CP-Algorithms Library
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cp-algo/math/multivar.hpp
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- Last update: 2025-05-31 15:35:36+02:00
- Include:
#include "cp-algo/math/multivar.hpp"
Depends on
- cp-algo/math/common.hpp
- cp-algo/math/cvector.hpp
- cp-algo/math/fft.hpp
- cp-algo/number_theory/modint.hpp
- cp-algo/random/rng.hpp
- cp-algo/util/big_alloc.hpp
- cp-algo/util/checkpoint.hpp
- cp-algo/util/complex.hpp
- cp-algo/util/simd.hpp
Verified with
Code
#ifndef CP_ALGO_MATH_MULTIVAR_HPP
#define CP_ALGO_MATH_MULTIVAR_HPP
#include "../util/big_alloc.hpp"
#include "../number_theory/modint.hpp"
#include "../math/fft.hpp"
namespace cp_algo::math::fft {
template<modint_type base>
struct multivar {
std::vector<base, cp_algo::big_alloc<base>> data;
std::vector<size_t, cp_algo::big_alloc<size_t>> ranks;
std::vector<size_t> dim;
size_t N;
size_t rank(size_t i) {
size_t cur = 1, res = 0, K = size(dim);
for(auto ni: dim) {
cur *= ni;
res += i / cur;
}
return res % K;
}
multivar(std::vector<size_t> const& dim): dim(dim), N(
std::ranges::fold_left(dim, 1, std::multiplies{})
) {
data.resize(N);
ranks.resize(N);
for(auto [i, x]: ranks | std::views::enumerate) {
x = rank(i);
}
checkpoint("multivar init");
}
void read() {
for(auto &it: data) {
std::cin >> it;
}
checkpoint("multivar read");
}
void print() {
for(auto &it: data) {
std::cout << it << " ";
}
std::cout << "\n";
checkpoint("multivar write");
}
void mul(multivar<base> const& b) {
assert(dim == b.dim);
size_t K = size(dim);
if(K == 0) {
data[0] *= b.data[0];
return;
}
std::vector<dft<base>> A, B;
size_t M = std::max(flen, std::bit_ceil(2 * N - 1) / 2);
for(size_t i = 0; i < K; i++) {
A.emplace_back(data | std::views::enumerate | std::views::transform(
[&](auto jx) {
auto [j, x] = jx;
return ranks[j] == i ? x : base(0);
}
), M, false);
B.emplace_back(b.data | std::views::enumerate | std::views::transform(
[&](auto jx) {
auto [j, x] = jx;
return ranks[j] == i ? x : base(0);
}
), M, false);
}
for(size_t i = 0; i < K; i++) {
dft<base> C(M);
cvector X = C.A;
for(size_t j = 0; j < K; j++) {
size_t tj = (i - j + K) % K;
A[j].template dot<false, false>(B[tj].A, B[tj].B, C.A, C.B, X);
}
checkpoint("dot");
std::vector<base, cp_algo::big_alloc<base>> res((N + flen - 1) / flen * flen);
C.A.template ifft<false>();
C.B.template ifft<false>();
X.template ifft<false>();
C.recover_mod(X, res, N);
for(size_t j = 0; j < N; j++) {
if(i == ranks[j]) {
data[j] = res[j];
}
}
checkpoint("store");
}
}
};
}
#endif // CP_ALGO_MATH_MULTIVAR_HPP#line 1 "cp-algo/math/multivar.hpp"
#line 1 "cp-algo/util/big_alloc.hpp"
#include <cstddef>
#include <iostream>
// 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, std::size_t Align = 32>
class big_alloc {
static_assert( Align >= alignof(void*), "Align must be at least pointer-size");
static_assert(std::popcount(Align) == 1, "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;
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);
madvise(raw, padded, MADV_POPULATE_WRITE);
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;
}
};
}
#line 1 "cp-algo/number_theory/modint.hpp"
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#include <cstdint>
#include <cassert>
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
auto bpow(auto const& x, auto n, auto const& one, auto op) {
if(n == 0) {
return one;
} else {
auto t = bpow(x, n / 2, one, op);
t = op(t, t);
if(n % 2) {
t = op(t, x);
}
return t;
}
}
auto bpow(auto x, auto n, auto ans) {
return bpow(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 Int 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()));
}
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 UInt modmod8() {return UInt(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;}
};
template<typename Int = int>
struct dynamic_modint: modint_base<dynamic_modint<Int>, Int> {
using Base = modint_base<dynamic_modint<Int>, Int>;
using Base::Base;
static Base::UInt m_reduce(Base::UInt2 ab) {
if(mod() % 2 == 0) [[unlikely]] {
return typename Base::UInt(ab % mod());
} else {
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 Int remod() {return 2 * m;}
static Base::UInt imod() {return im;}
static Base::UInt2 pw128() {return r2;}
static void switch_mod(Int nm) {
m = nm;
im = m % 2 ? inv2(-m) : 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 thread_local Int m;
static thread_local Base::UInt im, r2;
};
template<typename Int>
Int thread_local dynamic_modint<Int>::m = 1;
template<typename Int>
dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::im = -1;
template<typename Int>
dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::r2 = 0;
}
#line 1 "cp-algo/math/fft.hpp"
#line 1 "cp-algo/util/checkpoint.hpp"
#line 4 "cp-algo/util/checkpoint.hpp"
#include <chrono>
#include <string>
#include <map>
namespace cp_algo {
std::map<std::string, double> checkpoints;
template<bool final = false>
void checkpoint([[maybe_unused]] std::string const& msg = "") {
#ifdef CP_ALGO_CHECKPOINT
static double last = 0;
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
}
}
#line 1 "cp-algo/random/rng.hpp"
#line 4 "cp-algo/random/rng.hpp"
#include <random>
namespace cp_algo::random {
uint64_t rng() {
static std::mt19937_64 rng(
std::chrono::steady_clock::now().time_since_epoch().count()
);
return rng();
}
}
#line 1 "cp-algo/math/cvector.hpp"
#line 1 "cp-algo/util/simd.hpp"
#include <experimental/simd>
#line 6 "cp-algo/util/simd.hpp"
#include <memory>
namespace cp_algo {
template<typename T, size_t len>
using simd [[gnu::vector_size(len * sizeof(T))]] = T;
using i64x4 = simd<int64_t, 4>;
using u64x4 = simd<uint64_t, 4>;
using u32x8 = simd<uint32_t, 8>;
using i32x4 = simd<int32_t, 4>;
using u32x4 = simd<uint32_t, 4>;
using i16x4 = simd<int16_t, 4>;
using u8x32 = simd<uint8_t, 32>;
using dx4 = simd<double, 4>;
[[gnu::target("avx2")]] inline dx4 abs(dx4 a) {
return a < 0 ? -a : a;
}
// https://stackoverflow.com/a/77376595
// works for ints in (-2^51, 2^51)
static constexpr dx4 magic = dx4() + (3ULL << 51);
[[gnu::target("avx2")]] inline i64x4 lround(dx4 x) {
return i64x4(x + magic) - i64x4(magic);
}
[[gnu::target("avx2")]] inline dx4 to_double(i64x4 x) {
return dx4(x + i64x4(magic)) - magic;
}
[[gnu::target("avx2")]] inline dx4 round(dx4 a) {
return dx4{
std::nearbyint(a[0]),
std::nearbyint(a[1]),
std::nearbyint(a[2]),
std::nearbyint(a[3])
};
}
[[gnu::target("avx2")]] inline u64x4 low32(u64x4 x) {
return x & uint32_t(-1);
}
[[gnu::target("avx2")]] inline auto swap_bytes(auto x) {
return decltype(x)(__builtin_shufflevector(u32x8(x), u32x8(x), 1, 0, 3, 2, 5, 4, 7, 6));
}
[[gnu::target("avx2")]] inline u64x4 montgomery_reduce(u64x4 x, uint32_t mod, uint32_t imod) {
auto x_ninv = u64x4(_mm256_mul_epu32(__m256i(x), __m256i() + imod));
x += u64x4(_mm256_mul_epu32(__m256i(x_ninv), __m256i() + mod));
return swap_bytes(x);
}
[[gnu::target("avx2")]] inline u64x4 montgomery_mul(u64x4 x, u64x4 y, uint32_t mod, uint32_t imod) {
return montgomery_reduce(u64x4(_mm256_mul_epu32(__m256i(x), __m256i(y))), mod, imod);
}
[[gnu::target("avx2")]] 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)));
}
[[gnu::target("avx2")]] 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>
[[gnu::target("avx2")]] inline bool is_aligned(const auto* p) noexcept {
return (reinterpret_cast<std::uintptr_t>(p) % Align) == 0;
}
template<class Target>
[[gnu::target("avx2")]] inline Target& vector_cast(auto &&p) {
return *reinterpret_cast<Target*>(std::assume_aligned<alignof(Target)>(&p));
}
}
#line 1 "cp-algo/util/complex.hpp"
#line 4 "cp-algo/util/complex.hpp"
#include <cmath>
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;
constexpr complex(): x(), y() {}
constexpr complex(T x): x(x), y() {}
constexpr complex(T x, T y): x(x), y(y) {}
complex& operator *= (T t) {x *= t; y *= t; return *this;}
complex& operator /= (T t) {x /= t; y /= t; return *this;}
complex operator * (T t) const {return complex(*this) *= t;}
complex operator / (T t) const {return complex(*this) /= t;}
complex& operator += (complex t) {x += t.x; y += t.y; return *this;}
complex& operator -= (complex t) {x -= t.x; y -= t.y; return *this;}
complex operator * (complex t) const {return {x * t.x - y * t.y, x * t.y + y * t.x};}
complex operator / (complex t) const {return *this * t.conj() / t.norm();}
complex operator + (complex t) const {return complex(*this) += t;}
complex operator - (complex t) const {return complex(*this) -= t;}
complex& operator *= (complex t) {return *this = *this * t;}
complex& operator /= (complex t) {return *this = *this / t;}
complex operator - () const {return {-x, -y};}
complex conj() const {return {x, -y};}
T norm() const {return x * x + y * y;}
T abs() const {return std::sqrt(norm());}
T const real() const {return x;}
T const imag() const {return y;}
T& real() {return x;}
T& imag() {return y;}
static constexpr complex polar(T r, T theta) {return {T(r * cos(theta)), T(r * sin(theta))};}
auto operator <=> (complex const& t) const = default;
};
template<typename T>
complex<T> operator * (auto x, complex<T> y) {return y *= x;}
template<typename T> complex<T> conj(complex<T> x) {return x.conj();}
template<typename T> T norm(complex<T> x) {return x.norm();}
template<typename T> T abs(complex<T> x) {return x.abs();}
template<typename T> T& real(complex<T> &x) {return x.real();}
template<typename T> T& imag(complex<T> &x) {return x.imag();}
template<typename T> T const real(complex<T> const& x) {return x.real();}
template<typename T> T const imag(complex<T> const& x) {return x.imag();}
template<typename T>
constexpr complex<T> polar(T r, T theta) {
return complex<T>::polar(r, theta);
}
template<typename T>
std::ostream& operator << (std::ostream &out, complex<T> x) {
return out << x.real() << ' ' << x.imag();
}
}
#line 7 "cp-algo/math/cvector.hpp"
#include <ranges>
#include <bit>
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)};
}
struct cvector {
std::vector<vpoint, big_alloc<vpoint>> r;
cvector(size_t n) {
n = std::max(flen, std::bit_ceil(n));
r.resize(n / flen);
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>
void set(size_t k, pt 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>
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 {
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);
for(size_t i = 0; i < n; i++) {
callback(i, factor * eval_point(step * i));
}
}
void dot(cvector const& t) {
size_t n = this->size();
exec_on_evals<1>(n / flen, [&](size_t k, point rt) {
k *= flen;
auto [Ax, Ay] = at(k);
auto Bv = t.at(k);
vpoint res = vz;
for (size_t i = 0; i < flen; i++) {
res += vpoint(vz + Ax[i], vz + Ay[i]) * 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);
}
set(k, res);
});
checkpoint("dot");
}
template<bool partial = true>
void ifft() {
size_t n = size();
if constexpr (!partial) {
point pi(0, 1);
exec_on_evals<4>(n / 4, [&](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), [&](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;
});
}
for(size_t leaf = 3 * flen; leaf < n; leaf += 4 * flen) {
size_t level = std::countr_one(leaf + 3);
for(size_t lvl = 4 + parity; lvl <= level; lvl += 2) {
size_t i = (1 << lvl) / 4;
exec_on_eval<4>(n >> lvl, leaf >> lvl, [&](size_t k, point rt) {
k <<= lvl;
vpoint v1 = {vz + real(rt), vz - imag(rt)};
vpoint v2 = v1 * v1;
vpoint v3 = v1 * v2;
for(size_t j = k; j < k + i; j += flen) {
auto A = at(j);
auto B = at(j + i);
auto C = at(j + 2 * i);
auto D = at(j + 3 * i);
at(j) = ((A + B) + (C + D));
at(j + 2 * i) = ((A + B) - (C + D)) * v2;
at(j + i) = ((A - B) - vi(C - D)) * v1;
at(j + 3 * i) = ((A - B) + vi(C - D)) * v3;
}
});
}
}
checkpoint("ifft");
for(size_t k = 0; k < n; k += flen) {
if constexpr (partial) {
set(k, get<vpoint>(k) /= vz + ftype(n / flen));
} else {
set(k, get<vpoint>(k) /= vz + ftype(n));
}
}
}
template<bool partial = true>
void fft() {
size_t n = size();
bool parity = std::countr_zero(n) % 2;
for(size_t leaf = 0; leaf < n; leaf += 4 * flen) {
size_t level = std::countr_zero(n + leaf);
level -= level % 2 != parity;
for(size_t lvl = level; lvl >= 4; lvl -= 2) {
size_t i = (1 << lvl) / 4;
exec_on_eval<4>(n >> lvl, leaf >> lvl, [&](size_t k, point rt) {
k <<= lvl;
vpoint v1 = {vz + real(rt), vz + imag(rt)};
vpoint v2 = v1 * v1;
vpoint v3 = v1 * v2;
for(size_t j = k; j < k + i; j += flen) {
auto A = at(j);
auto B = at(j + i) * v1;
auto C = at(j + 2 * i) * v2;
auto D = at(j + 3 * i) * v3;
at(j) = (A + C) + (B + D);
at(j + i) = (A + C) - (B + D);
at(j + 2 * i) = (A - C) + vi(B - D);
at(j + 3 * i) = (A - C) - vi(B - D);
}
});
}
}
if(parity) {
exec_on_evals<2>(n / (2 * flen), [&](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) {
point pi(0, 1);
exec_on_evals<4>(n / 4, [&](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 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;
};
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;
}();
}
#line 9 "cp-algo/math/fft.hpp"
namespace cp_algo::math::fft {
template<modint_type base>
struct dft {
cvector A, B;
static base factor, ifactor;
using Int2 = base::Int2;
static bool _init;
static int split() {
static const int splt = int(std::sqrt(base::mod())) + 1;
return splt;
}
static uint32_t mod, imod;
static void init() {
if(!_init) {
factor = 1 + random::rng() % (base::mod() - 1);
ifactor = base(1) / factor;
mod = base::mod();
imod = -inv2<uint32_t>(base::mod());
_init = true;
}
}
dft(size_t n): A(n), B(n) {init();}
dft(auto const& a, size_t n, bool partial = true): A(n), B(n) {
init();
base b2x32 = bpow(base(2), 32);
u64x4 cur = {
(bpow(factor, 1) * b2x32).getr(),
(bpow(factor, 2) * b2x32).getr(),
(bpow(factor, 3) * b2x32).getr(),
(bpow(factor, 4) * b2x32).getr()
};
u64x4 step4 = u64x4{} + (bpow(factor, 4) * b2x32).getr();
u64x4 stepn = u64x4{} + (bpow(factor, n) * b2x32).getr();
for(size_t i = 0; i < std::min(n, std::size(a)); i += flen) {
auto splt = [&](size_t i, auto mul) {
if(i >= std::size(a)) {
return std::pair{vftype(), vftype()};
}
u64x4 au = {
i < std::size(a) ? a[i].getr() : 0,
i + 1 < std::size(a) ? a[i + 1].getr() : 0,
i + 2 < std::size(a) ? a[i + 2].getr() : 0,
i + 3 < std::size(a) ? a[i + 3].getr() : 0
};
au = montgomery_mul(au, mul, mod, imod);
au = au >= base::mod() ? au - base::mod() : au;
auto ai = to_double(i64x4(au >= base::mod() / 2 ? au - base::mod() : au));
auto quo = round(ai / split());
return std::pair{ai - quo * split(), quo};
};
auto [rai, qai] = splt(i, cur);
auto [rani, qani] = splt(n + i, montgomery_mul(cur, stepn, mod, imod));
A.at(i) = vpoint(rai, rani);
B.at(i) = vpoint(qai, qani);
cur = montgomery_mul(cur, step4, mod, imod);
}
checkpoint("dft init");
if(n) {
if(partial) {
A.fft();
B.fft();
} else {
A.template fft<false>();
B.template fft<false>();
}
}
}
template<bool overwrite = true, bool partial = true>
void dot(auto const& C, auto const& D, auto &Aout, auto &Bout, auto &Cout) const {
cvector::exec_on_evals<1>(A.size() / flen, [&](size_t k, point rt) {
k *= flen;
vpoint AC, AD, BC, BD;
AC = AD = BC = BD = vz;
auto Cv = C.at(k), Dv = D.at(k);
if constexpr(partial) {
auto [Ax, Ay] = A.at(k);
auto [Bx, By] = B.at(k);
for (size_t i = 0; i < flen; i++) {
vpoint Av = {vz + Ax[i], vz + Ay[i]}, Bv = {vz + Bx[i], vz + By[i]};
AC += Av * Cv; AD += Av * Dv;
BC += Bv * Cv; BD += Bv * Dv;
real(Cv) = rotate_right(real(Cv));
imag(Cv) = rotate_right(imag(Cv));
real(Dv) = rotate_right(real(Dv));
imag(Dv) = rotate_right(imag(Dv));
auto cx = real(Cv)[0], cy = imag(Cv)[0];
auto dx = real(Dv)[0], dy = imag(Dv)[0];
real(Cv)[0] = cx * real(rt) - cy * imag(rt);
imag(Cv)[0] = cx * imag(rt) + cy * real(rt);
real(Dv)[0] = dx * real(rt) - dy * imag(rt);
imag(Dv)[0] = dx * imag(rt) + dy * real(rt);
}
} else {
AC = A.at(k) * Cv;
AD = A.at(k) * Dv;
BC = B.at(k) * Cv;
BD = B.at(k) * Dv;
}
if constexpr (overwrite) {
Aout.at(k) = AC;
Cout.at(k) = AD + BC;
Bout.at(k) = BD;
} else {
Aout.at(k) += AC;
Cout.at(k) += AD + BC;
Bout.at(k) += BD;
}
});
checkpoint("dot");
}
void dot(auto &&C, auto const& D) {
dot(C, D, A, B, C);
}
void recover_mod(auto &&C, auto &res, size_t k) {
size_t check = (k + flen - 1) / flen * flen;
assert(res.size() >= check);
size_t n = A.size();
auto const splitsplit = base(split() * split()).getr();
base b2x32 = bpow(base(2), 32);
base b2x64 = bpow(base(2), 64);
u64x4 cur = {
(bpow(ifactor, 2) * b2x64).getr(),
(bpow(ifactor, 3) * b2x64).getr(),
(bpow(ifactor, 4) * b2x64).getr(),
(bpow(ifactor, 5) * b2x64).getr()
};
u64x4 step4 = u64x4{} + (bpow(ifactor, 4) * b2x32).getr();
u64x4 stepn = u64x4{} + (bpow(ifactor, n) * b2x32).getr();
for(size_t i = 0; i < std::min(n, k); i += flen) {
auto [Ax, Ay] = A.at(i);
auto [Bx, By] = B.at(i);
auto [Cx, Cy] = C.at(i);
auto set_i = [&](size_t i, auto A, auto B, auto C, auto mul) {
auto A0 = lround(A), A1 = lround(C), A2 = lround(B);
auto Ai = A0 + A1 * split() + A2 * splitsplit + uint64_t(base::modmod());
auto Au = montgomery_reduce(u64x4(Ai), mod, imod);
Au = montgomery_mul(Au, mul, mod, imod);
Au = Au >= base::mod() ? Au - base::mod() : Au;
for(size_t j = 0; j < flen; j++) {
res[i + j].setr(typename base::UInt(Au[j]));
}
};
set_i(i, Ax, Bx, Cx, cur);
if(i + n < k) {
set_i(i + n, Ay, By, Cy, montgomery_mul(cur, stepn, mod, imod));
}
cur = montgomery_mul(cur, step4, mod, imod);
}
checkpoint("recover mod");
}
void mul(auto &&C, auto const& D, auto &res, size_t k) {
assert(A.size() == C.size());
size_t n = A.size();
if(!n) {
res = {};
return;
}
dot(C, D);
A.ifft();
B.ifft();
C.ifft();
recover_mod(C, res, k);
}
void mul_inplace(auto &&B, auto& res, size_t k) {
mul(B.A, B.B, res, k);
}
void mul(auto const& B, auto& res, size_t k) {
mul(cvector(B.A), B.B, res, k);
}
std::vector<base, big_alloc<base>> operator *= (dft &B) {
std::vector<base, big_alloc<base>> res(2 * A.size());
mul_inplace(B, res, 2 * A.size());
return res;
}
std::vector<base, big_alloc<base>> operator *= (dft const& B) {
std::vector<base, big_alloc<base>> res(2 * A.size());
mul(B, res, 2 * A.size());
return res;
}
auto operator * (dft const& B) const {
return dft(*this) *= B;
}
point operator [](int i) const {return A.get(i);}
};
template<modint_type base> base dft<base>::factor = 1;
template<modint_type base> base dft<base>::ifactor = 1;
template<modint_type base> bool dft<base>::_init = false;
template<modint_type base> uint32_t dft<base>::mod = {};
template<modint_type base> uint32_t dft<base>::imod = {};
void mul_slow(auto &a, auto const& b, size_t k) {
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));
a.resize(k);
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];
}
}
}
}
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);
}
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;
}
auto n = std::max(flen, std::bit_ceil(
std::min(k, std::size(a)) + std::min(k, std::size(b)) - 1
) / 2);
auto A = dft<base>(a | std::views::take(k), n);
auto B = dft<base>(b | std::views::take(k), n);
a.resize((k + flen - 1) / flen * flen);
A.mul_inplace(B, a, k);
a.resize(k);
}
// store mod x^n-k in first half, x^n+k in second half
void mod_split(auto &&x, size_t n, auto k) {
using base = std::decay_t<decltype(k)>;
dft<base>::init();
assert(std::size(x) == 2 * n);
u64x4 cur = u64x4{} + (k * bpow(base(2), 32)).getr();
for(size_t i = 0; i < n; i += flen) {
u64x4 xl = {
x[i].getr(),
x[i + 1].getr(),
x[i + 2].getr(),
x[i + 3].getr()
};
u64x4 xr = {
x[n + i].getr(),
x[n + i + 1].getr(),
x[n + i + 2].getr(),
x[n + i + 3].getr()
};
xr = montgomery_mul(xr, cur, dft<base>::mod, dft<base>::imod);
xr = xr >= base::mod() ? xr - base::mod() : xr;
auto t = xr;
xr = xl - t;
xl += t;
xl = xl >= base::mod() ? xl - base::mod() : xl;
xr = xr >= base::mod() ? xr + base::mod() : xr;
for(size_t k = 0; k < flen; k++) {
x[i + k].setr(typename base::UInt(xl[k]));
x[n + i + k].setr(typename base::UInt(xr[k]));
}
}
cp_algo::checkpoint("mod split");
}
void cyclic_mul(auto &a, auto &&b, size_t k) {
assert(std::popcount(k) == 1);
assert(std::size(a) == std::size(b) && std::size(a) == k);
using base = std::decay_t<decltype(a[0])>;
dft<base>::init();
if(k <= (1 << 16)) {
auto ap = std::ranges::to<std::vector<base, big_alloc<base>>>(a);
mul_truncate(ap, b, 2 * k);
mod_split(ap, k, bpow(dft<base>::factor, k));
std::ranges::copy(ap | std::views::take(k), begin(a));
return;
}
k /= 2;
auto factor = bpow(dft<base>::factor, k);
mod_split(a, k, factor);
mod_split(b, k, factor);
auto la = std::span(a).first(k);
auto lb = std::span(b).first(k);
auto ra = std::span(a).last(k);
auto rb = std::span(b).last(k);
cyclic_mul(la, lb, k);
auto A = dft<base>(ra, k / 2);
auto B = dft<base>(rb, k / 2);
A.mul_inplace(B, ra, k);
base i2 = base(2).inv();
factor = factor.inv() * i2;
for(size_t i = 0; i < k; i++) {
auto t = (a[i] + a[i + k]) * i2;
a[i + k] = (a[i] - a[i + k]) * factor;
a[i] = t;
}
cp_algo::checkpoint("mod join");
}
auto make_copy(auto &&x) {
return x;
}
void cyclic_mul(auto &a, auto const& b, size_t k) {
return cyclic_mul(a, make_copy(b), k);
}
void mul(auto &a, auto &&b) {
size_t N = size(a) + size(b);
if(N > (1 << 20)) {
N--;
size_t NN = std::bit_ceil(N);
a.resize(NN);
b.resize(NN);
cyclic_mul(a, b, NN);
a.resize(N);
} else {
mul_truncate(a, b, N - 1);
}
}
void mul(auto &a, auto const& b) {
size_t N = size(a) + size(b);
if(N > (1 << 20)) {
mul(a, make_copy(b));
} else {
mul_truncate(a, b, N - 1);
}
}
}
#line 6 "cp-algo/math/multivar.hpp"
namespace cp_algo::math::fft {
template<modint_type base>
struct multivar {
std::vector<base, cp_algo::big_alloc<base>> data;
std::vector<size_t, cp_algo::big_alloc<size_t>> ranks;
std::vector<size_t> dim;
size_t N;
size_t rank(size_t i) {
size_t cur = 1, res = 0, K = size(dim);
for(auto ni: dim) {
cur *= ni;
res += i / cur;
}
return res % K;
}
multivar(std::vector<size_t> const& dim): dim(dim), N(
std::ranges::fold_left(dim, 1, std::multiplies{})
) {
data.resize(N);
ranks.resize(N);
for(auto [i, x]: ranks | std::views::enumerate) {
x = rank(i);
}
checkpoint("multivar init");
}
void read() {
for(auto &it: data) {
std::cin >> it;
}
checkpoint("multivar read");
}
void print() {
for(auto &it: data) {
std::cout << it << " ";
}
std::cout << "\n";
checkpoint("multivar write");
}
void mul(multivar<base> const& b) {
assert(dim == b.dim);
size_t K = size(dim);
if(K == 0) {
data[0] *= b.data[0];
return;
}
std::vector<dft<base>> A, B;
size_t M = std::max(flen, std::bit_ceil(2 * N - 1) / 2);
for(size_t i = 0; i < K; i++) {
A.emplace_back(data | std::views::enumerate | std::views::transform(
[&](auto jx) {
auto [j, x] = jx;
return ranks[j] == i ? x : base(0);
}
), M, false);
B.emplace_back(b.data | std::views::enumerate | std::views::transform(
[&](auto jx) {
auto [j, x] = jx;
return ranks[j] == i ? x : base(0);
}
), M, false);
}
for(size_t i = 0; i < K; i++) {
dft<base> C(M);
cvector X = C.A;
for(size_t j = 0; j < K; j++) {
size_t tj = (i - j + K) % K;
A[j].template dot<false, false>(B[tj].A, B[tj].B, C.A, C.B, X);
}
checkpoint("dot");
std::vector<base, cp_algo::big_alloc<base>> res((N + flen - 1) / flen * flen);
C.A.template ifft<false>();
C.B.template ifft<false>();
X.template ifft<false>();
C.recover_mod(X, res, N);
for(size_t j = 0; j < N; j++) {
if(i == ranks[j]) {
data[j] = res[j];
}
}
checkpoint("store");
}
}
};
}