#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#include <algorithm>
#include <cassert>
#include <cstdint>
#include "../../utilities/fast_io.hpp"
#include <random>
#include <utility>
#include <vector>
#include "../../matroid/graphic_matroid.hpp"
#include "../../matroid/linear_matroid.hpp"
#include "../../matroid/matroid_intersection.hpp"
#include "../../matroid/partition_matroid.hpp"
template <class Oracle1, class Oracle2>
int brute_force(int ground_size, const Oracle1& oracle1, const Oracle2& oracle2) {
int best = 0;
for (int mask = 0; mask < (1 << ground_size); mask++) {
std::vector<int> subset;
for (int i = 0; i < ground_size; i++) {
if (mask >> i & 1) subset.push_back(i);
}
if (oracle1(subset) && oracle2(subset)) best = std::max(best, int(subset.size()));
}
return best;
}
template <class Oracle1, class Oracle2>
void check(int ground_size, const Oracle1& oracle1, const Oracle2& oracle2) {
auto result = m1une::matroid::matroid_intersection(ground_size, oracle1, oracle2);
assert(std::is_sorted(result.begin(), result.end()));
assert(std::adjacent_find(result.begin(), result.end()) == result.end());
assert(oracle1(result));
assert(oracle2(result));
assert(int(result.size()) == brute_force(ground_size, oracle1, oracle2));
}
void test_augmenting_exchange() {
m1une::matroid::PartitionMatroid left(std::vector<int>{0, 0, 1});
m1une::matroid::PartitionMatroid right(std::vector<int>{0, 1, 0});
auto result = m1une::matroid::matroid_intersection(3, left, right);
assert((result == std::vector<int>{1, 2}));
}
void test_random_matroids() {
std::mt19937 random(123456789);
for (int test = 0; test < 300; test++) {
int ground_size = random() % 11;
int group_count = 1 + random() % 5;
std::vector<int> group(ground_size);
for (int& value : group) value = random() % group_count;
std::vector<int> capacity(group_count, 1);
m1une::matroid::PartitionMatroid partition(group, capacity);
int vertex_count = 1 + random() % 6;
std::vector<std::pair<int, int>> edge(ground_size);
for (auto& [u, v] : edge) {
u = random() % vertex_count;
v = random() % vertex_count;
}
m1une::matroid::GraphicMatroid graphic(vertex_count, edge);
check(ground_size, partition, graphic);
int second_group_count = 1 + random() % 5;
std::vector<int> second_group(ground_size);
for (int& value : second_group) value = random() % second_group_count;
m1une::matroid::PartitionMatroid second_partition(second_group);
check(ground_size, partition, second_partition);
std::vector<std::uint64_t> vector1(ground_size), vector2(ground_size);
for (int i = 0; i < ground_size; i++) {
vector1[i] = random() % 64;
vector2[i] = random() % 64;
}
m1une::matroid::BinaryLinearMatroid linear1(vector1), linear2(vector2);
check(ground_size, linear1, linear2);
}
}
int main() {
m1une::utilities::FastInput fast_input;
m1une::utilities::FastOutput fast_output;
test_augmenting_exchange();
test_random_matroids();
long long a, b;
fast_input >> a >> b;
fast_output << a + b << '\n';
}
#line 1 "verify/matroid/matroid_intersection.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#include <algorithm>
#include <cassert>
#include <cstdint>
#line 1 "utilities/fast_io.hpp"
#line 5 "utilities/fast_io.hpp"
#include <array>
#include <cerrno>
#include <charconv>
#include <cstddef>
#include <cstdio>
#include <cstdlib>
#line 12 "utilities/fast_io.hpp"
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#include <type_traits>
#include <utility>
#include <unistd.h>
#include <vector>
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 7 "verify/matroid/matroid_intersection.test.cpp"
#include <random>
#line 10 "verify/matroid/matroid_intersection.test.cpp"
#line 1 "matroid/graphic_matroid.hpp"
#line 5 "matroid/graphic_matroid.hpp"
#include <numeric>
#line 8 "matroid/graphic_matroid.hpp"
namespace m1une {
namespace matroid {
class GraphicMatroid {
private:
int _vertex_count;
std::vector<std::pair<int, int>> _edges;
public:
GraphicMatroid() : _vertex_count(0) {}
GraphicMatroid(int vertex_count, std::vector<std::pair<int, int>> edges)
: _vertex_count(vertex_count), _edges(std::move(edges)) {
assert(0 <= vertex_count);
#ifndef NDEBUG
for (auto [u, v] : _edges) {
assert(0 <= u && u < _vertex_count);
assert(0 <= v && v < _vertex_count);
}
#endif
}
int size() const {
return int(_edges.size());
}
int vertex_count() const {
return _vertex_count;
}
const std::vector<std::pair<int, int>>& edges() const {
return _edges;
}
bool independent(const std::vector<int>& subset) const {
std::vector<int> parent_or_size(_vertex_count, -1);
auto leader = [&](auto&& self, int v) -> int {
if (parent_or_size[v] < 0) return v;
return parent_or_size[v] = self(self, parent_or_size[v]);
};
for (int element : subset) {
assert(0 <= element && element < int(_edges.size()));
auto [u, v] = _edges[element];
u = leader(leader, u);
v = leader(leader, v);
if (u == v) return false;
if (-parent_or_size[u] < -parent_or_size[v]) std::swap(u, v);
parent_or_size[u] += parent_or_size[v];
parent_or_size[v] = u;
}
return true;
}
bool operator()(const std::vector<int>& subset) const {
return independent(subset);
}
};
} // namespace matroid
} // namespace m1une
#line 1 "matroid/linear_matroid.hpp"
#line 9 "matroid/linear_matroid.hpp"
namespace m1une {
namespace matroid {
template <class Field>
class LinearMatroid {
private:
int _dimension;
std::vector<std::vector<Field>> _vectors;
public:
LinearMatroid() : _dimension(0) {}
explicit LinearMatroid(std::vector<std::vector<Field>> vectors)
: _dimension(vectors.empty() ? 0 : int(vectors[0].size())),
_vectors(std::move(vectors)) {
#ifndef NDEBUG
for (const auto& vector : _vectors) assert(int(vector.size()) == _dimension);
#endif
}
int size() const {
return int(_vectors.size());
}
int dimension() const {
return _dimension;
}
const std::vector<std::vector<Field>>& vectors() const {
return _vectors;
}
bool independent(const std::vector<int>& subset) const {
if (int(subset.size()) > _dimension) return false;
std::vector<std::vector<Field>> basis(_dimension);
std::vector<char> has_pivot(_dimension, false);
for (int element : subset) {
assert(0 <= element && element < int(_vectors.size()));
std::vector<Field> vector = _vectors[element];
bool inserted = false;
for (int column = 0; column < _dimension; column++) {
if (vector[column] == Field(0)) continue;
if (!has_pivot[column]) {
Field inverse = Field(1) / vector[column];
for (int j = column; j < _dimension; j++) vector[j] *= inverse;
basis[column] = std::move(vector);
has_pivot[column] = true;
inserted = true;
break;
}
Field factor = vector[column];
for (int j = column; j < _dimension; j++) {
vector[j] -= factor * basis[column][j];
}
}
if (!inserted) return false;
}
return true;
}
bool operator()(const std::vector<int>& subset) const {
return independent(subset);
}
};
class BinaryLinearMatroid {
private:
std::vector<std::uint64_t> _vectors;
public:
BinaryLinearMatroid() = default;
explicit BinaryLinearMatroid(std::vector<std::uint64_t> vectors)
: _vectors(std::move(vectors)) {}
int size() const {
return int(_vectors.size());
}
int dimension() const {
return 64;
}
const std::vector<std::uint64_t>& vectors() const {
return _vectors;
}
bool independent(const std::vector<int>& subset) const {
if (subset.size() > 64) return false;
std::array<std::uint64_t, 64> basis = {};
for (int element : subset) {
assert(0 <= element && element < int(_vectors.size()));
std::uint64_t value = _vectors[element];
for (int bit = 63; bit >= 0; bit--) {
if ((value >> bit & 1) == 0) continue;
if (basis[bit] == 0) {
basis[bit] = value;
break;
}
value ^= basis[bit];
}
if (value == 0) return false;
}
return true;
}
bool operator()(const std::vector<int>& subset) const {
return independent(subset);
}
};
} // namespace matroid
} // namespace m1une
#line 1 "matroid/matroid_intersection.hpp"
#line 7 "matroid/matroid_intersection.hpp"
namespace m1une {
namespace matroid {
template <class IndependenceOracle1, class IndependenceOracle2>
std::vector<int> matroid_intersection(int ground_size, IndependenceOracle1 oracle1,
IndependenceOracle2 oracle2) {
assert(0 <= ground_size);
std::vector<char> selected(ground_size, false);
std::vector<int> elements;
std::vector<int> position(ground_size, -1);
while (true) {
std::vector<char> source(ground_size, false);
std::vector<char> sink(ground_size, false);
std::vector<int> distance(ground_size, -1);
std::vector<int> previous(ground_size, -1);
std::vector<int> queue;
queue.reserve(ground_size);
for (int x = 0; x < ground_size; x++) {
if (selected[x]) continue;
elements.push_back(x);
source[x] = oracle1(elements);
sink[x] = oracle2(elements);
elements.pop_back();
if (source[x]) {
distance[x] = 0;
queue.push_back(x);
}
}
int target = -1;
for (int head = 0; head < int(queue.size()) && target == -1; head++) {
int v = queue[head];
if (!selected[v] && sink[v]) {
target = v;
break;
}
if (selected[v]) {
int index = position[v];
assert(index != -1 && elements[index] == v);
for (int x = 0; x < ground_size; x++) {
if (selected[x] || distance[x] != -1) continue;
elements[index] = x;
bool independent = oracle1(elements);
elements[index] = v;
if (!independent) continue;
distance[x] = distance[v] + 1;
previous[x] = v;
queue.push_back(x);
}
} else {
for (int y : elements) {
if (distance[y] != -1) continue;
int index = position[y];
assert(index != -1 && elements[index] == y);
elements[index] = v;
bool independent = oracle2(elements);
elements[index] = y;
if (!independent) continue;
distance[y] = distance[v] + 1;
previous[y] = v;
queue.push_back(y);
}
}
}
if (target == -1) break;
for (int v = target; v != -1; v = previous[v]) selected[v] = !selected[v];
elements.clear();
std::fill(position.begin(), position.end(), -1);
for (int x = 0; x < ground_size; x++) {
if (!selected[x]) continue;
position[x] = int(elements.size());
elements.push_back(x);
}
#ifndef NDEBUG
assert(oracle1(elements));
assert(oracle2(elements));
#endif
}
return elements;
}
} // namespace matroid
} // namespace m1une
#line 1 "matroid/partition_matroid.hpp"
#line 8 "matroid/partition_matroid.hpp"
namespace m1une {
namespace matroid {
class PartitionMatroid {
private:
std::vector<int> _group;
std::vector<int> _capacity;
void validate() const {
#ifndef NDEBUG
for (int capacity : _capacity) assert(0 <= capacity);
for (int group : _group) assert(0 <= group && group < int(_capacity.size()));
#endif
}
public:
PartitionMatroid() = default;
explicit PartitionMatroid(std::vector<int> group) : _group(std::move(group)) {
int group_count = 0;
for (int value : _group) {
assert(0 <= value);
group_count = std::max(group_count, value + 1);
}
_capacity.assign(group_count, 1);
}
PartitionMatroid(std::vector<int> group, std::vector<int> capacity)
: _group(std::move(group)), _capacity(std::move(capacity)) {
validate();
}
int size() const {
return int(_group.size());
}
int group_count() const {
return int(_capacity.size());
}
const std::vector<int>& groups() const {
return _group;
}
const std::vector<int>& capacities() const {
return _capacity;
}
bool independent(const std::vector<int>& subset) const {
std::vector<int> count(_capacity.size(), 0);
for (int element : subset) {
assert(0 <= element && element < int(_group.size()));
int group = _group[element];
if (++count[group] > _capacity[group]) return false;
}
return true;
}
bool operator()(const std::vector<int>& subset) const {
return independent(subset);
}
};
} // namespace matroid
} // namespace m1une
#line 15 "verify/matroid/matroid_intersection.test.cpp"
template <class Oracle1, class Oracle2>
int brute_force(int ground_size, const Oracle1& oracle1, const Oracle2& oracle2) {
int best = 0;
for (int mask = 0; mask < (1 << ground_size); mask++) {
std::vector<int> subset;
for (int i = 0; i < ground_size; i++) {
if (mask >> i & 1) subset.push_back(i);
}
if (oracle1(subset) && oracle2(subset)) best = std::max(best, int(subset.size()));
}
return best;
}
template <class Oracle1, class Oracle2>
void check(int ground_size, const Oracle1& oracle1, const Oracle2& oracle2) {
auto result = m1une::matroid::matroid_intersection(ground_size, oracle1, oracle2);
assert(std::is_sorted(result.begin(), result.end()));
assert(std::adjacent_find(result.begin(), result.end()) == result.end());
assert(oracle1(result));
assert(oracle2(result));
assert(int(result.size()) == brute_force(ground_size, oracle1, oracle2));
}
void test_augmenting_exchange() {
m1une::matroid::PartitionMatroid left(std::vector<int>{0, 0, 1});
m1une::matroid::PartitionMatroid right(std::vector<int>{0, 1, 0});
auto result = m1une::matroid::matroid_intersection(3, left, right);
assert((result == std::vector<int>{1, 2}));
}
void test_random_matroids() {
std::mt19937 random(123456789);
for (int test = 0; test < 300; test++) {
int ground_size = random() % 11;
int group_count = 1 + random() % 5;
std::vector<int> group(ground_size);
for (int& value : group) value = random() % group_count;
std::vector<int> capacity(group_count, 1);
m1une::matroid::PartitionMatroid partition(group, capacity);
int vertex_count = 1 + random() % 6;
std::vector<std::pair<int, int>> edge(ground_size);
for (auto& [u, v] : edge) {
u = random() % vertex_count;
v = random() % vertex_count;
}
m1une::matroid::GraphicMatroid graphic(vertex_count, edge);
check(ground_size, partition, graphic);
int second_group_count = 1 + random() % 5;
std::vector<int> second_group(ground_size);
for (int& value : second_group) value = random() % second_group_count;
m1une::matroid::PartitionMatroid second_partition(second_group);
check(ground_size, partition, second_partition);
std::vector<std::uint64_t> vector1(ground_size), vector2(ground_size);
for (int i = 0; i < ground_size; i++) {
vector1[i] = random() % 64;
vector2[i] = random() % 64;
}
m1une::matroid::BinaryLinearMatroid linear1(vector1), linear2(vector2);
check(ground_size, linear1, linear2);
}
}
int main() {
m1une::utilities::FastInput fast_input;
m1une::utilities::FastOutput fast_output;
test_augmenting_exchange();
test_random_matroids();
long long a, b;
fast_input >> a >> b;
fast_output << a + b << '\n';
}
verify/matroid/matroid_intersection.test.cpp