
static constexpr int level = __builtin_ctzll(mod - 1);
mint dw[level], dy[level];
mint *buf1, *buf2;
constexpr NTT() {
setwy(level);
union raw_cast {
mint dat;
uint32_t _;
};
buf1 = &(((raw_cast *)(ntt_inner::_buf1))->dat);
buf2 = &(((raw_cast *)(ntt_inner::_buf2))->dat);
}
constexpr void setwy(int k) {
mint w[level], y[level];
w[k - 1] = mint(pr).pow((mod - 1) / (1 << k));
y[k - 1] = w[k - 1].inverse();
for (int i = k - 2; i > 0; --i)
w[i] = w[i + 1] * w[i + 1], y[i] = y[i + 1] * y[i + 1];
dw[0] = dy[0] = w[1] * w[1];
dw[1] = w[1], dy[1] = y[1], dw[2] = w[2], dy[2] = y[2];
for (int i = 3; i < k; ++i) {
dw[i] = dw[i - 1] * y[i - 2] * w[i];
dy[i] = dy[i - 1] * w[i - 2] * y[i];
}
}
__attribute__((target("avx2"))) void ntt(mint *a, int n) {
int k = n ? __builtin_ctz(n) : 0;
if (k == 0) return;
if (k == 1) {
mint a1 = a[1];
a[1] = a[0] - a[1];
a[0] = a[0] + a1;
return;
}
if (k & 1) {
int v = 1 << (k - 1);
if (v < 8) {
for (int j = 0; j < v; ++j) {
mint ajv = a[j + v];
a[j + v] = a[j] - ajv;
a[j] += ajv;
}
} else {
const __m256i m0 = _mm256_set1_epi32(0);
const __m256i m2 = _mm256_set1_epi32(mod + mod);
int j0 = 0;
int j1 = v;
for (; j0 < v; j0 += 8, j1 += 8) {
__m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
__m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
__m256i naj = montgomery_add_256(T0, T1, m2, m0);
__m256i najv = montgomery_sub_256(T0, T1, m2, m0);
_mm256_storeu_si256((__m256i *)(a + j0), naj);
_mm256_storeu_si256((__m256i *)(a + j1), najv);
}
}
}
int u = 1 << (2 + (k & 1));
int v = 1 << (k - 2 - (k & 1));
mint one = mint(1);
mint imag = dw[1];
while (v) {
if (v == 1) {
mint ww = one, xx = one, wx = one;
for (int jh = 0; jh < u;) {
ww = xx * xx, wx = ww * xx;
mint t0 = a[jh + 0], t1 = a[jh + 1] * xx;
mint t2 = a[jh + 2] * ww, t3 = a[jh + 3] * wx;
mint t0p2 = t0 + t2, t1p3 = t1 + t3;
mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
a[jh + 0] = t0p2 + t1p3, a[jh + 1] = t0p2 - t1p3;
a[jh + 2] = t0m2 + t1m3, a[jh + 3] = t0m2 - t1m3;
xx *= dw[__builtin_ctz((jh += 4))];
}
} else if (v == 4) {
const __m128i m0 = _mm_set1_epi32(0);
const __m128i m1 = _mm_set1_epi32(mod);
const __m128i m2 = _mm_set1_epi32(mod + mod);
const __m128i r = _mm_set1_epi32(mint::r);
const __m128i Imag = _mm_set1_epi32(imag.a);
mint ww = one, xx = one, wx = one;
for (int jh = 0; jh < u;) {
if (jh == 0) {
int j0 = 0;
int j1 = v;
int j2 = j1 + v;
int j3 = j2 + v;
int je = v;
for (; j0 < je; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
const __m128i T0P2 = montgomery_add_128(T0, T2, m2, m0);
const __m128i T1P3 = montgomery_add_128(T1, T3, m2, m0);
const __m128i T0M2 = montgomery_sub_128(T0, T2, m2, m0);
const __m128i T1M3 = montgomery_mul_128(
montgomery_sub_128(T1, T3, m2, m0), Imag, r, m1);
_mm_storeu_si128((__m128i *)(a + j0),
montgomery_add_128(T0P2, T1P3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j1),
montgomery_sub_128(T0P2, T1P3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j2),
montgomery_add_128(T0M2, T1M3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j3),
montgomery_sub_128(T0M2, T1M3, m2, m0));
}
} else {
ww = xx * xx, wx = ww * xx;
const __m128i WW = _mm_set1_epi32(ww.a);
const __m128i WX = _mm_set1_epi32(wx.a);
const __m128i XX = _mm_set1_epi32(xx.a);
int j0 = jh * v;
int j1 = j0 + v;
int j2 = j1 + v;
int j3 = j2 + v;
int je = j1;
for (; j0 < je; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
const __m128i MT1 = montgomery_mul_128(T1, XX, r, m1);
const __m128i MT2 = montgomery_mul_128(T2, WW, r, m1);
const __m128i MT3 = montgomery_mul_128(T3, WX, r, m1);
const __m128i T0P2 = montgomery_add_128(T0, MT2, m2, m0);
const __m128i T1P3 = montgomery_add_128(MT1, MT3, m2, m0);
const __m128i T0M2 = montgomery_sub_128(T0, MT2, m2, m0);
const __m128i T1M3 = montgomery_mul_128(
montgomery_sub_128(MT1, MT3, m2, m0), Imag, r, m1);
_mm_storeu_si128((__m128i *)(a + j0),
montgomery_add_128(T0P2, T1P3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j1),
montgomery_sub_128(T0P2, T1P3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j2),
montgomery_add_128(T0M2, T1M3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j3),
montgomery_sub_128(T0M2, T1M3, m2, m0));
}
}
xx *= dw[__builtin_ctz((jh += 4))];
}
} else {
const __m256i m0 = _mm256_set1_epi32(0);
const __m256i m1 = _mm256_set1_epi32(mod);
const __m256i m2 = _mm256_set1_epi32(mod + mod);
const __m256i r = _mm256_set1_epi32(mint::r);
const __m256i Imag = _mm256_set1_epi32(imag.a);
mint ww = one, xx = one, wx = one;
for (int jh = 0; jh < u;) {
if (jh == 0) {
int j0 = 0;
int j1 = v;
int j2 = j1 + v;
int j3 = j2 + v;
int je = v;
for (; j0 < je; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
const __m256i T0P2 = montgomery_add_256(T0, T2, m2, m0);
const __m256i T1P3 = montgomery_add_256(T1, T3, m2, m0);
const __m256i T0M2 = montgomery_sub_256(T0, T2, m2, m0);
const __m256i T1M3 = montgomery_mul_256(
montgomery_sub_256(T1, T3, m2, m0), Imag, r, m1);
_mm256_storeu_si256((__m256i *)(a + j0),
montgomery_add_256(T0P2, T1P3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j1),
montgomery_sub_256(T0P2, T1P3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j2),
montgomery_add_256(T0M2, T1M3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j3),
montgomery_sub_256(T0M2, T1M3, m2, m0));
}
} else {
ww = xx * xx, wx = ww * xx;
const __m256i WW = _mm256_set1_epi32(ww.a);
const __m256i WX = _mm256_set1_epi32(wx.a);
const __m256i XX = _mm256_set1_epi32(xx.a);
int j0 = jh * v;
int j1 = j0 + v;
int j2 = j1 + v;
int j3 = j2 + v;
int je = j1;
for (; j0 < je; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
const __m256i MT1 = montgomery_mul_256(T1, XX, r, m1);
const __m256i MT2 = montgomery_mul_256(T2, WW, r, m1);
const __m256i MT3 = montgomery_mul_256(T3, WX, r, m1);
const __m256i T0P2 = montgomery_add_256(T0, MT2, m2, m0);
const __m256i T1P3 = montgomery_add_256(MT1, MT3, m2, m0);
const __m256i T0M2 = montgomery_sub_256(T0, MT2, m2, m0);
const __m256i T1M3 = montgomery_mul_256(
montgomery_sub_256(MT1, MT3, m2, m0), Imag, r, m1);
_mm256_storeu_si256((__m256i *)(a + j0),
montgomery_add_256(T0P2, T1P3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j1),
montgomery_sub_256(T0P2, T1P3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j2),
montgomery_add_256(T0M2, T1M3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j3),
montgomery_sub_256(T0M2, T1M3, m2, m0));
}
}
xx *= dw[__builtin_ctz((jh += 4))];
}
}
u <<= 2;
v >>= 2;
}
}
__attribute__((target("avx2"))) void intt(mint *a, int n,
int normalize = true) {
int k = n ? __builtin_ctz(n) : 0;
if (k == 0) return;
if (k == 1) {
mint a1 = a[1];
a[1] = a[0] - a[1];
a[0] = a[0] + a1;
if (normalize) {
a[0] *= mint(2).inverse();
a[1] *= mint(2).inverse();
}
return;
}
int u = 1 << (k - 2);
int v = 1;
mint one = mint(1);
mint imag = dy[1];
while (u) {
if (v == 1) {
mint ww = one, xx = one, yy = one;
u <<= 2;
for (int jh = 0; jh < u;) {
ww = xx * xx, yy = xx * imag;
mint t0 = a[jh + 0], t1 = a[jh + 1];
mint t2 = a[jh + 2], t3 = a[jh + 3];
mint t0p1 = t0 + t1, t2p3 = t2 + t3;
mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy;
a[jh + 0] = t0p1 + t2p3, a[jh + 2] = (t0p1 - t2p3) * ww;
a[jh + 1] = t0m1 + t2m3, a[jh + 3] = (t0m1 - t2m3) * ww;
xx *= dy[__builtin_ctz(jh += 4)];
}
} else if (v == 4) {
const __m128i m0 = _mm_set1_epi32(0);
const __m128i m1 = _mm_set1_epi32(mod);
const __m128i m2 = _mm_set1_epi32(mod + mod);
const __m128i r = _mm_set1_epi32(mint::r);
const __m128i Imag = _mm_set1_epi32(imag.a);
mint ww = one, xx = one, yy = one;
u <<= 2;
for (int jh = 0; jh < u;) {
if (jh == 0) {
int j0 = 0;
int j1 = v;
int j2 = v + v;
int j3 = j2 + v;
for (; j0 < v; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
const __m128i T0P1 = montgomery_add_128(T0, T1, m2, m0);
const __m128i T2P3 = montgomery_add_128(T2, T3, m2, m0);
const __m128i T0M1 = montgomery_sub_128(T0, T1, m2, m0);
const __m128i T2M3 = montgomery_mul_128(
montgomery_sub_128(T2, T3, m2, m0), Imag, r, m1);
_mm_storeu_si128((__m128i *)(a + j0),
montgomery_add_128(T0P1, T2P3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j2),
montgomery_sub_128(T0P1, T2P3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j1),
montgomery_add_128(T0M1, T2M3, m2, m0));
_mm_storeu_si128((__m128i *)(a + j3),
montgomery_sub_128(T0M1, T2M3, m2, m0));
}
} else {
ww = xx * xx, yy = xx * imag;
const __m128i WW = _mm_set1_epi32(ww.a);
const __m128i XX = _mm_set1_epi32(xx.a);
const __m128i YY = _mm_set1_epi32(yy.a);
int j0 = jh * v;
int j1 = j0 + v;
int j2 = j1 + v;
int j3 = j2 + v;
int je = j1;
for (; j0 < je; j0 += 4, j1 += 4, j2 += 4, j3 += 4) {
const __m128i T0 = _mm_loadu_si128((__m128i *)(a + j0));
const __m128i T1 = _mm_loadu_si128((__m128i *)(a + j1));
const __m128i T2 = _mm_loadu_si128((__m128i *)(a + j2));
const __m128i T3 = _mm_loadu_si128((__m128i *)(a + j3));
const __m128i T0P1 = montgomery_add_128(T0, T1, m2, m0);
const __m128i T2P3 = montgomery_add_128(T2, T3, m2, m0);
const __m128i T0M1 = montgomery_mul_128(
montgomery_sub_128(T0, T1, m2, m0), XX, r, m1);
__m128i T2M3 = montgomery_mul_128(
montgomery_sub_128(T2, T3, m2, m0), YY, r, m1);
_mm_storeu_si128((__m128i *)(a + j0),
montgomery_add_128(T0P1, T2P3, m2, m0));
_mm_storeu_si128(
(__m128i *)(a + j2),
montgomery_mul_128(montgomery_sub_128(T0P1, T2P3, m2, m0), WW,
r, m1));
_mm_storeu_si128((__m128i *)(a + j1),
montgomery_add_128(T0M1, T2M3, m2, m0));
_mm_storeu_si128(
(__m128i *)(a + j3),
montgomery_mul_128(montgomery_sub_128(T0M1, T2M3, m2, m0), WW,
r, m1));
}
}
xx *= dy[__builtin_ctz(jh += 4)];
}
} else {
const __m256i m0 = _mm256_set1_epi32(0);
const __m256i m1 = _mm256_set1_epi32(mod);
const __m256i m2 = _mm256_set1_epi32(mod + mod);
const __m256i r = _mm256_set1_epi32(mint::r);
const __m256i Imag = _mm256_set1_epi32(imag.a);
mint ww = one, xx = one, yy = one;
u <<= 2;
for (int jh = 0; jh < u;) {
if (jh == 0) {
int j0 = 0;
int j1 = v;
int j2 = v + v;
int j3 = j2 + v;
for (; j0 < v; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
const __m256i T0P1 = montgomery_add_256(T0, T1, m2, m0);
const __m256i T2P3 = montgomery_add_256(T2, T3, m2, m0);
const __m256i T0M1 = montgomery_sub_256(T0, T1, m2, m0);
const __m256i T2M3 = montgomery_mul_256(
montgomery_sub_256(T2, T3, m2, m0), Imag, r, m1);
_mm256_storeu_si256((__m256i *)(a + j0),
montgomery_add_256(T0P1, T2P3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j2),
montgomery_sub_256(T0P1, T2P3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j1),
montgomery_add_256(T0M1, T2M3, m2, m0));
_mm256_storeu_si256((__m256i *)(a + j3),
montgomery_sub_256(T0M1, T2M3, m2, m0));
}
} else {
ww = xx * xx, yy = xx * imag;
const __m256i WW = _mm256_set1_epi32(ww.a);
const __m256i XX = _mm256_set1_epi32(xx.a);
const __m256i YY = _mm256_set1_epi32(yy.a);
int j0 = jh * v;
int j1 = j0 + v;
int j2 = j1 + v;
int j3 = j2 + v;
int je = j1;
for (; j0 < je; j0 += 8, j1 += 8, j2 += 8, j3 += 8) {
const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
const __m256i T2 = _mm256_loadu_si256((__m256i *)(a + j2));
const __m256i T3 = _mm256_loadu_si256((__m256i *)(a + j3));
const __m256i T0P1 = montgomery_add_256(T0, T1, m2, m0);
const __m256i T2P3 = montgomery_add_256(T2, T3, m2, m0);
const __m256i T0M1 = montgomery_mul_256(
montgomery_sub_256(T0, T1, m2, m0), XX, r, m1);
const __m256i T2M3 = montgomery_mul_256(
montgomery_sub_256(T2, T3, m2, m0), YY, r, m1);
_mm256_storeu_si256((__m256i *)(a + j0),
montgomery_add_256(T0P1, T2P3, m2, m0));
_mm256_storeu_si256(
(__m256i *)(a + j2),
montgomery_mul_256(montgomery_sub_256(T0P1, T2P3, m2, m0), WW,
r, m1));
_mm256_storeu_si256((__m256i *)(a + j1),
montgomery_add_256(T0M1, T2M3, m2, m0));
_mm256_storeu_si256(
(__m256i *)(a + j3),
montgomery_mul_256(montgomery_sub_256(T0M1, T2M3, m2, m0), WW,
r, m1));
}
}
xx *= dy[__builtin_ctz(jh += 4)];
}
}
u >>= 4;
v <<= 2;
}
if (k & 1) {
v = 1 << (k - 1);
if (v < 8) {
for (int j = 0; j < v; ++j) {
mint ajv = a[j] - a[j + v];
a[j] += a[j + v];
a[j + v] = ajv;
}
} else {
const __m256i m0 = _mm256_set1_epi32(0);
const __m256i m2 = _mm256_set1_epi32(mod + mod);
int j0 = 0;
int j1 = v;
for (; j0 < v; j0 += 8, j1 += 8) {
const __m256i T0 = _mm256_loadu_si256((__m256i *)(a + j0));
const __m256i T1 = _mm256_loadu_si256((__m256i *)(a + j1));
__m256i naj = montgomery_add_256(T0, T1, m2, m0);
__m256i najv = montgomery_sub_256(T0, T1, m2, m0);
_mm256_storeu_si256((__m256i *)(a + j0), naj);
_mm256_storeu_si256((__m256i *)(a + j1), najv);
}
}
}
if (normalize) {
mint invn = mint(n).inverse();
for (int i = 0; i < n; i++) a[i] *= invn;
}
}
__attribute__((target("avx2"))) void inplace_multiply(
int l1, int l2, int zero_padding = true) {
int l = l1 + l2 - 1;
int M = 4;
while (M < l) M <<= 1;
if (zero_padding) {
for (int i = l1; i < M; i++) ntt_inner::_buf1[i] = 0;
for (int i = l2; i < M; i++) ntt_inner::_buf2[i] = 0;
}
const __m256i m0 = _mm256_set1_epi32(0);
const __m256i m1 = _mm256_set1_epi32(mod);
const __m256i r = _mm256_set1_epi32(mint::r);
const __m256i N2 = _mm256_set1_epi32(mint::n2);
for (int i = 0; i < l1; i += 8) {
__m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf1 + i));
__m256i b = montgomery_mul_256(a, N2, r, m1);
_mm256_storeu_si256((__m256i *)(ntt_inner::_buf1 + i), b);
}
for (int i = 0; i < l2; i += 8) {
__m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf2 + i));
__m256i b = montgomery_mul_256(a, N2, r, m1);
_mm256_storeu_si256((__m256i *)(ntt_inner::_buf2 + i), b);
}
ntt(buf1, M);
ntt(buf2, M);
for (int i = 0; i < M; i += 8) {
__m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf1 + i));
__m256i b = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf2 + i));
__m256i c = montgomery_mul_256(a, b, r, m1);
_mm256_storeu_si256((__m256i *)(ntt_inner::_buf1 + i), c);
}
intt(buf1, M, false);
const __m256i INVM = _mm256_set1_epi32((mint(M).inverse()).a);
for (int i = 0; i < l; i += 8) {
__m256i a = _mm256_loadu_si256((__m256i *)(ntt_inner::_buf1 + i));
__m256i b = montgomery_mul_256(a, INVM, r, m1);
__m256i c = my256_mulhi_epu32(my256_mullo_epu32(b, r), m1);
__m256i d = _mm256_and_si256(_mm256_cmpgt_epi32(c, m0), m1);
__m256i e = _mm256_sub_epi32(d, c);
_mm256_storeu_si256((__m256i *)(ntt_inner::_buf1 + i), e);
}
}
void ntt(vector<mint> &a) {
int M = (int)a.size();
for (int i = 0; i < M; i++) buf1[i].a = a[i].a;
ntt(buf1, M);
for (int i = 0; i < M; i++) a[i].a = buf1[i].a;
}
void intt(vector<mint> &a) {
int M = (int)a.size();
for (int i = 0; i < M; i++) buf1[i].a = a[i].a;
intt(buf1, M, true);
for (int i = 0; i < M; i++) a[i].a = buf1[i].a;
}
vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) {
if (a.size() == 0 && b.size() == 0) return vector<mint>{};
int l = a.size() + b.size() - 1;
if (min<int>(a.size(), b.size()) <= 40) {
vector<mint> s(l);
for (int i = 0; i < (int)a.size(); ++i)
for (int j = 0; j < (int)b.size(); ++j) s[i + j] += a[i] * b[j];
return s;
}
assert(l <= ntt_inner::SZ_FFT_BUF);
int M = 4;
while (M < l) M <<= 1;
for (int i = 0; i < (int)a.size(); ++i) buf1[i].a = a[i].a;
for (int i = (int)a.size(); i < M; ++i) buf1[i].a = 0;
for (int i = 0; i < (int)b.size(); ++i) buf2[i].a = b[i].a;
for (int i = (int)b.size(); i < M; ++i) buf2[i].a = 0;
ntt(buf1, M);
ntt(buf2, M);
for (int i = 0; i < M; ++i)
buf1[i].a = mint::reduce(uint64_t(buf1[i].a) * buf2[i].a);
intt(buf1, M, false);
vector<mint> s(l);
mint invm = mint(M).inverse();
for (int i = 0; i < l; ++i) s[i] = buf1[i] * invm;
return s;
}
void ntt_doubling(vector<mint> &a) {
int M = (int)a.size();
for (int i = 0; i < M; i++) buf1[i].a = a[i].a;
intt(buf1, M);
mint r = 1, zeta = mint(pr).pow((mint::get_mod() - 1) / (M << 1));
for (int i = 0; i < M; i++) buf1[i] *= r, r *= zeta;
ntt(buf1, M);
a.resize(2 * M);
for (int i = 0; i < M; i++) a[M + i].a = buf1[i].a;
}
};
#line 5 "fps/ntt-friendly-fps.hpp"
template <typename mint>
void FormalPowerSeries<mint>::set_fft() {
if (!ntt_ptr) ntt_ptr = new NTT<mint>;
}
template <typename mint>
FormalPowerSeries<mint>& FormalPowerSeries<mint>::operator*=(
const FormalPowerSeries<mint>& r) {
if (this->empty() || r.empty()) {
this->clear();
return *this;
}
set_fft();
auto ret = static_cast<NTT<mint>*>(ntt_ptr)->multiply(*this, r);
return *this = FormalPowerSeries<mint>(ret.begin(), ret.end());
}
template <typename mint>
void FormalPowerSeries<mint>::ntt() {
set_fft();
static_cast<NTT<mint>*>(ntt_ptr)->ntt(*this);
}
template <typename mint>
void FormalPowerSeries<mint>::intt() {
set_fft();
static_cast<NTT<mint>*>(ntt_ptr)->intt(*this);
}
template <typename mint>
void FormalPowerSeries<mint>::ntt_doubling() {
set_fft();
static_cast<NTT<mint>*>(ntt_ptr)->ntt_doubling(*this);
}
template <typename mint>
int FormalPowerSeries<mint>::ntt_pr() {
set_fft();
return static_cast<NTT<mint>*>(ntt_ptr)->pr;
}
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::inv(int deg) const {
assert((*this)[0] != mint(0));
if (deg == -1) deg = (int)this->size();
FormalPowerSeries<mint> res(deg);
res[0] = {mint(1) / (*this)[0]};
for (int d = 1; d < deg; d <<= 1) {
FormalPowerSeries<mint> f(2 * d), g(2 * d);
for (int j = 0; j < min((int)this->size(), 2 * d); j++) f[j] = (*this)[j];
for (int j = 0; j < d; j++) g[j] = res[j];
f.ntt();
g.ntt();
for (int j = 0; j < 2 * d; j++) f[j] *= g[j];
f.intt();
for (int j = 0; j < d; j++) f[j] = 0;
f.ntt();
for (int j = 0; j < 2 * d; j++) f[j] *= g[j];
f.intt();
for (int j = d; j < min(2 * d, deg); j++) res[j] = -f[j];
}
return res.pre(deg);
}
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::exp(int deg) const {
using fps = FormalPowerSeries<mint>;
assert((*this).size() == 0 || (*this)[0] == mint(0));
if (deg == -1) deg = this->size();
fps inv;
inv.reserve(deg + 1);
inv.push_back(mint(0));
inv.push_back(mint(1));
auto inplace_integral = [&](fps& F) -> void {
const int n = (int)F.size();
auto mod = mint::get_mod();
while ((int)inv.size() <= n) {
int i = inv.size();
inv.push_back((-inv[mod % i]) * (mod / i));
}
F.insert(begin(F), mint(0));
for (int i = 1; i <= n; i++) F[i] *= inv[i];
};
auto inplace_diff = [](fps& F) -> void {
if (F.empty()) return;
F.erase(begin(F));
mint coeff = 1, one = 1;
for (int i = 0; i < (int)F.size(); i++) {
F[i] *= coeff;
coeff += one;
}
};
fps b{1, 1 < (int)this->size() ? (*this)[1] : 0}, c{1}, z1, z2{1, 1};
for (int m = 2; m < deg; m *= 2) {
auto y = b;
y.resize(2 * m);
y.ntt();
z1 = z2;
fps z(m);
for (int i = 0; i < m; ++i) z[i] = y[i] * z1[i];
z.intt();
fill(begin(z), begin(z) + m / 2, mint(0));
z.ntt();
for (int i = 0; i < m; ++i) z[i] *= -z1[i];
z.intt();
c.insert(end(c), begin(z) + m / 2, end(z));
z2 = c;
z2.resize(2 * m);
z2.ntt();
fps x(begin(*this), begin(*this) + min<int>(this->size(), m));
x.resize(m);
inplace_diff(x);
x.push_back(mint(0));
x.ntt();
for (int i = 0; i < m; ++i) x[i] *= y[i];
x.intt();
x -= b.diff();
x.resize(2 * m);
for (int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = mint(0);
x.ntt();
for (int i = 0; i < 2 * m; ++i) x[i] *= z2[i];
x.intt();
x.pop_back();
inplace_integral(x);
for (int i = m; i < min<int>(this->size(), 2 * m); ++i) x[i] += (*this)[i];
fill(begin(x), begin(x) + m, mint(0));
x.ntt();
for (int i = 0; i < 2 * m; ++i) x[i] *= y[i];
x.intt();
b.insert(end(b), begin(x) + m, end(x));
}
return fps{begin(b), begin(b) + deg};
}
/**
* @brief NTT mod用FPSライブラリ
* @docs docs/fps/ntt-friendly-fps.md
*/
#line 2 "modint/montgomery-modint.hpp"
template <uint32_t mod>
struct LazyMontgomeryModInt {
using mint = LazyMontgomeryModInt;
using i32 = int32_t;
using u32 = uint32_t;
using u64 = uint64_t;
static constexpr u32 get_r() {
u32 ret = mod;
for (i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret;
return ret;
}
static constexpr u32 r = get_r();
static constexpr u32 n2 = -u64(mod) % mod;
static_assert(r * mod == 1, "invalid, r * mod != 1");
static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30");
static_assert((mod & 1) == 1, "invalid, mod % 2 == 0");
u32 a;
constexpr LazyMontgomeryModInt() : a(0) {}
constexpr LazyMontgomeryModInt(const int64_t &b)
: a(reduce(u64(b % mod + mod) * n2)){};
static constexpr u32 reduce(const u64 &b) {
return (b + u64(u32(b) * u32(-r)) * mod) >> 32;
}
constexpr mint &operator+=(const mint &b) {
if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod;
return *this;
}
constexpr mint &operator-=(const mint &b) {
if (i32(a -= b.a) < 0) a += 2 * mod;
return *this;
}
constexpr mint &operator*=(const mint &b) {
a = reduce(u64(a) * b.a);
return *this;
}
constexpr mint &operator/=(const mint &b) {
*this *= b.inverse();
return *this;
}
constexpr mint operator+(const mint &b) const { return mint(*this) += b; }
constexpr mint operator-(const mint &b) const { return mint(*this) -= b; }
constexpr mint operator*(const mint &b) const { return mint(*this) *= b; }
constexpr mint operator/(const mint &b) const { return mint(*this) /= b; }
constexpr bool operator==(const mint &b) const {
return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
}
constexpr bool operator!=(const mint &b) const {
return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
}
constexpr mint operator-() const { return mint() - mint(*this); }
constexpr mint pow(u64 n) const {
mint ret(1), mul(*this);
while (n > 0) {
if (n & 1) ret *= mul;
mul *= mul;
n >>= 1;
}
return ret;
}
constexpr mint inverse() const { return pow(mod - 2); }
friend ostream &operator<<(ostream &os, const mint &b) {
return os << b.get();
}
friend istream &operator>>(istream &is, mint &b) {
int64_t t;
is >> t;
b = LazyMontgomeryModInt<mod>(t);
return (is);
}
constexpr u32 get() const {
u32 ret = reduce(a);
return ret >= mod ? ret - mod : ret;
}
static constexpr u32 get_mod() { return mod; }
};
#line 2 "fps/polynomial-interpolation.hpp"
#line 5 "fps/polynomial-interpolation.hpp"
template <class mint>
FormalPowerSeries<mint> PolynomialInterpolation(const vector<mint> &xs,
const vector<mint> &ys) {
using fps = FormalPowerSeries<mint>;
assert(xs.size() == ys.size());
ProductTree<mint> ptree(xs);
fps w = ptree.buf[1].diff();
vector<mint> vs = InnerMultipointEvaluation<mint>(w, xs, ptree);
auto rec = [&](auto self, int idx) -> fps {
if (idx >= ptree.N) {
if (idx - ptree.N < (int)xs.size())
return {ys[idx - ptree.N] / vs[idx - ptree.N]};
else
return {mint(1)};
}
if (ptree.buf[idx << 1 | 0].empty())
return {};
else if (ptree.buf[idx << 1 | 1].empty())
return self(self, idx << 1 | 0);
return self(self, idx << 1 | 0) * ptree.buf[idx << 1 | 1] +
self(self, idx << 1 | 1) * ptree.buf[idx << 1 | 0];
};
return rec(rec, 1);
}
// https://nyaannyaan.github.io/library/verify/verify-yosupo-fps/yosupo-interpolation.test.cpp
using mint = LazyMontgomeryModInt<998244353>;
using fps = FormalPowerSeries<mint>;
int main(){
LL(n,X);
fps x(n),y(n);
rep(i,n)cin >> x[i] >> y[i];
fps res(n-1);
rep(i,n){
fps a,b;
rep(j,n){
if(i==j)continue;
a.push_back(x[j]);
b.push_back(y[j]);
}
auto X = PolynomialInterpolation(a,b);
rep(i,X.size()){
res[i] += X[i];
}
}
cout << res.eval(X) << endl;
return 0;
}