CP-Algorithms Library
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Matrix Determinant (verify/linalg/det.test.cpp)
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- Last update: 2026-10-16 14:16:08+02:00
- Problem: https://judge.yosupo.jp/problem/matrix_det
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 Matrix Determinant
#define PROBLEM "https://judge.yosupo.jp/problem/matrix_det"
#pragma GCC optimize("O3,unroll-loops")
#pragma GCC target("avx2")
#include <bits/stdc++.h>
//#include "blazingio/blazingio.min.hpp"
#define CP_ALGO_CHECKPOINT
#include "cp-algo/linalg/matrix.hpp"
using namespace std;
using namespace cp_algo::linalg;
using namespace cp_algo::math;
const int64_t mod = 998244353;
void solve() {
int n;
cin >> n;
matrix<modint<mod>> a(n, n);
a.read();
cout << a.det() << "\n";
}
signed main() {
//freopen("input.txt", "r", stdin);
ios::sync_with_stdio(0);
cin.tie(0);
int t = 1;
while(t--) {
solve();
}
}#line 1 "verify/linalg/det.test.cpp"
// @brief Matrix Determinant
#define PROBLEM "https://judge.yosupo.jp/problem/matrix_det"
#pragma GCC optimize("O3,unroll-loops")
#pragma GCC target("avx2")
#include <bits/stdc++.h>
//#include "blazingio/blazingio.min.hpp"
#define CP_ALGO_CHECKPOINT
#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 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 UInt remod() {
return modint::remod();
}
constexpr static UInt2 modmod() {
return UInt2(mod()) * mod();
}
constexpr modint_base() = default;
constexpr modint_base(Int2 rr) {
to_modint().setr(UInt((rr + modmod()) % mod()));
}
constexpr modint inv() const {
return bpow(to_modint(), mod() - 2);
}
modint operator-() const {
modint neg;
neg.r = std::min(-r, remod() - r);
return neg;
}
modint& operator/=(const modint& t) {
return to_modint() *= t.inv();
}
modint& operator*=(const modint& t) {
r = UInt(UInt2(r) * t.r % mod());
return to_modint();
}
modint& operator+=(const modint& t) {
r += t.r;
r = std::min(r, r - remod());
return to_modint();
}
modint& operator-=(const modint& t) {
r -= t.r;
r = std::min(r, r + remod());
return to_modint();
}
modint operator+(const modint& t) const { return modint(to_modint()) += t; }
modint operator-(const modint& t) const { return modint(to_modint()) -= t; }
modint operator*(const modint& t) const { return modint(to_modint()) *= t; }
modint operator/(const modint& t) const { return modint(to_modint()) /= t; }
// Why <=> doesn't work?..
auto operator==(const modint& t) const { return to_modint().getr() == t.getr(); }
auto operator!=(const modint& t) const { return to_modint().getr() != t.getr(); }
auto operator<=(const modint& t) const { return to_modint().getr() <= t.getr(); }
auto operator>=(const modint& t) const { return to_modint().getr() >= t.getr(); }
auto operator<(const modint& t) const { return to_modint().getr() < t.getr(); }
auto operator>(const modint& t) const { return to_modint().getr() > t.getr(); }
Int rem() const {
UInt R = to_modint().getr();
return R - (R > (UInt)mod() / 2) * mod();
}
constexpr void setr(UInt rr) {
r = rr;
}
constexpr UInt getr() const {
return r;
}
// Only use these if you really know what you're doing!
static uint64_t modmod8() { return uint64_t(8 * modmod()); }
void add_unsafe(UInt t) { r += t; }
void pseudonormalize() { r = std::min(r, r - modmod8()); }
modint const& normalize() {
if(r >= (UInt)mod()) {
r %= mod();
}
return to_modint();
}
void setr_direct(UInt rr) { r = rr; }
UInt getr_direct() const { return r; }
protected:
UInt r;
private:
constexpr modint& to_modint() { return static_cast<modint&>(*this); }
constexpr modint const& to_modint() const { return static_cast<modint const&>(*this); }
};
template<typename modint>
concept modint_type = std::is_base_of_v<modint_base<modint, typename modint::Int>, modint>;
template<modint_type modint>
decltype(std::cin)& operator>>(decltype(std::cin)& in, modint& x) {
typename modint::UInt r;
auto &res = in >> r;
x.setr(r);
return res;
}
template<modint_type modint>
decltype(std::cout)& operator<<(decltype(std::cout)& out, modint const& x) {
return out << x.getr();
}
template<auto m>
struct modint: modint_base<modint<m>, decltype(m)> {
using Base = modint_base<modint<m>, decltype(m)>;
using Base::Base;
static constexpr Base::Int mod() { return m; }
static constexpr Base::UInt remod() { return m; }
auto getr() const { return Base::r; }
};
// Odd moduli up to a quarter of the unsigned word keep Montgomery residues lazily in
// [0, 2 mod): remod() = 2 mod, and both 4 mod and ab + q * mod fit. Any other modulus keeps
// fully reduced residues, remod() = mod, which is what the sums of modint_base need then:
// an even one without the Montgomery form, a wide odd one with a reduction that subtracts
// high words instead of adding double words that would overflow.
template<typename Int = int>
struct dynamic_modint: modint_base<dynamic_modint<Int>, Int> {
using Base = modint_base<dynamic_modint<Int>, Int>;
using Base::Base;
// Out of line, so that the hot path stays as small as it was.
[[gnu::noinline, gnu::cold]] static Base::UInt m_reduce_reduced(Base::UInt2 ab) {
if(mod() % 2 == 0) {
return typename Base::UInt(ab % mod());
}
// q * mod has the low word of ab, so the difference of the high words is exact.
typename Base::UInt q = -(typename Base::UInt(ab) * inverse);
auto high = typename Base::UInt(ab >> Base::bits);
auto low = typename Base::UInt(typename Base::UInt2(q) * typename Base::UInt(mod()) >> Base::bits);
return high >= low ? high - low : high - low + mod();
}
static Base::UInt m_reduce(Base::UInt2 ab) {
if(imod() == 0) [[unlikely]] {
return m_reduce_reduced(ab);
}
typename Base::UInt2 m = typename Base::UInt(ab) * imod();
return typename Base::UInt((ab + m * mod()) >> Base::bits);
}
static Base::UInt m_transform(Base::UInt a) {
if(mod() % 2 == 0) [[unlikely]] {
return a;
} else {
return m_reduce(a * pw128());
}
}
dynamic_modint& operator*=(const dynamic_modint& t) {
Base::r = m_reduce(typename Base::UInt2(Base::r) * t.r);
return *this;
}
void setr(Base::UInt rr) {
Base::r = m_transform(rr);
}
Base::UInt getr() const {
typename Base::UInt res = m_reduce(Base::r);
return std::min(res, res - mod());
}
static Int mod() { return m; }
static Base::UInt remod() { return rm; }
static Base::UInt imod() { return im; }
static Base::UInt2 pw128() { return r2; }
static void switch_mod(Int nm) {
m = nm;
bool lazy = m % 2 && typename Base::UInt(m) <= typename Base::UInt(-1) / 4;
rm = typename Base::UInt(m) * (lazy ? 2 : 1);
inverse = m % 2 ? inv2(-m) : 0;
im = lazy ? inverse : 0;
r2 = static_cast<Base::UInt>(static_cast<Base::UInt2>(-1) % m + 1);
}
// Wrapper for temp switching
auto static with_mod(Int tmp, auto callback) {
struct scoped {
Int prev = mod();
~scoped() { switch_mod(prev); }
} _;
switch_mod(tmp);
return callback();
}
private:
static inline thread_local Int m = 1;
// im: -1 / mod modulo 2^bits for lazy residues and 0 for reduced ones; inverse: the same
// for every odd mod; rm: the value of remod().
static inline thread_local Base::UInt im = -1, r2 = 0, inverse = -1, rm = 2;
};
}
#line 1 "cp-algo/util/big_alloc.hpp"
#line 15 "cp-algo/util/big_alloc.hpp"
// Single macro to detect POSIX platforms (Linux, Unix, macOS)
#if defined(__linux__) || defined(__unix__) || (defined(__APPLE__) && defined(__MACH__))
#define CP_ALGO_USE_MMAP 1
#include <sys/mman.h>
#else
#define CP_ALGO_USE_MMAP 0
#endif
namespace cp_algo {
template<typename T, size_t Align = 32>
class big_alloc {
static_assert(Align >= alignof(void*), "Align must be at least pointer-size");
static_assert(std::has_single_bit(Align), "Align must be a power of two");
public:
using value_type = T;
template<class U>
struct rebind {
using other = big_alloc<U, Align>;
};
constexpr bool operator==(const big_alloc&) const = default;
big_alloc() noexcept = default;
template<typename U, std::size_t A>
big_alloc(const big_alloc<U, A>&) noexcept {}
[[nodiscard]] T* allocate(std::size_t n) {
std::size_t padded = round_up(n * sizeof(T));
std::size_t align = std::max<std::size_t>(alignof(T), Align);
#if CP_ALGO_USE_MMAP
if(padded >= MEGABYTE) {
void* raw = mmap(nullptr, padded, PROT_READ | PROT_WRITE, MAP_PRIVATE | MAP_ANONYMOUS, -1, 0);
madvise(raw, padded, MADV_HUGEPAGE);
return static_cast<T*>(raw);
}
#endif
return static_cast<T*>(::operator new(padded, std::align_val_t(align)));
}
void deallocate(T* p, std::size_t n) noexcept {
if(!p) {
return;
}
std::size_t padded = round_up(n * sizeof(T));
std::size_t align = std::max<std::size_t>(alignof(T), Align);
#if CP_ALGO_USE_MMAP
if(padded >= MEGABYTE) {
munmap(p, padded);
return;
}
#endif
::operator delete(p, padded, std::align_val_t(align));
}
private:
static constexpr std::size_t MEGABYTE = 1 << 20;
static constexpr std::size_t round_up(std::size_t x) noexcept {
return (x + Align - 1) / Align * Align;
}
};
template<typename T>
using big_vector = std::vector<T, big_alloc<T>>;
template<typename T>
using big_basic_string = std::basic_string<T, std::char_traits<T>, big_alloc<T>>;
template<typename T>
using big_deque = std::deque<T, big_alloc<T>>;
template<typename T>
using big_stack = std::stack<T, big_deque<T>>;
template<typename T>
using big_queue = std::queue<T, big_deque<T>>;
template<typename T>
using big_priority_queue = std::priority_queue<T, big_vector<T>>;
template<typename T>
using big_forward_list = std::forward_list<T, big_alloc<T>>;
using big_string = big_basic_string<char>;
template<typename Key, typename Value, typename Compare = std::less<Key>>
using big_map = std::map<Key, Value, Compare, big_alloc<std::pair<const Key, Value>>>;
template<typename T, typename Compare = std::less<T>>
using big_multiset = std::multiset<T, Compare, big_alloc<T>>;
template<typename T, typename Compare = std::less<T>>
using big_set = std::set<T, Compare, big_alloc<T>>;
}
#line 1 "cp-algo/util/simd.hpp"
#include <experimental/simd>
#line 7 "cp-algo/util/simd.hpp"
#if defined(__x86_64__) && !defined(CP_ALGO_DISABLE_AVX2)
#define CP_ALGO_SIMD_AVX2_TARGET _Pragma("GCC target(\"avx2,fma\")")
#else
#define CP_ALGO_SIMD_AVX2_TARGET
#endif
#define CP_ALGO_SIMD_PRAGMA_PUSH \
_Pragma("GCC push_options") \
CP_ALGO_SIMD_AVX2_TARGET
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo {
template<typename T, size_t len>
using simd [[gnu::vector_size(len * sizeof(T))]] = T;
using u64x8 = simd<uint64_t, 8>;
using u32x16 = simd<uint32_t, 16>;
using i64x4 = simd<int64_t, 4>;
using u64x4 = simd<uint64_t, 4>;
using u32x8 = simd<uint32_t, 8>;
using u16x16 = simd<uint16_t, 16>;
using i32x4 = simd<int32_t, 4>;
using u32x4 = simd<uint32_t, 4>;
using u16x8 = simd<uint16_t, 8>;
using u16x4 = simd<uint16_t, 4>;
using i16x4 = simd<int16_t, 4>;
using u8x32 = simd<uint8_t, 32>;
using u8x16 = simd<uint8_t, 16>;
using u8x8 = simd<uint8_t, 8>;
using u8x4 = simd<uint8_t, 4>;
using dx4 = simd<double, 4>;
inline dx4 abs(dx4 a) {
return dx4{
std::abs(a[0]),
std::abs(a[1]),
std::abs(a[2]),
std::abs(a[3])
};
}
// https://stackoverflow.com/a/77376595
// works for ints in (-2^51, 2^51)
static constexpr dx4 magic = dx4() + (3ULL << 51);
inline i64x4 lround(dx4 x) {
return i64x4(x + magic) - i64x4(magic);
}
inline dx4 to_double(i64x4 x) {
return dx4(x + i64x4(magic)) - magic;
}
inline dx4 round(dx4 a) {
return dx4{
std::nearbyint(a[0]),
std::nearbyint(a[1]),
std::nearbyint(a[2]),
std::nearbyint(a[3])
};
}
inline u64x4 low32(u64x4 x) {
return x & uint32_t(-1);
}
inline auto swap_bytes(auto x) {
return decltype(x)(__builtin_shufflevector(u32x8(x), u32x8(x), 1, 0, 3, 2, 5, 4, 7, 6));
}
inline u64x4 montgomery_reduce(u64x4 x, uint32_t mod, uint32_t imod) {
#ifdef __AVX2__
auto x_ninv = u64x4(_mm256_mul_epu32(__m256i(x), __m256i() + imod));
x += u64x4(_mm256_mul_epu32(__m256i(x_ninv), __m256i() + mod));
#else
auto x_ninv = u64x4(u32x8(low32(x)) * imod);
x += x_ninv * uint64_t(mod);
#endif
return swap_bytes(x);
}
// x - mod where x >= mod, for x in [0, 2 mod). Unsigned, so it holds up to mod < 2^31;
// zero upper halves of 64-bit lanes stay zero.
inline u32x8 reduce_once(u32x8 x, uint32_t mod) {
auto y = x - mod;
return x < y ? x : y;
}
inline u64x4 reduce_once(u64x4 x, uint32_t mod) {
return u64x4(reduce_once(u32x8(x), mod));
}
inline u64x4 montgomery_mul(u64x4 x, u64x4 y, uint32_t mod, uint32_t imod) {
#ifdef __AVX2__
return montgomery_reduce(u64x4(_mm256_mul_epu32(__m256i(x), __m256i(y))), mod, imod);
#else
return montgomery_reduce(x * y, mod, imod);
#endif
}
inline u32x8 montgomery_mul(u32x8 x, u32x8 y, uint32_t mod, uint32_t imod) {
return u32x8(montgomery_mul(u64x4(x), u64x4(y), mod, imod)) |
u32x8(swap_bytes(montgomery_mul(u64x4(swap_bytes(x)), u64x4(swap_bytes(y)), mod, imod)));
}
inline dx4 rotate_right(dx4 x) {
static constexpr u64x4 shuffler = {3, 0, 1, 2};
return __builtin_shuffle(x, shuffler);
}
template<std::size_t Align = 32>
inline bool is_aligned(const auto *p) noexcept {
return (reinterpret_cast<std::uintptr_t>(p) % Align) == 0;
}
template<class Target>
inline Target& vector_cast(auto &&p) {
return *reinterpret_cast<Target*>(std::assume_aligned<alignof(Target)>(&p));
}
}
#pragma GCC pop_options
#line 1 "cp-algo/util/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 17 "cp-algo/linalg/vector.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::linalg {
template<typename base, class Alloc = big_alloc<base>>
struct vec: std::basic_string<base, std::char_traits<base>, Alloc> {
using Base = std::basic_string<base, std::char_traits<base>, Alloc>;
using Base::Base;
vec(Base const& t): Base(t) {}
vec(Base&& t): Base(std::move(t)) {}
vec(size_t n): Base(n, base()) {}
vec(auto &&r): Base(r.begin(), r.end()) {}
static vec ei(size_t n, size_t i) {
vec res(n);
res[i] = 1;
return res;
}
auto operator-() const {
return *this | std::views::transform([](auto x) { return -x; });
}
auto operator*(base t) const {
return *this | std::views::transform([t](auto x) { return x * t; });
}
vec& operator*=(base t) {
for(auto &it: *this) {
it *= t;
}
return *this;
}
virtual void add_scaled(vec const& b, base scale, size_t i = 0) {
if(scale != base(0)) {
for(; i < size(*this); i++) {
(*this)[i] += scale * b[i];
}
}
}
virtual vec const& normalize() {
return static_cast<vec&>(*this);
}
virtual base normalize(size_t i) {
return (*this)[i];
}
void read() {
for(auto &it: *this) {
std::cin >> it;
}
}
void print() const {
for(auto &it: *this) {
std::cout << it << ' ';
}
std::cout << '\n';
}
static vec random(size_t n) {
vec res(n);
std::ranges::generate(res, random::rng);
return res;
}
// Concatenate vectors
vec operator|(vec const& t) const {
return std::views::join(std::array{std::views::all(*this), std::views::all(t)});
}
// Generally, vec shouldn't be modified
// after its pivot index is set
std::pair<size_t, base> find_pivot() {
if(pivot == size_t(-1)) {
pivot = 0;
while(pivot < size(*this) && normalize(pivot) == base(0)) {
pivot++;
}
if(pivot < size(*this)) {
pivot_inv = base(1) / (*this)[pivot];
}
}
return {pivot, pivot_inv};
}
void reduce_by(vec& t) {
auto [pivot, pinv] = t.find_pivot();
if(pivot < size(*this)) {
add_scaled(t, -normalize(pivot) * pinv, pivot);
}
}
private:
size_t pivot = -1;
base pivot_inv;
};
template<math::modint_type base, class Alloc = big_alloc<base>>
struct modint_vec: vec<base, Alloc> {
using Base = vec<base, Alloc>;
using Base::Base;
modint_vec(Base const& t): Base(t) {}
modint_vec(Base&& t): Base(std::move(t)) {}
void add_scaled(Base const& b, base scale, size_t i = 0) override {
static_assert(base::bits >= 64, "Only wide modint types for linalg");
if(scale != base(0)) {
assert(Base::size() == b.size());
size_t n = size(*this);
u64x4 scaler = u64x4() + scale.getr();
bool aligned = is_aligned(&(*this)[0]) && is_aligned(&b[0]);
if(aligned) {
i -= i % 4;
}
bool reduce = ++counter == accumulation_period();
if(reduce) {
counter = 0;
for(size_t j = 0; j < i; j++) {
(*this)[j].pseudonormalize();
}
}
if(aligned) {
for(; i + 3 < n; i += 4) {
auto &ai = vector_cast<u64x4>((*this)[i]);
auto bi = vector_cast<u64x4 const>(b[i]);
#ifdef __AVX2__
ai += u64x4(_mm256_mul_epu32(__m256i(scaler), __m256i(bi)));
#else
ai += scaler * bi;
#endif
if(reduce) {
ai = shrink(ai);
}
}
}
for(; i < n; i++) {
(*this)[i].add_unsafe(b[i].getr_direct() * scale.getr());
if(reduce) {
(*this)[i].pseudonormalize();
}
}
}
}
Base const& normalize() override {
for(auto &it: *this) {
it.normalize();
}
return *this;
}
base normalize(size_t i) override {
return (*this)[i].normalize();
}
private:
template<typename, typename>
friend struct matrix;
static size_t accumulation_period() {
// Eight canonical products keep the accumulator below 16 * mod^2.
// Montgomery residues can be wider, so retain four updates there.
return base::remod() == typename base::UInt(base::mod()) && base::mod() < (1LL << 30) ? 8 : 4;
}
static u64x4 mul(u64x4 a, u64x4 b) {
#ifdef __AVX2__
return u64x4(_mm256_mul_epu32(__m256i(a), __m256i(b)));
#else
return a * b;
#endif
}
static u64x4 shrink(u64x4 a) {
auto b = a - (u64x4() + base::modmod8());
return a < b ? a : b;
}
// Two source contributions to two distinct rows; sources must be normalized.
static void add_scaled_pair(modint_vec& x, modint_vec& y, Base const& p, Base const& q, std::array<base, 4> c,
size_t first = 0) {
if(std::ranges::find(c, base(0)) != c.end()) {
x.add_scaled(p, c[0], first);
x.add_scaled(q, c[1], first);
y.add_scaled(p, c[2], first);
y.add_scaled(q, c[3], first);
return;
}
size_t n = x.size();
assert(y.size() == n && p.size() == n && q.size() == n && first <= n);
// Before first, sources are zero for elimination or destinations are unused for Pfaffians.
if(is_aligned(x.data()) && is_aligned(y.data()) && is_aligned(p.data()) && is_aligned(q.data())) {
first -= first % 4;
}
size_t period = accumulation_period();
auto prepare = [&](modint_vec& a) {
// A pair must not cross the accumulator's reduction boundary.
if(a.counter + 2 > period) {
for(auto &v: a) {
v.pseudonormalize();
}
a.counter = 0;
}
a.counter += 2;
if(a.counter != period) {
return false;
}
a.counter = 0;
for(size_t i = 0; i < first; i++) {
a[i].pseudonormalize();
}
return true;
};
bool nx = prepare(x), ny = prepare(y);
auto *__restrict__ dx = x.data();
auto *__restrict__ dy = y.data();
auto const* __restrict__ sp = p.data();
auto const* __restrict__ sq = q.data();
uint64_t xp = c[0].getr(), xq = c[1].getr(), yp = c[2].getr(), yq = c[3].getr();
u64x4 xp4 = u64x4() + xp, xq4 = u64x4() + xq;
u64x4 yp4 = u64x4() + yp, yq4 = u64x4() + yq;
size_t i = first;
for(; i + 4 <= n; i += 4) {
u64x4 vx, vy, vp, vq;
std::memcpy(&vx, dx + i, sizeof vx);
std::memcpy(&vy, dy + i, sizeof vy);
std::memcpy(&vp, sp + i, sizeof vp);
std::memcpy(&vq, sq + i, sizeof vq);
vx += mul(xp4, vp) + mul(xq4, vq);
vy += mul(yp4, vp) + mul(yq4, vq);
if(nx) {
vx = shrink(vx);
}
if(ny) {
vy = shrink(vy);
}
std::memcpy(dx + i, &vx, sizeof vx);
std::memcpy(dy + i, &vy, sizeof vy);
}
for(; i < n; i++) {
dx[i].add_unsafe(xp * sp[i].getr_direct() + xq * sq[i].getr_direct());
dy[i].add_unsafe(yp * sp[i].getr_direct() + yq * sq[i].getr_direct());
if(nx) {
dx[i].pseudonormalize();
}
if(ny) {
dy[i].pseudonormalize();
}
}
}
size_t counter = 0;
};
// Narrow rows keep canonical residues; small batches accumulate in wide registers.
template<typename base>
requires(base::bits <= 32)
struct modint_vec<base>: vec<base> {
using Base = vec<base>;
using Base::Base;
modint_vec(Base const& t): Base(t) {}
modint_vec(Base&& t): Base(std::move(t)) {}
void add_scaled(Base const& b, base scale, size_t first = 0) override {
if(scale == base(0)) {
return;
}
if(&b == this) {
Base::add_scaled(b, scale, first);
} else {
add_scaled_batch<1, 1>({this}, {&b}, {scale}, first);
}
}
private:
template<typename, typename>
friend struct matrix;
static constexpr size_t batch_size = 8;
static constexpr bool use_simd = [] {
if constexpr(requires {
std::integral_constant<uint32_t, base::mod()>{};
std::integral_constant<uint32_t, base::remod()>{};
}) {
return sizeof(base) == sizeof(uint32_t) && base::mod() > 1 && base::mod() % 2 &&
base::mod() < (1U << 30) && base::remod() == typename base::UInt(base::mod());
} else {
return false;
}
}();
static u64x4 mul(u64x4 a, u64x4 b) {
#ifdef __AVX2__
return u64x4(_mm256_mul_epu32(__m256i(a), __m256i(b)));
#else
return low32(a) * low32(b);
#endif
}
// Sources are normalized and do not alias the distinct destination rows.
template<size_t count, size_t rows>
static void add_scaled_batch(std::array<modint_vec*, rows> const& dst,
std::array<Base const*, count> const& src, std::array<base, rows * count> const& c,
size_t first = 0) {
static_assert(count <= batch_size && (rows == 1 || rows == 2));
if(std::ranges::all_of(c, [](base value) { return value == base(0); })) {
return;
}
if constexpr(use_simd) {
constexpr uint32_t mod = base::mod(), inv = math::inv2(uint32_t(-mod));
std::array<uint32_t, rows * count> scale;
for(size_t t = 0; t < rows * count; t++) {
scale[t] = uint32_t((uint64_t(c[t].getr()) << 32) % mod);
}
auto *__restrict__ dx = dst[0]->data();
auto *__restrict__ dy = rows == 2 ? dst[1]->data() : nullptr;
size_t n = dst[0]->size();
for(; first + 8 <= n; first += 8) {
u64x4 acc[rows][2]{};
#pragma GCC unroll 1
for(size_t t = 0; t < count; t++) {
u64x4 p;
std::memcpy(&p, src[t]->data() + first, sizeof p);
auto q = p >> 32;
for(size_t row = 0; row < rows; row++) {
auto v = u64x4(u32x8() + scale[row * count + t]);
acc[row][0] += mul(p, v);
acc[row][1] += mul(q, v);
}
}
for(size_t row = 0; row < rows; row++) {
// At most eight products: reduction gives <3*mod, then add the old residue.
for(auto &v: acc[row]) {
v = montgomery_reduce(v, mod, inv);
}
auto *out = (row ? dy : dx) + first;
u32x8 old;
std::memcpy(&old, out, sizeof old);
auto z = old + u32x8(acc[row][0] | (acc[row][1] << 32));
z = z < z - 2 * mod ? z : z - 2 * mod;
z = z < z - mod ? z : z - mod;
std::memcpy(out, &z, sizeof z);
}
}
}
for(size_t t = 0; t < count; t++) {
for(size_t row = 0; row < rows; row++) {
dst[row]->Base::add_scaled(*src[t], c[row * count + t], first);
}
}
}
static void add_scaled_pair(modint_vec& x, modint_vec& y, Base const& p, Base const& q, std::array<base, 4> c,
size_t first = 0) {
add_scaled_batch<2, 2>({&x, &y}, {&p, &q}, c, first);
}
};
}
#pragma GCC pop_options
#line 1 "cp-algo/linalg/strassen.hpp"
#line 4 "cp-algo/linalg/strassen.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::linalg::impl {
template<class row>
constexpr bool use_strassen = false;
template<auto mod>
constexpr bool use_strassen<modint_vec<math::modint<mod>>> = mod > 1 && mod % 2 && mod < (1LL << 30);
// Internal packed storage; the public matrix keeps its usual row representation.
template<uint32_t mod>
struct strassen_product {
struct view {
uint32_t* data;
size_t stride;
uint32_t* operator[](size_t i) const { return data + i * stride; }
};
static u64x4 mul(u64x4 a, u64x4 b) {
#ifdef __AVX2__
return u64x4(_mm256_mul_epu32(__m256i(a), __m256i(b)));
#else
return low32(a) * low32(b);
#endif
}
static u64x4 shrink(u64x4 x) {
// x < 4*mod*2^32; reduce the high word modulo 2*mod.
auto words = u32x8(x);
auto bound = u32x8(u64x4() + (uint64_t(2) * mod << 32));
#ifdef __AVX2__
return u64x4(_mm256_min_epu32(__m256i(words), __m256i(words - bound)));
#else
return u64x4(words < words - bound ? words : words - bound);
#endif
}
template<size_t dim = 0>
[[gnu::noinline]] static void leaf(view a, view b, view c, size_t n, size_t m, size_t k) {
if constexpr(dim) {
n = m = k = dim;
a.stride = b.stride = c.stride = dim;
}
for(size_t i = 0; i < n; i += 4) {
for(size_t j = 0; j < k; j += 8) {
u64x4 acc[4][2]{};
for(size_t first = 0; first < m; first += 8) {
#pragma GCC unroll 1
for(size_t z = first; z < first + 8; z++) {
u64x4 x;
std::memcpy(&x, b[z] + j, sizeof x);
u64x4 scales[4];
for(size_t t = 0; t < 4; t++) {
scales[t] = u64x4(u32x8() + a[i + t][z]);
}
for(size_t t = 0; t < 4; t++) {
acc[t][0] += mul(scales[t], x);
}
x >>= 32;
for(size_t t = 0; t < 4; t++) {
acc[t][1] += mul(scales[t], x);
}
}
for(auto &row: acc) {
for(auto &x: row) {
x = shrink(x);
}
}
}
for(size_t t = 0; t < 4; t++) {
constexpr uint32_t inv = math::inv2(uint32_t(-mod));
// x < 2*mod*2^32, so Montgomery reduction yields a value below 3*mod.
for(auto &x: acc[t]) {
x = montgomery_reduce(x, mod, inv);
}
u32x8 out = u32x8(acc[t][0] | (acc[t][1] << 32));
out = out < out - mod ? out : out - mod;
out = out < out - mod ? out : out - mod;
std::memcpy(c[i + t] + j, &out, sizeof out);
}
}
}
}
// c = a +/- b; c may alias a.
template<bool subtract = false>
static void combine(uint32_t* a, uint32_t* b, uint32_t* c, size_t n, size_t m) {
for(size_t j = 0; j < n * m; j += 8) {
u32x8 x, y;
std::memcpy(&x, a + j, sizeof x);
std::memcpy(&y, b + j, sizeof y);
u32x8 z = subtract ? x + mod - y : x + y;
z = z < z - mod ? z : z - mod;
std::memcpy(c + j, &z, sizeof z);
}
}
static bool can_split(size_t n, size_t m, size_t k) {
return std::min({n, m, k}) > 64 && n % 16 == 0 && m % 16 == 0 && k % 16 == 0;
}
[[gnu::noinline]] static void multiply(uint32_t* a, uint32_t* b, uint32_t* c, size_t n, size_t m, size_t k,
uint32_t* work) {
// Every leaf dimension must remain a multiple of eight.
if(!can_split(n, m, k)) {
if(n == 64 && m == 64 && k == 64) {
leaf<64>({a, m}, {b, k}, {c, k}, n, m, k);
} else {
leaf({a, m}, {b, k}, {c, k}, n, m, k);
}
return;
}
n /= 2;
m /= 2;
k /= 2;
auto s = work, t = s + n * m, p = t + m * k;
work += n * m + m * k + n * k;
auto a00 = a, a01 = a + n * m, a10 = a + 2 * n * m, a11 = a + 3 * n * m;
auto b00 = b, b01 = b + m * k, b10 = b + 2 * m * k, b11 = b + 3 * m * k;
auto c00 = c, c01 = c + n * k, c10 = c + 2 * n * k, c11 = c + 3 * n * k;
// Winograd's schedule uses seven products and fifteen additions.
multiply(a00, b00, c11, n, m, k, work); // P1
multiply(a01, b10, c00, n, m, k, work); // P2
combine(c00, c11, c00, n, k); // C00 = P1 + P2
combine(a10, a11, s, n, m);
combine<true>(b01, b00, t, m, k); // S1, T1
multiply(s, t, c01, n, m, k, work); // P5
combine<true>(s, a00, s, n, m);
combine<true>(b11, t, t, m, k); // S2, T2
multiply(s, t, c10, n, m, k, work); // P6
combine(c11, c10, c10, n, k); // U2 = P1 + P6
combine<true>(a01, s, s, n, m); // S4
multiply(s, b11, p, n, m, k, work); // P3
combine(c10, c01, c11, n, k); // U4 = U2 + P5
combine(c11, p, c01, n, k); // C01 = U4 + P3
combine<true>(t, b10, t, m, k); // T4
multiply(a11, t, p, n, m, k, work); // P4
combine<true>(c10, p, c10, n, k);
combine<true>(a00, a10, s, n, m);
combine<true>(b11, b01, t, m, k); // S3, T3
multiply(s, t, p, n, m, k, work); // P7
combine(c10, p, c10, n, k);
combine(c11, p, c11, n, k); // C10, C11
}
static u32x8 encode(u32x8 x) {
// Scale by 2^32 with a fixed reciprocal; its quotient is off by at most one.
constexpr uint32_t scale = (uint64_t(1) << 32) % mod;
constexpr uint32_t quotient = (uint64_t(scale) << 32) / mod;
auto packed = u64x4(x), q = u64x4() + quotient;
auto lo = mul(packed, q) >> 32, hi = mul(packed >> 32, q);
auto approx = u32x8(lo | (hi & (~uint64_t(0) << 32)));
auto out = x * scale - approx * mod;
return out < out - mod ? out : out - mod;
}
// Copy between matrix rows and contiguous recursive quadrants.
template<bool unpack = false, int add = 0, class matrix>
static void copy(matrix& a, uint32_t* ptr, size_t n, size_t m, size_t depth, size_t row = 0, size_t col = 0) {
if(depth) {
n /= 2;
m /= 2;
for(size_t q = 0; q < 4; q++) {
copy<unpack, add>(a, ptr + q * n * m, n, m, depth - 1, row + q / 2 * n, col + q % 2 * m);
}
return;
}
if(col >= a.m()) {
return;
}
size_t width = std::min(m, a.m() - col);
for(size_t i = row; i < std::min(row + n, a.n()); i++) {
auto data = a[i].data() + col;
auto packed = ptr + (i - row) * m;
for(size_t j = 0; j < width; j++) {
if constexpr(unpack && add != 0) {
typename matrix::base value;
value.setr(packed[j]);
if constexpr(add == 1) {
data[j] += value;
} else {
data[j] -= value;
}
} else if constexpr(unpack) {
data[j].setr(packed[j]);
} else {
packed[j] = uint32_t(data[j].getr());
}
}
}
}
template<int add = 0, class matrix>
static matrix product(matrix const& a, matrix const& b, matrix res = {}) {
static_assert(add >= -1 && add <= 1);
auto pad = [](size_t x) {
return (x + 31) / 32 * 32;
};
size_t n = pad(a.n()), m = pad(a.m()), k = pad(b.m());
// Each recursion level needs a quarter as much scratch; children reuse it.
size_t entries = n * m + m * k + n * k;
// Small products avoid repeated mmap/madvise setup for temporary storage.
std::vector<uint32_t> small;
big_vector<uint32_t> large;
size_t count = entries + entries / 3;
auto ap = count < (1 << 21) ? (small.resize(count), small.data()) : (large.resize(count), large.data());
auto bp = ap + n * m, cp = bp + m * k;
auto scratch = cp + n * k;
// A, B, and C must use the same depth, including rectangular products.
size_t depth = 0;
for(size_t x = n, y = m, z = k; can_split(x, y, z); x /= 2, y /= 2, z /= 2) {
depth++;
}
copy(a, ap, n, m, depth);
copy(b, bp, m, k, depth);
// Only A is scaled; each leaf's Montgomery reduction removes the factor.
for(size_t i = 0; i < n * m; i += 8) {
u32x8 x;
std::memcpy(&x, ap + i, sizeof x);
x = encode(x);
std::memcpy(ap + i, &x, sizeof x);
}
multiply(ap, bp, cp, n, m, k, scratch);
if constexpr(add == 0) {
res = matrix(a.n(), b.m());
}
assert(res.n() == a.n() && res.m() == b.m());
copy<true, add>(res, cp, n, k, depth);
return res;
}
};
}
#pragma GCC pop_options
#line 12 "cp-algo/linalg/matrix.hpp"
CP_ALGO_SIMD_PRAGMA_PUSH
namespace cp_algo::linalg {
enum gauss_mode { normal, reverse };
template<typename base_t,
class _vec_t = std::conditional_t<math::modint_type<base_t>, modint_vec<base_t>, vec<base_t>>>
struct matrix: big_vector<_vec_t> {
using vec_t = _vec_t;
using base = base_t;
using Base = big_vector<vec_t>;
using Base::Base;
matrix(size_t n): matrix(n, n) {}
matrix(size_t n, size_t m) {
Base::reserve(n);
for(size_t i = 0; i < n; i++) {
Base::emplace_back(m);
}
}
matrix(Base const& t): Base(t) {}
matrix(Base&& t): Base(std::move(t)) {}
template<std::ranges::input_range R>
matrix(R&& r): Base(r.begin(), r.end()) {}
size_t n() const { return size(*this); }
size_t m() const { return n() ? size(row(0)) : 0; }
void resize(size_t n, size_t m) {
Base::resize(n);
for(auto &it: *this) {
it.resize(m);
}
}
auto &row(this auto&& self, size_t i) { return self[i]; }
auto elements(this auto&& self) { return self | std::views::join; }
matrix operator-() const {
return *this | std::views::transform([](auto const& x) { return vec_t(-x); });
}
matrix& operator+=(matrix const& t) {
for(auto [a, b]: std::views::zip(*this, t)) {
for(auto [x, y]: std::views::zip(a, b)) {
x += y;
}
}
return *this;
}
matrix& operator-=(matrix const& t) {
for(auto [a, b]: std::views::zip(*this, t)) {
for(auto [x, y]: std::views::zip(a, b)) {
x -= y;
}
}
return *this;
}
matrix operator+(matrix const& t) const { return matrix(*this) += t; }
matrix operator-(matrix const& t) const { return matrix(*this) -= t; }
matrix& operator*=(base t) {
for(auto &it: *this) {
it *= t;
}
return *this;
}
matrix operator*(base t) const { return matrix(*this) *= t; }
matrix& operator/=(base t) { return *this *= base(1) / t; }
matrix operator/(base t) const { return matrix(*this) /= t; }
// Make sure the result is matrix, not Base
matrix& operator*=(matrix const& t) { return *this = *this * t; }
void read_transposed() {
for(size_t j = 0; j < m(); j++) {
for(size_t i = 0; i < n(); i++) {
std::cin >> (*this)[i][j];
}
}
}
void read() {
for(auto &it: *this) {
it.read();
}
}
void print() const {
for(auto const& it: *this) {
it.print();
}
}
static matrix block_diagonal(big_vector<matrix> const& blocks) {
size_t n = 0;
for(auto &it: blocks) {
assert(it.n() == it.m());
n += it.n();
}
matrix res(n);
n = 0;
for(auto &it: blocks) {
for(size_t i = 0; i < it.n(); i++) {
std::ranges::copy(it[i], begin(res[n + i]) + n);
}
n += it.n();
}
return res;
}
static matrix random(size_t n, size_t m) {
matrix res(n, m);
std::ranges::generate(res, [m] { return vec_t::random(m); });
return res;
}
static matrix random(size_t n) {
return random(n, n);
}
static matrix eye(size_t n) {
matrix res(n);
for(size_t i = 0; i < n; i++) {
res[i][i] = 1;
}
return res;
}
// Concatenate matrices
matrix operator|(matrix const& b) const {
assert(n() == b.n());
matrix res(n(), 0);
for(size_t i = 0; i < n(); i++) {
res[i] = row(i) | b[i];
}
return res;
}
void assign_submatrix(auto viewx, auto viewy, matrix const& t) {
for(auto [a, b]: std::views::zip(*this | viewx, t)) {
std::ranges::copy(b, begin(a | viewy));
}
}
auto submatrix(auto viewx, auto viewy) const {
return *this | viewx | std::views::transform([viewy](auto const& y) {
return y | viewy;
});
}
matrix T() const {
matrix res(m(), n());
constexpr size_t block = 128;
for(size_t first = 0; first < m(); first += block) {
size_t last = std::min(first + block, m());
for(size_t i = 0; i < n(); i++) {
auto const& src = row(i);
for(size_t j = first; j < last; j++) {
res[j][i] = src[j];
}
}
}
return res;
}
matrix operator*(matrix const& b) const {
assert(m() == b.n());
if constexpr(impl::use_strassen<vec_t>) {
if(std::min({n(), m(), b.m()}) >= (base::bits <= 32 ? 64 : 512)) {
return impl::strassen_product<base::mod()>::product(*this, b);
}
}
matrix res(n(), b.m());
if constexpr(requires { requires vec_t::use_simd; }) {
if(n() == 1) {
res[0] = b.apply(row(0));
return res;
}
}
constexpr size_t block = 32;
for(size_t first = 0; first < m(); first += block) {
size_t last = std::min(first + block, m());
size_t i = 0;
for(; i + 1 < n(); i += 2) {
size_t j = first;
if constexpr(requires { requires vec_t::use_simd; }) {
constexpr size_t batch = vec_t::batch_size;
if(b.m() >= 8) {
for(; j + batch <= last; j += batch) {
std::array<typename vec_t::Base const*, batch> src;
std::array<base, 2 * batch> c;
for(size_t t = 0; t < batch; t++) {
src[t] = &b[j + t];
c[t] = row(i)[j + t];
c[batch + t] = row(i + 1)[j + t];
}
vec_t::template add_scaled_batch<batch, 2>({&res[i], &res[i + 1]}, src, c);
}
}
}
for(; j + 1 < last; j += 2) {
add_scaled_pair(res[i], res[i + 1], b[j], b[j + 1],
{row(i)[j], row(i)[j + 1], row(i + 1)[j], row(i + 1)[j + 1]});
}
if(j < last) {
res[i].add_scaled(b[j], row(i)[j]);
res[i + 1].add_scaled(b[j], row(i + 1)[j]);
}
}
for(; i < n(); i++) {
for(size_t j = first; j < last; j++) {
res[i].add_scaled(b[j], row(i)[j]);
}
}
}
res.normalize();
return res;
}
vec_t apply(vec_t const& x) const {
assert(x.size() == n());
vec_t res(m());
size_t i = 0;
if constexpr(requires { requires vec_t::use_simd; }) {
constexpr size_t batch = vec_t::batch_size;
for(; i + batch <= n(); i += batch) {
std::array<typename vec_t::Base const*, batch> src;
std::array<base, batch> c;
for(size_t t = 0; t < batch; t++) {
src[t] = &row(i + t);
c[t] = x[i + t];
}
vec_t::template add_scaled_batch<batch, 1>({&res}, src, c);
}
}
for(; i < n(); i++) {
res.add_scaled(row(i), x[i]);
}
res.normalize();
return res;
}
matrix pow(uint64_t k) const {
assert(n() == m());
return bpow<3>(*this, k, eye(n()));
}
matrix& normalize() {
for(auto &it: *this) {
it.normalize();
}
return *this;
}
template<gauss_mode mode = normal>
void eliminate(size_t i, size_t k) {
auto kinv = base(1) / row(i).normalize()[k];
for(size_t j = (mode == normal) * i; j < n(); j++) {
if(j != i) {
row(j).add_scaled(row(i), -row(j).normalize(k) * kinv);
}
}
}
template<gauss_mode mode = normal>
void eliminate(size_t i) {
row(i).normalize();
for(size_t j = (mode == normal) * i; j < n(); j++) {
if(j != i) {
row(j).reduce_by(row(i));
}
}
}
template<gauss_mode mode = normal>
matrix& gauss() {
return gauss_to<mode>(m());
}
private:
// Restrict pivot columns, while updating every column of each row.
template<gauss_mode mode>
matrix& gauss_to(size_t lim) {
constexpr size_t block = 32;
for(size_t first = 0; first < n(); first += block) {
size_t last = std::min(first + block, n());
// Reduce the pivot block before applying it to the other rows.
for(size_t i = first; i < last; i++) {
row(i).normalize();
if(row(i).find_pivot().first >= lim) {
continue;
}
for(size_t j = mode == normal ? i + 1 : first; j < last; j++) {
if(j != i) {
row(j).reduce_by(row(i));
}
}
}
if constexpr(mode == reverse) {
// Later pivots may have changed earlier rows in this block.
for(size_t i = first; i < last; i++) {
row(i).normalize();
}
}
for(size_t j = mode == normal ? last : 0; j < n(); j++) {
if(j >= first && j < last) {
continue;
}
bool pair = j + 1 < n() && j + 1 != first;
size_t i = first;
if constexpr(requires { requires vec_t::use_simd; }) {
constexpr size_t batch = vec_t::batch_size;
if(pair) {
for(; i + batch <= last; i += batch) {
reduce_batch<mode, batch>(j, i, lim);
}
}
}
if(pair) {
for(; i + 1 < last; i += 2) {
reduce_pair<mode>(j, i, lim);
}
}
for(; i < last; i++) {
if(row(i).find_pivot().first >= lim) {
continue;
}
row(j).reduce_by(row(i));
if(pair) {
row(j + 1).reduce_by(row(i));
}
}
j += pair;
}
}
// Forward pivots are normalized before use and never modified afterward.
return mode == reverse ? normalize() : *this;
}
public:
template<gauss_mode mode = normal>
auto echelonize(size_t lim) {
return gauss<mode>().sort_classify(lim);
}
template<gauss_mode mode = normal>
auto echelonize() {
return echelonize<mode>(m());
}
size_t rank() const {
auto A = n() > m() ? T() : *this;
A.gauss();
return std::ranges::count_if(A, [&](auto &row) {
return row.find_pivot().first < A.m();
});
}
base det() const {
assert(n() == m());
matrix b = *this;
b.gauss();
base res = 1;
for(size_t i = 0; i < n(); i++) {
for(size_t p = b[i].find_pivot().first; p != i; p = b[i].find_pivot().first) {
if(p == n()) {
return 0;
}
std::swap(b[i], b[p]);
res = -res;
}
res *= b[i][i];
}
return res;
}
// Pfaffian of an alternating matrix over a field.
base pfaffian() const {
assert(n() == m() && n() % 2 == 0);
matrix b = *this;
base res = 1;
for(size_t k = 0; k < n(); k += 2) {
size_t q = k + 1, p = q;
while(p < n() && b[k].normalize(p) == base(0)) {
p++;
}
if(p == n()) {
return base(0);
}
if(p != q) {
res = -res;
// Normalize values before moving them between row accumulators.
b[q].normalize();
b[p].normalize();
// Symmetric permutation, storing only the upper triangle.
for(size_t i = 0; i < q; i++) {
std::swap(b[i][q], b[i][p]);
}
for(size_t i = q + 1; i < p; i++) {
std::swap(b[q][i], b[i][p]);
b[q][i] = -b[q].normalize(i);
b[i][p] = -b[i].normalize(p);
}
for(size_t i = p + 1; i < n(); i++) {
std::swap(b[q][i], b[p][i]);
}
b[q][p] = -b[q].normalize(p);
}
b[k].normalize();
b[q].normalize();
res *= b[k][q];
base inv = base(1) / b[k][q];
for(size_t i = k + 2; i + 1 < n(); i += 2) {
add_scaled_pair(b[i], b[i + 1], b[k], b[q],
{b[q][i] * inv, -b[k][i] * inv, b[q][i + 1] * inv, -b[k][i + 1] * inv}, i + 1);
}
}
return res;
}
std::pair<base, matrix> inv() const {
assert(n() == m());
matrix b(n(), 2 * n());
for(size_t i = 0; i < n(); i++) {
std::ranges::copy(row(i), begin(b[i]));
b[i][n() + i] = 1;
}
// Pivots in the appended identity cannot make A invertible.
auto [pivots, free] = b.template gauss_to<normal>(n()).sort_classify(n());
if(size(pivots) < n()) {
return {0, {}};
}
base det = 1;
for(size_t i = 0; i < n(); i++) {
det *= b[i][i];
}
matrix res = b.submatrix(std::views::all, std::views::drop(n()));
back_substitute(b, pivots, res);
return {det, std::move(res)};
}
auto kernel() const {
return kernel_of(*this);
}
// [solution, basis], transposed
std::optional<std::array<matrix, 2>> solve(matrix const& t) const {
matrix sols = kernel_of(*this | t);
if(sols.n() < t.m() ||
matrix(sols.submatrix(std::views::drop(sols.n() - t.m()), std::views::drop(m()))) != -eye(t.m())) {
return std::nullopt;
} else {
for(auto &row: sols) {
row.resize(m());
}
auto first = begin(sols) + (sols.n() - t.m());
matrix sol(std::make_move_iterator(first), std::make_move_iterator(end(sols)));
sols.erase(first, end(sols));
return std::array{std::move(sol), std::move(sols)};
}
}
// To be called after a gaussian elimination run
// Sorts rows by pivots and classifies
// variables into pivots and free
auto sort_classify(size_t lim) {
size_t rk = 0;
big_vector<size_t> free, pivots;
for(size_t j = 0; j < lim; j++) {
for(size_t i = rk + 1; i < n() && row(rk)[j] == base(0); i++) {
if(row(i)[j] != base(0)) {
std::swap(row(i), row(rk));
row(rk) = -row(rk);
}
}
if(rk < n() && row(rk)[j] != base(0)) {
pivots.push_back(j);
rk++;
} else {
free.push_back(j);
}
}
return std::array{std::move(pivots), std::move(free)};
}
private:
// Own the working matrix so augmented temporaries do not need a copy.
static matrix kernel_of(matrix A) {
auto [pivots, free] = A.echelonize();
matrix rhs(size(pivots), size(free));
for(size_t i = 0; i < size(pivots); i++) {
for(size_t j = 0; j < size(free); j++) {
rhs[i][j] = A[i][free[j]];
}
}
back_substitute(A, pivots, rhs);
matrix sols(size(free), A.m());
for(size_t j = 0; j < size(pivots); j++) {
for(size_t i = 0; i < size(free); i++) {
sols[i][pivots[j]] = rhs[j][i];
}
}
for(size_t i = 0; i < size(free); i++) {
sols[i][free[i]] = -1;
}
return sols;
}
// Solve an upper-echelon system for the supplied right-hand sides.
static void back_substitute(matrix const& A, auto const& pivots, matrix& b) {
for(size_t i = size(pivots); i-- > 0;) {
b[i].normalize();
b[i] *= base(1) / A[i][pivots[i]];
for(size_t j = 0; j < i; j++) {
b[j].add_scaled(b[i], -A[j][pivots[i]]);
}
}
}
static void add_scaled_pair(vec_t& x, vec_t& y, vec_t const& p, vec_t const& q, std::array<base, 4> c,
size_t first = 0) {
if constexpr(requires { vec_t::add_scaled_pair(x, y, p, q, c, first); }) {
vec_t::add_scaled_pair(x, y, p, q, c, first);
} else {
x.add_scaled(p, c[0], first);
x.add_scaled(q, c[1], first);
y.add_scaled(p, c[2], first);
y.add_scaled(q, c[3], first);
}
}
// Determine the sequential pivot coefficients before updating the full rows.
template<gauss_mode mode, size_t count>
void reduce_batch(size_t dst, size_t src, size_t lim) {
static_assert(count <= 8);
std::array<typename vec_t::Base const*, count> sources;
std::array<size_t, count> pivots;
std::array<base, count> inverses;
size_t first = lim;
for(size_t t = 0; t < count; t++) {
sources[t] = &row(src + t);
auto [p, inv] = row(src + t).find_pivot();
pivots[t] = p;
inverses[t] = p < lim ? inv : base(0);
first = std::min(first, p);
}
if(first == lim) {
return;
}
std::array<base, 2 * count> c{};
for(size_t r = 0; r < 2; r++) {
for(size_t t = 0; t < count; t++) {
if(pivots[t] >= lim) {
continue;
}
base value = row(dst + r).normalize(pivots[t]);
if constexpr(mode == normal) {
// Fewer than eight canonical products fit in a 64-bit accumulator.
uint64_t sum = value.getr();
for(size_t h = 0; h < t; h++) {
sum += uint64_t(c[r * count + h].getr()) * (*sources[h])[pivots[t]].getr();
}
value.setr(sum % base::mod());
}
c[r * count + t] = -value * inverses[t];
}
}
vec_t::template add_scaled_batch<count, 2>({&row(dst), &row(dst + 1)}, sources, c, first);
}
// Fuse two sequential reductions, accounting for the first one's effect
// on the second pivot before updating either destination row.
template<gauss_mode mode>
void reduce_pair(size_t dst, size_t src, size_t lim) {
auto &p = row(src), &q = row(src + 1);
auto [u, pu] = p.find_pivot();
auto [v, qv] = q.find_pivot();
if(u >= lim || v >= lim) {
for(size_t j = dst; j < dst + 2; j++) {
if(u < lim) {
row(j).reduce_by(p);
}
if(v < lim) {
row(j).reduce_by(q);
}
}
return;
}
// Canonical reduction coefficients fit in a 64-bit product.
auto mul = [](base x, base y) {
if constexpr(impl::use_strassen<vec_t>) {
x.setr(uint64_t(x.getr()) * y.getr() % base::mod());
return x;
} else {
return x * y;
}
};
auto scales = [&](vec_t& a) {
base s = mul(-a.normalize(u), pu);
base t = -a.normalize(v);
// Reverse elimination has already cleared p[v] within the pivot block.
if constexpr(mode == normal) {
t -= mul(s, p[v]);
}
t = mul(t, qv);
return std::array{s, t};
};
auto a = scales(row(dst)), b = scales(row(dst + 1));
add_scaled_pair(row(dst), row(dst + 1), p, q, {a[0], a[1], b[0], b[1]}, std::min(u, v));
}
};
template<typename base_t>
auto operator*(base_t t, matrix<base_t> const& A) {
return A * t;
}
}
#pragma GCC pop_options
#line 9 "verify/linalg/det.test.cpp"
using namespace std;
using namespace cp_algo::linalg;
using namespace cp_algo::math;
const int64_t mod = 998244353;
void solve() {
int n;
cin >> n;
matrix<modint<mod>> a(n, n);
a.read();
cout << a.det() << "\n";
}
signed main() {
//freopen("input.txt", "r", stdin);
ios::sync_with_stdio(0);
cin.tie(0);
int t = 1;
while(t--) {
solve();
}
}
Test cases
| Env | Name | Status | Elapsed | Memory |
|---|---|---|---|---|
| g++ | example_00 |
AC |
2 ms | 4 MB |
| g++ | example_01 |
AC |
2 ms | 4 MB |
| g++ | example_02 |
AC |
2 ms | 4 MB |
| g++ | lowrank_max_random_00 |
AC |
14 ms | 7 MB |
| g++ | lowrank_max_random_01 |
AC |
15 ms | 7 MB |
| g++ | lowrank_max_random_02 |
AC |
15 ms | 8 MB |
| g++ | lowrank_max_random_03 |
AC |
14 ms | 7 MB |
| g++ | lowrank_max_random_04 |
AC |
17 ms | 7 MB |
| g++ | max_random_00 |
AC |
15 ms | 7 MB |
| g++ | max_random_01 |
AC |
15 ms | 7 MB |
| g++ | max_random_02 |
AC |
15 ms | 7 MB |
| g++ | max_random_03 |
AC |
14 ms | 7 MB |
| g++ | max_random_04 |
AC |
15 ms | 7 MB |
| g++ | perm_max_random_00 |
AC |
11 ms | 7 MB |
| g++ | perm_max_random_01 |
AC |
11 ms | 7 MB |
| g++ | perm_max_random_02 |
AC |
11 ms | 7 MB |
| g++ | perm_max_random_03 |
AC |
11 ms | 7 MB |
| g++ | perm_max_random_04 |
AC |
11 ms | 7 MB |
| g++ | random_00 |
AC |
4 ms | 4 MB |
| g++ | random_01 |
AC |
4 ms | 4 MB |
| g++ | random_02 |
AC |
3 ms | 4 MB |
| g++ | random_03 |
AC |
4 ms | 4 MB |
| g++ | random_04 |
AC |
2 ms | 4 MB |
| g++ | signed_overflow_00 |
AC |
2 ms | 4 MB |
| g++ | unsigned_overflow_00 |
AC |
2 ms | 4 MB |