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:heavy_check_mark: verify/convex/slope_trick.test.cpp

Depends on

Code

#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"

#include <algorithm>
#include <cassert>
#include "../../utilities/fast_io.hpp"
#include <limits>
#include <vector>

#include "../../convex/slope_trick.hpp"

using SlopeTrick = m1une::convex::SlopeTrick<long long>;

constexpr int coordinate_limit = 300;
constexpr long long inf = std::numeric_limits<long long>::max() / 4;

int index_of(int x) {
    return x + coordinate_limit;
}

void check_values(const SlopeTrick& slope, const std::vector<long long>& value) {
    long long expected_minimum = *std::min_element(value.begin(), value.end());
    assert(slope.minimum() == expected_minimum);
    for (int x = -100; x <= 100; x++) {
        assert(slope.evaluate(x) == value[index_of(x)]);
    }

    auto range = slope.argmin();
    int first = -coordinate_limit;
    while (first <= coordinate_limit && value[index_of(first)] != expected_minimum) first++;
    int last = coordinate_limit;
    while (last >= -coordinate_limit && value[index_of(last)] != expected_minimum) last--;
    if (range.left.has_value()) assert(*range.left == first);
    if (range.right.has_value()) assert(*range.right == last);
}

void test_basic() {
    SlopeTrick slope;
    assert(slope.minimum() == 0);
    assert(!slope.argmin().left.has_value());
    assert(!slope.argmin().right.has_value());

    slope.add_abs(3);
    slope.add_x_minus_a(-2);
    slope.add_a_minus_x(7);
    slope.add_constant(5);
    assert(slope.minimum() == 14);
    assert(slope.evaluate(3) == 14);
    auto range = slope.argmin();
    assert(range.left == std::optional<long long>(3));
    assert(range.right == std::optional<long long>(3));
    assert(slope.breakpoint_count() == 4);

    SlopeTrick right_hinge;
    right_hinge.add_x_minus_a(5);
    assert(!right_hinge.argmin().left.has_value());
    assert(right_hinge.argmin().right == std::optional<long long>(5));

    SlopeTrick left_hinge;
    left_hinge.add_a_minus_x(-4);
    assert(left_hinge.argmin().left == std::optional<long long>(-4));
    assert(!left_hinge.argmin().right.has_value());
}

void test_operations_against_explicit_function() {
    for (int test = 0; test < 80; test++) {
        SlopeTrick slope;
        std::vector<long long> value(coordinate_limit * 2 + 1, 0);

        for (int operation = 0; operation < 80; operation++) {
            int type = (test * 11 + operation * 7) % 8;
            int a = (test * 17 + operation * 13) % 61 - 30;

            if (type == 0) {
                long long constant = (test + operation * 3) % 17 - 8;
                slope.add_constant(constant);
                for (auto& current : value) current += constant;
            } else if (type == 1) {
                slope.add_x_minus_a(a);
                for (int x = -coordinate_limit; x <= coordinate_limit; x++) {
                    value[index_of(x)] += std::max(0, x - a);
                }
            } else if (type == 2) {
                slope.add_a_minus_x(a);
                for (int x = -coordinate_limit; x <= coordinate_limit; x++) {
                    value[index_of(x)] += std::max(0, a - x);
                }
            } else if (type == 3) {
                slope.add_abs(a);
                for (int x = -coordinate_limit; x <= coordinate_limit; x++) {
                    value[index_of(x)] += std::abs(x - a);
                }
            } else if (type == 4) {
                int delta = (test + operation) % 7 - 3;
                slope.shift(delta);
                std::vector<long long> next(value.size(), inf);
                for (int x = -coordinate_limit; x <= coordinate_limit; x++) {
                    int source = x - delta;
                    if (-coordinate_limit <= source && source <= coordinate_limit) {
                        next[index_of(x)] = value[index_of(source)];
                    }
                }
                value.swap(next);
            } else if (type == 5) {
                int left = (test + operation * 2) % 4 - 3;
                int right = left + 1 + (test * 3 + operation) % 4;
                slope.shift(left, right);
                std::vector<long long> next(value.size(), inf);
                for (int x = -coordinate_limit; x <= coordinate_limit; x++) {
                    for (int y = x - right; y <= x - left; y++) {
                        if (-coordinate_limit <= y && y <= coordinate_limit) {
                            next[index_of(x)] = std::min(next[index_of(x)], value[index_of(y)]);
                        }
                    }
                }
                value.swap(next);
            } else if (type == 6) {
                slope.prefix_minimum();
                long long best = inf;
                for (int x = -coordinate_limit; x <= coordinate_limit; x++) {
                    best = std::min(best, value[index_of(x)]);
                    value[index_of(x)] = best;
                }
            } else {
                slope.suffix_minimum();
                long long best = inf;
                for (int x = coordinate_limit; x >= -coordinate_limit; x--) {
                    best = std::min(best, value[index_of(x)]);
                    value[index_of(x)] = best;
                }
            }
            check_values(slope, value);
        }
    }
}

void test_merge() {
    for (int test = 0; test < 100; test++) {
        SlopeTrick first;
        SlopeTrick second;
        for (int i = 0; i < 20; i++) {
            int a = (test * 7 + i * 11) % 41 - 20;
            int b = (test * 13 + i * 5) % 41 - 20;
            if (i % 3 == 0) {
                first.add_abs(a);
                second.add_abs(b);
            } else if (i % 3 == 1) {
                first.add_x_minus_a(a);
                second.add_a_minus_x(b);
            } else {
                first.add_a_minus_x(a);
                second.add_x_minus_a(b);
            }
        }
        first.add_constant(test - 50);
        second.add_constant(30 - test);
        first.shift(-2, 1);
        second.shift(3);
        if (test % 2 == 0) first.clear_right();
        if (test % 3 == 0) second.clear_left();

        std::vector<long long> expected(201);
        for (int x = -100; x <= 100; x++) {
            expected[x + 100] = first.evaluate(x) + second.evaluate(x);
        }
        first.merge(second);
        for (int x = -100; x <= 100; x++) {
            assert(first.evaluate(x) == expected[x + 100]);
        }
    }
}

void test_min_plus_convolution() {
    constexpr int brute_limit = 500;

    for (int test = 0; test < 120; test++) {
        SlopeTrick first;
        SlopeTrick second;

        for (int i = 0; i < 18; i++) {
            int a = (test * 17 + i * 11) % 51 - 25;
            int b = (test * 23 + i * 7) % 51 - 25;

            if (i % 4 == 0) {
                first.add_abs(a);
                second.add_abs(b);
            } else if (i % 4 == 1) {
                first.add_x_minus_a(a);
                second.add_a_minus_x(b);
            } else if (i % 4 == 2) {
                first.add_a_minus_x(a);
                second.add_x_minus_a(b);
            } else {
                first.add_constant(a - b);
                second.add_constant(b - a / 2);
            }

            if (i % 7 == 3) first.shift(-2, 3);
            if (i % 7 == 5) second.shift(1);
        }

        if (test % 4 == 0) first.prefix_minimum();
        if (test % 5 == 0) first.suffix_minimum();
        if (test % 6 == 0) second.prefix_minimum();
        if (test % 7 == 0) second.suffix_minimum();

        SlopeTrick assigned = first;
        assigned.min_plus_convolve(second);
        SlopeTrick returned = m1une::convex::min_plus_convolution(first, second);

        assert(assigned.minimum() == first.minimum() + second.minimum());
        auto first_range = first.argmin();
        auto second_range = second.argmin();
        auto result_range = assigned.argmin();
        if (first_range.left && second_range.left) {
            assert(result_range.left);
            assert(*result_range.left == *first_range.left + *second_range.left);
        } else {
            assert(!result_range.left);
        }
        if (first_range.right && second_range.right) {
            assert(result_range.right);
            assert(*result_range.right == *first_range.right + *second_range.right);
        } else {
            assert(!result_range.right);
        }

        for (int x = -100; x <= 100; x++) {
            long long expected = inf;
            for (int y = -brute_limit; y <= brute_limit; y++) {
                expected =
                    std::min(expected, first.evaluate(y) + second.evaluate(x - y));
            }
            assert(assigned.evaluate(x) == expected);
            assert(returned.evaluate(x) == expected);
        }
    }
}

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    test_basic();
    test_operations_against_explicit_function();
    test_merge();
    test_min_plus_convolution();

    long long a, b;
    fast_input >> a >> b;
    fast_output << a + b << '\n';
}
#line 1 "verify/convex/slope_trick.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"

#include <algorithm>
#include <cassert>
#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 6 "verify/convex/slope_trick.test.cpp"
#include <limits>
#line 8 "verify/convex/slope_trick.test.cpp"

#line 1 "convex/slope_trick.hpp"



#line 5 "convex/slope_trick.hpp"
#include <functional>
#include <optional>
#include <queue>
#line 11 "convex/slope_trick.hpp"

namespace m1une {
namespace convex {

template <class T>
struct SlopeTrickArgmin {
    std::optional<T> left;
    std::optional<T> right;
};

template <class T>
class SlopeTrick {
    static_assert(std::is_arithmetic_v<T> && std::is_signed_v<T>);

    T _minimum = T();
    T _left_offset = T();
    T _right_offset = T();
    std::priority_queue<T> _left;
    std::priority_queue<T, std::vector<T>, std::greater<T>> _right;

    T left_top() const {
        return _left.top() + _left_offset;
    }

    T right_top() const {
        return _right.top() + _right_offset;
    }

    void push_left(T value) {
        _left.push(value - _left_offset);
    }

    void push_right(T value) {
        _right.push(value - _right_offset);
    }

   public:
    SlopeTrick() = default;

    T minimum() const {
        return _minimum;
    }

    int breakpoint_count() const {
        return int(_left.size() + _right.size());
    }

    SlopeTrickArgmin<T> argmin() const {
        SlopeTrickArgmin<T> result;
        if (!_left.empty()) result.left = left_top();
        if (!_right.empty()) result.right = right_top();
        return result;
    }

    void add_constant(T value) {
        _minimum += value;
    }

    void add_x_minus_a(T a) {
        if (!_left.empty() && left_top() > a) {
            T old = left_top();
            _minimum += old - a;
            _left.pop();
            push_left(a);
            push_right(old);
        } else {
            push_right(a);
        }
    }

    void add_a_minus_x(T a) {
        if (!_right.empty() && right_top() < a) {
            T old = right_top();
            _minimum += a - old;
            _right.pop();
            push_right(a);
            push_left(old);
        } else {
            push_left(a);
        }
    }

    void add_abs(T a) {
        add_a_minus_x(a);
        add_x_minus_a(a);
    }

    void clear_left() {
        _left = std::priority_queue<T>();
    }

    void clear_right() {
        _right = std::priority_queue<T, std::vector<T>, std::greater<T>>();
    }

    void prefix_minimum() {
        clear_right();
    }

    void suffix_minimum() {
        clear_left();
    }

    void shift(T delta) {
        _left_offset += delta;
        _right_offset += delta;
    }

    void shift(T left_delta, T right_delta) {
        assert(left_delta <= right_delta);
        _left_offset += left_delta;
        _right_offset += right_delta;
    }

    T evaluate(T x) const {
        T result = _minimum;
        auto left = _left;
        while (!left.empty()) {
            T breakpoint = left.top() + _left_offset;
            if (breakpoint > x) result += breakpoint - x;
            left.pop();
        }

        auto right = _right;
        while (!right.empty()) {
            T breakpoint = right.top() + _right_offset;
            if (x > breakpoint) result += x - breakpoint;
            right.pop();
        }
        return result;
    }

    void merge(SlopeTrick other) {
        add_constant(other._minimum);
        while (!other._left.empty()) {
            add_a_minus_x(other.left_top());
            other._left.pop();
        }
        while (!other._right.empty()) {
            add_x_minus_a(other.right_top());
            other._right.pop();
        }
    }

    void min_plus_convolve(SlopeTrick other) {
        SlopeTrick result;
        result._minimum = _minimum + other._minimum;

        while (!_left.empty() && !other._left.empty()) {
            result.push_left(left_top() + other.left_top());
            _left.pop();
            other._left.pop();
        }
        while (!_right.empty() && !other._right.empty()) {
            result.push_right(right_top() + other.right_top());
            _right.pop();
            other._right.pop();
        }
        *this = std::move(result);
    }
};

template <class T>
SlopeTrick<T> min_plus_convolution(SlopeTrick<T> first,
                                   SlopeTrick<T> second) {
    first.min_plus_convolve(std::move(second));
    return first;
}

}  // namespace convex
}  // namespace m1une


#line 10 "verify/convex/slope_trick.test.cpp"

using SlopeTrick = m1une::convex::SlopeTrick<long long>;

constexpr int coordinate_limit = 300;
constexpr long long inf = std::numeric_limits<long long>::max() / 4;

int index_of(int x) {
    return x + coordinate_limit;
}

void check_values(const SlopeTrick& slope, const std::vector<long long>& value) {
    long long expected_minimum = *std::min_element(value.begin(), value.end());
    assert(slope.minimum() == expected_minimum);
    for (int x = -100; x <= 100; x++) {
        assert(slope.evaluate(x) == value[index_of(x)]);
    }

    auto range = slope.argmin();
    int first = -coordinate_limit;
    while (first <= coordinate_limit && value[index_of(first)] != expected_minimum) first++;
    int last = coordinate_limit;
    while (last >= -coordinate_limit && value[index_of(last)] != expected_minimum) last--;
    if (range.left.has_value()) assert(*range.left == first);
    if (range.right.has_value()) assert(*range.right == last);
}

void test_basic() {
    SlopeTrick slope;
    assert(slope.minimum() == 0);
    assert(!slope.argmin().left.has_value());
    assert(!slope.argmin().right.has_value());

    slope.add_abs(3);
    slope.add_x_minus_a(-2);
    slope.add_a_minus_x(7);
    slope.add_constant(5);
    assert(slope.minimum() == 14);
    assert(slope.evaluate(3) == 14);
    auto range = slope.argmin();
    assert(range.left == std::optional<long long>(3));
    assert(range.right == std::optional<long long>(3));
    assert(slope.breakpoint_count() == 4);

    SlopeTrick right_hinge;
    right_hinge.add_x_minus_a(5);
    assert(!right_hinge.argmin().left.has_value());
    assert(right_hinge.argmin().right == std::optional<long long>(5));

    SlopeTrick left_hinge;
    left_hinge.add_a_minus_x(-4);
    assert(left_hinge.argmin().left == std::optional<long long>(-4));
    assert(!left_hinge.argmin().right.has_value());
}

void test_operations_against_explicit_function() {
    for (int test = 0; test < 80; test++) {
        SlopeTrick slope;
        std::vector<long long> value(coordinate_limit * 2 + 1, 0);

        for (int operation = 0; operation < 80; operation++) {
            int type = (test * 11 + operation * 7) % 8;
            int a = (test * 17 + operation * 13) % 61 - 30;

            if (type == 0) {
                long long constant = (test + operation * 3) % 17 - 8;
                slope.add_constant(constant);
                for (auto& current : value) current += constant;
            } else if (type == 1) {
                slope.add_x_minus_a(a);
                for (int x = -coordinate_limit; x <= coordinate_limit; x++) {
                    value[index_of(x)] += std::max(0, x - a);
                }
            } else if (type == 2) {
                slope.add_a_minus_x(a);
                for (int x = -coordinate_limit; x <= coordinate_limit; x++) {
                    value[index_of(x)] += std::max(0, a - x);
                }
            } else if (type == 3) {
                slope.add_abs(a);
                for (int x = -coordinate_limit; x <= coordinate_limit; x++) {
                    value[index_of(x)] += std::abs(x - a);
                }
            } else if (type == 4) {
                int delta = (test + operation) % 7 - 3;
                slope.shift(delta);
                std::vector<long long> next(value.size(), inf);
                for (int x = -coordinate_limit; x <= coordinate_limit; x++) {
                    int source = x - delta;
                    if (-coordinate_limit <= source && source <= coordinate_limit) {
                        next[index_of(x)] = value[index_of(source)];
                    }
                }
                value.swap(next);
            } else if (type == 5) {
                int left = (test + operation * 2) % 4 - 3;
                int right = left + 1 + (test * 3 + operation) % 4;
                slope.shift(left, right);
                std::vector<long long> next(value.size(), inf);
                for (int x = -coordinate_limit; x <= coordinate_limit; x++) {
                    for (int y = x - right; y <= x - left; y++) {
                        if (-coordinate_limit <= y && y <= coordinate_limit) {
                            next[index_of(x)] = std::min(next[index_of(x)], value[index_of(y)]);
                        }
                    }
                }
                value.swap(next);
            } else if (type == 6) {
                slope.prefix_minimum();
                long long best = inf;
                for (int x = -coordinate_limit; x <= coordinate_limit; x++) {
                    best = std::min(best, value[index_of(x)]);
                    value[index_of(x)] = best;
                }
            } else {
                slope.suffix_minimum();
                long long best = inf;
                for (int x = coordinate_limit; x >= -coordinate_limit; x--) {
                    best = std::min(best, value[index_of(x)]);
                    value[index_of(x)] = best;
                }
            }
            check_values(slope, value);
        }
    }
}

void test_merge() {
    for (int test = 0; test < 100; test++) {
        SlopeTrick first;
        SlopeTrick second;
        for (int i = 0; i < 20; i++) {
            int a = (test * 7 + i * 11) % 41 - 20;
            int b = (test * 13 + i * 5) % 41 - 20;
            if (i % 3 == 0) {
                first.add_abs(a);
                second.add_abs(b);
            } else if (i % 3 == 1) {
                first.add_x_minus_a(a);
                second.add_a_minus_x(b);
            } else {
                first.add_a_minus_x(a);
                second.add_x_minus_a(b);
            }
        }
        first.add_constant(test - 50);
        second.add_constant(30 - test);
        first.shift(-2, 1);
        second.shift(3);
        if (test % 2 == 0) first.clear_right();
        if (test % 3 == 0) second.clear_left();

        std::vector<long long> expected(201);
        for (int x = -100; x <= 100; x++) {
            expected[x + 100] = first.evaluate(x) + second.evaluate(x);
        }
        first.merge(second);
        for (int x = -100; x <= 100; x++) {
            assert(first.evaluate(x) == expected[x + 100]);
        }
    }
}

void test_min_plus_convolution() {
    constexpr int brute_limit = 500;

    for (int test = 0; test < 120; test++) {
        SlopeTrick first;
        SlopeTrick second;

        for (int i = 0; i < 18; i++) {
            int a = (test * 17 + i * 11) % 51 - 25;
            int b = (test * 23 + i * 7) % 51 - 25;

            if (i % 4 == 0) {
                first.add_abs(a);
                second.add_abs(b);
            } else if (i % 4 == 1) {
                first.add_x_minus_a(a);
                second.add_a_minus_x(b);
            } else if (i % 4 == 2) {
                first.add_a_minus_x(a);
                second.add_x_minus_a(b);
            } else {
                first.add_constant(a - b);
                second.add_constant(b - a / 2);
            }

            if (i % 7 == 3) first.shift(-2, 3);
            if (i % 7 == 5) second.shift(1);
        }

        if (test % 4 == 0) first.prefix_minimum();
        if (test % 5 == 0) first.suffix_minimum();
        if (test % 6 == 0) second.prefix_minimum();
        if (test % 7 == 0) second.suffix_minimum();

        SlopeTrick assigned = first;
        assigned.min_plus_convolve(second);
        SlopeTrick returned = m1une::convex::min_plus_convolution(first, second);

        assert(assigned.minimum() == first.minimum() + second.minimum());
        auto first_range = first.argmin();
        auto second_range = second.argmin();
        auto result_range = assigned.argmin();
        if (first_range.left && second_range.left) {
            assert(result_range.left);
            assert(*result_range.left == *first_range.left + *second_range.left);
        } else {
            assert(!result_range.left);
        }
        if (first_range.right && second_range.right) {
            assert(result_range.right);
            assert(*result_range.right == *first_range.right + *second_range.right);
        } else {
            assert(!result_range.right);
        }

        for (int x = -100; x <= 100; x++) {
            long long expected = inf;
            for (int y = -brute_limit; y <= brute_limit; y++) {
                expected =
                    std::min(expected, first.evaluate(y) + second.evaluate(x - y));
            }
            assert(assigned.evaluate(x) == expected);
            assert(returned.evaluate(x) == expected);
        }
    }
}

int main() {
    m1une::utilities::FastInput fast_input;
    m1une::utilities::FastOutput fast_output;

    test_basic();
    test_operations_against_explicit_function();
    test_merge();
    test_min_plus_convolution();

    long long a, b;
    fast_input >> a >> b;
    fast_output << a + b << '\n';
}
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