結果
| 問題 | No.2166 Paint and Fill |
| ユーザー |
 xiaodao
|
| 提出日時 | 2023-05-10 15:13:12 |
| 言語 | C++14 (gcc 12.3.0 + boost 1.83.0) |
| 結果 |
TLE
|
| 実行時間 | - |
| コード長 | 45,680 bytes |
| コンパイル時間 | 4,631 ms |
| コンパイル使用メモリ | 201,408 KB |
| 実行使用メモリ | 21,612 KB |
| 最終ジャッジ日時 | 2024-05-05 03:19:43 |
| 合計ジャッジ時間 | 26,380 ms |
|
ジャッジサーバーID （参考情報） |
judge4 / judge1 |
（要ログイン）
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 | 22 ms
16,128 KB |
| testcase_01 | AC | 146 ms
21,612 KB |
| testcase_02 | AC | 3,867 ms
17,920 KB |
| testcase_03 | AC | 86 ms
16,128 KB |
| testcase_04 | AC | 84 ms
16,128 KB |
| testcase_05 | AC | 81 ms
16,000 KB |
| testcase_06 | AC | 82 ms
16,128 KB |
| testcase_07 | AC | 77 ms
16,128 KB |
| testcase_08 | AC | 845 ms
16,256 KB |
| testcase_09 | AC | 812 ms
16,128 KB |
| testcase_10 | AC | 799 ms
16,128 KB |
| testcase_11 | AC | 812 ms
16,128 KB |
| testcase_12 | AC | 821 ms
16,128 KB |
| testcase_13 | TLE | - |
| 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 | -- | - |
| testcase_28 | -- | - |
| testcase_29 | -- | - |
| testcase_30 | -- | - |
| testcase_31 | -- | - |
| testcase_32 | -- | - |
| testcase_33 | -- | - |
| testcase_34 | -- | - |
| testcase_35 | -- | - |
| testcase_36 | -- | - |
| testcase_37 | -- | - |
| testcase_38 | -- | - |
| testcase_39 | -- | - |
ソースコード
#pragma GCC optimize ("Ofast")
#pragma GCC optimize ("unroll-loops")
#pragma GCC target ("avx")
#include <cassert>
#include <cmath>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <algorithm>
#include <bitset>
#include <complex>
#include <deque>
#include <functional>
#include <iostream>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <sstream>
#include <string>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
using namespace std;
using Int = long long;
template <class T1, class T2> ostream &operator<<(ostream &os, const pair<T1, T2> &a) { return os << "(" << a.first << ", " << a.second << ")"; };
template <class T> ostream &operator<<(ostream &os, const vector<T> &as) { const int sz = as.size(); os << "["; for (int i = 0; i < sz; ++i) { if (i >= 256) { os << ", ..."; break; } if (i > 0) { os << ", "; } os << as[i]; } return os << "]"; }
template <class T> void pv(T a, T b) { for (T i = a; i != b; ++i) cerr << *i << " "; cerr << endl; }
template <class T> bool chmin(T &t, const T &f) { if (t > f) { t = f; return true; } return false; }
template <class T> bool chmax(T &t, const T &f) { if (t < f) { t = f; return true; } return false; }
////////////////////////////////////////////////////////////////////////////////
template <unsigned M_> struct ModInt {
static constexpr unsigned M = M_;
unsigned x;
constexpr ModInt() : x(0U) {}
constexpr ModInt(unsigned x_) : x(x_ % M) {}
constexpr ModInt(unsigned long long x_) : x(x_ % M) {}
constexpr ModInt(int x_) : x(((x_ %= static_cast<int>(M)) < 0) ? (x_ + static_cast<int>(M)) : x_) {}
constexpr ModInt(long long x_) : x(((x_ %= static_cast<long long>(M)) < 0) ? (x_ + static_cast<long long>(M)) : x_) {}
ModInt &operator+=(const ModInt &a) { x = ((x += a.x) >= M) ? (x - M) : x; return *this; }
ModInt &operator-=(const ModInt &a) { x = ((x -= a.x) >= M) ? (x + M) : x; return *this; }
ModInt &operator*=(const ModInt &a) { x = (static_cast<unsigned long long>(x) * a.x) % M; return *this; }
ModInt &operator/=(const ModInt &a) { return (*this *= a.inv()); }
ModInt pow(long long e) const {
if (e < 0) return inv().pow(-e);
ModInt a = *this, b = 1U; for (; e; e >>= 1) { if (e & 1) b *= a; a *= a; } return b;
}
ModInt inv() const {
unsigned a = M, b = x; int y = 0, z = 1;
for (; b; ) { const unsigned q = a / b; const unsigned c = a - q * b; a = b; b = c; const int w = y - static_cast<int>(q) * z; y = z; z = w; }
assert(a == 1U); return ModInt(y);
}
ModInt operator+() const { return *this; }
ModInt operator-() const { ModInt a; a.x = x ? (M - x) : 0U; return a; }
ModInt operator+(const ModInt &a) const { return (ModInt(*this) += a); }
ModInt operator-(const ModInt &a) const { return (ModInt(*this) -= a); }
ModInt operator*(const ModInt &a) const { return (ModInt(*this) *= a); }
ModInt operator/(const ModInt &a) const { return (ModInt(*this) /= a); }
template <class T> friend ModInt operator+(T a, const ModInt &b) { return (ModInt(a) += b); }
template <class T> friend ModInt operator-(T a, const ModInt &b) { return (ModInt(a) -= b); }
template <class T> friend ModInt operator*(T a, const ModInt &b) { return (ModInt(a) *= b); }
template <class T> friend ModInt operator/(T a, const ModInt &b) { return (ModInt(a) /= b); }
explicit operator bool() const { return x; }
bool operator==(const ModInt &a) const { return (x == a.x); }
bool operator!=(const ModInt &a) const { return (x != a.x); }
friend std::ostream &operator<<(std::ostream &os, const ModInt &a) { return os << a.x; }
};
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
constexpr unsigned MO = 998244353U;
constexpr unsigned MO2 = 2U * MO;
constexpr int FFT_MAX = 23;
using Mint = ModInt<MO>;
constexpr Mint FFT_ROOTS[FFT_MAX + 1] = {1U, 998244352U, 911660635U, 372528824U, 929031873U, 452798380U, 922799308U, 781712469U, 476477967U, 166035806U, 258648936U, 584193783U, 63912897U, 350007156U, 666702199U, 968855178U, 629671588U, 24514907U, 996173970U, 363395222U, 565042129U, 733596141U, 267099868U, 15311432U};
constexpr Mint INV_FFT_ROOTS[FFT_MAX + 1] = {1U, 998244352U, 86583718U, 509520358U, 337190230U, 87557064U, 609441965U, 135236158U, 304459705U, 685443576U, 381598368U, 335559352U, 129292727U, 358024708U, 814576206U, 708402881U, 283043518U, 3707709U, 121392023U, 704923114U, 950391366U, 428961804U, 382752275U, 469870224U};
constexpr Mint FFT_RATIOS[FFT_MAX] = {911660635U, 509520358U, 369330050U, 332049552U, 983190778U, 123842337U, 238493703U, 975955924U, 603855026U, 856644456U, 131300601U, 842657263U, 730768835U, 942482514U, 806263778U, 151565301U, 510815449U, 503497456U, 743006876U, 741047443U, 56250497U, 867605899U};
constexpr Mint INV_FFT_RATIOS[FFT_MAX] = {86583718U, 372528824U, 373294451U, 645684063U, 112220581U, 692852209U, 155456985U, 797128860U, 90816748U, 860285882U, 927414960U, 354738543U, 109331171U, 293255632U, 535113200U, 308540755U, 121186627U, 608385704U, 438932459U, 359477183U, 824071951U, 103369235U};
// as[rev(i)] <- \sum_j \zeta^(ij) as[j]
void fft(Mint *as, int n) {
assert(!(n & (n - 1))); assert(1 <= n); assert(n <= 1 << FFT_MAX);
int m = n;
if (m >>= 1) {
for (int i = 0; i < m; ++i) {
const unsigned x = as[i + m].x; // < MO
as[i + m].x = as[i].x + MO - x; // < 2 MO
as[i].x += x; // < 2 MO
}
}
if (m >>= 1) {
Mint prod = 1U;
for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) {
for (int i = i0; i < i0 + m; ++i) {
const unsigned x = (prod * as[i + m]).x; // < MO
as[i + m].x = as[i].x + MO - x; // < 3 MO
as[i].x += x; // < 3 MO
}
prod *= FFT_RATIOS[__builtin_ctz(++h)];
}
}
for (; m; ) {
if (m >>= 1) {
Mint prod = 1U;
for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) {
for (int i = i0; i < i0 + m; ++i) {
const unsigned x = (prod * as[i + m]).x; // < MO
as[i + m].x = as[i].x + MO - x; // < 4 MO
as[i].x += x; // < 4 MO
}
prod *= FFT_RATIOS[__builtin_ctz(++h)];
}
}
if (m >>= 1) {
Mint prod = 1U;
for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) {
for (int i = i0; i < i0 + m; ++i) {
const unsigned x = (prod * as[i + m]).x; // < MO
as[i].x = (as[i].x >= MO2) ? (as[i].x - MO2) : as[i].x; // < 2 MO
as[i + m].x = as[i].x + MO - x; // < 3 MO
as[i].x += x; // < 3 MO
}
prod *= FFT_RATIOS[__builtin_ctz(++h)];
}
}
}
for (int i = 0; i < n; ++i) {
as[i].x = (as[i].x >= MO2) ? (as[i].x - MO2) : as[i].x; // < 2 MO
as[i].x = (as[i].x >= MO) ? (as[i].x - MO) : as[i].x; // < MO
}
}
// as[i] <- (1/n) \sum_j \zeta^(-ij) as[rev(j)]
void invFft(Mint *as, int n) {
assert(!(n & (n - 1))); assert(1 <= n); assert(n <= 1 << FFT_MAX);
int m = 1;
if (m < n >> 1) {
Mint prod = 1U;
for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) {
for (int i = i0; i < i0 + m; ++i) {
const unsigned long long y = as[i].x + MO - as[i + m].x; // < 2 MO
as[i].x += as[i + m].x; // < 2 MO
as[i + m].x = (prod.x * y) % MO; // < MO
}
prod *= INV_FFT_RATIOS[__builtin_ctz(++h)];
}
m <<= 1;
}
for (; m < n >> 1; m <<= 1) {
Mint prod = 1U;
for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) {
for (int i = i0; i < i0 + (m >> 1); ++i) {
const unsigned long long y = as[i].x + MO2 - as[i + m].x; // < 4 MO
as[i].x += as[i + m].x; // < 4 MO
as[i].x = (as[i].x >= MO2) ? (as[i].x - MO2) : as[i].x; // < 2 MO
as[i + m].x = (prod.x * y) % MO; // < MO
}
for (int i = i0 + (m >> 1); i < i0 + m; ++i) {
const unsigned long long y = as[i].x + MO - as[i + m].x; // < 2 MO
as[i].x += as[i + m].x; // < 2 MO
as[i + m].x = (prod.x * y) % MO; // < MO
}
prod *= INV_FFT_RATIOS[__builtin_ctz(++h)];
}
}
if (m < n) {
for (int i = 0; i < m; ++i) {
const unsigned y = as[i].x + MO2 - as[i + m].x; // < 4 MO
as[i].x += as[i + m].x; // < 4 MO
as[i + m].x = y; // < 4 MO
}
}
const Mint invN = Mint(n).inv();
for (int i = 0; i < n; ++i) {
as[i] *= invN;
}
}
void fft(vector<Mint> &as) {
fft(as.data(), as.size());
}
void invFft(vector<Mint> &as) {
invFft(as.data(), as.size());
}
vector<Mint> convolve(vector<Mint> as, vector<Mint> bs) {
if (as.empty() || bs.empty()) return {};
const int len = as.size() + bs.size() - 1;
int n = 1;
for (; n < len; n <<= 1) {}
as.resize(n); fft(as);
bs.resize(n); fft(bs);
for (int i = 0; i < n; ++i) as[i] *= bs[i];
invFft(as);
as.resize(len);
return as;
}
vector<Mint> square(vector<Mint> as) {
if (as.empty()) return {};
const int len = as.size() + as.size() - 1;
int n = 1;
for (; n < len; n <<= 1) {}
as.resize(n); fft(as);
for (int i = 0; i < n; ++i) as[i] *= as[i];
invFft(as);
as.resize(len);
return as;
}
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// inv: log, exp, pow
// fac: shift
// invFac: shift
constexpr int LIM_INV = 1 << 20; // @
Mint inv[LIM_INV], fac[LIM_INV], invFac[LIM_INV];
struct ModIntPreparator {
ModIntPreparator() {
inv[1] = 1;
for (int i = 2; i < LIM_INV; ++i) inv[i] = -((Mint::M / i) * inv[Mint::M % i]);
fac[0] = 1;
for (int i = 1; i < LIM_INV; ++i) fac[i] = fac[i - 1] * i;
invFac[0] = 1;
for (int i = 1; i < LIM_INV; ++i) invFac[i] = invFac[i - 1] * inv[i];
}
} preparator;
// polyWork0: *, inv, div, divAt, log, exp, pow, sqrt, shift
// polyWork1: inv, div, divAt, log, exp, pow, sqrt, shift
// polyWork2: divAt, exp, pow, sqrt
// polyWork3: exp, pow, sqrt
static constexpr int LIM_POLY = 1 << 20; // @
static_assert(LIM_POLY <= 1 << FFT_MAX, "Poly: LIM_POLY <= 1 << FFT_MAX must hold.");
static Mint polyWork0[LIM_POLY], polyWork1[LIM_POLY], polyWork2[LIM_POLY], polyWork3[LIM_POLY];
struct Poly : public vector<Mint> {
Poly() {}
explicit Poly(int n) : vector<Mint>(n) {}
Poly(const vector<Mint> &vec) : vector<Mint>(vec) {}
Poly(std::initializer_list<Mint> il) : vector<Mint>(il) {}
int size() const { return vector<Mint>::size(); }
Mint at(long long k) const { return (0 <= k && k < size()) ? (*this)[k] : 0U; }
int ord() const { for (int i = 0; i < size(); ++i) if ((*this)[i]) return i; return -1; }
int deg() const { for (int i = size(); --i >= 0; ) if ((*this)[i]) return i; return -1; }
Poly mod(int n) const { return Poly(vector<Mint>(data(), data() + min(n, size()))); }
friend std::ostream &operator<<(std::ostream &os, const Poly &fs) {
os << "[";
for (int i = 0; i < fs.size(); ++i) { if (i > 0) os << ", "; os << fs[i]; }
return os << "]";
}
Poly &operator+=(const Poly &fs) {
if (size() < fs.size()) resize(fs.size());
for (int i = 0; i < fs.size(); ++i) (*this)[i] += fs[i];
return *this;
}
Poly &operator-=(const Poly &fs) {
if (size() < fs.size()) resize(fs.size());
for (int i = 0; i < fs.size(); ++i) (*this)[i] -= fs[i];
return *this;
}
// 3 E(|t| + |f|)
Poly &operator*=(const Poly &fs) {
if (empty() || fs.empty()) return *this = {};
const int nt = size(), nf = fs.size();
int n = 1;
for (; n < nt + nf - 1; n <<= 1) {}
assert(n <= LIM_POLY);
resize(n);
fft(data(), n); // 1 E(n)
memcpy(polyWork0, fs.data(), nf * sizeof(Mint));
memset(polyWork0 + nf, 0, (n - nf) * sizeof(Mint));
fft(polyWork0, n); // 1 E(n)
for (int i = 0; i < n; ++i) (*this)[i] *= polyWork0[i];
invFft(data(), n); // 1 E(n)
resize(nt + nf - 1);
return *this;
}
// 13 E(deg(t) - deg(f) + 1)
// rev(t) = rev(f) rev(q) + x^(deg(t)-deg(f)+1) rev(r)
Poly &operator/=(const Poly &fs) {
const int m = deg(), n = fs.deg();
assert(n != -1);
if (m < n) return *this = {};
Poly tsRev(m - n + 1), fsRev(min(m - n, n) + 1);
for (int i = 0; i <= m - n; ++i) tsRev[i] = (*this)[m - i];
for (int i = 0, i0 = min(m - n, n); i <= i0; ++i) fsRev[i] = fs[n - i];
const Poly qsRev = tsRev.div(fsRev, m - n + 1); // 13 E(m - n + 1)
resize(m - n + 1);
for (int i = 0; i <= m - n; ++i) (*this)[i] = qsRev[m - n - i];
return *this;
}
// 13 E(deg(t) - deg(f) + 1) + 3 E(|t|)
Poly &operator%=(const Poly &fs) {
const Poly qs = *this / fs; // 13 E(deg(t) - deg(f) + 1)
*this -= fs * qs; // 3 E(|t|)
resize(deg() + 1);
return *this;
}
Poly &operator*=(const Mint &a) {
for (int i = 0; i < size(); ++i) (*this)[i] *= a;
return *this;
}
Poly &operator/=(const Mint &a) {
const Mint b = a.inv();
for (int i = 0; i < size(); ++i) (*this)[i] *= b;
return *this;
}
Poly operator+() const { return *this; }
Poly operator-() const {
Poly fs(size());
for (int i = 0; i < size(); ++i) fs[i] = -(*this)[i];
return fs;
}
Poly operator+(const Poly &fs) const { return (Poly(*this) += fs); }
Poly operator-(const Poly &fs) const { return (Poly(*this) -= fs); }
Poly operator*(const Poly &fs) const { return (Poly(*this) *= fs); }
Poly operator/(const Poly &fs) const { return (Poly(*this) /= fs); }
Poly operator%(const Poly &fs) const { return (Poly(*this) %= fs); }
Poly operator*(const Mint &a) const { return (Poly(*this) *= a); }
Poly operator/(const Mint &a) const { return (Poly(*this) /= a); }
friend Poly operator*(const Mint &a, const Poly &fs) { return fs * a; }
// 10 E(n)
// f <- f - (t f - 1) f
Poly inv(int n) const {
assert(!empty()); assert((*this)[0]); assert(1 <= n);
assert(n == 1 || 1 << (32 - __builtin_clz(n - 1)) <= LIM_POLY);
Poly fs(n);
fs[0] = (*this)[0].inv();
for (int m = 1; m < n; m <<= 1) {
memcpy(polyWork0, data(), min(m << 1, size()) * sizeof(Mint));
memset(polyWork0 + min(m << 1, size()), 0, ((m << 1) - min(m << 1, size())) * sizeof(Mint));
fft(polyWork0, m << 1); // 2 E(n)
memcpy(polyWork1, fs.data(), min(m << 1, n) * sizeof(Mint));
memset(polyWork1 + min(m << 1, n), 0, ((m << 1) - min(m << 1, n)) * sizeof(Mint));
fft(polyWork1, m << 1); // 2 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i];
invFft(polyWork0, m << 1); // 2 E(n)
memset(polyWork0, 0, m * sizeof(Mint));
fft(polyWork0, m << 1); // 2 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i];
invFft(polyWork0, m << 1); // 2 E(n)
for (int i = m, i0 = min(m << 1, n); i < i0; ++i) fs[i] = -polyWork0[i];
}
return fs;
}
// 9 E(n)
// Need (4 m)-th roots of unity to lift from (mod x^m) to (mod x^(2m)).
// f <- f - (t f - 1) f
// (t f^2) mod ((x^(2m) - 1) (x^m - 1^(1/4)))
/*
Poly inv(int n) const {
assert(!empty()); assert((*this)[0]); assert(1 <= n);
assert(n == 1 || 3 << (31 - __builtin_clz(n - 1)) <= LIM_POLY);
assert(n <= 1 << (FFT_MAX - 1));
Poly fs(n);
fs[0] = (*this)[0].inv();
for (int h = 2, m = 1; m < n; ++h, m <<= 1) {
const Mint a = FFT_ROOTS[h], b = INV_FFT_ROOTS[h];
memcpy(polyWork0, data(), min(m << 1, size()) * sizeof(Mint));
memset(polyWork0 + min(m << 1, size()), 0, ((m << 1) - min(m << 1, size())) * sizeof(Mint));
{
Mint aa = 1;
for (int i = 0; i < m; ++i) { polyWork0[(m << 1) + i] = aa * polyWork0[i]; aa *= a; }
for (int i = 0; i < m; ++i) { polyWork0[(m << 1) + i] += aa * polyWork0[m + i]; aa *= a; }
}
fft(polyWork0, m << 1); // 2 E(n)
fft(polyWork0 + (m << 1), m); // 1 E(n)
memcpy(polyWork1, fs.data(), min(m << 1, n) * sizeof(Mint));
memset(polyWork1 + min(m << 1, n), 0, ((m << 1) - min(m << 1, n)) * sizeof(Mint));
{
Mint aa = 1;
for (int i = 0; i < m; ++i) { polyWork1[(m << 1) + i] = aa * polyWork1[i]; aa *= a; }
for (int i = 0; i < m; ++i) { polyWork1[(m << 1) + i] += aa * polyWork1[m + i]; aa *= a; }
}
fft(polyWork1, m << 1); // 2 E(n)
fft(polyWork1 + (m << 1), m); // 1 E(n)
for (int i = 0; i < (m << 1) + m; ++i) polyWork0[i] *= polyWork1[i] * polyWork1[i];
invFft(polyWork0, m << 1); // 2 E(n)
invFft(polyWork0 + (m << 1), m); // 1 E(n)
// 2 f0 + (-f2), (-f1) + (-f3), 1^(1/4) (-f1) - (-f2) - 1^(1/4) (-f3)
{
Mint bb = 1;
for (int i = 0, i0 = min(m, n - m); i < i0; ++i) {
unsigned x = polyWork0[i].x + (bb * polyWork0[(m << 1) + i]).x + MO2 - (fs[i].x << 1); // < 4 MO
fs[m + i] = Mint(static_cast<unsigned long long>(FFT_ROOTS[2].x) * x) - polyWork0[m + i];
fs[m + i].x = ((fs[m + i].x & 1) ? (fs[m + i].x + MO) : fs[m + i].x) >> 1;
bb *= b;
}
}
}
return fs;
}
*/
// 13 E(n)
// g = (1 / f) mod x^m
// h <- h - (f h - t) g
Poly div(const Poly &fs, int n) const {
assert(!fs.empty()); assert(fs[0]); assert(1 <= n);
if (n == 1) return {at(0) / fs[0]};
// m < n <= 2 m
const int m = 1 << (31 - __builtin_clz(n - 1));
assert(m << 1 <= LIM_POLY);
Poly gs = fs.inv(m); // 5 E(n)
gs.resize(m << 1);
fft(gs.data(), m << 1); // 1 E(n)
memcpy(polyWork0, data(), min(m, size()) * sizeof(Mint));
memset(polyWork0 + min(m, size()), 0, ((m << 1) - min(m, size())) * sizeof(Mint));
fft(polyWork0, m << 1); // 1 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] *= gs[i];
invFft(polyWork0, m << 1); // 1 E(n)
Poly hs(n);
memcpy(hs.data(), polyWork0, m * sizeof(Mint));
memset(polyWork0 + m, 0, m * sizeof(Mint));
fft(polyWork0, m << 1); // 1 E(n)
memcpy(polyWork1, fs.data(), min(m << 1, fs.size()) * sizeof(Mint));
memset(polyWork1 + min(m << 1, fs.size()), 0, ((m << 1) - min(m << 1, fs.size())) * sizeof(Mint));
fft(polyWork1, m << 1); // 1 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i];
invFft(polyWork0, m << 1); // 1 E(n)
memset(polyWork0, 0, m * sizeof(Mint));
for (int i = m, i0 = min(m << 1, size()); i < i0; ++i) polyWork0[i] -= (*this)[i];
fft(polyWork0, m << 1); // 1 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] *= gs[i];
invFft(polyWork0, m << 1); // 1 E(n)
for (int i = m; i < n; ++i) hs[i] = -polyWork0[i];
return hs;
}
// (4 (floor(log_2 k) - ceil(log_2 |f|)) + 16) E(|f|) for |t| < |f|
// [x^k] (t(x) / f(x)) = [x^k] ((t(x) f(-x)) / (f(x) f(-x))
// polyWork0: half of (2 m)-th roots of unity, inversed, bit-reversed
Mint divAt(const Poly &fs, long long k) const {
assert(k >= 0);
if (size() >= fs.size()) {
const Poly qs = *this / fs; // 13 E(deg(t) - deg(f) + 1)
Poly rs = *this - fs * qs; // 3 E(|t|)
rs.resize(rs.deg() + 1);
return qs.at(k) + rs.divAt(fs, k);
}
int h = 0, m = 1;
for (; m < fs.size(); ++h, m <<= 1) {}
if (k < m) {
const Poly gs = fs.inv(k + 1); // 10 E(|f|)
Mint sum;
for (int i = 0, i0 = min<int>(k + 1, size()); i < i0; ++i) sum += (*this)[i] * gs[k - i];
return sum;
}
assert(m << 1 <= LIM_POLY);
polyWork0[0] = Mint(2U).inv();
for (int hh = 0; hh < h; ++hh) for (int i = 0; i < 1 << hh; ++i) polyWork0[1 << hh | i] = polyWork0[i] * INV_FFT_ROOTS[hh + 2];
const Mint a = FFT_ROOTS[h + 1];
memcpy(polyWork2, data(), size() * sizeof(Mint));
memset(polyWork2 + size(), 0, ((m << 1) - size()) * sizeof(Mint));
fft(polyWork2, m << 1); // 2 E(|f|)
memcpy(polyWork1, fs.data(), fs.size() * sizeof(Mint));
memset(polyWork1 + fs.size(), 0, ((m << 1) - fs.size()) * sizeof(Mint));
fft(polyWork1, m << 1); // 2 E(|f|)
for (; ; ) {
if (k & 1) {
for (int i = 0; i < m; ++i) polyWork2[i] = polyWork0[i] * (polyWork2[i << 1 | 0] * polyWork1[i << 1 | 1] - polyWork2[i << 1 | 1] * polyWork1[i << 1 | 0]);
} else {
for (int i = 0; i < m; ++i) {
polyWork2[i] = polyWork2[i << 1 | 0] * polyWork1[i << 1 | 1] + polyWork2[i << 1 | 1] * polyWork1[i << 1 | 0];
polyWork2[i].x = ((polyWork2[i].x & 1) ? (polyWork2[i].x + MO) : polyWork2[i].x) >> 1;
}
}
for (int i = 0; i < m; ++i) polyWork1[i] = polyWork1[i << 1 | 0] * polyWork1[i << 1 | 1];
if ((k >>= 1) < m) {
invFft(polyWork2, m); // 1 E(|f|)
invFft(polyWork1, m); // 1 E(|f|)
// Poly::inv does not use polyWork2
const Poly gs = Poly(vector<Mint>(polyWork1, polyWork1 + k + 1)).inv(k + 1); // 10 E(|f|)
Mint sum;
for (int i = 0; i <= k; ++i) sum += polyWork2[i] * gs[k - i];
return sum;
}
memcpy(polyWork2 + m, polyWork2, m * sizeof(Mint));
invFft(polyWork2 + m, m); // (floor(log_2 k) - ceil(log_2 |f|)) E(|f|)
memcpy(polyWork1 + m, polyWork1, m * sizeof(Mint));
invFft(polyWork1 + m, m); // (floor(log_2 k) - ceil(log_2 |f|)) E(|f|)
Mint aa = 1;
for (int i = m; i < m << 1; ++i) { polyWork2[i] *= aa; polyWork1[i] *= aa; aa *= a; }
fft(polyWork2 + m, m); // (floor(log_2 k) - ceil(log_2 |f|)) E(|f|)
fft(polyWork1 + m, m); // (floor(log_2 k) - ceil(log_2 |f|)) E(|f|)
}
}
// 13 E(n)
// D log(t) = (D t) / t
Poly log(int n) const {
assert(!empty()); assert((*this)[0].x == 1U); assert(n <= LIM_INV);
Poly fs = mod(n);
for (int i = 0; i < fs.size(); ++i) fs[i] *= i;
fs = fs.div(*this, n);
for (int i = 1; i < n; ++i) fs[i] *= ::inv[i];
return fs;
}
// (16 + 1/2) E(n)
// f = exp(t) mod x^m ==> (D f) / f == D t (mod x^m)
// g = (1 / exp(t)) mod x^m
// f <- f - (log f - t) / (1 / f)
// = f - (I ((D f) / f) - t) f
// == f - (I ((D f) / f + (f g - 1) ((D f) / f - D (t mod x^m))) - t) f (mod x^(2m))
// = f - (I (g (D f - f D (t mod x^m)) + D (t mod x^m)) - t) f
// g <- g - (f g - 1) g
// polyWork1: DFT(f, 2 m), polyWork2: g, polyWork3: DFT(g, 2 m)
Poly exp(int n) const {
assert(!empty()); assert(!(*this)[0]); assert(1 <= n);
assert(n == 1 || 1 << (32 - __builtin_clz(n - 1)) <= min(LIM_INV, LIM_POLY));
if (n == 1) return {1U};
if (n == 2) return {1U, at(1)};
Poly fs(n);
fs[0].x = polyWork1[0].x = polyWork1[1].x = polyWork2[0].x = 1U;
int m;
for (m = 1; m << 1 < n; m <<= 1) {
for (int i = 0, i0 = min(m, size()); i < i0; ++i) polyWork0[i] = i * (*this)[i];
memset(polyWork0 + min(m, size()), 0, (m - min(m, size())) * sizeof(Mint));
fft(polyWork0, m); // (1/2) E(n)
for (int i = 0; i < m; ++i) polyWork0[i] *= polyWork1[i];
invFft(polyWork0, m); // (1/2) E(n)
for (int i = 0; i < m; ++i) polyWork0[i] -= i * fs[i];
memset(polyWork0 + m, 0, m * sizeof(Mint));
fft(polyWork0, m << 1); // 1 E(n)
memcpy(polyWork3, polyWork2, m * sizeof(Mint));
memset(polyWork3 + m, 0, m * sizeof(Mint));
fft(polyWork3, m << 1); // 1 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork3[i];
invFft(polyWork0, m << 1); // 1 E(n)
for (int i = 0; i < m; ++i) polyWork0[i] *= ::inv[m + i];
for (int i = 0, i0 = min(m, size() - m); i < i0; ++i) polyWork0[i] += (*this)[m + i];
memset(polyWork0 + m, 0, m * sizeof(Mint));
fft(polyWork0, m << 1); // 1 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i];
invFft(polyWork0, m << 1); // 1 E(n)
memcpy(fs.data() + m, polyWork0, m * sizeof(Mint));
memcpy(polyWork1, fs.data(), (m << 1) * sizeof(Mint));
memset(polyWork1 + (m << 1), 0, (m << 1) * sizeof(Mint));
fft(polyWork1, m << 2); // 2 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] = polyWork1[i] * polyWork3[i];
invFft(polyWork0, m << 1); // 1 E(n)
memset(polyWork0, 0, m * sizeof(Mint));
fft(polyWork0, m << 1); // 1 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork3[i];
invFft(polyWork0, m << 1); // 1 E(n)
for (int i = m; i < m << 1; ++i) polyWork2[i] = -polyWork0[i];
}
for (int i = 0, i0 = min(m, size()); i < i0; ++i) polyWork0[i] = i * (*this)[i];
memset(polyWork0 + min(m, size()), 0, (m - min(m, size())) * sizeof(Mint));
fft(polyWork0, m); // (1/2) E(n)
for (int i = 0; i < m; ++i) polyWork0[i] *= polyWork1[i];
invFft(polyWork0, m); // (1/2) E(n)
for (int i = 0; i < m; ++i) polyWork0[i] -= i * fs[i];
memcpy(polyWork0 + m, polyWork0 + (m >> 1), (m >> 1) * sizeof(Mint));
memset(polyWork0 + (m >> 1), 0, (m >> 1) * sizeof(Mint));
memset(polyWork0 + m + (m >> 1), 0, (m >> 1) * sizeof(Mint));
fft(polyWork0, m); // (1/2) E(n)
fft(polyWork0 + m, m); // (1/2) E(n)
memcpy(polyWork3 + m, polyWork2 + (m >> 1), (m >> 1) * sizeof(Mint));
memset(polyWork3 + m + (m >> 1), 0, (m >> 1) * sizeof(Mint));
fft(polyWork3 + m, m); // (1/2) E(n)
for (int i = 0; i < m; ++i) polyWork0[m + i] = polyWork0[i] * polyWork3[m + i] + polyWork0[m + i] * polyWork3[i];
for (int i = 0; i < m; ++i) polyWork0[i] *= polyWork3[i];
invFft(polyWork0, m); // (1/2) E(n)
invFft(polyWork0 + m, m); // (1/2) E(n)
for (int i = 0; i < m >> 1; ++i) polyWork0[(m >> 1) + i] += polyWork0[m + i];
for (int i = 0; i < m; ++i) polyWork0[i] *= ::inv[m + i];
for (int i = 0, i0 = min(m, size() - m); i < i0; ++i) polyWork0[i] += (*this)[m + i];
memset(polyWork0 + m, 0, m * sizeof(Mint));
fft(polyWork0, m << 1); // 1 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i];
invFft(polyWork0, m << 1); // 1 E(n)
memcpy(fs.data() + m, polyWork0, (n - m) * sizeof(Mint));
return fs;
}
// (29 + 1/2) E(n)
// g <- g - (log g - a log t) g
Poly pow(Mint a, int n) const {
assert(!empty()); assert((*this)[0].x == 1U); assert(1 <= n);
return (a * log(n)).exp(n); // 13 E(n) + (16 + 1/2) E(n)
}
// (29 + 1/2) E(n - a ord(t))
Poly pow(long long a, int n) const {
assert(a >= 0); assert(1 <= n);
if (a == 0) { Poly gs(n); gs[0].x = 1U; return gs; }
const int o = ord();
if (o == -1 || o > (n - 1) / a) return Poly(n);
const Mint b = (*this)[o].inv(), c = (*this)[o].pow(a);
const int ntt = min<int>(n - a * o, size() - o);
Poly tts(ntt);
for (int i = 0; i < ntt; ++i) tts[i] = b * (*this)[o + i];
tts = tts.pow(Mint(a), n - a * o); // (29 + 1/2) E(n - a ord(t))
Poly gs(n);
for (int i = 0; i < n - a * o; ++i) gs[a * o + i] = c * tts[i];
return gs;
}
// (10 + 1/2) E(n)
// f = t^(1/2) mod x^m, g = 1 / t^(1/2) mod x^m
// f <- f - (f^2 - h) g / 2
// g <- g - (f g - 1) g
// polyWork1: DFT(f, m), polyWork2: g, polyWork3: DFT(g, 2 m)
Poly sqrt(int n) const {
assert(!empty()); assert((*this)[0].x == 1U); assert(1 <= n);
assert(n == 1 || 1 << (32 - __builtin_clz(n - 1)) <= LIM_POLY);
if (n == 1) return {1U};
if (n == 2) return {1U, at(1) / 2};
Poly fs(n);
fs[0].x = polyWork1[0].x = polyWork2[0].x = 1U;
int m;
for (m = 1; m << 1 < n; m <<= 1) {
for (int i = 0; i < m; ++i) polyWork1[i] *= polyWork1[i];
invFft(polyWork1, m); // (1/2) E(n)
for (int i = 0, i0 = min(m, size()); i < i0; ++i) polyWork1[i] -= (*this)[i];
for (int i = 0, i0 = min(m, size() - m); i < i0; ++i) polyWork1[i] -= (*this)[m + i];
memset(polyWork1 + m, 0, m * sizeof(Mint));
fft(polyWork1, m << 1); // 1 E(n)
memcpy(polyWork3, polyWork2, m * sizeof(Mint));
memset(polyWork3 + m, 0, m * sizeof(Mint));
fft(polyWork3, m << 1); // 1 E(n)
for (int i = 0; i < m << 1; ++i) polyWork1[i] *= polyWork3[i];
invFft(polyWork1, m << 1); // 1 E(n)
for (int i = 0; i < m; ++i) { polyWork1[i] = -polyWork1[i]; fs[m + i].x = ((polyWork1[i].x & 1) ? (polyWork1[i].x + MO) : polyWork1[i].x) >> 1; }
memcpy(polyWork1, fs.data(), (m << 1) * sizeof(Mint));
fft(polyWork1, m << 1); // 1 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] = polyWork1[i] * polyWork3[i];
invFft(polyWork0, m << 1); // 1 E(n)
memset(polyWork0, 0, m * sizeof(Mint));
fft(polyWork0, m << 1); // 1 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork3[i];
invFft(polyWork0, m << 1); // 1 E(n)
for (int i = m; i < m << 1; ++i) polyWork2[i] = -polyWork0[i];
}
for (int i = 0; i < m; ++i) polyWork1[i] *= polyWork1[i];
invFft(polyWork1, m); // (1/2) E(n)
for (int i = 0, i0 = min(m, size()); i < i0; ++i) polyWork1[i] -= (*this)[i];
for (int i = 0, i0 = min(m, size() - m); i < i0; ++i) polyWork1[i] -= (*this)[m + i];
memcpy(polyWork1 + m, polyWork1 + (m >> 1), (m >> 1) * sizeof(Mint));
memset(polyWork1 + (m >> 1), 0, (m >> 1) * sizeof(Mint));
memset(polyWork1 + m + (m >> 1), 0, (m >> 1) * sizeof(Mint));
fft(polyWork1, m); // (1/2) E(n)
fft(polyWork1 + m, m); // (1/2) E(n)
memcpy(polyWork3 + m, polyWork2 + (m >> 1), (m >> 1) * sizeof(Mint));
memset(polyWork3 + m + (m >> 1), 0, (m >> 1) * sizeof(Mint));
fft(polyWork3 + m, m); // (1/2) E(n)
// for (int i = 0; i < m << 1; ++i) polyWork1[i] *= polyWork3[i];
for (int i = 0; i < m; ++i) polyWork1[m + i] = polyWork1[i] * polyWork3[m + i] + polyWork1[m + i] * polyWork3[i];
for (int i = 0; i < m; ++i) polyWork1[i] *= polyWork3[i];
invFft(polyWork1, m); // (1/2) E(n)
invFft(polyWork1 + m, m); // (1/2) E(n)
for (int i = 0; i < m >> 1; ++i) polyWork1[(m >> 1) + i] += polyWork1[m + i];
for (int i = 0; i < n - m; ++i) { polyWork1[i] = -polyWork1[i]; fs[m + i].x = ((polyWork1[i].x & 1) ? (polyWork1[i].x + MO) : polyWork1[i].x) >> 1; }
return fs;
}
// (10 + 1/2) E(n)
// modSqrt must return a quadratic residue if exists, or anything otherwise.
// Return {} if *this does not have a square root.
template <class F> Poly sqrt(int n, F modSqrt) const {
assert(1 <= n);
const int o = ord();
if (o == -1) return Poly(n);
if (o & 1) return {};
const Mint c = modSqrt((*this)[o]);
if (c * c != (*this)[o]) return {};
if (o >> 1 >= n) return Poly(n);
const Mint b = (*this)[o].inv();
const int ntt = min(n - (o >> 1), size() - o);
Poly tts(ntt);
for (int i = 0; i < ntt; ++i) tts[i] = b * (*this)[o + i];
tts = tts.sqrt(n - (o >> 1)); // (10 + 1/2) E(n)
Poly gs(n);
for (int i = 0; i < n - (o >> 1); ++i) gs[(o >> 1) + i] = c * tts[i];
return gs;
}
// 6 E(|t|)
// x -> x + a
Poly shift(const Mint &a) const {
if (empty()) return {};
const int n = size();
int m = 1;
for (; m < n; m <<= 1) {}
for (int i = 0; i < n; ++i) polyWork0[i] = fac[i] * (*this)[i];
memset(polyWork0 + n, 0, ((m << 1) - n) * sizeof(Mint));
fft(polyWork0, m << 1); // 2 E(|t|)
{
Mint aa = 1;
for (int i = 0; i < n; ++i) { polyWork1[n - 1 - i] = invFac[i] * aa; aa *= a; }
}
memset(polyWork1 + n, 0, ((m << 1) - n) * sizeof(Mint));
fft(polyWork1, m << 1); // 2 E(|t|)
for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i];
invFft(polyWork0, m << 1); // 2 E(|t|)
Poly fs(n);
for (int i = 0; i < n; ++i) fs[i] = invFac[i] * polyWork0[n - 1 + i];
return fs;
}
};
Mint linearRecurrenceAt(const vector<Mint> &as, const vector<Mint> &cs, long long k) {
assert(!cs.empty()); assert(cs[0]);
const int d = cs.size() - 1;
assert(as.size() >= static_cast<size_t>(d));
return (Poly(vector<Mint>(as.begin(), as.begin() + d)) * cs).mod(d).divAt(cs, k);
}
struct SubproductTree {
int logN, n, nn;
vector<Mint> xs;
// [DFT_4((X-xs[0])(X-xs[1])(X-xs[2])(X-xs[3]))] [(X-xs[0])(X-xs[1])(X-xs[2])(X-xs[3])mod X^4]
// [ DFT_4((X-xs[0])(X-xs[1])) ] [ DFT_4((X-xs[2])(X-xs[3])) ]
// [ DFT_2(X-xs[0]) ] [ DFT_2(X-xs[1]) ] [ DFT_2(X-xs[2]) ] [ DFT_2(X-xs[3]) ]
vector<Mint> buf;
vector<Mint *> gss;
// (1 - xs[0] X) ... (1 - xs[nn-1] X)
Poly all;
SubproductTree() {}
// (ceil(log_2 n) + O(1)) E(n)
SubproductTree(const vector<Mint> &xs_) {
n = xs_.size();
for (logN = 0, nn = 1; nn < n; ++logN, nn <<= 1) {}
xs.assign(nn, 0U);
memcpy(xs.data(), xs_.data(), n * sizeof(Mint));
buf.assign((logN + 1) * (nn << 1), 0U);
gss.assign(nn << 1, nullptr);
for (int h = 0; h <= logN; ++h) for (int u = 1 << h; u < 1 << (h + 1); ++u) {
gss[u] = buf.data() + (h * (nn << 1) + ((u - (1 << h)) << (logN - h + 1)));
}
for (int i = 0; i < nn; ++i) {
gss[nn + i][0] = -xs[i] + 1;
gss[nn + i][1] = -xs[i] - 1;
}
if (nn == 1) gss[1][1] += 2;
for (int h = logN; --h >= 0; ) {
const int m = 1 << (logN - h);
for (int u = 1 << (h + 1); --u >= 1 << h; ) {
for (int i = 0; i < m; ++i) gss[u][i] = gss[u << 1][i] * gss[u << 1 | 1][i];
memcpy(gss[u] + m, gss[u], m * sizeof(Mint));
invFft(gss[u] + m, m); // ((1/2) ceil(log_2 n) + O(1)) E(n)
if (h > 0) {
gss[u][m] -= 2;
const Mint a = FFT_ROOTS[logN - h + 1];
Mint aa = 1;
for (int i = m; i < m << 1; ++i) { gss[u][i] *= aa; aa *= a; };
fft(gss[u] + m, m); // ((1/2) ceil(log_2 n) + O(1)) E(n)
}
}
}
all.resize(nn + 1);
all[0] = 1;
for (int i = 1; i < nn; ++i) all[i] = gss[1][nn + nn - i];
all[nn] = gss[1][nn] - 1;
}
// ((3/2) ceil(log_2 n) + O(1)) E(n) + 10 E(|f|) + 3 E(|f| + 2^(ceil(log_2 n)))
vector<Mint> multiEval(const Poly &fs) const {
vector<Mint> work0(nn), work1(nn), work2(nn);
{
const int m = max(fs.size(), 1);
auto invAll = all.inv(m); // 10 E(|f|)
std::reverse(invAll.begin(), invAll.end());
int mm;
for (mm = 1; mm < m - 1 + nn; mm <<= 1) {}
invAll.resize(mm, 0U);
fft(invAll); // E(|f| + 2^(ceil(log_2 n)))
vector<Mint> ffs(mm, 0U);
memcpy(ffs.data(), fs.data(), fs.size() * sizeof(Mint));
fft(ffs); // E(|f| + 2^(ceil(log_2 n)))
for (int i = 0; i < mm; ++i) ffs[i] *= invAll[i];
invFft(ffs); // E(|f| + 2^(ceil(log_2 n)))
memcpy(((logN & 1) ? work1 : work0).data(), ffs.data() + m - 1, nn * sizeof(Mint));
}
for (int h = 0; h < logN; ++h) {
const int m = 1 << (logN - h);
for (int u = 1 << h; u < 1 << (h + 1); ++u) {
Mint *hs = (((logN - h) & 1) ? work1 : work0).data() + ((u - (1 << h)) << (logN - h));
Mint *hs0 = (((logN - h) & 1) ? work0 : work1).data() + ((u - (1 << h)) << (logN - h));
Mint *hs1 = hs0 + (m >> 1);
fft(hs, m); // ((1/2) ceil(log_2 n) + O(1)) E(n)
for (int i = 0; i < m; ++i) work2[i] = gss[u << 1 | 1][i] * hs[i];
invFft(work2.data(), m); // ((1/2) ceil(log_2 n) + O(1)) E(n)
memcpy(hs0, work2.data() + (m >> 1), (m >> 1) * sizeof(Mint));
for (int i = 0; i < m; ++i) work2[i] = gss[u << 1][i] * hs[i];
invFft(work2.data(), m); // ((1/2) ceil(log_2 n) + O(1)) E(n)
memcpy(hs1, work2.data() + (m >> 1), (m >> 1) * sizeof(Mint));
}
}
work0.resize(n);
return work0;
}
// ((5/2) ceil(log_2 n) + O(1)) E(n)
Poly interpolate(const vector<Mint> &ys) const {
assert(static_cast<int>(ys.size()) == n);
Poly gs(n);
for (int i = 0; i < n; ++i) gs[i] = (i + 1) * all[n - (i + 1)];
const vector<Mint> denoms = multiEval(gs); // ((3/2) ceil(log_2 n) + O(1)) E(n)
vector<Mint> work(nn << 1, 0U);
for (int i = 0; i < n; ++i) {
// xs[0], ..., xs[n - 1] are not distinct
assert(denoms[i]);
work[i << 1] = work[i << 1 | 1] = ys[i] / denoms[i];
}
for (int h = logN; --h >= 0; ) {
const int m = 1 << (logN - h);
for (int u = 1 << (h + 1); --u >= 1 << h; ) {
Mint *hs = work.data() + ((u - (1 << h)) << (logN - h + 1));
for (int i = 0; i < m; ++i) hs[i] = gss[u << 1 | 1][i] * hs[i] + gss[u << 1][i] * hs[m + i];
if (h > 0) {
memcpy(hs + m, hs, m * sizeof(Mint));
invFft(hs + m, m); // ((1/2) ceil(log_2 n) + O(1)) E(n)
const Mint a = FFT_ROOTS[logN - h + 1];
Mint aa = 1;
for (int i = m; i < m << 1; ++i) { hs[i] *= aa; aa *= a; };
fft(hs + m, m); // ((1/2) ceil(log_2 n) + O(1)) E(n)
}
}
}
invFft(work.data(), nn); // E(n)
return Poly(vector<Mint>(work.data() + nn - n, work.data() + nn));
}
};
////////////////////////////////////////////////////////////////////////////////
// a(xL) ... a(xR-1)
// a: d * d matrix of poly entry of size <= e
// O(d^2 e^2 + d^2 sqrt(e (xR-xL)) log(e (xR-xL)) + d^3 sqrt(e (xR-xL))) time
// Needs inv, fac, invFac for [0, e block), where
// block: min power of 2 s.t. e block^2 >= xR-xL
vector<vector<Mint>> polyMatrixProduct(const vector<vector<Poly>> &a,
long long xL, long long xR) {
const int d = a.size();
for (int i = 0; i < d; ++i) assert(static_cast<size_t>(d) == a[i].size());
assert(xL <= xR);
int e = 1;
for (int i = 0; i < d; ++i) for (int j = 0; j < d; ++j) {
for (; e < a[i][j].deg() + 1; e <<= 1) {}
}
long long block = 1;
for (; e * block * block < xR - xL; block <<= 1) {}
assert(e * block <= LIM_INV);
const Mint invBlock = Mint(block).inv();
// O(d^2 e^2) (more precisely, O(e \sum[i,j] a[i][j].size()))
vector<Mint> b(d * d * e, 0);
for (int i = 0; i < d; ++i) for (int j = 0; j < d; ++j) {
for (int l = 0; l < e; ++l) {
const Mint x = xL + block * l;
Mint &y = b[(i * d + j) * e + l];
for (int m = a[i][j].size(); --m >= 0; ) (y *= x) += a[i][j][m];
}
}
// O(d^2 e block log(e block) + d^3 e block)
for (int w = 1; w < block; w <<= 1) {
// b[i][j]: product of w factors,
// evaluated at (xL, xL + block, ..., xL + block (ew-1))
// for f: poly of size ew, given [f(0), ..., f(ew-1)], find:
// - [f(ew), ..., f(2ew-1)]
// - [f(w/block), ..., f(2ew-1+w/block)]
const int ew = e * w;
const int ew1 = ew << 1, ew2 = ew << 2;
vector<Mint> workExp0(ew1, 0), workExp1(ew2, 0);
vector<Mint> workInvExp0(ew1, 0);
vector<Mint> workFall0(ew1, 0), workFall1(ew1, 0);
for (int l = 0; l < ew; ++l) workExp0[l] = invFac[l];
for (int l = 0; l < ew1; ++l) workExp1[l] = invFac[l];
for (int l = 0; l < ew; ++l) workInvExp0[l] = (l & 1) ? -invFac[l] : invFac[l];
workFall0[ew - 1] = workFall1[ew - 1] = 1;
for (int l = 1; l < ew; ++l) workFall0[ew - 1 - l] = workFall0[ew - l] * (ew - (l - 1)) * inv[l];
for (int l = 1; l < ew; ++l) workFall1[ew - 1 - l] = workFall1[ew - l] * (invBlock * w - (l - 1)) * inv[l];
fft(workExp0);
fft(workExp1);
fft(workInvExp0);
fft(workFall0);
fft(workFall1);
vector<Mint> b0(d * d * ew), b1(d * d * ew1);
for (int i = 0; i < d; ++i) for (int j = 0; j < d; ++j) {
const Mint *bij = b.data() + (i * d + j) * ew;
Mint *b0ij = b0.data() + (i * d + j) * ew;
Mint *b1ij = b1.data() + (i * d + j) * ew1;
vector<Mint> ys0(ew1);
for (int l = 0; l < ew; ++l) ys0[l] = invFac[l] * bij[l];
fft(ys0);
for (int l = 0; l < ew1; ++l) ys0[l] *= workInvExp0[l];
invFft(ys0);
for (int l = 0; l < ew1; ++l) ys0[l] *= fac[l];
memset(ys0.data() + ew, 0, ew * sizeof(Mint));
fft(ys0);
vector<Mint> ys1 = ys0;
for (int l = 0; l < ew1; ++l) ys0[l] *= workFall0[l];
for (int l = 0; l < ew1; ++l) ys1[l] *= workFall1[l];
invFft(ys0);
invFft(ys1);
ys0.erase(ys0.begin(), ys0.begin() + (ew - 1));
ys1.erase(ys1.begin(), ys1.begin() + (ew - 1));
for (int l = 0; l < ew; ++l) ys0[l] *= invFac[l];
for (int l = 0; l < ew; ++l) ys1[l] *= invFac[l];
ys0.resize(ew1, 0);
ys1.resize(ew2, 0);
fft(ys0);
fft(ys1);
for (int l = 0; l < ew1; ++l) ys0[l] *= workExp0[l];
for (int l = 0; l < ew2; ++l) ys1[l] *= workExp1[l];
invFft(ys0);
invFft(ys1);
for (int l = 0; l < ew; ++l) b0ij[l] = fac[l] * ys0[l];
for (int l = 0; l < ew1; ++l) b1ij[l] = fac[l] * ys1[l];
}
vector<Mint> bb(d * d * ew1, 0);
for (int i = 0; i < d; ++i) for (int k = 0; k < d; ++k) for (int j = 0; j < d; ++j) {
Mint *bbij = bb.data() + ((i * d + j) * ew1);
const Mint *bik = b.data() + ((i * d + k) * ew);
const Mint *b0ik = b0.data() + ((i * d + k) * ew);
const Mint *b1kj = b1.data() + ((k * d + j) * ew1);
for (int l = 0; l < ew; ++l) bbij[l] += bik[l] * b1kj[l];
for (int l = 0; l < ew; ++l) bbij[ew + l] += b0ik[l] * b1kj[ew + l];
}
b = bb;
}
vector<Mint> c(d * d, 0);
for (int i = 0; i < d; ++i) c[i * d + i] = 1;
long long x = xL;
// O(d^3 (xR-xL)/block) <= O(d^3 e block)
for (int l = 0; x + block <= xR; ++l, x += block) {
vector<Mint> cc(d * d, 0);
for (int i = 0; i < d; ++i) for (int k = 0; k < d; ++k) for (int j = 0; j < d; ++j) {
cc[i * d + j] += c[i * d + k] * b[(k * d + j) * e * block + l];
}
c = cc;
}
// O(d^3 block + d^2 e block)
for (; x < xR; ++x) {
const Mint x_ = x;
vector<Mint> ax(d * d, 0), cc(d * d, 0);
for (int i = 0; i < d; ++i) for (int j = 0; j < d; ++j) {
Mint &y = ax[i * d + j];
for (int m = a[i][j].size(); --m >= 0; ) (y *= x_) += a[i][j][m];
}
for (int i = 0; i < d; ++i) for (int k = 0; k < d; ++k) for (int j = 0; j < d; ++j) {
cc[i * d + j] += c[i * d + k] * ax[k * d + j];
}
c = cc;
}
vector<vector<Mint>> ret(d, vector<Mint>(d));
for (int i = 0; i < d; ++i) for (int j = 0; j < d; ++j) ret[i][j] = c[i * d + j];
return ret;
}
////////////////////////////////////////////////////////////////////////////////
/*
f(n, k) := k! [x^k] (1 + A x + B x^2)^n
f(n, k) = A (n + 1 - k) f(n, k-1) + B (2 n + 2 - k) (k - 1) f(n, k-2)
(f(n, 0) = 1, f(n, -1) := 0)
f(n, k+1) = A (n - k) f(n, k) + B (2 n + 1 - k) k f(n, k-1)
[ f(n, k+1) f(n, k) ] = [ A (n - k) 1 ]
[ B (2 n + 1 - k) k 0 ]
*/
const Mint A = 2;
const Mint B = Mint(2).inv();
Mint brute(Int N, Int K) {
Mint f = 1, g = 0;
for (Int k = 0; k < K; ++k) {
const Mint h = g;
g = f;
f = A * (N - k) * g + B * (2 * N + 1 - k) * k * h;
}
return f;
}
int T;
vector<Int> N, K;
vector<Mint> ans;
namespace small {
struct Mat {
Poly x[2][2];
Mat() : x{} {}
friend ostream &operator<<(ostream &os, const Mat &a) {
return os << "[" << a.x[0][0] << ", " << a.x[0][1] << "; " << a.x[1][0] << ", " << a.x[1][1] << "]";
}
};
Mat operator*(const Mat &a, const Mat &b) {
Mat c;
for (int i = 0; i < 2; ++i) for (int k = 0; k < 2; ++k) for (int j = 0; j < 2; ++j) {
c.x[i][j] += a.x[i][k] * b.x[k][j];
}
return c;
}
vector<pair<int, int>> kts;
vector<Poly> fss;
vector<Mat> ms;
inline int id(int l, int r) {
return (l + 1 == r) ? l : (T + (l + r) / 2);
}
Mat sub(int kL, int kR) {
if (kL == kR) {
Mat ret;
ret.x[0][0] = ret.x[1][1] = {1};
return ret;
} else if (kL + 1 == kR) {
const Mint k = kL;
Mat ret;
ret.x[0][0] = {A * -k, A};
ret.x[0][1] = {1};
ret.x[1][0] = {B * (1 - k) * k, B * 2 * k};
return ret;
} else {
const int kMid = (kL + kR) / 2;
return sub(kL, kMid) * sub(kMid, kR);
}
}
void dfs(int l, int r) {
if (l + 1 == r) {
fss[id(l, r)] = Poly{-N[kts[l].second], 1};
ms[id(l, r)] = sub(l ? kts[l - 1].first : 0, kts[l].first);
} else {
const int mid = (l + r) / 2;
dfs(l, mid);
dfs(mid, r);
fss[id(l, r)] = fss[id(l, mid)] * fss[id(mid, r)];
ms[id(l, r)] = ms[id(l, mid)] * ms[id(mid, r)];
}
// cerr<<"dfs "<<l<<" "<<r<<" "<<fss[id(l,r)]<<" "<<ms[id(l,r)]<<endl;
}
void DFS(int l, int r, const Mat &above_) {
Mat above;
for (int i = 0; i < 2; ++i) for (int j = 0; j < 2; ++j) {
above.x[i][j] = above_.x[i][j] % fss[id(l, r)];
}
// cerr<<"DFS "<<l<<" "<<r<<" "<<above<<endl;
if (l + 1 == r) {
const int t = kts[l].second;
const Mint n = N[t];
Mint as[2];
for (int j = 0; j < 2; ++j) {
assert(above.x[0][j].size() <= 1);
as[j] = above.x[0][j].at(0);
}
ans[t] = 0;
for (int k = 0; k < 2; ++k) {
const Poly &mk = ms[id(l, r)].x[k][0];
Mint tmp = 0;
for (int h = mk.size(); --h >= 0; ) {
(tmp *= n) += mk[h];
}
ans[t] += as[k] * tmp;
}
} else {
const int mid = (l + r) / 2;
DFS(l, mid, above);
DFS(mid, r, above * ms[id(l, mid)]);
}
}
void run() {
kts.resize(T);
for (int t = 0; t < T; ++t) {
kts[t] = make_pair(K[t], t);
}
sort(kts.begin(), kts.end());
vector<Mint> ns(T);
for (int i = 0; i < T; ++i) {
ns[i] = N[kts[i].second];
}
// cerr<<"kts = "<<kts<<endl;
// cerr<<"ns = "<<ns<<endl;
fss.resize(2 * T);
ms.resize(2 * T);
dfs(0, T);
// cerr<<"DONE dfs"<<endl;
Mat ini;
ini.x[0][0] = ini.x[1][1] = {1};
DFS(0, T, ini);
#ifdef LOCAL
vector<Mint>brt(T);
for(int t=0;t<T;++t)brt[t]=brute(N[t],K[t]);
cerr<<"brt = "<<brt<<endl;
#endif
}
} // small
namespace large {
void run() {
for (int t = 0; t < T; ++t) if (K[t] < MO) {
const Mint n = N[t];
const vector<vector<Poly>> a{
{{A * n, -A}, {1}},
{{0, B * (2 * n + 1), -B}, {}},
};
const auto res = polyMatrixProduct(a, 0, K[t]);
ans[t] = res[0][0];
}
}
} // large
int main() {
for (; ~scanf("%d", &T); ) {
N.resize(T);
K.resize(T);
for (int t = 0; t < T; ++t) {
scanf("%lld%lld", &N[t], &K[t]);
}
ans.assign(T, 0);
const Int maxK = *max_element(K.begin(), K.end());
large::run();
for (int t = 0; t < T; ++t) {
printf("%u\n", ans[t].x);
}
}
return 0;
}