m1une's library

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:heavy_check_mark: verify/algo/dp/knapsack.test.cpp

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Code

#define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ALDS1_5_A"

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

#include "../../../algo/dp/knapsack.hpp"

std::vector<char> naive_subset_sum(const std::vector<int>& weights, int limit) {
    std::vector<char> reachable(limit + 1, 0);
    reachable[0] = 1;
    for (int weight : weights) {
        for (int sum = limit; weight <= sum; --sum) {
            if (reachable[sum - weight]) reachable[sum] = 1;
        }
    }
    return reachable;
}

int naive_subset_sum_max_value(const std::vector<int>& weights, int limit) {
    const std::vector<char> reachable = naive_subset_sum(weights, limit);
    for (int sum = limit; 0 <= sum; --sum) {
        if (reachable[sum]) return sum;
    }
    return 0;
}

std::vector<long long> naive_zero_one_max_value(
    const std::vector<int>& weights,
    const std::vector<long long>& values,
    int capacity
) {
    std::vector<long long> result(capacity + 1, 0);
    const int n = int(weights.size());
    for (int mask = 0; mask < (1 << n); ++mask) {
        int weight = 0;
        long long value = 0;
        for (int i = 0; i < n; ++i) {
            if ((mask >> i) & 1) {
                weight += weights[i];
                value += values[i];
            }
        }
        if (weight <= capacity) {
            for (int current = weight; current <= capacity; ++current) {
                result[current] = std::max(result[current], value);
            }
        }
    }
    return result;
}

std::vector<long long> naive_bounded_max_value(
    const std::vector<int>& weights,
    const std::vector<long long>& values,
    const std::vector<int>& counts,
    int capacity
) {
    std::vector<long long> result(capacity + 1, 0);
    const int n = int(weights.size());

    auto dfs = [&](auto& self, int item, int weight, long long value) -> void {
        if (item == n) {
            if (weight <= capacity) {
                for (int current = weight; current <= capacity; ++current) {
                    result[current] = std::max(result[current], value);
                }
            }
            return;
        }
        for (int take = 0; take <= counts[item]; ++take) {
            self(self, item + 1, weight + weights[item] * take, value + values[item] * take);
        }
    };
    dfs(dfs, 0, 0, 0);
    return result;
}

std::vector<long long> naive_min_weight_for_value(
    const std::vector<long long>& weights,
    const std::vector<int>& values,
    int value_limit
) {
    const long long inf = std::numeric_limits<long long>::max() / 4;
    std::vector<long long> result(value_limit + 1, inf);
    const int n = int(weights.size());
    for (int mask = 0; mask < (1 << n); ++mask) {
        long long weight = 0;
        int value = 0;
        for (int i = 0; i < n; ++i) {
            if ((mask >> i) & 1) {
                weight += weights[i];
                value += values[i];
            }
        }
        if (value <= value_limit) {
            result[value] = std::min(result[value], weight);
        }
    }
    return result;
}

void test_subset_sum() {
    const std::vector<int> limits = {0, 1, 30, 63, 64, 65, 127, 128, 129};
    for (int limit : limits) {
        for (int n = 0; n <= 8; ++n) {
            std::vector<int> weights(n);
            for (int seed = 0; seed < 60; ++seed) {
                for (int i = 0; i < n; ++i) {
                    weights[i] = (seed * 13 + i * 67) % 140;
                }
                const std::vector<char> expected = naive_subset_sum(weights, limit);
                assert(m1une::algo::subset_sum_reachable(weights, limit) == expected);
                assert(
                    m1une::algo::subset_sum_max_value(weights, limit) ==
                    naive_subset_sum_max_value(weights, limit)
                );
            }
        }
    }
}

void test_zero_one_knapsack() {
    std::vector<int> weights = {2, 3, 4, 5};
    std::vector<long long> values = {4, 5, 7, 8};
    assert(
        m1une::algo::zero_one_knapsack_max_value(weights, values, 10)
        == naive_zero_one_max_value(weights, values, 10)
    );
}

void test_bounded_knapsack() {
    std::vector<int> weights = {0, 2, 3};
    std::vector<long long> values = {5, 3, 4};
    std::vector<int> counts = {2, 3, 2};
    assert(
        m1une::algo::bounded_knapsack_max_value(weights, values, counts, 10)
        == naive_bounded_max_value(weights, values, counts, 10)
    );
}

void test_min_weight_for_value() {
    std::vector<long long> weights = {3, 2, 5, 7};
    std::vector<int> values = {4, 3, 6, 8};
    assert(
        m1une::algo::zero_one_knapsack_min_weight_for_value(weights, values, 20)
        == naive_min_weight_for_value(weights, values, 20)
    );
}

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

    test_subset_sum();
    test_zero_one_knapsack();
    test_bounded_knapsack();
    test_min_weight_for_value();

    int n;
    fast_input >> n;
    std::vector<int> weights(n);
    for (int& weight : weights) fast_input >> weight;

    int query_count;
    fast_input >> query_count;
    while (query_count--) {
        int target;
        fast_input >> target;
        const bool reachable =
            m1une::algo::subset_sum_max_value(weights, target) == target;
        fast_output << (reachable ? "yes" : "no") << '\n';
    }
}
#line 1 "verify/algo/dp/knapsack.test.cpp"
#define PROBLEM "https://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=ALDS1_5_A"

#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/algo/dp/knapsack.test.cpp"
#include <limits>
#line 8 "verify/algo/dp/knapsack.test.cpp"

#line 1 "algo/dp/knapsack.hpp"



#line 5 "algo/dp/knapsack.hpp"
#include <bit>
#line 8 "algo/dp/knapsack.hpp"
#include <deque>
#line 11 "algo/dp/knapsack.hpp"

namespace m1une {
namespace algo {

namespace internal {

using SubsetSumWord = unsigned long long;

inline std::vector<SubsetSumWord> subset_sum_reachability_bits(
    const std::vector<int>& weights,
    int limit
) {
    assert(0 <= limit);
    using Word = SubsetSumWord;
    constexpr int word_bits = std::numeric_limits<Word>::digits;

    const std::size_t bit_count = std::size_t(limit) + 1;
    std::vector<Word> bits((bit_count + word_bits - 1) / word_bits, Word(0));
    bits[0] = Word(1);

    auto trim = [&]() {
        const int extra = int(bit_count % word_bits);
        if (extra != 0) {
            bits.back() &= (Word(1) << extra) - Word(1);
        }
    };

    for (int weight : weights) {
        assert(0 <= weight);
        if (weight == 0 || limit < weight) continue;

        const std::size_t word_shift = std::size_t(weight / word_bits);
        const int bit_shift = weight % word_bits;
        for (std::size_t i = bits.size() - word_shift; i-- > 0;) {
            const Word source = bits[i];
            if (source == Word(0)) continue;
            const std::size_t target = i + word_shift;
            bits[target] |= source << bit_shift;
            if (bit_shift != 0 && target + 1 < bits.size()) {
                bits[target + 1] |= source >> (word_bits - bit_shift);
            }
        }
        trim();
    }
    return bits;
}

}  // namespace internal

inline std::vector<char> subset_sum_reachable(const std::vector<int>& weights, int limit) {
    using Word = internal::SubsetSumWord;
    constexpr int word_bits = std::numeric_limits<Word>::digits;
    const std::vector<Word> bits =
        internal::subset_sum_reachability_bits(weights, limit);

    std::vector<char> reachable(std::size_t(limit) + 1, 0);
    for (int sum = 0; sum <= limit; ++sum) {
        reachable[sum] = char((bits[std::size_t(sum / word_bits)] >> (sum % word_bits)) & Word(1));
    }
    return reachable;
}

// Returns the maximum subset sum not exceeding limit.
inline int subset_sum_max_value(const std::vector<int>& weights, int limit) {
    using Word = internal::SubsetSumWord;
    constexpr int word_bits = std::numeric_limits<Word>::digits;
    const std::vector<Word> bits =
        internal::subset_sum_reachability_bits(weights, limit);

    for (std::size_t i = bits.size(); i-- > 0;) {
        if (bits[i] != Word(0)) {
            return int(i * word_bits + std::bit_width(bits[i]) - 1);
        }
    }
    return 0;
}

template <typename Value = long long>
std::vector<Value> zero_one_knapsack_max_value(
    const std::vector<int>& weights,
    const std::vector<Value>& values,
    int capacity,
    Value neg_inf = std::numeric_limits<Value>::lowest() / Value(4)
) {
    assert(weights.size() == values.size());
    assert(0 <= capacity);

    std::vector<Value> dp(std::size_t(capacity) + 1, neg_inf);
    dp[0] = Value{};
    for (std::size_t item = 0; item < weights.size(); ++item) {
        const int weight = weights[item];
        assert(0 <= weight);
        for (int current = capacity; weight <= current; --current) {
            if (dp[current - weight] == neg_inf) continue;
            dp[current] = std::max(dp[current], dp[current - weight] + values[item]);
        }
    }

    for (int current = 1; current <= capacity; ++current) {
        dp[current] = std::max(dp[current], dp[current - 1]);
    }
    return dp;
}

template <typename Value = long long>
std::vector<Value> bounded_knapsack_max_value(
    const std::vector<int>& weights,
    const std::vector<Value>& values,
    const std::vector<int>& counts,
    int capacity,
    Value neg_inf = std::numeric_limits<Value>::lowest() / Value(4)
) {
    assert(weights.size() == values.size());
    assert(weights.size() == counts.size());
    assert(0 <= capacity);

    std::vector<Value> dp(std::size_t(capacity) + 1, neg_inf);
    dp[0] = Value{};

    for (std::size_t item = 0; item < weights.size(); ++item) {
        const int weight = weights[item];
        const Value value = values[item];
        const int count = counts[item];
        assert(0 <= weight);
        assert(0 <= count);
        if (count == 0) continue;

        if (weight == 0) {
            if (Value{} < value) {
                const Value gain = value * Value(count);
                for (Value& current : dp) {
                    if (current != neg_inf) current += gain;
                }
            }
            continue;
        }

        std::vector<Value> next = dp;
        for (int residue = 0; residue < weight && residue <= capacity; ++residue) {
            std::deque<int> indices;
            std::deque<Value> bases;
            int k = 0;
            for (int current = residue; current <= capacity; current += weight, ++k) {
                if (dp[current] != neg_inf) {
                    const Value base = dp[current] - Value(k) * value;
                    while (!bases.empty() && bases.back() <= base) {
                        bases.pop_back();
                        indices.pop_back();
                    }
                    bases.push_back(base);
                    indices.push_back(k);
                }

                while (!indices.empty() && indices.front() < k - count) {
                    indices.pop_front();
                    bases.pop_front();
                }
                if (!bases.empty()) {
                    next[current] = std::max(next[current], bases.front() + Value(k) * value);
                }
            }
        }
        dp.swap(next);
    }

    for (int current = 1; current <= capacity; ++current) {
        dp[current] = std::max(dp[current], dp[current - 1]);
    }
    return dp;
}

template <typename Weight = long long>
std::vector<Weight> zero_one_knapsack_min_weight_for_value(
    const std::vector<Weight>& weights,
    const std::vector<int>& values,
    int value_limit,
    Weight inf = std::numeric_limits<Weight>::max() / Weight(4)
) {
    assert(weights.size() == values.size());
    assert(0 <= value_limit);

    std::vector<Weight> dp(std::size_t(value_limit) + 1, inf);
    dp[0] = Weight{};
    for (std::size_t item = 0; item < weights.size(); ++item) {
        assert(Weight{} <= weights[item]);
        assert(0 <= values[item]);
        for (int value = value_limit; values[item] <= value; --value) {
            if (dp[value - values[item]] == inf) continue;
            dp[value] = std::min(dp[value], dp[value - values[item]] + weights[item]);
        }
    }
    return dp;
}

}  // namespace algo
}  // namespace m1une


#line 10 "verify/algo/dp/knapsack.test.cpp"

std::vector<char> naive_subset_sum(const std::vector<int>& weights, int limit) {
    std::vector<char> reachable(limit + 1, 0);
    reachable[0] = 1;
    for (int weight : weights) {
        for (int sum = limit; weight <= sum; --sum) {
            if (reachable[sum - weight]) reachable[sum] = 1;
        }
    }
    return reachable;
}

int naive_subset_sum_max_value(const std::vector<int>& weights, int limit) {
    const std::vector<char> reachable = naive_subset_sum(weights, limit);
    for (int sum = limit; 0 <= sum; --sum) {
        if (reachable[sum]) return sum;
    }
    return 0;
}

std::vector<long long> naive_zero_one_max_value(
    const std::vector<int>& weights,
    const std::vector<long long>& values,
    int capacity
) {
    std::vector<long long> result(capacity + 1, 0);
    const int n = int(weights.size());
    for (int mask = 0; mask < (1 << n); ++mask) {
        int weight = 0;
        long long value = 0;
        for (int i = 0; i < n; ++i) {
            if ((mask >> i) & 1) {
                weight += weights[i];
                value += values[i];
            }
        }
        if (weight <= capacity) {
            for (int current = weight; current <= capacity; ++current) {
                result[current] = std::max(result[current], value);
            }
        }
    }
    return result;
}

std::vector<long long> naive_bounded_max_value(
    const std::vector<int>& weights,
    const std::vector<long long>& values,
    const std::vector<int>& counts,
    int capacity
) {
    std::vector<long long> result(capacity + 1, 0);
    const int n = int(weights.size());

    auto dfs = [&](auto& self, int item, int weight, long long value) -> void {
        if (item == n) {
            if (weight <= capacity) {
                for (int current = weight; current <= capacity; ++current) {
                    result[current] = std::max(result[current], value);
                }
            }
            return;
        }
        for (int take = 0; take <= counts[item]; ++take) {
            self(self, item + 1, weight + weights[item] * take, value + values[item] * take);
        }
    };
    dfs(dfs, 0, 0, 0);
    return result;
}

std::vector<long long> naive_min_weight_for_value(
    const std::vector<long long>& weights,
    const std::vector<int>& values,
    int value_limit
) {
    const long long inf = std::numeric_limits<long long>::max() / 4;
    std::vector<long long> result(value_limit + 1, inf);
    const int n = int(weights.size());
    for (int mask = 0; mask < (1 << n); ++mask) {
        long long weight = 0;
        int value = 0;
        for (int i = 0; i < n; ++i) {
            if ((mask >> i) & 1) {
                weight += weights[i];
                value += values[i];
            }
        }
        if (value <= value_limit) {
            result[value] = std::min(result[value], weight);
        }
    }
    return result;
}

void test_subset_sum() {
    const std::vector<int> limits = {0, 1, 30, 63, 64, 65, 127, 128, 129};
    for (int limit : limits) {
        for (int n = 0; n <= 8; ++n) {
            std::vector<int> weights(n);
            for (int seed = 0; seed < 60; ++seed) {
                for (int i = 0; i < n; ++i) {
                    weights[i] = (seed * 13 + i * 67) % 140;
                }
                const std::vector<char> expected = naive_subset_sum(weights, limit);
                assert(m1une::algo::subset_sum_reachable(weights, limit) == expected);
                assert(
                    m1une::algo::subset_sum_max_value(weights, limit) ==
                    naive_subset_sum_max_value(weights, limit)
                );
            }
        }
    }
}

void test_zero_one_knapsack() {
    std::vector<int> weights = {2, 3, 4, 5};
    std::vector<long long> values = {4, 5, 7, 8};
    assert(
        m1une::algo::zero_one_knapsack_max_value(weights, values, 10)
        == naive_zero_one_max_value(weights, values, 10)
    );
}

void test_bounded_knapsack() {
    std::vector<int> weights = {0, 2, 3};
    std::vector<long long> values = {5, 3, 4};
    std::vector<int> counts = {2, 3, 2};
    assert(
        m1une::algo::bounded_knapsack_max_value(weights, values, counts, 10)
        == naive_bounded_max_value(weights, values, counts, 10)
    );
}

void test_min_weight_for_value() {
    std::vector<long long> weights = {3, 2, 5, 7};
    std::vector<int> values = {4, 3, 6, 8};
    assert(
        m1une::algo::zero_one_knapsack_min_weight_for_value(weights, values, 20)
        == naive_min_weight_for_value(weights, values, 20)
    );
}

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

    test_subset_sum();
    test_zero_one_knapsack();
    test_bounded_knapsack();
    test_min_weight_for_value();

    int n;
    fast_input >> n;
    std::vector<int> weights(n);
    for (int& weight : weights) fast_input >> weight;

    int query_count;
    fast_input >> query_count;
    while (query_count--) {
        int target;
        fast_input >> target;
        const bool reachable =
            m1une::algo::subset_sum_max_value(weights, target) == target;
        fast_output << (reachable ? "yes" : "no") << '\n';
    }
}
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