CP-Algorithms Library
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Pow of Matrix (Frobenius) (verify/linalg/pow_fast.test.cpp)
- View this file on GitHub
- Last update: 2026-10-16 14:16:08+02:00
- Problem: https://judge.yosupo.jp/problem/pow_of_matrix
Depends on
- cp-algo/linalg/frobenius.hpp
- cp-algo/linalg/matrix.hpp
- cp-algo/linalg/strassen.hpp
- cp-algo/linalg/vector.hpp
- cp-algo/math/affine.hpp
- cp-algo/math/combinatorics.hpp
- cp-algo/math/common.hpp
- cp-algo/math/cvector.hpp
- cp-algo/math/factorials.hpp
- cp-algo/math/fft.hpp
- cp-algo/math/poly/base.hpp
- cp-algo/math/poly/calculus.hpp
- cp-algo/math/poly/div.hpp
- cp-algo/math/poly/impl/div.hpp
- cp-algo/math/poly/powmod.hpp
- cp-algo/math/poly/series.hpp
- cp-algo/math/poly/series/exp.hpp
- cp-algo/math/poly/series/inv.hpp
- cp-algo/math/poly/series/log.hpp
- cp-algo/math/poly/series/pow.hpp
- cp-algo/math/ring.hpp
- cp-algo/number_theory/discrete_sqrt.hpp
- cp-algo/number_theory/modint.hpp
- cp-algo/number_theory/primality.hpp
- cp-algo/random/rng.hpp
- cp-algo/util/big_alloc.hpp
- cp-algo/util/bump_alloc.hpp
- cp-algo/util/checkpoint.hpp
- cp-algo/util/complex.hpp
- cp-algo/util/simd.hpp
Code
// @brief Pow of Matrix (Frobenius)
#define PROBLEM "https://judge.yosupo.jp/problem/pow_of_matrix"
#include <bits/stdc++.h>
#pragma GCC optimize("O3,unroll-loops")
#pragma GCC target("avx2")
#define CP_ALGO_MAXN 256
#include "blazingio/blazingio.min.hpp"
#include "cp-algo/linalg/frobenius.hpp"
using namespace std;
using namespace cp_algo::math;
using namespace cp_algo::linalg;
const int64_t mod = 998244353;
using base = modint<mod>;
using polyn = poly_t<base>;
void solve() {
size_t n;
uint64_t k;
cin >> n >> k;
matrix<base> A(n);
A.read();
frobenius_pow(A, k).print();
}
signed main() {
//freopen("input.txt", "r", stdin);
ios::sync_with_stdio(0);
cin.tie(0);
int t = 1;
//cin >> t;
while(t--) {
solve();
}
}
#line 1 "verify/linalg/pow_fast.test.cpp"
// @brief Pow of Matrix (Frobenius)
#define PROBLEM "https://judge.yosupo.jp/problem/pow_of_matrix"
#include <bits/stdc++.h>
#pragma GCC optimize("O3,unroll-loops")
#pragma GCC target("avx2")
#define CP_ALGO_MAXN 256
#line 1 "blazingio/blazingio.min.hpp"
// NOLINTBEGIN
// clang-format off
// DO NOT REMOVE THIS MESSAGE. The mess that follows is a minified build of
// https://github.com/purplesyringa/blazingio. Refer to the repository for
// a human-readable version and documentation.
// Options: cbfoiedrhWLMXaIaAn
#define M$(x,...)_mm256_##x##_epi8(__VA_ARGS__)
#define $u(...)__VA_ARGS__
#if __APPLE__
#define $m(A,B)A
#else
#define $m(A,B)B
#endif
#if _WIN32
#define $w(A,B)A
#else
#define $w(A,B)B
#endif
#if __i386__|_M_IX86
#define $H(A,B)A
#else
#define $H(A,B)B
#endif
#if __aarch64__
#define $a(A,B)A
#else
#define $a(A,B)B
#endif
#define $P(x)void F(x K){
#define $T template<$c T
#define $c class
#define $C constexpr
#define $R return
#define $O operator
#define u$ uint64_t
#define $r $R*this;
#line 41 "blazingio/blazingio.min.hpp"
#include $a(<arm_neon.h>,<immintrin.h>)
#line 43 "blazingio/blazingio.min.hpp"
#include $w(<windows.h>,<sys/mman.h>)
#include<sys/stat.h>
#include $w(<io.h>,<unistd.h>)
#include $w(<ios>,<sys/resource.h>)
#if _MSC_VER
#define __builtin_add_overflow(a,b,c)_addcarry_u64(0,a,b,c)
#define $s
#else
$H(,u$ _umul128(u$ a,u$ b,u$*D){auto x=(__uint128_t)a*b;*D=u$(x>>64);$R(u$)x;})
#define $s $a(,__attribute__((target("avx2"))))
#endif
#define $z $a(16,32)
#define $t $a(uint8x16_t,__m256i)
#define $I $w(__forceinline,__attribute__((always_inline)))
#define $F M(),
#define E$(x)if(!(x))abort();
$w(LONG WINAPI $x(_EXCEPTION_POINTERS*);,)namespace $f{using namespace std;struct B{enum $c A:char{}c;B&$O=(char x){c=A{x};$r}$O char(){$R(char)c;}};$C u$ C=~0ULL/255;struct D{string&K;};static B E[65568];template<int F>struct G{B*H,*S;void K(off_t C){$w(char*D=(char*)VirtualAlloc(0,(C+8191)&-4096,8192,1);E$(D)E$(VirtualFree(D,0,32768))DWORD A=C&-65536;E$(!A||MapViewOfFileEx(CreateFileMapping(GetStdHandle(-10),0,2,0,A,0),4,0,0,0,D)==D)E$(VirtualAlloc(D+A,65536,12288,4)==D+A)E$(~_lseek(0,A,0))DWORD E=0;ReadFile(GetStdHandle(-10),D+A,65536,&E,0);,int A=getpagesize();char*D=(char*)mmap(0,C+A,3,2,0,0);E$(D!=(void*)-1)E$(mmap(D+((C+A-1)&-A),A,3,$m(4114,50),-1,0)!=(void*)-1))H=(B*)D+C;*H=10;H[1]=48;H[2]=0;S=(B*)D;}void L(){H=S=E;}$I void M(){if(F&&S==H){$w(DWORD A=0;ReadFile(GetStdHandle(-10),S=E,65536,&A,0);,$a($u(register long A asm("x0")=0,D asm("x1")=(long)E,G asm("x2")=65536,C asm($m("x16","x8"))=$m(3,63);asm volatile("svc 0" $m("x80",):"+r"(A),"+r"(D):"r"(C),"r"(G));S=launder(E);),off_t A=$H(3,$m(33554435,0));B*D=E;asm volatile($H("int $128","syscall"):"+a"(A),$H("+c"(D):"b","+S"(D):"D")(0),"d"(65536)$H(,$u(:"rcx","r11")));S=D;))H=S+A;*H=10;if(!A)E[1]=48,E[2]=0;}}$T>$I void N(T&x){while($F(*S&240)==48)x=T(x*10+(*S++-48));}$T>$I decltype((void)~T{1})O(T&x){M();int A=is_signed_v<T>&&*S==45;S+=A;N(x=0);x=A?1+~x:x;}$T>$I decltype((void)T{1.})O(T&x){M();int A=*S==45;S+=A;$F S+=*S==43;u$ n=0;int i=0;for(;i<18&&($F*S&240)==48;i++)n=n*10+*S++-48;int B=20;int C=*S==46;S+=C;for(;i<18&&($F*S&240)==48;i++)n=n*10+*S++-48,B-=C;x=(T)n;while(($F*S&240)==48)x=x*10+*S++-48,B-=C;if(*S==46)S++,C=1;while(($F*S&240)==48)x=x*10+*S++-48,B-=C;int D;if((*S|32)==101)S++,$F S+=*S==43,O(D),B+=D;static $C auto E=[](){array<T,41>E{};T x=1;for(int i=21;i--;)E[40-i]=x,E[i]=1/x,x*=10;$R E;}();while(B>40)x*=(T)1e10,B-=10;while(B<0)x*=(T)1e-10,B+=10;x*=E[B];x=A?-x:x;}$I void O(bool&x){$F x=*S++==49;}$I void O(char&x){$F x=*S++;}$I void O(uint8_t&x){$F x=*S++;}$I void O(int8_t&x){$F x=*S++;}$T>$s void P(string&K,T C){M();B*G=S;C();K.assign((char*)G,S-G);while(F&&S==H&&($F H!=E)){C();K.append(E,S);}}$s void O(string&K){P(K,[&]()$s{B*p=S;$w(ULONG R;,)$t x;$a(uint64x2_t A;while(memcpy(&x,p,16),A=uint64x2_t(x<33),!(A[0]|A[1]))p+=16;S=p+(A[0]?0:8)+$w((_BitScanForward64(&R,A[0]?A[0]:A[1]),R),__builtin_ctzll(A[0]?A[0]:A[1]))/8;,int J;$t C=M$(set1,32);while(memcpy(&x,p,32),!(J=M$(movemask,M$(cmpeq,C,_mm256_max_epu8(C,x)))))p+=32;S=p+$w((_BitScanForward(&R,J),R),__builtin_ctz(J));)});}$s void O(D&A){P(A.K,[&](){S=(B*)memchr(S,10,H-S+1);});if(A.K.size()&&A.K.back()==13)A.K.pop_back();if(A.K.empty()||S<H)S+=*S==10;}$T>$I void O(complex<T>&K){T A,B{};if($F*S==40){S++;O(A);if($F*S++==44)Q(B),S++;}else O(A);K={A,B};}template<size_t N>$s void O(bitset<N>&K){if(N>4095&&!*this)$R;ptrdiff_t i=N;while(i)if($F i%$z||H-S<$z)K[--i]=*S++==49;else{B*p=S;for(int64_t j=0;j<min(i,H-S)/$z;j++){i-=$z;$t x;memcpy(&x,p,$z);$a(auto B=(uint8x16_t)vdupq_n_u64(~2ULL/254)&(48-x);auto C=vzip_u8(vget_high_u8(B),vget_low_u8(B));auto y=vaddvq_u16((uint16x8_t)vcombine_u8(C.val[0],C.val[1]));,u$ a=~0ULL/65025;auto y=$w(_byteswap_ulong,__builtin_bswap32)(M$(movemask,M$(shuffle,_mm256_slli_epi32(x,7),_mm256_set_epi64x(a+C*24,a+C*16,a+C*8,a))));)p+=$z;memcpy((char*)&K+i/8,&y,$z/8);}S=p;}}$T>$I void Q(T&K){if(!is_same_v<T,D>)while($F(uint8_t)*S<33)S++;O(K);}$O bool(){$R!!*this;}bool $O!(){$R S>H;}};struct U{G<0>A;G<1>B;U(){struct stat D;E$(~fstat(0,&D))(D.st_mode>>12)==8?A.K(D.st_size):B.L();}U*tie(nullptr_t){$R this;}void sync_with_stdio(bool){}$T>$I U&$O>>(T&K){A.S?A.Q(K):B.Q(K);$r}$O bool(){$R!!*this;}bool $O!(){$R A.S?!A:!B;}};short A[100];char L[64]{1};struct
V{char*D;B*S;int J;V(){$w(E$(D=(char*)VirtualAlloc(0,536870912,8192,4))E$(VirtualAlloc(D,4096,4096,260))AddVectoredExceptionHandler(1,$x);,size_t C=536870912;$m(,rlimit E;getrlimit(RLIMIT_AS,&E);if(~E.rlim_cur)C=25165824;)D=(char*)mmap(0,C,3,$m(4162,16418),-1,0);E$(D!=(void*)-1))S=(B*)D;for(int i=0;i<100;i++)A[i]=short((48+i/10)|((48+i%10)<<8));for(int i=1;i<64;i++)L[i]=L[i-1]+(0x8922489224892249>>i&1);}~V(){flush($w(!J,));}void flush($w(int F=0,)){$w(J=1;auto E=GetStdHandle(-11);auto C=F?ReOpenFile(E,1073741824,7,2684354560):(void*)-1;DWORD A;E$(C==(void*)-1?WriteFile(E,D,DWORD((char*)S-D),&A,0):(WriteFile(C,D,DWORD(((char*)S-D+4095)&-4096),&A,0)&&~_chsize(1,int((char*)S-D)))),auto G=D;ssize_t A;while((A=write(1,G,(char*)S-G))>0)G+=A;E$(~A))S=(B*)D;}$P(char)*S++=K;}$P(uint8_t)*S++=K;}$P(int8_t)*S++=K;}$P(bool)*S++=48+K;}$T>decltype((void)~T{1})F(T K){using D=make_unsigned_t<T>;D C=K;if(K<0)F('-'),C=1+~C;static $C auto N=[](){array<D,5*sizeof(T)/2>N{};D n=1;for(size_t i=1;i<N.size();i++)n*=10,N[i]=n;$R N;}();$w(ULONG M;,)int G=L[$w(($H(_BitScanReverse(&M,ULONG((int64_t)C>>32))?M+=32:_BitScanReverse(&M,(ULONG)C|1),_BitScanReverse64(&M,C|1)),M),63^__builtin_clzll(C|1))];G-=C<N[G-1];short H[20];if $C(sizeof(T)==2){auto n=33555U*C-C/2;u$ H=A[n>>25];n=(n&33554431)*25;H|=A[n>>23]<<16;H|=u$(48+((n&8388607)*5>>22))<<32;H>>=40-G*8;memcpy(S,&H,8);}else if $C(sizeof(T)==4){auto n=1441151881ULL*C;$H(n>>=25;n++;for(int i=0;i<5;i++){H[i]=A[n>>32];n=(n&~0U)*100;},int K=57;auto J=~0ULL>>7;for(int i=0;i<5;i++){H[i]=A[n>>K];n=(n&J)*25;K-=2;J/=4;})memcpy(S,(B*)H+10-G,16);}else{$H($u(if(C<(1ULL<<32)){$R F((uint32_t)C);}auto J=(u$)1e10;auto x=C/J,y=C%J;int K=100000,b[]{int(x/K),int(x%K),int(y/K),int(y%K)};B H[40];for(int i=0;i<4;i++){int n=int((429497ULL*b[i]>>7)+1);B*p=H+i*5;*p=48+char(n>>25);n=(n&~0U>>7)*25;memcpy(p+1,A+(n>>23),2);memcpy(p+3,A+((n&~0U>>9)*25>>21),2);}),$u(u$ D,E=_umul128(18,C,&D),F;_umul128(0x725dd1d243aba0e8,C,&F);D+=__builtin_add_overflow(E,F+1,&E);for(int i=0;i<10;i++)H[i]=A[D],E=_umul128(100,E,&D);))memcpy(S,(B*)H+20-G,20);}S+=G;}$T>decltype((void)T{1.})F(T K){if(K<0)F('-'),K=-K;auto G=[&](){auto x=u$(K*1e12);$H($u(x-=x>999999999999;uint32_t n[]{uint32_t(x/1000000*429497>>7)+1,uint32_t(x%1000000*429497>>7)+1};int K=25,J=~0U>>7;for(int i=0;i<3;i++){for(int j=0;j<2;j++)memcpy(S+i*2+j*6,A+(n[j]>>K),2),n[j]=(n[j]&J)*25;K-=2;J/=4;}S+=12;),$u(u$ D,E=_umul128(472236648287,x,&D)>>8;E|=D<<56;D>>=8;E++;for(int i=0;i<6;i++)memcpy(S,A+D,2),S+=2,E=_umul128(100,E,&D);))};if(K==0)$R F('0');if(K>=1e16){K*=(T)1e-16;int B=16;while(K>=1)K*=(T).1,B++;F("0.");G();F('e');F(B);}else if(K>=1){auto B=(u$)K;F(B);if((K-=(T)B)>0)F('.'),G();}else F("0."),G();}$P(const char*)$w(size_t A=strlen(K);memcpy((char*)S,K,A);S+=A;,S=(B*)stpcpy((char*)S,K);)}$P(const uint8_t*)F((char*)K);}$P(const int8_t*)F((char*)K);}$P(string_view)memcpy(S,K.data(),K.size());S+=K.size();}$T>$P(complex<T>)*this<<'('<<K.real()<<','<<K.imag()<<')';}template<size_t N>$s $P(const bitset<N>&)auto i=N;while(i%$z)*S++=48+K[--i];B*p=S;while(i){i-=$z;$a(short,int)x;memcpy(&x,(char*)&K+i/8,$z/8);$a(auto A=(uint8x8_t)vdup_n_u16(x);vst1q_u8((uint8_t*)p,48-vtstq_u8(vcombine_u8(vuzp2_u8(A,A),vuzp1_u8(A,A)),(uint8x16_t)vdupq_n_u64(~2ULL/254)));,auto b=_mm256_set1_epi64x(~2ULL/254);_mm256_storeu_si256(($t*)p,M$(sub,M$(set1,48),M$(cmpeq,_mm256_and_si256(M$(shuffle,_mm256_set1_epi32(x),_mm256_set_epi64x(0,C,C*2,C*3)),b),b)));)p+=$z;}S=p;}$T>V&$O<<(const T&K){F(K);$r}V&$O<<(V&(*A)(V&)){$R A(*this);}};struct W{$T>W&$O<<(const T&K){$r}W&$O<<(W&(*A)(W&)){$R A(*this);}};}namespace std{$f::U i$;$f::V o$;$f::W e$;$f::U&getline($f::U&B,string&K){$f::D A{K};$R B>>A;}$f::V&flush($f::V&B){if(!i$.A.S)B.flush();$R B;}$f::V&endl($f::V&B){$R B<<'\n'<<flush;}$f::W&endl($f::W&B){$R B;}$f::W&flush($f::W&B){$R B;}}$w(LONG WINAPI $x(_EXCEPTION_POINTERS*A){auto C=A->ExceptionRecord;auto B=C->ExceptionInformation[1];if(C->ExceptionCode==2147483649&&B-(ULONG_PTR)std::o$.D<0x40000000){E$(VirtualAlloc((char*)B,16777216,4096,4)&&VirtualAlloc((char*)(B+16777216),4096,4096,260))$R-1;}$R 0;},)
#define freopen(...)if(freopen(__VA_ARGS__)==stdin)std::i$=$f::U{}
#define cin i$
#define cout o$
#ifdef ONLINE_JUDGE
#define cerr e$
#define clog e$
#endif
// End of blazingio
// NOLINTEND
// clang-format on
#line 1 "cp-algo/linalg/frobenius.hpp"
#line 1 "cp-algo/math/poly/powmod.hpp"
#line 1 "cp-algo/math/poly/div.hpp"
#line 1 "cp-algo/math/poly/impl/div.hpp"
#line 1 "cp-algo/math/poly/series/inv.hpp"
#line 1 "cp-algo/math/poly/base.hpp"
#line 1 "cp-algo/math/fft.hpp"
#line 1 "cp-algo/math/ring.hpp"
#line 1 "cp-algo/number_theory/discrete_sqrt.hpp"
#line 1 "cp-algo/number_theory/modint.hpp"
#line 1 "cp-algo/math/common.hpp"
#line 9 "cp-algo/math/common.hpp"
namespace cp_algo::math {
#ifdef CP_ALGO_MAXN
const int maxn = CP_ALGO_MAXN;
#else
const int maxn = 1 << 19;
#endif
const int magic = 64; // threshold for sizes to run the naive algo
// Nonnegative 64-bit exponents, with an associative operation and its identity.
// Windows >1 precompute odd powers only when that saves operations.
template<int window = 1>
auto bpow(auto const& x, auto n, auto const& one, auto op) {
static_assert(window >= 1 && window <= 6);
if constexpr(window > 1) {
if(n == 0) {return one;}
int bits = std::bit_width(uint64_t(n));
auto low_bit = [&](int high) {
int low = std::max(0, high - window + 1);
while(!((n >> low) & 1)) {low++;}
return low;
};
int first = low_bit(bits - 1);
int cost = (1 << (window - 1)) + first;
for(int j = first - 1; j >= 0;) {
if(!((n >> j) & 1)) {j--;}
else {cost++; j = low_bit(j) - 1;}
}
// Do not pay for the table when binary powering uses fewer operations.
if(cost >= bits + std::popcount(uint64_t(n)) - 2) {return bpow<1>(x, n, one, op);}
using T = std::decay_t<decltype(x)>;
std::vector<T> odd;
odd.reserve(1 << (window - 1));
odd.push_back(x);
auto square = op(x, x);
while(odd.size() < size_t(1 << (window - 1))) {odd.push_back(op(odd.back(), square));}
auto ans = odd[(n >> first) / 2];
for(int j = first - 1; j >= 0;) {
if(!((n >> j) & 1)) {ans = op(ans, ans); j--;}
else {
int low = low_bit(j), length = j - low + 1;
auto digit = (n >> low) & ((1u << length) - 1);
for(int i = 0; i < length; i++) {ans = op(ans, ans);}
ans = op(ans, odd[digit / 2]);
j = low - 1;
}
}
return ans;
} else {
if (n == 0) {
return one;
}
auto ans = x;
for(int j = std::bit_width<uint64_t>(n) - 2; ~j; j--) {
ans = op(ans, ans);
if((n >> j) & 1) {
ans = op(ans, x);
}
}
return ans;
}
}
template<int window = 1>
auto bpow(auto x, auto n, auto ans) {
return bpow<window>(x, n, ans, std::multiplies{});
}
template<typename T>
T bpow(T const& x, auto n) {
return bpow(x, n, T(1));
}
inline constexpr auto inv2(auto x) {
assert(x % 2);
std::make_unsigned_t<decltype(x)> y = 1;
while(y * x != 1) {
y *= 2 - x * y;
}
return y;
}
}
#line 6 "cp-algo/number_theory/modint.hpp"
namespace cp_algo::math {
template<typename modint, typename _Int>
struct modint_base {
using Int = _Int;
using UInt = std::make_unsigned_t<Int>;
static constexpr size_t bits = sizeof(Int) * 8;
using Int2 = std::conditional_t<bits <= 32, int64_t, __int128_t>;
using UInt2 = std::conditional_t<bits <= 32, uint64_t, __uint128_t>;
constexpr static Int mod() {
return modint::mod();
}
constexpr static UInt remod() {
return modint::remod();
}
constexpr static UInt2 modmod() {
return UInt2(mod()) * mod();
}
constexpr modint_base() = default;
constexpr modint_base(Int2 rr) {
to_modint().setr(UInt((rr + modmod()) % mod()));
}
constexpr modint inv() const {
return bpow(to_modint(), mod() - 2);
}
modint operator-() const {
modint neg;
neg.r = std::min(-r, remod() - r);
return neg;
}
modint& operator/=(const modint& t) {
return to_modint() *= t.inv();
}
modint& operator*=(const modint& t) {
r = UInt(UInt2(r) * t.r % mod());
return to_modint();
}
modint& operator+=(const modint& t) {
r += t.r;
r = std::min(r, r - remod());
return to_modint();
}
modint& operator-=(const modint& t) {
r -= t.r;
r = std::min(r, r + remod());
return to_modint();
}
modint operator+(const modint& t) const { return modint(to_modint()) += t; }
modint operator-(const modint& t) const { return modint(to_modint()) -= t; }
modint operator*(const modint& t) const { return modint(to_modint()) *= t; }
modint operator/(const modint& t) const { return modint(to_modint()) /= t; }
// Why <=> doesn't work?..
auto operator==(const modint& t) const { return to_modint().getr() == t.getr(); }
auto operator!=(const modint& t) const { return to_modint().getr() != t.getr(); }
auto operator<=(const modint& t) const { return to_modint().getr() <= t.getr(); }
auto operator>=(const modint& t) const { return to_modint().getr() >= t.getr(); }
auto operator<(const modint& t) const { return to_modint().getr() < t.getr(); }
auto operator>(const modint& t) const { return to_modint().getr() > t.getr(); }
Int rem() const {
UInt R = to_modint().getr();
return R - (R > (UInt)mod() / 2) * mod();
}
constexpr void setr(UInt rr) {
r = rr;
}
constexpr UInt getr() const {
return r;
}
// Only use these if you really know what you're doing!
static uint64_t modmod8() { return uint64_t(8 * modmod()); }
void add_unsafe(UInt t) { r += t; }
void pseudonormalize() { r = std::min(r, r - modmod8()); }
modint const& normalize() {
if(r >= (UInt)mod()) {
r %= mod();
}
return to_modint();
}
void setr_direct(UInt rr) { r = rr; }
UInt getr_direct() const { return r; }
protected:
UInt r;
private:
constexpr modint& to_modint() { return static_cast<modint&>(*this); }
constexpr modint const& to_modint() const { return static_cast<modint const&>(*this); }
};
template<typename modint>
concept modint_type = std::is_base_of_v<modint_base<modint, typename modint::Int>, modint>;
template<modint_type modint>
decltype(std::cin)& operator>>(decltype(std::cin)& in, modint& x) {
typename modint::UInt r;
auto &res = in >> r;
x.setr(r);
return res;
}
template<modint_type modint>
decltype(std::cout)& operator<<(decltype(std::cout)& out, modint const& x) {
return out << x.getr();
}
template<auto m>
struct modint: modint_base<modint<m>, decltype(m)> {
using Base = modint_base<modint<m>, decltype(m)>;
using Base::Base;
static constexpr Base::Int mod() { return m; }
static constexpr Base::UInt remod() { return m; }
auto getr() const { return Base::r; }
};
// Odd moduli up to a quarter of the unsigned word keep Montgomery residues lazily in
// [0, 2 mod): remod() = 2 mod, and both 4 mod and ab + q * mod fit. Any other modulus keeps
// fully reduced residues, remod() = mod, which is what the sums of modint_base need then:
// an even one without the Montgomery form, a wide odd one with a reduction that subtracts
// high words instead of adding double words that would overflow.
template<typename Int = int>
struct dynamic_modint: modint_base<dynamic_modint<Int>, Int> {
using Base = modint_base<dynamic_modint<Int>, Int>;
using Base::Base;
// Out of line, so that the hot path stays as small as it was.
[[gnu::noinline, gnu::cold]] static Base::UInt m_reduce_reduced(Base::UInt2 ab) {
if(mod() % 2 == 0) {
return typename Base::UInt(ab % mod());
}
// q * mod has the low word of ab, so the difference of the high words is exact.
typename Base::UInt q = -(typename Base::UInt(ab) * inverse);
auto high = typename Base::UInt(ab >> Base::bits);
auto low = typename Base::UInt(typename Base::UInt2(q) * typename Base::UInt(mod()) >> Base::bits);
return high >= low ? high - low : high - low + mod();
}
static Base::UInt m_reduce(Base::UInt2 ab) {
if(imod() == 0) [[unlikely]] {
return m_reduce_reduced(ab);
}
typename Base::UInt2 m = typename Base::UInt(ab) * imod();
return typename Base::UInt((ab + m * mod()) >> Base::bits);
}
static Base::UInt m_transform(Base::UInt a) {
if(mod() % 2 == 0) [[unlikely]] {
return a;
} else {
return m_reduce(a * pw128());
}
}
dynamic_modint& operator*=(const dynamic_modint& t) {
Base::r = m_reduce(typename Base::UInt2(Base::r) * t.r);
return *this;
}
void setr(Base::UInt rr) {
Base::r = m_transform(rr);
}
Base::UInt getr() const {
typename Base::UInt res = m_reduce(Base::r);
return std::min(res, res - mod());
}
static Int mod() { return m; }
static Base::UInt remod() { return rm; }
static Base::UInt imod() { return im; }
static Base::UInt2 pw128() { return r2; }
static void switch_mod(Int nm) {
m = nm;
bool lazy = m % 2 && typename Base::UInt(m) <= typename Base::UInt(-1) / 4;
rm = typename Base::UInt(m) * (lazy ? 2 : 1);
inverse = m % 2 ? inv2(-m) : 0;
im = lazy ? inverse : 0;
r2 = static_cast<Base::UInt>(static_cast<Base::UInt2>(-1) % m + 1);
}
// Wrapper for temp switching
auto static with_mod(Int tmp, auto callback) {
struct scoped {
Int prev = mod();
~scoped() { switch_mod(prev); }
} _;
switch_mod(tmp);
return callback();
}
private:
static inline thread_local Int m = 1;
// im: -1 / mod modulo 2^bits for lazy residues and 0 for reduced ones; inverse: the same
// for every odd mod; rm: the value of remod().
static inline thread_local Base::UInt im = -1, r2 = 0, inverse = -1, rm = 2;
};
}
#line 1 "cp-algo/random/rng.hpp"
#line 5 "cp-algo/random/rng.hpp"
namespace cp_algo::random {
std::mt19937_64 gen(
std::chrono::steady_clock::now().time_since_epoch().count()
);
uint64_t rng() {
return gen();
}
}
#line 1 "cp-algo/math/affine.hpp"
#line 7 "cp-algo/math/affine.hpp"
namespace cp_algo::math {
// a * x + b
template<typename base>
struct lin {
base a = 1, b = 0;
std::optional<base> c;
lin() {}
lin(base b): a(0), b(b) {}
lin(base a, base b): a(a), b(b) {}
lin(base a, base b, base _c): a(a), b(b), c(_c) {}
// polynomial product modulo x^2 - c
lin operator * (const lin& t) {
assert(c && t.c && *c == *t.c);
return {a * t.b + b * t.a, b * t.b + a * t.a * (*c), *c};
}
// a * (t.a * x + t.b) + b
lin apply(lin const& t) const {
return {a * t.a, a * t.b + b};
}
void prepend(lin const& t) {
*this = t.apply(*this);
}
base eval(base x) const {
return a * x + b;
}
};
// (ax+b) / (cx+d)
template<typename base>
struct linfrac {
base a, b, c, d;
linfrac(): a(1), b(0), c(0), d(1) {} // x, identity for composition
linfrac(base a): a(a), b(1), c(1), d(0) {} // a + 1/x, for continued fractions
linfrac(base a, base b, base c, base d): a(a), b(b), c(c), d(d) {}
// composition of two linfracs
linfrac operator * (linfrac t) const {
return t.prepend(linfrac(*this));
}
linfrac operator-() const {
return {-a, -b, -c, -d};
}
linfrac adj() const {
return {d, -b, -c, a};
}
linfrac& prepend(linfrac const& t) {
t.apply(a, c);
t.apply(b, d);
return *this;
}
// apply linfrac to A/B
void apply(base &A, base &B) const {
std::tie(A, B) = std::pair{a * A + b * B, c * A + d * B};
}
};
}
#line 6 "cp-algo/number_theory/discrete_sqrt.hpp"
namespace cp_algo::math {
// https://en.wikipedia.org/wiki/Berlekamp-Rabin_algorithm
template<modint_type base>
std::optional<base> sqrt(base b) {
if(b == base(0)) {
return base(0);
} else if(bpow(b, (b.mod() - 1) / 2) != base(1)) {
return std::nullopt;
} else {
while(true) {
base z = random::rng();
if(z * z == b) {
return z;
}
lin<base> x(1, z, b); // x + z (mod x^2 - b)
x = bpow(x, (b.mod() - 1) / 2, lin<base>(0, 1, b));
if(x.a != base(0)) {
return x.a.inv();
}
}
}
}
}
#line 1 "cp-algo/number_theory/primality.hpp"
#line 6 "cp-algo/number_theory/primality.hpp"
namespace cp_algo::math {
// https://en.wikipedia.org/wiki/Miller–Rabin_primality_test
template<typename _Int>
bool is_prime(_Int m) {
using Int = std::make_signed_t<_Int>;
using UInt = std::make_unsigned_t<Int>;
if(m == 1 || m % 2 == 0) {
return m == 2;
}
// m - 1 = 2^s * d
int s = std::countr_zero(UInt(m - 1));
auto d = (m - 1) >> s;
using base = dynamic_modint<Int>;
auto test = [&](base x) {
x = bpow(x, d);
if(std::abs(x.rem()) <= 1) {
return true;
}
for(int i = 1; i < s && x != -1; i++) {
x *= x;
}
return x == -1;
};
return base::with_mod(m, [&]() {
#ifdef CP_ALGO_NUMBER_THEORY_PRIMALITY_BASES_HPP
uint16_t base2 = 7, base3 = 61;
if (m != uint32_t(m)) {
base2 = base_table1[uint32_t(m * 0xAD625B89) >> 18];
base3 = base_table2[base2 >> 13];
}
return test(2) && test(base2) && test(base3);
#else
return std::ranges::all_of(std::array{2, 325, 9375, 28178, 450775, 9780504, 1795265022}, test);
#endif
});
}
}
#line 1 "cp-algo/util/checkpoint.hpp"
#line 1 "cp-algo/util/big_alloc.hpp"
#line 15 "cp-algo/util/big_alloc.hpp"
// Single macro to detect POSIX platforms (Linux, Unix, macOS)
#if defined(__linux__) || defined(__unix__) || (defined(__APPLE__) && defined(__MACH__))
#define CP_ALGO_USE_MMAP 1
#include <sys/mman.h>
#else
#define CP_ALGO_USE_MMAP 0
#endif
namespace cp_algo {
template<typename T, size_t Align = 32>
class big_alloc {
static_assert(Align >= alignof(void*), "Align must be at least pointer-size");
static_assert(std::has_single_bit(Align), "Align must be a power of two");
public:
using value_type = T;
template<class U>
struct rebind {
using other = big_alloc<U, Align>;
};
constexpr bool operator==(const big_alloc&) const = default;
big_alloc() noexcept = default;
template<typename U, std::size_t A>
big_alloc(const big_alloc<U, A>&) noexcept {}
[[nodiscard]] T* allocate(std::size_t n) {
std::size_t padded = round_up(n * sizeof(T));
std::size_t align = std::max<std::size_t>(alignof(T), Align);
#if CP_ALGO_USE_MMAP
if(padded >= MEGABYTE) {
void* raw = mmap(nullptr, padded, PROT_READ | PROT_WRITE, MAP_PRIVATE | MAP_ANONYMOUS, -1, 0);
madvise(raw, padded, MADV_HUGEPAGE);
return static_cast<T*>(raw);
}
#endif
return static_cast<T*>(::operator new(padded, std::align_val_t(align)));
}
void deallocate(T* p, std::size_t n) noexcept {
if(!p) {
return;
}
std::size_t padded = round_up(n * sizeof(T));
std::size_t align = std::max<std::size_t>(alignof(T), Align);
#if CP_ALGO_USE_MMAP
if(padded >= MEGABYTE) {
munmap(p, padded);
return;
}
#endif
::operator delete(p, padded, std::align_val_t(align));
}
private:
static constexpr std::size_t MEGABYTE = 1 << 20;
static constexpr std::size_t round_up(std::size_t x) noexcept {
return (x + Align - 1) / Align * Align;
}
};
template<typename T>
using big_vector = std::vector<T, big_alloc<T>>;
template<typename T>
using big_basic_string = std::basic_string<T, std::char_traits<T>, big_alloc<T>>;
template<typename T>
using big_deque = std::deque<T, big_alloc<T>>;
template<typename T>
using big_stack = std::stack<T, big_deque<T>>;
template<typename T>
using big_queue = std::queue<T, big_deque<T>>;
template<typename T>
using big_priority_queue = std::priority_queue<T, big_vector<T>>;
template<typename T>
using big_forward_list = std::forward_list<T, big_alloc<T>>;
using big_string = big_basic_string<char>;
template<typename Key, typename Value, typename Compare = std::less<Key>>
using big_map = std::map<Key, Value, Compare, big_alloc<std::pair<const Key, Value>>>;
template<typename T, typename Compare = std::less<T>>
using big_multiset = std::multiset<T, Compare, big_alloc<T>>;
template<typename T, typename Compare = std::less<T>>
using big_set = std::set<T, Compare, big_alloc<T>>;
}
#line 8 "cp-algo/util/checkpoint.hpp"
namespace cp_algo {
#ifdef CP_ALGO_CHECKPOINT
big_map<big_string, double> checkpoints;
double last;
#endif
template<bool final = false>
void checkpoint([[maybe_unused]] auto const& _msg) {
#ifdef CP_ALGO_CHECKPOINT
big_string msg = _msg;
double now = (double)clock() / CLOCKS_PER_SEC;
double delta = now - last;
last = now;
if(msg.size() && !final) {
checkpoints[msg] += delta;
}
if(final) {
for(auto const& [key, value] : checkpoints) {
std::cerr << key << ": " << value * 1000 << " ms\n";
}
std::cerr << "Total: " << now * 1000 << " ms\n";
}
#endif
}
template<bool final = false>
void checkpoint() {
checkpoint<final>("");
}
}
#line 1 "cp-algo/math/cvector.hpp"
#line 1 "cp-algo/util/simd.hpp"
#include <experimental/simd>
#line 7 "cp-algo/util/simd.hpp"
#if defined(__x86_64__) && !defined(CP_ALGO_DISABLE_AVX2)
#define CP_ALGO_SIMD_AVX2_TARGET _Pragma("GCC target(\"avx2,fma\")")
#else
#define CP_ALGO_SIMD_AVX2_TARGET
#endif
#define CP_ALGO_SIMD_PRAGMA_PUSH \
_Pragma("GCC push_options") \
CP_ALGO_SIMD_AVX2_TARGET
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo {
template<typename T, size_t len>
using simd [[gnu::vector_size(len * sizeof(T))]] = T;
using u64x8 = simd<uint64_t, 8>;
using u32x16 = simd<uint32_t, 16>;
using i64x4 = simd<int64_t, 4>;
using u64x4 = simd<uint64_t, 4>;
using u32x8 = simd<uint32_t, 8>;
using u16x16 = simd<uint16_t, 16>;
using i32x4 = simd<int32_t, 4>;
using u32x4 = simd<uint32_t, 4>;
using u16x8 = simd<uint16_t, 8>;
using u16x4 = simd<uint16_t, 4>;
using i16x4 = simd<int16_t, 4>;
using u8x32 = simd<uint8_t, 32>;
using u8x16 = simd<uint8_t, 16>;
using u8x8 = simd<uint8_t, 8>;
using u8x4 = simd<uint8_t, 4>;
using dx4 = simd<double, 4>;
inline dx4 abs(dx4 a) {
return dx4{
std::abs(a[0]),
std::abs(a[1]),
std::abs(a[2]),
std::abs(a[3])
};
}
// https://stackoverflow.com/a/77376595
// works for ints in (-2^51, 2^51)
static constexpr dx4 magic = dx4() + (3ULL << 51);
inline i64x4 lround(dx4 x) {
return i64x4(x + magic) - i64x4(magic);
}
inline dx4 to_double(i64x4 x) {
return dx4(x + i64x4(magic)) - magic;
}
inline dx4 round(dx4 a) {
return dx4{
std::nearbyint(a[0]),
std::nearbyint(a[1]),
std::nearbyint(a[2]),
std::nearbyint(a[3])
};
}
inline u64x4 low32(u64x4 x) {
return x & uint32_t(-1);
}
inline auto swap_bytes(auto x) {
return decltype(x)(__builtin_shufflevector(u32x8(x), u32x8(x), 1, 0, 3, 2, 5, 4, 7, 6));
}
inline u64x4 montgomery_reduce(u64x4 x, uint32_t mod, uint32_t imod) {
#ifdef __AVX2__
auto x_ninv = u64x4(_mm256_mul_epu32(__m256i(x), __m256i() + imod));
x += u64x4(_mm256_mul_epu32(__m256i(x_ninv), __m256i() + mod));
#else
auto x_ninv = u64x4(u32x8(low32(x)) * imod);
x += x_ninv * uint64_t(mod);
#endif
return swap_bytes(x);
}
// x - mod where x >= mod, for x in [0, 2 mod). Unsigned, so it holds up to mod < 2^31;
// zero upper halves of 64-bit lanes stay zero.
inline u32x8 reduce_once(u32x8 x, uint32_t mod) {
auto y = x - mod;
return x < y ? x : y;
}
inline u64x4 reduce_once(u64x4 x, uint32_t mod) {
return u64x4(reduce_once(u32x8(x), mod));
}
inline u64x4 montgomery_mul(u64x4 x, u64x4 y, uint32_t mod, uint32_t imod) {
#ifdef __AVX2__
return montgomery_reduce(u64x4(_mm256_mul_epu32(__m256i(x), __m256i(y))), mod, imod);
#else
return montgomery_reduce(x * y, mod, imod);
#endif
}
inline u32x8 montgomery_mul(u32x8 x, u32x8 y, uint32_t mod, uint32_t imod) {
return u32x8(montgomery_mul(u64x4(x), u64x4(y), mod, imod)) |
u32x8(swap_bytes(montgomery_mul(u64x4(swap_bytes(x)), u64x4(swap_bytes(y)), mod, imod)));
}
inline dx4 rotate_right(dx4 x) {
static constexpr u64x4 shuffler = {3, 0, 1, 2};
return __builtin_shuffle(x, shuffler);
}
template<std::size_t Align = 32>
inline bool is_aligned(const auto *p) noexcept {
return (reinterpret_cast<std::uintptr_t>(p) % Align) == 0;
}
template<class Target>
inline Target& vector_cast(auto &&p) {
return *reinterpret_cast<Target*>(std::assume_aligned<alignof(Target)>(&p));
}
}
#pragma GCC pop_options
#line 1 "cp-algo/util/complex.hpp"
#line 7 "cp-algo/util/complex.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo {
// Custom implementation, since std::complex is UB on non-floating types
template<typename T>
struct complex {
using value_type = T;
T x, y;
inline constexpr complex(): x(), y() {}
inline constexpr complex(T const& x): x(x), y() {}
inline constexpr complex(T const& x, T const& y): x(x), y(y) {}
inline complex& operator *= (T const& t) {x *= t; y *= t; return *this;}
inline complex& operator /= (T const& t) {x /= t; y /= t; return *this;}
inline complex operator * (T const& t) const {return complex(*this) *= t;}
inline complex operator / (T const& t) const {return complex(*this) /= t;}
inline complex& operator += (complex const& t) {x += t.x; y += t.y; return *this;}
inline complex& operator -= (complex const& t) {x -= t.x; y -= t.y; return *this;}
inline complex operator * (complex const& t) const {return {x * t.x - y * t.y, x * t.y + y * t.x};}
inline complex operator / (complex const& t) const {return *this * t.conj() / t.norm();}
inline complex operator + (complex const& t) const {return complex(*this) += t;}
inline complex operator - (complex const& t) const {return complex(*this) -= t;}
inline complex& operator *= (complex const& t) {return *this = *this * t;}
inline complex& operator /= (complex const& t) {return *this = *this / t;}
inline complex operator - () const {return {-x, -y};}
inline complex conj() const {return {x, -y};}
inline T norm() const {return x * x + y * y;}
inline T abs() const {return std::sqrt(norm());}
inline T const real() const {return x;}
inline T const imag() const {return y;}
inline T& real() {return x;}
inline T& imag() {return y;}
inline static constexpr complex polar(T r, T theta) {return {T(r * cos(theta)), T(r * sin(theta))};}
inline auto operator <=> (complex const& t) const = default;
};
template<typename T> inline complex<T> conj(complex<T> const& x) {return x.conj();}
template<typename T> inline T norm(complex<T> const& x) {return x.norm();}
template<typename T> inline T abs(complex<T> const& x) {return x.abs();}
template<typename T> inline T& real(complex<T> &x) {return x.real();}
template<typename T> inline T& imag(complex<T> &x) {return x.imag();}
template<typename T> inline T const real(complex<T> const& x) {return x.real();}
template<typename T> inline T const imag(complex<T> const& x) {return x.imag();}
template<typename T>
inline constexpr complex<T> polar(T r, T theta) {
return complex<T>::polar(r, theta);
}
template<typename T>
inline std::ostream& operator << (std::ostream &out, complex<T> const& x) {
return out << x.real() << ' ' << x.imag();
}
}
#pragma GCC pop_options
#line 7 "cp-algo/math/cvector.hpp"
#include <immintrin.h>
#line 13 "cp-algo/math/cvector.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace stdx = std::experimental;
namespace cp_algo::math::fft {
static constexpr size_t flen = 4;
using ftype = double;
using vftype = dx4;
using point = complex<ftype>;
using vpoint = complex<vftype>;
static constexpr vftype vz = {};
vpoint vi(vpoint const& r) {
return {-imag(r), real(r)};
}
// Spectrum storage skips zero-filling on resize; every user writes before reading.
template<class T>
struct spectrum_alloc: big_alloc<T> {
using big_alloc<T>::big_alloc;
template<class U>
struct rebind {
using other = spectrum_alloc<U>;
};
template<class U>
requires(std::is_same_v<U, vpoint>)
void construct(U*) noexcept {}
};
using spectrum_vector = std::vector<vpoint, spectrum_alloc<vpoint>>;
struct cvector {
spectrum_vector r;
cvector(size_t n) {
n = std::max(flen, std::bit_ceil(n));
r.assign(n / flen, vpoint{});
prepare_roots(n / 16);
checkpoint("cvector create");
}
vpoint& at(size_t k) { return r[k / flen]; }
vpoint at(size_t k) const { return r[k / flen]; }
template<class pt = point>
inline void set(size_t k, pt const& t) {
if constexpr(std::is_same_v<pt, point>) {
real(r[k / flen])[k % flen] = real(t);
imag(r[k / flen])[k % flen] = imag(t);
} else {
at(k) = t;
}
}
template<class pt = point>
inline pt get(size_t k) const {
if constexpr(std::is_same_v<pt, point>) {
return {real(r[k / flen])[k % flen], imag(r[k / flen])[k % flen]};
} else {
return at(k);
}
}
size_t size() const {
return flen * r.size();
}
static constexpr size_t eval_arg(size_t n) {
if(n < pre_evals) {
return eval_args[n];
} else {
return eval_arg(n / 2) | (n & 1) << (std::bit_width(n) - 1);
}
}
static constexpr point eval_point(size_t n) {
if(n % 2) {
return -eval_point(n - 1);
} else if(n % 4) {
return eval_point(n - 2) * point(0, 1);
} else if(n / 4 < pre_evals) {
return evalp[n / 4];
} else if(n / 4 - pre_evals < extra.size()) {
return extra[n / 4 - pre_evals];
} else {
return polar<ftype>(1., std::numbers::pi / (ftype)std::bit_floor(n) * (ftype)eval_arg(n));
}
}
static constexpr std::array<point, 32> roots = []() {
std::array<point, 32> res;
for(size_t i = 2; i < 32; i++) {
res[i] = polar<ftype>(1., std::numbers::pi / (1ull << (i - 2)));
}
return res;
}();
static constexpr point root(size_t n) {
return roots[std::bit_width(n)];
}
template<int step>
static void exec_on_eval(size_t n, size_t k, auto &&callback) {
callback(k, root(4 * step * n) * eval_point(step * k));
}
template<int step>
static void exec_on_evals(size_t n, auto &&callback) {
point factor = root(4 * step * n);
if constexpr(step == 1 || step == 2 || step == 4) {
prepare_roots((step * n + 3) / 4);
size_t i = 0;
if constexpr(step == 1) {
for(; i + 4 <= n; i += 4) {
size_t k = i / 4;
point e = k < pre_evals ? evalp[k] : extra[k - pre_evals];
point v = factor * e;
callback(i, v);
callback(i + 1, -v);
point iv(-imag(v), real(v));
callback(i + 2, iv);
callback(i + 3, -iv);
}
} else if constexpr(step == 2) {
for(; i + 2 <= n; i += 2) {
size_t k = i / 2;
point e = k < pre_evals ? evalp[k] : extra[k - pre_evals];
point v = factor * e;
callback(i, v);
callback(i + 1, point(-imag(v), real(v)));
}
}
for(; i < n; i++) {
size_t index = step * i, k = index / 4;
point e = k < pre_evals ? evalp[k] : extra[k - pre_evals];
if(index & 2) {
e = e * point(0, 1);
}
if(index & 1) {
e = -e;
}
callback(i, factor * e);
}
} else {
for(size_t i = 0; i < n; i++) {
callback(i, factor * eval_point(step * i));
}
}
}
static void do_dot_iter(point rt, vpoint& Bv, vpoint const& Av, vpoint& res) {
res += Av * Bv;
real(Bv) = rotate_right(real(Bv));
imag(Bv) = rotate_right(imag(Bv));
auto x = real(Bv)[0], y = imag(Bv)[0];
real(Bv)[0] = x * real(rt) - y * imag(rt);
imag(Bv)[0] = x * imag(rt) + y * real(rt);
}
template<size_t fixed = 0>
void dot(cvector const& t) {
size_t n = fixed ? fixed : this->size();
exec_on_evals<1>(n / flen, [&] [[gnu::always_inline]] (size_t k, point rt) {
k *= flen;
auto [Ax, Ay] = at(k);
auto Cv = t.at(k);
vpoint vrt = {vz + real(rt), vz + imag(rt)};
auto Cr = Cv * vrt;
vpoint res = vz;
auto iter = [&]<int i> [[gnu::always_inline]] () {
auto wrap = [&](vftype original, vftype rotated) {
if constexpr(i == 0) {
return original;
} else {
return __builtin_shufflevector(rotated, original, 4 - i, 5 - i, 6 - i, 7 - i);
}
};
vpoint Cw = {wrap(real(Cv), real(Cr)), wrap(imag(Cv), imag(Cr))};
vpoint Av = {vz + Ax[i], vz + Ay[i]};
return Av * Cw;
};
auto p0 = iter.template operator()<0>(), p1 = iter.template operator()<1>();
auto p2 = iter.template operator()<2>(), p3 = iter.template operator()<3>();
res = (p0 + p1) + (p2 + p3);
set(k, res);
});
checkpoint("dot");
}
// normalize=false leaves the inverse-transform scale for the caller.
template<bool partial = true, bool normalize = true, size_t fixed = 0>
void ifft() {
size_t n = fixed ? fixed : size();
if constexpr(!partial) {
prepare_roots(n / 4);
point pi(0, 1);
exec_on_evals<4>(n / 4, [&] [[gnu::always_inline]] (size_t k, point rt) {
k *= 4;
point v1 = conj(rt);
point v2 = v1 * v1;
point v3 = v1 * v2;
auto A = get(k);
auto B = get(k + 1);
auto C = get(k + 2);
auto D = get(k + 3);
set(k, (A + B) + (C + D));
set(k + 2, ((A + B) - (C + D)) * v2);
set(k + 1, ((A - B) - pi * (C - D)) * v1);
set(k + 3, ((A - B) + pi * (C - D)) * v3);
});
}
bool parity = std::countr_zero(n) % 2;
if(parity) {
exec_on_evals<2>(n / (2 * flen), [&] [[gnu::always_inline]] (size_t k, point rt) {
k *= 2 * flen;
vpoint cvrt = {vz + real(rt), vz - imag(rt)};
auto B = at(k) - at(k + flen);
at(k) += at(k + flen);
at(k + flen) = B * cvrt;
});
}
transform<true, fixed>(n, parity);
checkpoint("ifft");
if constexpr(normalize) {
auto scale = vz + ftype(partial ? flen : 1) / ftype(n);
for(size_t k = 0; k < n; k += flen) {
set(k, get<vpoint>(k) * scale);
}
}
}
template<bool partial = true, size_t fixed = 0>
void fft() {
size_t n = fixed ? fixed : size();
prepare_roots(n / (partial ? 16 : 4));
bool parity = std::countr_zero(n) % 2;
transform<false, fixed>(n, parity);
if(parity) {
exec_on_evals<2>(n / (2 * flen), [&] [[gnu::always_inline]] (size_t k, point rt) {
k *= 2 * flen;
vpoint vrt = {vz + real(rt), vz + imag(rt)};
auto t = at(k + flen) * vrt;
at(k + flen) = at(k) - t;
at(k) += t;
});
}
if constexpr(!partial) {
prepare_roots(n / 4);
point pi(0, 1);
exec_on_evals<4>(n / 4, [&] [[gnu::always_inline]] (size_t k, point rt) {
k *= 4;
point v1 = rt;
point v2 = v1 * v1;
point v3 = v1 * v2;
auto A = get(k);
auto B = get(k + 1) * v1;
auto C = get(k + 2) * v2;
auto D = get(k + 3) * v3;
set(k, (A + C) + (B + D));
set(k + 1, (A + C) - (B + D));
set(k + 2, (A - C) + pi * (B - D));
set(k + 3, (A - C) - pi * (B - D));
});
}
checkpoint("fft");
}
static std::array<vpoint, 4> transpose(std::array<vpoint, 4> const& a) {
auto half = [](auto part) {
auto a0 = __m256d(part(0)), a1 = __m256d(part(1)), a2 = __m256d(part(2)), a3 = __m256d(part(3));
auto t0 = _mm256_unpacklo_pd(a0, a1), t1 = _mm256_unpackhi_pd(a0, a1), t2 = _mm256_unpacklo_pd(a2, a3),
t3 = _mm256_unpackhi_pd(a2, a3);
return std::array<vftype, 4>{
vftype(_mm256_permute2f128_pd(t0, t2, 0x20)), vftype(_mm256_permute2f128_pd(t1, t3, 0x20)),
vftype(_mm256_permute2f128_pd(t0, t2, 0x31)), vftype(_mm256_permute2f128_pd(t1, t3, 0x31))};
};
auto re = half([&](int k) { return real(a[k]); }), im = half([&](int k) { return imag(a[k]); });
return {vpoint{re[0], im[0]}, vpoint{re[1], im[1]}, vpoint{re[2], im[2]}, vpoint{re[3], im[3]}};
}
// Multiply groups of 16 points as polynomials modulo x^16 - rt: one radix-4 level on
// each operand, a lane-parallel 4-coefficient Karatsuba product, one inverse level.
// The weights rt, rt^2, rt^3 of a tile of groups are computed four groups per SIMD
// operation ahead of the tile, so the product loop only broadcasts them from memory.
static constexpr size_t dot_tile = 64;
template<size_t fixed, bool square = false>
void dot_fused16(cvector const& t, size_t offset, size_t length) {
constexpr size_t T = dot_tile;
const size_t n = fixed ? fixed : size();
point factor = root(n);
auto fr = _mm256_set1_pd(real(factor)), fi = _mm256_set1_pd(imag(factor));
alignas(32) double tw[6][T];
const bool ahead = offset + length < n;
auto cmadd = [] [[gnu::always_inline]] (vpoint x, vpoint y, vpoint c) {
auto re = _mm256_fmadd_pd(__m256d(real(x)), __m256d(real(y)),
_mm256_fnmadd_pd(__m256d(imag(x)), __m256d(imag(y)), __m256d(real(c))));
auto im = _mm256_fmadd_pd(__m256d(real(x)), __m256d(imag(y)),
_mm256_fmadd_pd(__m256d(imag(x)), __m256d(real(y)), __m256d(imag(c))));
return vpoint{vftype(re), vftype(im)};
};
auto cmcross = [] [[gnu::always_inline]] (vpoint x, vpoint y, vpoint p, vpoint q) {
auto re = _mm256_sub_pd(_mm256_fmsub_pd(__m256d(real(x)), __m256d(real(y)), __m256d(real(p))),
_mm256_fmadd_pd(__m256d(imag(x)), __m256d(imag(y)), __m256d(real(q))));
auto im = _mm256_add_pd(_mm256_fmsub_pd(__m256d(real(x)), __m256d(imag(y)), __m256d(imag(p))),
_mm256_fmsub_pd(__m256d(imag(x)), __m256d(real(y)), __m256d(imag(q))));
return vpoint{vftype(re), vftype(im)};
};
// z * conj(v)
auto mulconj = [] [[gnu::always_inline]] (vpoint z, vpoint v) {
auto re = _mm256_fmadd_pd(__m256d(real(z)), __m256d(real(v)),
_mm256_mul_pd(__m256d(imag(z)), __m256d(imag(v))));
auto im = _mm256_fmsub_pd(__m256d(imag(z)), __m256d(real(v)),
_mm256_mul_pd(__m256d(real(z)), __m256d(imag(v))));
return vpoint{vftype(re), vftype(im)};
};
for(size_t tile = offset; tile < offset + length; tile += 16 * T) {
size_t k0 = tile / 16;
auto const* e = reinterpret_cast<double const*>(k0 < pre_evals ? evalp.data() + k0
: extra.data() + (k0 - pre_evals));
for(size_t g = 0; g < T; g += 4) {
auto x = _mm256_loadu_pd(e + 2 * g), y = _mm256_loadu_pd(e + 2 * g + 4);
auto er = _mm256_permute4x64_pd(_mm256_unpacklo_pd(x, y), 0xD8),
ei = _mm256_permute4x64_pd(_mm256_unpackhi_pd(x, y), 0xD8);
auto r1 = _mm256_fmsub_pd(fr, er, _mm256_mul_pd(fi, ei)),
i1 = _mm256_fmadd_pd(fr, ei, _mm256_mul_pd(fi, er));
auto r2 = _mm256_fmsub_pd(r1, r1, _mm256_mul_pd(i1, i1)),
i2 = _mm256_fmadd_pd(r1, i1, _mm256_mul_pd(i1, r1));
auto r3 = _mm256_fmsub_pd(r1, r2, _mm256_mul_pd(i1, i2)),
i3 = _mm256_fmadd_pd(r1, i2, _mm256_mul_pd(i1, r2));
_mm256_store_pd(tw[0] + g, r1);
_mm256_store_pd(tw[1] + g, i1);
_mm256_store_pd(tw[2] + g, r2);
_mm256_store_pd(tw[3] + g, i2);
_mm256_store_pd(tw[4] + g, r3);
_mm256_store_pd(tw[5] + g, i3);
}
for(size_t g = 0; g < T; g++) {
size_t pos = tile + 16 * g;
if(ahead) {
// Pull the next block of both operands towards the cache while this one is multiplied.
auto const* pa = reinterpret_cast<char const*>(r.data() + (pos + length) / flen);
auto const* pb = reinterpret_cast<char const*>(t.r.data() + (pos + length) / flen);
for(size_t c = 0; c < 4; c++) {
_mm_prefetch(pa + 64 * c, _MM_HINT_T2);
_mm_prefetch(pb + 64 * c, _MM_HINT_T2);
}
}
auto bc = [&] [[gnu::always_inline]] (size_t c) {
return vftype(_mm256_broadcast_sd(tw[c] + g));
};
vpoint v1 = {bc(0), bc(1)}, v2 = {bc(2), bc(3)}, v3 = {bc(4), bc(5)};
auto forward = [&] [[gnu::always_inline]] (cvector const& a) {
auto A = a.at(pos), B = a.at(pos + 4) * v1, C = a.at(pos + 8) * v2, D = a.at(pos + 12) * v3;
return std::array<vpoint, 4>{(A + C) + (B + D), (A + C) - (B + D), (A - C) + vi(B - D),
(A - C) - vi(B - D)};
};
auto a = transpose(forward(*this));
auto b = [&] {
if constexpr(square) {
return a;
} else {
return transpose(forward(t));
}
}();
// The four residues are taken modulo x^4 - rt * {1, -1, i, -i}.
const auto flip_re = _mm256_set_pd(0., -0., -0., 0.), flip_im = _mm256_set_pd(-0., 0., -0., 0.);
vpoint w = {
vftype(_mm256_xor_pd(_mm256_blend_pd(__m256d(real(v1)), __m256d(imag(v1)), 0b1100), flip_re)),
vftype(_mm256_xor_pd(_mm256_blend_pd(__m256d(imag(v1)), __m256d(real(v1)), 0b1100), flip_im))};
auto mul = [&] [[gnu::always_inline]] (vpoint a0, vpoint a1, vpoint b0, vpoint b1) {
auto p = a0 * b0, q = a1 * b1;
return std::array<vpoint, 2>{cmadd(w, q, p), cmcross(a0 + a1, b0 + b1, p, q)};
};
auto p = mul(a[0], a[2], b[0], b[2]), q = mul(a[1], a[3], b[1], b[3]);
auto m = mul(a[0] + a[1], a[2] + a[3], b[0] + b[1], b[2] + b[3]);
auto c = transpose({cmadd(w, q[1], p[0]), (m[0] - p[0]) - q[0], p[1] + q[0], (m[1] - p[1]) - q[1]});
auto A = c[0], B = c[1], C = c[2], D = c[3];
at(pos) = (A + B) + (C + D);
at(pos + 8) = mulconj((A + B) - (C + D), v2);
at(pos + 4) = mulconj((A - B) - vi(C - D), v1);
at(pos + 12) = mulconj((A - B) + vi(C - D), v3);
}
}
}
// A product in two steps, this <- this * t modulo x^n - i up to the factor n / flen:
// forward() on both operands, then multiply(t), which may be *this for a square.
// The forward transform stops at groups of 16 points and dot_fused16 multiplies those,
// which saves a pass over each operand and one over the result. That needs powers of
// four, so a length 2 * 4^k takes its radix-two level at the top, modulo
// x^(n/2) -+ sqrt(i); short lengths take the complete transform and the 4-point product.
void forward() {
if(!fused_leaves()) {
return fft();
}
size_t n = size(), part = fused_part();
if(part < n) {
vpoint vrt = {vz + real(roots[4]), vz + imag(roots[4])};
for(size_t k = 0; k < part; k += flen) {
auto t = at(k + part) * vrt;
at(k + part) = at(k) - t;
at(k) += t;
}
}
for(size_t offset = 0; offset < n; offset += part) {
transform<false, 0, 0, true>(n, false, offset, part);
}
checkpoint("fft");
}
void multiply(cvector const& t) {
if(!fused_leaves()) {
dot(t);
return ifft<true, false>();
}
size_t n = size(), part = fused_part();
dot_fused16<0>(t, 0, n);
checkpoint("dot");
for(size_t offset = 0; offset < n; offset += part) {
transform<true, 0, 0, true>(n, false, offset, part);
}
if(part < n) {
vpoint cvrt = {vz + real(roots[4]), vz - imag(roots[4])};
for(size_t k = 0; k < part; k += flen) {
auto t = at(k) - at(k + part);
at(k) += at(k + part);
at(k + part) = t * cvrt;
}
}
checkpoint("ifft");
}
// Radix-64 out-of-cache pass for n = 2^24, run as two tiled radix-8 stages.
// With Fuse=1 (forward only) the first stage lifts u32 residues to Gaussian
// coordinates on load, so the spectrum buffer is written once and never re-read
// before the in-cache block phase. The index-keyed noise makes both branches see
// the same representatives.
struct fuse_args {
const uint32_t* src = nullptr;
size_t count = 0;
uint64_t seed = 0;
double a = 0, b = 0, a_over_p = 0, b_over_p = 0;
};
static constexpr size_t sweep_tile = 256;
template<bool inverse, int Mode, size_t Tile, int Fuse = 0, bool Neg = false>
void sweep8(fuse_args const& fa = {}) {
constexpr size_t n = 1 << 24;
static const std::array<std::array<point, 7>, 9> weights = []() {
std::array<std::array<point, 7>, 9> table;
for(size_t count: {size_t(1), size_t(8)}) {
for(size_t k = 0; k < count; k++) {
size_t rev = 0, x = k;
for(size_t c = count; c > 1; c >>= 1) {
rev = (rev << 1) | (x & 1);
x >>= 1;
}
long double angle = std::numbers::pi_v<long double> * (1 + 4 * rev) / (16 * count);
if(k & 1) {
angle -= std::numbers::pi_v<long double> / 4;
}
for(size_t j = 1; j < 8; j++) {
long double a = j * angle;
if constexpr(Mode == 0) {
table[(count - 1) / 7 + k][j - 1] = {double(cosl(a)), double(sinl(a))};
} else {
table[(count - 1) / 7 + k][j - 1] = {double(sinl(a)), double(tanl(a / 2))};
}
}
}
}
return table;
}();
auto lift = [&] [[gnu::always_inline]] (size_t idx) -> vpoint {
i32x4 bits{};
if(idx + 4 <= fa.count) {
std::memcpy(&bits, fa.src + idx, sizeof(bits));
} else if(idx < fa.count) {
for(size_t j = 0; j < fa.count - idx; j++) {
bits[j] = int32_t(fa.src[idx + j]);
}
} else {
return vpoint{vz, vz};
}
auto x = __builtin_convertvector(bits, vftype);
u32x4 h =
u32x4{uint32_t(idx), uint32_t(idx + 1), uint32_t(idx + 2), uint32_t(idx + 3)} ^ uint32_t(fa.seed);
h *= 0x9E3779B1u;
h ^= h >> 15;
h *= 0x85EBCA77u;
h ^= h >> 13;
h *= 0xC2B2AE3Du;
h ^= h >> 16;
auto noise = __builtin_convertvector(i32x4(h), vftype) * 0x1p-32;
auto q = round(x * fa.a_over_p + noise), t = round(x * fa.b_over_p + noise);
auto re = x - q * fa.a - t * fa.b, im = t * fa.a - q * fa.b;
return vpoint{re, Neg ? -im : im};
};
auto stage = [&]<bool top> [[gnu::always_inline]] (size_t offset, size_t length, size_t begin, size_t end) {
size_t step = length / 8, k = offset / length;
auto const& w = weights[(n / length - 1) / 7 + k];
auto rot = [&]<size_t J> [[gnu::always_inline]] (vpoint z) {
auto c = w[J - 1];
if constexpr(Mode == 0) {
return z * vpoint{vz + real(c), inverse ? vz - imag(c) : vz + imag(c)};
} else {
auto s = inverse ? vz - real(c) : vz + real(c), t = inverse ? vz - imag(c) : vz + imag(c);
auto x = vftype(_mm256_fnmadd_pd(__m256d(t), __m256d(imag(z)), __m256d(real(z))));
auto y = vftype(_mm256_fmadd_pd(__m256d(s), __m256d(x), __m256d(imag(z))));
return vpoint{vftype(_mm256_fnmadd_pd(__m256d(t), __m256d(y), __m256d(x))), y};
}
};
auto add = [] [[gnu::always_inline]] (vpoint a, vpoint b) {
if constexpr(Mode != 2) {
return a + b;
} else {
return vpoint{vftype(_mm256_fmadd_pd(__m256d(real(a)), _mm256_set1_pd(1), __m256d(real(b)))),
vftype(_mm256_fmadd_pd(__m256d(imag(a)), _mm256_set1_pd(1), __m256d(imag(b))))};
}
};
auto sub = [] [[gnu::always_inline]] (vpoint a, vpoint b) {
if constexpr(Mode != 2) {
return a - b;
} else {
return vpoint{vftype(_mm256_fmsub_pd(__m256d(real(a)), _mm256_set1_pd(1), __m256d(real(b)))),
vftype(_mm256_fmsub_pd(__m256d(imag(a)), _mm256_set1_pd(1), __m256d(imag(b))))};
}
};
auto d4 = [] [[gnu::always_inline]] (vpoint a, vpoint b, vpoint c, vpoint d) {
auto s = a + c, t = a - c, u = b + d, v = vi(b - d);
if constexpr(inverse) {
return std::array<vpoint, 4>{s + u, t - v, s - u, t + v};
} else {
return std::array<vpoint, 4>{s + u, t + v, s - u, t - v};
}
};
constexpr double q = 0.707106781186547524400844362104849039;
auto r1 = [] [[gnu::always_inline]] (vpoint z) {
if constexpr(inverse) {
return vpoint{(real(z) + imag(z)) * q, (imag(z) - real(z)) * q};
} else {
return vpoint{(real(z) - imag(z)) * q, (real(z) + imag(z)) * q};
}
};
auto r3 = [] [[gnu::always_inline]] (vpoint z) {
if constexpr(inverse) {
return vpoint{(imag(z) - real(z)) * q, (-imag(z) - real(z)) * q};
} else {
return vpoint{(-real(z) - imag(z)) * q, (real(z) - imag(z)) * q};
}
};
std::array<vpoint*, 8> input, output;
for(size_t j = 0; j < 8; j++) {
input[j] = output[j] = r.data() + (offset + begin + j * step) / flen;
}
if(k & 1) {
constexpr std::array<size_t, 8> perm = {7, 6, 4, 5, 0, 1, 2, 3};
for(size_t j = 0; j < 8; j++) {
if constexpr(inverse) {
input[j] = output[perm[j]];
} else {
output[j] = input[perm[j]];
}
}
}
constexpr bool fused_in = top && Fuse == 1 && !inverse;
for(size_t j = 0; j < (end - begin) / flen; j++) {
size_t base = offset + begin + j * flen;
auto in = [&]<size_t S> [[gnu::always_inline]] () {
if constexpr(fused_in) {
return lift(base + S * step);
} else {
return input[S][j];
}
};
if constexpr(!inverse) {
auto E =
d4(in.template operator()<0>(), rot.template operator()<2>(in.template operator()<2>()),
rot.template operator()<4>(in.template operator()<4>()),
rot.template operator()<6>(in.template operator()<6>()));
auto O = d4(rot.template operator()<1>(in.template operator()<1>()),
rot.template operator()<3>(in.template operator()<3>()),
rot.template operator()<5>(in.template operator()<5>()),
rot.template operator()<7>(in.template operator()<7>()));
auto a = O[0], b = r1(O[1]), c = vi(O[2]), d = r3(O[3]);
output[0][j] = add(E[0], a);
output[1][j] = sub(E[0], a);
output[2][j] = add(E[2], c);
output[3][j] = sub(E[2], c);
output[4][j] = add(E[1], b);
output[5][j] = sub(E[1], b);
output[6][j] = add(E[3], d);
output[7][j] = sub(E[3], d);
} else {
auto E = d4(add(input[0][j], input[1][j]), add(input[4][j], input[5][j]),
add(input[2][j], input[3][j]), add(input[6][j], input[7][j]));
auto O = d4(sub(input[0][j], input[1][j]), r1(sub(input[4][j], input[5][j])),
-vi(sub(input[2][j], input[3][j])), r3(sub(input[6][j], input[7][j])));
output[0][j] = E[0];
output[1][j] = rot.template operator()<1>(O[0]);
output[2][j] = rot.template operator()<2>(E[1]);
output[3][j] = rot.template operator()<3>(O[1]);
output[4][j] = rot.template operator()<4>(E[2]);
output[5][j] = rot.template operator()<5>(O[2]);
output[6][j] = rot.template operator()<6>(E[3]);
output[7][j] = rot.template operator()<7>(O[3]);
}
}
};
constexpr size_t h = n / 64;
for(size_t j = 0; j < h; j += Tile) {
size_t end = std::min(h, j + Tile);
auto first = [&]() {
for(size_t t = 0; t < 8; t++) {
stage.template operator()<true>(0, n, j + t * h, end + t * h);
}
};
auto second = [&]() {
for(size_t k = 0; k < 8; k++) {
stage.template operator()<false>(k * n / 8, n / 8, j, end);
}
};
if constexpr(inverse) {
second();
first();
} else {
first();
second();
}
}
}
// Product for n = 2^24: the top three stages split each input into 64 independent
// blocks. Both forward transforms read the u32 inputs directly; each block then
// completes its two forward transforms, the product, and the inverse before the
// final three inverse stages combine the results.
// Fusing the Gaussian lift into the first pass saves a write and a read of each
// spectrum, but makes that pass compute-bound on the judge (measured slower there),
// so it stays opt-in; by default both spectra are filled first and swept in place.
static constexpr bool fuse_forward = false;
template<bool Neg, bool Fused = fuse_forward, bool Square = false>
void cache_product(cvector& b, fuse_args const& fa = {}, fuse_args const& fb = {}) {
constexpr size_t n = 1 << 24, block = 1 << 18;
prepare_roots(n / 16);
prepare_shear_roots();
if constexpr(Fused) {
sweep8<false, 2, sweep_tile, 1, Neg>(fa);
if constexpr(!Square) {
b.sweep8<false, 2, sweep_tile, 1, Neg>(fb);
}
} else {
sweep8<false, 2, sweep_tile>();
if constexpr(!Square) {
b.sweep8<false, 2, sweep_tile>();
}
}
checkpoint("sweep forward");
for(size_t offset = 0; offset < n; offset += block) {
transform<false, n, block, true>(n, false, offset, block);
if constexpr(!Square) {
b.transform<false, n, block, true>(n, false, offset, block);
}
dot_fused16<n, Square>(b, offset, block);
transform<true, n, block, true>(n, false, offset, block);
}
checkpoint("blocks");
sweep8<true, 2, sweep_tile>();
checkpoint("sweep inverse");
}
static constexpr size_t pre_evals = 1 << 16;
static const std::array<size_t, pre_evals> eval_args;
static const std::array<point, pre_evals> evalp;
private:
// The power of four that the fused product works on: all of n = 4^k, half of n = 2 * 4^k.
size_t fused_part() const {
return size() >> (std::countr_zero(size()) % 2);
}
bool fused_leaves() const {
return fused_part() >= 16 * dot_tile;
}
// Tile two radix-four stages together before descending into each child.
template<bool inverse, size_t fixed = 0, size_t range_fixed = 0, bool omit16 = false>
void transform(size_t input_n, bool parity, size_t range_offset = 0, size_t range_length = 0) {
if constexpr(range_fixed) {
range_length = range_fixed;
}
const size_t n = fixed ? fixed : input_n;
if constexpr(!range_fixed) {
prepare_roots(n / 16);
if constexpr(fixed == (1 << 24)) {
prepare_shear_roots();
}
}
size_t log_n = std::countr_zero(n);
auto butterfly = [&] [[gnu::always_inline]] (size_t offset, size_t length, size_t begin, size_t end) {
if constexpr(omit16) {
if(length == 16) {
return;
}
}
size_t step = length / 4, log_length = std::countr_zero(length), k = offset >> log_length;
auto *p0 = r.data() + (offset + begin) / flen, *p1 = r.data() + (offset + begin + step) / flen,
*p2 = r.data() + (offset + begin + 2 * step) / flen,
*p3 = r.data() + (offset + begin + 3 * step) / flen;
auto run = [&]<bool shear> [[gnu::always_inline]] () {
vpoint v1, v2, v3;
vftype t1{}, t2{}, t3{};
if constexpr(shear && fixed == (1 << 24)) {
auto const& c = shear_roots[((n >> log_length) - 1) / 3 + k];
v1 = {vz, inverse ? vz - real(c[0]) : vz + real(c[0])};
v2 = {vz, inverse ? vz - real(c[1]) : vz + real(c[1])};
v3 = {vz, inverse ? vz - real(c[2]) : vz + real(c[2])};
t1 = inverse ? vz - imag(c[0]) : vz + imag(c[0]);
t2 = inverse ? vz - imag(c[1]) : vz + imag(c[1]);
t3 = inverse ? vz - imag(c[2]) : vz + imag(c[2]);
} else {
point e = k < pre_evals ? evalp[k] : extra[k - pre_evals];
point rt = roots[log_n + 5 - log_length] * e;
v1 = {vz + real(rt), inverse ? vz - imag(rt) : vz + imag(rt)};
if constexpr(shear) {
if(k & 1) {
if constexpr(inverse) {
v1 = vi(v1);
} else {
v1 = -vi(v1);
}
}
}
v2 = v1 * v1;
v3 = v1 * v2;
if constexpr(shear) {
t1 = imag(v1) / (vz + 1.0 + real(v1));
t2 = imag(v2) / (vz + 1.0 + real(v2));
t3 = imag(v3) / (vz + 1.0 + real(v3));
}
}
auto rotate = [] [[gnu::always_inline]] (vpoint z, vpoint v, vftype t) {
if constexpr(!shear) {
return z * v;
} else {
auto x = vftype(_mm256_fnmadd_pd(__m256d(t), __m256d(imag(z)), __m256d(real(z))));
auto y = vftype(_mm256_fmadd_pd(__m256d(imag(v)), __m256d(x), __m256d(imag(z))));
return vpoint{vftype(_mm256_fnmadd_pd(__m256d(t), __m256d(y), __m256d(x))), y};
}
};
auto *i0 = p0, *i1 = p1, *i2 = p2, *i3 = p3, *o0 = p0, *o1 = p1, *o2 = p2, *o3 = p3;
if constexpr(shear) {
if(k & 1) {
if constexpr(inverse) {
i0 = p3;
i1 = p2;
i2 = p0;
i3 = p1;
} else {
o0 = p3;
o1 = p2;
o2 = p0;
o3 = p1;
}
}
}
for(size_t j = 0; j < (end - begin) / flen; j++) {
auto A = i0[j], B = i1[j], C = i2[j], D = i3[j];
if constexpr(inverse) {
o0[j] = (A + B) + (C + D);
o2[j] = rotate((A + B) - (C + D), v2, t2);
o1[j] = rotate((A - B) - vi(C - D), v1, t1);
o3[j] = rotate((A - B) + vi(C - D), v3, t3);
} else {
B = rotate(B, v1, t1);
C = rotate(C, v2, t2);
D = rotate(D, v3, t3);
o0[j] = (A + C) + (B + D);
o1[j] = (A + C) - (B + D);
o2[j] = (A - C) + vi(B - D);
o3[j] = (A - C) - vi(B - D);
}
}
};
if(length >= shear_min<fixed>) {
run.template operator()<true>();
} else {
run.template operator()<false>();
}
};
auto recurse = [&](this auto const& self, size_t offset, size_t length) -> void {
if(length < 4 * flen) {
return;
}
if(length >= (1 << 15)) {
size_t step = length / 16;
if constexpr(inverse) {
for(size_t t = 0; t < 16; t++) {
self(offset + t * step, step);
}
}
for(size_t j = 0; j < step; j += 256) {
size_t end = std::min(step, j + 256);
if constexpr(inverse) {
for(size_t t = 0; t < 4; t++) {
butterfly(offset + t * length / 4, length / 4, j, end);
}
for(size_t t = 0; t < 4; t++) {
butterfly(offset, length, j + t * step, end + t * step);
}
} else {
for(size_t t = 0; t < 4; t++) {
butterfly(offset, length, j + t * step, end + t * step);
}
for(size_t t = 0; t < 4; t++) {
butterfly(offset + t * length / 4, length / 4, j, end);
}
}
}
if constexpr(!inverse) {
for(size_t t = 0; t < 16; t++) {
self(offset + t * step, step);
}
}
} else if(length >= (size_t(1) << (6 + parity))) {
auto finish = [&]<bool par>() {
constexpr size_t chunk = size_t(1) << (6 + par);
constexpr size_t bottom = size_t(1) << (4 + par);
auto small = [&]<size_t L> [[gnu::always_inline]] (this auto const& self, size_t pos) -> void {
if constexpr(inverse && L > bottom) {
for(size_t q = 0; q < 4; q++) {
self.template operator()<L / 4>(pos + q * (L / 4));
}
}
butterfly(pos, L, 0, L / 4);
if constexpr(!inverse && L > bottom) {
for(size_t q = 0; q < 4; q++) {
self.template operator()<L / 4>(pos + q * (L / 4));
}
}
};
for(size_t leaf = offset; leaf < offset + length; leaf += chunk) {
if constexpr(inverse) {
small.template operator()<chunk>(leaf);
size_t level =
std::min<size_t>(std::countr_one(leaf + chunk - 1), std::countr_zero(length));
for(size_t lvl = 6 + 2 + par; lvl <= level; lvl += 2) {
size_t len = size_t(1) << lvl;
butterfly(leaf & ~(len - 1), len, 0, len / 4);
}
} else {
size_t level = std::min<size_t>(std::countr_zero(n + leaf), std::countr_zero(length));
level -= level % 2 != par;
for(size_t lvl = level; lvl >= 6 + 2 + par; lvl -= 2) {
size_t len = size_t(1) << lvl;
butterfly(leaf & ~(len - 1), len, 0, len / 4);
}
small.template operator()<chunk>(leaf);
}
}
};
if(parity) {
finish.template operator()<true>();
} else {
finish.template operator()<false>();
}
} else {
if constexpr(inverse) {
for(size_t leaf = offset + 3 * flen; leaf < offset + length; leaf += 4 * flen) {
size_t level = std::min<size_t>(std::countr_one(leaf + 3), std::countr_zero(length));
for(size_t lvl = 4 + parity; lvl <= level; lvl += 2) {
size_t len = size_t(1) << lvl;
butterfly(leaf & ~(len - 1), len, 0, len / 4);
}
}
} else {
for(size_t leaf = offset; leaf < offset + length; leaf += 4 * flen) {
size_t level = std::min<size_t>(std::countr_zero(n + leaf), std::countr_zero(length));
level -= level % 2 != parity;
for(size_t lvl = level; lvl >= 4; lvl -= 2) {
size_t len = size_t(1) << lvl;
butterfly(leaf & ~(len - 1), len, 0, len / 4);
}
}
}
}
};
// Radix two is performed separately at the leaves.
recurse(range_offset, range_length ? range_length : n);
}
// Shortest butterfly that uses the shear rotation; the fixed 2^24 transform reads its
// shear roots from a table, so they are worth using down to the last in-block level.
template<size_t fixed>
static constexpr size_t shear_min = fixed == (1 << 24) ? 64 : 256;
static big_vector<std::array<point, 3>> shear_roots;
static void prepare_shear_roots() {
constexpr size_t n = 1 << 24;
if(!shear_roots.empty()) {
return;
}
shear_roots.resize((n / 16 - 1) / 3, std::array<point, 3>{});
for(size_t len = n; len >= shear_min<n>; len /= 4) {
size_t count = n / len, base = (count - 1) / 3;
point factor = roots[29 - std::countr_zero(len)];
for(size_t k = 0; k < count; k++) {
point rt = factor * (k < pre_evals ? evalp[k] : extra[k - pre_evals]);
vpoint v1 = {vz + real(rt), vz + imag(rt)};
if(k & 1) {
v1 = -vi(v1);
}
vpoint v2 = v1 * v1, v3 = v1 * v2;
vftype t1 = imag(v1) / (vz + 1.0 + real(v1)), t2 = imag(v2) / (vz + 1.0 + real(v2)),
t3 = imag(v3) / (vz + 1.0 + real(v3));
shear_roots[base + k] = {point(imag(v1)[0], t1[0]), point(imag(v2)[0], t2[0]),
point(imag(v3)[0], t3[0])};
}
}
}
static big_vector<point> extra;
// Keep the usual table small; cache additional roots for large transforms.
static void prepare_roots(size_t n) {
if(n <= pre_evals + extra.size()) {
return;
}
size_t old = extra.size();
extra.resize(std::bit_ceil(n) - pre_evals);
static const std::array<point, 256> coarse = []() {
std::array<point, 256> out;
for(size_t i = 0; i < 256; i++) {
out[i] = polar<ftype>(1., std::numbers::pi * double((eval_args[256 + i] - 1) / 2) / 512.0);
}
return out;
}();
for(size_t h = pre_evals + old; h < pre_evals + extra.size(); h *= 2) {
for(size_t i = h; i < 2 * h; i += 256) {
point fine = polar<ftype>(1., std::numbers::pi / double(4 * h) * double(eval_arg(4 * i)));
for(size_t j = 0; j < 256; j++) {
extra[i + j - pre_evals] = coarse[j] * fine;
}
}
}
}
};
big_vector<std::array<point, 3>> cvector::shear_roots;
big_vector<point> cvector::extra;
const std::array<size_t, cvector::pre_evals> cvector::eval_args = []() {
std::array<size_t, pre_evals> res = {};
for(size_t i = 1; i < pre_evals; i++) {
res[i] = res[i >> 1] | (i & 1) << (std::bit_width(i) - 1);
}
return res;
}();
const std::array<point, cvector::pre_evals> cvector::evalp = []() {
std::array<point, pre_evals> res = {};
res[0] = 1;
for(size_t n = 1; n < pre_evals; n++) {
res[n] = polar<ftype>(1., std::numbers::pi * ftype(eval_args[n]) / ftype(4 * std::bit_floor(n)));
}
return res;
}();
}
#pragma GCC pop_options
#line 12 "cp-algo/math/ring.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math::fft {
size_t com_size(size_t as, size_t bs) {
if(!as || !bs) {
return 0;
}
return std::max(flen, std::bit_ceil(as + bs - 1) / 2);
}
// Whether a product of these sizes exceeds half of its padded length by a tail short enough
// to be corrected naively, which lets mul_truncate halve its transform.
constexpr size_t short_tail = 32;
bool has_short_tail(size_t as, size_t bs) {
size_t n = com_size(as, bs);
return as + bs - 1 - n <= short_tail && as <= n && bs <= n;
}
// Convolution through an imaginary quadratic ring, for a prime modulus p < 2^31.
// Let d be the smallest positive integer such that -d is a quadratic residue (d = 1 exactly
// when p = 1 mod 4) and root^2 = -d. A residue x is written as re + im*sqrt(-d) with
// re + im*root = x, (re, im) reduced by the lattice of representations of zero under the norm
// re^2 + d*im^2, so that |re| and sqrt(d)*|im| are of order sqrt(p), and is embedded as the
// complex number re + i*sqrt(d)*im. The product is then one complex convolution instead of
// three real ones. For d = 1 (Gaussian integers) it is computed modulo x^n - i and x^n + i
// (the latter through conjugation) and recombined modulo p, which needs i to lie in the ring;
// for any other d it is a single transform of the full product length.
constexpr uint64_t pow_mod(uint64_t x, uint64_t e, uint64_t p) {
uint64_t res = 1;
for(x %= p; e; e >>= 1, x = x * x % p) {
if(e & 1) {
res = res * x % p;
}
}
return res;
}
// Smallest d > 0 with -d a quadratic residue modulo the odd prime p < 2^31, or 0.
constexpr uint32_t least_negated_residue(uint64_t p) {
for(uint32_t d = 1; d < 256 && d < p; d++) {
if(pow_mod(p - d, (p - 1) / 2, p) == 1) {
return d;
}
}
return 0;
}
template<modint_type base>
struct quadratic {
static inline bool ready = false, available = false;
static inline uint32_t d = 0, root = 0, prime = 0;
// (a, b) and (a2, b2): reduced basis of the pairs (re, im) that represent zero.
static inline int32_t a = 0, b = 0, a2 = 0, b2 = 0;
// A fixed modulus settles d at compile time, so only the path in use is instantiated.
static constexpr bool fixed_mod = requires { typename std::bool_constant<(base::mod(), true)>; };
static constexpr uint32_t fixed_d = [] {
if constexpr(fixed_mod) {
return base::mod() > 2 && base::mod() % 2 ? least_negated_residue(base::mod()) : 0u;
} else {
return 0u;
}
}();
static void init() {
if(ready && prime == uint32_t(base::mod())) {
return;
}
ready = true;
available = false;
uint64_t p = base::mod();
prime = uint32_t(p);
if(p < 3 || p >= (uint64_t(1) << 31) || !is_prime(p)) {
return;
}
d = least_negated_residue(p);
auto s = d ? cp_algo::math::sqrt(base(-int64_t(d))) : std::nullopt;
if(!s) {
return;
}
root = s->getr();
// Lagrange reduction of (p, 0), (-root, 1) under the norm x^2 + d*y^2.
using wide = __int128;
wide x1 = p, y1 = 0, x2 = -wide(root), y2 = 1;
auto norm = [&](wide x, wide y) {
return x * x + d * y * y;
};
while(true) {
if(norm(x1, y1) > norm(x2, y2)) {
std::swap(x1, x2);
std::swap(y1, y2);
}
wide dot = x1 * x2 + d * y1 * y2, len = norm(x1, y1);
wide m = (2 * dot + (dot >= 0 ? len : -len)) / (2 * len);
if(m == 0 || norm(x2 - m * x1, y2 - m * y1) >= norm(x2, y2)) {
break;
}
x2 -= m * x1;
y2 -= m * y1;
}
a = int32_t(x1);
b = int32_t(y1);
// In the Gaussian integers i*(a, b) = (-b, a) lies in the lattice as well.
if(d == 1) {
a2 = b;
b2 = -a;
} else {
a2 = int32_t(x2);
b2 = int32_t(y2);
}
assert(base(a) + base(b) * base(root) == base(0) && base(a2) + base(b2) * base(root) == base(0));
assert(std::abs(int64_t(a) * b2 - int64_t(a2) * b) == int64_t(p));
available = true;
}
// Points per transform: two branches of half the padded product length for d = 1, one
// transform of the whole length otherwise. A short tail beyond that is computed naively
// and taken back out, as in mul_truncate, which halves the transform: over the residues
// for d = 1, where the branches together work modulo x^(2n) + 1, and in ring coordinates
// for the single transform, which works modulo x^n - i.
static size_t length(size_t as, size_t bs) {
return (d == 1 ? com_size(as, bs) : 2 * com_size(as, bs)) >> has_short_tail(as, bs);
}
// Whether this path is used for operands of these sizes. With n = com_size(as, bs) it
// takes six transforms of n points (three of 2n for d > 1), four for a square, where the
// split representation takes seven and five, so it is preferred whenever it is exact,
// unless a transform exceeds 2^24 points, the only length with a kernel tiled for data
// outside the caches (which in turn does not let an operand wrap around).
// Empirical rounding budget. Reusing a spectrum for a tiled square correlates its
// errors, so that optimization uses a stricter budget below and otherwise keeps
// independent lifts of the two operands.
static bool usable(size_t as, size_t bs) {
init();
if(!available || std::min(as, bs) < size_t(magic)) {
return false;
}
size_t need = as + bs - 1, n = length(as, bs);
if(n > (1 << 24) || (n == (1 << 24) && d == 1 && std::max(as, bs) > n)) {
return false;
}
using wide = unsigned __int128;
return wide(need) * d * base::mod() * base::mod() <= wide(1) << 90;
}
// Lattice constants as doubles, hoisted out of the hot loops (the statics are 32-bit
// integers and could alias the 32-bit output stream).
struct lattice {
// (c, e): the basis vector with the larger second coordinate, used to shrink im.
double a, b, a2, b2, to_q, to_t, c, e, inv_e, root, scale, inv_scale, p;
lattice(): a(quadratic::a), b(quadratic::b), a2(quadratic::a2), b2(quadratic::b2), p(base::mod()) {
double det = a * b2 - a2 * b;
to_q = b2 / det;
to_t = -b / det;
bool first = std::abs(b) >= std::abs(b2);
c = first ? a : a2;
e = first ? b : b2;
inv_e = 1.0 / e;
root = quadratic::root;
scale = std::sqrt(double(d));
inv_scale = 1.0 / scale;
}
};
// Map a rounded coordinate pair back to the residue re + root*im.
template<bool unit>
static u32x4 project(vpoint value, bool negative, lattice const& L) {
const double p = L.p;
auto R = round(real(value)), I = negative ? -imag(value) : imag(value);
if constexpr(unit) {
I = round(I);
} else {
I = round(I * L.inv_scale);
}
auto q = round(I * L.inv_e);
auto U = vftype(_mm256_fnmadd_pd(__m256d(q), _mm256_set1_pd(L.c), __m256d(R)));
auto V = vftype(_mm256_fnmadd_pd(__m256d(q), _mm256_set1_pd(L.e), __m256d(I)));
auto h = vftype(_mm256_fmadd_pd(__m256d(V), _mm256_set1_pd(L.root), __m256d(U)));
q = round(h * (1.0 / p));
auto out = vftype(_mm256_fnmadd_pd(__m256d(q), _mm256_set1_pd(p), __m256d(h)));
out = out < 0 ? out + p : out;
return u32x4(_mm256_cvttpd_epi32(__m256d(out)));
}
// Lift residues to ring coordinates with stochastic rounding.
// The xorshift stream is advanced once per 8 residues and restarted from the same
// seed for both branches, so both see the same representatives.
// With stream set the spectrum is written with non-temporal stores and left for
// cache_product to transform: for n = 2^24 it is far larger than the caches and is next
// read by a separate pass, so this saves the read-for-ownership of every destination line.
// Coefficients from n on (d = 1 only) wrap around: x^n = i in the branch modulo x^n - i,
// and the conjugated branch modulo x^n + i sees conj(-i * z) = i * conj(z).
template<bool stream, bool unit>
static void fill(cvector& c, auto const& x, auto const& upper, size_t n, bool negative, u64x4 state,
bool transform = true) {
const lattice L;
auto const* src = reinterpret_cast<const uint32_t*>(std::data(x));
size_t count = std::size(x);
assert(count <= n && (std::empty(upper) || (unit && !stream && count == n)));
c.r.resize(n / flen);
auto *dst = reinterpret_cast<double*>(c.r.data());
u32x8 words{};
auto emit = [&]<bool wrap = false> [[gnu::always_inline]] (size_t i, i32x4 bits) {
auto v = __builtin_convertvector(bits, vftype);
if(i % 8 == 0) {
state ^= state << 13;
state ^= state >> 7;
state ^= state << 17;
words = u32x8(state);
}
i32x4 small = i % 8 ? i32x4(__builtin_shufflevector(words, words, 1, 3, 5, 7))
: i32x4(__builtin_shufflevector(words, words, 0, 2, 4, 6));
auto noise = __builtin_convertvector(small, vftype) * 0x1p-32;
auto q = round(v * L.to_q + noise), t = round(v * L.to_t + noise);
auto re = v - q * L.a - t * L.a2, im = -(q * L.b) - t * L.b2;
if constexpr(!unit) {
im = im * L.scale;
}
if constexpr(stream) {
_mm256_stream_pd(dst + 2 * i, __m256d(re));
_mm256_stream_pd(dst + 2 * i + flen, __m256d(negative ? -im : im));
} else if constexpr(wrap) {
c.r[(i - n) / flen] += vpoint{negative ? im : -im, re};
} else {
c.r[i / flen] = vpoint{re, negative ? -im : im};
}
};
size_t full = count / flen * flen;
for(size_t i = 0; i < full; i += flen) {
i32x4 bits;
std::memcpy(&bits, src + i, sizeof(bits));
emit(i, bits);
}
if(full < count) {
i32x4 bits{};
for(size_t j = full; j < count; j++) {
bits[j - full] = int32_t(src[j]);
}
emit(full, bits);
}
if constexpr(stream) {
_mm_sfence();
}
std::fill(c.r.begin() + (count + flen - 1) / flen, c.r.end(), vpoint{});
auto const* wrapped = reinterpret_cast<const uint32_t*>(std::data(upper));
for(size_t i = 0; i < std::size(upper); i += flen) {
i32x4 bits{};
for(size_t j = i; j < std::min(i + flen, std::size(upper)); j++) {
bits[j - i] = int32_t(wrapped[j]);
}
emit.template operator()<true>(n + i, bits);
}
checkpoint("quadratic init");
if constexpr(!stream) {
if(transform) {
c.forward();
}
}
}
// Lift a range of residues to ring coordinates, one point per coefficient, for the
// reusable transforms below. Unlike fill it reads through the modint interface, so it
// serves views and Montgomery storage alike, and it neither wraps nor streams.
template<bool unit>
static void lift(cvector& c, auto const& x, size_t n, bool negative, u64x4 state) {
const lattice L;
size_t total = std::size(x), count = std::min(n, total);
assert(total <= 2 * n && (d == 1 || total <= n));
c.r.resize(n / flen);
u32x8 words{};
size_t step = 0;
auto emit = [&]<bool wrap> [[gnu::always_inline]] (size_t i, i32x4 bits) {
auto v = __builtin_convertvector(bits, vftype);
if(step % 2 == 0) {
state ^= state << 13;
state ^= state >> 7;
state ^= state << 17;
words = u32x8(state);
}
i32x4 small = step++ % 2 ? i32x4(__builtin_shufflevector(words, words, 1, 3, 5, 7))
: i32x4(__builtin_shufflevector(words, words, 0, 2, 4, 6));
auto noise = __builtin_convertvector(small, vftype) * 0x1p-32;
auto q = round(v * L.to_q + noise), t = round(v * L.to_t + noise);
auto re = v - q * L.a - t * L.a2, im = -(q * L.b) - t * L.b2;
if constexpr(!unit) {
im = im * L.scale;
}
// Coefficients from n on wrap around: x^n is i in this branch, -i in the other.
if constexpr(wrap) {
c.r[(i - n) / flen] += vpoint{negative ? im : -im, re};
} else {
c.r[i / flen] = vpoint{re, negative ? -im : im};
}
};
// Residues are raw words exactly when the modulus is a compile-time constant; wider
// storage than the residue needs is narrowed on the way in.
constexpr bool plain = fixed_mod && std::ranges::contiguous_range<std::decay_t<decltype(x)>>;
constexpr bool raw = plain && sizeof(base) == 4;
constexpr bool wide = plain && sizeof(base) == 8;
auto load = [&](size_t i, size_t upto) {
i32x4 bits{};
if constexpr(raw || wide) {
if(i + flen <= upto) {
if constexpr(raw) {
std::memcpy(&bits, reinterpret_cast<const uint32_t*>(std::data(x)) + i, sizeof(bits));
} else {
u64x4 words;
std::memcpy(&words, reinterpret_cast<const uint64_t*>(std::data(x)) + i, sizeof(words));
bits = __builtin_convertvector(words, i32x4);
}
return bits;
}
}
for(size_t j = i; j < std::min(i + flen, upto); j++) {
bits[j - i] = int32_t(x[j].getr());
}
return bits;
};
for(size_t i = 0; i < count; i += flen) {
emit.template operator()<false>(i, load(i, count));
}
std::fill(c.r.begin() + (count + flen - 1) / flen, c.r.end(), vpoint{});
for(size_t i = n; i < total; i += flen) {
emit.template operator()<true>(i, load(i, total));
}
checkpoint("quadratic init");
}
// Read a product out of its transformed branches, applying the inverse-transform scale.
// For d = 1 the branches hold the product modulo x^n - i and modulo x^n + i, so the low
// half of the result is their half-sum and the high half their half-difference over i.
// The half of the result that the branches are recombined into is parked in the output
// itself, which is always long enough: the high half is only wanted where the low half
// has already been read back.
static void recover(std::array<cvector, 2> const& parts, size_t n, double factor, auto &out, size_t k) {
// Residues are raw 32-bit words exactly when the modulus is a compile-time constant,
// which is what lets the recombination stay vectorized.
constexpr bool plain = fixed_mod && std::ranges::contiguous_range<std::decay_t<decltype(out)>>;
constexpr bool raw = plain && sizeof(base) == 4;
constexpr bool wide = plain && sizeof(base) == 8;
const lattice L;
auto scale = vz + factor;
size_t low = std::min(k, n);
auto project_at = [&](cvector const& part, size_t i, bool negative) {
if(d == 1) {
return project<true>(part.at(i) * scale, negative, L);
} else {
return project<false>(part.at(i) * scale, negative, L);
}
};
// Eight coefficients at a time, the width the recombination works in.
auto project8 = [&](cvector const& part, size_t i, bool negative, size_t count) {
auto lo8 = project_at(part, i, negative);
auto hi8 = count > flen ? project_at(part, i + flen, negative) : u32x4{};
return __builtin_shufflevector(lo8, hi8, 0, 1, 2, 3, 4, 5, 6, 7);
};
auto store8 = [&](size_t idx, u32x8 v, size_t count) {
if constexpr(raw) {
if(count == 8) {
std::memcpy(reinterpret_cast<uint32_t*>(std::data(out)) + idx, &v, sizeof(v));
return;
}
} else if constexpr(wide) {
if(count == 8) {
auto words = reinterpret_cast<uint64_t*>(std::data(out)) + idx;
auto lo4 = __builtin_convertvector(u32x4{v[0], v[1], v[2], v[3]}, u64x4);
auto hi4 = __builtin_convertvector(u32x4{v[4], v[5], v[6], v[7]}, u64x4);
std::memcpy(words, &lo4, sizeof(lo4));
std::memcpy(words + flen, &hi4, sizeof(hi4));
return;
}
}
for(size_t l = 0; l < count; l++) {
out[idx + l].setr(typename base::UInt(v[l]));
}
};
auto load8 = [&](size_t idx, size_t count) {
u32x8 v{};
if constexpr(raw) {
if(count == 8) {
std::memcpy(&v, reinterpret_cast<uint32_t const*>(std::data(out)) + idx, sizeof(v));
return v;
}
} else if constexpr(wide) {
if(count == 8) {
auto words = reinterpret_cast<uint64_t const*>(std::data(out)) + idx;
u64x4 lo4, hi4;
std::memcpy(&lo4, words, sizeof(lo4));
std::memcpy(&hi4, words + flen, sizeof(hi4));
auto lo = __builtin_convertvector(lo4, u32x4), hi = __builtin_convertvector(hi4, u32x4);
return u32x8(__builtin_shufflevector(lo, hi, 0, 1, 2, 3, 4, 5, 6, 7));
}
}
for(size_t l = 0; l < count; l++) {
v[l] = uint32_t(out[idx + l].getr());
}
return v;
};
for(size_t i = 0; i < low; i += 8) {
store8(i, project8(parts[0], i, false, std::min<size_t>(8, low - i)), std::min<size_t>(8, low - i));
}
if(d > 1) {
checkpoint("quadratic recover");
return;
}
const uint32_t mod32 = uint32_t(base::mod()), imod32 = -inv2<uint32_t>(base::mod());
auto highmul = u32x8{} + uint32_t(((base(2) * base(root)).inv() * bpow(base(2), 32)).getr());
for(size_t i = 0; i < low; i += 8) {
size_t count = std::min<size_t>(8, low - i);
auto minus = project8(parts[1], i, true, count);
auto plus = load8(i, count);
auto lo8 = reduce_once(plus + minus, mod32);
lo8 = (lo8 + (lo8 & 1) * mod32) >> 1;
auto hi8 = reduce_once(montgomery_mul(plus + mod32 - minus, highmul, mod32, imod32), mod32);
store8(i, lo8, count);
if(n + i < k) {
store8(n + i, hi8, std::min<size_t>(count, k - n - i));
}
}
checkpoint("quadratic recover");
}
// Cyclic product modulo x^k - 1, in place over a, for operands of exactly k coefficients.
// The transform evaluates at the k-th roots of i, so twisting coefficient j by w^j with
// w^k = -i turns the product it computes, modulo x^k - i, into the cyclic one; the
// inverse twist is folded into the readback. A fold of the linear product would need
// twice the transform.
static void cyclic(auto &a, auto const& b, size_t k) {
init();
assert(available && std::popcount(k) == 1 && std::size(a) == k && std::size(b) == k);
bool square = (void const*)std::data(a) == (void const*)std::data(b);
// w^j from a table of every fourth power, built the way the root tables of the
// transform are, and four consecutive powers per vector by one broadcast multiply.
size_t groups = k / flen;
size_t fine_bits = std::min<size_t>(8, std::countr_zero(std::max<size_t>(groups, 1)));
size_t fine = size_t(1) << fine_bits;
big_vector<point> low(fine), high((groups + fine - 1) >> fine_bits), step(groups);
auto w = [&](size_t j) {
return polar<ftype>(1., -std::numbers::pi * ftype(j) / ftype(2 * k));
};
for(size_t t = 0; t < low.size(); t++) {
low[t] = w(flen * t);
}
for(size_t c = 0; c < high.size(); c++) {
high[c] = w(flen * (c << fine_bits));
}
for(size_t c = 0; c < groups; c++) {
step[c] = high[c >> fine_bits] * low[c & (fine - 1)];
}
vpoint quarter, quarter_conj;
for(size_t l = 0; l < flen; l++) {
point v = w(l);
real(quarter)[l] = real(v);
imag(quarter)[l] = imag(v);
real(quarter_conj)[l] = real(v);
imag(quarter_conj)[l] = -imag(v);
}
auto twiddle = [&](size_t j, bool inverse, ftype scale) {
point t = step[j / flen];
vpoint head = {vz + real(t) * scale, vz + (inverse ? -imag(t) : imag(t)) * scale};
return head * (inverse ? quarter_conj : quarter);
};
auto run = [&]<bool unit>() {
const lattice L;
cvector A(k), B(0);
lift<unit>(A, a, k, false, seed());
if(!square) {
lift<unit>(B, b, k, false, seed());
}
for(size_t j = 0; j < k; j += flen) {
auto w = twiddle(j, false, 1);
A.at(j) *= w;
if(!square) {
B.at(j) *= w;
}
}
A.forward();
if(!square) {
B.forward();
}
A.multiply(square ? A : B);
for(size_t j = 0; j < k; j += flen) {
auto v = project<unit>(A.at(j) * twiddle(j, true, ftype(flen) / ftype(k)), false, L);
for(size_t l = 0; l < flen; l++) {
a[j + l].setr(typename base::UInt(v[l]));
}
}
};
if(d == 1) {
run.template operator()<true>();
} else {
run.template operator()<false>();
}
checkpoint("quadratic recover");
}
static u64x4 seed() {
return u64x4{random::rng() | 1, random::rng() | 1, random::rng() | 1, random::rng() | 1};
}
// a <- a * b for d = 1, or a <- a * a with square set (b is not read then).
// The first branch is stored in the upper half of a, so the part of a that wraps around
// is set aside first.
static void mul_branches(auto &a, auto const& b, bool square) {
size_t as = std::size(a), bs = square ? as : std::size(b), need = as + bs - 1;
size_t n = length(as, bs);
assert(available && d == 1);
// Coefficients from 2n on come back negated at the bottom; there are few of them.
std::array<uint32_t, short_tail> high{};
size_t tail = need > 2 * n ? need - 2 * n : 0;
for(size_t i = 0; i < tail; i++) {
auto const* x = reinterpret_cast<const uint32_t*>(std::data(a));
auto const* y = square ? x : reinterpret_cast<const uint32_t*>(std::data(b));
uint64_t sum = 0;
for(size_t j = 2 * n + i - bs + 1; j < as; j++) {
sum = (sum + uint64_t(x[j]) * y[2 * n + i - j]) % prime;
}
high[i] = uint32_t(sum);
}
// Montgomery constants for the recombination of the two branches.
const uint32_t mod32 = uint32_t(base::mod()), imod32 = -inv2<uint32_t>(base::mod());
base r32 = bpow(base(2), 32);
auto highmul = u32x8{} + uint32_t(((base(2) * base(root)).inv() * r32).getr());
u64x4 seed_a = seed(), seed_b = seed();
const lattice L;
big_vector<base> a_upper(std::begin(a) + std::min(as, n), std::end(a));
a.resize(2 * n);
std::span<base const> a_lower = std::span(a).first(std::min(as, n));
// A square may come with b aliasing the storage that a has just left.
auto b_lower = square ? a_lower : std::span<base const>(b).first(std::min(bs, n));
auto b_upper = square ? std::span<base const>(a_upper) : std::span<base const>(b).subspan(b_lower.size());
cvector A(0), B(0);
auto *out = reinterpret_cast<uint32_t*>(std::data(a));
for(bool negative: {false, true}) {
if(n == (1 << 24)) {
if constexpr(cvector::fuse_forward) {
cvector::fuse_args fa{
out, as, seed_a[0], L.a, L.b, L.a / double(base::mod()), L.b / double(base::mod())};
cvector::fuse_args fb{reinterpret_cast<const uint32_t*>(std::data(b)),
bs,
seed_b[0],
fa.a,
fa.b,
fa.a_over_p,
fa.b_over_p};
if(negative) {
A.template cache_product<true>(B, fa, fb);
} else {
A.template cache_product<false>(B, fa, fb);
}
} else {
fill<true, true>(A, a_lower, a_upper, n, negative, seed_a);
if(square && __uint128_t(need) * d * prime * prime <= (__uint128_t(1) << 88)) {
if(negative) {
A.template cache_product<true, false, true>(A);
} else {
A.template cache_product<false, false, true>(A);
}
} else {
fill<true, true>(B, b_lower, b_upper, n, negative, seed_b);
if(negative) {
A.template cache_product<true>(B);
} else {
A.template cache_product<false>(B);
}
}
}
} else {
fill<false, true>(A, a_lower, a_upper, n, negative, seed_a);
if(!square) {
fill<false, true>(B, b_lower, b_upper, n, negative, seed_b);
}
A.multiply(square ? A : B);
}
auto scale = vz + double(flen) / double(n);
for(size_t i = 0; i < n; i += 8) {
auto sum0 = project<true>(A.at(i) * scale, negative, L);
auto sum1 = project<true>(A.at(i + 4) * scale, negative, L);
auto sum = __builtin_shufflevector(sum0, sum1, 0, 1, 2, 3, 4, 5, 6, 7);
if(negative) {
u32x8 plus;
std::memcpy(&plus, out + n + i, sizeof(plus));
auto lo = plus + sum;
// Unsigned reduction: these sums pass 2^31 for moduli above 2^30.
lo = reduce_once(lo, mod32);
lo = (lo + (lo & 1) * base::mod()) >> 1;
auto hi = reduce_once(montgomery_mul(plus + base::mod() - sum, highmul, mod32, imod32), mod32);
std::memcpy(out + i, &lo, sizeof(lo));
std::memcpy(out + n + i, &hi, sizeof(hi));
} else {
std::memcpy(out + n + i, &sum, sizeof(sum));
}
}
checkpoint("quadratic recover");
}
a.resize(need);
out = reinterpret_cast<uint32_t*>(std::data(a));
for(size_t i = 0; i < tail; i++) {
uint32_t sum = out[i] + high[i];
out[i] = std::min(sum, sum - prime);
out[2 * n + i] = high[i];
}
}
// a <- a * b for d > 1, or a <- a * a with square set: i is not in the ring, so two
// branches could only be recombined as complex numbers, which is slower and less exact
// than one full transform.
static void mul_single(auto &a, auto const& b, bool square) {
size_t as = std::size(a), bs = square ? as : std::size(b), need = as + bs - 1;
size_t n = length(as, bs), tail = need > n ? need - n : 0;
assert(available && d > 1);
const lattice L;
cvector A(0), B(0);
std::span<base const> none;
// Coefficients from n on, in ring coordinates: they come back multiplied by i.
std::array<point, short_tail> high{};
auto wrapped = [&](cvector const& rhs) {
for(size_t i = 0; i < tail; i++) {
for(size_t j = n + i - bs + 1; j < as; j++) {
high[i] += A.template get<point>(j) * rhs.template get<point>(n + i - j);
}
}
};
if(n == (1 << 24)) {
fill<true, false>(A, a, none, n, false, seed());
if(square && __uint128_t(need) * d * prime * prime <= (__uint128_t(1) << 88)) {
wrapped(A);
A.template cache_product<false, false, true>(A);
} else {
if(square) {
fill<true, false>(B, a, none, n, false, seed());
} else {
fill<true, false>(B, b, none, n, false, seed());
}
wrapped(B);
A.template cache_product<false>(B);
}
} else {
fill<false, false>(A, a, none, n, false, seed(), !tail);
if(!square) {
fill<false, false>(B, b, none, n, false, seed(), !tail);
}
if(tail) {
wrapped(square ? A : B);
A.forward();
if(!square) {
B.forward();
}
}
A.multiply(square ? A : B);
}
for(size_t i = 0; i < tail; i++) {
A.set(i, A.template get<point>(i) - point(0, 1) * high[i] * (double(n) / double(flen)));
}
a.resize(n);
auto *out = reinterpret_cast<uint32_t*>(std::data(a));
auto scale = vz + double(flen) / double(n);
for(size_t i = 0; i < n; i += flen) {
auto sum = project<false>(A.at(i) * scale, false, L);
std::memcpy(out + i, &sum, sizeof(sum));
}
checkpoint("quadratic recover");
a.resize(need);
out = reinterpret_cast<uint32_t*>(std::data(a));
for(size_t i = 0; i < tail; i += flen) {
vpoint lanes = {vz, vz};
for(size_t j = i; j < std::min(tail, i + flen); j++) {
real(lanes)[j - i] = real(high[j]);
imag(lanes)[j - i] = imag(high[j]);
}
auto sum = project<false>(lanes, false, L);
for(size_t j = i; j < std::min(tail, i + flen); j++) {
out[n + j] = sum[j - i];
}
}
}
// Both routines read and write the storage as plain residues, which it is for modint<m>.
// A runtime-modulus type keeps another form, so its operands are converted on the way.
static void mul(auto &a, auto const& b, bool square) {
static_assert(sizeof(std::decay_t<decltype(a[0])>) == 4);
auto run = [&](auto const& rhs) {
if constexpr(fixed_d == 1) {
mul_branches(a, rhs, square);
} else if constexpr(fixed_d > 1) {
mul_single(a, rhs, square);
} else if(d == 1) {
mul_branches(a, rhs, square);
} else {
mul_single(a, rhs, square);
}
};
if constexpr(fixed_mod) {
run(b);
} else {
big_vector<base> plain;
if(!square) {
plain.assign(std::begin(b), std::end(b));
}
for(auto &x: plain) {
x.setr_direct(x.getr());
}
for(auto &x: a) {
x.setr_direct(x.getr());
}
run(plain);
for(auto &x: a) {
x.setr(x.getr_direct());
}
}
}
};
// A polynomial transformed for repeated multiplication in the ring of quadratic<base>.
//
// capacity() is the number of product coefficients the transform represents, and both
// operands of a product must fit in it. For d = 1 the polynomial is held as two conjugate
// branches of capacity/2 points, modulo x^n - i and modulo x^n + i, whose moduli multiply
// to x^(2n) + 1; for larger d as one transform of capacity points modulo x^n - i. Either
// way a product of at most capacity coefficients does not wrap around.
//
// complete selects the layout. The default stops one radix level short, which the fused
// product of multiply() finishes; the complete transform makes a product pointwise, which
// is what an accumulated sum of products needs.
//
// A transform costs two doubles per coefficient, so the type is move-only and every copy is
// spelled clone(). multiply() and square() consume the object and only read the other
// operand, which is what lets one transform serve several products.
//
// A transform belongs to the modulus it was built under; a runtime modulus must not change
// while one is alive.
template<modint_type base, bool complete = false>
struct spectrum {
using ring = quadratic<base>;
spectrum(auto const& a, size_t capacity): cap(std::max(2 * flen, std::bit_ceil(capacity))) {
ring::init();
assert(ring::available && std::size(a) <= cap);
auto state = ring::seed();
for(size_t j = 0; j < branches(); j++) {
if(ring::d == 1) {
ring::template lift<true>(parts[j], a, points(), j == 1, state);
} else {
ring::template lift<false>(parts[j], a, points(), false, state);
}
if constexpr(complete) {
parts[j].template fft<false>();
} else {
parts[j].forward();
}
}
}
spectrum(spectrum&&) = default;
spectrum& operator=(spectrum&&) = default;
spectrum clone() const { return *this; }
size_t capacity() const { return cap; }
// out[0..k) = *this * other. Consumes *this and only reads other.
void multiply(spectrum const& other, auto &out, size_t k) && requires(!complete) {
assert(other.cap == cap && k <= cap);
for(size_t j = 0; j < branches(); j++) {
parts[j].multiply(other.parts[j]);
}
ring::recover(parts, points(), double(flen) / double(points()), out, k);
}
// out[0..k) = *this * *this, consuming *this.
void square(auto &out, size_t k) && requires(!complete) {
std::move(*this).multiply(*this, out, k);
}
private:
spectrum(spectrum const&) = default;
size_t branches() const { return ring::d == 1 ? 2 : 1; }
size_t points() const { return ring::d == 1 ? cap / 2 : cap; }
template<modint_type>
friend struct product;
std::array<cvector, 2> parts = {cvector(0), cvector(0)};
size_t cap;
};
// A sum of products of transforms, read back once. Every term costs one pointwise pass and
// the sum one inverse transform, which is what makes it worth keeping the operands around.
template<modint_type base>
struct product {
using ring = quadratic<base>;
using operand = spectrum<base, true>;
explicit product(size_t capacity): cap(std::max(2 * flen, std::bit_ceil(capacity))) {
ring::init();
assert(ring::available);
for(size_t j = 0; j < branches(); j++) {
acc[j] = cvector(points());
}
}
// *this += x * y.
void add(operand const& x, operand const& y) {
assert(x.cap == cap && y.cap == cap);
for(size_t j = 0; j < branches(); j++) {
for(size_t i = 0; i < points(); i += flen) {
acc[j].at(i) += x.parts[j].at(i) * y.parts[j].at(i);
}
}
checkpoint("dot");
}
// out[0..k) = the accumulated sum, consuming *this.
void recover(auto &out, size_t k) && {
assert(k <= cap);
for(size_t j = 0; j < branches(); j++) {
acc[j].template ifft<false>();
}
ring::recover(acc, points(), 1.0, out, k);
}
private:
size_t branches() const { return ring::d == 1 ? 2 : 1; }
size_t points() const { return ring::d == 1 ? cap / 2 : cap; }
std::array<cvector, 2> acc = {cvector(0), cvector(0)};
size_t cap;
};
}
#pragma GCC pop_options
#line 6 "cp-algo/math/fft.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math::fft {
void mul_slow(auto &a, auto const& b, size_t k) {
using base = std::decay_t<decltype(a[0])>;
if(!std::empty(a) && std::data(a) == std::data(b)) {
size_t n = std::min(k, std::size(a)), m = std::min(k, std::size(b));
if(!m) {
a.clear();
return;
}
a.resize(k);
// Descending output only reads original coefficients at indices <=j.
for(size_t j = k; j-- > 0;) {
base sum = 0;
size_t lo = j >= n ? j + 1 - n : 0, hi = std::min(j + 1, m);
for(size_t i = lo; i < hi; i++) {
if(n == m && i > j - i) {
break;
}
auto term = a[i] * a[j - i];
sum += n == m && i != j - i ? term + term : term;
}
a[j] = sum;
}
return;
}
if(std::empty(a) || std::empty(b)) {
a.clear();
} else {
size_t n = std::min(k, std::size(a));
size_t m = std::min(k, std::size(b));
size_t had = std::size(a);
a.resize(k);
// The loop below reads every coefficient it writes, so the growth is zeroed here
// rather than relying on the caller's allocator to do it.
if(k > had) {
std::fill(std::begin(a) + had, std::end(a), base(0));
}
for(int j = int(k - 1); j >= 0; j--) {
a[j] *= b[0];
for(int i = std::max(j - (int)n, 0) + 1; i < std::min(j + 1, (int)m); i++) {
a[j] += a[j - i] * b[i];
}
}
}
}
// Whether two ranges are backed by the same storage, which makes a product a square.
bool same_storage(auto const& a, auto const& b) {
if constexpr(std::ranges::contiguous_range<decltype(b)>) {
return !std::empty(a) && (void const*)std::data(a) == (void const*)std::data(b);
} else {
return false;
}
}
// The product of the truncated operands, keeping k coefficients. Short operands are cheaper
// to multiply naively; every other product goes through the ring.
void mul_truncate(auto &a, auto const& b, size_t k) {
using base = std::decay_t<decltype(a[0])>;
if(std::min({k, std::size(a), std::size(b)}) < magic) {
mul_slow(a, b, k);
return;
}
size_t as = std::min(k, std::size(a)), bs = std::min(k, std::size(b));
assert(quadratic<base>::usable(as, bs) &&
"the ring needs an odd prime modulus below 2^31 and a product within its rounding bound");
bool aliased = same_storage(a, b), square = aliased && as == bs;
// Everything past the true length of the product is zero, written here rather than
// left to the caller's allocator.
size_t need = as + bs - 1, keep = std::min(k, need);
auto pad = [&] {
if(k > need) {
std::fill(std::begin(a) + need, std::end(a), base(0));
}
};
if constexpr(sizeof(base) == 4 && std::ranges::contiguous_range<decltype(b)>) {
if(aliased && !square) {
big_vector<base> copy(std::data(b), std::data(b) + bs);
a.resize(as);
quadratic<base>::mul(a, copy, false);
} else {
auto prefix = std::span<base const>(std::data(b), bs);
a.resize(as);
quadratic<base>::mul(a, prefix, square);
}
a.resize(k);
pad();
} else {
// Wider storage or a non-contiguous operand: the reusable transform reads and writes
// through the modint interface instead of the raw residues.
size_t cap = std::bit_ceil(need);
auto A = spectrum<base>(a | std::views::take(as), cap);
a.resize(k);
if(square) {
std::move(A).square(a, keep);
} else {
std::move(A).multiply(spectrum<base>(b | std::views::take(bs), cap), a, keep);
}
pad();
}
}
// Cyclic product modulo x^k - 1, in place over a.
void cyclic_mul(auto &a, auto const& b, size_t k) {
using base = std::decay_t<decltype(a[0])>;
quadratic<base>::cyclic(a, b, k);
}
namespace impl {
// Overlap-add for a short fixed operand; every block reuses its transform.
void mul_unbalanced(auto &a, auto const& b) {
using base = std::decay_t<decltype(a[0])>;
auto x = std::span<base const>(a), y = std::span<base const>(b);
if(x.size() < y.size()) {
std::swap(x, y);
}
constexpr size_t length = 1 << 15;
size_t step = length - y.size() + 1;
auto fixed = spectrum<base>(y, length);
std::decay_t<decltype(a)> result;
// Accumulated into, so the zeros are written rather than assumed.
result.assign(x.size() + y.size() - 1, base(0));
big_vector<base> work(length);
for(size_t start = 0; start < x.size(); start += step) {
size_t count = std::min(step, x.size() - start);
size_t need = count + y.size() - 1;
spectrum<base>(x.subspan(start, count), length).multiply(fixed, work, need);
for(size_t i = 0; i < need; i++) {
result[start + i] += work[i];
}
}
a = std::move(result);
}
}
void mul(auto &a, auto const& b) {
if(std::empty(a) || std::empty(b)) {
a.clear();
return;
}
size_t small = std::min(std::size(a), std::size(b));
size_t large = std::max(std::size(a), std::size(b));
if(small >= magic && small <= 4096 && large >= (1 << 20) && large / small >= 64) {
return impl::mul_unbalanced(a, b);
}
mul_truncate(a, b, std::size(a) + std::size(b) - 1);
}
}
#pragma GCC pop_options
#line 7 "cp-algo/math/poly/base.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
template<typename T> struct poly_t;
template<typename T>
std::array<poly_t<T>, 2> divmod(poly_t<T> p, poly_t<T> const& q);
template<typename T>
struct poly_t {
using Vector = big_vector<T>;
using base = T;
Vector a;
poly_t& normalize() {
while(deg() >= 0 && lead() == base(0)) {
a.pop_back();
}
return *this;
}
poly_t() = default;
poly_t(T a0): a{a0} {normalize();}
poly_t(Vector const& t): a(t) {normalize();}
poly_t(Vector &&t): a(std::move(t)) {normalize();}
poly_t& negate_inplace() {
std::ranges::transform(a, begin(a), std::negate{});
return *this;
}
friend poly_t operator -(poly_t p) {p.negate_inplace(); return p;}
poly_t& operator += (poly_t const& t) {
a.resize(std::max(size(a), size(t.a)));
std::ranges::transform(a, t.a, begin(a), std::plus{});
return normalize();
}
poly_t& operator -= (poly_t const& t) {
a.resize(std::max(size(a), size(t.a)));
std::ranges::transform(a, t.a, begin(a), std::minus{});
return normalize();
}
friend poly_t operator + (poly_t p, poly_t const& t) {p += t; return p;}
friend poly_t operator - (poly_t p, poly_t const& t) {p -= t; return p;}
poly_t& mod_xk_inplace(size_t k) {
a.resize(std::min(size(a), k));
return normalize();
}
poly_t& mul_xk_inplace(size_t k) {
if(is_zero()) {return *this;}
a.insert(begin(a), k, T(0));
return normalize();
}
poly_t& div_xk_inplace(int64_t k) {
if(k < 0) {
return mul_xk_inplace(-k);
}
a.erase(begin(a), begin(a) + std::min<size_t>(k, size(a)));
return normalize();
}
poly_t &substr_inplace(size_t l, size_t k) {
return mod_xk_inplace(l + k).div_xk_inplace(l);
}
poly_t mod_xk(size_t k) const & {return substr(0, k);}
poly_t mod_xk(size_t k) && {mod_xk_inplace(k); return std::move(*this);}
poly_t mul_xk(size_t k) const & {auto p = *this; p.mul_xk_inplace(k); return p;}
poly_t mul_xk(size_t k) && {mul_xk_inplace(k); return std::move(*this);}
poly_t div_xk(int64_t k) const & {return k < 0 ? mul_xk(-k) : substr(k, a.size());}
poly_t div_xk(int64_t k) && {div_xk_inplace(k); return std::move(*this);}
poly_t substr(size_t l, size_t k) const & {
l = std::min(l, a.size());
k = std::min(k, a.size() - l);
return Vector(begin(a) + l, begin(a) + l + k);
}
poly_t substr(size_t l, size_t k) && {substr_inplace(l, k); return std::move(*this);}
poly_t& operator *= (const poly_t &t) {fft::mul(a, t.a); normalize(); return *this;}
friend poly_t operator * (poly_t p, const poly_t &t) {p *= t; return p;}
poly_t& operator /= (const poly_t &t) {
assert(!t.is_zero());
if(this == &t) {return *this = T(1);}
auto [q, r] = divmod(std::move(*this), t);
return *this = std::move(q);
}
poly_t& operator %= (const poly_t &t) {
assert(!t.is_zero());
if(this == &t) {a.clear(); return *this;}
auto [q, r] = divmod(std::move(*this), t);
return *this = std::move(r);
}
friend poly_t operator / (poly_t p, poly_t const& t) {p /= t; return p;}
friend poly_t operator % (poly_t p, poly_t const& t) {p %= t; return p;}
poly_t& operator *= (T const& x) {
for(auto &it: a) {
it *= x;
}
return normalize();
}
poly_t& operator /= (T const& x) {return *this *= x.inv();}
friend poly_t operator * (poly_t p, T const& x) {p *= x; return p;}
friend poly_t operator / (poly_t p, T const& x) {p /= x; return p;}
poly_t& reverse(size_t n) {
a.resize(n);
std::ranges::reverse(a);
return normalize();
}
poly_t& reverse() {return reverse(size(a));}
poly_t reversed(size_t n) const & {auto p = *this; p.reverse(n); return p;}
poly_t reversed(size_t n) && {reverse(n); return std::move(*this);}
poly_t reversed() const & {return reversed(a.size());}
poly_t reversed() && {reverse(); return std::move(*this);}
friend poly_t operator * (T const& x, poly_t p) {p *= x; return p;}
poly_t negx() const { // A(x) -> A(-x)
auto res = *this;
for(int i = 1; i <= deg(); i += 2) {
res.a[i] = -res[i];
}
return res;
}
void print(int n) const {
for(int i = 0; i < n; i++) {
std::cout << (*this)[i] << ' ';
}
std::cout << "\n";
}
void print() const {
print(deg() + 1);
}
T eval(T x) const { // evaluates in single point x
T res(0);
for(int i = deg(); i >= 0; i--) {
res *= x;
res += a[i];
}
return res;
}
T lead() const { // leading coefficient
assert(!is_zero());
return a.back();
}
int deg() const { // degree, -1 for P(x) = 0
return (int)a.size() - 1;
}
bool is_zero() const {
return a.empty();
}
T operator [](int idx) const {
return idx < 0 || idx > deg() ? T(0) : a[idx];
}
T& coef(size_t idx) { // mutable reference at coefficient
return a[idx];
}
bool operator == (const poly_t &t) const = default;
size_t trailing_xk() const { // Let p(x) = x^k * t(x), return k
if(is_zero()) {
return -1;
}
int res = 0;
while(a[res] == T(0)) {
res++;
}
return res;
}
poly_t& mul_truncate(poly_t const& t, size_t k) {
fft::mul_truncate(a, t.a, k);
return normalize();
}
static poly_t xk(size_t n) { // P(x) = x^n
return poly_t(T(1)).mul_xk(n);
}
static poly_t ones(size_t n) { // P(x) = 1 + x + ... + x^{n-1}
return Vector(n, 1);
}
poly_t x2() const { // P(x) -> P(x^2)
Vector res(2 * a.size());
for(size_t i = 0; i < a.size(); i++) {
res[2 * i] = a[i];
}
return res;
}
// Return {P0, P1}, where P(x) = P0(x^2) + xP1(x^2)
std::array<poly_t, 2> bisect(size_t n) const {
n = std::min(n, size(a));
Vector res[2];
for(size_t i = 0; i < n; i++) {
res[i % 2].push_back(a[i]);
}
return {std::move(res[0]), std::move(res[1])};
}
std::array<poly_t, 2> bisect() const {
return bisect(size(a));
}
};
}
#pragma GCC pop_options
#line 4 "cp-algo/math/poly/series/inv.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math::poly::impl {
template<typename poly>
poly& inv_inplace(poly& p, size_t n) {
using base = poly::base;
if(n == 0) {
p.a.clear();
return p;
}
assert(p[0] != base(0));
if(n < magic) {
typename poly::Vector q(n);
q[0] = base(1) / p[0];
for(size_t i = 1; i < n; i++) {
for(size_t j = 1; j <= std::min(i, p.a.size() - 1); j++) {
q[i] -= p.a[j] * q[i - j];
}
q[i] *= q[0];
}
return p = std::move(q);
}
size_t m = std::bit_floor(size_t(magic - 1));
auto q = p.mod_xk(m);
inv_inplace(q, m);
for(; m < n; m *= 2) {
size_t k = std::min(2 * m, n);
typename poly::Vector error((k + fft::flen - 1) / fft::flen * fft::flen);
auto Q = fft::spectrum<base>(q.a, 2 * m);
// Wrapping modulo x^(2m) + 1 only changes the discarded low half.
fft::spectrum<base>(p.a | std::views::take(k), 2 * m).multiply(Q, error, k);
auto E = fft::spectrum<base>(error | std::views::drop(m) | std::views::take(k - m), 2 * m);
std::move(Q).multiply(E, error, k - m);
q.a.resize(k);
for(size_t i = m; i < k; i++) {
q.a[i] = -error[i - m];
}
}
p = std::move(q);
p.normalize();
return p;
}
}
namespace cp_algo::math {
// Inverse modulo x^n; the constant coefficient must be invertible.
template<typename T>
poly_t<T> inv(poly_t<T> p, size_t n) {
poly::impl::inv_inplace(p, n);
return p;
}
}
#pragma GCC pop_options
#line 4 "cp-algo/math/poly/impl/div.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math::poly::impl {
template<typename T>
std::array<poly_t<T>, 2> divmod_slow(poly_t<T> p, poly_t<T> const& q) {
poly_t<T> d;
auto qi = q.lead() == T(1) ? T(1) : q.lead().inv();
while(p.deg() >= q.deg()) {
d.a.push_back(p.lead() * qi);
if(d.lead() != T(0)) {
for(size_t i = 1; i <= q.a.size(); i++) {
p.a[p.a.size() - i] -= d.lead() * q.a[q.a.size() - i];
}
}
p.a.pop_back();
}
std::ranges::reverse(d.a);
p.normalize();
return {std::move(d), std::move(p)};
}
template<typename T>
std::array<poly_t<T>, 2> divmod_hint(poly_t<T> p, poly_t<T> const& q, poly_t<T> const& qri) {
assert(!q.is_zero());
int n = p.deg() - q.deg();
if(std::min(n, q.deg()) < magic) {
return divmod_slow(std::move(p), q);
}
poly_t<T> d(typename poly_t<T>::Vector(p.a.rbegin(), p.a.rbegin() + n + 1));
d.mul_truncate(qri, n + 1).reverse(n + 1);
// Only coefficients below deg(q) survive in the remainder.
auto low = d.mod_xk(q.deg());
low.mul_truncate(q, q.deg());
p.mod_xk_inplace(q.deg());
p -= low;
return {std::move(d), std::move(p)};
}
}
#pragma GCC pop_options
#line 4 "cp-algo/math/poly/div.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
// Quotient and remainder; q must be nonzero.
template<typename T>
std::array<poly_t<T>, 2> divmod(poly_t<T> p, poly_t<T> const& q) {
assert(!q.is_zero());
int n = p.deg() - q.deg();
if(std::min(n, q.deg()) < magic) {
return poly::impl::divmod_slow(std::move(p), q);
}
auto qi = inv(q.reversed(), n + 1);
return poly::impl::divmod_hint(std::move(p), q, qi);
}
}
#pragma GCC pop_options
#line 1 "cp-algo/math/poly/series.hpp"
#line 1 "cp-algo/math/poly/series/log.hpp"
#line 1 "cp-algo/math/poly/calculus.hpp"
#line 1 "cp-algo/math/factorials.hpp"
#line 1 "cp-algo/util/bump_alloc.hpp"
#line 5 "cp-algo/util/bump_alloc.hpp"
namespace cp_algo {
template<class T, size_t max_len>
struct bump_alloc {
static inline char* buf = big_alloc<char>().allocate(max_len * sizeof(T));
static inline size_t buf_ind = max_len * sizeof(T);
using value_type = T;
template <class U> struct rebind { using other = bump_alloc<U, max_len>; };
constexpr bool operator==(const bump_alloc&) const = default;
bump_alloc() = default;
template<class U> bump_alloc(const U&) {}
T* allocate(size_t n) {
buf_ind -= n * sizeof(T);
buf_ind &= 0 - alignof(T);
return (T*)(buf + buf_ind);
}
void deallocate(T*, size_t) {}
};
}
#line 1 "cp-algo/math/combinatorics.hpp"
#line 7 "cp-algo/math/combinatorics.hpp"
namespace cp_algo::math {
// fact/rfact/small_inv are caching
// Beware of usage with dynamic mod
template<typename T>
T fact(auto n) {
static const auto F = [] {
big_vector<T> F(maxn);
F[0] = T(1);
for(int i = 1; i < maxn; i++) {
F[i] = F[i - 1] * T(i);
}
return F;
}();
return F[n];
}
// Only works for modint types
template<typename T>
T rfact(auto n) {
static const auto F = [] {
big_vector<T> F(maxn);
int t = (int)std::min<int64_t>(T::mod(), maxn) - 1;
F[t] = T(1) / fact<T>(t);
for(int i = t - 1; i >= 0; i--) {
F[i] = F[i + 1] * T(i + 1);
}
return F;
}();
return F[n];
}
template<typename T, int base>
T pow_fixed(int n) {
static const auto prec = [] {
big_vector<T> prec_low(1 << 16), prec_high(1 << 16);
prec_low[0] = prec_high[0] = T(1);
T step_low = T(base);
T step_high = bpow(T(base), 1 << 16);
for(int i = 1; i < (1 << 16); i++) {
prec_low[i] = prec_low[i - 1] * step_low;
prec_high[i] = prec_high[i - 1] * step_high;
}
return std::pair{std::move(prec_low), std::move(prec_high)};
}();
return prec.first[n & 0xFFFF] * prec.second[n >> 16];
}
template<typename T>
big_vector<T> bulk_invs(auto const& args) {
big_vector<T> res(std::size(args), args[0]);
for(size_t i = 1; i < std::size(args); i++) {
res[i] = res[i - 1] * args[i];
}
auto all_invs = T(1) / res.back();
for(size_t i = std::size(args) - 1; i > 0; i--) {
res[i] = all_invs * res[i - 1];
all_invs *= args[i];
}
res[0] = all_invs;
return res;
}
template<typename T>
T small_inv(auto n) {
static auto F = bulk_invs<T>(std::views::iota(1, maxn));
return F[n - 1];
}
template<typename T>
T binom_large(T n, auto r) {
assert(r < maxn);
T ans = 1;
for(decltype(r) i = 0; i < r; i++) {
ans = ans * T(n - i) * small_inv<T>(i + 1);
}
return ans;
}
template<typename T>
T binom(auto n, auto r) {
if(r < 0 || r > n) {
return T(0);
} else if(n >= maxn) {
return binom_large(T(n), r);
} else {
return fact<T>(n) * rfact<T>(r) * rfact<T>(n - r);
}
}
}
#line 9 "cp-algo/math/factorials.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
template<bool use_bump_alloc = false, int maxn = -1>
auto facts(auto const& args) {
static_assert(!use_bump_alloc || maxn > 0, "maxn must be set if use_bump_alloc is true");
constexpr int max_mod = 1'000'000'000;
constexpr int accum = 4;
constexpr int simd_size = 8;
constexpr int block = 1 << 18;
constexpr int subblock = block / simd_size;
using base = std::decay_t<decltype(args[0])>;
static_assert(modint_type<base>, "Base type must be a modint type");
using T = std::array<int, 2>;
using alloc = std::conditional_t<use_bump_alloc,
bump_alloc<T, 30 * maxn>,
big_alloc<T>>;
std::basic_string<T, std::char_traits<T>, alloc> odd_args_per_block[max_mod / subblock];
std::basic_string<T, std::char_traits<T>, alloc> reg_args_per_block[max_mod / subblock];
constexpr int limit_reg = max_mod / 64;
int limit_odd = 0;
big_vector<base> res(size(args), 1);
const int mod = base::mod();
const int imod = -math::inv2(mod);
for(auto [i, xy]: std::views::zip(args, res) | std::views::enumerate) {
auto [x, y] = xy;
int t = x.getr();
if(t >= mod / 2) {
t = mod - t - 1;
y = t % 2 ? 1 : mod-1;
}
auto pw = 32ull * (t + 1);
while(t > limit_reg) {
limit_odd = std::max(limit_odd, (t - 1) / 2);
odd_args_per_block[(t - 1) / 2 / subblock].push_back({int(i), (t - 1) / 2});
t /= 2;
pw += t;
}
reg_args_per_block[t / subblock].push_back({int(i), t});
y *= pow_fixed<base, 2>(int(pw % (mod - 1)));
}
checkpoint("init");
base bi2x32 = pow_fixed<base, 2>(32).inv();
auto process = [&](int limit, auto &args_per_block, auto step, auto &&proj) {
base fact = 1;
for(int b = 0; b <= limit; b += accum * block) {
u32x8 cur[accum];
static std::array<u32x8, subblock> prods[accum];
for(int z = 0; z < accum; z++) {
for(int j = 0; j < simd_size; j++) {
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wmaybe-uninitialized"
cur[z][j] = uint32_t(b + z * block + j * subblock);
cur[z][j] = proj(cur[z][j]);
prods[z][0][j] = cur[z][j] + !cur[z][j];
prods[z][0][j] = uint32_t(uint64_t(prods[z][0][j]) * bi2x32.getr() % mod);
#pragma GCC diagnostic pop
}
}
for(int i = 1; i < block / simd_size; i++) {
for(int z = 0; z < accum; z++) {
cur[z] += step;
prods[z][i] = montgomery_mul(prods[z][i - 1], cur[z], mod, imod);
}
}
checkpoint("inner loop");
for(int z = 0; z < accum; z++) {
for(int j = 0; j < simd_size; j++) {
int bl = b + z * block + j * subblock;
for(auto [i, x]: args_per_block[bl / subblock]) {
res[i] *= fact * prods[z][x - bl][j];
}
fact *= base(prods[z].back()[j]);
}
}
checkpoint("mul ans");
}
};
process(limit_reg, reg_args_per_block, 1, std::identity{});
process(limit_odd, odd_args_per_block, 2, [](uint32_t x) {return 2 * x + 1;});
auto invs = bulk_invs<base>(res);
for(auto [i, x]: res | std::views::enumerate) {
if (args[i] >= mod / 2) {
x = invs[i];
}
}
checkpoint("inv ans");
return res;
}
}
#pragma GCC pop_options
#line 5 "cp-algo/math/poly/calculus.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
// k-th derivative; consumes temporaries and moved polynomials.
template<typename T>
poly_t<T> deriv(poly_t<T> p, int k = 1) {
assert(k >= 0);
if(k > p.deg()) {return k == 0 ? p : poly_t<T>{};}
if(k == 0) {return p;}
if(k == 1) {
for(int i = 1; i <= p.deg(); i++) {p.a[i - 1] = T(i) * p.a[i];}
p.a.pop_back();
p.normalize();
return p;
}
for(int i = k; i <= p.deg(); i++) {
p.a[i - k] = fact<T>(i) * rfact<T>(i - k) * p.a[i];
}
p.a.resize(p.a.size() - k);
p.normalize();
return p;
}
// Antiderivative with zero constant coefficient.
template<typename T>
poly_t<T> integr(poly_t<T> p) {
if(p.is_zero()) {return p;}
p.a.push_back(0);
for(int i = p.deg() - 1; i >= 0; i--) {
p.a[i + 1] = p.a[i] * small_inv<T>(i + 1);
}
p.a[0] = 0;
return p;
}
}
#pragma GCC pop_options
#line 5 "cp-algo/math/poly/series/log.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
// log(p) modulo x^n, for p[0] = 1.
template<typename T>
poly_t<T> log(poly_t<T> p, size_t n) {
if(n == 0) {
return {};
}
assert(p[0] == T(1));
p.mod_xk_inplace(n);
auto dp = deriv(p);
size_t k = n - 1;
if(k < magic) {
poly::impl::inv_inplace(p, k);
p.mul_truncate(dp, k);
return integr(std::move(p));
}
// Solve p*q = p' in two halves, avoiding a full-precision reciprocal.
size_t m = std::bit_floor(k - 1), t = k - m;
auto r = inv(p, m);
auto R = fft::spectrum<T>(r.a, 2 * m);
typename poly_t<T>::Vector work(2 * m);
fft::spectrum<T>(dp.a | std::views::take(m), 2 * m).multiply(R, work, m);
poly_t<T> q(typename poly_t<T>::Vector(begin(work), begin(work) + m));
{
auto Q = fft::spectrum<T>(q.a, 2 * m);
fft::spectrum<T>(p.a | std::views::take(k), 2 * m).multiply(Q, work, k);
}
// Cyclic wraparound only affects the discarded low half.
for(size_t i = 0; i < t; i++) {
work[m + i] = dp[int(m + i)] - work[m + i];
}
fft::spectrum<T>(work | std::views::drop(m) | std::views::take(t), 2 * m).multiply(R, work, t);
q.a.resize(k);
std::copy_n(begin(work), t, begin(q.a) + m);
q.normalize();
return integr(std::move(q));
}
}
#pragma GCC pop_options
#line 1 "cp-algo/math/poly/series/exp.hpp"
#line 5 "cp-algo/math/poly/series/exp.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
// exp(p) modulo x^n, for p[0] = 0.
template<typename T>
poly_t<T> exp(poly_t<T> p, size_t n) {
if(n == 0) {
return {};
}
assert(p[0] == T(0));
p.mod_xk_inplace(n);
if(p.is_zero()) {
return T(1);
}
size_t m = std::min(n, std::bit_floor(size_t(magic - 1)));
typename poly_t<T>::Vector seed(m);
seed[0] = 1;
for(size_t i = 1; i < m; i++) {
for(size_t j = 1; j <= i && j < p.a.size(); j++) {
seed[i] += T(j) * p.a[j] * seed[i - j];
}
seed[i] *= small_inv<T>(i);
}
poly_t<T> q(std::move(seed));
if(m == n) {
return q;
}
auto r = inv(q, m), dp = deriv(p);
for(; m < n; m *= 2) {
size_t k = std::min(2 * m, n), t = k - m;
auto Q = fft::spectrum<T>(q.a, 2 * m), R = fft::spectrum<T>(r.a, 2 * m);
typename poly_t<T>::Vector work(2 * m);
fft::spectrum<T>(dp.a | std::views::take(k - 1), 2 * m).multiply(Q, work, k - 1);
// p' q - q' vanishes below m-1. Cyclic wrap only touches <m-1.
fft::spectrum<T>(work | std::views::drop(m - 1) | std::views::take(t), 2 * m).multiply(R, work, t);
for(size_t i = 0; i < t; i++) {
work[i] *= small_inv<T>(m + i);
}
// d = (p - log(q)) / x^m. Keep it for the reciprocal correction.
auto d = typename poly_t<T>::Vector(begin(work), begin(work) + t);
fft::spectrum<T>(d, 2 * m).multiply(Q, work, t);
q.a.resize(k);
std::copy_n(begin(work), t, begin(q.a) + m);
if(k == n) {
break;
}
// (q*(1+x^m*d))^-1 = q^-1*(1-x^m*d) modulo x^(2m).
std::move(Q).multiply(R, work, k);
for(size_t i = 0; i < t; i++) {
work[m + i] += d[i];
}
fft::spectrum<T>(work | std::views::drop(m) | std::views::take(t), 2 * m).multiply(R, work, t);
r.a.resize(k);
for(size_t i = 0; i < t; i++) {
r.a[m + i] = -work[i];
}
}
q.normalize();
return q;
}
}
#pragma GCC pop_options
#line 1 "cp-algo/math/poly/series/pow.hpp"
#line 5 "cp-algo/math/poly/series/pow.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
namespace poly::impl {
// O(deg(p) * n), using p q' = k p' q.
template<typename T>
poly_t<T> pow_slow(poly_t<T> const& p, int64_t k, size_t n) {
typename poly_t<T>::Vector q(n);
q[0] = bpow(p[0], k);
auto a0inv = p[0].inv();
for(int i = 1; i < (int)n; i++) {
for(int j = 1; j <= std::min(p.deg(), i); j++) {
q[i] += p[j] * q[i - j] * (T(k) * T(j) - T(i - j));
}
q[i] *= small_inv<T>(i) * a0inv;
}
return q;
}
}
// Nonnegative integer power modulo x^n.
template<typename T>
poly_t<T> pow(poly_t<T> p, int64_t k, size_t n) {
assert(k >= 0);
if(n == 0) {return {};}
if(k == 0) {return T(1);}
p.mod_xk_inplace(n);
if(p.is_zero()) {return p;}
size_t shift = p.trailing_xk();
if(shift) {
if(uint64_t(k) > (n - 1) / shift) {return {};}
p.div_xk_inplace(shift);
return pow(std::move(p), k, n - shift * k).mul_xk(shift * k);
}
if(std::min(p.deg(), (int)n) <= magic) {
return poly::impl::pow_slow(p, k, n);
}
if(k <= magic) {
auto t = pow(p, k / 2, n);
t.mul_truncate(t, n);
if(k % 2) {t.mul_truncate(p, n);}
return t;
}
T c = p[0];
p /= c;
return bpow(c, k) * exp(log(std::move(p), n) * T(k), n);
}
}
#pragma GCC pop_options
#line 6 "cp-algo/math/poly/series.hpp"
#line 5 "cp-algo/math/poly/powmod.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
// Reduce modulo x^m - 1.
template<typename T>
poly_t<T> circular_closure(poly_t<T> p, size_t m) {
assert(m > 0);
for(size_t i = p.a.size(); i > m; --i) {p.a[i - 1 - m] += p.a[i - 1];}
p.mod_xk_inplace(m);
return p;
}
template<typename T>
poly_t<T> powmod_circular(poly_t<T> p, int64_t k, size_t m) {
assert(k >= 0 && m > 0);
p = circular_closure(std::move(p), m);
return bpow(p, k, poly_t<T>(1), [m](auto const& a, auto const& b) {
auto product = a;
product *= &a == &b ? product : b;
return circular_closure(std::move(product), m);
});
}
// Nonnegative integer power modulo a nonzero polynomial.
template<typename T>
poly_t<T> powmod(poly_t<T> p, int64_t k, poly_t<T> const& md) {
assert(k >= 0 && !md.is_zero());
int d = md.deg();
if(d == 0) {return {};}
if(md == poly_t<T>::xk(d)) {return pow(std::move(p), k, d);}
if(md == poly_t<T>::xk(d) - poly_t<T>(1)) {return powmod_circular(std::move(p), k, d);}
auto mdri = inv(md.reversed(), d + 1);
return bpow(p % md, k, poly_t<T>(1), [&](auto const& a, auto const& b) {
auto product = a;
product *= &a == &b ? product : b;
auto [q, r] = poly::impl::divmod_hint(std::move(product), md, mdri);
return r;
});
}
}
#pragma GCC pop_options
#line 1 "cp-algo/linalg/matrix.hpp"
#line 1 "cp-algo/linalg/vector.hpp"
#line 17 "cp-algo/linalg/vector.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::linalg {
template<typename base, class Alloc = big_alloc<base>>
struct vec: std::basic_string<base, std::char_traits<base>, Alloc> {
using Base = std::basic_string<base, std::char_traits<base>, Alloc>;
using Base::Base;
vec(Base const& t): Base(t) {}
vec(Base&& t): Base(std::move(t)) {}
vec(size_t n): Base(n, base()) {}
vec(auto &&r): Base(r.begin(), r.end()) {}
static vec ei(size_t n, size_t i) {
vec res(n);
res[i] = 1;
return res;
}
auto operator-() const {
return *this | std::views::transform([](auto x) { return -x; });
}
auto operator*(base t) const {
return *this | std::views::transform([t](auto x) { return x * t; });
}
vec& operator*=(base t) {
for(auto &it: *this) {
it *= t;
}
return *this;
}
virtual void add_scaled(vec const& b, base scale, size_t i = 0) {
if(scale != base(0)) {
for(; i < size(*this); i++) {
(*this)[i] += scale * b[i];
}
}
}
virtual vec const& normalize() {
return static_cast<vec&>(*this);
}
virtual base normalize(size_t i) {
return (*this)[i];
}
void read() {
for(auto &it: *this) {
std::cin >> it;
}
}
void print() const {
for(auto &it: *this) {
std::cout << it << ' ';
}
std::cout << '\n';
}
static vec random(size_t n) {
vec res(n);
std::ranges::generate(res, random::rng);
return res;
}
// Concatenate vectors
vec operator|(vec const& t) const {
return std::views::join(std::array{std::views::all(*this), std::views::all(t)});
}
// Generally, vec shouldn't be modified
// after its pivot index is set
std::pair<size_t, base> find_pivot() {
if(pivot == size_t(-1)) {
pivot = 0;
while(pivot < size(*this) && normalize(pivot) == base(0)) {
pivot++;
}
if(pivot < size(*this)) {
pivot_inv = base(1) / (*this)[pivot];
}
}
return {pivot, pivot_inv};
}
void reduce_by(vec& t) {
auto [pivot, pinv] = t.find_pivot();
if(pivot < size(*this)) {
add_scaled(t, -normalize(pivot) * pinv, pivot);
}
}
private:
size_t pivot = -1;
base pivot_inv;
};
template<math::modint_type base, class Alloc = big_alloc<base>>
struct modint_vec: vec<base, Alloc> {
using Base = vec<base, Alloc>;
using Base::Base;
modint_vec(Base const& t): Base(t) {}
modint_vec(Base&& t): Base(std::move(t)) {}
void add_scaled(Base const& b, base scale, size_t i = 0) override {
static_assert(base::bits >= 64, "Only wide modint types for linalg");
if(scale != base(0)) {
assert(Base::size() == b.size());
size_t n = size(*this);
u64x4 scaler = u64x4() + scale.getr();
bool aligned = is_aligned(&(*this)[0]) && is_aligned(&b[0]);
if(aligned) {
i -= i % 4;
}
bool reduce = ++counter == accumulation_period();
if(reduce) {
counter = 0;
for(size_t j = 0; j < i; j++) {
(*this)[j].pseudonormalize();
}
}
if(aligned) {
for(; i + 3 < n; i += 4) {
auto &ai = vector_cast<u64x4>((*this)[i]);
auto bi = vector_cast<u64x4 const>(b[i]);
#ifdef __AVX2__
ai += u64x4(_mm256_mul_epu32(__m256i(scaler), __m256i(bi)));
#else
ai += scaler * bi;
#endif
if(reduce) {
ai = shrink(ai);
}
}
}
for(; i < n; i++) {
(*this)[i].add_unsafe(b[i].getr_direct() * scale.getr());
if(reduce) {
(*this)[i].pseudonormalize();
}
}
}
}
Base const& normalize() override {
for(auto &it: *this) {
it.normalize();
}
return *this;
}
base normalize(size_t i) override {
return (*this)[i].normalize();
}
private:
template<typename, typename>
friend struct matrix;
static size_t accumulation_period() {
// Eight canonical products keep the accumulator below 16 * mod^2.
// Montgomery residues can be wider, so retain four updates there.
return base::remod() == typename base::UInt(base::mod()) && base::mod() < (1LL << 30) ? 8 : 4;
}
static u64x4 mul(u64x4 a, u64x4 b) {
#ifdef __AVX2__
return u64x4(_mm256_mul_epu32(__m256i(a), __m256i(b)));
#else
return a * b;
#endif
}
static u64x4 shrink(u64x4 a) {
auto b = a - (u64x4() + base::modmod8());
return a < b ? a : b;
}
// Two source contributions to two distinct rows; sources must be normalized.
static void add_scaled_pair(modint_vec& x, modint_vec& y, Base const& p, Base const& q, std::array<base, 4> c,
size_t first = 0) {
if(std::ranges::find(c, base(0)) != c.end()) {
x.add_scaled(p, c[0], first);
x.add_scaled(q, c[1], first);
y.add_scaled(p, c[2], first);
y.add_scaled(q, c[3], first);
return;
}
size_t n = x.size();
assert(y.size() == n && p.size() == n && q.size() == n && first <= n);
// Before first, sources are zero for elimination or destinations are unused for Pfaffians.
if(is_aligned(x.data()) && is_aligned(y.data()) && is_aligned(p.data()) && is_aligned(q.data())) {
first -= first % 4;
}
size_t period = accumulation_period();
auto prepare = [&](modint_vec& a) {
// A pair must not cross the accumulator's reduction boundary.
if(a.counter + 2 > period) {
for(auto &v: a) {
v.pseudonormalize();
}
a.counter = 0;
}
a.counter += 2;
if(a.counter != period) {
return false;
}
a.counter = 0;
for(size_t i = 0; i < first; i++) {
a[i].pseudonormalize();
}
return true;
};
bool nx = prepare(x), ny = prepare(y);
auto *__restrict__ dx = x.data();
auto *__restrict__ dy = y.data();
auto const* __restrict__ sp = p.data();
auto const* __restrict__ sq = q.data();
uint64_t xp = c[0].getr(), xq = c[1].getr(), yp = c[2].getr(), yq = c[3].getr();
u64x4 xp4 = u64x4() + xp, xq4 = u64x4() + xq;
u64x4 yp4 = u64x4() + yp, yq4 = u64x4() + yq;
size_t i = first;
for(; i + 4 <= n; i += 4) {
u64x4 vx, vy, vp, vq;
std::memcpy(&vx, dx + i, sizeof vx);
std::memcpy(&vy, dy + i, sizeof vy);
std::memcpy(&vp, sp + i, sizeof vp);
std::memcpy(&vq, sq + i, sizeof vq);
vx += mul(xp4, vp) + mul(xq4, vq);
vy += mul(yp4, vp) + mul(yq4, vq);
if(nx) {
vx = shrink(vx);
}
if(ny) {
vy = shrink(vy);
}
std::memcpy(dx + i, &vx, sizeof vx);
std::memcpy(dy + i, &vy, sizeof vy);
}
for(; i < n; i++) {
dx[i].add_unsafe(xp * sp[i].getr_direct() + xq * sq[i].getr_direct());
dy[i].add_unsafe(yp * sp[i].getr_direct() + yq * sq[i].getr_direct());
if(nx) {
dx[i].pseudonormalize();
}
if(ny) {
dy[i].pseudonormalize();
}
}
}
size_t counter = 0;
};
// Narrow rows keep canonical residues; small batches accumulate in wide registers.
template<typename base>
requires(base::bits <= 32)
struct modint_vec<base>: vec<base> {
using Base = vec<base>;
using Base::Base;
modint_vec(Base const& t): Base(t) {}
modint_vec(Base&& t): Base(std::move(t)) {}
void add_scaled(Base const& b, base scale, size_t first = 0) override {
if(scale == base(0)) {
return;
}
if(&b == this) {
Base::add_scaled(b, scale, first);
} else {
add_scaled_batch<1, 1>({this}, {&b}, {scale}, first);
}
}
private:
template<typename, typename>
friend struct matrix;
static constexpr size_t batch_size = 8;
static constexpr bool use_simd = [] {
if constexpr(requires {
std::integral_constant<uint32_t, base::mod()>{};
std::integral_constant<uint32_t, base::remod()>{};
}) {
return sizeof(base) == sizeof(uint32_t) && base::mod() > 1 && base::mod() % 2 &&
base::mod() < (1U << 30) && base::remod() == typename base::UInt(base::mod());
} else {
return false;
}
}();
static u64x4 mul(u64x4 a, u64x4 b) {
#ifdef __AVX2__
return u64x4(_mm256_mul_epu32(__m256i(a), __m256i(b)));
#else
return low32(a) * low32(b);
#endif
}
// Sources are normalized and do not alias the distinct destination rows.
template<size_t count, size_t rows>
static void add_scaled_batch(std::array<modint_vec*, rows> const& dst,
std::array<Base const*, count> const& src, std::array<base, rows * count> const& c,
size_t first = 0) {
static_assert(count <= batch_size && (rows == 1 || rows == 2));
if(std::ranges::all_of(c, [](base value) { return value == base(0); })) {
return;
}
if constexpr(use_simd) {
constexpr uint32_t mod = base::mod(), inv = math::inv2(uint32_t(-mod));
std::array<uint32_t, rows * count> scale;
for(size_t t = 0; t < rows * count; t++) {
scale[t] = uint32_t((uint64_t(c[t].getr()) << 32) % mod);
}
auto *__restrict__ dx = dst[0]->data();
auto *__restrict__ dy = rows == 2 ? dst[1]->data() : nullptr;
size_t n = dst[0]->size();
for(; first + 8 <= n; first += 8) {
u64x4 acc[rows][2]{};
#pragma GCC unroll 1
for(size_t t = 0; t < count; t++) {
u64x4 p;
std::memcpy(&p, src[t]->data() + first, sizeof p);
auto q = p >> 32;
for(size_t row = 0; row < rows; row++) {
auto v = u64x4(u32x8() + scale[row * count + t]);
acc[row][0] += mul(p, v);
acc[row][1] += mul(q, v);
}
}
for(size_t row = 0; row < rows; row++) {
// At most eight products: reduction gives <3*mod, then add the old residue.
for(auto &v: acc[row]) {
v = montgomery_reduce(v, mod, inv);
}
auto *out = (row ? dy : dx) + first;
u32x8 old;
std::memcpy(&old, out, sizeof old);
auto z = old + u32x8(acc[row][0] | (acc[row][1] << 32));
z = z < z - 2 * mod ? z : z - 2 * mod;
z = z < z - mod ? z : z - mod;
std::memcpy(out, &z, sizeof z);
}
}
}
for(size_t t = 0; t < count; t++) {
for(size_t row = 0; row < rows; row++) {
dst[row]->Base::add_scaled(*src[t], c[row * count + t], first);
}
}
}
static void add_scaled_pair(modint_vec& x, modint_vec& y, Base const& p, Base const& q, std::array<base, 4> c,
size_t first = 0) {
add_scaled_batch<2, 2>({&x, &y}, {&p, &q}, c, first);
}
};
}
#pragma GCC pop_options
#line 1 "cp-algo/linalg/strassen.hpp"
#line 4 "cp-algo/linalg/strassen.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::linalg::impl {
template<class row>
constexpr bool use_strassen = false;
template<auto mod>
constexpr bool use_strassen<modint_vec<math::modint<mod>>> = mod > 1 && mod % 2 && mod < (1LL << 30);
// Internal packed storage; the public matrix keeps its usual row representation.
template<uint32_t mod>
struct strassen_product {
struct view {
uint32_t* data;
size_t stride;
uint32_t* operator[](size_t i) const { return data + i * stride; }
};
static u64x4 mul(u64x4 a, u64x4 b) {
#ifdef __AVX2__
return u64x4(_mm256_mul_epu32(__m256i(a), __m256i(b)));
#else
return low32(a) * low32(b);
#endif
}
static u64x4 shrink(u64x4 x) {
// x < 4*mod*2^32; reduce the high word modulo 2*mod.
auto words = u32x8(x);
auto bound = u32x8(u64x4() + (uint64_t(2) * mod << 32));
#ifdef __AVX2__
return u64x4(_mm256_min_epu32(__m256i(words), __m256i(words - bound)));
#else
return u64x4(words < words - bound ? words : words - bound);
#endif
}
template<size_t dim = 0>
[[gnu::noinline]] static void leaf(view a, view b, view c, size_t n, size_t m, size_t k) {
if constexpr(dim) {
n = m = k = dim;
a.stride = b.stride = c.stride = dim;
}
for(size_t i = 0; i < n; i += 4) {
for(size_t j = 0; j < k; j += 8) {
u64x4 acc[4][2]{};
for(size_t first = 0; first < m; first += 8) {
#pragma GCC unroll 1
for(size_t z = first; z < first + 8; z++) {
u64x4 x;
std::memcpy(&x, b[z] + j, sizeof x);
u64x4 scales[4];
for(size_t t = 0; t < 4; t++) {
scales[t] = u64x4(u32x8() + a[i + t][z]);
}
for(size_t t = 0; t < 4; t++) {
acc[t][0] += mul(scales[t], x);
}
x >>= 32;
for(size_t t = 0; t < 4; t++) {
acc[t][1] += mul(scales[t], x);
}
}
for(auto &row: acc) {
for(auto &x: row) {
x = shrink(x);
}
}
}
for(size_t t = 0; t < 4; t++) {
constexpr uint32_t inv = math::inv2(uint32_t(-mod));
// x < 2*mod*2^32, so Montgomery reduction yields a value below 3*mod.
for(auto &x: acc[t]) {
x = montgomery_reduce(x, mod, inv);
}
u32x8 out = u32x8(acc[t][0] | (acc[t][1] << 32));
out = out < out - mod ? out : out - mod;
out = out < out - mod ? out : out - mod;
std::memcpy(c[i + t] + j, &out, sizeof out);
}
}
}
}
// c = a +/- b; c may alias a.
template<bool subtract = false>
static void combine(uint32_t* a, uint32_t* b, uint32_t* c, size_t n, size_t m) {
for(size_t j = 0; j < n * m; j += 8) {
u32x8 x, y;
std::memcpy(&x, a + j, sizeof x);
std::memcpy(&y, b + j, sizeof y);
u32x8 z = subtract ? x + mod - y : x + y;
z = z < z - mod ? z : z - mod;
std::memcpy(c + j, &z, sizeof z);
}
}
static bool can_split(size_t n, size_t m, size_t k) {
return std::min({n, m, k}) > 64 && n % 16 == 0 && m % 16 == 0 && k % 16 == 0;
}
[[gnu::noinline]] static void multiply(uint32_t* a, uint32_t* b, uint32_t* c, size_t n, size_t m, size_t k,
uint32_t* work) {
// Every leaf dimension must remain a multiple of eight.
if(!can_split(n, m, k)) {
if(n == 64 && m == 64 && k == 64) {
leaf<64>({a, m}, {b, k}, {c, k}, n, m, k);
} else {
leaf({a, m}, {b, k}, {c, k}, n, m, k);
}
return;
}
n /= 2;
m /= 2;
k /= 2;
auto s = work, t = s + n * m, p = t + m * k;
work += n * m + m * k + n * k;
auto a00 = a, a01 = a + n * m, a10 = a + 2 * n * m, a11 = a + 3 * n * m;
auto b00 = b, b01 = b + m * k, b10 = b + 2 * m * k, b11 = b + 3 * m * k;
auto c00 = c, c01 = c + n * k, c10 = c + 2 * n * k, c11 = c + 3 * n * k;
// Winograd's schedule uses seven products and fifteen additions.
multiply(a00, b00, c11, n, m, k, work); // P1
multiply(a01, b10, c00, n, m, k, work); // P2
combine(c00, c11, c00, n, k); // C00 = P1 + P2
combine(a10, a11, s, n, m);
combine<true>(b01, b00, t, m, k); // S1, T1
multiply(s, t, c01, n, m, k, work); // P5
combine<true>(s, a00, s, n, m);
combine<true>(b11, t, t, m, k); // S2, T2
multiply(s, t, c10, n, m, k, work); // P6
combine(c11, c10, c10, n, k); // U2 = P1 + P6
combine<true>(a01, s, s, n, m); // S4
multiply(s, b11, p, n, m, k, work); // P3
combine(c10, c01, c11, n, k); // U4 = U2 + P5
combine(c11, p, c01, n, k); // C01 = U4 + P3
combine<true>(t, b10, t, m, k); // T4
multiply(a11, t, p, n, m, k, work); // P4
combine<true>(c10, p, c10, n, k);
combine<true>(a00, a10, s, n, m);
combine<true>(b11, b01, t, m, k); // S3, T3
multiply(s, t, p, n, m, k, work); // P7
combine(c10, p, c10, n, k);
combine(c11, p, c11, n, k); // C10, C11
}
static u32x8 encode(u32x8 x) {
// Scale by 2^32 with a fixed reciprocal; its quotient is off by at most one.
constexpr uint32_t scale = (uint64_t(1) << 32) % mod;
constexpr uint32_t quotient = (uint64_t(scale) << 32) / mod;
auto packed = u64x4(x), q = u64x4() + quotient;
auto lo = mul(packed, q) >> 32, hi = mul(packed >> 32, q);
auto approx = u32x8(lo | (hi & (~uint64_t(0) << 32)));
auto out = x * scale - approx * mod;
return out < out - mod ? out : out - mod;
}
// Copy between matrix rows and contiguous recursive quadrants.
template<bool unpack = false, int add = 0, class matrix>
static void copy(matrix& a, uint32_t* ptr, size_t n, size_t m, size_t depth, size_t row = 0, size_t col = 0) {
if(depth) {
n /= 2;
m /= 2;
for(size_t q = 0; q < 4; q++) {
copy<unpack, add>(a, ptr + q * n * m, n, m, depth - 1, row + q / 2 * n, col + q % 2 * m);
}
return;
}
if(col >= a.m()) {
return;
}
size_t width = std::min(m, a.m() - col);
for(size_t i = row; i < std::min(row + n, a.n()); i++) {
auto data = a[i].data() + col;
auto packed = ptr + (i - row) * m;
for(size_t j = 0; j < width; j++) {
if constexpr(unpack && add != 0) {
typename matrix::base value;
value.setr(packed[j]);
if constexpr(add == 1) {
data[j] += value;
} else {
data[j] -= value;
}
} else if constexpr(unpack) {
data[j].setr(packed[j]);
} else {
packed[j] = uint32_t(data[j].getr());
}
}
}
}
template<int add = 0, class matrix>
static matrix product(matrix const& a, matrix const& b, matrix res = {}) {
static_assert(add >= -1 && add <= 1);
auto pad = [](size_t x) {
return (x + 31) / 32 * 32;
};
size_t n = pad(a.n()), m = pad(a.m()), k = pad(b.m());
// Each recursion level needs a quarter as much scratch; children reuse it.
size_t entries = n * m + m * k + n * k;
// Small products avoid repeated mmap/madvise setup for temporary storage.
std::vector<uint32_t> small;
big_vector<uint32_t> large;
size_t count = entries + entries / 3;
auto ap = count < (1 << 21) ? (small.resize(count), small.data()) : (large.resize(count), large.data());
auto bp = ap + n * m, cp = bp + m * k;
auto scratch = cp + n * k;
// A, B, and C must use the same depth, including rectangular products.
size_t depth = 0;
for(size_t x = n, y = m, z = k; can_split(x, y, z); x /= 2, y /= 2, z /= 2) {
depth++;
}
copy(a, ap, n, m, depth);
copy(b, bp, m, k, depth);
// Only A is scaled; each leaf's Montgomery reduction removes the factor.
for(size_t i = 0; i < n * m; i += 8) {
u32x8 x;
std::memcpy(&x, ap + i, sizeof x);
x = encode(x);
std::memcpy(ap + i, &x, sizeof x);
}
multiply(ap, bp, cp, n, m, k, scratch);
if constexpr(add == 0) {
res = matrix(a.n(), b.m());
}
assert(res.n() == a.n() && res.m() == b.m());
copy<true, add>(res, cp, n, k, depth);
return res;
}
};
}
#pragma GCC pop_options
#line 12 "cp-algo/linalg/matrix.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::linalg {
enum gauss_mode { normal, reverse };
template<typename base_t,
class _vec_t = std::conditional_t<math::modint_type<base_t>, modint_vec<base_t>, vec<base_t>>>
struct matrix: big_vector<_vec_t> {
using vec_t = _vec_t;
using base = base_t;
using Base = big_vector<vec_t>;
using Base::Base;
matrix(size_t n): matrix(n, n) {}
matrix(size_t n, size_t m) {
Base::reserve(n);
for(size_t i = 0; i < n; i++) {
Base::emplace_back(m);
}
}
matrix(Base const& t): Base(t) {}
matrix(Base&& t): Base(std::move(t)) {}
template<std::ranges::input_range R>
matrix(R&& r): Base(r.begin(), r.end()) {}
size_t n() const { return size(*this); }
size_t m() const { return n() ? size(row(0)) : 0; }
void resize(size_t n, size_t m) {
Base::resize(n);
for(auto &it: *this) {
it.resize(m);
}
}
auto &row(this auto&& self, size_t i) { return self[i]; }
auto elements(this auto&& self) { return self | std::views::join; }
matrix operator-() const {
return *this | std::views::transform([](auto const& x) { return vec_t(-x); });
}
matrix& operator+=(matrix const& t) {
for(auto [a, b]: std::views::zip(*this, t)) {
for(auto [x, y]: std::views::zip(a, b)) {
x += y;
}
}
return *this;
}
matrix& operator-=(matrix const& t) {
for(auto [a, b]: std::views::zip(*this, t)) {
for(auto [x, y]: std::views::zip(a, b)) {
x -= y;
}
}
return *this;
}
matrix operator+(matrix const& t) const { return matrix(*this) += t; }
matrix operator-(matrix const& t) const { return matrix(*this) -= t; }
matrix& operator*=(base t) {
for(auto &it: *this) {
it *= t;
}
return *this;
}
matrix operator*(base t) const { return matrix(*this) *= t; }
matrix& operator/=(base t) { return *this *= base(1) / t; }
matrix operator/(base t) const { return matrix(*this) /= t; }
// Make sure the result is matrix, not Base
matrix& operator*=(matrix const& t) { return *this = *this * t; }
void read_transposed() {
for(size_t j = 0; j < m(); j++) {
for(size_t i = 0; i < n(); i++) {
std::cin >> (*this)[i][j];
}
}
}
void read() {
for(auto &it: *this) {
it.read();
}
}
void print() const {
for(auto const& it: *this) {
it.print();
}
}
static matrix block_diagonal(big_vector<matrix> const& blocks) {
size_t n = 0;
for(auto &it: blocks) {
assert(it.n() == it.m());
n += it.n();
}
matrix res(n);
n = 0;
for(auto &it: blocks) {
for(size_t i = 0; i < it.n(); i++) {
std::ranges::copy(it[i], begin(res[n + i]) + n);
}
n += it.n();
}
return res;
}
static matrix random(size_t n, size_t m) {
matrix res(n, m);
std::ranges::generate(res, [m] { return vec_t::random(m); });
return res;
}
static matrix random(size_t n) {
return random(n, n);
}
static matrix eye(size_t n) {
matrix res(n);
for(size_t i = 0; i < n; i++) {
res[i][i] = 1;
}
return res;
}
// Concatenate matrices
matrix operator|(matrix const& b) const {
assert(n() == b.n());
matrix res(n(), 0);
for(size_t i = 0; i < n(); i++) {
res[i] = row(i) | b[i];
}
return res;
}
void assign_submatrix(auto viewx, auto viewy, matrix const& t) {
for(auto [a, b]: std::views::zip(*this | viewx, t)) {
std::ranges::copy(b, begin(a | viewy));
}
}
auto submatrix(auto viewx, auto viewy) const {
return *this | viewx | std::views::transform([viewy](auto const& y) {
return y | viewy;
});
}
matrix T() const {
matrix res(m(), n());
constexpr size_t block = 128;
for(size_t first = 0; first < m(); first += block) {
size_t last = std::min(first + block, m());
for(size_t i = 0; i < n(); i++) {
auto const& src = row(i);
for(size_t j = first; j < last; j++) {
res[j][i] = src[j];
}
}
}
return res;
}
matrix operator*(matrix const& b) const {
assert(m() == b.n());
if constexpr(impl::use_strassen<vec_t>) {
if(std::min({n(), m(), b.m()}) >= (base::bits <= 32 ? 64 : 512)) {
return impl::strassen_product<base::mod()>::product(*this, b);
}
}
matrix res(n(), b.m());
if constexpr(requires { requires vec_t::use_simd; }) {
if(n() == 1) {
res[0] = b.apply(row(0));
return res;
}
}
constexpr size_t block = 32;
for(size_t first = 0; first < m(); first += block) {
size_t last = std::min(first + block, m());
size_t i = 0;
for(; i + 1 < n(); i += 2) {
size_t j = first;
if constexpr(requires { requires vec_t::use_simd; }) {
constexpr size_t batch = vec_t::batch_size;
if(b.m() >= 8) {
for(; j + batch <= last; j += batch) {
std::array<typename vec_t::Base const*, batch> src;
std::array<base, 2 * batch> c;
for(size_t t = 0; t < batch; t++) {
src[t] = &b[j + t];
c[t] = row(i)[j + t];
c[batch + t] = row(i + 1)[j + t];
}
vec_t::template add_scaled_batch<batch, 2>({&res[i], &res[i + 1]}, src, c);
}
}
}
for(; j + 1 < last; j += 2) {
add_scaled_pair(res[i], res[i + 1], b[j], b[j + 1],
{row(i)[j], row(i)[j + 1], row(i + 1)[j], row(i + 1)[j + 1]});
}
if(j < last) {
res[i].add_scaled(b[j], row(i)[j]);
res[i + 1].add_scaled(b[j], row(i + 1)[j]);
}
}
for(; i < n(); i++) {
for(size_t j = first; j < last; j++) {
res[i].add_scaled(b[j], row(i)[j]);
}
}
}
res.normalize();
return res;
}
vec_t apply(vec_t const& x) const {
assert(x.size() == n());
vec_t res(m());
size_t i = 0;
if constexpr(requires { requires vec_t::use_simd; }) {
constexpr size_t batch = vec_t::batch_size;
for(; i + batch <= n(); i += batch) {
std::array<typename vec_t::Base const*, batch> src;
std::array<base, batch> c;
for(size_t t = 0; t < batch; t++) {
src[t] = &row(i + t);
c[t] = x[i + t];
}
vec_t::template add_scaled_batch<batch, 1>({&res}, src, c);
}
}
for(; i < n(); i++) {
res.add_scaled(row(i), x[i]);
}
res.normalize();
return res;
}
matrix pow(uint64_t k) const {
assert(n() == m());
return bpow<3>(*this, k, eye(n()));
}
matrix& normalize() {
for(auto &it: *this) {
it.normalize();
}
return *this;
}
template<gauss_mode mode = normal>
void eliminate(size_t i, size_t k) {
auto kinv = base(1) / row(i).normalize()[k];
for(size_t j = (mode == normal) * i; j < n(); j++) {
if(j != i) {
row(j).add_scaled(row(i), -row(j).normalize(k) * kinv);
}
}
}
template<gauss_mode mode = normal>
void eliminate(size_t i) {
row(i).normalize();
for(size_t j = (mode == normal) * i; j < n(); j++) {
if(j != i) {
row(j).reduce_by(row(i));
}
}
}
template<gauss_mode mode = normal>
matrix& gauss() {
return gauss_to<mode>(m());
}
private:
// Restrict pivot columns, while updating every column of each row.
template<gauss_mode mode>
matrix& gauss_to(size_t lim) {
constexpr size_t block = 32;
for(size_t first = 0; first < n(); first += block) {
size_t last = std::min(first + block, n());
// Reduce the pivot block before applying it to the other rows.
for(size_t i = first; i < last; i++) {
row(i).normalize();
if(row(i).find_pivot().first >= lim) {
continue;
}
for(size_t j = mode == normal ? i + 1 : first; j < last; j++) {
if(j != i) {
row(j).reduce_by(row(i));
}
}
}
if constexpr(mode == reverse) {
// Later pivots may have changed earlier rows in this block.
for(size_t i = first; i < last; i++) {
row(i).normalize();
}
}
for(size_t j = mode == normal ? last : 0; j < n(); j++) {
if(j >= first && j < last) {
continue;
}
bool pair = j + 1 < n() && j + 1 != first;
size_t i = first;
if constexpr(requires { requires vec_t::use_simd; }) {
constexpr size_t batch = vec_t::batch_size;
if(pair) {
for(; i + batch <= last; i += batch) {
reduce_batch<mode, batch>(j, i, lim);
}
}
}
if(pair) {
for(; i + 1 < last; i += 2) {
reduce_pair<mode>(j, i, lim);
}
}
for(; i < last; i++) {
if(row(i).find_pivot().first >= lim) {
continue;
}
row(j).reduce_by(row(i));
if(pair) {
row(j + 1).reduce_by(row(i));
}
}
j += pair;
}
}
// Forward pivots are normalized before use and never modified afterward.
return mode == reverse ? normalize() : *this;
}
public:
template<gauss_mode mode = normal>
auto echelonize(size_t lim) {
return gauss<mode>().sort_classify(lim);
}
template<gauss_mode mode = normal>
auto echelonize() {
return echelonize<mode>(m());
}
size_t rank() const {
auto A = n() > m() ? T() : *this;
A.gauss();
return std::ranges::count_if(A, [&](auto &row) {
return row.find_pivot().first < A.m();
});
}
base det() const {
assert(n() == m());
matrix b = *this;
b.gauss();
base res = 1;
for(size_t i = 0; i < n(); i++) {
for(size_t p = b[i].find_pivot().first; p != i; p = b[i].find_pivot().first) {
if(p == n()) {
return 0;
}
std::swap(b[i], b[p]);
res = -res;
}
res *= b[i][i];
}
return res;
}
// Pfaffian of an alternating matrix over a field.
base pfaffian() const {
assert(n() == m() && n() % 2 == 0);
matrix b = *this;
base res = 1;
for(size_t k = 0; k < n(); k += 2) {
size_t q = k + 1, p = q;
while(p < n() && b[k].normalize(p) == base(0)) {
p++;
}
if(p == n()) {
return base(0);
}
if(p != q) {
res = -res;
// Normalize values before moving them between row accumulators.
b[q].normalize();
b[p].normalize();
// Symmetric permutation, storing only the upper triangle.
for(size_t i = 0; i < q; i++) {
std::swap(b[i][q], b[i][p]);
}
for(size_t i = q + 1; i < p; i++) {
std::swap(b[q][i], b[i][p]);
b[q][i] = -b[q].normalize(i);
b[i][p] = -b[i].normalize(p);
}
for(size_t i = p + 1; i < n(); i++) {
std::swap(b[q][i], b[p][i]);
}
b[q][p] = -b[q].normalize(p);
}
b[k].normalize();
b[q].normalize();
res *= b[k][q];
base inv = base(1) / b[k][q];
for(size_t i = k + 2; i + 1 < n(); i += 2) {
add_scaled_pair(b[i], b[i + 1], b[k], b[q],
{b[q][i] * inv, -b[k][i] * inv, b[q][i + 1] * inv, -b[k][i + 1] * inv}, i + 1);
}
}
return res;
}
std::pair<base, matrix> inv() const {
assert(n() == m());
matrix b(n(), 2 * n());
for(size_t i = 0; i < n(); i++) {
std::ranges::copy(row(i), begin(b[i]));
b[i][n() + i] = 1;
}
// Pivots in the appended identity cannot make A invertible.
auto [pivots, free] = b.template gauss_to<normal>(n()).sort_classify(n());
if(size(pivots) < n()) {
return {0, {}};
}
base det = 1;
for(size_t i = 0; i < n(); i++) {
det *= b[i][i];
}
matrix res = b.submatrix(std::views::all, std::views::drop(n()));
back_substitute(b, pivots, res);
return {det, std::move(res)};
}
auto kernel() const {
return kernel_of(*this);
}
// [solution, basis], transposed
std::optional<std::array<matrix, 2>> solve(matrix const& t) const {
matrix sols = kernel_of(*this | t);
if(sols.n() < t.m() ||
matrix(sols.submatrix(std::views::drop(sols.n() - t.m()), std::views::drop(m()))) != -eye(t.m())) {
return std::nullopt;
} else {
for(auto &row: sols) {
row.resize(m());
}
auto first = begin(sols) + (sols.n() - t.m());
matrix sol(std::make_move_iterator(first), std::make_move_iterator(end(sols)));
sols.erase(first, end(sols));
return std::array{std::move(sol), std::move(sols)};
}
}
// To be called after a gaussian elimination run
// Sorts rows by pivots and classifies
// variables into pivots and free
auto sort_classify(size_t lim) {
size_t rk = 0;
big_vector<size_t> free, pivots;
for(size_t j = 0; j < lim; j++) {
for(size_t i = rk + 1; i < n() && row(rk)[j] == base(0); i++) {
if(row(i)[j] != base(0)) {
std::swap(row(i), row(rk));
row(rk) = -row(rk);
}
}
if(rk < n() && row(rk)[j] != base(0)) {
pivots.push_back(j);
rk++;
} else {
free.push_back(j);
}
}
return std::array{std::move(pivots), std::move(free)};
}
private:
// Own the working matrix so augmented temporaries do not need a copy.
static matrix kernel_of(matrix A) {
auto [pivots, free] = A.echelonize();
matrix rhs(size(pivots), size(free));
for(size_t i = 0; i < size(pivots); i++) {
for(size_t j = 0; j < size(free); j++) {
rhs[i][j] = A[i][free[j]];
}
}
back_substitute(A, pivots, rhs);
matrix sols(size(free), A.m());
for(size_t j = 0; j < size(pivots); j++) {
for(size_t i = 0; i < size(free); i++) {
sols[i][pivots[j]] = rhs[j][i];
}
}
for(size_t i = 0; i < size(free); i++) {
sols[i][free[i]] = -1;
}
return sols;
}
// Solve an upper-echelon system for the supplied right-hand sides.
static void back_substitute(matrix const& A, auto const& pivots, matrix& b) {
for(size_t i = size(pivots); i-- > 0;) {
b[i].normalize();
b[i] *= base(1) / A[i][pivots[i]];
for(size_t j = 0; j < i; j++) {
b[j].add_scaled(b[i], -A[j][pivots[i]]);
}
}
}
static void add_scaled_pair(vec_t& x, vec_t& y, vec_t const& p, vec_t const& q, std::array<base, 4> c,
size_t first = 0) {
if constexpr(requires { vec_t::add_scaled_pair(x, y, p, q, c, first); }) {
vec_t::add_scaled_pair(x, y, p, q, c, first);
} else {
x.add_scaled(p, c[0], first);
x.add_scaled(q, c[1], first);
y.add_scaled(p, c[2], first);
y.add_scaled(q, c[3], first);
}
}
// Determine the sequential pivot coefficients before updating the full rows.
template<gauss_mode mode, size_t count>
void reduce_batch(size_t dst, size_t src, size_t lim) {
static_assert(count <= 8);
std::array<typename vec_t::Base const*, count> sources;
std::array<size_t, count> pivots;
std::array<base, count> inverses;
size_t first = lim;
for(size_t t = 0; t < count; t++) {
sources[t] = &row(src + t);
auto [p, inv] = row(src + t).find_pivot();
pivots[t] = p;
inverses[t] = p < lim ? inv : base(0);
first = std::min(first, p);
}
if(first == lim) {
return;
}
std::array<base, 2 * count> c{};
for(size_t r = 0; r < 2; r++) {
for(size_t t = 0; t < count; t++) {
if(pivots[t] >= lim) {
continue;
}
base value = row(dst + r).normalize(pivots[t]);
if constexpr(mode == normal) {
// Fewer than eight canonical products fit in a 64-bit accumulator.
uint64_t sum = value.getr();
for(size_t h = 0; h < t; h++) {
sum += uint64_t(c[r * count + h].getr()) * (*sources[h])[pivots[t]].getr();
}
value.setr(sum % base::mod());
}
c[r * count + t] = -value * inverses[t];
}
}
vec_t::template add_scaled_batch<count, 2>({&row(dst), &row(dst + 1)}, sources, c, first);
}
// Fuse two sequential reductions, accounting for the first one's effect
// on the second pivot before updating either destination row.
template<gauss_mode mode>
void reduce_pair(size_t dst, size_t src, size_t lim) {
auto &p = row(src), &q = row(src + 1);
auto [u, pu] = p.find_pivot();
auto [v, qv] = q.find_pivot();
if(u >= lim || v >= lim) {
for(size_t j = dst; j < dst + 2; j++) {
if(u < lim) {
row(j).reduce_by(p);
}
if(v < lim) {
row(j).reduce_by(q);
}
}
return;
}
// Canonical reduction coefficients fit in a 64-bit product.
auto mul = [](base x, base y) {
if constexpr(impl::use_strassen<vec_t>) {
x.setr(uint64_t(x.getr()) * y.getr() % base::mod());
return x;
} else {
return x * y;
}
};
auto scales = [&](vec_t& a) {
base s = mul(-a.normalize(u), pu);
base t = -a.normalize(v);
// Reverse elimination has already cleared p[v] within the pivot block.
if constexpr(mode == normal) {
t -= mul(s, p[v]);
}
t = mul(t, qv);
return std::array{s, t};
};
auto a = scales(row(dst)), b = scales(row(dst + 1));
add_scaled_pair(row(dst), row(dst + 1), p, q, {a[0], a[1], b[0], b[1]}, std::min(u, v));
}
};
template<typename base_t>
auto operator*(base_t t, matrix<base_t> const& A) {
return A * t;
}
}
#pragma GCC pop_options
#line 8 "cp-algo/linalg/frobenius.hpp"
namespace cp_algo::linalg {
enum frobenius_mode { blocks, full };
template<frobenius_mode mode = blocks>
auto frobenius_form(auto const& A) {
using matrix = std::decay_t<decltype(A)>;
using vec_t = matrix::vec_t;
using base = matrix::base;
using polyn = math::poly_t<base>;
assert(A.n() == A.m());
size_t n = A.n();
big_vector<polyn> charps;
big_vector<vec_t> basis, basis_init;
while(size(basis) < n) {
size_t start = size(basis);
// Record y_i = x_i + sum_j reductions[i][j] * y_j modulo earlier blocks.
big_vector<typename polyn::Vector> reductions;
auto generate_block = [&](auto x) {
while(true) {
vec_t y = x;
typename polyn::Vector coefficients;
if constexpr(mode == full) {
y.reserve(2 * n + 1);
} else {
coefficients.resize(size(basis) - start);
}
for(size_t i = 0; i < size(basis); i++) {
auto &it = basis[i];
if constexpr(mode == full) {
y.resize(it.size());
y.reduce_by(it);
} else {
auto [p, inv] = it.find_pivot();
base scale = -y.normalize(p) * inv;
y.add_scaled(it, scale, p);
if(i >= start) {
coefficients[i - start] = scale;
}
}
}
if constexpr(mode == full) {
y.push_back(1);
}
y.normalize();
if(y.find_pivot().first >= n) {
if constexpr(mode == full) {
return polyn(typename polyn::Vector(begin(y) + n, end(y)));
} else {
// Expand the dependency through the triangular reduction history.
coefficients.push_back(1);
for(size_t i = size(reductions); i-- > 0;) {
for(size_t j = 0; j < i; j++) {
coefficients[j] += coefficients[i] * reductions[i][j];
}
}
return polyn(std::move(coefficients));
}
} else {
if constexpr(mode == blocks) {
reductions.push_back(std::move(coefficients));
}
basis.push_back(std::move(y));
if constexpr(mode == full) {
basis_init.push_back(std::move(x));
x = A.apply(basis_init.back());
} else {
x = A.apply(x);
}
}
}
};
auto x = vec_t::random(n);
// Choose a random representative already reduced against earlier blocks.
for(auto &row: basis) {
x[row.find_pivot().first] = 0;
}
auto full_rec = generate_block(std::move(x));
// Extra trimming to make it block-diagonal (expensive)
if constexpr(mode == full) {
if(full_rec.mod_xk(start) != polyn()) {
auto charp = full_rec.div_xk(start);
auto x = basis_init[start];
auto shift = full_rec / charp;
for(int j = 0; j < shift.deg(); j++) {
x.add_scaled(basis_init[j], shift[j]);
}
basis.resize(start);
basis_init.resize(start);
full_rec = generate_block(x.normalize());
}
}
charps.push_back(full_rec.div_xk(mode == full ? start : 0));
}
// Find transform matrices while we're at it...
if constexpr(mode == full) {
for(auto &row: basis) {
row.resize(2 * n + 1);
}
for(size_t i = 0; i < n; i++) {
for(size_t j = i + 1; j < n; j++) {
basis[i].reduce_by(basis[j]);
}
basis[i].normalize();
}
auto T = matrix(std::move(basis_init));
auto Tinv = matrix(std::move(basis));
std::ignore = Tinv.sort_classify(n);
for(size_t i = 0; i < n; i++) {
Tinv[i] = vec_t(Tinv[i] | std::views::drop(n) | std::views::take(n)) * (base(1) / Tinv[i][i]);
}
return std::tuple{std::move(T), std::move(Tinv), std::move(charps)};
} else {
return charps;
}
}
template<typename base>
auto with_frobenius(matrix<base> const& A, auto &&callback) {
auto [T, Tinv, charps] = frobenius_form<full>(A);
// Apply each diagonal block to its rows of T without forming S.
matrix<base> ST(A.n());
size_t start = 0;
for(auto const& charp: charps) {
size_t d = charp.deg();
auto xk = callback(charp);
for(size_t i = 0; i < d; i++) {
for(size_t j = 0; j < xk.a.size(); j++) {
ST[start + i].add_scaled(T[start + j], xk[j]);
}
if(i + 1 < d) {
xk = xk.mul_xk(1) % charp;
}
}
start += d;
}
return Tinv * ST.normalize();
}
template<typename base>
auto frobenius_pow(matrix<base> const& A, uint64_t k) {
if(k <= 2) {
return A.pow(k);
}
return with_frobenius(A, [k](auto const& charp) {
if(charp.deg() == 1) {
return math::poly_t<base>(bpow(-charp[0] / charp[1], k));
}
return math::powmod(math::poly_t<base>::xk(1), k, charp);
});
}
};
#line 9 "verify/linalg/pow_fast.test.cpp"
using namespace std;
using namespace cp_algo::math;
using namespace cp_algo::linalg;
const int64_t mod = 998244353;
using base = modint<mod>;
using polyn = poly_t<base>;
void solve() {
size_t n;
uint64_t k;
cin >> n >> k;
matrix<base> A(n);
A.read();
frobenius_pow(A, k).print();
}
signed main() {
//freopen("input.txt", "r", stdin);
ios::sync_with_stdio(0);
cin.tie(0);
int t = 1;
//cin >> t;
while(t--) {
solve();
}
}
Test cases
| Env | Name | Status | Elapsed | Memory |
|---|---|---|---|---|
| g++ | example_00 |
AC |
3 ms | 4 MB |
| g++ | example_01 |
AC |
2 ms | 4 MB |
| g++ | example_02 |
AC |
2 ms | 4 MB |
| g++ | frobenius_hack_00 |
AC |
2 ms | 4 MB |
| g++ | lowrank_max_random_00 |
AC |
14 ms | 7 MB |
| g++ | lowrank_max_random_01 |
AC |
14 ms | 7 MB |
| g++ | max_random_00 |
AC |
14 ms | 6 MB |
| g++ | max_random_01 |
AC |
14 ms | 6 MB |
| g++ | max_random_02 |
AC |
14 ms | 7 MB |
| g++ | max_random_03 |
AC |
15 ms | 7 MB |
| g++ | max_random_worst_00 |
AC |
15 ms | 7 MB |
| g++ | max_random_worst_01 |
AC |
14 ms | 7 MB |
| g++ | max_random_worst_02 |
AC |
14 ms | 7 MB |
| g++ | max_random_worst_03 |
AC |
14 ms | 6 MB |
| g++ | nontrivial_frobenius_form_00 |
AC |
10 ms | 6 MB |
| g++ | nontrivial_frobenius_form_01 |
AC |
14 ms | 7 MB |
| g++ | nontrivial_frobenius_form_02 |
AC |
16 ms | 7 MB |
| g++ | nontrivial_frobenius_form_03 |
AC |
15 ms | 7 MB |
| g++ | nontrivial_frobenius_form_04 |
AC |
15 ms | 7 MB |
| g++ | nontrivial_frobenius_form_05 |
AC |
13 ms | 7 MB |
| g++ | nontrivial_frobenius_form_06 |
AC |
16 ms | 7 MB |
| g++ | nontrivial_frobenius_form_07 |
AC |
17 ms | 7 MB |
| g++ | nontrivial_frobenius_form_08 |
AC |
17 ms | 7 MB |
| g++ | nontrivial_frobenius_form_09 |
AC |
15 ms | 7 MB |
| g++ | perm_max_random_00 |
AC |
11 ms | 6 MB |
| g++ | perm_max_random_01 |
AC |
11 ms | 6 MB |
| g++ | random_00 |
AC |
11 ms | 6 MB |
| g++ | random_01 |
AC |
13 ms | 7 MB |
| g++ | random_02 |
AC |
4 ms | 5 MB |
| g++ | signed_overflow_00 |
AC |
3 ms | 4 MB |
| g++ | small_00 |
AC |
2 ms | 4 MB |
| g++ | small_01 |
AC |
2 ms | 4 MB |
| g++ | small_02 |
AC |
2 ms | 4 MB |
| g++ | small_03 |
AC |
2 ms | 4 MB |
| g++ | small_04 |
AC |
2 ms | 4 MB |
| g++ | small_05 |
AC |
2 ms | 4 MB |
| g++ | small_06 |
AC |
2 ms | 4 MB |
| g++ | small_07 |
AC |
2 ms | 4 MB |
| g++ | small_08 |
AC |
2 ms | 4 MB |
| g++ | small_09 |
AC |
2 ms | 4 MB |
| g++ | small_10 |
AC |
2 ms | 4 MB |
| g++ | small_11 |
AC |
2 ms | 4 MB |
| g++ | unsigned_overflow_00 |
AC |
2 ms | 4 MB |