結果
| 問題 | No.1938 Lagrange Sum |
| ユーザー |
 noshi91
|
| 提出日時 | 2022-05-13 22:47:41 |
| 言語 | C++17 (gcc 12.3.0 + boost 1.83.0) |
| 結果 |
AC
|
| 実行時間 | 108 ms / 3,000 ms |
| コード長 | 24,683 bytes |
| コンパイル時間 | 3,302 ms |
| コンパイル使用メモリ | 228,224 KB |
| 実行使用メモリ | 7,680 KB |
| 最終ジャッジ日時 | 2024-07-22 02:59:51 |
| 合計ジャッジ時間 | 5,860 ms |
|
ジャッジサーバーID （参考情報） |
judge1 / judge2 |
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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 | AC | 108 ms
7,552 KB |
| testcase_01 | AC | 107 ms
7,552 KB |
| testcase_02 | AC | 107 ms
7,680 KB |
| testcase_03 | AC | 2 ms
6,940 KB |
| testcase_04 | AC | 2 ms
6,940 KB |
| testcase_05 | AC | 77 ms
6,944 KB |
| testcase_06 | AC | 94 ms
7,296 KB |
| testcase_07 | AC | 3 ms
6,940 KB |
| testcase_08 | AC | 35 ms
6,944 KB |
| testcase_09 | AC | 18 ms
6,944 KB |
| testcase_10 | AC | 106 ms
7,680 KB |
| testcase_11 | AC | 36 ms
6,940 KB |
| testcase_12 | AC | 107 ms
7,552 KB |
| testcase_13 | AC | 107 ms
7,552 KB |
| testcase_14 | AC | 58 ms
6,940 KB |
| testcase_15 | AC | 14 ms
6,940 KB |
| testcase_16 | AC | 105 ms
7,544 KB |
| testcase_17 | AC | 56 ms
6,940 KB |
| testcase_18 | AC | 56 ms
6,944 KB |
| testcase_19 | AC | 105 ms
7,552 KB |
| testcase_20 | AC | 106 ms
7,680 KB |
| testcase_21 | AC | 9 ms
6,940 KB |
| testcase_22 | AC | 2 ms
6,940 KB |
| testcase_23 | AC | 9 ms
6,944 KB |
| testcase_24 | AC | 105 ms
7,540 KB |
| testcase_25 | AC | 2 ms
6,940 KB |
| testcase_26 | AC | 2 ms
6,940 KB |
| testcase_27 | AC | 2 ms
6,944 KB |
ソースコード
#include <bits/stdc++.h>
#pragma GCC target("popcnt")
#include <array>
#include <cstddef>
#ifndef __GNUC__
int __builtin_ctz(const unsigned int c) noexcept {
static constexpr std::array<std::uint8_t, 32> table = {
0, 1, 2, 6, 3, 11, 7, 16, 4, 14, 12, 21, 8, 23, 17, 26,
31, 5, 10, 15, 13, 20, 22, 25, 30, 9, 19, 24, 29, 18, 28, 27 };
return table[(c & ~c + 1) * 0x4653ADF >> 27 & 0x1F];
}
int __builtin_ctzll(const unsigned long long c) noexcept {
static constexpr std::array<std::uint8_t, 64> table = {
0, 1, 2, 7, 3, 13, 8, 27, 4, 33, 14, 36, 9, 49, 28, 19,
5, 25, 34, 17, 15, 53, 37, 55, 10, 46, 50, 39, 29, 42, 20, 57,
63, 6, 12, 26, 32, 35, 48, 18, 24, 16, 52, 54, 45, 38, 41, 56,
62, 11, 31, 47, 23, 51, 44, 40, 61, 30, 22, 43, 60, 21, 59, 58 };
return table[(c & ~c + 1) * 0x218A7A392DD9ABFULL >> 58 & 0x3F];
}
int __builtin_clz(unsigned int c) noexcept {
static constexpr std::array<std::uint8_t, 32> table = {
0, 1, 2, 6, 3, 11, 7, 16, 4, 14, 12, 21, 8, 23, 17, 26,
31, 5, 10, 15, 13, 20, 22, 25, 30, 9, 19, 24, 29, 18, 28, 27 };
c |= c >> 1;
c |= c >> 2;
c |= c >> 4;
c |= c >> 8;
c |= c >> 16;
return table[((c >> 1) + 1) * 0x4653ADF >> 27 & 0x1F];
}
int __builtin_clzll(unsigned long long c) noexcept {
static constexpr std::array<std::uint8_t, 64> table = {
0, 1, 2, 7, 3, 13, 8, 27, 4, 33, 14, 36, 9, 49, 28, 19,
5, 25, 34, 17, 15, 53, 37, 55, 10, 46, 50, 39, 29, 42, 20, 57,
63, 6, 12, 26, 32, 35, 48, 18, 24, 16, 52, 54, 45, 38, 41, 56,
62, 11, 31, 47, 23, 51, 44, 40, 61, 30, 22, 43, 60, 21, 59, 58 };
c |= c >> 1;
c |= c >> 2;
c |= c >> 4;
c |= c >> 8;
c |= c >> 16;
c |= c >> 32;
return table[((c >> 1) + 1) * 0x218A7A392DD9ABFULL >> 58 & 0x3F];
}
constexpr int __builtin_popcount(unsigned int c) noexcept {
c -= c >> 1 & 0x55555555;
c = (c & 0x33333333) + (c >> 2 & 0x33333333);
c = (c + (c >> 4)) & 0x0F0F0F0F;
return c * 0x01010101 >> 24 & 0x3F;
}
constexpr int __builtin_popcountll(unsigned long long c) noexcept {
c -= c >> 1 & 0x5555555555555555;
c = (c & 0x3333333333333333) + (c >> 2 & 0x3333333333333333);
c = (c + (c >> 4)) & 0x0F0F0F0F0F0F0F0F;
return c * 0x0101010101010101 >> 56 & 0x7f;
}
constexpr bool __builtin_parity(unsigned int c) noexcept {
c ^= c >> 1;
c ^= c >> 2;
return ((c & 0x11111111) * 0x11111111 >> 28 & 0x1) != 0;
}
constexpr bool __builtin_parityll(unsigned long long c) noexcept {
c ^= c >> 1;
c ^= c >> 2;
return ((c & 0x1111111111111111) * 0x1111111111111111 >> 60 & 0x1) != 0;
}
#endif
using namespace std;
// https://ei1333.github.io/library/test/verify/yosupo-polynomial-interpolation.test.cpp
// #line 1 "test/verify/yosupo-polynomial-interpolation.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/polynomial_interpolation"
// #line 1 "template/template.cpp"
#include <bits/stdc++.h>
using namespace std;
using int64 = long long;
const int mod = 1e9 + 7;
const int64 infll = (1LL << 62) - 1;
const int inf = (1 << 30) - 1;
struct IoSetup {
IoSetup() {
cin.tie(nullptr);
ios::sync_with_stdio(false);
cout << fixed << setprecision(10);
cerr << fixed << setprecision(10);
}
} iosetup;
template <typename T1, typename T2>
ostream& operator<<(ostream& os, const pair<T1, T2>& p) {
os << p.first << " " << p.second;
return os;
}
template <typename T1, typename T2>
istream& operator>>(istream& is, pair<T1, T2>& p) {
is >> p.first >> p.second;
return is;
}
template <typename T> ostream& operator<<(ostream& os, const vector<T>& v) {
for (int i = 0; i < (int)v.size(); i++) {
os << v[i] << (i + 1 != v.size() ? " " : "");
}
return os;
}
template <typename T> istream& operator>>(istream& is, vector<T>& v) {
for (T& in : v)
is >> in;
return is;
}
template <typename T1, typename T2> inline bool chmax(T1& a, T2 b) {
return a < b && (a = b, true);
}
template <typename T1, typename T2> inline bool chmin(T1& a, T2 b) {
return a > b && (a = b, true);
}
template <typename T = int64> vector<T> make_v(size_t a) {
return vector<T>(a);
}
template <typename T, typename... Ts> auto make_v(size_t a, Ts... ts) {
return vector<decltype(make_v<T>(ts...))>(a, make_v<T>(ts...));
}
template <typename T, typename V>
typename enable_if<is_class<T>::value == 0>::type fill_v(T& t, const V& v) {
t = v;
}
template <typename T, typename V>
typename enable_if<is_class<T>::value != 0>::type fill_v(T& t, const V& v) {
for (auto& e : t)
fill_v(e, v);
}
template <typename F> struct FixPoint : F {
explicit FixPoint(F&& f) : F(forward<F>(f)) {}
template <typename... Args> decltype(auto) operator()(Args &&... args) const {
return F::operator()(*this, forward<Args>(args)...);
}
};
template <typename F> inline decltype(auto) MFP(F&& f) {
return FixPoint<F>{forward<F>(f)};
}
// #line 4 "test/verify/yosupo-polynomial-interpolation.test.cpp"
// #line 1 "math/combinatorics/mod-int.cpp"
template <int mod> struct ModInt {
int x;
ModInt() : x(0) {}
ModInt(int64_t y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {}
ModInt& operator+=(const ModInt& p) {
if ((x += p.x) >= mod)
x -= mod;
return *this;
}
ModInt& operator-=(const ModInt& p) {
if ((x += mod - p.x) >= mod)
x -= mod;
return *this;
}
ModInt& operator*=(const ModInt& p) {
x = (int)(1LL * x * p.x % mod);
return *this;
}
ModInt& operator/=(const ModInt& p) {
*this *= p.inverse();
return *this;
}
ModInt operator-() const { return ModInt(-x); }
ModInt operator+(const ModInt& p) const { return ModInt(*this) += p; }
ModInt operator-(const ModInt& p) const { return ModInt(*this) -= p; }
ModInt operator*(const ModInt& p) const { return ModInt(*this) *= p; }
ModInt operator/(const ModInt& p) const { return ModInt(*this) /= p; }
bool operator==(const ModInt& p) const { return x == p.x; }
bool operator!=(const ModInt& p) const { return x != p.x; }
ModInt inverse() const {
int a = x, b = mod, u = 1, v = 0, t;
while (b > 0) {
t = a / b;
swap(a -= t * b, b);
swap(u -= t * v, v);
}
return ModInt(u);
}
ModInt pow(int64_t n) const {
ModInt ret(1), mul(x);
while (n > 0) {
if (n & 1)
ret *= mul;
mul *= mul;
n >>= 1;
}
return ret;
}
friend ostream& operator<<(ostream& os, const ModInt& p) { return os << p.x; }
friend istream& operator>>(istream& is, ModInt& a) {
int64_t t;
is >> t;
a = ModInt<mod>(t);
return (is);
}
static int get_mod() { return mod; }
};
using modint = ModInt<mod>;
// #line 1 "math/fft/number-theoretic-transform-friendly-mod-int.cpp"
/**
* @brief Number Theoretic Transform Friendly ModInt
*/
template <typename Mint> struct NumberTheoreticTransformFriendlyModInt {
static vector<Mint> roots, iroots, rate3, irate3;
static int max_base;
NumberTheoreticTransformFriendlyModInt() = default;
static void init() {
if (roots.empty()) {
const unsigned mod = Mint::get_mod();
assert(mod >= 3 && mod % 2 == 1);
auto tmp = mod - 1;
max_base = 0;
while (tmp % 2 == 0)
tmp >>= 1, max_base++;
Mint root = 2;
while (root.pow((mod - 1) >> 1) == 1) {
root += 1;
}
assert(root.pow(mod - 1) == 1);
roots.resize(max_base + 1);
iroots.resize(max_base + 1);
rate3.resize(max_base + 1);
irate3.resize(max_base + 1);
roots[max_base] = root.pow((mod - 1) >> max_base);
iroots[max_base] = Mint(1) / roots[max_base];
for (int i = max_base - 1; i >= 0; i--) {
roots[i] = roots[i + 1] * roots[i + 1];
iroots[i] = iroots[i + 1] * iroots[i + 1];
}
{
Mint prod = 1, iprod = 1;
for (int i = 0; i <= max_base - 3; i++) {
rate3[i] = roots[i + 3] * prod;
irate3[i] = iroots[i + 3] * iprod;
prod *= iroots[i + 3];
iprod *= roots[i + 3];
}
}
}
}
static void ntt(vector<Mint>& a) {
init();
const int n = (int)a.size();
assert((n & (n - 1)) == 0);
int h = __builtin_ctz(n);
assert(h <= max_base);
int len = 0;
Mint imag = roots[2];
if (h & 1) {
int p = 1 << (h - 1);
Mint rot = 1;
for (int i = 0; i < p; i++) {
auto r = a[i + p];
a[i + p] = a[i] - r;
a[i] += r;
}
len++;
}
for (; len + 1 < h; len += 2) {
int p = 1 << (h - len - 2);
{ // s = 0
for (int i = 0; i < p; i++) {
auto a0 = a[i];
auto a1 = a[i + p];
auto a2 = a[i + 2 * p];
auto a3 = a[i + 3 * p];
auto a1na3imag = (a1 - a3) * imag;
auto a0a2 = a0 + a2;
auto a1a3 = a1 + a3;
auto a0na2 = a0 - a2;
a[i] = a0a2 + a1a3;
a[i + 1 * p] = a0a2 - a1a3;
a[i + 2 * p] = a0na2 + a1na3imag;
a[i + 3 * p] = a0na2 - a1na3imag;
}
}
Mint rot = rate3[0];
for (int s = 1; s < (1 << len); s++) {
int offset = s << (h - len);
Mint rot2 = rot * rot;
Mint rot3 = rot2 * rot;
for (int i = 0; i < p; i++) {
auto a0 = a[i + offset];
auto a1 = a[i + offset + p] * rot;
auto a2 = a[i + offset + 2 * p] * rot2;
auto a3 = a[i + offset + 3 * p] * rot3;
auto a1na3imag = (a1 - a3) * imag;
auto a0a2 = a0 + a2;
auto a1a3 = a1 + a3;
auto a0na2 = a0 - a2;
a[i + offset] = a0a2 + a1a3;
a[i + offset + 1 * p] = a0a2 - a1a3;
a[i + offset + 2 * p] = a0na2 + a1na3imag;
a[i + offset + 3 * p] = a0na2 - a1na3imag;
}
rot *= rate3[__builtin_ctz(~s)];
}
}
}
static void intt(vector<Mint>& a, bool f = true) {
init();
const int n = (int)a.size();
assert((n & (n - 1)) == 0);
int h = __builtin_ctz(n);
assert(h <= max_base);
int len = h;
Mint iimag = iroots[2];
for (; len > 1; len -= 2) {
int p = 1 << (h - len);
{ // s = 0
for (int i = 0; i < p; i++) {
auto a0 = a[i];
auto a1 = a[i + 1 * p];
auto a2 = a[i + 2 * p];
auto a3 = a[i + 3 * p];
auto a2na3iimag = (a2 - a3) * iimag;
auto a0na1 = a0 - a1;
auto a0a1 = a0 + a1;
auto a2a3 = a2 + a3;
a[i] = a0a1 + a2a3;
a[i + 1 * p] = (a0na1 + a2na3iimag);
a[i + 2 * p] = (a0a1 - a2a3);
a[i + 3 * p] = (a0na1 - a2na3iimag);
}
}
Mint irot = irate3[0];
for (int s = 1; s < (1 << (len - 2)); s++) {
int offset = s << (h - len + 2);
Mint irot2 = irot * irot;
Mint irot3 = irot2 * irot;
for (int i = 0; i < p; i++) {
auto a0 = a[i + offset];
auto a1 = a[i + offset + 1 * p];
auto a2 = a[i + offset + 2 * p];
auto a3 = a[i + offset + 3 * p];
auto a2na3iimag = (a2 - a3) * iimag;
auto a0na1 = a0 - a1;
auto a0a1 = a0 + a1;
auto a2a3 = a2 + a3;
a[i + offset] = a0a1 + a2a3;
a[i + offset + 1 * p] = (a0na1 + a2na3iimag) * irot;
a[i + offset + 2 * p] = (a0a1 - a2a3) * irot2;
a[i + offset + 3 * p] = (a0na1 - a2na3iimag) * irot3;
}
irot *= irate3[__builtin_ctz(~s)];
}
}
if (len >= 1) {
int p = 1 << (h - 1);
for (int i = 0; i < p; i++) {
auto ajp = a[i] - a[i + p];
a[i] += a[i + p];
a[i + p] = ajp;
}
}
if (f) {
Mint inv_sz = Mint(1) / n;
for (int i = 0; i < n; i++)
a[i] *= inv_sz;
}
}
static vector<Mint> multiply(vector<Mint> a, vector<Mint> b) {
int need = a.size() + b.size() - 1;
int nbase = 1;
while ((1 << nbase) < need)
nbase++;
int sz = 1 << nbase;
a.resize(sz, 0);
b.resize(sz, 0);
ntt(a);
ntt(b);
Mint inv_sz = Mint(1) / sz;
for (int i = 0; i < sz; i++)
a[i] *= b[i] * inv_sz;
intt(a, false);
a.resize(need);
return a;
}
};
template <typename Mint>
vector<Mint>
NumberTheoreticTransformFriendlyModInt<Mint>::roots = vector<Mint>();
template <typename Mint>
vector<Mint>
NumberTheoreticTransformFriendlyModInt<Mint>::iroots = vector<Mint>();
template <typename Mint>
vector<Mint>
NumberTheoreticTransformFriendlyModInt<Mint>::rate3 = vector<Mint>();
template <typename Mint>
vector<Mint>
NumberTheoreticTransformFriendlyModInt<Mint>::irate3 = vector<Mint>();
template <typename Mint>
int NumberTheoreticTransformFriendlyModInt<Mint>::max_base = 0;
// #line 2 "math/fps/formal-power-series-friendly-ntt.cpp"
/**
* @brief Formal Power Series Friendly NTT(NTTmod用形式的冪級数)
* @docs docs/formal-power-series-friendly-ntt.md
*/
template <typename T> struct FormalPowerSeriesFriendlyNTT : vector<T> {
using vector<T>::vector;
using P = FormalPowerSeriesFriendlyNTT;
using NTT = NumberTheoreticTransformFriendlyModInt<T>;
P pre(int deg) const {
return P(begin(*this), begin(*this) + min((int)this->size(), deg));
}
P rev(int deg = -1) const {
P ret(*this);
if (deg != -1)
ret.resize(deg, T(0));
reverse(begin(ret), end(ret));
return ret;
}
void shrink() {
while (this->size() && this->back() == T(0))
this->pop_back();
}
P operator+(const P& r) const { return P(*this) += r; }
P operator+(const T& v) const { return P(*this) += v; }
P operator-(const P& r) const { return P(*this) -= r; }
P operator-(const T& v) const { return P(*this) -= v; }
P operator*(const P& r) const { return P(*this) *= r; }
P operator*(const T& v) const { return P(*this) *= v; }
P operator/(const P& r) const { return P(*this) /= r; }
P operator%(const P& r) const { return P(*this) %= r; }
P& operator+=(const P& 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;
}
P& operator-=(const P& 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;
}
// https://judge.yosupo.jp/problem/convolution_mod
P& operator*=(const P& r) {
if (this->empty() || r.empty()) {
this->clear();
return *this;
}
auto ret = NTT::multiply(*this, r);
return *this = { begin(ret), end(ret) };
}
P& operator/=(const P& r) {
if (this->size() < r.size()) {
this->clear();
return *this;
}
int n = this->size() - r.size() + 1;
return *this = (rev().pre(n) * r.rev().inv(n)).pre(n).rev(n);
}
P& operator%=(const P& r) {
*this -= *this / r * r;
shrink();
return *this;
}
// https://judge.yosupo.jp/problem/division_of_polynomials
pair<P, P> div_mod(const P& r) {
P q = *this / r;
P x = *this - q * r;
x.shrink();
return make_pair(q, x);
}
P operator-() const {
P ret(this->size());
for (int i = 0; i < (int)this->size(); i++)
ret[i] = -(*this)[i];
return ret;
}
P& operator+=(const T& r) {
if (this->empty())
this->resize(1);
(*this)[0] += r;
return *this;
}
P& operator-=(const T& r) {
if (this->empty())
this->resize(1);
(*this)[0] -= r;
return *this;
}
P& operator*=(const T& v) {
for (int i = 0; i < (int)this->size(); i++)
(*this)[i] *= v;
return *this;
}
P dot(P r) const {
P ret(min(this->size(), r.size()));
for (int i = 0; i < (int)ret.size(); i++)
ret[i] = (*this)[i] * r[i];
return ret;
}
P operator>>(int sz) const {
if ((int)this->size() <= sz)
return {};
P ret(*this);
ret.erase(ret.begin(), ret.begin() + sz);
return ret;
}
P operator<<(int sz) const {
P ret(*this);
ret.insert(ret.begin(), sz, T(0));
return ret;
}
T operator()(T x) const {
T r = 0, w = 1;
for (auto& v : *this) {
r += w * v;
w *= x;
}
return r;
}
P diff() const {
const int n = (int)this->size();
P ret(max(0, n - 1));
for (int i = 1; i < n; i++)
ret[i - 1] = (*this)[i] * T(i);
return ret;
}
P integral() const {
const int n = (int)this->size();
P ret(n + 1);
ret[0] = T(0);
for (int i = 0; i < n; i++)
ret[i + 1] = (*this)[i] / T(i + 1);
return ret;
}
// https://judge.yosupo.jp/problem/inv_of_formal_power_series
// F(0) must not be 0
P inv(int deg = -1) const {
assert(((*this)[0]) != T(0));
const int n = (int)this->size();
if (deg == -1)
deg = n;
P res(deg);
res[0] = { T(1) / (*this)[0] };
for (int d = 1; d < deg; d <<= 1) {
P f(2 * d), g(2 * d);
for (int j = 0; j < min(n, 2 * d); j++)
f[j] = (*this)[j];
for (int j = 0; j < d; j++)
g[j] = res[j];
NTT::ntt(f);
NTT::ntt(g);
f = f.dot(g);
NTT::intt(f);
for (int j = 0; j < d; j++)
f[j] = 0;
NTT::ntt(f);
for (int j = 0; j < 2 * d; j++)
f[j] *= g[j];
NTT::intt(f);
for (int j = d; j < min(2 * d, deg); j++)
res[j] = -f[j];
}
return res;
}
// https://judge.yosupo.jp/problem/log_of_formal_power_series
// F(0) must be 1
P log(int deg = -1) const {
assert((*this)[0] == T(1));
const int n = (int)this->size();
if (deg == -1)
deg = n;
return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
}
// https://judge.yosupo.jp/problem/sqrt_of_formal_power_series
P sqrt(int deg = -1,
const function<T(T)>& get_sqrt = [](T) { return T(1); }) const {
const int n = (int)this->size();
if (deg == -1)
deg = n;
if ((*this)[0] == T(0)) {
for (int i = 1; i < n; i++) {
if ((*this)[i] != T(0)) {
if (i & 1)
return {};
if (deg - i / 2 <= 0)
break;
auto ret = (*this >> i).sqrt(deg - i / 2, get_sqrt);
if (ret.empty())
return {};
ret = ret << (i / 2);
if ((int)ret.size() < deg)
ret.resize(deg, T(0));
return ret;
}
}
return P(deg, 0);
}
auto sqr = T(get_sqrt((*this)[0]));
if (sqr * sqr != (*this)[0])
return {};
P ret{ sqr };
T inv2 = T(1) / T(2);
for (int i = 1; i < deg; i <<= 1) {
ret = (ret + pre(i << 1) * ret.inv(i << 1)) * inv2;
}
return ret.pre(deg);
}
P sqrt(const function<T(T)>& get_sqrt, int deg = -1) const {
return sqrt(deg, get_sqrt);
}
// https://judge.yosupo.jp/problem/exp_of_formal_power_series
// F(0) must be 0
P exp(int deg = -1) const {
if (deg == -1)
deg = this->size();
assert((*this)[0] == T(0));
P inv;
inv.reserve(deg + 1);
inv.push_back(T(0));
inv.push_back(T(1));
auto inplace_integral = [&](P& F) -> void {
const int n = (int)F.size();
auto mod = T::get_mod();
while ((int)inv.size() <= n) {
int i = inv.size();
inv.push_back((-inv[mod % i]) * (mod / i));
}
F.insert(begin(F), T(0));
for (int i = 1; i <= n; i++)
F[i] *= inv[i];
};
auto inplace_diff = [](P& F) -> void {
if (F.empty())
return;
F.erase(begin(F));
T coeff = 1, one = 1;
for (int i = 0; i < (int)F.size(); i++) {
F[i] *= coeff;
coeff += one;
}
};
P 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);
NTT::ntt(y);
z1 = z2;
P z(m);
for (int i = 0; i < m; ++i)
z[i] = y[i] * z1[i];
NTT::intt(z);
fill(begin(z), begin(z) + m / 2, T(0));
NTT::ntt(z);
for (int i = 0; i < m; ++i)
z[i] *= -z1[i];
NTT::intt(z);
c.insert(end(c), begin(z) + m / 2, end(z));
z2 = c;
z2.resize(2 * m);
NTT::ntt(z2);
P x(begin(*this), begin(*this) + min<int>(this->size(), m));
inplace_diff(x);
x.push_back(T(0));
NTT::ntt(x);
for (int i = 0; i < m; ++i)
x[i] *= y[i];
NTT::intt(x);
x -= b.diff();
x.resize(2 * m);
for (int i = 0; i < m - 1; ++i)
x[m + i] = x[i], x[i] = T(0);
NTT::ntt(x);
for (int i = 0; i < 2 * m; ++i)
x[i] *= z2[i];
NTT::intt(x);
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, T(0));
NTT::ntt(x);
for (int i = 0; i < 2 * m; ++i)
x[i] *= y[i];
NTT::intt(x);
b.insert(end(b), begin(x) + m, end(x));
}
return P{ begin(b), begin(b) + deg };
}
// https://judge.yosupo.jp/problem/pow_of_formal_power_series
P 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] != T(0)) {
T rev = T(1) / (*this)[i];
P ret = (((*this * rev) >> i).log() * k).exp() * ((*this)[i].pow(k));
if (i * k > deg)
return P(deg, T(0));
ret = (ret << (i * k)).pre(deg);
if ((int)ret.size() < deg)
ret.resize(deg, T(0));
return ret;
}
}
return *this;
}
P mod_pow(int64_t k, P g) const {
P modinv = g.rev().inv();
auto get_div = [&](P base) {
if (base.size() < g.size()) {
base.clear();
return base;
}
int n = base.size() - g.size() + 1;
return (base.rev().pre(n) * modinv.pre(n)).pre(n).rev(n);
};
P x(*this), ret{ 1 };
while (k > 0) {
if (k & 1) {
ret *= x;
ret -= get_div(ret) * g;
ret.shrink();
}
x *= x;
x -= get_div(x) * g;
x.shrink();
k >>= 1;
}
return ret;
}
// https://judge.yosupo.jp/problem/polynomial_taylor_shift
P taylor_shift(T c) const {
int n = (int)this->size();
vector<T> fact(n), rfact(n);
fact[0] = rfact[0] = T(1);
for (int i = 1; i < n; i++)
fact[i] = fact[i - 1] * T(i);
rfact[n - 1] = T(1) / fact[n - 1];
for (int i = n - 1; i > 1; i--)
rfact[i - 1] = rfact[i] * T(i);
P p(*this);
for (int i = 0; i < n; i++)
p[i] *= fact[i];
p = p.rev();
P bs(n, T(1));
for (int i = 1; i < n; i++)
bs[i] = bs[i - 1] * c * rfact[i] * fact[i - 1];
p = (p * bs).pre(n);
p = p.rev();
for (int i = 0; i < n; i++)
p[i] *= rfact[i];
return p;
}
};
template <typename Mint> using FPS = FormalPowerSeriesFriendlyNTT<Mint>;
// #line 1 "math/fps/subproduct-tree.cpp"
/**
* @brief Subproduct Tree
*/
template <template <typename> class FPS, typename Mint>
vector<FPS<Mint>> subproduct_tree(const FPS<Mint>& xs) {
int n = (int)xs.size();
int k = 1;
while (k < n)
k <<= 1;
vector<FPS<Mint>> g(2 * k, { 1 });
for (int i = 0; i < n; i++)
g[k + i] = { -xs[i], Mint(1) };
for (int i = k; i-- > 1;)
g[i] = g[i << 1] * g[i << 1 | 1];
return g;
}
// #line 2 "math/fps/polynomial-interpolation.cpp"
/**
* @brief Polynomial Interpolation(多項式補間)
*/
template <template <typename> class FPS, typename Mint>
FPS<Mint> polynomial_interpolation(const FPS<Mint>& xs, const FPS<Mint>& ys) {
assert(xs.size() == ys.size());
auto mul = subproduct_tree(xs);
int n = (int)xs.size(), k = (int)mul.size() / 2;
vector<FPS<Mint>> g(2 * k);
g[1] = mul[1].diff() % mul[1];
for (int i = 2; i < k + n; i++)
g[i] = g[i >> 1] % mul[i];
for (int i = 0; i < n; i++)
g[k + i] = { ys[i] / g[k + i][0] };
for (int i = k; i-- > 1;)
g[i] = g[i << 1] * mul[i << 1 | 1] + g[i << 1 | 1] * mul[i << 1];
return g[1];
}
// #line 8 "test/verify/yosupo-polynomial-interpolation.test.cpp"
const int MOD = 998244353;
using mint = ModInt<MOD>;
int main() {
int N;
mint X;
cin >> N >> X;
FPS<mint> xs(N), ys(N);
for (int i = 0; i < N; i++) {
cin >> xs[i] >> ys[i];
}
auto f = polynomial_interpolation(xs, ys);
mint c = f[N - 1];
for (int i = 0; i < N; i++) {
if (X == xs[i]) {
mint ans = mint(N) * ys[i];
mint prod = c;
for (int j = 0; j < N; j++) {
if (j != i) {
prod *= X - xs[j];
}
}
cout << ans - prod << "\n";
return 0;
}
}
mint ans = mint(N) * f(X);
mint sum = 0;
for (int i = 0; i < N; i++) {
sum += mint(1) / (X - xs[i]);
}
for (int i = 0; i < N; i++) {
sum *= X - xs[i];
}
cout << ans - c * sum << "\n";
return 0;
}