CP-Algorithms Library
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Pfaffian of Matrix (verify/linalg/pfaffian.test.cpp)
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- Last update: 2026-09-10 02:45:56+02:00
- Problem: https://judge.yosupo.jp/problem/pfaffian_of_matrix
Depends on
- cp-algo/linalg/matrix.hpp
- cp-algo/linalg/strassen.hpp
- cp-algo/linalg/vector.hpp
- cp-algo/math/common.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/simd.hpp
Code
// @brief Pfaffian of Matrix
#define PROBLEM "https://judge.yosupo.jp/problem/pfaffian_of_matrix"
#include <bits/stdc++.h>
#include "cp-algo/linalg/matrix.hpp"
int main() {
std::ios::sync_with_stdio(false);
std::cin.tie(nullptr);
size_t n;
std::cin >> n;
using base = cp_algo::math::modint<998244353LL>;
cp_algo::linalg::matrix<base> a(2 * n);
a.read();
std::cout << a.pfaffian() << '\n';
}
#line 1 "verify/linalg/pfaffian.test.cpp"
// @brief Pfaffian of Matrix
#define PROBLEM "https://judge.yosupo.jp/problem/pfaffian_of_matrix"
#include <bits/stdc++.h>
#line 1 "cp-algo/linalg/matrix.hpp"
#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/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 1 "cp-algo/linalg/vector.hpp"
#line 1 "cp-algo/number_theory/modint.hpp"
#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/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/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/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 14 "cp-algo/linalg/vector.hpp"
#include <ranges>
#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() == 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);
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() == 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 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, 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>(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) data[j].setr(packed[j]);
else packed[j] = uint32_t(data[j].getr());
}
}
}
template<class matrix>
static matrix product(matrix const& a, matrix const& b) {
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);
matrix res(a.n(), b.m());
copy<true>(res, cp, n, k, depth);
return res;
}
};
}
#pragma GCC pop_options
#line 8 "cp-algo/linalg/matrix.hpp"
#include <optional>
#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): Base(n, vec_t(n)) {}
matrix(size_t n, size_t m): Base(n, vec_t(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(size_t i) {return (*this)[i];}
auto const& row(size_t i) const {return (*this)[i];}
auto elements() {return *this | std::views::join;}
auto elements() const {return *this | 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(elements(), t.elements())) {
a += b;
}
return *this;
}
matrix& operator -=(matrix const& t) {
for(auto [a, b]: std::views::zip(elements(), t.elements())) {
a -= b;
}
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, std::bind(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(), m()+b.m());
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() {
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();
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);
}
if(pair) for(; i + 1 < last; i += 2) reduce_pair<mode>(j, i);
for(; i < last; i++) {
row(j).reduce_by(row(i));
if(pair) row(j + 1).reduce_by(row(i));
}
j += pair;
}
}
return normalize();
}
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 {
if(n() > m()) {
return T().rank();
}
auto A = *this;
A.gauss();
return std::ranges::count_if(A, [&](auto &row) {
return row.find_pivot().first < m();
});
}
base det() const {
assert(n() == m());
matrix b = *this;
b.echelonize();
base res = 1;
for(size_t i = 0; i < n(); i++) {
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 i = 1; i < n(); i++) {
for(size_t j = i + 1; j < n() && b[i].normalize(i - 1) == base(0); j++) {
if(b[j].normalize(i - 1) != base(0)) {
std::swap(b[i], b[j]);
for(size_t k = i; k < n(); k++) {
std::swap(b[k][i], b[k][j]);
}
res = -res;
}
}
b[i].normalize();
if(i % 2) {
res *= -b[i][i - 1];
if(res == base(0)) return res;
}
if(b[i][i - 1] == base(0)) continue;
base inv = base(1) / b[i][i - 1];
for(size_t j = i + 1; j < n(); j++) {
b[j].add_scaled(b[i], -b[j].normalize(i - 1) * inv, i);
}
}
return res;
}
std::pair<base, matrix> inv() const {
assert(n() == m());
matrix b = *this | eye(n());
if(size(b.echelonize<reverse>(n())[0]) < n()) {
return {0, {}};
}
base det = 1;
for(size_t i = 0; i < n(); i++) {
det *= b[i][i];
b[i] *= base(1) / b[i][i];
}
return {det, b.submatrix(std::views::all, std::views::drop(n()))};
}
// Can also just run gauss on T() | eye(m)
// but it would be slower :(
auto kernel() const {
auto A = *this;
auto [pivots, free] = A.template echelonize<reverse>();
matrix sols(size(free), m());
for(size_t j = 0; j < size(pivots); j++) {
base scale = A[j].find_pivot().second;
for(size_t i = 0; i < size(free); i++) {
sols[i][pivots[j]] = A[j][free[i]] * scale;
}
}
for(size_t i = 0; i < size(free); i++) {
sols[i][free[i]] = -1;
}
return sols;
}
// [solution, basis], transposed
std::optional<std::array<matrix, 2>> solve(matrix t) const {
matrix sols = (*this | t).kernel();
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 {
return std::array{
matrix(sols.submatrix(std::views::drop(sols.n() - t.m()), std::views::take(m()))),
matrix(sols.submatrix(std::views::take(sols.n() - t.m()), std::views::take(m())))
};
}
}
// 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:
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) {
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 = m();
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 < m() ? inv : base(0);
first = std::min(first, p);
}
if(first == m()) 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] == m()) 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) {
auto &p = row(src), &q = row(src + 1);
auto [u, pu] = p.find_pivot();
auto [v, qv] = q.find_pivot();
if(u == m() || v == m()) {
for(size_t j = dst; j < dst + 2; j++) {
row(j).reduce_by(p); row(j).reduce_by(q);
}
return;
}
auto scales = [&](vec_t &a) {
base s = -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 -= s * p[v];
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 5 "verify/linalg/pfaffian.test.cpp"
int main() {
std::ios::sync_with_stdio(false);
std::cin.tie(nullptr);
size_t n;
std::cin >> n;
using base = cp_algo::math::modint<998244353LL>;
cp_algo::linalg::matrix<base> a(2 * n);
a.read();
std::cout << a.pfaffian() << '\n';
}
Test cases
| Env | Name | Status | Elapsed | Memory |
|---|---|---|---|---|
| g++ | example_00 |
AC |
2 ms | 4 MB |
| g++ | example_01 |
AC |
2 ms | 4 MB |
| g++ | max_random_00 |
AC |
44 ms | 9 MB |
| g++ | max_random_01 |
AC |
44 ms | 9 MB |
| g++ | max_random_02 |
AC |
45 ms | 9 MB |
| g++ | max_random_03 |
AC |
42 ms | 9 MB |
| g++ | max_random_04 |
AC |
44 ms | 9 MB |
| g++ | random_00 |
AC |
13 ms | 6 MB |
| g++ | random_01 |
AC |
16 ms | 6 MB |
| g++ | random_02 |
AC |
4 ms | 4 MB |
| g++ | random_03 |
AC |
15 ms | 6 MB |
| g++ | random_04 |
AC |
2 ms | 4 MB |
| g++ | small_00 |
AC |
2 ms | 4 MB |
| g++ | small_01 |
AC |
3 ms | 4 MB |
| g++ | small_02 |
AC |
4 ms | 4 MB |
| g++ | small_03 |
AC |
3 ms | 4 MB |
| g++ | small_04 |
AC |
2 ms | 4 MB |
| g++ | sparse_00 |
AC |
26 ms | 9 MB |
| g++ | sparse_01 |
AC |
26 ms | 9 MB |
| g++ | sparse_02 |
AC |
27 ms | 9 MB |
| g++ | sparse_03 |
AC |
25 ms | 9 MB |
| g++ | sparse_04 |
AC |
26 ms | 9 MB |
| g++ | sparse_05 |
AC |
27 ms | 9 MB |
| g++ | sparse_06 |
AC |
25 ms | 9 MB |
| g++ | sparse_07 |
AC |
26 ms | 9 MB |
| g++ | sparse_08 |
AC |
27 ms | 9 MB |
| g++ | sparse_09 |
AC |
25 ms | 9 MB |