#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#include <algorithm>
#include <cassert>
#include <functional>
#include "../../../utilities/fast_io.hpp"
#include <limits>
#include <vector>
#include "../../../convex/monge/all.hpp"
template <class Value, class Compare>
std::vector<int> brute_row_optima(int rows, int columns, Value value, Compare compare) {
std::vector<int> result(rows, -1);
if (columns == 0) return result;
for (int row = 0; row < rows; row++) {
result[row] = 0;
for (int column = 1; column < columns; column++) {
if (compare(value(row, column), value(row, result[row]))) {
result[row] = column;
}
}
}
return result;
}
void test_smawk_monge() {
for (int rows = 0; rows <= 40; rows++) {
for (int columns = 0; columns <= 40; columns++) {
auto value = [&](int row, int column) {
long long difference = row * 3LL - column * 2LL;
return difference * difference + row * 7LL + column * 5LL;
};
auto expected = brute_row_optima(rows, columns, value, std::less<>());
assert(m1une::convex::smawk_row_argmin(rows, columns, value) == expected);
}
}
}
void test_smawk_totally_monotone_and_ties() {
for (int rows = 1; rows <= 50; rows++) {
for (int columns = 1; columns <= 50; columns++) {
std::vector<int> threshold(rows);
for (int row = 0; row < rows; row++) {
threshold[row] = std::min(columns - 1, (row * 7 + rows) / 5);
}
auto value = [&](int row, int column) {
int difference = column - threshold[row];
return difference * difference;
};
auto result = m1une::convex::smawk_row_argmin(rows, columns, value);
assert(result == threshold);
}
}
auto constant = [](int, int) { return 0; };
auto result = m1une::convex::smawk_row_argmin(20, 30, constant);
assert(result == std::vector<int>(20, 0));
}
void test_smawk_max() {
auto value = [](int row, int column) {
long long difference = row - column;
return -difference * difference;
};
auto expected = brute_row_optima(30, 25, value, std::greater<>());
assert(m1une::convex::smawk_row_argmax(30, 25, value) == expected);
}
void test_smawk_matrix_overload_and_evaluations() {
std::vector<std::vector<long long>> matrix(17, std::vector<long long>(23));
for (int row = 0; row < 17; row++) {
for (int column = 0; column < 23; column++) {
long long difference = row * 2LL - column;
matrix[row][column] = difference * difference;
}
}
auto expected = brute_row_optima(
17, 23, [&](int row, int column) { return matrix[row][column]; }, std::less<>());
assert(m1une::convex::smawk_row_argmin(matrix) == expected);
for (auto sizes : std::vector<std::pair<int, int>>{
std::pair<int, int>{10, 1000},
std::pair<int, int>{1000, 10},
std::pair<int, int>{700, 900},
}) {
int evaluations = 0;
auto value = [&](int row, int column) {
evaluations++;
long long difference = row * 3LL - column * 2LL;
return difference * difference;
};
auto result = m1une::convex::smawk_row_argmin(sizes.first, sizes.second, value);
assert(int(result.size()) == sizes.first);
assert(evaluations <= 20 * (sizes.first + sizes.second));
}
}
void test_monotone_minima() {
for (int rows = 1; rows <= 40; rows++) {
for (int columns = 1; columns <= 40; columns++) {
std::vector<int> optimum(rows);
for (int row = 0; row < rows; row++) {
optimum[row] = std::min(columns - 1, row * columns / rows);
}
auto value = [&](int row, int column) {
if (column == optimum[row]) return 0;
return 1000 + (row * 97 + column * 53) % 101;
};
auto result = m1une::convex::monotone_row_argmin(rows, columns, value);
assert(result == optimum);
}
}
auto empty = m1une::convex::monotone_row_argmin(5, 0, [](int, int) { return 0; });
assert(empty == std::vector<int>(5, -1));
std::vector<std::vector<int>> matrix;
matrix.emplace_back(std::vector<int>{0, 5, 7});
matrix.emplace_back(std::vector<int>{4, 0, 8});
matrix.emplace_back(std::vector<int>{9, 3, 0});
assert(m1une::convex::monotone_row_argmin(matrix) == std::vector<int>({0, 1, 2}));
}
void test_monge_checks() {
std::vector<std::vector<long long>> matrix(8, std::vector<long long>(11));
for (int row = 0; row < 8; row++) {
for (int column = 0; column < 11; column++) {
long long difference = row - column;
matrix[row][column] = difference * difference;
}
}
assert(m1une::convex::is_monge(matrix));
assert(!m1une::convex::is_anti_monge(matrix));
for (auto& row : matrix) {
for (auto& value : row) value = -value;
}
assert(!m1une::convex::is_monge(matrix));
assert(m1une::convex::is_anti_monge(matrix));
std::vector<std::vector<int>> empty;
assert(m1une::convex::is_monge(empty));
assert(m1une::convex::is_anti_monge(empty));
}
template <class T, class Compare>
std::vector<T> brute_convolution(const std::vector<T>& first, const std::vector<T>& second,
Compare compare) {
if (first.empty() || second.empty()) return {};
std::vector<T> result(first.size() + second.size() - 1);
for (int index = 0; index < int(result.size()); index++) {
int left = std::max(0, index - int(second.size()) + 1);
int right = std::min(int(first.size()), index + 1);
result[index] = first[left] + second[index - left];
for (int i = left + 1; i < right; i++) {
T value = first[i] + second[index - i];
if (compare(value, result[index])) result[index] = value;
}
}
return result;
}
template <class T, class Compare>
std::vector<T> brute_convolution_with_infinity(const std::vector<T>& first,
const std::vector<T>& second,
Compare compare,
const T& infinity) {
if (first.empty() || second.empty()) return {};
std::vector<T> result(first.size() + second.size() - 1, infinity);
for (int i = 0; i < int(first.size()); i++) {
for (int j = 0; j < int(second.size()); j++) {
if (first[i] == infinity || second[j] == infinity) continue;
T value = first[i] + second[j];
if (result[i + j] == infinity || compare(value, result[i + j])) {
result[i + j] = value;
}
}
}
return result;
}
void test_structured_convolutions() {
const long long infinity = 2'000'000'000'000'000'000LL;
const long long negative_infinity = -infinity;
for (int first_size = 0; first_size <= 35; first_size++) {
for (int second_size = 0; second_size <= 35; second_size++) {
for (int test = 0; test < 8; test++) {
std::vector<long long> arbitrary(first_size);
for (int i = 0; i < first_size; i++) {
arbitrary[i] = (test * 17 + i * 31 + first_size * 7) % 61 - 30;
}
std::vector<long long> convex(second_size);
long long difference = -10 + test;
for (int i = 1; i < second_size; i++) {
difference += (test * 3 + i * 5) % 4;
convex[i] = convex[i - 1] + difference;
}
assert(m1une::convex::is_convex_sequence(convex));
auto expected_min = brute_convolution(arbitrary, convex, std::less<>());
assert(m1une::convex::min_plus_convolution_convex(arbitrary, convex) ==
expected_min);
std::vector<long long> arbitrary_extended = arbitrary;
for (int i = 0; i < first_size; i++) {
if ((i + test) % 7 == 0) arbitrary_extended[i] = infinity;
}
std::vector<long long> convex_extended(second_size + 2, infinity);
std::copy(convex.begin(), convex.end(), convex_extended.begin() + 1);
assert(m1une::convex::is_convex_sequence(convex_extended, infinity));
auto expected_extended_min = brute_convolution_with_infinity(
arbitrary_extended, convex_extended, std::less<>(), infinity);
assert(m1une::convex::min_plus_convolution_convex(
arbitrary_extended, convex_extended, infinity) ==
expected_extended_min);
std::vector<long long> first_convex(first_size);
difference = -12 - test;
for (int i = 1; i < first_size; i++) {
difference += (test * 5 + i * 7) % 5;
first_convex[i] = first_convex[i - 1] + difference;
}
assert(m1une::convex::is_convex_sequence(first_convex));
auto expected_convex =
brute_convolution(first_convex, convex, std::less<>());
assert(m1une::convex::min_plus_convolution_convex(first_convex, convex) ==
expected_convex);
assert(m1une::convex::min_plus_convolution_convex(convex, first_convex) ==
expected_convex);
assert(m1une::convex::min_plus_convolution_convex_convex(first_convex,
convex) ==
expected_convex);
assert(m1une::convex::min_plus_convolution_convex_convex(convex,
first_convex) ==
expected_convex);
std::vector<long long> first_convex_extended(first_size + 2,
infinity);
std::copy(first_convex.begin(), first_convex.end(),
first_convex_extended.begin() + 1);
auto expected_extended_convex = brute_convolution_with_infinity(
first_convex_extended, convex_extended, std::less<>(), infinity);
assert(m1une::convex::min_plus_convolution_convex_convex(
first_convex_extended, convex_extended, infinity) ==
expected_extended_convex);
std::vector<long long> concave = convex;
for (auto& value : concave) value = -value;
assert(m1une::convex::is_concave_sequence(concave));
auto expected_max = brute_convolution(arbitrary, concave, std::greater<>());
assert(m1une::convex::max_plus_convolution_concave(arbitrary, concave) ==
expected_max);
std::vector<long long> first_concave = first_convex;
for (auto& value : first_concave) value = -value;
assert(m1une::convex::is_concave_sequence(first_concave));
auto expected_concave =
brute_convolution(first_concave, concave, std::greater<>());
assert(m1une::convex::max_plus_convolution_concave(first_concave, concave) ==
expected_concave);
assert(m1une::convex::max_plus_convolution_concave(concave, first_concave) ==
expected_concave);
assert(m1une::convex::max_plus_convolution_concave_concave(first_concave,
concave) ==
expected_concave);
assert(m1une::convex::max_plus_convolution_concave_concave(concave,
first_concave) ==
expected_concave);
std::vector<long long> arbitrary_max_extended = arbitrary_extended;
for (long long& value : arbitrary_max_extended) {
value = value == infinity ? negative_infinity : -value;
}
std::vector<long long> concave_extended = convex_extended;
for (long long& value : concave_extended) {
value = value == infinity ? negative_infinity : -value;
}
auto expected_extended_max = brute_convolution_with_infinity(
arbitrary_max_extended, concave_extended, std::greater<>(),
negative_infinity);
assert(m1une::convex::max_plus_convolution_concave(
arbitrary_max_extended, concave_extended,
negative_infinity) == expected_extended_max);
std::vector<long long> first_concave_extended =
first_convex_extended;
for (long long& value : first_concave_extended) {
value = value == infinity ? negative_infinity : -value;
}
auto expected_extended_concave = brute_convolution_with_infinity(
first_concave_extended, concave_extended, std::greater<>(),
negative_infinity);
assert(m1une::convex::max_plus_convolution_concave_concave(
first_concave_extended, concave_extended,
negative_infinity) == expected_extended_concave);
}
}
}
assert(!m1une::convex::is_convex_sequence(std::vector<int>{0, 2, 1}));
assert(!m1une::convex::is_concave_sequence(std::vector<int>{0, -2, -1}));
assert(!m1une::convex::is_convex_sequence(
std::vector<long long>{1, infinity, infinity}));
assert(m1une::convex::is_convex_sequence(
std::vector<long long>{1, infinity, infinity}, infinity));
assert(m1une::convex::is_convex_sequence(
std::vector<long long>{infinity, 0, 1, 4, infinity}, infinity));
assert(!m1une::convex::is_convex_sequence(
std::vector<long long>{0, infinity, 1}, infinity));
std::vector<long long> arbitrary = {infinity, 3, infinity, -2};
std::vector<long long> convex = {infinity, 0, 1, 4, infinity, infinity};
auto expected_min =
brute_convolution_with_infinity(arbitrary, convex, std::less<>(), infinity);
assert(m1une::convex::min_plus_convolution_convex(arbitrary, convex, infinity) ==
expected_min);
std::vector<long long> first_convex = {infinity, 2, 2, 3, infinity};
expected_min = brute_convolution_with_infinity(first_convex, convex,
std::less<>(), infinity);
assert(m1une::convex::min_plus_convolution_convex_convex(
first_convex, convex, infinity) == expected_min);
long long unordered_infinity = 7;
std::vector<long long> large_arbitrary = {10, 20};
std::vector<long long> small_convex = {0, 1};
assert(m1une::convex::min_plus_convolution_convex(
large_arbitrary, small_convex, unordered_infinity) ==
std::vector<long long>({10, 11, 21}));
std::vector<long long> arbitrary_max = {negative_infinity, 3,
negative_infinity, -2};
std::vector<long long> concave = {negative_infinity, 0, -1, -4,
negative_infinity};
assert(m1une::convex::is_concave_sequence(concave, negative_infinity));
auto expected_max = brute_convolution_with_infinity(
arbitrary_max, concave, std::greater<>(), negative_infinity);
assert(m1une::convex::max_plus_convolution_concave(
arbitrary_max, concave, negative_infinity) == expected_max);
std::vector<long long> first_concave = {negative_infinity, 2, 2, 1,
negative_infinity};
expected_max = brute_convolution_with_infinity(
first_concave, concave, std::greater<>(), negative_infinity);
assert(m1une::convex::max_plus_convolution_concave_concave(
first_concave, concave, negative_infinity) == expected_max);
std::vector<long long> all_infinity(3, infinity);
assert(m1une::convex::is_convex_sequence(all_infinity, infinity));
assert(m1une::convex::min_plus_convolution_convex_convex(
all_infinity, convex, infinity) ==
std::vector<long long>(all_infinity.size() + convex.size() - 1,
infinity));
}
int main() {
m1une::utilities::FastInput fast_input;
m1une::utilities::FastOutput fast_output;
test_smawk_monge();
test_smawk_totally_monotone_and_ties();
test_smawk_max();
test_smawk_matrix_overload_and_evaluations();
test_monotone_minima();
test_monge_checks();
test_structured_convolutions();
long long a, b;
fast_input >> a >> b;
fast_output << a + b << '\n';
}
#line 1 "verify/convex/monge/monge_algorithms.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#include <algorithm>
#include <cassert>
#include <functional>
#line 1 "utilities/fast_io.hpp"
#line 5 "utilities/fast_io.hpp"
#include <array>
#include <cerrno>
#include <charconv>
#include <cstddef>
#include <cstdio>
#include <cstdlib>
#include <cstdint>
#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/convex/monge/monge_algorithms.test.cpp"
#include <limits>
#line 9 "verify/convex/monge/monge_algorithms.test.cpp"
#line 1 "convex/monge/all.hpp"
#line 1 "convex/monge/check.hpp"
#line 6 "convex/monge/check.hpp"
namespace m1une {
namespace convex {
template <class Value>
bool is_monge(int row_count, int column_count, Value value) {
assert(row_count >= 0);
assert(column_count >= 0);
for (int row = 0; row + 1 < row_count; row++) {
for (int column = 0; column + 1 < column_count; column++) {
if (value(row, column) + value(row + 1, column + 1) >
value(row, column + 1) + value(row + 1, column)) {
return false;
}
}
}
return true;
}
template <class Value>
bool is_anti_monge(int row_count, int column_count, Value value) {
assert(row_count >= 0);
assert(column_count >= 0);
for (int row = 0; row + 1 < row_count; row++) {
for (int column = 0; column + 1 < column_count; column++) {
if (value(row, column) + value(row + 1, column + 1) <
value(row, column + 1) + value(row + 1, column)) {
return false;
}
}
}
return true;
}
template <class T>
bool is_monge(const std::vector<std::vector<T>>& matrix) {
int row_count = int(matrix.size());
int column_count = row_count == 0 ? 0 : int(matrix[0].size());
for (const auto& row : matrix) assert(int(row.size()) == column_count);
return is_monge(row_count, column_count,
[&](int row, int column) -> const T& { return matrix[row][column]; });
}
template <class T>
bool is_anti_monge(const std::vector<std::vector<T>>& matrix) {
int row_count = int(matrix.size());
int column_count = row_count == 0 ? 0 : int(matrix[0].size());
for (const auto& row : matrix) assert(int(row.size()) == column_count);
return is_anti_monge(
row_count, column_count,
[&](int row, int column) -> const T& { return matrix[row][column]; });
}
} // namespace convex
} // namespace m1une
#line 1 "convex/monge/divide_and_conquer_optimization.hpp"
#line 7 "convex/monge/divide_and_conquer_optimization.hpp"
#line 1 "convex/monge/monotone_minima.hpp"
#line 7 "convex/monge/monotone_minima.hpp"
namespace m1une {
namespace convex {
namespace monotone_minima_detail {
template <class Value, class Compare>
void solve(int row_left, int row_right, int column_left, int column_right,
const Value& value, const Compare& compare, std::vector<int>& answer) {
if (row_left == row_right) return;
int row = (row_left + row_right) / 2;
int best = column_left;
for (int column = column_left + 1; column < column_right; column++) {
if (compare(value(row, column), value(row, best))) best = column;
}
answer[row] = best;
solve(row_left, row, column_left, best + 1, value, compare, answer);
solve(row + 1, row_right, best, column_right, value, compare, answer);
}
} // namespace monotone_minima_detail
template <class Value, class Compare = std::less<>>
std::vector<int> monotone_row_optima(int row_count, int column_count, Value value,
Compare compare = Compare()) {
assert(row_count >= 0);
assert(column_count >= 0);
std::vector<int> answer(row_count, -1);
if (row_count == 0 || column_count == 0) return answer;
monotone_minima_detail::solve(0, row_count, 0, column_count, value, compare, answer);
return answer;
}
template <class Value>
std::vector<int> monotone_row_argmin(int row_count, int column_count, Value value) {
return monotone_row_optima(row_count, column_count, value, std::less<>());
}
template <class Value>
std::vector<int> monotone_row_argmax(int row_count, int column_count, Value value) {
return monotone_row_optima(row_count, column_count, value, std::greater<>());
}
template <class T>
std::vector<int> monotone_row_argmin(const std::vector<std::vector<T>>& matrix) {
int row_count = int(matrix.size());
int column_count = row_count == 0 ? 0 : int(matrix[0].size());
for (const auto& row : matrix) assert(int(row.size()) == column_count);
return monotone_row_argmin(
row_count, column_count,
[&](int row, int column) -> const T& { return matrix[row][column]; });
}
template <class T>
std::vector<int> monotone_row_argmax(const std::vector<std::vector<T>>& matrix) {
int row_count = int(matrix.size());
int column_count = row_count == 0 ? 0 : int(matrix[0].size());
for (const auto& row : matrix) assert(int(row.size()) == column_count);
return monotone_row_argmax(
row_count, column_count,
[&](int row, int column) -> const T& { return matrix[row][column]; });
}
} // namespace convex
} // namespace m1une
#line 9 "convex/monge/divide_and_conquer_optimization.hpp"
namespace m1une {
namespace convex {
template <class T>
struct DivideAndConquerDpResult {
std::vector<T> value;
std::vector<int> argmin;
};
template <class Value>
auto divide_and_conquer_dp(int state_count, int candidate_count, Value value)
-> DivideAndConquerDpResult<
std::decay_t<std::invoke_result_t<Value, int, int>>> {
using T = std::decay_t<std::invoke_result_t<Value, int, int>>;
DivideAndConquerDpResult<T> result;
result.argmin = monotone_row_argmin(state_count, candidate_count, value);
result.value.resize(state_count);
for (int state = 0; state < state_count; state++) {
if (result.argmin[state] != -1) {
result.value[state] = value(state, result.argmin[state]);
}
}
return result;
}
template <class T, class Cost>
auto divide_and_conquer_transition(const std::vector<T>& previous, int state_count,
Cost cost)
-> DivideAndConquerDpResult<
std::decay_t<decltype(std::declval<T>() + cost(0, 0))>> {
using Result = std::decay_t<decltype(std::declval<T>() + cost(0, 0))>;
return divide_and_conquer_dp(
state_count, int(previous.size()),
[&](int state, int candidate) -> Result {
return previous[candidate] + cost(candidate, state);
});
}
} // namespace convex
} // namespace m1une
#line 1 "convex/monge/knuth_optimization.hpp"
#line 8 "convex/monge/knuth_optimization.hpp"
namespace m1une {
namespace convex {
template <class T>
struct KnuthOptimizationResult {
std::vector<std::vector<T>> value;
std::vector<std::vector<int>> split;
T optimum() const {
return value[0].back();
}
};
template <class IntervalCost>
auto knuth_optimization(int element_count, IntervalCost interval_cost)
-> KnuthOptimizationResult<
std::decay_t<std::invoke_result_t<IntervalCost, int, int>>> {
assert(element_count >= 0);
using T = std::decay_t<std::invoke_result_t<IntervalCost, int, int>>;
KnuthOptimizationResult<T> result;
result.value.assign(element_count + 1, std::vector<T>(element_count + 1, T()));
result.split.assign(element_count + 1, std::vector<int>(element_count + 1, -1));
for (int left = 0; left <= element_count; left++) result.split[left][left] = left;
for (int left = 0; left < element_count; left++) result.split[left][left + 1] = left + 1;
for (int length = 2; length <= element_count; length++) {
for (int left = 0; left + length <= element_count; left++) {
int right = left + length;
int first = std::max(left + 1, result.split[left][right - 1]);
int last = std::min(right - 1, result.split[left + 1][right]);
assert(first <= last);
int best = first;
T best_value = result.value[left][best] + result.value[best][right];
for (int split = first + 1; split <= last; split++) {
T candidate = result.value[left][split] + result.value[split][right];
if (candidate < best_value) {
best = split;
best_value = candidate;
}
}
result.value[left][right] = best_value + interval_cost(left, right);
result.split[left][right] = best;
}
}
return result;
}
} // namespace convex
} // namespace m1une
#line 1 "convex/monge/larsch.hpp"
#line 6 "convex/monge/larsch.hpp"
#include <memory>
#line 10 "convex/monge/larsch.hpp"
namespace m1une {
namespace convex {
template <class T>
class Larsch {
struct ReduceColumn;
struct ReduceRow {
int size;
std::function<T(int, int)> value;
int current_row = 0;
int boundary = 0;
std::unique_ptr<ReduceColumn> recursive;
explicit ReduceRow(int size_) : size(size_) {
if (size / 2 != 0) recursive = std::make_unique<ReduceColumn>(size / 2);
}
void set_value(std::function<T(int, int)> value_) {
value = std::move(value_);
if (recursive) {
recursive->set_value(
[&](int row, int column) { return value(row * 2 + 1, column); });
}
}
int next_argmin() {
int row = current_row++;
if (row % 2 == 0) {
int previous = boundary;
int next = row + 1 == size ? size - 1 : recursive->next_argmin();
boundary = next;
int best = previous;
for (int column = previous + 1; column <= next; column++) {
if (value(row, column) < value(row, best)) best = column;
}
return best;
}
return value(row, boundary) <= value(row, row) ? boundary : row;
}
};
struct ReduceColumn {
int size;
std::function<T(int, int)> value;
int current_row = 0;
std::vector<int> columns;
ReduceRow recursive;
explicit ReduceColumn(int size_) : size(size_), recursive(size_) {}
void set_value(std::function<T(int, int)> value_) {
value = std::move(value_);
recursive.set_value(
[&](int row, int column) { return value(row, columns[column]); });
}
int next_argmin() {
int row = current_row++;
int first = row == 0 ? 0 : row * 2 - 1;
int last = row * 2;
for (int column = first; column <= last; column++) {
while (int(columns.size()) != row &&
value(int(columns.size()) - 1, columns.back()) >
value(int(columns.size()) - 1, column)) {
columns.pop_back();
}
if (int(columns.size()) != size) columns.push_back(column);
}
return columns[recursive.next_argmin()];
}
};
int _size;
int _processed = 0;
std::unique_ptr<ReduceRow> _base;
public:
template <class Value>
explicit Larsch(int size, Value value)
: _size(size), _base(std::make_unique<ReduceRow>(size)) {
assert(size >= 0);
_base->set_value(std::function<T(int, int)>(std::move(value)));
}
int size() const {
return _size;
}
int processed_rows() const {
return _processed;
}
bool finished() const {
return _processed == _size;
}
int next_argmin() {
assert(!finished());
_processed++;
return _base->next_argmin();
}
};
template <class T>
struct LarschShortestPathResult {
std::vector<T> distance;
std::vector<int> parent;
};
template <class Cost>
auto larsch_shortest_path(int vertex_count, Cost cost)
-> LarschShortestPathResult<
std::decay_t<std::invoke_result_t<Cost, int, int>>> {
using T = std::decay_t<std::invoke_result_t<Cost, int, int>>;
assert(vertex_count >= 0);
LarschShortestPathResult<T> result;
result.distance.assign(vertex_count, T());
result.parent.assign(vertex_count, -1);
if (vertex_count <= 1) return result;
Larsch<T> optimizer(vertex_count - 1, [&](int row, int column) {
return result.distance[column] + cost(column, row + 1);
});
for (int vertex = 1; vertex < vertex_count; vertex++) {
int parent = optimizer.next_argmin();
result.parent[vertex] = parent;
result.distance[vertex] = result.distance[parent] + cost(parent, vertex);
}
return result;
}
} // namespace convex
} // namespace m1une
#line 1 "convex/monge/min_plus_convolution.hpp"
#line 7 "convex/monge/min_plus_convolution.hpp"
#line 1 "convex/monge/smawk.hpp"
#line 6 "convex/monge/smawk.hpp"
#include <numeric>
#line 8 "convex/monge/smawk.hpp"
namespace m1une {
namespace convex {
namespace smawk_detail {
template <class Select>
void solve(const std::vector<int>& rows, const std::vector<int>& columns,
const Select& select, std::vector<int>& answer) {
if (rows.empty()) return;
std::vector<int> reduced;
reduced.reserve(columns.size());
for (int column : columns) {
while (!reduced.empty()) {
int row = rows[int(reduced.size()) - 1];
if (!select(row, reduced.back(), column)) break;
reduced.pop_back();
}
if (reduced.size() < rows.size()) reduced.push_back(column);
}
std::vector<int> odd_rows;
odd_rows.reserve(rows.size() / 2);
for (int i = 1; i < int(rows.size()); i += 2) odd_rows.push_back(rows[i]);
solve(odd_rows, reduced, select, answer);
int left = 0;
int right = 0;
for (int i = 0; i < int(rows.size()); i += 2) {
if (i + 1 < int(rows.size())) {
while (reduced[right] != answer[rows[i + 1]]) right++;
} else {
right = int(reduced.size()) - 1;
}
int best = left;
for (int j = left + 1; j <= right; j++) {
if (select(rows[i], reduced[best], reduced[j])) {
best = j;
}
}
answer[rows[i]] = reduced[best];
left = right;
}
}
template <class Select>
std::vector<int> row_optima(int row_count, int column_count, const Select& select) {
std::vector<int> answer(row_count, -1);
if (row_count == 0 || column_count == 0) return answer;
std::vector<int> rows(row_count), columns(column_count);
std::iota(rows.begin(), rows.end(), 0);
std::iota(columns.begin(), columns.end(), 0);
solve(rows, columns, select, answer);
return answer;
}
} // namespace smawk_detail
template <class Value, class Compare = std::less<>>
std::vector<int> smawk_row_optima(int row_count, int column_count, Value value,
Compare compare = Compare()) {
assert(row_count >= 0);
assert(column_count >= 0);
return smawk_detail::row_optima(
row_count, column_count,
[&](int row, int current, int candidate) {
return compare(value(row, candidate), value(row, current));
});
}
template <class Value>
std::vector<int> smawk_row_argmin(int row_count, int column_count, Value value) {
return smawk_row_optima(row_count, column_count, value, std::less<>());
}
template <class Value>
std::vector<int> smawk_row_argmax(int row_count, int column_count, Value value) {
return smawk_row_optima(row_count, column_count, value, std::greater<>());
}
template <class T>
std::vector<int> smawk_row_argmin(const std::vector<std::vector<T>>& matrix) {
int row_count = int(matrix.size());
int column_count = row_count == 0 ? 0 : int(matrix[0].size());
for (const auto& row : matrix) assert(int(row.size()) == column_count);
return smawk_row_argmin(
row_count, column_count,
[&](int row, int column) -> const T& { return matrix[row][column]; });
}
template <class T>
std::vector<int> smawk_row_argmax(const std::vector<std::vector<T>>& matrix) {
int row_count = int(matrix.size());
int column_count = row_count == 0 ? 0 : int(matrix[0].size());
for (const auto& row : matrix) assert(int(row.size()) == column_count);
return smawk_row_argmax(
row_count, column_count,
[&](int row, int column) -> const T& { return matrix[row][column]; });
}
} // namespace convex
} // namespace m1une
#line 9 "convex/monge/min_plus_convolution.hpp"
namespace m1une {
namespace convex {
namespace convolution_detail {
template <class T, class Compare, class Add>
std::vector<T> structured_convolution(const std::vector<T>& arbitrary,
const std::vector<T>& structured,
Compare compare, Add add) {
if (arbitrary.empty() || structured.empty()) return {};
int first_size = int(arbitrary.size());
int second_size = int(structured.size());
int result_size = first_size + second_size - 1;
auto select = [&](int index, int current, int candidate) {
if (index < candidate) return false;
if (index - current >= second_size) return true;
T current_value = add(arbitrary[current], structured[index - current]);
T candidate_value = add(arbitrary[candidate], structured[index - candidate]);
return !compare(current_value, candidate_value);
};
std::vector<int> optima =
smawk_detail::row_optima(result_size, first_size, select);
std::vector<T> result;
result.reserve(result_size);
for (int index = 0; index < result_size; index++) {
int first_index = optima[index];
result.emplace_back(add(arbitrary[first_index],
structured[index - first_index]));
}
return result;
}
template <class T>
std::pair<int, int> finite_interval(const std::vector<T>& sequence,
const T& infinity) {
int left = 0;
while (left < int(sequence.size()) && sequence[left] == infinity) left++;
int right = int(sequence.size());
while (right > left && sequence[right - 1] == infinity) right--;
return {left, right};
}
template <class T, class Compare>
std::vector<T> structured_convolution_with_infinity(
const std::vector<T>& arbitrary, const std::vector<T>& structured,
const T& infinity, Compare compare) {
if (arbitrary.empty() || structured.empty()) return {};
auto [left, right] = finite_interval(structured, infinity);
int result_size = int(arbitrary.size() + structured.size() - 1);
std::vector<T> result(result_size, infinity);
if (left == right) return result;
std::vector<int> columns;
columns.reserve(arbitrary.size());
for (int i = 0; i < int(arbitrary.size()); i++) {
if (arbitrary[i] != infinity) columns.push_back(i);
}
if (columns.empty()) return result;
int finite_size = right - left;
int middle_size = int(arbitrary.size()) + finite_size - 1;
std::vector<int> rows;
rows.reserve(middle_size);
int active = 0;
for (int row = 0; row < middle_size; row++) {
if (row < int(arbitrary.size()) && arbitrary[row] != infinity) active++;
if (row >= finite_size && arbitrary[row - finite_size] != infinity) active--;
if (active > 0) rows.push_back(row);
}
auto select = [&](int index, int current, int candidate) {
if (index < candidate) return false;
if (index - current >= finite_size) return true;
T current_value =
arbitrary[current] + structured[left + index - current];
T candidate_value =
arbitrary[candidate] + structured[left + index - candidate];
return !compare(current_value, candidate_value);
};
std::vector<int> optima(middle_size, -1);
smawk_detail::solve(rows, columns, select, optima);
for (int row : rows) {
int first_index = optima[row];
result[left + row] =
arbitrary[first_index] + structured[left + row - first_index];
}
return result;
}
template <class T, class Compare>
std::vector<T> linear_structured_convolution(const std::vector<T>& first,
const std::vector<T>& second,
Compare compare) {
if (first.empty() || second.empty()) return {};
int first_size = int(first.size());
int second_size = int(second.size());
std::vector<T> result(first_size + second_size - 1);
result[0] = first[0] + second[0];
int first_index = 1;
int second_index = 1;
int result_index = 1;
while (first_index < first_size && second_index < second_size) {
T first_difference = first[first_index] - first[first_index - 1];
T second_difference = second[second_index] - second[second_index - 1];
if (compare(second_difference, first_difference)) {
result[result_index] = result[result_index - 1] + second_difference;
second_index++;
} else {
result[result_index] = result[result_index - 1] + first_difference;
first_index++;
}
result_index++;
}
while (first_index < first_size) {
T difference = first[first_index] - first[first_index - 1];
result[result_index] = result[result_index - 1] + difference;
first_index++;
result_index++;
}
while (second_index < second_size) {
T difference = second[second_index] - second[second_index - 1];
result[result_index] = result[result_index - 1] + difference;
second_index++;
result_index++;
}
return result;
}
template <class T, class Compare>
std::vector<T> linear_structured_convolution_with_infinity(
const std::vector<T>& first, const std::vector<T>& second,
const T& infinity, Compare compare) {
if (first.empty() || second.empty()) return {};
auto [first_left, first_right] = finite_interval(first, infinity);
auto [second_left, second_right] = finite_interval(second, infinity);
int result_size = int(first.size() + second.size() - 1);
std::vector<T> result(result_size, infinity);
if (first_left == first_right || second_left == second_right) return result;
int offset = first_left + second_left;
result[offset] = first[first_left] + second[second_left];
int first_index = first_left + 1;
int second_index = second_left + 1;
int result_index = offset + 1;
while (first_index < first_right && second_index < second_right) {
T first_difference = first[first_index] - first[first_index - 1];
T second_difference = second[second_index] - second[second_index - 1];
if (compare(second_difference, first_difference)) {
result[result_index] = result[result_index - 1] + second_difference;
second_index++;
} else {
result[result_index] = result[result_index - 1] + first_difference;
first_index++;
}
result_index++;
}
while (first_index < first_right) {
T difference = first[first_index] - first[first_index - 1];
result[result_index] = result[result_index - 1] + difference;
first_index++;
result_index++;
}
while (second_index < second_right) {
T difference = second[second_index] - second[second_index - 1];
result[result_index] = result[result_index - 1] + difference;
second_index++;
result_index++;
}
return result;
}
template <class T, class Compare>
bool is_structured_sequence_with_infinity(const std::vector<T>& sequence,
const T& infinity, Compare violation) {
auto [left, right] = finite_interval(sequence, infinity);
for (int i = left; i < right; i++) {
if (sequence[i] == infinity) return false;
}
for (int i = left + 1; i + 1 < right; i++) {
T first_difference = sequence[i] - sequence[i - 1];
T second_difference = sequence[i + 1] - sequence[i];
if (violation(first_difference, second_difference)) return false;
}
return true;
}
} // namespace convolution_detail
template <class T>
bool is_convex_sequence(const std::vector<T>& sequence) {
for (int i = 1; i + 1 < int(sequence.size()); i++) {
if (sequence[i] - sequence[i - 1] > sequence[i + 1] - sequence[i]) {
return false;
}
}
return true;
}
template <class T>
bool is_convex_sequence(const std::vector<T>& sequence, const T& infinity) {
return convolution_detail::is_structured_sequence_with_infinity(
sequence, infinity, std::greater<>());
}
template <class T>
bool is_concave_sequence(const std::vector<T>& sequence) {
for (int i = 1; i + 1 < int(sequence.size()); i++) {
if (sequence[i] - sequence[i - 1] < sequence[i + 1] - sequence[i]) {
return false;
}
}
return true;
}
template <class T>
bool is_concave_sequence(const std::vector<T>& sequence,
const T& negative_infinity) {
return convolution_detail::is_structured_sequence_with_infinity(
sequence, negative_infinity, std::less<>());
}
template <class T>
std::vector<T> min_plus_convolution_convex(const std::vector<T>& arbitrary,
const std::vector<T>& convex) {
auto add = [](const T& first, const T& second) { return first + second; };
return convolution_detail::structured_convolution(arbitrary, convex,
std::less<>(), add);
}
template <class T>
std::vector<T> min_plus_convolution_convex(const std::vector<T>& arbitrary,
const std::vector<T>& convex,
const T& infinity) {
return convolution_detail::structured_convolution_with_infinity(
arbitrary, convex, infinity, std::less<>());
}
template <class T>
std::vector<T> min_plus_convolution_convex_convex(const std::vector<T>& first,
const std::vector<T>& second) {
return convolution_detail::linear_structured_convolution(first, second, std::less<>());
}
template <class T>
std::vector<T> min_plus_convolution_convex_convex(
const std::vector<T>& first, const std::vector<T>& second,
const T& infinity) {
return convolution_detail::linear_structured_convolution_with_infinity(
first, second, infinity, std::less<>());
}
template <class T>
std::vector<T> max_plus_convolution_concave(const std::vector<T>& arbitrary,
const std::vector<T>& concave) {
auto add = [](const T& first, const T& second) { return first + second; };
return convolution_detail::structured_convolution(arbitrary, concave,
std::greater<>(), add);
}
template <class T>
std::vector<T> max_plus_convolution_concave(const std::vector<T>& arbitrary,
const std::vector<T>& concave,
const T& negative_infinity) {
return convolution_detail::structured_convolution_with_infinity(
arbitrary, concave, negative_infinity, std::greater<>());
}
template <class T>
std::vector<T> max_plus_convolution_concave_concave(const std::vector<T>& first,
const std::vector<T>& second) {
return convolution_detail::linear_structured_convolution(first, second, std::greater<>());
}
template <class T>
std::vector<T> max_plus_convolution_concave_concave(
const std::vector<T>& first, const std::vector<T>& second,
const T& negative_infinity) {
return convolution_detail::linear_structured_convolution_with_infinity(
first, second, negative_infinity, std::greater<>());
}
} // namespace convex
} // namespace m1une
#line 11 "convex/monge/all.hpp"
#line 11 "verify/convex/monge/monge_algorithms.test.cpp"
template <class Value, class Compare>
std::vector<int> brute_row_optima(int rows, int columns, Value value, Compare compare) {
std::vector<int> result(rows, -1);
if (columns == 0) return result;
for (int row = 0; row < rows; row++) {
result[row] = 0;
for (int column = 1; column < columns; column++) {
if (compare(value(row, column), value(row, result[row]))) {
result[row] = column;
}
}
}
return result;
}
void test_smawk_monge() {
for (int rows = 0; rows <= 40; rows++) {
for (int columns = 0; columns <= 40; columns++) {
auto value = [&](int row, int column) {
long long difference = row * 3LL - column * 2LL;
return difference * difference + row * 7LL + column * 5LL;
};
auto expected = brute_row_optima(rows, columns, value, std::less<>());
assert(m1une::convex::smawk_row_argmin(rows, columns, value) == expected);
}
}
}
void test_smawk_totally_monotone_and_ties() {
for (int rows = 1; rows <= 50; rows++) {
for (int columns = 1; columns <= 50; columns++) {
std::vector<int> threshold(rows);
for (int row = 0; row < rows; row++) {
threshold[row] = std::min(columns - 1, (row * 7 + rows) / 5);
}
auto value = [&](int row, int column) {
int difference = column - threshold[row];
return difference * difference;
};
auto result = m1une::convex::smawk_row_argmin(rows, columns, value);
assert(result == threshold);
}
}
auto constant = [](int, int) { return 0; };
auto result = m1une::convex::smawk_row_argmin(20, 30, constant);
assert(result == std::vector<int>(20, 0));
}
void test_smawk_max() {
auto value = [](int row, int column) {
long long difference = row - column;
return -difference * difference;
};
auto expected = brute_row_optima(30, 25, value, std::greater<>());
assert(m1une::convex::smawk_row_argmax(30, 25, value) == expected);
}
void test_smawk_matrix_overload_and_evaluations() {
std::vector<std::vector<long long>> matrix(17, std::vector<long long>(23));
for (int row = 0; row < 17; row++) {
for (int column = 0; column < 23; column++) {
long long difference = row * 2LL - column;
matrix[row][column] = difference * difference;
}
}
auto expected = brute_row_optima(
17, 23, [&](int row, int column) { return matrix[row][column]; }, std::less<>());
assert(m1une::convex::smawk_row_argmin(matrix) == expected);
for (auto sizes : std::vector<std::pair<int, int>>{
std::pair<int, int>{10, 1000},
std::pair<int, int>{1000, 10},
std::pair<int, int>{700, 900},
}) {
int evaluations = 0;
auto value = [&](int row, int column) {
evaluations++;
long long difference = row * 3LL - column * 2LL;
return difference * difference;
};
auto result = m1une::convex::smawk_row_argmin(sizes.first, sizes.second, value);
assert(int(result.size()) == sizes.first);
assert(evaluations <= 20 * (sizes.first + sizes.second));
}
}
void test_monotone_minima() {
for (int rows = 1; rows <= 40; rows++) {
for (int columns = 1; columns <= 40; columns++) {
std::vector<int> optimum(rows);
for (int row = 0; row < rows; row++) {
optimum[row] = std::min(columns - 1, row * columns / rows);
}
auto value = [&](int row, int column) {
if (column == optimum[row]) return 0;
return 1000 + (row * 97 + column * 53) % 101;
};
auto result = m1une::convex::monotone_row_argmin(rows, columns, value);
assert(result == optimum);
}
}
auto empty = m1une::convex::monotone_row_argmin(5, 0, [](int, int) { return 0; });
assert(empty == std::vector<int>(5, -1));
std::vector<std::vector<int>> matrix;
matrix.emplace_back(std::vector<int>{0, 5, 7});
matrix.emplace_back(std::vector<int>{4, 0, 8});
matrix.emplace_back(std::vector<int>{9, 3, 0});
assert(m1une::convex::monotone_row_argmin(matrix) == std::vector<int>({0, 1, 2}));
}
void test_monge_checks() {
std::vector<std::vector<long long>> matrix(8, std::vector<long long>(11));
for (int row = 0; row < 8; row++) {
for (int column = 0; column < 11; column++) {
long long difference = row - column;
matrix[row][column] = difference * difference;
}
}
assert(m1une::convex::is_monge(matrix));
assert(!m1une::convex::is_anti_monge(matrix));
for (auto& row : matrix) {
for (auto& value : row) value = -value;
}
assert(!m1une::convex::is_monge(matrix));
assert(m1une::convex::is_anti_monge(matrix));
std::vector<std::vector<int>> empty;
assert(m1une::convex::is_monge(empty));
assert(m1une::convex::is_anti_monge(empty));
}
template <class T, class Compare>
std::vector<T> brute_convolution(const std::vector<T>& first, const std::vector<T>& second,
Compare compare) {
if (first.empty() || second.empty()) return {};
std::vector<T> result(first.size() + second.size() - 1);
for (int index = 0; index < int(result.size()); index++) {
int left = std::max(0, index - int(second.size()) + 1);
int right = std::min(int(first.size()), index + 1);
result[index] = first[left] + second[index - left];
for (int i = left + 1; i < right; i++) {
T value = first[i] + second[index - i];
if (compare(value, result[index])) result[index] = value;
}
}
return result;
}
template <class T, class Compare>
std::vector<T> brute_convolution_with_infinity(const std::vector<T>& first,
const std::vector<T>& second,
Compare compare,
const T& infinity) {
if (first.empty() || second.empty()) return {};
std::vector<T> result(first.size() + second.size() - 1, infinity);
for (int i = 0; i < int(first.size()); i++) {
for (int j = 0; j < int(second.size()); j++) {
if (first[i] == infinity || second[j] == infinity) continue;
T value = first[i] + second[j];
if (result[i + j] == infinity || compare(value, result[i + j])) {
result[i + j] = value;
}
}
}
return result;
}
void test_structured_convolutions() {
const long long infinity = 2'000'000'000'000'000'000LL;
const long long negative_infinity = -infinity;
for (int first_size = 0; first_size <= 35; first_size++) {
for (int second_size = 0; second_size <= 35; second_size++) {
for (int test = 0; test < 8; test++) {
std::vector<long long> arbitrary(first_size);
for (int i = 0; i < first_size; i++) {
arbitrary[i] = (test * 17 + i * 31 + first_size * 7) % 61 - 30;
}
std::vector<long long> convex(second_size);
long long difference = -10 + test;
for (int i = 1; i < second_size; i++) {
difference += (test * 3 + i * 5) % 4;
convex[i] = convex[i - 1] + difference;
}
assert(m1une::convex::is_convex_sequence(convex));
auto expected_min = brute_convolution(arbitrary, convex, std::less<>());
assert(m1une::convex::min_plus_convolution_convex(arbitrary, convex) ==
expected_min);
std::vector<long long> arbitrary_extended = arbitrary;
for (int i = 0; i < first_size; i++) {
if ((i + test) % 7 == 0) arbitrary_extended[i] = infinity;
}
std::vector<long long> convex_extended(second_size + 2, infinity);
std::copy(convex.begin(), convex.end(), convex_extended.begin() + 1);
assert(m1une::convex::is_convex_sequence(convex_extended, infinity));
auto expected_extended_min = brute_convolution_with_infinity(
arbitrary_extended, convex_extended, std::less<>(), infinity);
assert(m1une::convex::min_plus_convolution_convex(
arbitrary_extended, convex_extended, infinity) ==
expected_extended_min);
std::vector<long long> first_convex(first_size);
difference = -12 - test;
for (int i = 1; i < first_size; i++) {
difference += (test * 5 + i * 7) % 5;
first_convex[i] = first_convex[i - 1] + difference;
}
assert(m1une::convex::is_convex_sequence(first_convex));
auto expected_convex =
brute_convolution(first_convex, convex, std::less<>());
assert(m1une::convex::min_plus_convolution_convex(first_convex, convex) ==
expected_convex);
assert(m1une::convex::min_plus_convolution_convex(convex, first_convex) ==
expected_convex);
assert(m1une::convex::min_plus_convolution_convex_convex(first_convex,
convex) ==
expected_convex);
assert(m1une::convex::min_plus_convolution_convex_convex(convex,
first_convex) ==
expected_convex);
std::vector<long long> first_convex_extended(first_size + 2,
infinity);
std::copy(first_convex.begin(), first_convex.end(),
first_convex_extended.begin() + 1);
auto expected_extended_convex = brute_convolution_with_infinity(
first_convex_extended, convex_extended, std::less<>(), infinity);
assert(m1une::convex::min_plus_convolution_convex_convex(
first_convex_extended, convex_extended, infinity) ==
expected_extended_convex);
std::vector<long long> concave = convex;
for (auto& value : concave) value = -value;
assert(m1une::convex::is_concave_sequence(concave));
auto expected_max = brute_convolution(arbitrary, concave, std::greater<>());
assert(m1une::convex::max_plus_convolution_concave(arbitrary, concave) ==
expected_max);
std::vector<long long> first_concave = first_convex;
for (auto& value : first_concave) value = -value;
assert(m1une::convex::is_concave_sequence(first_concave));
auto expected_concave =
brute_convolution(first_concave, concave, std::greater<>());
assert(m1une::convex::max_plus_convolution_concave(first_concave, concave) ==
expected_concave);
assert(m1une::convex::max_plus_convolution_concave(concave, first_concave) ==
expected_concave);
assert(m1une::convex::max_plus_convolution_concave_concave(first_concave,
concave) ==
expected_concave);
assert(m1une::convex::max_plus_convolution_concave_concave(concave,
first_concave) ==
expected_concave);
std::vector<long long> arbitrary_max_extended = arbitrary_extended;
for (long long& value : arbitrary_max_extended) {
value = value == infinity ? negative_infinity : -value;
}
std::vector<long long> concave_extended = convex_extended;
for (long long& value : concave_extended) {
value = value == infinity ? negative_infinity : -value;
}
auto expected_extended_max = brute_convolution_with_infinity(
arbitrary_max_extended, concave_extended, std::greater<>(),
negative_infinity);
assert(m1une::convex::max_plus_convolution_concave(
arbitrary_max_extended, concave_extended,
negative_infinity) == expected_extended_max);
std::vector<long long> first_concave_extended =
first_convex_extended;
for (long long& value : first_concave_extended) {
value = value == infinity ? negative_infinity : -value;
}
auto expected_extended_concave = brute_convolution_with_infinity(
first_concave_extended, concave_extended, std::greater<>(),
negative_infinity);
assert(m1une::convex::max_plus_convolution_concave_concave(
first_concave_extended, concave_extended,
negative_infinity) == expected_extended_concave);
}
}
}
assert(!m1une::convex::is_convex_sequence(std::vector<int>{0, 2, 1}));
assert(!m1une::convex::is_concave_sequence(std::vector<int>{0, -2, -1}));
assert(!m1une::convex::is_convex_sequence(
std::vector<long long>{1, infinity, infinity}));
assert(m1une::convex::is_convex_sequence(
std::vector<long long>{1, infinity, infinity}, infinity));
assert(m1une::convex::is_convex_sequence(
std::vector<long long>{infinity, 0, 1, 4, infinity}, infinity));
assert(!m1une::convex::is_convex_sequence(
std::vector<long long>{0, infinity, 1}, infinity));
std::vector<long long> arbitrary = {infinity, 3, infinity, -2};
std::vector<long long> convex = {infinity, 0, 1, 4, infinity, infinity};
auto expected_min =
brute_convolution_with_infinity(arbitrary, convex, std::less<>(), infinity);
assert(m1une::convex::min_plus_convolution_convex(arbitrary, convex, infinity) ==
expected_min);
std::vector<long long> first_convex = {infinity, 2, 2, 3, infinity};
expected_min = brute_convolution_with_infinity(first_convex, convex,
std::less<>(), infinity);
assert(m1une::convex::min_plus_convolution_convex_convex(
first_convex, convex, infinity) == expected_min);
long long unordered_infinity = 7;
std::vector<long long> large_arbitrary = {10, 20};
std::vector<long long> small_convex = {0, 1};
assert(m1une::convex::min_plus_convolution_convex(
large_arbitrary, small_convex, unordered_infinity) ==
std::vector<long long>({10, 11, 21}));
std::vector<long long> arbitrary_max = {negative_infinity, 3,
negative_infinity, -2};
std::vector<long long> concave = {negative_infinity, 0, -1, -4,
negative_infinity};
assert(m1une::convex::is_concave_sequence(concave, negative_infinity));
auto expected_max = brute_convolution_with_infinity(
arbitrary_max, concave, std::greater<>(), negative_infinity);
assert(m1une::convex::max_plus_convolution_concave(
arbitrary_max, concave, negative_infinity) == expected_max);
std::vector<long long> first_concave = {negative_infinity, 2, 2, 1,
negative_infinity};
expected_max = brute_convolution_with_infinity(
first_concave, concave, std::greater<>(), negative_infinity);
assert(m1une::convex::max_plus_convolution_concave_concave(
first_concave, concave, negative_infinity) == expected_max);
std::vector<long long> all_infinity(3, infinity);
assert(m1une::convex::is_convex_sequence(all_infinity, infinity));
assert(m1une::convex::min_plus_convolution_convex_convex(
all_infinity, convex, infinity) ==
std::vector<long long>(all_infinity.size() + convex.size() - 1,
infinity));
}
int main() {
m1une::utilities::FastInput fast_input;
m1une::utilities::FastOutput fast_output;
test_smawk_monge();
test_smawk_totally_monotone_and_ties();
test_smawk_max();
test_smawk_matrix_overload_and_evaluations();
test_monotone_minima();
test_monge_checks();
test_structured_convolutions();
long long a, b;
fast_input >> a >> b;
fast_output << a + b << '\n';
}
verify/convex/monge/monge_algorithms.test.cpp