#line 1 "verify/math/matrix/pfaffian.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/pfaffian_of_matrix"
#line 1 "math/matrix/linear_algebra.hpp"
#include <optional>
#include <type_traits>
#include <vector>
#line 1 "math/matrix/matrix.hpp"
#include <cassert>
#include <cstddef>
#include <cstdint>
#include <utility>
#line 9 "math/matrix/matrix.hpp"
namespace m1une {
namespace matrix {
template <class T>
class Matrix {
private:
int _rows;
int _cols;
std::vector<T> _data;
static std::size_t storage_size(int rows, int cols) {
assert(rows >= 0);
assert(cols >= 0);
return std::size_t(rows) * std::size_t(cols);
}
public:
using value_type = T;
Matrix() : _rows(0), _cols(0) {}
Matrix(int rows, int cols, const T& value = T())
: _rows(rows), _cols(cols), _data(storage_size(rows, cols), value) {}
Matrix(int rows, int cols, std::vector<T> values)
: _rows(rows), _cols(cols), _data(std::move(values)) {
assert(rows >= 0);
assert(cols >= 0);
assert(_data.size() == std::size_t(rows) * std::size_t(cols));
}
explicit Matrix(const std::vector<std::vector<T>>& values)
: _rows(int(values.size())), _cols(values.empty() ? 0 : int(values[0].size())),
_data(storage_size(_rows, _cols)) {
for (int row = 0; row < _rows; row++) {
assert(int(values[std::size_t(row)].size()) == _cols);
for (int col = 0; col < _cols; col++) {
(*this)[row][col] = values[std::size_t(row)][std::size_t(col)];
}
}
}
int rows() const {
return _rows;
}
int cols() const {
return _cols;
}
bool empty() const {
return _rows == 0 || _cols == 0;
}
std::vector<T>& data() {
return _data;
}
const std::vector<T>& data() const {
return _data;
}
T* operator[](int row) {
assert(0 <= row && row < _rows);
return _data.data() + std::size_t(row) * std::size_t(_cols);
}
const T* operator[](int row) const {
assert(0 <= row && row < _rows);
return _data.data() + std::size_t(row) * std::size_t(_cols);
}
T& operator()(int row, int col) {
assert(0 <= col && col < _cols);
return (*this)[row][col];
}
const T& operator()(int row, int col) const {
assert(0 <= col && col < _cols);
return (*this)[row][col];
}
static Matrix identity(int size) {
assert(size >= 0);
Matrix result(size, size);
for (int i = 0; i < size; i++) result[i][i] = T(1);
return result;
}
Matrix transposed() const {
Matrix result(_cols, _rows);
for (int row = 0; row < _rows; row++) {
for (int col = 0; col < _cols; col++) {
result[col][row] = (*this)[row][col];
}
}
return result;
}
void swap_rows(int first, int second) {
assert(0 <= first && first < _rows);
assert(0 <= second && second < _rows);
if (first == second) return;
for (int col = 0; col < _cols; col++) {
std::swap((*this)[first][col], (*this)[second][col]);
}
}
Matrix& operator+=(const Matrix& rhs) {
assert(_rows == rhs._rows && _cols == rhs._cols);
for (std::size_t i = 0; i < _data.size(); i++) _data[i] += rhs._data[i];
return *this;
}
Matrix& operator-=(const Matrix& rhs) {
assert(_rows == rhs._rows && _cols == rhs._cols);
for (std::size_t i = 0; i < _data.size(); i++) _data[i] -= rhs._data[i];
return *this;
}
Matrix& operator*=(const T& scalar) {
for (T& value : _data) value *= scalar;
return *this;
}
Matrix& operator/=(const T& scalar) {
for (T& value : _data) value /= scalar;
return *this;
}
Matrix& operator*=(const Matrix& rhs) {
return *this = *this * rhs;
}
Matrix operator+() const {
return *this;
}
Matrix operator-() const {
Matrix result = *this;
for (T& value : result._data) value = T() - value;
return result;
}
friend Matrix operator+(Matrix lhs, const Matrix& rhs) {
return lhs += rhs;
}
friend Matrix operator-(Matrix lhs, const Matrix& rhs) {
return lhs -= rhs;
}
friend Matrix operator*(Matrix lhs, const T& rhs) {
return lhs *= rhs;
}
friend Matrix operator*(const T& lhs, Matrix rhs) {
return rhs *= lhs;
}
friend Matrix operator/(Matrix lhs, const T& rhs) {
return lhs /= rhs;
}
friend Matrix operator*(const Matrix& lhs, const Matrix& rhs) {
assert(lhs._cols == rhs._rows);
Matrix result(lhs._rows, rhs._cols);
for (int row = 0; row < lhs._rows; row++) {
T* output = result[row];
for (int middle = 0; middle < lhs._cols; middle++) {
const T coefficient = lhs[row][middle];
if (coefficient == T()) continue;
const T* input = rhs[middle];
for (int col = 0; col < rhs._cols; col++) {
output[col] += coefficient * input[col];
}
}
}
return result;
}
friend std::vector<T> operator*(const Matrix& lhs, const std::vector<T>& rhs) {
assert(lhs._cols == int(rhs.size()));
std::vector<T> result(std::size_t(lhs._rows));
for (int row = 0; row < lhs._rows; row++) {
T value = T();
for (int col = 0; col < lhs._cols; col++) {
value += lhs[row][col] * rhs[std::size_t(col)];
}
result[std::size_t(row)] = value;
}
return result;
}
friend std::vector<T> operator*(const std::vector<T>& lhs, const Matrix& rhs) {
assert(int(lhs.size()) == rhs._rows);
std::vector<T> result(std::size_t(rhs._cols));
for (int row = 0; row < rhs._rows; row++) {
if (lhs[std::size_t(row)] == T()) continue;
for (int col = 0; col < rhs._cols; col++) {
result[std::size_t(col)] += lhs[std::size_t(row)] * rhs[row][col];
}
}
return result;
}
bool operator==(const Matrix& rhs) const {
return _rows == rhs._rows && _cols == rhs._cols && _data == rhs._data;
}
bool operator!=(const Matrix& rhs) const {
return !(*this == rhs);
}
Matrix pow(std::uint64_t exponent) const {
assert(_rows == _cols);
Matrix result = identity(_rows);
Matrix base = *this;
while (exponent > 0) {
if (exponent & 1) result *= base;
exponent >>= 1;
if (exponent > 0) base *= base;
}
return result;
}
};
} // namespace matrix
} // namespace m1une
#line 9 "math/matrix/linear_algebra.hpp"
namespace m1une {
namespace matrix {
template <class T>
constexpr T default_epsilon() {
if constexpr (std::is_floating_point_v<T>) {
return T(1e-10);
} else {
return T();
}
}
namespace detail {
template <class T>
T matrix_abs(T value) {
return value < T() ? T() - value : value;
}
template <class T>
bool is_zero(const T& value, const T& eps) {
if constexpr (std::is_floating_point_v<T>) {
return matrix_abs(value) <= eps;
} else {
(void)eps;
return value == T();
}
}
template <class T>
int choose_pivot(const Matrix<T>& matrix, int first_row, int col, const T& eps) {
int pivot = -1;
if constexpr (std::is_floating_point_v<T>) {
for (int row = first_row; row < matrix.rows(); row++) {
if (is_zero(matrix[row][col], eps)) continue;
if (pivot == -1 || matrix_abs(matrix[pivot][col]) < matrix_abs(matrix[row][col])) {
pivot = row;
}
}
} else {
for (int row = first_row; row < matrix.rows(); row++) {
if (!is_zero(matrix[row][col], eps)) {
pivot = row;
break;
}
}
}
return pivot;
}
template <class T>
std::vector<int> row_reduce(Matrix<T>& matrix, int pivot_col_limit, const T& eps,
bool reduced) {
std::vector<int> pivot_columns;
int pivot_row = 0;
for (int col = 0; col < pivot_col_limit && pivot_row < matrix.rows(); col++) {
int pivot = choose_pivot(matrix, pivot_row, col, eps);
if (pivot == -1) continue;
matrix.swap_rows(pivot_row, pivot);
const T pivot_value = matrix[pivot_row][col];
if (reduced) {
for (int j = col; j < matrix.cols(); j++) matrix[pivot_row][j] /= pivot_value;
}
const int first_row = reduced ? 0 : pivot_row + 1;
for (int row = first_row; row < matrix.rows(); row++) {
if (row == pivot_row || is_zero(matrix[row][col], eps)) continue;
T factor = matrix[row][col];
if (!reduced) factor /= pivot_value;
matrix[row][col] = T();
for (int j = col + 1; j < matrix.cols(); j++) {
matrix[row][j] -= factor * matrix[pivot_row][j];
}
}
pivot_columns.push_back(col);
pivot_row++;
}
if constexpr (std::is_floating_point_v<T>) {
for (T& value : matrix.data()) {
if (is_zero(value, eps)) value = T();
}
}
return pivot_columns;
}
} // namespace detail
template <class T>
struct RowReduction {
Matrix<T> matrix;
std::vector<int> pivot_columns;
int rank() const {
return int(pivot_columns.size());
}
};
template <class T>
RowReduction<T> reduced_row_echelon_form(Matrix<T> matrix,
T eps = default_epsilon<T>()) {
RowReduction<T> result;
result.pivot_columns = detail::row_reduce(matrix, matrix.cols(), eps, true);
result.matrix = std::move(matrix);
return result;
}
template <class T>
int matrix_rank(Matrix<T> matrix, T eps = default_epsilon<T>()) {
return int(detail::row_reduce(matrix, matrix.cols(), eps, false).size());
}
template <class T>
T determinant(Matrix<T> matrix, T eps = default_epsilon<T>()) {
assert(matrix.rows() == matrix.cols());
const int size = matrix.rows();
T result = T(1);
bool negate = false;
for (int col = 0; col < size; col++) {
int pivot = detail::choose_pivot(matrix, col, col, eps);
if (pivot == -1) return T();
if (pivot != col) {
matrix.swap_rows(pivot, col);
negate = !negate;
}
const T pivot_value = matrix[col][col];
result *= pivot_value;
for (int row = col + 1; row < size; row++) {
if (detail::is_zero(matrix[row][col], eps)) continue;
const T factor = matrix[row][col] / pivot_value;
matrix[row][col] = T();
for (int j = col + 1; j < size; j++) {
matrix[row][j] -= factor * matrix[col][j];
}
}
}
return negate ? T() - result : result;
}
template <class T>
std::optional<Matrix<T>> inverse(const Matrix<T>& matrix,
T eps = default_epsilon<T>()) {
assert(matrix.rows() == matrix.cols());
const int size = matrix.rows();
Matrix<T> augmented(size, size * 2);
for (int row = 0; row < size; row++) {
for (int col = 0; col < size; col++) {
augmented[row][col] = matrix[row][col];
}
augmented[row][size + row] = T(1);
}
const std::vector<int> pivots = detail::row_reduce(augmented, size, eps, true);
if (int(pivots.size()) != size) return std::nullopt;
Matrix<T> result(size, size);
for (int row = 0; row < size; row++) {
for (int col = 0; col < size; col++) {
result[row][col] = augmented[row][size + col];
}
}
return result;
}
template <class T>
struct LinearSystemResult {
bool consistent = false;
std::vector<T> particular_solution;
std::vector<std::vector<T>> nullspace_basis;
std::vector<int> pivot_columns;
int rank() const {
return int(pivot_columns.size());
}
int nullity() const {
return consistent ? int(nullspace_basis.size()) : 0;
}
bool has_unique_solution() const {
return consistent && nullspace_basis.empty();
}
};
template <class T>
LinearSystemResult<T> solve_linear_system(const Matrix<T>& coefficients,
const std::vector<T>& constants,
T eps = default_epsilon<T>()) {
assert(coefficients.rows() == int(constants.size()));
const int equation_count = coefficients.rows();
const int variable_count = coefficients.cols();
Matrix<T> augmented(equation_count, variable_count + 1);
for (int row = 0; row < equation_count; row++) {
for (int col = 0; col < variable_count; col++) {
augmented[row][col] = coefficients[row][col];
}
augmented[row][variable_count] = constants[std::size_t(row)];
}
LinearSystemResult<T> result;
result.pivot_columns =
detail::row_reduce(augmented, variable_count, eps, true);
for (int row = result.rank(); row < equation_count; row++) {
bool zero_left = true;
for (int col = 0; col < variable_count; col++) {
if (!detail::is_zero(augmented[row][col], eps)) {
zero_left = false;
break;
}
}
if (zero_left && !detail::is_zero(augmented[row][variable_count], eps)) {
return result;
}
}
result.consistent = true;
result.particular_solution.assign(std::size_t(variable_count), T());
std::vector<bool> is_pivot(std::size_t(variable_count), false);
for (int row = 0; row < result.rank(); row++) {
const int col = result.pivot_columns[std::size_t(row)];
is_pivot[std::size_t(col)] = true;
result.particular_solution[std::size_t(col)] = augmented[row][variable_count];
}
for (int free_col = 0; free_col < variable_count; free_col++) {
if (is_pivot[std::size_t(free_col)]) continue;
std::vector<T> direction(static_cast<std::size_t>(variable_count));
direction[std::size_t(free_col)] = T(1);
for (int row = 0; row < result.rank(); row++) {
const int pivot_col = result.pivot_columns[std::size_t(row)];
direction[std::size_t(pivot_col)] = T() - augmented[row][free_col];
}
result.nullspace_basis.push_back(std::move(direction));
}
return result;
}
} // namespace matrix
} // namespace m1une
#line 1 "math/matrix/pfaffian.hpp"
#line 6 "math/matrix/pfaffian.hpp"
#line 8 "math/matrix/pfaffian.hpp"
namespace m1une {
namespace matrix {
// Returns the Pfaffian of an even-dimensional alternating matrix over a field.
template <class T>
T pfaffian(Matrix<T> matrix) {
assert(matrix.rows() == matrix.cols());
const int size = matrix.rows();
assert(size % 2 == 0);
#ifndef NDEBUG
for (int row = 0; row < size; row++) {
assert(matrix[row][row] == T());
for (int col = row + 1; col < size; col++) {
assert(matrix[row][col] == T() - matrix[col][row]);
}
}
#endif
T result = T(1);
for (int first = 0; first < size; first += 2) {
int pivot = first + 1;
while (pivot < size && matrix[first][pivot] == T()) pivot++;
if (pivot == size) return T();
if (pivot != first + 1) {
matrix.swap_rows(pivot, first + 1);
for (int row = 0; row < size; row++) {
std::swap(matrix[row][pivot], matrix[row][first + 1]);
}
result = T() - result;
}
const int second = first + 1;
const T pivot_value = matrix[first][second];
result *= pivot_value;
const T inverse_pivot = T(1) / pivot_value;
for (int row = second + 1; row < size; row++) {
for (int col = row + 1; col < size; col++) {
matrix[row][col] +=
(matrix[second][row] * matrix[first][col] -
matrix[first][row] * matrix[second][col]) *
inverse_pivot;
matrix[col][row] = T() - matrix[row][col];
}
}
}
return result;
}
} // namespace matrix
} // namespace m1une
#line 1 "math/modint.hpp"
#line 6 "math/modint.hpp"
#include <iostream>
#line 9 "math/modint.hpp"
namespace m1une {
namespace math {
template <uint32_t Modulus>
struct ModInt {
static_assert(0 < Modulus, "Modulus must be positive");
private:
uint32_t _v;
public:
static constexpr uint32_t mod() {
return Modulus;
}
static constexpr ModInt raw(uint32_t v) noexcept {
ModInt x;
x._v = v;
return x;
}
constexpr ModInt() noexcept : _v(0) {}
template <class Integer, std::enable_if_t<std::is_integral_v<Integer>, int> = 0>
constexpr ModInt(Integer v) noexcept {
if constexpr (std::is_signed_v<Integer>) {
int64_t x = static_cast<int64_t>(v) % static_cast<int64_t>(Modulus);
if (x < 0) x += Modulus;
_v = static_cast<uint32_t>(x);
} else {
_v = static_cast<uint32_t>(static_cast<uint64_t>(v) % Modulus);
}
}
constexpr uint32_t val() const noexcept {
return _v;
}
constexpr ModInt& operator++() noexcept {
_v++;
if (_v == Modulus) _v = 0;
return *this;
}
constexpr ModInt& operator--() noexcept {
if (_v == 0) _v = Modulus;
_v--;
return *this;
}
constexpr ModInt operator++(int) noexcept {
ModInt res = *this;
++*this;
return res;
}
constexpr ModInt operator--(int) noexcept {
ModInt res = *this;
--*this;
return res;
}
constexpr ModInt& operator+=(const ModInt& rhs) noexcept {
_v += rhs._v;
if (_v >= Modulus) _v -= Modulus;
return *this;
}
constexpr ModInt& operator-=(const ModInt& rhs) noexcept {
_v -= rhs._v;
if (_v >= Modulus) _v += Modulus;
return *this;
}
constexpr ModInt& operator*=(const ModInt& rhs) noexcept {
uint64_t z = _v;
z *= rhs._v;
_v = static_cast<uint32_t>(z % Modulus);
return *this;
}
constexpr ModInt& operator/=(const ModInt& rhs) noexcept {
return *this *= rhs.inv();
}
constexpr ModInt operator+(const ModInt& rhs) const noexcept {
return ModInt(*this) += rhs;
}
constexpr ModInt operator-(const ModInt& rhs) const noexcept {
return ModInt(*this) -= rhs;
}
constexpr ModInt operator*(const ModInt& rhs) const noexcept {
return ModInt(*this) *= rhs;
}
constexpr ModInt operator/(const ModInt& rhs) const noexcept {
return ModInt(*this) /= rhs;
}
constexpr bool operator==(const ModInt& rhs) const noexcept {
return _v == rhs._v;
}
constexpr bool operator!=(const ModInt& rhs) const noexcept {
return _v != rhs._v;
}
constexpr ModInt pow(long long n) const noexcept {
ModInt res = raw(1 % Modulus);
ModInt x = n < 0 ? inv() : *this;
uint64_t exponent = n < 0 ? uint64_t(-(n + 1)) + 1 : uint64_t(n);
while (exponent > 0) {
if (exponent & 1) res *= x;
x *= x;
exponent >>= 1;
}
return res;
}
constexpr ModInt inv() const noexcept {
int64_t a = _v, b = Modulus, u = 1, v = 0;
while (b) {
int64_t t = a / b;
a -= t * b;
std::swap(a, b);
u -= t * v;
std::swap(u, v);
}
assert(a == 1);
u %= Modulus;
if (u < 0) u += Modulus;
return raw(static_cast<uint32_t>(u));
}
friend std::ostream& operator<<(std::ostream& os, const ModInt& rhs) {
return os << rhs._v;
}
friend std::istream& operator>>(std::istream& is, ModInt& rhs) {
long long v;
is >> v;
rhs = ModInt(v);
return is;
}
};
using modint998244353 = ModInt<998244353>;
using modint1000000007 = ModInt<1000000007>;
template <int Id = 0>
struct DynamicModInt {
private:
uint32_t _v;
inline static uint32_t _mod = 1;
public:
static uint32_t mod() noexcept {
return _mod;
}
static void set_mod(uint32_t modulus) noexcept {
assert(modulus > 0);
assert(modulus <= uint32_t(1) << 31);
_mod = modulus;
}
static DynamicModInt raw(uint32_t v) noexcept {
assert(v < _mod);
DynamicModInt x;
x._v = v;
return x;
}
DynamicModInt() noexcept : _v(0) {}
template <class Integer, std::enable_if_t<std::is_integral_v<Integer>, int> = 0>
DynamicModInt(Integer v) noexcept {
if constexpr (std::is_signed_v<Integer>) {
int64_t x = static_cast<int64_t>(v) % static_cast<int64_t>(_mod);
if (x < 0) x += _mod;
_v = static_cast<uint32_t>(x);
} else {
_v = static_cast<uint32_t>(static_cast<uint64_t>(v) % _mod);
}
}
uint32_t val() const noexcept {
return _v;
}
DynamicModInt& operator++() noexcept {
_v++;
if (_v == _mod) _v = 0;
return *this;
}
DynamicModInt& operator--() noexcept {
if (_v == 0) _v = _mod;
_v--;
return *this;
}
DynamicModInt operator++(int) noexcept {
DynamicModInt result = *this;
++*this;
return result;
}
DynamicModInt operator--(int) noexcept {
DynamicModInt result = *this;
--*this;
return result;
}
DynamicModInt& operator+=(const DynamicModInt& rhs) noexcept {
_v += rhs._v;
if (_v >= _mod) _v -= _mod;
return *this;
}
DynamicModInt& operator-=(const DynamicModInt& rhs) noexcept {
_v -= rhs._v;
if (_v >= _mod) _v += _mod;
return *this;
}
DynamicModInt& operator*=(const DynamicModInt& rhs) noexcept {
_v = static_cast<uint32_t>(uint64_t(_v) * rhs._v % _mod);
return *this;
}
DynamicModInt& operator/=(const DynamicModInt& rhs) noexcept {
return *this *= rhs.inv();
}
DynamicModInt operator+(const DynamicModInt& rhs) const noexcept {
return DynamicModInt(*this) += rhs;
}
DynamicModInt operator-(const DynamicModInt& rhs) const noexcept {
return DynamicModInt(*this) -= rhs;
}
DynamicModInt operator*(const DynamicModInt& rhs) const noexcept {
return DynamicModInt(*this) *= rhs;
}
DynamicModInt operator/(const DynamicModInt& rhs) const noexcept {
return DynamicModInt(*this) /= rhs;
}
bool operator==(const DynamicModInt& rhs) const noexcept {
return _v == rhs._v;
}
bool operator!=(const DynamicModInt& rhs) const noexcept {
return _v != rhs._v;
}
DynamicModInt pow(long long exponent) const noexcept {
DynamicModInt result = raw(1 % _mod);
DynamicModInt base = exponent < 0 ? inv() : *this;
uint64_t magnitude =
exponent < 0 ? uint64_t(-(exponent + 1)) + 1 : uint64_t(exponent);
while (magnitude > 0) {
if (magnitude & 1) result *= base;
base *= base;
magnitude >>= 1;
}
return result;
}
DynamicModInt inv() const noexcept {
int64_t a = _v, b = _mod, u = 1, v = 0;
while (b) {
int64_t quotient = a / b;
a -= quotient * b;
std::swap(a, b);
u -= quotient * v;
std::swap(u, v);
}
assert(a == 1);
u %= _mod;
if (u < 0) u += _mod;
return raw(static_cast<uint32_t>(u));
}
friend std::ostream& operator<<(std::ostream& os, const DynamicModInt& rhs) {
return os << rhs._v;
}
friend std::istream& operator>>(std::istream& is, DynamicModInt& rhs) {
long long value;
is >> value;
rhs = DynamicModInt(value);
return is;
}
};
} // namespace math
} // namespace m1une
#line 6 "verify/math/matrix/pfaffian.test.cpp"
#line 1 "utilities/fast_io.hpp"
#include <algorithm>
#include <array>
#include <cerrno>
#include <charconv>
#line 9 "utilities/fast_io.hpp"
#include <cstdio>
#include <cstdlib>
#line 12 "utilities/fast_io.hpp"
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#line 18 "utilities/fast_io.hpp"
#include <unistd.h>
#line 20 "utilities/fast_io.hpp"
namespace m1une {
namespace utilities {
struct FastOutput;
namespace internal {
// Shared with the convenience helpers in template.hpp.
inline FastOutput* standard_output_instance = nullptr;
// Detect std::begin(x), std::end(x).
template <class T, class = void>
struct is_range : std::false_type {};
template <class T>
struct is_range<T, std::void_t<
decltype(std::begin(std::declval<T&>())),
decltype(std::end(std::declval<T&>()))
>> : std::true_type {};
template <class T>
inline constexpr bool is_range_v = is_range<T>::value;
template <class T>
using range_reference_t = decltype(*std::begin(std::declval<T&>()));
template <class T>
using range_value_t = std::remove_cv_t<std::remove_reference_t<range_reference_t<T>>>;
template <class T, class = void>
struct range_stored_value {
using type = range_value_t<T>;
};
template <class T>
struct range_stored_value<T, std::void_t<typename std::remove_cv_t<std::remove_reference_t<T>>::value_type>> {
using type = typename std::remove_cv_t<std::remove_reference_t<T>>::value_type;
};
template <class T>
using range_stored_value_t = typename range_stored_value<T>::type;
// Treat strings and C strings as scalar output objects, not as ranges.
template <class T>
struct is_char_array : std::false_type {};
template <class T, std::size_t N>
struct is_char_array<T[N]>
: std::bool_constant<std::is_same_v<std::remove_cv_t<T>, char>> {};
template <class T>
struct is_string_like
: std::bool_constant<
std::is_same_v<std::decay_t<T>, std::string>
|| std::is_same_v<std::decay_t<T>, const char*>
|| std::is_same_v<std::decay_t<T>, char*>
|| is_char_array<std::remove_reference_t<T>>::value
> {};
template <class T>
inline constexpr bool is_string_like_v = is_string_like<T>::value;
// ModInt-like type: x.val() is printable, and x can be assigned from long long.
template <class T, class = void>
struct has_val_method : std::false_type {};
template <class T>
struct has_val_method<T, std::void_t<decltype(std::declval<const T&>().val())>>
: std::true_type {};
template <class T>
inline constexpr bool has_val_method_v = has_val_method<T>::value;
template <class T, class = void>
struct has_static_mod_raw : std::false_type {};
template <class T>
struct has_static_mod_raw<
T, std::void_t<decltype(T::mod()), decltype(T::raw(std::declval<uint32_t>()))>>
: std::true_type {};
template <class T>
inline constexpr bool has_static_mod_raw_v = has_static_mod_raw<T>::value;
// libstdc++ before GCC 16 does not classify __int128 as an integral type in
// strict ISO modes such as -std=c++23. Keep the fast-I/O interface independent
// of that implementation detail.
template <class T>
inline constexpr bool is_integral_v =
std::is_integral_v<T>
|| std::is_same_v<std::remove_cv_t<T>, __int128_t>
|| std::is_same_v<std::remove_cv_t<T>, __uint128_t>;
template <class T>
inline constexpr bool is_signed_v =
std::is_signed_v<T>
|| std::is_same_v<std::remove_cv_t<T>, __int128_t>;
template <class T>
struct make_unsigned {
using type = std::make_unsigned_t<T>;
};
template <>
struct make_unsigned<__int128_t> {
using type = __uint128_t;
};
template <>
struct make_unsigned<__uint128_t> {
using type = __uint128_t;
};
template <class T>
using make_unsigned_t = typename make_unsigned<std::remove_cv_t<T>>::type;
} // namespace internal
struct FastInput {
static constexpr int buffer_size = 1 << 20;
private:
std::FILE* _stream;
char _buffer[buffer_size];
int _position;
int _length;
int _file_descriptor;
bool _streaming;
bool refill() {
_position = 0;
if (_streaming) {
ssize_t length;
do {
length = ::read(_file_descriptor, _buffer, buffer_size);
} while (length < 0 && errno == EINTR);
if (length <= 0) {
_length = 0;
return false;
}
_length = int(length);
} else {
_length = int(std::fread(_buffer, 1, buffer_size, _stream));
}
return _length != 0;
}
template <class T>
bool read_integer_from_stream(T& value) {
if (!skip_spaces()) return false;
int c = read_char_raw();
bool negative = false;
if (c == '-') {
negative = true;
c = read_char_raw();
}
if constexpr (internal::is_signed_v<T>) {
T result = 0;
while ('0' <= c && c <= '9') {
result = negative ? result * 10 - (c - '0')
: result * 10 + (c - '0');
c = read_char_raw();
}
value = result;
} else {
T result = 0;
while ('0' <= c && c <= '9') {
result = result * 10 + T(c - '0');
c = read_char_raw();
}
value = negative ? T(0) - result : result;
}
return true;
}
bool prepare_number() {
if (_length - _position >= 64) return true;
const int remaining = _length - _position;
if (remaining > 0) std::memmove(_buffer, _buffer + _position, remaining);
const int added = int(std::fread(_buffer + remaining, 1, buffer_size - remaining, _stream));
_position = 0;
_length = remaining + added;
if (_length < buffer_size) _buffer[_length] = '\0';
return _length != 0;
}
public:
explicit FastInput(std::FILE* stream = stdin)
: _stream(stream),
_position(0),
_length(0),
_file_descriptor(::fileno(stream)),
_streaming([&] {
struct stat status;
return _file_descriptor >= 0
&& ::fstat(_file_descriptor, &status) == 0
&& !S_ISREG(status.st_mode);
}()) {}
FastInput(const FastInput&) = delete;
FastInput& operator=(const FastInput&) = delete;
int read_char_raw() {
if (_position == _length && !refill()) return EOF;
return _buffer[_position++];
}
bool skip_spaces() {
int c = read_char_raw();
while (c != EOF && c <= ' ') c = read_char_raw();
if (c == EOF) return false;
--_position;
return true;
}
bool read(char& value) {
if (!skip_spaces()) return false;
value = char(read_char_raw());
return true;
}
bool read(std::string& value) {
if (!skip_spaces()) return false;
value.clear();
while (true) {
const int begin = _position;
while (_position < _length &&
static_cast<unsigned char>(_buffer[_position]) > ' ') {
++_position;
}
value.append(_buffer + begin, _position - begin);
if (_position < _length) {
++_position;
return true;
}
if (!refill()) return true;
}
}
bool read(bool& value) {
int x;
if (!read(x)) return false;
value = x != 0;
return true;
}
template <class T>
std::enable_if_t<
internal::is_integral_v<T>
&& !std::is_same_v<std::remove_cv_t<T>, bool>
&& !std::is_same_v<std::remove_cv_t<T>, char>,
bool
>
read(T& value) {
if (_streaming) return read_integer_from_stream(value);
if (!prepare_number()) return false;
int c = static_cast<unsigned char>(_buffer[_position++]);
while (c <= ' ') c = static_cast<unsigned char>(_buffer[_position++]);
bool negative = false;
if (c == '-') {
negative = true;
c = static_cast<unsigned char>(_buffer[_position++]);
}
if constexpr (internal::is_signed_v<T>) {
T result = 0;
while ('0' <= c && c <= '9') {
const int first = c - '0';
const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
if (0 <= second && second <= 9) {
result = negative ? result * 100 - (first * 10 + second)
: result * 100 + (first * 10 + second);
++_position;
} else {
result = negative ? result * 10 - first : result * 10 + first;
}
c = static_cast<unsigned char>(_buffer[_position++]);
}
value = result;
} else {
T result = 0;
while ('0' <= c && c <= '9') {
const unsigned first = unsigned(c - '0');
const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
if (0 <= second && second <= 9) {
result = result * 100 + T(first * 10 + unsigned(second));
++_position;
} else {
result = result * 10 + T(first);
}
c = static_cast<unsigned char>(_buffer[_position++]);
}
value = negative ? T(0) - result : result;
}
if (_position > _length) _position = _length;
return true;
}
template <class T>
std::enable_if_t<std::is_floating_point_v<T>, bool>
read(T& value) {
if (!skip_spaces()) return false;
int c = read_char_raw();
bool negative = false;
if (c == '-' || c == '+') {
negative = c == '-';
c = read_char_raw();
}
long double result = 0;
while ('0' <= c && c <= '9') {
result = result * 10 + (c - '0');
c = read_char_raw();
}
if (c == '.') {
long double place = 0.1L;
c = read_char_raw();
while ('0' <= c && c <= '9') {
result += (c - '0') * place;
place *= 0.1L;
c = read_char_raw();
}
}
if (c == 'e' || c == 'E') {
c = read_char_raw();
bool exponent_negative = false;
if (c == '-' || c == '+') {
exponent_negative = c == '-';
c = read_char_raw();
}
int exponent = 0;
while ('0' <= c && c <= '9') {
exponent = exponent * 10 + (c - '0');
c = read_char_raw();
}
long double scale = 1;
long double power = 10;
while (exponent > 0) {
if (exponent & 1) scale *= power;
power *= power;
exponent >>= 1;
}
result = exponent_negative ? result / scale : result * scale;
}
value = static_cast<T>(negative ? -result : result);
return true;
}
template <class T>
std::enable_if_t<
internal::has_val_method_v<T>
&& !internal::is_integral_v<T>
&& !internal::is_range_v<T>,
bool
>
read(T& value) {
long long x;
if (!read(x)) return false;
if constexpr (internal::has_static_mod_raw_v<T>) {
if (x >= 0 && uint64_t(x) < uint64_t(T::mod())) {
value = T::raw(uint32_t(x));
} else {
value = T(x);
}
} else {
value = T(x);
}
return true;
}
template <class First, class Second>
bool read(std::pair<First, Second>& value) {
if (!read(value.first)) return false;
return read(value.second);
}
template <class Range>
std::enable_if_t<
internal::is_range_v<Range>
&& !internal::is_string_like_v<Range>,
bool
>
read(Range& range) {
using StoredValue = internal::range_stored_value_t<Range>;
constexpr bool nested = internal::is_range_v<StoredValue>
&& !internal::is_string_like_v<StoredValue>;
for (auto&& value : range) {
if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
bool x;
if (!read(x)) return false;
value = x;
} else {
if (!read(value)) return false;
}
}
return true;
}
template <class First, class Second, class... Rest>
bool read(First& first, Second& second, Rest&... rest) {
if (!read(first)) return false;
return read(second, rest...);
}
template <class T>
FastInput& operator>>(T& value) {
if (!read(value)) std::abort();
return *this;
}
};
struct FastOutput {
static constexpr int buffer_size = 1 << 20;
private:
inline static const auto digit_quads = [] {
std::array<char, 40000> result{};
for (int i = 0; i < 10000; i++) {
int value = i;
for (int j = 3; j >= 0; j--) {
result[4 * i + j] = char('0' + value % 10);
value /= 10;
}
}
return result;
}();
std::FILE* _stream;
char _buffer[buffer_size];
int _position;
int _precision;
std::chars_format _float_format;
char _range_separator;
std::string* _capture = nullptr;
template <class T>
std::string format_cell(const T& value) {
std::string result;
struct CaptureGuard {
std::string*& target;
std::string* previous;
~CaptureGuard() { target = previous; }
} guard{_capture, _capture};
_capture = &result;
write(value);
return result;
}
template <class Matrix>
void write_aligned_matrix(const Matrix& matrix) {
std::vector<std::vector<std::string>> rows;
std::vector<std::size_t> widths;
for (const auto& row : matrix) {
auto& cells = rows.emplace_back();
std::size_t column = 0;
for (const auto& value : row) {
cells.push_back(format_cell(value));
if (column == widths.size()) widths.push_back(0);
widths[column] = std::max(widths[column], cells.back().size());
++column;
}
}
bool first = true;
for (const auto& row : rows) {
if (!first) write_char('\n');
first = false;
for (std::size_t column = 0; column < row.size(); ++column) {
if (column != 0) write_char(_range_separator);
for (std::size_t padding = row[column].size();
padding < widths[column]; ++padding) {
write_char(' ');
}
write(row[column]);
}
}
}
public:
explicit FastOutput(std::FILE* stream = stdout)
: _stream(stream),
_position(0),
_precision(6),
_float_format(std::chars_format::general),
_range_separator(' ') {
if (_stream == stdout
&& internal::standard_output_instance == nullptr) {
internal::standard_output_instance = this;
}
}
FastOutput(const FastOutput&) = delete;
FastOutput& operator=(const FastOutput&) = delete;
~FastOutput() {
flush();
if (internal::standard_output_instance == this) {
internal::standard_output_instance = nullptr;
}
}
void flush() {
if (_position != 0) {
std::fwrite(_buffer, 1, _position, _stream);
_position = 0;
}
std::fflush(_stream);
}
void write_char(char c) {
if (_capture != nullptr) {
_capture->push_back(c);
return;
}
if (_position == buffer_size) flush();
_buffer[_position++] = c;
}
void write(const char* s) {
while (*s != '\0') write_char(*s++);
}
void write(const std::string& s) {
if (_capture != nullptr) {
_capture->append(s);
return;
}
std::size_t position = 0;
while (position < s.size()) {
if (_position == buffer_size) flush();
const std::size_t copied =
std::min<std::size_t>(buffer_size - _position, s.size() - position);
std::memcpy(_buffer + _position, s.data() + position, copied);
_position += int(copied);
position += copied;
}
}
void write(char c) {
write_char(c);
}
void write(bool value) {
write_char(value ? '1' : '0');
}
template <class T>
std::enable_if_t<std::is_floating_point_v<T>>
write(T value) {
char digits[128];
auto [end, error] = std::to_chars(
digits,
digits + sizeof(digits),
value,
_float_format,
_precision
);
if (error != std::errc()) std::abort();
for (const char* pointer = digits; pointer != end; pointer++) {
write_char(*pointer);
}
}
template <class T>
std::enable_if_t<
internal::is_integral_v<T>
&& !std::is_same_v<std::remove_cv_t<T>, bool>
&& !std::is_same_v<std::remove_cv_t<T>, char>
>
write(T value) {
using Raw = std::remove_cv_t<T>;
using Unsigned = internal::make_unsigned_t<Raw>;
Unsigned magnitude;
if constexpr (internal::is_signed_v<Raw>) {
if (value < 0) {
write_char('-');
magnitude = Unsigned(0) - Unsigned(value);
} else {
magnitude = Unsigned(value);
}
} else {
magnitude = value;
}
if (magnitude == 0) {
write_char('0');
return;
}
unsigned chunks[16];
int count = 0;
while (magnitude >= 10000) {
const Unsigned quotient = magnitude / 10000;
chunks[count++] = unsigned(magnitude - quotient * 10000);
magnitude = quotient;
}
if (_capture == nullptr && _position > buffer_size - 64) flush();
char captured[64];
char* const begin = _capture != nullptr ? captured : _buffer + _position;
char* destination = begin;
const unsigned leading = unsigned(magnitude);
const char* first = digit_quads.data() + 4 * leading;
int skip = leading < 10 ? 3 : leading < 100 ? 2 : leading < 1000 ? 1 : 0;
for (; skip < 4; skip++) *destination++ = first[skip];
while (count--) {
const char* digits = digit_quads.data() + 4 * chunks[count];
std::memcpy(destination, digits, 4);
destination += 4;
}
if (_capture != nullptr) {
_capture->append(begin, destination - begin);
} else {
_position += int(destination - begin);
}
}
template <class T>
std::enable_if_t<
internal::has_val_method_v<T>
&& !internal::is_integral_v<T>
&& !internal::is_range_v<T>
>
write(const T& value) {
write(value.val());
}
template <class First, class Second>
void write(const std::pair<First, Second>& value) {
write(value.first);
write_char(' ');
write(value.second);
}
template <class Range>
std::enable_if_t<
internal::is_range_v<Range>
&& !internal::is_string_like_v<Range>
>
write(const Range& range) {
using StoredValue = internal::range_stored_value_t<const Range>;
constexpr bool nested = internal::is_range_v<StoredValue>
&& !internal::is_string_like_v<StoredValue>;
bool first = true;
for (const auto& value : range) {
if (!first) write_char(nested ? '\n' : _range_separator);
first = false;
if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
write(static_cast<bool>(value));
} else {
write(value);
}
}
}
template <class First, class... Rest>
void print(const First& first, const Rest&... rest) {
write(first);
((write_char(' '), write(rest)), ...);
}
void println() {
write_char('\n');
}
void set_precision(int precision) {
_precision = precision;
}
void set_fixed(int precision = 6) {
_float_format = std::chars_format::fixed;
_precision = precision;
}
void set_general(int precision = 6) {
_float_format = std::chars_format::general;
_precision = precision;
}
void set_range_separator(char separator) {
_range_separator = separator;
}
template <class Matrix>
void write_aligned(const Matrix& matrix) {
using Row = internal::range_stored_value_t<const Matrix>;
using Cell = internal::range_stored_value_t<const Row>;
static_assert(internal::is_range_v<Row> && !internal::is_string_like_v<Row>,
"write_aligned requires a two-dimensional range");
static_assert(!internal::is_range_v<Cell> || internal::is_string_like_v<Cell>,
"write_aligned requires scalar cells");
write_aligned_matrix(matrix);
}
template <class Matrix>
void println_aligned(const Matrix& matrix) {
write_aligned(matrix);
write_char('\n');
}
template <class... Args>
void println(const Args&... args) {
print(args...);
write_char('\n');
}
template <class T>
FastOutput& operator<<(const T& value) {
write(value);
return *this;
}
};
} // namespace utilities
} // namespace m1une
#line 11 "verify/math/matrix/pfaffian.test.cpp"
namespace {
using mint = m1une::math::modint998244353;
using Matrix = m1une::matrix::Matrix<mint>;
mint naive_pfaffian(const Matrix& matrix) {
const int size = matrix.rows();
std::vector<mint> memo(std::size_t(1 << size));
std::vector<char> calculated(std::size_t(1 << size), false);
auto solve = [&](auto&& self, int mask) -> mint {
if (mask == 0) return mint(1);
if (calculated[mask]) return memo[mask];
calculated[mask] = true;
const int first = __builtin_ctz(unsigned(mask));
mint result = 0;
for (int second = first + 1; second < size; second++) {
if ((mask >> second & 1) == 0) continue;
int next = mask ^ (1 << first) ^ (1 << second);
mint term = matrix[first][second] * self(self, next);
int position = __builtin_popcount(unsigned(mask & ((1 << second) - 1)));
if (position % 2 == 1) {
result += term;
} else {
result -= term;
}
}
return memo[mask] = result;
};
return solve(solve, (1 << size) - 1);
}
void test_edge_cases() {
assert(m1une::matrix::pfaffian(Matrix(0, 0)) == mint(1));
Matrix pair(2, 2);
pair[0][1] = 7;
pair[1][0] = mint() - pair[0][1];
assert(m1une::matrix::pfaffian(pair) == mint(7));
Matrix zero(6, 6);
assert(m1une::matrix::pfaffian(zero) == mint(0));
}
void test_randomized() {
std::uint64_t state = 0x8aed2a6bb7e15162ULL;
auto random = [&]() {
state ^= state << 7;
state ^= state >> 9;
return state;
};
for (int trial = 0; trial < 500; trial++) {
int size = 2 * int(random() % 6);
Matrix matrix(size, size);
for (int row = 0; row < size; row++) {
for (int col = row + 1; col < size; col++) {
matrix[row][col] = int(random() % 21) - 10;
matrix[col][row] = mint() - matrix[row][col];
}
}
mint actual = m1une::matrix::pfaffian(matrix);
assert(actual == naive_pfaffian(matrix));
assert(actual * actual == m1une::matrix::determinant(matrix));
}
}
} // namespace
int main() {
m1une::utilities::FastInput fast_input;
m1une::utilities::FastOutput fast_output;
test_edge_cases();
test_randomized();
int half_size;
fast_input >> half_size;
const int size = 2 * half_size;
Matrix matrix(size, size);
for (int row = 0; row < size; row++) {
for (int col = 0; col < size; col++) fast_input >> matrix[row][col];
}
fast_output << m1une::matrix::pfaffian(std::move(matrix)) << '\n';
}
verify/math/matrix/pfaffian.test.cpp