結果
| 問題 |
No.2166 Paint and Fill
|
| コンテスト | |
| ユーザー |
 Nachia
|
| 提出日時 | 2023-01-03 03:59:20 |
| 言語 | C++17 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 2,945 ms / 10,000 ms |
| コード長 | 35,336 bytes |
| コンパイル時間 | 3,464 ms |
| コンパイル使用メモリ | 153,816 KB |
| 最終ジャッジ日時 | 2025-02-09 23:08:25 |
|
ジャッジサーバーID （参考情報） |
judge1 / judge3 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 2 |
| other | AC * 38 |
ソースコード
#line 2 "nachia\\fps\\formal-power-series-struct.hpp"
#include <vector>
#include <algorithm>
#include <string>
#include <cassert>
#include <iostream>
#line 3 "nachia\\math-modulo\\modulo-primitive-root.hpp"
#include <utility>
namespace nachia{
template<unsigned int MOD>
struct PrimitiveRoot{
static constexpr unsigned long long powm(unsigned long long a, unsigned long long i) {
unsigned long long res = 1, aa = a;
while(i){
if(i & 1) res = res * aa % MOD;
aa = aa * aa % MOD;
i /= 2;
}
return res;
}
static constexpr bool ExamineVal(unsigned int g){
unsigned int t = MOD - 1;
for(unsigned long long d=2; d*d<=t; d++) if(t % d == 0){
if(powm(g, (MOD - 1) / d) == 1) return false;
while(t % d == 0) t /= d;
}
if(t != 1) if(powm(g, (MOD - 1) / t) == 1) return false;
return true;
}
static constexpr unsigned int GetVal(){
for(unsigned int x=2; x<MOD; x++) if(ExamineVal(x)) return x;
return 0;
}
static const unsigned int val = GetVal();
};
} // namespace nachia
#line 3 "nachia\\math\\combination.hpp"
namespace nachia{
template<class Modint>
class Comb{
private:
std::vector<Modint> F;
std::vector<Modint> iF;
public:
void extend(int newN){
int prevN = (int)F.size() - 1;
if(prevN >= newN) return;
F.resize(newN+1);
iF.resize(newN+1);
for(int i=prevN+1; i<=newN; i++) F[i] = F[i-1] * Modint::raw(i);
iF[newN] = F[newN].inv();
for(int i=newN; i>prevN; i--) iF[i-1] = iF[i] * Modint::raw(i);
}
Comb(int n = 1){
F.assign(2, Modint(1));
iF.assign(2, Modint(1));
extend(n);
}
Modint factorial(int n) const { return F[n]; }
Modint invFactorial(int n) const { return iF[n]; }
Modint invOf(int n) const { return iF[n] * F[n-1]; }
Modint comb(int n, int r) const {
if(n < 0 || n < r || r < 0) return Modint(0);
return F[n] * iF[r] * iF[n-r];
}
Modint invComb(int n, int r) const {
if(n < 0 || n < r || r < 0) return Modint(0);
return iF[n] * F[r] * F[n-r];
}
Modint perm(int n, int r) const {
if(n < 0 || n < r || r < 0) return Modint(0);
return F[n] * iF[n-r];
}
Modint invPerm(int n, int r) const {
if(n < 0 || n < r || r < 0) return Modint(0);
return iF[n] * F[n-r];
}
Modint operator()(int n, int r) const { return comb(n,r); }
};
} // namespace nachia
#line 4 "nachia\\misc\\bit-operations.hpp"
namespace nachia{
int Popcount(unsigned long long c) noexcept {
#ifdef __GNUC__
return __builtin_popcountll(c);
#else
c = (c & (~0ull/3)) + ((c >> 1) & (~0ull/3));
c = (c & (~0ull/5)) + ((c >> 2) & (~0ull/5));
c = (c & (~0ull/17)) + ((c >> 4) & (~0ull/17));
c = (c * (~0ull/257)) >> 56;
return c;
#endif
}
// please ensure x != 0
int MsbIndex(unsigned long long x) noexcept {
#ifdef __GNUC__
return 63 - __builtin_clzll(x);
#else
int res = 0;
for(int d=32; d>0; d>>=1) if(x >> d){ res |= d; x >>= d; }
return res;
#endif
}
// please ensure x != 0
int LsbIndex(unsigned long long x) noexcept {
#ifdef __GNUC__
return __builtin_ctzll(x);
#else
return MsbIndex(x & -x);
#endif
}
}
#line 2 "nachia\\fps\\ntt-interface.hpp"
namespace nachia {
template<class mint>
struct NttInterface{
template<class Iter>
void Butterfly(Iter, int) const {}
template<class Iter>
void IButterfly(Iter, int) const {}
template<class Iter>
void BitReversal(Iter a, int N) const {
for(int i=0, j=0; j<N; j++){
if(i < j) std::swap(a[i], a[j]);
for(int k = N>>1; k > (i^=k); k>>=1);
}
}
};
} // namespace nachia
#line 5 "nachia\\fps\\ntt-acl.hpp"
#include <iterator>
#line 8 "nachia\\fps\\ntt-acl.hpp"
#include <array>
namespace nachia{
constexpr int bsf_constexpr(unsigned int n) {
int x = 0;
while (!(n & (1 << x))) x++;
return x;
}
template <class mint>
struct NttFromAcl : NttInterface<mint> {
using u32 = unsigned int;
using u64 = unsigned long long;
static int ceil_pow2(int n) {
int x = 0;
while ((1U << x) < (u32)(n)) x++;
return x;
}
struct fft_info {
static constexpr u32 g = nachia::PrimitiveRoot<mint::mod()>::val;
static constexpr int rank2 = bsf_constexpr(mint::mod()-1);
std::array<mint, rank2+1> root;
std::array<mint, rank2+1> iroot;
std::array<mint, std::max(0, rank2-1)> rate2;
std::array<mint, std::max(0, rank2-1)> irate2;
std::array<mint, std::max(0, rank2-2)> rate3;
std::array<mint, std::max(0, rank2-2)> irate3;
fft_info(){
root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2);
iroot[rank2] = root[rank2].inv();
for(int i=rank2-1; i>=0; i--){
root[i] = root[i+1] * root[i+1];
iroot[i] = iroot[i+1] * iroot[i+1];
}
mint prod = 1, iprod = 1;
for(int i=0; i<=rank2-2; i++){
rate2[i] = root[i+2] * prod;
irate2[i] = iroot[i+2] * iprod;
prod *= iroot[i+2];
iprod *= root[i+2];
}
prod = 1; iprod = 1;
for(int i=0; i<=rank2-3; i++){
rate3[i] = root[i+3] * prod;
irate3[i] = iroot[i+3] * iprod;
prod *= iroot[i+3];
iprod *= root[i+3];
}
}
};
template<class RandomAccessIterator>
void Butterfly(RandomAccessIterator a, int n) const {
int h = ceil_pow2(n);
static const fft_info info;
int len = 0;
while(len < h){
if(h-len == 1){
int p = 1 << (h-len-1);
mint rot = 1;
for(int s=0; s<(1<<len); s++){
int offset = s << (h-len);
for(int i=0; i<p; i++){
auto l = a[i+offset];
auto r = a[i+offset+p] * rot;
a[i+offset] = l+r;
a[i+offset+p] = l-r;
}
if(s+1 != (1<<len)) rot *= info.rate2[LsbIndex(~(u32)(s))];
}
len++;
} else {
int p = 1 << (h-len-2);
mint rot = 1, imag = info.root[2];
for(int s=0; s<(1<<len); s++){
mint rot2 = rot * rot;
mint rot3 = rot2 * rot;
int offset = s << (h-len);
for(int i=0; i<p; i++){
auto mod2 = 1ULL * mint::mod() * mint::mod();
auto a0 = 1ULL * a[i+offset].val();
auto a1 = 1ULL * a[i+offset+p].val() * rot.val();
auto a2 = 1ULL * a[i+offset+2*p].val() * rot2.val();
auto a3 = 1ULL * a[i+offset+3*p].val() * rot3.val();
auto a1na3imag = 1ULL * mint(a1 + mod2 - a3).val() * imag.val();
auto na2 = mod2 - a2;
a[i+offset] = a0 + a2 + a1 + a3;
a[i+offset+1*p] = a0 + a2 + (2 * mod2 - (a1 + a3));
a[i+offset+2*p] = a0 + na2 + a1na3imag;
a[i+offset+3*p] = a0 + na2 + (mod2 - a1na3imag);
}
if(s+1 != (1<<len)) rot *= info.rate3[LsbIndex(~(u32)(s))];
}
len += 2;
}
}
}
template<class RandomAccessIterator>
void IButterfly(RandomAccessIterator a, int n) const {
int h = ceil_pow2(n);
static const fft_info info;
constexpr int MOD = mint::mod();
int len = h;
while(len){
if(len == 1){
int p = 1 << (h-len);
mint irot = 1;
for(int s=0; s<(1<<(len-1)); s++){
int offset = s << (h-len+1);
for(int i=0; i<p; i++){
auto l = a[i+offset];
auto r = a[i+offset+p];
a[i+offset] = l+r;
a[i+offset+p] = (u64)(MOD + l.val() - r.val()) * irot.val();
}
if(s+1 != (1<<(len-1))) irot *= info.irate2[LsbIndex(~(u32)(s))];
}
len--;
} else {
int p = 1 << (h-len);
mint irot = 1, iimag = info.iroot[2];
for(int s=0; s<(1<<(len-2)); s++){
mint irot2 = irot * irot;
mint irot3 = irot2 * irot;
int offset = s << (h-len+2);
for(int i=0; i<p; i++){
auto a0 = 1ULL * a[i+offset+0*p].val();
auto a1 = 1ULL * a[i+offset+1*p].val();
auto a2 = 1ULL * a[i+offset+2*p].val();
auto a3 = 1ULL * a[i+offset+3*p].val();
auto a2na3iimag = 1ULL * mint((MOD + a2 - a3) * iimag.val()).val();
a[i+offset] = a0 + a1 + a2 + a3;
a[i+offset+1*p] = (a0 + (MOD - a1) + a2na3iimag) * irot.val();
a[i+offset+2*p] = (a0 + a1 + (MOD - a2) + (MOD - a3)) * irot2.val();
a[i+offset+3*p] = (a0 + (MOD - a1) + (MOD - a2na3iimag)) * irot3.val();
}
if(s+1 != (1<<(len-2))) irot *= info.irate3[LsbIndex(~(u32)(s))];
}
len -= 2;
}
}
}
};
} // namespace nachia
#line 11 "nachia\\fps\\formal-power-series-struct.hpp"
namespace nachia {
template<class Elem, class NttInst = NttFromAcl<Elem>>
struct FormalPowerSeriesNTT {
public:
using Fps = FormalPowerSeriesNTT;
using ElemTy = Elem;
static constexpr unsigned int MOD = Elem::mod();
static constexpr int CONV_THRES = 30;
static const NttInst nttInst;
static const unsigned int zeta = nachia::PrimitiveRoot<MOD>::GetVal();
private:
using u32 = unsigned int;
static Elem ZeroElem() noexcept { return Elem(0); }
static Elem OneElem() noexcept { return Elem(1); }
static Comb<Elem> comb;
std::vector<Elem> a;
public:
int size() const noexcept { return a.size(); }
Elem& operator[](int x) noexcept { return a[x]; }
const Elem& operator[](int x) const noexcept { return a[x]; }
Elem getCoeff(int x) const noexcept { return (0 <= x && x < size()) ? a[x] : ZeroElem(); }
static Comb<Elem>& GetComb() { return comb; }
static int BestNttSize(int x) noexcept { assert(x); return 1 << MsbIndex(x*2-1); }
Fps move(){ return std::move(*this); }
Fps& set(int i, Elem c){ a[i] = c; return *this; }
Fps& removeLeadingZeros(){
int newsz = size();
while(newsz && a[newsz-1].val() == 0) newsz--;
a.resize(newsz);
if((int)a.capacity() / 4 > newsz) a.shrink_to_fit();
return *this;
}
FormalPowerSeriesNTT(){}
FormalPowerSeriesNTT(int sz) : a(sz, ZeroElem()) {}
FormalPowerSeriesNTT(std::vector<Elem>&& src) : a(std::move(src)) {}
FormalPowerSeriesNTT(const std::vector<Elem>& src) : a(src) {}
Fps& ntt() {
capSize(BestNttSize(size()));
nttInst.Butterfly(a.begin(), size());
return *this;
}
Fps& intt() {
nttInst.IButterfly(a.begin(), a.size());
return times(Elem::raw(size()).inv());
}
Fps nttDouble(Fps vanilla) const {
int n = size();
assert(n == (n&-n)); // n is a power of 2
Elem q = Elem::raw(zeta).pow((Elem::mod() - 1) / (n*2));
Elem qq = Elem::raw(1);
for(int i=0; i<n; i++){ vanilla[i] *= qq; qq *= q; }
vanilla.ntt();
Fps res = clip(0, n*2);
for(int i=0; i<n; i++) res[n+i] = vanilla[i];
return res;
}
Fps nttDouble() const { return nttDouble(clip().intt().move()); }
// Fps res(resSz);
// for(int j=0; j<resSz-destL && j+srcL < srcR; j++) res[j+destL] = a.getCoeff(j+srcL)
// if srcR is unspecified -> srcR = max(srcL, size());
// if resSz is unspecified -> resSz = destL + srcR - srcL
Fps clip(int srcL, int srcR = -1, int destL = 0, int resSz = -1) const {
if(srcR < 0) srcR = std::max(srcL, size());
if(resSz < 0) resSz = destL + srcR - srcL;
if(srcR > size()) srcR = size();
Fps res(resSz);
for(int j=std::max(0, -srcL); j+destL < resSz && j+srcL < srcR; j++) res[j+destL] = a[j+srcL];
return res;
}
Fps clip() const { return *this; }
Fps& capSize(int l, int r) {
if(r <= (int)size()) a.resize(r);
if(size() <= l) a.resize(l, ZeroElem());
return *this;
}
Fps& capSize(int z){ a.resize(z, ZeroElem()); return *this; }
Fps& times(Elem x){ for(int i=0; i<size(); i++){ a[i] *= x; } return *this; }
Fps& clrRange(int l, int r){ for(int i=l; i<r; i++){ a[i] = ZeroElem(); } return *this; }
Fps& negate(){ for(auto& e : a){ e = -e; } return *this; }
Fps& mulEach(const Fps& other, size_t maxi = ~(size_t)0){
maxi = std::min(maxi, (size_t)std::min(size(), other.size()));
for(size_t i=0; i<maxi; i++) a[i] *= other[i];
return *this;
}
Fps& reverse(){ std::reverse(a.begin(), a.end()); return *this; }
static Fps convolution(const Fps& a, const Fps& b, int sz = -1){
if(std::min(a.size(), b.size()) <= CONV_THRES){
if(a.size() > b.size()) return convolution(b, a, sz);
if(sz < 0) sz = std::max(0, a.size() + b.size() - 1);
std::vector<Elem> res(sz);
for(int i=0; i<a.size(); i++) for(int j=0; j<b.size() && i+j<sz; j++) res[i+j] += a[i] * b[j];
return res;
}
int Z = BestNttSize(a.size() + b.size() - 1);
if(sz == -1) sz = Z;
return a.clip(0, Z).ntt().mulEach(b.clip(0, Z).ntt()).intt().capSize(sz).move();
}
Fps convolve(const Fps& r, int sz = -1) const { return convolution(*this, r, sz); }
// 1
// ----- = 1 + f + f^2 + f^3 + ...
// 1-f
Fps powerSum(int sz) const {
if(sz < 0) sz = size();
if(sz == 0) return {};
int q = std::min(sz, 32);
Fps x = Fps(q).set(0, OneElem()).move();
for(int i=1; i<q; i++) for(int j=1; j<=std::min(i,(int)a.size()-1); j++) x[i] += x[i-j] * a[j];
while(x.size() < sz){
int hN = x.size(), N = hN*2;
Fps a = x.clip(0, hN, 0, N).ntt().move();
Fps b = clip(0, N).ntt().mulEach(a).intt().clrRange(0,hN).ntt().mulEach(a).intt().move();
for(int i=0; i<hN; i++) b[i] = x[i];
std::swap(b, x);
}
return x.capSize(sz).move();
}
Fps inv(int sz = -1) const {
if(sz < 0) sz = size();
Elem iA0 = a[0].inv();
return clip(0, std::min(sz, size())).times(-iA0).move().set(0, ZeroElem()).powerSum(sz).times(iA0).move();
}
Fps& difference(){
if(size() == 0) return *this;
for(int i=0; i+1<size(); i++) a[i] = a[i+1] * Elem::raw(i+1);
capSize(0, size()-1);
return *this;
}
Fps& integral(){
if(size() == 0) return capSize(1);
capSize(size()+1);
comb.extend(size());
for(int i=size()-1; i>=1; i--) a[i] = a[i-1] * comb.invOf(i);
return set(0, ZeroElem());
}
Fps log(int sz = -1){
if(sz < 0) sz = size();
assert(sz != 0);
assert(a[0].val() == 1);
return convolution(inv(sz), clip().difference(), sz-1).integral();
}
Fps exp(int sz = -1){
if(sz < 0) sz = size();
Fps res = Fps(1).set(0, OneElem());
while(res.size() < sz){
auto z = res.size();
auto tmp = res.capSize(z*2).log().set(0, -OneElem()).move();
for(int i=0; i<z*2 && i<size(); i++) tmp[i] -= a[i];
auto resntt = res.clip().ntt().mulEach(tmp.ntt()).intt().move();
for(int i=z; i<z*2; i++) res[i] = -resntt[i];
}
return res.capSize(0, sz).move();
}
Fps pow(unsigned long long k, int sz = -1){
int n = sz < 0 ? size() : sz;
if(k == 0) return Fps(n).set(0, OneElem()).move();
int ctz = 0;
while(ctz<n && a[ctz].val() == 0) ctz++;
if((unsigned long long)ctz >= (n-1) / k + 1) return Fps(n);
Fps res = clip(ctz, ctz+n-ctz*k);
Elem a0 = res[0];
ctz *= k; n -= ctz;
return res.times(a0.inv()).log(n).times(Elem(k)).exp(n).times(a0.pow(k)).clip(0, n, ctz);
}
auto begin(){ return a.begin(); }
auto end(){ return a.end(); }
auto begin() const { return a.begin(); }
auto end() const { return a.end(); }
std::string toString(std::string beg = "[ ", std::string delim = " ", std::string en = " ]") const {
std::string res = beg;
bool f = false;
for(auto x : a){ if(f){ res += delim; } f = true; res += std::to_string(x.val()); }
res += en;
return res;
}
std::vector<Elem> getVectorMoved(){ return std::move(a); }
Fps& operator+=(const Fps& r){
capSize(std::max(size(), r.size()));
for(int i=0; i<r.size(); i++) a[i] += r[i];
return *this;
}
Fps& operator-=(const Fps& r){
capSize(std::max(size(), r.size()));
for(int i=0; i<r.size(); i++) a[i] -= r[i];
return *this;
}
Fps operator+(const Fps& r) const { return (clip(0, std::max(size(), r.size())) += r).move(); }
Fps operator-(const Fps& r) const { return (clip(0, std::max(size(), r.size())) -= r).move(); }
Fps operator-() const { return (clip().negate()).move(); }
Fps operator*(const Fps& r) const { return convolve(r).removeLeadingZeros().move(); }
Fps& operator*=(const Fps& r){ return (*this) = operator*(r); }
Fps& operator*=(Elem m){ return times(m); }
Fps operator*(Elem m) const { return (clip() *= m).move(); }
Elem eval(Elem x) const {
Elem res = 0;
for(int i=size()-1; i>=0; i--) res = res * x + a[i];
return res;
}
};
template<class Elem, class NttInst> Comb<Elem> FormalPowerSeriesNTT<Elem, NttInst>::comb;
template<class Elem, class NttInst> const NttInst FormalPowerSeriesNTT<Elem, NttInst>::nttInst;
} // namespace nachia
#line 5 "nachia\\linear\\matrix-on-ring.hpp"
namespace nachia{
template<class Elem>
struct MatrixOnRing{
private:
int h;
int w;
std::vector<Elem> elems;
public:
MatrixOnRing(int new_h=0, int new_w=0){ h = new_h; w = new_w; elems.resize(h * w); }
MatrixOnRing(MatrixOnRing const&) = default;
int numRow() const { return h; }
int numColumn() const { return w; }
int height() const { return numRow(); }
int width() const { return numColumn(); }
typename std::vector<Elem>::iterator operator[](int y){ return elems.begin() + (y*w); }
typename std::vector<Elem>::const_iterator operator[](int y) const { return elems.begin() + (y*w); }
static MatrixOnRing Identity(int idx, Elem One){ auto res = MatrixOnRing(idx, idx); for(int i=0; i<idx; i++) res[i][i] = One; return res; }
void swapColumns(int x1, int x2){
assert(0 <= x1 && x1 < numColumn());
assert(0 <= x2 && x2 < numColumn());
for(int y=0; y<numRow(); y++) std::swap((*this)[y][x1], (*this)[y][x2]);
}
void swapRows(int y1, int y2){
assert(0 <= y1 && y1 < numRow());
assert(0 <= y2 && y2 < numRow());
for(int x=0; x<numColumn(); x++) std::swap((*this)[y1][x], (*this)[y2][x]);
}
MatrixOnRing operator*(const MatrixOnRing& r) const {
assert(width() == r.height());
auto res = MatrixOnRing(h, r.w);
for(int i=0; i<h; i++) for(int j=0; j<w; j++) for(int k=0; k<r.w; k++) res[i][k] = res[i][k] + (*this)[i][j] * r[j][k];
return res;
}
MatrixOnRing pow(unsigned long long i){
auto a = *this;
auto res = Identity(height());
while(i){
if(i % 2 == 1) res = res * a;
a = a * a;
i /= 2;
}
return res;
}
};
} // namespace nachia
#line 4 "nachia\\fps\\p-recursive-matrix-product.hpp"
namespace nachia{
template<class Elem>
MatrixOnRing<Elem> PRecursiveMatrixProduct(
MatrixOnRing<FormalPowerSeriesNTT<Elem>> p,
unsigned long long idx
){
struct ShiftOfSamplingPointsOfPolynomialUpdate{
using Fps = FormalPowerSeriesNTT<Elem>;
int n;
int N2;
Fps iF, F, iFI, iFIntt1, iFntt;
std::vector<Fps> iFIntt2s;
ShiftOfSamplingPointsOfPolynomialUpdate(int n, std::vector<Elem> sh){
auto& comb = Fps::GetComb();
comb.extend(n);
this->n = n;
N2 = Fps::BestNttSize(n*2);
iF = F = iFI = Fps(n);
for(int i=0; i<n; i++) F[i] = comb.factorial(i);
for(int i=0; i<n; i++) iF[i] = comb.invFactorial(i);
for(int i=0; i<n; i++) iFI[i] = (i%2) ? -iF[i] : iF[i];
iFIntt1 = iFI.clip(0, N2); iFIntt1.ntt();
iFntt = iF.clip(0, N2); iFntt.ntt();
for(size_t shi=0; shi<sh.size(); shi++){
Elem q = 1;
Fps T(N2); T[0] = Elem(1);
for(int i=1; i<n; i++) T[i] = iF[i] * (q *= (sh[shi] - Elem::raw(i-1)));
iFIntt2s.push_back(T.ntt().move());
}
}
std::vector<std::vector<Elem>> calc(const std::vector<Elem>& points){
Fps P(N2);
for(int i=0; i<n; i++) P[i] = points[i] * iF[i];
P.ntt().mulEach(iFIntt1).intt().clrRange(n, N2).mulEach(F, n).reverse().ntt();
std::vector<std::vector<Elem>> res2(iFIntt2s.size());
for(size_t shi=0; shi<iFIntt2s.size(); shi++){
res2[shi] = P.clip().mulEach(iFIntt2s[shi]).intt()
.reverse().clrRange(n, N2).mulEach(iF, n).ntt()
.mulEach(iFntt).intt().mulEach(F, n).clip(0, n).getVectorMoved();
}
return res2;
}
};
using u64 = unsigned long long;
int h = p.height();
std::vector<std::vector<Elem>> res;
res.resize(h*h);
for(auto& a : res) a.resize(h);
u64 a = 1, b = 1;
for(int i=0; i<h; i++) for(int j=0; j<h; j++) while(b < (u64)p[i][j].size()) b <<= 1;
u64 maxA = 1, maxB = b;
while(maxA * maxB <= idx){ maxB <<= 1; maxA <<= 1; }
for(int i=0; i<h; i++) for(int j=0; j<h; j++){
res[i*h+j].resize(b);
for(u64 k=0; k<b; k++) res[i*h+j][k] = p[i][j].eval(Elem(maxA) * Elem(k));
}
auto EvalP = [p, h](Elem val) -> MatrixOnRing<Elem> {
MatrixOnRing<Elem> res(h, h);
for(int y=0; y<h; y++) for(int x=0; x<h; x++) res[y][x] = p[y][x].eval(val);
return res;
};
auto EvalL = [&res, h](u64 idx) -> MatrixOnRing<Elem> {
MatrixOnRing<Elem> g(h, h);
for(int y=0; y<h; y++) for(int x=0; x<h; x++) g[y][x] = res[y*h+x][idx];
return g;
};
while(b < maxB){
std::vector<Elem> sh(3);
sh[0] = Elem(b);
sh[1] = Elem(a) / Elem(maxA);
sh[2] = sh[0] + sh[1];
std::vector<std::vector<std::vector<Elem>>> shbuf(h*h);
auto shman = ShiftOfSamplingPointsOfPolynomialUpdate(b, sh);
for(int i=0; i<h*h; i++) shbuf[i] = shman.calc(res[i]);
std::vector<std::vector<Elem>> resbuf;
resbuf.assign(h*h, std::vector<Elem>(b*2));
for(int i=0; i<h; i++) for(int j=0; j<h; j++) for(int k=0; k<h; k++){
auto Lbeg1 = shbuf[i*h+j][1].begin();
auto Rbeg1 = res[j*h+k].begin();
auto destbeg1 = resbuf[i*h+k].begin();
for(u64 id=0; id<b; id++) destbeg1[id] += Lbeg1[id] * Rbeg1[id];
auto Lbeg2 = shbuf[i*h+j][2].begin();
auto Rbeg2 = shbuf[j*h+k][0].begin();
auto destbeg2 = resbuf[i*h+k].begin() + b;
for(u64 id=0; id<b; id++) destbeg2[id] += Lbeg2[id] * Rbeg2[id];
}
std::swap(res, resbuf);
a *= 2;
b *= 2;
}
u64 pos = 0;
MatrixOnRing<Elem> ans = MatrixOnRing<Elem>::Identity(h, Elem::raw(1));
while(pos + maxA <= idx){ ans = EvalL(pos / maxA) * ans; pos += maxA; }
while(pos < idx){ ans = EvalP(pos++) * ans; }
return ans;
}
} // namespace nachia
#line 2 "nachia\\fps\\ntt-setup-manager.hpp"
namespace nachia{
template<class Elem>
class FpsNttSetupManager {
using ElemTy = Elem;
using Fps = typename nachia::FormalPowerSeriesNTT<Elem>;
using MyType = FpsNttSetupManager;
Fps raw;
mutable Fps ntt;
static const int THRESH = 30;
FpsNttSetupManager(Fps _raw) : raw(_raw.move()) , ntt() {}
public:
FpsNttSetupManager() : FpsNttSetupManager(Fps()) {}
FpsNttSetupManager(Fps _raw, Fps _ntt) : raw(_raw.move()) , ntt(_ntt.move()) {}
const Fps& getRaw() const { return raw; }
int size() const { return raw.size(); }
int Least(){ return Fps::BestNttSize(raw.size()); }
static MyType FromRaw(Fps _raw){ return FpsNttSetupManager(_raw.move()); }
static MyType FromNtt(Fps _ntt){
Fps x = _ntt.clip();
return MyType(x.intt().removeLeadingZeros().move(), _ntt.move());
}
void doubling() const {
if(ntt.size() == 0) ntt = raw.clip(0, Fps::BestNttSize(raw.size())).ntt().move();
else ntt = ntt.nttDouble(raw.clip(0, ntt.size()));
}
Fps& ensureNtt(int sz) const {
if(sz / 8 >= ntt.size()) ntt = raw.clip(0, sz).ntt().move();
while(ntt.size() < sz) doubling();
return ntt;
}
Fps nttClip(int sz) const { return ensureNtt(sz).clip(0,sz); }
std::pair<Fps, Fps> destruct(){ return std::make_pair(raw.move(), ntt.move()); }
MyType operator+(const MyType& r) const {
Fps nntt;
int z1 = std::min(ntt.size(), r.ntt.size());
if(z1 >= std::max(size(), r.size())){
nntt.capSize(std::min(ntt.size(), r.ntt.size()));
for(int i=0; i<nntt.size(); i++) nntt[i] = ntt[i] + r.ntt[i];
}
return FpsNttSetupManager(raw + r.raw, nntt.move());
}
MyType operator*(const MyType& r) const {
if(std::min(size(), r.size()) <= THRESH) return FromRaw(raw * r.raw);
int sz = Fps::BestNttSize(size() + r.size() - 1);
return FromNtt(nttClip(sz).mulEach(r.ensureNtt(sz)).move());
}
};
} // namespace nachia
#line 2 "nachia\\misc\\fastio.hpp"
#include <cstdio>
#include <cctype>
#include <cstdint>
#line 6 "nachia\\misc\\fastio.hpp"
namespace nachia{
struct CInStream{
private:
static const unsigned int INPUT_BUF_SIZE = 1 << 17;
unsigned int p = INPUT_BUF_SIZE;
static char Q[INPUT_BUF_SIZE];
public:
using MyType = CInStream;
char seekChar() noexcept {
if(p == INPUT_BUF_SIZE){
size_t len = fread(Q, 1, INPUT_BUF_SIZE, stdin);
if(len != INPUT_BUF_SIZE) Q[len] = '\0';
p = 0;
}
return Q[p];
}
void skipSpace() noexcept { while(isspace(seekChar())) p++; }
uint32_t nextU32() noexcept {
skipSpace();
uint32_t buf = 0;
while(true){
char tmp = seekChar();
if('9' < tmp || tmp < '0') break;
buf = buf * 10 + (tmp - '0');
p++;
}
return buf;
}
int32_t nextI32() noexcept {
skipSpace();
if(seekChar() == '-'){ p++; return (int32_t)(-nextU32()); }
return (int32_t)nextU32();
}
uint64_t nextU64() noexcept {
skipSpace();
uint64_t buf = 0;
while(true){
char tmp = seekChar();
if('9' < tmp || tmp < '0') break;
buf = buf * 10 + (tmp - '0');
p++;
}
return buf;
}
int64_t nextI64() noexcept {
skipSpace();
if(seekChar() == '-'){ p++; return (int64_t)(-nextU64()); }
return (int64_t)nextU64();
}
char nextChar() noexcept { skipSpace(); char buf = seekChar(); p++; return buf; }
std::string nextToken(){
skipSpace();
std::string buf;
while(true){
char ch = seekChar();
if(isspace(ch) || ch == '\0') break;
buf.push_back(ch);
p++;
}
return buf;
}
MyType& operator>>(unsigned int& dest) noexcept { dest = nextU32(); return *this; }
MyType& operator>>(int& dest) noexcept { dest = nextI32(); return *this; }
MyType& operator>>(unsigned long& dest) noexcept { dest = nextU64(); return *this; }
MyType& operator>>(long& dest) noexcept { dest = nextI64(); return *this; }
MyType& operator>>(unsigned long long& dest) noexcept { dest = nextU64(); return *this; }
MyType& operator>>(long long& dest) noexcept { dest = nextI64(); return *this; }
MyType& operator>>(std::string& dest){ dest = nextToken(); return *this; }
MyType& operator>>(char& dest) noexcept { dest = nextChar(); return *this; }
} cin;
struct FastOutputTable{
char LZ[1000][4] = {};
char NLZ[1000][4] = {};
constexpr FastOutputTable(){
using u32 = uint_fast32_t;
for(u32 d=0; d<1000; d++){
LZ[d][0] = ('0' + d / 100 % 10);
LZ[d][1] = ('0' + d / 10 % 10);
LZ[d][2] = ('0' + d / 1 % 10);
LZ[d][3] = '\0';
}
for(u32 d=0; d<1000; d++){
u32 i = 0;
if(d >= 100) NLZ[d][i++] = ('0' + d / 100 % 10);
if(d >= 10) NLZ[d][i++] = ('0' + d / 10 % 10);
if(d >= 1) NLZ[d][i++] = ('0' + d / 1 % 10);
NLZ[d][i++] = '\0';
}
}
};
struct COutStream{
private:
using u32 = uint32_t;
using u64 = uint64_t;
using MyType = COutStream;
static const u32 OUTPUT_BUF_SIZE = 1 << 17;
static char Q[OUTPUT_BUF_SIZE];
static constexpr FastOutputTable TB = FastOutputTable();
u32 p = 0;
static constexpr u32 P10(u32 d){ return d ? P10(d-1)*10 : 1; }
static constexpr u64 P10L(u32 d){ return d ? P10L(d-1)*10 : 1; }
template<class T, class U> static void Fil(T& m, U& l, U x) noexcept { m = l/x; l -= m*x; }
void next_dig9(u32 x){
u32 y;
Fil(y, x, P10(6));
nextCstr(TB.LZ[y]);
Fil(y, x, P10(3));
nextCstr(TB.LZ[y]); nextCstr(TB.LZ[x]);
}
public:
void nextChar(char c){
Q[p++] = c;
if(p == OUTPUT_BUF_SIZE){ fwrite(Q, p, 1, stdout); p = 0; }
}
void nextEoln(){ nextChar('\n'); }
void nextCstr(const char* s){ while(*s) nextChar(*(s++)); }
void nextU32(uint32_t x){
u32 y = 0;
if(x >= P10(9)){
Fil(y, x, P10(9));
nextCstr(TB.NLZ[y]); next_dig9(x);
}
else if(x >= P10(6)){
Fil(y, x, P10(6));
nextCstr(TB.NLZ[y]);
Fil(y, x, P10(3));
nextCstr(TB.LZ[y]); nextCstr(TB.LZ[x]);
}
else if(x >= P10(3)){
Fil(y, x, P10(3));
nextCstr(TB.NLZ[y]); nextCstr(TB.LZ[x]);
}
else if(x >= 1) nextCstr(TB.NLZ[x]);
else nextChar('0');
}
void nextI32(int32_t x){
if(x >= 0) nextU32(x);
else{ nextChar('-'); nextU32((u32)-x); }
}
void nextU64(uint64_t x){
u32 y = 0;
if(x >= P10L(18)){
Fil(y, x, P10L(18));
nextU32(y);
Fil(y, x, P10L(9));
next_dig9(y); next_dig9(x);
}
else if(x >= P10L(9)){
Fil(y, x, P10L(9));
nextU32(y); next_dig9(x);
}
else nextU32(x);
}
void nextI64(int64_t x){
if(x >= 0) nextU64(x);
else{ nextChar('-'); nextU64((u64)-x); }
}
void writeToFile(bool flush = false){
fwrite(Q, p, 1, stdout);
if(flush) fflush(stdout);
p = 0;
}
COutStream(){ Q[0] = 0; }
~COutStream(){ writeToFile(); }
MyType& operator<<(unsigned int tg){ nextU32(tg); return *this; }
MyType& operator<<(unsigned long tg){ nextU64(tg); return *this; }
MyType& operator<<(unsigned long long tg){ nextU64(tg); return *this; }
MyType& operator<<(int tg){ nextI32(tg); return *this; }
MyType& operator<<(long tg){ nextI64(tg); return *this; }
MyType& operator<<(long long tg){ nextI64(tg); return *this; }
MyType& operator<<(const std::string& tg){ nextCstr(tg.c_str()); return *this; }
MyType& operator<<(const char* tg){ nextCstr(tg); return *this; }
MyType& operator<<(char tg){ nextChar(tg); return *this; }
} cout;
char CInStream::Q[INPUT_BUF_SIZE];
char COutStream::Q[OUTPUT_BUF_SIZE];
} // namespace nachia
#line 5 "Main.cpp"
#include <atcoder/modint>
#line 7 "Main.cpp"
int main(){
using Modint = atcoder::static_modint<998244353>;
using Polynomial = nachia::FormalPowerSeriesNTT<Modint>;
using PolynomialMat = nachia::MatrixOnRing<Polynomial>;
using NttSetup = nachia::FpsNttSetupManager<Modint>;
using NttSetupMat = nachia::MatrixOnRing<NttSetup>;
using nachia::cin, nachia::cout;
auto MatMod = [&](NttSetupMat mat, NttSetup mod) -> NttSetupMat {
int n = mat.height();
NttSetupMat res(n, n);
int maxlen = 0;
for(int i=0; i<n; i++) for(int j=0; j<n; j++) maxlen = std::max(maxlen, mat[i][j].size());
int deg = mod.size();
if(maxlen < deg) return mat;
auto K = mod.getRaw().clip().reverse().inv(maxlen - deg + 1);
for(int i=0; i<n; i++) for(int j=0; j<n; j++){
auto buf = mat[i][j];
if(buf.size() < mod.size()){ res[i][j] = std::move(buf); continue; }
int divlen = buf.size() + 1 - deg;
auto div = std::move(buf.getRaw().clip(deg-1).reverse().convolve(K.clip(0, divlen), divlen).reverse());
res[i][j] = NttSetup::FromRaw((mat[i][j].getRaw() - (NttSetup::FromRaw(div) * mod).getRaw()).clip(0, deg-1));
}
return res;
};
int T; cin >> T;
if(T <= 5){
for(int t=0; t<T; t++){
unsigned long long N, K; cin >> N >> K;
if(K >= 998244353){ cout << "0\n"; continue; }
PolynomialMat M_nX = PolynomialMat(2,2);
M_nX[0][0] = std::vector<Modint>{ Modint(N) * 2 , -Modint(2) }; // 2N - 2k
M_nX[0][1] = std::vector<Modint>{ 0, (Modint(N)*2+1) / 2, -Modint(1) / 2 }; // (2N+1)k/2 - k^2/2
M_nX[1][0] = std::vector<Modint>{ 1 };
M_nX[1][1] = std::vector<Modint>{};
auto ansMat = nachia::PRecursiveMatrixProduct(M_nX, K);
Modint ans = ansMat[0][0];
cout << ans.val() << '\n';
}
}
else{
int MAX_K = 100000;
int MATRIX_QUERY = 1001001001;
std::vector<std::pair<unsigned long long, int>> NK(T);
for(auto& nk : NK) cin >> nk.first >> nk.second;
std::vector<std::pair<int, int>> queries;
for(int k=0; k<MAX_K; k++) queries.emplace_back(k, MATRIX_QUERY);
for(int t=0; t<T; t++) queries.emplace_back(NK[t].second, t);
std::sort(queries.begin(), queries.end());
int segN = 1;
while(segN < (int)queries.size()) segN *= 2;
auto ConstructNttSetup = [](std::vector<Modint> x){ return NttSetup::FromRaw(x); };
std::vector<NttSetupMat> FX;
std::vector<NttSetup> KX;
FX.assign(segN*2, NttSetupMat::Identity(2, ConstructNttSetup({1})));
KX.assign(segN*2, ConstructNttSetup({1}));
for(int q=0; q<(int)queries.size(); q++){
if(queries[q].second == MATRIX_QUERY){
int k = queries[q].first;
FX[segN+q][0][0] = ConstructNttSetup({ -Modint(k)*2, Modint(2) }); // 2N - 2k
FX[segN+q][0][1] = ConstructNttSetup({ Modint(k)*(1-k) / 2, Modint(k) }); // Nk + k(1-k)/2
FX[segN+q][1][0] = ConstructNttSetup({ 1 });
FX[segN+q][1][1] = ConstructNttSetup({});
}
else{
unsigned long long N = NK[queries[q].second].first;
KX[segN+q] = ConstructNttSetup({ -Modint(N), 1 }); // x - N
}
}
for(int i=segN-1; i>=1; i--) FX[i] = FX[i*2+1] * FX[i*2];
for(int i=segN-1; i>=1; i--) KX[i] = KX[i*2+1] * KX[i*2];
std::vector<NttSetupMat> FXmodKX(segN*2);
FXmodKX[1] = MatMod(NttSetupMat::Identity(2, ConstructNttSetup({1})), std::move(KX[1]));
for(int i=1; i<=segN-1; i++){
FXmodKX[i*2+1] = MatMod(FX[i*2] * FXmodKX[i], std::move(KX[i*2+1]));
FXmodKX[i*2] = MatMod(std::move(FXmodKX[i]), std::move(KX[i*2]));
}
std::vector<Modint> ans(T);
for(int q=0; q<(int)queries.size(); q++){
if(queries[q].second != MATRIX_QUERY){
ans[queries[q].second] = FXmodKX[segN+q][0][0].getRaw().eval(0);
}
}
for(int i=0; i<T; i++) cout << ans[i].val() << '\n';
}
return 0;
}