CP-Algorithms Library
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tests/subset.cpp
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- Last update: 2026-09-10 02:45:56+02:00
Depends on
cp-algo/math/common.hpp- cp-algo/math/subset_convolution.hpp
- cp-algo/number_theory/modint.hpp
- cp-algo/util/big_alloc.hpp
- cp-algo/util/bit.hpp
- cp-algo/util/checkpoint.hpp
- cp-algo/util/simd.hpp
Code
#define _GLIBCXX_ASSERTIONS
#include <bits/stdc++.h>
#include "cp-algo/number_theory/modint.hpp"
#include "cp-algo/math/subset_convolution.hpp"
using namespace cp_algo;
using namespace cp_algo::math;
template<class T>
big_vector<T> naive(big_vector<T> const& a, big_vector<T> const& b) {
big_vector<T> c(a.size());
for(size_t s = 0; s < a.size(); s++) {
for(size_t t = s;; t = (t - 1) & s) {
c[s] += a[t] * b[s ^ t];
if(!t) break;
}
}
return c;
}
template<class T> void check() {
std::mt19937 rng(87234);
for(int rep = 0; rep < 80; rep++) {
size_t n = size_t(1) << (rep % 8);
big_vector<T> a(n), b(n);
for(auto &x: a) x = rng() % T::mod();
for(auto &x: b) x = rng() % T::mod();
b[0] = rep + 1;
auto c = naive(a, b);
assert(subset_div<T>(c, b) == a);
// The multiplication kernel supports the usual primes below 2^30.
if(T::mod() < (1LL << 30)) {
assert(subset_convolution<T>(a, b) == c);
a[0] = 0;
auto e = subset_exp<T>(a);
assert(subset_log<T>(e) == a);
if(n <= 32) {
big_vector<T> f{3, 2, 7, 4}, powers(n), composed(n), projection(f.size());
powers[0] = 1;
for(size_t k = 0; k < f.size(); k++) {
for(size_t i = 0; i < n; i++) {
composed[i] += f[k] * powers[i];
projection[k] += b[i] * powers[i];
}
powers = naive(powers, a);
}
assert(subset_compose<T>(f, a) == composed);
assert(subset_power_projection<T>(a, b, f.size()) == projection);
}
}
}
}
int main() {
check<modint<998244353>>();
check<modint<1000000007>>();
check<modint<2147483647>>();
dynamic_modint<>::with_mod(998244353, [] {check<dynamic_modint<>>();});
dynamic_modint<>::with_mod(1000000007, [] {check<dynamic_modint<>>();});
using T = modint<998244353>;
big_vector<T> coefficients{1, 2, 3, 4}, constant{2}, weight{7};
assert(subset_compose<T>(coefficients, constant) == big_vector<T>{49});
auto projected = subset_power_projection<T>(constant, weight, 6);
assert(projected == (big_vector<T>{7, 14, 28, 56, 112, 224}));
big_vector<T> f(1 << 20), g(f.size());
for(size_t i = 0; i < f.size(); i++) {
auto n = std::popcount(i);
f[i] = bpow(T(-2), n);
g[i] = bpow(T(-1), n);
}
assert(subset_div<T>(f, g) == g);
std::cout << "Subset operations passed naive, changing-divisor/modulus and full-rank checks\n";
}
#line 1 "tests/subset.cpp"
#define _GLIBCXX_ASSERTIONS
#include <bits/stdc++.h>
#line 1 "cp-algo/number_theory/modint.hpp"
#line 1 "cp-algo/math/common.hpp"
#line 6 "cp-algo/math/common.hpp"
#include <bit>
#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 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()));
}
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;}
};
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/subset_convolution.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\")")
#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);
}
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/big_alloc.hpp"
#line 14 "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::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);
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 1 "cp-algo/util/bit.hpp"
#line 8 "cp-algo/util/bit.hpp"
#if defined(__x86_64__) && !defined(CP_ALGO_DISABLE_AVX2)
#define CP_ALGO_BIT_OPS_TARGET _Pragma("GCC target(\"avx2,bmi,bmi2,lzcnt,popcnt\")")
#else
#define CP_ALGO_BIT_OPS_TARGET _Pragma("GCC target(\"bmi,bmi2,lzcnt,popcnt\")")
#endif
#define CP_ALGO_BIT_PRAGMA_PUSH \
_Pragma("GCC push_options") \
CP_ALGO_BIT_OPS_TARGET
CP_ALGO_BIT_PRAGMA_PUSH
namespace cp_algo {
template<typename Uint>
constexpr size_t bit_width = sizeof(Uint) * 8;
// n < 64
uint64_t mask(size_t n) {
return (1ULL << n) - 1;
}
size_t order_of_bit(auto x, size_t k) {
return k ? std::popcount(x << (bit_width<decltype(x)> - k)) : 0;
}
inline size_t kth_set_bit(uint64_t x, size_t k) {
return std::countr_zero(_pdep_u64(1ULL << k, x));
}
template<int fl = 0>
void with_bit_floor(size_t n, auto &&callback) {
if constexpr (fl >= 63) {
return;
} else if (n >> (fl + 1)) {
with_bit_floor<fl + 1>(n, callback);
} else {
callback.template operator()<1ULL << fl>();
}
}
void with_bit_ceil(size_t n, auto &&callback) {
with_bit_floor(n, [&]<size_t N>() {
if(N == n) {
callback.template operator()<N>();
} else {
callback.template operator()<N << 1>();
}
});
}
inline uint32_t read_bits(char const* p) {
return _mm256_movemask_epi8(__m256i(vector_cast<u8x32 const>(p[0]) + (127 - '0')));
}
inline uint64_t read_bits64(char const* p) {
return read_bits(p) | (uint64_t(read_bits(p + 32)) << 32);
}
inline void write_bits(char *p, uint32_t bits) {
static constexpr u8x32 shuffler = {
0, 0, 0, 0, 0, 0, 0, 0,
1, 1, 1, 1, 1, 1, 1, 1,
2, 2, 2, 2, 2, 2, 2, 2,
3, 3, 3, 3, 3, 3, 3, 3
};
auto shuffled = u8x32(_mm256_shuffle_epi8(__m256i() + bits, __m256i(shuffler)));
static constexpr u8x32 mask = {
1, 2, 4, 8, 16, 32, 64, 128,
1, 2, 4, 8, 16, 32, 64, 128,
1, 2, 4, 8, 16, 32, 64, 128,
1, 2, 4, 8, 16, 32, 64, 128
};
for(int z = 0; z < 32; z++) {
p[z] = shuffled[z] & mask[z] ? '1' : '0';
}
}
inline void write_bits64(char *p, uint64_t bits) {
write_bits(p, uint32_t(bits));
write_bits(p + 32, uint32_t(bits >> 32));
}
}
#pragma GCC pop_options
#line 1 "cp-algo/util/checkpoint.hpp"
#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 8 "cp-algo/math/subset_convolution.hpp"
#include <ranges>
#line 12 "cp-algo/math/subset_convolution.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math {
#ifndef CP_ALGO_SUBSET_CONVOLUTION_MAX_LOGN
#define CP_ALGO_SUBSET_CONVOLUTION_MAX_LOGN 20
#endif
const size_t max_logn = CP_ALGO_SUBSET_CONVOLUTION_MAX_LOGN;
template<auto N>
inline void xor_transform(auto &&a) {
if constexpr (N >> max_logn) {
throw std::runtime_error("N too large for xor_transform");
} else if constexpr (N <= 32) {
for (size_t i = 1; i < N; i *= 2) {
for (size_t j = 0; j < N; j += 2 * i) {
for (size_t k = j; k < j + i; k++) {
for (size_t z = 0; z < max_logn; z++) {
auto x = a[k][z] + a[k + i][z];
auto y = a[k][z] - a[k + i][z];
a[k][z] = x;
a[k + i][z] = y;
}
}
}
}
} else {
auto add = [&](auto &a, auto &b) __attribute__((always_inline)) {
auto x = a + b, y = a - b;
a = x, b = y;
};
constexpr auto quar = N / 4;
for (size_t i = 0; i < (size_t)quar; i++) {
auto x0 = a[i + (size_t)quar * 0];
auto x1 = a[i + (size_t)quar * 1];
auto x2 = a[i + (size_t)quar * 2];
auto x3 = a[i + (size_t)quar * 3];
#pragma GCC unroll max_logn
for (size_t z = 0; z < max_logn; z++) {
add(x0[z], x2[z]);
add(x1[z], x3[z]);
}
#pragma GCC unroll max_logn
for (size_t z = 0; z < max_logn; z++) {
add(x0[z], x1[z]);
add(x2[z], x3[z]);
}
a[i + (size_t)quar * 0] = x0;
a[i + (size_t)quar * 1] = x1;
a[i + (size_t)quar * 2] = x2;
a[i + (size_t)quar * 3] = x3;
}
xor_transform<quar>(&a[quar * 0]);
xor_transform<quar>(&a[quar * 1]);
xor_transform<quar>(&a[quar * 2]);
xor_transform<quar>(&a[quar * 3]);
}
}
inline void xor_transform(auto &&a, auto n) {
with_bit_floor(n, [&]<auto NN>() {
assert(NN == n);
xor_transform<NN>(a);
});
}
inline void xor_transform(auto &&a) {
xor_transform(a, std::size(a));
}
// Generic rank vectors processor with variadic inputs
// Assumes output[0] = 0, caller is responsible for handling rank 0
// Returns the output array
auto on_rank_vectors(auto &&cb, auto const& ...inputs) {
static_assert(sizeof...(inputs) >= 1, "on_rank_vectors requires at least one input");
// Create tuple of input references once
auto input_tuple = std::forward_as_tuple(inputs...);
auto const& first_input = std::get<0>(input_tuple);
using base = std::decay_t<decltype(first_input[0])>;
big_vector<base> out(std::size(first_input));
auto N = std::size(first_input);
constexpr size_t K = 4;
N = std::max(N, 2 * K);
const size_t n = std::bit_width(N) - 1;
const size_t T = std::min<size_t>(n - 3, 2);
const size_t bottoms = 1 << (n - T - 1);
const auto M = std::size(first_input);
// Create array buffers for each input
auto create_buffers = [bottoms]<typename... Args>(const Args&...) {
return std::make_tuple(
big_vector<std::array<typename std::decay_t<Args>::value_type, max_logn>>(bottoms)...
);
};
auto buffers = std::apply(create_buffers, input_tuple);
checkpoint("alloc buffers");
big_vector<uint32_t> counts(2 * bottoms);
for(size_t i = 1; i < 2 * bottoms; i++) {
counts[i] = (uint32_t)std::popcount(i);
}
checkpoint("prepare");
for(size_t top = 0; top < N / 2; top += bottoms) {
// Clear all buffers
std::apply([bottoms](auto&... bufs) {
(..., memset(bufs.data(), 0, sizeof(bufs[0]) * bottoms));
}, buffers);
checkpoint("memset");
// Initialize buffers from inputs
std::apply([&](auto const&... inps) {
std::apply([&](auto&... bufs) {
auto init_one = [&](auto const& inp, auto& buf) {
for(size_t i = 0; i < M; i += 2 * bottoms) {
bool parity = __builtin_parity(uint32_t((i >> 1) & top));
size_t limit = std::min(M, i + 2 * bottoms) - i;
uint32_t count = (uint32_t)std::popcount(i) - 1;
for(size_t bottom = (i == 0); bottom < limit; bottom++) {
if (parity) {
buf[bottom >> 1][count + counts[bottom]] -= inp[i + bottom];
} else {
buf[bottom >> 1][count + counts[bottom]] += inp[i + bottom];
}
}
}
};
(init_one(inps, bufs), ...);
}, buffers);
}, input_tuple);
checkpoint("init");
std::apply([](auto&... bufs) {
(..., xor_transform(bufs));
}, buffers);
checkpoint("transform");
assert(bottoms % K == 0);
for(size_t i = 0; i < bottoms; i += K) {
std::apply([&](auto&... bufs) {
auto extract_one = [&](auto& buf) {
std::array<u64x4, max_logn> aa;
for(size_t j = 0; j < max_logn; j++) {
for(size_t z = 0; z < K; z++) {
aa[j][z] = buf[i + z][j].getr();
}
}
return aa;
};
auto aa_tuple = std::make_tuple(extract_one(bufs)...);
std::apply(cb, aa_tuple);
// Write results back: only first array needs to be written
auto& first_buf = std::get<0>(std::forward_as_tuple(bufs...));
const auto& first_aa = std::get<0>(aa_tuple);
for(size_t j = 0; j < max_logn; j++) {
for(size_t z = 0; z < K; z++) {
first_buf[i + z][j].setr((uint32_t)first_aa[j][z]);
}
}
}, buffers);
}
checkpoint("dot");
auto& first_buf = std::get<0>(buffers);
xor_transform(first_buf);
checkpoint("transform");
// Gather results from first buffer
for(size_t i = 0; i < M; i += 2 * bottoms) {
bool parity = __builtin_parity(uint32_t((i >> 1) & top));
size_t limit = std::min(M, i + 2 * bottoms) - i;
uint32_t count = (uint32_t)std::popcount(i) - 1;
for(size_t bottom = (i == 0); bottom < limit; bottom++) {
if (parity) {
out[i + bottom] -= first_buf[bottom >> 1][count + counts[bottom]];
} else {
out[i + bottom] += first_buf[bottom >> 1][count + counts[bottom]];
}
}
}
checkpoint("gather");
}
const base ni = base(N / 2).inv();
for(auto& x : out) {x *= ni;}
return out;
}
template<typename value_type>
big_vector<std::remove_const_t<value_type>> subset_convolution(std::span<value_type> f, std::span<value_type> g) {
using base = std::remove_const_t<value_type>;
big_vector<base> outpa;
const size_t lgn = std::min<size_t>(max_logn, std::bit_width(f.size()) - 1);
outpa = on_rank_vectors([lgn](auto &a, auto const& b) {
std::decay_t<decltype(a)> res = {};
const auto mod = base::mod();
const auto imod = math::inv2(-mod);
const auto r4 = u64x4() + uint64_t(-1) % mod + 1;
auto add = [&](size_t i) {
for(size_t j = 0; i + j + 1 < lgn; j++) {
res[i + j + 1] += (u64x4)_mm256_mul_epu32(__m256i(a[i]), __m256i(b[j]));
}
if (i == 15) {
for(size_t k = 0; k < lgn; k++) {
res[k] -= (res[k] >= base::modmod8()) & base::modmod8();
}
}
};
for(size_t i = 0; i < lgn; i++) { add(i); }
for(size_t k = 0; k < max_logn; k++) {
res[k] = montgomery_reduce(res[k], mod, imod);
res[k] = montgomery_mul(res[k], r4, mod, imod);
a[k] = res[k] >= mod ? res[k] - mod : res[k];
}
}, f, g);
outpa[0] = f[0] * g[0];
for(size_t i = 1; i < std::size(f); i++) {
outpa[i] += f[i] * g[0] + f[0] * g[i];
}
checkpoint("fix 0");
return outpa;
}
template<typename base>
big_vector<base> subset_div(std::span<base> f, std::span<base> g) {
big_vector<base> outpa;
constexpr size_t lgn = max_logn;
auto inv = g[0].inv();
auto f0 = (f[0] * inv).getr(), gi = inv.getr();
const auto mod = base::mod();
const auto imod = math::inv2(-mod);
const auto gir4 = u64x4() + (uint64_t(-1) % mod + 1) * gi % mod;
// Eight products and the subsequent Montgomery reduction fit below 2^64.
const size_t period = mod < (1u << 30) ? 8 : 1;
const uint64_t bound = uint64_t(period * base::modmod());
outpa = on_rank_vectors([=](auto &a, auto const& b) {
for(size_t k = 0; k < lgn; k++) {
for(size_t i = 0; i < k; i++) {
a[k] -= (u64x4)_mm256_mul_epu32(__m256i(a[i]), __m256i(b[k - 1 - i]));
if(i % period == period - 1 || i + 1 == k) {
a[k] = a[k] >= bound ? a[k] + bound : a[k];
}
}
a[k] -= (u64x4)_mm256_mul_epu32(__m256i() + f0, __m256i(b[k]));
a[k] = a[k] >= bound ? a[k] + bound : a[k];
a[k] = montgomery_reduce(a[k], mod, imod);
a[k] = montgomery_mul(a[k], gir4, mod, imod);
a[k] = a[k] >= mod ? a[k] - mod : a[k];
}
}, f, g);
outpa[0] = f0;
checkpoint("fix 0");
return outpa;
}
template<typename base>
big_vector<base> subset_log(std::span<base> g) {
if (size(g) == 1) {
assert(g[0] == base(1));
return big_vector<base>{0};
}
size_t N = std::size(g);
auto out0 = subset_log(std::span(g).first(N / 2));
auto out1 = subset_div<base>(std::span(g).last(N / 2), std::span(g).first(N / 2));
out0.insert(end(out0), begin(out1), end(out1));
cp_algo::checkpoint("extend out");
return out0;
}
template<typename base>
big_vector<base> subset_exp(std::span<base> g) {
if (size(g) == 1) {
assert(g[0] == base(0));
return big_vector<base>{1};
}
size_t N = std::size(g);
auto out0 = subset_exp(std::span(g).first(N / 2));
auto out1 = subset_convolution<base>(out0, std::span(g).last(N / 2));
out0.insert(end(out0), begin(out1), end(out1));
cp_algo::checkpoint("extend out");
return out0;
}
template<typename base>
big_vector<big_vector<base>> subset_compose(std::span<base> f, std::span<base> g, size_t n) {
if (size(g) == 1) {
size_t M = size(f);
big_vector res(n, big_vector<base>{0});
big_vector<base> pw(std::max(n, M) + 1);
pw[0] = 1;
for (size_t j = 1; j < M; j++) {
pw[j] = pw[j - 1] * g[0];
}
for (size_t i = 0; i < n; i++) {
for (size_t j = 0; j < M; j++) {
res[i][0] += pw[j] * f[j];
}
for (size_t j = M; j > i; j--) {
pw[j] = pw[j - 1] * base(j);
}
pw[i] = 0;
}
cp_algo::checkpoint("base case");
return res;
}
size_t N = std::size(g);
auto deeper = subset_compose(f, std::span(g).first(N / 2), n + 1);
for(size_t i = 0; i + 1 < size(deeper); i++) {
auto next = subset_convolution<base>(deeper[i + 1], std::span(g).last(N / 2));
deeper[i].insert(end(deeper[i]), begin(next), end(next));
}
deeper.pop_back();
cp_algo::checkpoint("combine");
return deeper;
}
template<typename base>
big_vector<base> subset_compose(std::span<base> f, std::span<base> g) {
return subset_compose(f, g, 1)[0];
}
// Transpose of f -> f * g = h
template<typename base>
big_vector<base> subset_conv_transpose(std::span<base> h, std::span<base> g) {
std::ranges::reverse(h);
auto res = subset_convolution<base>(h, g);
std::ranges::reverse(h);
std::ranges::reverse(res);
return res;
}
template<typename base>
big_vector<base> subset_power_projection(big_vector<big_vector<base>> &&fg, std::span<base> g, size_t M) {
if (size(g) == 1) {
size_t n = size(fg);
big_vector<base> res(M);
big_vector<base> pw(std::max(n, M) + 1);
pw[0] = 1;
for (size_t j = 1; j < M; j++) {
pw[j] = pw[j - 1] * g[0];
}
for (size_t i = 0; i < size(fg); i++) {
for (size_t j = 0; j < M; j++) {
res[j] += pw[j] * fg[i][0];
}
for (size_t j = M; j > i; j--) {
pw[j] = pw[j - 1] * base(j);
}
pw[i] = 0;
}
cp_algo::checkpoint("base case");
return res;
}
size_t N = std::size(g);
fg.emplace_back(N / 2);
for(auto&& [i, h]: fg | std::views::enumerate | std::views::reverse | std::views::drop(1)) {
auto prev = subset_conv_transpose<base>(std::span(h).last(N / 2), std::span(g).last(N / 2));
for (size_t j = 0; j < N / 2; j++) {
fg[i + 1][j] += prev[j];
}
fg[i + 1].resize(N / 2);
}
fg[0].resize(N / 2);
cp_algo::checkpoint("decombine");
return subset_power_projection(std::move(fg), std::span(g).first(N / 2), M);
}
template<typename base>
big_vector<base> subset_power_projection(std::span<base> g, std::span<base> w, size_t M) {
return subset_power_projection({{begin(w), end(w)}}, g, M);
}
}
#pragma GCC pop_options
#line 5 "tests/subset.cpp"
using namespace cp_algo;
using namespace cp_algo::math;
template<class T>
big_vector<T> naive(big_vector<T> const& a, big_vector<T> const& b) {
big_vector<T> c(a.size());
for(size_t s = 0; s < a.size(); s++) {
for(size_t t = s;; t = (t - 1) & s) {
c[s] += a[t] * b[s ^ t];
if(!t) break;
}
}
return c;
}
template<class T> void check() {
std::mt19937 rng(87234);
for(int rep = 0; rep < 80; rep++) {
size_t n = size_t(1) << (rep % 8);
big_vector<T> a(n), b(n);
for(auto &x: a) x = rng() % T::mod();
for(auto &x: b) x = rng() % T::mod();
b[0] = rep + 1;
auto c = naive(a, b);
assert(subset_div<T>(c, b) == a);
// The multiplication kernel supports the usual primes below 2^30.
if(T::mod() < (1LL << 30)) {
assert(subset_convolution<T>(a, b) == c);
a[0] = 0;
auto e = subset_exp<T>(a);
assert(subset_log<T>(e) == a);
if(n <= 32) {
big_vector<T> f{3, 2, 7, 4}, powers(n), composed(n), projection(f.size());
powers[0] = 1;
for(size_t k = 0; k < f.size(); k++) {
for(size_t i = 0; i < n; i++) {
composed[i] += f[k] * powers[i];
projection[k] += b[i] * powers[i];
}
powers = naive(powers, a);
}
assert(subset_compose<T>(f, a) == composed);
assert(subset_power_projection<T>(a, b, f.size()) == projection);
}
}
}
}
int main() {
check<modint<998244353>>();
check<modint<1000000007>>();
check<modint<2147483647>>();
dynamic_modint<>::with_mod(998244353, [] {check<dynamic_modint<>>();});
dynamic_modint<>::with_mod(1000000007, [] {check<dynamic_modint<>>();});
using T = modint<998244353>;
big_vector<T> coefficients{1, 2, 3, 4}, constant{2}, weight{7};
assert(subset_compose<T>(coefficients, constant) == big_vector<T>{49});
auto projected = subset_power_projection<T>(constant, weight, 6);
assert(projected == (big_vector<T>{7, 14, 28, 56, 112, 224}));
big_vector<T> f(1 << 20), g(f.size());
for(size_t i = 0; i < f.size(); i++) {
auto n = std::popcount(i);
f[i] = bpow(T(-2), n);
g[i] = bpow(T(-1), n);
}
assert(subset_div<T>(f, g) == g);
std::cout << "Subset operations passed naive, changing-divisor/modulus and full-rank checks\n";
}
#line 1 "tests/subset.cpp"
#define _GLIBCXX_ASSERTIONS
#include <bits/stdc++.h>
#line 1 "cp-algo/number_theory/modint.hpp"
#line 1 "cp-algo/math/common.hpp"
#line 6 "cp-algo/math/common.hpp"
#include <bit>
#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;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;}}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 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()));}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;}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;}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;}};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);}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/subset_convolution.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\")")
#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])};}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);}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/big_alloc.hpp"
#line 14 "cp-algo/util/big_alloc.hpp"
#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::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);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 1 "cp-algo/util/bit.hpp"
#line 8 "cp-algo/util/bit.hpp"
#if defined(__x86_64__) && !defined(CP_ALGO_DISABLE_AVX2)
#define CP_ALGO_BIT_OPS_TARGET _Pragma("GCC target(\"avx2,bmi,bmi2,lzcnt,popcnt\")")
#else
#define CP_ALGO_BIT_OPS_TARGET _Pragma("GCC target(\"bmi,bmi2,lzcnt,popcnt\")")
#endif
#define CP_ALGO_BIT_PRAGMA_PUSH _Pragma("GCC push_options") CP_ALGO_BIT_OPS_TARGET
CP_ALGO_BIT_PRAGMA_PUSH
namespace cp_algo{template<typename Uint>constexpr size_t bit_width=sizeof(Uint)*8;uint64_t mask(size_t n){return(1ULL<<n)-1;}size_t order_of_bit(auto x,size_t k){return k?std::popcount(x<<(bit_width<decltype(x)>-k)):0;}inline size_t kth_set_bit(uint64_t x,size_t k){return std::countr_zero(_pdep_u64(1ULL<<k,x));}template<int fl=0>void with_bit_floor(size_t n,auto&&callback){if constexpr(fl>=63){return;}else if(n>>(fl+1)){with_bit_floor<fl+1>(n,callback);}else{callback.template operator()<1ULL<<fl>();}}void with_bit_ceil(size_t n,auto&&callback){with_bit_floor(n,[&]<size_t N>(){if(N==n){callback.template operator()<N>();}else{callback.template operator()<N<<1>();}});}inline uint32_t read_bits(char const*p){return _mm256_movemask_epi8(__m256i(vector_cast<u8x32 const>(p[0])+(127-'0')));}inline uint64_t read_bits64(char const*p){return read_bits(p)|(uint64_t(read_bits(p+32))<<32);}inline void write_bits(char*p,uint32_t bits){static constexpr u8x32 shuffler={0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3};auto shuffled=u8x32(_mm256_shuffle_epi8(__m256i()+bits,__m256i(shuffler)));static constexpr u8x32 mask={1,2,4,8,16,32,64,128,1,2,4,8,16,32,64,128,1,2,4,8,16,32,64,128,1,2,4,8,16,32,64,128};for(int z=0;z<32;z++){p[z]=shuffled[z]&mask[z]?'1':'0';}}inline void write_bits64(char*p,uint64_t bits){write_bits(p,uint32_t(bits));write_bits(p+32,uint32_t(bits>>32));}}
#pragma GCC pop_options
#line 1 "cp-algo/util/checkpoint.hpp"
#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 8 "cp-algo/math/subset_convolution.hpp"
#include <ranges>
#line 12 "cp-algo/math/subset_convolution.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::math{
#ifndef CP_ALGO_SUBSET_CONVOLUTION_MAX_LOGN
#define CP_ALGO_SUBSET_CONVOLUTION_MAX_LOGN 20
#endif
const size_t max_logn=CP_ALGO_SUBSET_CONVOLUTION_MAX_LOGN;template<auto N>inline void xor_transform(auto&&a){if constexpr(N>>max_logn){throw std::runtime_error("N too large for xor_transform");}else if constexpr(N<=32){for(size_t i=1;i<N;i*=2){for(size_t j=0;j<N;j+=2*i){for(size_t k=j;k<j+i;k++){for(size_t z=0;z<max_logn;z++){auto x=a[k][z]+a[k+i][z];auto y=a[k][z]-a[k+i][z];a[k][z]=x;a[k+i][z]=y;}}}}}else{auto add=[&](auto&a,auto&b)__attribute__((always_inline)){auto x=a+b,y=a-b;a=x,b=y;};constexpr auto quar=N/4;for(size_t i=0;i<(size_t)quar;i++){auto x0=a[i+(size_t)quar*0];auto x1=a[i+(size_t)quar*1];auto x2=a[i+(size_t)quar*2];auto x3=a[i+(size_t)quar*3];
#pragma GCC unroll max_logn
for(size_t z=0;z<max_logn;z++){add(x0[z],x2[z]);add(x1[z],x3[z]);}
#pragma GCC unroll max_logn
for(size_t z=0;z<max_logn;z++){add(x0[z],x1[z]);add(x2[z],x3[z]);}a[i+(size_t)quar*0]=x0;a[i+(size_t)quar*1]=x1;a[i+(size_t)quar*2]=x2;a[i+(size_t)quar*3]=x3;}xor_transform<quar>(&a[quar*0]);xor_transform<quar>(&a[quar*1]);xor_transform<quar>(&a[quar*2]);xor_transform<quar>(&a[quar*3]);}}inline void xor_transform(auto&&a,auto n){with_bit_floor(n,[&]<auto NN>(){assert(NN==n);xor_transform<NN>(a);});}inline void xor_transform(auto&&a){xor_transform(a,std::size(a));}auto on_rank_vectors(auto&&cb,auto const&...inputs){static_assert(sizeof...(inputs)>=1,"on_rank_vectors requires at least one input");auto input_tuple=std::forward_as_tuple(inputs...);auto const&first_input=std::get<0>(input_tuple);using base=std::decay_t<decltype(first_input[0])>;big_vector<base>out(std::size(first_input));auto N=std::size(first_input);constexpr size_t K=4;N=std::max(N,2*K);const size_t n=std::bit_width(N)-1;const size_t T=std::min<size_t>(n-3,2);const size_t bottoms=1<<(n-T-1);const auto M=std::size(first_input);auto create_buffers=[bottoms]<typename... Args>(const Args&...){return std::make_tuple(big_vector<std::array<typename std::decay_t<Args>::value_type,max_logn>>(bottoms)...);};auto buffers=std::apply(create_buffers,input_tuple);checkpoint("alloc buffers");big_vector<uint32_t>counts(2*bottoms);for(size_t i=1;i<2*bottoms;i++){counts[i]=(uint32_t)std::popcount(i);}checkpoint("prepare");for(size_t top=0;top<N/2;top+=bottoms){std::apply([bottoms](auto&... bufs){(...,memset(bufs.data(),0,sizeof(bufs[0])*bottoms));},buffers);checkpoint("memset");std::apply([&](auto const&... inps){std::apply([&](auto&... bufs){auto init_one=[&](auto const&inp,auto&buf){for(size_t i=0;i<M;i+=2*bottoms){bool parity=__builtin_parity(uint32_t((i>>1)&top));size_t limit=std::min(M,i+2*bottoms)-i;uint32_t count=(uint32_t)std::popcount(i)-1;for(size_t bottom=(i==0);bottom<limit;bottom++){if(parity){buf[bottom>>1][count+counts[bottom]]-=inp[i+bottom];}else{buf[bottom>>1][count+counts[bottom]]+=inp[i+bottom];}}}};(init_one(inps,bufs),...);},buffers);},input_tuple);checkpoint("init");std::apply([](auto&... bufs){(...,xor_transform(bufs));},buffers);checkpoint("transform");assert(bottoms%K==0);for(size_t i=0;i<bottoms;i+=K){std::apply([&](auto&... bufs){auto extract_one=[&](auto&buf){std::array<u64x4,max_logn>aa;for(size_t j=0;j<max_logn;j++){for(size_t z=0;z<K;z++){aa[j][z]=buf[i+z][j].getr();}}return aa;};auto aa_tuple=std::make_tuple(extract_one(bufs)...);std::apply(cb,aa_tuple);auto&first_buf=std::get<0>(std::forward_as_tuple(bufs...));const auto&first_aa=std::get<0>(aa_tuple);for(size_t j=0;j<max_logn;j++){for(size_t z=0;z<K;z++){first_buf[i+z][j].setr((uint32_t)first_aa[j][z]);}}},buffers);}checkpoint("dot");auto&first_buf=std::get<0>(buffers);xor_transform(first_buf);checkpoint("transform");for(size_t i=0;i<M;i+=2*bottoms){bool parity=__builtin_parity(uint32_t((i>>1)&top));size_t limit=std::min(M,i+2*bottoms)-i;uint32_t count=(uint32_t)std::popcount(i)-1;for(size_t bottom=(i==0);bottom<limit;bottom++){if(parity){out[i+bottom]-=first_buf[bottom>>1][count+counts[bottom]];}else{out[i+bottom]+=first_buf[bottom>>1][count+counts[bottom]];}}}checkpoint("gather");}const base ni=base(N/2).inv();for(auto&x:out){x*=ni;}return out;}template<typename value_type>big_vector<std::remove_const_t<value_type>>subset_convolution(std::span<value_type>f,std::span<value_type>g){using base=std::remove_const_t<value_type>;big_vector<base>outpa;const size_t lgn=std::min<size_t>(max_logn,std::bit_width(f.size())-1);outpa=on_rank_vectors([lgn](auto&a,auto const&b){std::decay_t<decltype(a)>res={};const auto mod=base::mod();const auto imod=math::inv2(-mod);const auto r4=u64x4()+uint64_t(-1)%mod+1;auto add=[&](size_t i){for(size_t j=0;i+j+1<lgn;j++){res[i+j+1]+=(u64x4)_mm256_mul_epu32(__m256i(a[i]),__m256i(b[j]));}if(i==15){for(size_t k=0;k<lgn;k++){res[k]-=(res[k]>=base::modmod8())&base::modmod8();}}};for(size_t i=0;i<lgn;i++){add(i);}for(size_t k=0;k<max_logn;k++){res[k]=montgomery_reduce(res[k],mod,imod);res[k]=montgomery_mul(res[k],r4,mod,imod);a[k]=res[k]>=mod?res[k]-mod:res[k];}},f,g);outpa[0]=f[0]*g[0];for(size_t i=1;i<std::size(f);i++){outpa[i]+=f[i]*g[0]+f[0]*g[i];}checkpoint("fix 0");return outpa;}template<typename base>big_vector<base>subset_div(std::span<base>f,std::span<base>g){big_vector<base>outpa;constexpr size_t lgn=max_logn;auto inv=g[0].inv();auto f0=(f[0]*inv).getr(),gi=inv.getr();const auto mod=base::mod();const auto imod=math::inv2(-mod);const auto gir4=u64x4()+(uint64_t(-1)%mod+1)*gi%mod;const size_t period=mod<(1u<<30)?8:1;const uint64_t bound=uint64_t(period*base::modmod());outpa=on_rank_vectors([=](auto&a,auto const&b){for(size_t k=0;k<lgn;k++){for(size_t i=0;i<k;i++){a[k]-=(u64x4)_mm256_mul_epu32(__m256i(a[i]),__m256i(b[k-1-i]));if(i%period==period-1||i+1==k){a[k]=a[k]>=bound?a[k]+bound:a[k];}}a[k]-=(u64x4)_mm256_mul_epu32(__m256i()+f0,__m256i(b[k]));a[k]=a[k]>=bound?a[k]+bound:a[k];a[k]=montgomery_reduce(a[k],mod,imod);a[k]=montgomery_mul(a[k],gir4,mod,imod);a[k]=a[k]>=mod?a[k]-mod:a[k];}},f,g);outpa[0]=f0;checkpoint("fix 0");return outpa;}template<typename base>big_vector<base>subset_log(std::span<base>g){if(size(g)==1){assert(g[0]==base(1));return big_vector<base>{0};}size_t N=std::size(g);auto out0=subset_log(std::span(g).first(N/2));auto out1=subset_div<base>(std::span(g).last(N/2),std::span(g).first(N/2));out0.insert(end(out0),begin(out1),end(out1));cp_algo::checkpoint("extend out");return out0;}template<typename base>big_vector<base>subset_exp(std::span<base>g){if(size(g)==1){assert(g[0]==base(0));return big_vector<base>{1};}size_t N=std::size(g);auto out0=subset_exp(std::span(g).first(N/2));auto out1=subset_convolution<base>(out0,std::span(g).last(N/2));out0.insert(end(out0),begin(out1),end(out1));cp_algo::checkpoint("extend out");return out0;}template<typename base>big_vector<big_vector<base>>subset_compose(std::span<base>f,std::span<base>g,size_t n){if(size(g)==1){size_t M=size(f);big_vector res(n,big_vector<base>{0});big_vector<base>pw(std::max(n,M)+1);pw[0]=1;for(size_t j=1;j<M;j++){pw[j]=pw[j-1]*g[0];}for(size_t i=0;i<n;i++){for(size_t j=0;j<M;j++){res[i][0]+=pw[j]*f[j];}for(size_t j=M;j>i;j--){pw[j]=pw[j-1]*base(j);}pw[i]=0;}cp_algo::checkpoint("base case");return res;}size_t N=std::size(g);auto deeper=subset_compose(f,std::span(g).first(N/2),n+1);for(size_t i=0;i+1<size(deeper);i++){auto next=subset_convolution<base>(deeper[i+1],std::span(g).last(N/2));deeper[i].insert(end(deeper[i]),begin(next),end(next));}deeper.pop_back();cp_algo::checkpoint("combine");return deeper;}template<typename base>big_vector<base>subset_compose(std::span<base>f,std::span<base>g){return subset_compose(f,g,1)[0];}template<typename base>big_vector<base>subset_conv_transpose(std::span<base>h,std::span<base>g){std::ranges::reverse(h);auto res=subset_convolution<base>(h,g);std::ranges::reverse(h);std::ranges::reverse(res);return res;}template<typename base>big_vector<base>subset_power_projection(big_vector<big_vector<base>>&&fg,std::span<base>g,size_t M){if(size(g)==1){size_t n=size(fg);big_vector<base>res(M);big_vector<base>pw(std::max(n,M)+1);pw[0]=1;for(size_t j=1;j<M;j++){pw[j]=pw[j-1]*g[0];}for(size_t i=0;i<size(fg);i++){for(size_t j=0;j<M;j++){res[j]+=pw[j]*fg[i][0];}for(size_t j=M;j>i;j--){pw[j]=pw[j-1]*base(j);}pw[i]=0;}cp_algo::checkpoint("base case");return res;}size_t N=std::size(g);fg.emplace_back(N/2);for(auto&&[i,h]:fg|std::views::enumerate|std::views::reverse|std::views::drop(1)){auto prev=subset_conv_transpose<base>(std::span(h).last(N/2),std::span(g).last(N/2));for(size_t j=0;j<N/2;j++){fg[i+1][j]+=prev[j];}fg[i+1].resize(N/2);}fg[0].resize(N/2);cp_algo::checkpoint("decombine");return subset_power_projection(std::move(fg),std::span(g).first(N/2),M);}template<typename base>big_vector<base>subset_power_projection(std::span<base>g,std::span<base>w,size_t M){return subset_power_projection({{begin(w),end(w)}},g,M);}}
#pragma GCC pop_options
#line 5 "tests/subset.cpp"
using namespace cp_algo;using namespace cp_algo::math;template<class T>big_vector<T>naive(big_vector<T>const&a,big_vector<T>const&b){big_vector<T>c(a.size());for(size_t s=0;s<a.size();s++){for(size_t t=s;;t=(t-1)&s){c[s]+=a[t]*b[s^t];if(!t)break;}}return c;}template<class T>void check(){std::mt19937 rng(87234);for(int rep=0;rep<80;rep++){size_t n=size_t(1)<<(rep%8);big_vector<T>a(n),b(n);for(auto&x:a)x=rng()%T::mod();for(auto&x:b)x=rng()%T::mod();b[0]=rep+1;auto c=naive(a,b);assert(subset_div<T>(c,b)==a);if(T::mod()<(1LL<<30)){assert(subset_convolution<T>(a,b)==c);a[0]=0;auto e=subset_exp<T>(a);assert(subset_log<T>(e)==a);if(n<=32){big_vector<T>f{3,2,7,4},powers(n),composed(n),projection(f.size());powers[0]=1;for(size_t k=0;k<f.size();k++){for(size_t i=0;i<n;i++){composed[i]+=f[k]*powers[i];projection[k]+=b[i]*powers[i];}powers=naive(powers,a);}assert(subset_compose<T>(f,a)==composed);assert(subset_power_projection<T>(a,b,f.size())==projection);}}}}int main(){check<modint<998244353>>();check<modint<1000000007>>();check<modint<2147483647>>();dynamic_modint<>::with_mod(998244353,[]{check<dynamic_modint<>>();});dynamic_modint<>::with_mod(1000000007,[]{check<dynamic_modint<>>();});using T=modint<998244353>;big_vector<T>coefficients{1,2,3,4},constant{2},weight{7};assert(subset_compose<T>(coefficients,constant)==big_vector<T>{49});auto projected=subset_power_projection<T>(constant,weight,6);assert(projected==(big_vector<T>{7,14,28,56,112,224}));big_vector<T>f(1<<20),g(f.size());for(size_t i=0;i<f.size();i++){auto n=std::popcount(i);f[i]=bpow(T(-2),n);g[i]=bpow(T(-1),n);}assert(subset_div<T>(f,g)==g);std::cout<<"Subset operations passed naive, changing-divisor/modulus and full-rank checks\n";}