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:heavy_check_mark: verify/ds/bst/mex_multiset.test.cpp

Depends on

Code

#define PROBLEM "https://yukicoder.me/problems/no/3075"

#include "../../../algo/sequence/mex.hpp"
#include "../../../ds/bst/mex_multiset.hpp"
#include "../../../utilities/fast_io.hpp"

#include <algorithm>
#include <cassert>
#include <cstdint>
#include <limits>
#include <numeric>
#include <random>
#include <set>
#include <vector>

namespace {

template <class T>
int naive_mex(const std::vector<T>& values) {
    std::set<T> present(values.begin(), values.end());
    int answer = 0;
    while (present.contains(T(answer))) ++answer;
    return answer;
}

void test_static() {
    using m1une::algo::mex;
    assert(mex(std::vector<int>()) == 0);
    assert(mex(std::vector<int>{0, 1, 1, 3}) == 2);
    assert(mex(std::vector<int>{1, 2, 3}) == 0);
    assert(mex(std::vector<long long>{
        std::numeric_limits<long long>::min(), 0, 1,
        std::numeric_limits<long long>::max()
    }) == 2);
    assert(mex(std::vector<std::uint64_t>{
        0, 1, std::numeric_limits<std::uint64_t>::max()
    }) == 2);
    assert(mex(std::vector<bool>{false, true}) == 2);

    // Exhaust all short vectors over an alphabet containing a negative value.
    int cases = 1;
    for (int n = 0; n <= 6; ++n, cases *= 6) {
        for (int code = 0; code < cases; ++code) {
            int remaining = code;
            std::vector<int> values(n);
            for (int& value : values) {
                value = remaining % 6 - 1;
                remaining /= 6;
            }
            const auto original = values;
            assert(mex(values) == naive_mex(values));
            assert(values == original);
        }
    }

    std::mt19937_64 random(1602176623);
    for (int iteration = 0; iteration < 5000; ++iteration) {
        const int n = int(random() % 200);
        std::vector<long long> values(n);
        for (long long& value : values) {
            value = static_cast<long long>(random() % (2 * n + 3)) - n / 2;
        }
        assert(mex(values) == naive_mex(values));
        std::vector<std::uint64_t> unsigned_values(n);
        for (auto& value : unsigned_values) {
            value = random() % 3 == 0 ? random() : random() % (n + 1);
        }
        assert(mex(unsigned_values) == naive_mex(unsigned_values));
    }
}

void test_boundaries() {
    using m1une::ds::MexMultiset;
    MexMultiset empty;
    assert(empty.universe_size() == 0);
    empty.insert(0);
    empty.insert(-1);
    assert(!empty.erase(0));
    assert(empty.count(0) == 0);
    assert(empty.mex() == 0);

    for (int n : {0, 1, 2, 63, 64, 65, 127, 128, 129, 4095, 4096, 4097, 262145}) {
        MexMultiset values(n);
        const MexMultiset& query = values;
        for (int value = 0; value < n; ++value) {
            assert(query.mex() == value);
            values.insert(value);
            values.insert(value);
            assert(query.count(value) == 2);
            assert(query.mex() == value + 1);
            assert(values.erase(value));
            assert(query.mex() == value + 1);
        }
        assert(query.mex() == n);
        values.insert(std::numeric_limits<long long>::min());
        values.insert(std::numeric_limits<long long>::max());
        values.insert(n);
        assert(!values.erase(-1));
        assert(!values.erase(n));
        assert(query.count(-1) == 0);
        assert(query.count(n) == 0);
        assert(query.mex() == n);

        const MexMultiset original = values;
        for (int hole : {0, 63, 64, 127, 128, 4095, 4096, 262143, 262144}) {
            if (hole >= n) continue;
            assert(values.erase(hole));
            assert(!values.erase(hole));
            assert(query.mex() == hole);
            assert(original.mex() == n);
            values.insert(hole);
            assert(query.mex() == n);
        }
        for (int value = n - 1; value >= 0; --value) {
            assert(values.erase(value));
            assert(query.mex() == value);
        }

        std::vector<int> permutation(n);
        std::iota(permutation.begin(), permutation.end(), 0);
        const MexMultiset built(permutation);
        assert(built.universe_size() == n);
        assert(built.mex() == n);
    }

    const MexMultiset wide(std::vector<std::uint64_t>{
        0, 1, std::numeric_limits<std::uint64_t>::max()
    });
    assert(wide.mex() == 2);
    assert(wide.count(0) == 1);
    assert(wide.count(1) == 1);
    assert(wide.count(2) == 0);
    assert(MexMultiset(std::vector<bool>{false, true}).mex() == 2);
}

void test_randomized_updates() {
    std::mt19937 random(314159265);
    for (int iteration = 0; iteration < 100; ++iteration) {
        const int n = int(random() % 257);
        std::vector<long long> initial(n);
        std::vector<int> counts(n, 0);
        for (long long& value : initial) {
            value = int(random() % (n + 7)) - 3;
            if (0 <= value && value < n) ++counts[int(value)];
        }
        m1une::ds::MexMultiset values(initial);
        assert(values.mex() == naive_mex(initial));
        for (int step = 0; step < 2000; ++step) {
            const int key = int(random() % (n + 7)) - 3;
            if (random() % 2 == 0) {
                values.insert(key);
                if (0 <= key && key < n) ++counts[key];
            } else {
                const bool exists = 0 <= key && key < n && counts[key] != 0;
                assert(values.erase(key) == exists);
                if (exists) --counts[key];
            }
            int expected = 0;
            while (expected < n && counts[expected] != 0) ++expected;
            assert(values.mex() == expected);
            assert(values.count(key) == (0 <= key && key < n ? counts[key] : 0));
        }
    }
}

long long recurrence_term(std::vector<long long> window, long long index) {
    const int n = int(window.size());
    if (index <= n) return window[int(index - 1)];
    m1une::ds::MexMultiset values(window);
    assert(values.mex() == m1une::algo::mex(window));
    std::vector<int> period(n + 1);
    for (int step = 0; step <= n; ++step) {
        const int next = values.mex();
        period[step] = next;
        values.erase(window[step % n]);
        values.insert(next);
        window[step % n] = next;
    }
    // Consecutive generated terms cannot repeat within N positions.
    // These N+1 terms are therefore a permutation of [0, N], which repeats.
    return period[static_cast<std::size_t>((index - n - 1) % (n + 1))];
}

void test_recurrence() {
    std::mt19937 random(271828182);
    for (int iteration = 0; iteration < 300; ++iteration) {
        const int n = 1 + int(random() % 30);
        std::vector<long long> initial(n);
        for (auto& value : initial) value = random() % (2 * n + 5);
        std::vector<long long> sequence = initial;
        for (int index = n; index < 5 * n + 10; ++index) {
            std::vector<long long> window(sequence.end() - n, sequence.end());
            sequence.push_back(naive_mex(window));
        }
        for (int index = 1; index <= int(sequence.size()); ++index) {
            assert(recurrence_term(initial, index) == sequence[index - 1]);
        }
        const long long distant = 1000000000000000000LL;
        const int equivalent = n + 1 + int((distant - n - 1) % (n + 1));
        assert(recurrence_term(initial, distant) == sequence[equivalent - 1]);
    }
}

}  // namespace

int main() {
    test_static();
    test_boundaries();
    test_randomized_updates();
    test_recurrence();

    m1une::utilities::FastInput input;
    m1une::utilities::FastOutput output;
    int n = 0;
    long long index = 0;
    input.read(n, index);
    std::vector<long long> initial(n);
    for (auto& value : initial) input.read(value);
    output.println(recurrence_term(initial, index));
}
#line 1 "verify/ds/bst/mex_multiset.test.cpp"
#define PROBLEM "https://yukicoder.me/problems/no/3075"

#line 1 "algo/sequence/mex.hpp"



#include <cassert>
#include <cstdint>
#include <limits>
#include <type_traits>
#include <vector>

namespace m1une {
namespace algo {

// Returns the smallest nonnegative integer absent from values.
template <class T>
int mex(const std::vector<T>& values) {
    static_assert(
        std::is_integral_v<T> && sizeof(T) <= sizeof(std::uintmax_t),
        "mex requires standard integral values"
    );
    assert(values.size() <= static_cast<std::size_t>(std::numeric_limits<int>::max()));
    const int n = int(values.size());
    std::vector<unsigned char> present(n, 0);
    for (T value : values) {
        if constexpr (std::is_signed_v<T>) {
            if (value < 0) continue;
        }
        if (static_cast<std::uintmax_t>(value) < static_cast<std::uintmax_t>(n)) {
            present[int(value)] = 1;
        }
    }
    int answer = 0;
    while (answer < n && present[answer]) ++answer;
    return answer;
}

}  // namespace algo
}  // namespace m1une


#line 1 "ds/bst/mex_multiset.hpp"



#line 7 "ds/bst/mex_multiset.hpp"
#include <string>
#line 10 "ds/bst/mex_multiset.hpp"

#line 1 "ds/bst/predecessor_set.hpp"



#include <bit>
#line 8 "ds/bst/predecessor_set.hpp"
#include <string_view>
#line 10 "ds/bst/predecessor_set.hpp"

namespace m1une {
namespace ds {

// Fixed-universe integer set with predecessor and successor queries.
struct PredecessorSet {
   private:
    static constexpr int word_bits = 64;

    int _universe_size;
    int _size;
    std::vector<std::vector<std::uint64_t>> _levels;

    static int checked_size(std::string_view membership) {
        assert(
            membership.size()
            <= static_cast<std::size_t>(std::numeric_limits<int>::max())
        );
        return int(membership.size());
    }

    int next_index(int index) const {
        if (index >= _universe_size) return _universe_size;
        for (int level = 0; level < int(_levels.size()); level++) {
            if (index / word_bits >= int(_levels[level].size())) break;
            std::uint64_t word =
                _levels[level][index / word_bits] >> (index % word_bits);
            if (word == 0) {
                index = index / word_bits + 1;
                continue;
            }
            index += int(std::countr_zero(word));
            for (int lower = level - 1; lower >= 0; lower--) {
                index *= word_bits;
                std::uint64_t lower_word =
                    _levels[lower][index / word_bits];
                index += int(std::countr_zero(lower_word));
            }
            return index;
        }
        return _universe_size;
    }

    int previous_index(int index) const {
        if (_universe_size == 0 || index < 0) return -1;
        if (index >= _universe_size) index = _universe_size - 1;
        for (int level = 0; level < int(_levels.size()); level++) {
            int offset = index % word_bits;
            std::uint64_t word = _levels[level][index / word_bits];
            if (offset != word_bits - 1) {
                word &= (std::uint64_t(1) << (offset + 1)) - 1;
            }
            if (word == 0) {
                index = index / word_bits - 1;
                if (index < 0) break;
                continue;
            }
            index += word_bits - 1 - int(std::countl_zero(word)) - offset;
            for (int lower = level - 1; lower >= 0; lower--) {
                index *= word_bits;
                std::uint64_t lower_word =
                    _levels[lower][index / word_bits];
                index += word_bits - 1 - int(std::countl_zero(lower_word));
            }
            return index;
        }
        return -1;
    }

    static int not_found_if_end(int index, int universe_size) {
        return index == universe_size ? -1 : index;
    }

   public:
    PredecessorSet() : PredecessorSet(0) {}

    explicit PredecessorSet(int universe_size)
        : _universe_size(universe_size), _size(0) {
        assert(universe_size >= 0);
        int length = universe_size == 0 ? 1 : universe_size;
        do {
            int words = int((std::int64_t(length) + word_bits - 1) / word_bits);
            _levels.emplace_back(words, 0);
            length = words;
        } while (length > 1);
    }

    explicit PredecessorSet(std::string_view membership)
        : PredecessorSet(checked_size(membership)) {
        for (int index = 0; index < _universe_size; index++) {
            assert(membership[index] == '0' || membership[index] == '1');
            if (membership[index] == '1') {
                _levels[0][index / word_bits]
                    |= std::uint64_t(1) << (index % word_bits);
                _size++;
            }
        }
        for (int level = 1; level < int(_levels.size()); level++) {
            for (int index = 0; index < int(_levels[level - 1].size()); index++) {
                if (_levels[level - 1][index] != 0) {
                    _levels[level][index / word_bits]
                        |= std::uint64_t(1) << (index % word_bits);
                }
            }
        }
    }

    int universe_size() const {
        return _universe_size;
    }

    int size() const {
        return _size;
    }

    bool empty() const {
        return _size == 0;
    }

    bool contains(int key) const {
        assert(0 <= key && key < _universe_size);
        return ((_levels[0][key / word_bits] >> (key % word_bits)) & 1U) != 0;
    }

    bool insert(int key) {
        assert(0 <= key && key < _universe_size);
        if (contains(key)) return false;
        int index = key;
        for (auto& level : _levels) {
            std::uint64_t& word = level[index / word_bits];
            bool was_empty = word == 0;
            word |= std::uint64_t(1) << (index % word_bits);
            if (!was_empty) break;
            index /= word_bits;
        }
        _size++;
        return true;
    }

    bool erase(int key) {
        assert(0 <= key && key < _universe_size);
        if (!contains(key)) return false;
        int index = key;
        for (auto& level : _levels) {
            std::uint64_t& word = level[index / word_bits];
            word &= ~(std::uint64_t(1) << (index % word_bits));
            if (word != 0) break;
            index /= word_bits;
        }
        _size--;
        return true;
    }

    // Returns the smallest key greater than or equal to key, or -1.
    int successor(int key) const {
        assert(0 <= key && key < _universe_size);
        return not_found_if_end(next_index(key), _universe_size);
    }

    // Returns the largest key less than or equal to key, or -1.
    int predecessor(int key) const {
        assert(0 <= key && key < _universe_size);
        return previous_index(key);
    }

    int min_ge(int key) const {
        return successor(key);
    }

    int min_gt(int key) const {
        assert(0 <= key && key < _universe_size);
        return not_found_if_end(next_index(key + 1), _universe_size);
    }

    int max_le(int key) const {
        return predecessor(key);
    }

    int max_lt(int key) const {
        assert(0 <= key && key < _universe_size);
        return previous_index(key - 1);
    }

    int min() const {
        return not_found_if_end(next_index(0), _universe_size);
    }

    int max() const {
        return previous_index(_universe_size - 1);
    }
};

}  // namespace ds
}  // namespace m1une


#line 12 "ds/bst/mex_multiset.hpp"

namespace m1une {
namespace ds {

// Tracks multiplicities in [0, U) and returns min(actual mex, U).
struct MexMultiset {
   private:
    std::vector<int> _count;
    PredecessorSet _missing;
    int _mex;

    static int checked_universe_size(int universe_size) {
        assert(universe_size >= 0);
        return universe_size;
    }

    static int checked_size(std::size_t size) {
        assert(size <= static_cast<std::size_t>(std::numeric_limits<int>::max()));
        return int(size);
    }

   public:
    MexMultiset() : MexMultiset(0) {}

    explicit MexMultiset(int universe_size)
        : _count(checked_universe_size(universe_size), 0),
          _missing(std::string(universe_size, '1')), _mex(0) {}

    template <class T>
    explicit MexMultiset(const std::vector<T>& values)
        : _count(checked_size(values.size()), 0), _missing(0), _mex(0) {
        static_assert(
            std::is_integral_v<T> && sizeof(T) <= sizeof(std::uintmax_t),
            "MexMultiset requires standard integral values"
        );
        const int n = universe_size();
        for (T value : values) {
            if constexpr (std::is_signed_v<T>) {
                if (value < 0) continue;
            }
            if (static_cast<std::uintmax_t>(value) < static_cast<std::uintmax_t>(n)) {
                ++_count[int(value)];
            }
        }
        std::string membership(n, '1');
        for (int value = 0; value < n; ++value) {
            if (_count[value] != 0) membership[value] = '0';
        }
        _missing = PredecessorSet(membership);
        const int first = _missing.min();
        _mex = first == -1 ? n : first;
    }

    int universe_size() const {
        return int(_count.size());
    }

    int count(long long value) const {
        if (value < 0 || value >= universe_size()) return 0;
        return _count[int(value)];
    }

    void insert(long long value) {
        if (value < 0 || value >= universe_size()) return;
        const int key = int(value);
        assert(_count[key] < std::numeric_limits<int>::max());
        if (_count[key]++ != 0) return;
        _missing.erase(key);
        if (key == _mex) {
            const int first = _missing.min();
            _mex = first == -1 ? universe_size() : first;
        }
    }

    // Removes one occurrence; returns false for absent or untracked values.
    bool erase(long long value) {
        if (value < 0 || value >= universe_size()) return false;
        const int key = int(value);
        if (_count[key] == 0) return false;
        if (--_count[key] == 0) {
            _missing.insert(key);
            if (key < _mex) _mex = key;
        }
        return true;
    }

    int mex() const {
        return _mex;
    }
};

}  // namespace ds
}  // namespace m1une


#line 1 "utilities/fast_io.hpp"



#include <algorithm>
#include <array>
#include <cerrno>
#include <charconv>
#include <cstddef>
#include <cstdio>
#include <cstdlib>
#line 12 "utilities/fast_io.hpp"
#include <cstring>
#include <iterator>
#line 15 "utilities/fast_io.hpp"
#include <sys/stat.h>
#line 17 "utilities/fast_io.hpp"
#include <utility>
#include <unistd.h>
#line 20 "utilities/fast_io.hpp"

namespace m1une {
namespace utilities {

struct FastOutput;

namespace internal {

// Shared with the convenience helpers in template.hpp.
inline FastOutput* standard_output_instance = nullptr;

// Detect std::begin(x), std::end(x).
template <class T, class = void>
struct is_range : std::false_type {};

template <class T>
struct is_range<T, std::void_t<
    decltype(std::begin(std::declval<T&>())),
    decltype(std::end(std::declval<T&>()))
>> : std::true_type {};

template <class T>
inline constexpr bool is_range_v = is_range<T>::value;

template <class T>
using range_reference_t = decltype(*std::begin(std::declval<T&>()));

template <class T>
using range_value_t = std::remove_cv_t<std::remove_reference_t<range_reference_t<T>>>;

template <class T, class = void>
struct range_stored_value {
    using type = range_value_t<T>;
};

template <class T>
struct range_stored_value<T, std::void_t<typename std::remove_cv_t<std::remove_reference_t<T>>::value_type>> {
    using type = typename std::remove_cv_t<std::remove_reference_t<T>>::value_type;
};

template <class T>
using range_stored_value_t = typename range_stored_value<T>::type;

// Treat strings and C strings as scalar output objects, not as ranges.
template <class T>
struct is_char_array : std::false_type {};

template <class T, std::size_t N>
struct is_char_array<T[N]>
    : std::bool_constant<std::is_same_v<std::remove_cv_t<T>, char>> {};

template <class T>
struct is_string_like
    : std::bool_constant<
          std::is_same_v<std::decay_t<T>, std::string>
          || std::is_same_v<std::decay_t<T>, const char*>
          || std::is_same_v<std::decay_t<T>, char*>
          || is_char_array<std::remove_reference_t<T>>::value
      > {};

template <class T>
inline constexpr bool is_string_like_v = is_string_like<T>::value;

// ModInt-like type: x.val() is printable, and x can be assigned from long long.
template <class T, class = void>
struct has_val_method : std::false_type {};

template <class T>
struct has_val_method<T, std::void_t<decltype(std::declval<const T&>().val())>>
    : std::true_type {};

template <class T>
inline constexpr bool has_val_method_v = has_val_method<T>::value;

template <class T, class = void>
struct has_static_mod_raw : std::false_type {};

template <class T>
struct has_static_mod_raw<
    T, std::void_t<decltype(T::mod()), decltype(T::raw(std::declval<uint32_t>()))>>
    : std::true_type {};

template <class T>
inline constexpr bool has_static_mod_raw_v = has_static_mod_raw<T>::value;

// libstdc++ before GCC 16 does not classify __int128 as an integral type in
// strict ISO modes such as -std=c++23. Keep the fast-I/O interface independent
// of that implementation detail.
template <class T>
inline constexpr bool is_integral_v =
    std::is_integral_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>
    || std::is_same_v<std::remove_cv_t<T>, __uint128_t>;

template <class T>
inline constexpr bool is_signed_v =
    std::is_signed_v<T>
    || std::is_same_v<std::remove_cv_t<T>, __int128_t>;

template <class T>
struct make_unsigned {
    using type = std::make_unsigned_t<T>;
};

template <>
struct make_unsigned<__int128_t> {
    using type = __uint128_t;
};

template <>
struct make_unsigned<__uint128_t> {
    using type = __uint128_t;
};

template <class T>
using make_unsigned_t = typename make_unsigned<std::remove_cv_t<T>>::type;

}  // namespace internal

struct FastInput {
    static constexpr int buffer_size = 1 << 20;

   private:
    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _length;
    int _file_descriptor;
    bool _streaming;

    bool refill() {
        _position = 0;
        if (_streaming) {
            ssize_t length;
            do {
                length = ::read(_file_descriptor, _buffer, buffer_size);
            } while (length < 0 && errno == EINTR);
            if (length <= 0) {
                _length = 0;
                return false;
            }
            _length = int(length);
        } else {
            _length = int(std::fread(_buffer, 1, buffer_size, _stream));
        }
        return _length != 0;
    }

    template <class T>
    bool read_integer_from_stream(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = read_char_raw();
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = negative ? result * 10 - (c - '0')
                                  : result * 10 + (c - '0');
                c = read_char_raw();
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                result = result * 10 + T(c - '0');
                c = read_char_raw();
            }
            value = negative ? T(0) - result : result;
        }
        return true;
    }

    bool prepare_number() {
        if (_length - _position >= 64) return true;
        const int remaining = _length - _position;
        if (remaining > 0) std::memmove(_buffer, _buffer + _position, remaining);
        const int added = int(std::fread(_buffer + remaining, 1, buffer_size - remaining, _stream));
        _position = 0;
        _length = remaining + added;
        if (_length < buffer_size) _buffer[_length] = '\0';
        return _length != 0;
    }

   public:
    explicit FastInput(std::FILE* stream = stdin)
        : _stream(stream),
          _position(0),
          _length(0),
          _file_descriptor(::fileno(stream)),
          _streaming([&] {
              struct stat status;
              return _file_descriptor >= 0
                     && ::fstat(_file_descriptor, &status) == 0
                     && !S_ISREG(status.st_mode);
          }()) {}

    FastInput(const FastInput&) = delete;
    FastInput& operator=(const FastInput&) = delete;

    int read_char_raw() {
        if (_position == _length && !refill()) return EOF;
        return _buffer[_position++];
    }

    bool skip_spaces() {
        int c = read_char_raw();
        while (c != EOF && c <= ' ') c = read_char_raw();
        if (c == EOF) return false;
        --_position;
        return true;
    }

    bool read(char& value) {
        if (!skip_spaces()) return false;
        value = char(read_char_raw());
        return true;
    }

    bool read(std::string& value) {
        if (!skip_spaces()) return false;
        value.clear();
        while (true) {
            const int begin = _position;
            while (_position < _length &&
                   static_cast<unsigned char>(_buffer[_position]) > ' ') {
                ++_position;
            }
            value.append(_buffer + begin, _position - begin);
            if (_position < _length) {
                ++_position;
                return true;
            }
            if (!refill()) return true;
        }
    }

    bool read(bool& value) {
        int x;
        if (!read(x)) return false;
        value = x != 0;
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>,
        bool
    >
    read(T& value) {
        if (_streaming) return read_integer_from_stream(value);
        if (!prepare_number()) return false;
        int c = static_cast<unsigned char>(_buffer[_position++]);
        while (c <= ' ') c = static_cast<unsigned char>(_buffer[_position++]);

        bool negative = false;
        if (c == '-') {
            negative = true;
            c = static_cast<unsigned char>(_buffer[_position++]);
        }

        if constexpr (internal::is_signed_v<T>) {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const int first = c - '0';
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = negative ? result * 100 - (first * 10 + second)
                                      : result * 100 + (first * 10 + second);
                    ++_position;
                } else {
                    result = negative ? result * 10 - first : result * 10 + first;
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = result;
        } else {
            T result = 0;
            while ('0' <= c && c <= '9') {
                const unsigned first = unsigned(c - '0');
                const int second = static_cast<unsigned char>(_buffer[_position]) - '0';
                if (0 <= second && second <= 9) {
                    result = result * 100 + T(first * 10 + unsigned(second));
                    ++_position;
                } else {
                    result = result * 10 + T(first);
                }
                c = static_cast<unsigned char>(_buffer[_position++]);
            }
            value = negative ? T(0) - result : result;
        }
        if (_position > _length) _position = _length;
        return true;
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>, bool>
    read(T& value) {
        if (!skip_spaces()) return false;
        int c = read_char_raw();
        bool negative = false;
        if (c == '-' || c == '+') {
            negative = c == '-';
            c = read_char_raw();
        }

        long double result = 0;
        while ('0' <= c && c <= '9') {
            result = result * 10 + (c - '0');
            c = read_char_raw();
        }
        if (c == '.') {
            long double place = 0.1L;
            c = read_char_raw();
            while ('0' <= c && c <= '9') {
                result += (c - '0') * place;
                place *= 0.1L;
                c = read_char_raw();
            }
        }
        if (c == 'e' || c == 'E') {
            c = read_char_raw();
            bool exponent_negative = false;
            if (c == '-' || c == '+') {
                exponent_negative = c == '-';
                c = read_char_raw();
            }
            int exponent = 0;
            while ('0' <= c && c <= '9') {
                exponent = exponent * 10 + (c - '0');
                c = read_char_raw();
            }
            long double scale = 1;
            long double power = 10;
            while (exponent > 0) {
                if (exponent & 1) scale *= power;
                power *= power;
                exponent >>= 1;
            }
            result = exponent_negative ? result / scale : result * scale;
        }
        value = static_cast<T>(negative ? -result : result);
        return true;
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>,
        bool
    >
    read(T& value) {
        long long x;
        if (!read(x)) return false;
        if constexpr (internal::has_static_mod_raw_v<T>) {
            if (x >= 0 && uint64_t(x) < uint64_t(T::mod())) {
                value = T::raw(uint32_t(x));
            } else {
                value = T(x);
            }
        } else {
            value = T(x);
        }
        return true;
    }

    template <class First, class Second>
    bool read(std::pair<First, Second>& value) {
        if (!read(value.first)) return false;
        return read(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>,
        bool
    >
    read(Range& range) {
        using StoredValue = internal::range_stored_value_t<Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        for (auto&& value : range) {
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                bool x;
                if (!read(x)) return false;
                value = x;
            } else {
                if (!read(value)) return false;
            }
        }
        return true;
    }

    template <class First, class Second, class... Rest>
    bool read(First& first, Second& second, Rest&... rest) {
        if (!read(first)) return false;
        return read(second, rest...);
    }

    template <class T>
    FastInput& operator>>(T& value) {
        if (!read(value)) std::abort();
        return *this;
    }
};

struct FastOutput {
    static constexpr int buffer_size = 1 << 20;

   private:
    inline static const auto digit_quads = [] {
        std::array<char, 40000> result{};
        for (int i = 0; i < 10000; i++) {
            int value = i;
            for (int j = 3; j >= 0; j--) {
                result[4 * i + j] = char('0' + value % 10);
                value /= 10;
            }
        }
        return result;
    }();

    std::FILE* _stream;
    char _buffer[buffer_size];
    int _position;
    int _precision;
    std::chars_format _float_format;
    char _range_separator;
    std::string* _capture = nullptr;

    template <class T>
    std::string format_cell(const T& value) {
        std::string result;
        struct CaptureGuard {
            std::string*& target;
            std::string* previous;
            ~CaptureGuard() { target = previous; }
        } guard{_capture, _capture};
        _capture = &result;
        write(value);
        return result;
    }

    template <class Matrix>
    void write_aligned_matrix(const Matrix& matrix) {
        std::vector<std::vector<std::string>> rows;
        std::vector<std::size_t> widths;
        for (const auto& row : matrix) {
            auto& cells = rows.emplace_back();
            std::size_t column = 0;
            for (const auto& value : row) {
                cells.push_back(format_cell(value));
                if (column == widths.size()) widths.push_back(0);
                widths[column] = std::max(widths[column], cells.back().size());
                ++column;
            }
        }
        bool first = true;
        for (const auto& row : rows) {
            if (!first) write_char('\n');
            first = false;
            for (std::size_t column = 0; column < row.size(); ++column) {
                if (column != 0) write_char(_range_separator);
                for (std::size_t padding = row[column].size();
                     padding < widths[column]; ++padding) {
                    write_char(' ');
                }
                write(row[column]);
            }
        }
    }

   public:
    explicit FastOutput(std::FILE* stream = stdout)
        : _stream(stream),
          _position(0),
          _precision(6),
          _float_format(std::chars_format::general),
          _range_separator(' ') {
        if (_stream == stdout
            && internal::standard_output_instance == nullptr) {
            internal::standard_output_instance = this;
        }
    }

    FastOutput(const FastOutput&) = delete;
    FastOutput& operator=(const FastOutput&) = delete;

    ~FastOutput() {
        flush();
        if (internal::standard_output_instance == this) {
            internal::standard_output_instance = nullptr;
        }
    }

    void flush() {
        if (_position != 0) {
            std::fwrite(_buffer, 1, _position, _stream);
            _position = 0;
        }
        std::fflush(_stream);
    }

    void write_char(char c) {
        if (_capture != nullptr) {
            _capture->push_back(c);
            return;
        }
        if (_position == buffer_size) flush();
        _buffer[_position++] = c;
    }

    void write(const char* s) {
        while (*s != '\0') write_char(*s++);
    }

    void write(const std::string& s) {
        if (_capture != nullptr) {
            _capture->append(s);
            return;
        }
        std::size_t position = 0;
        while (position < s.size()) {
            if (_position == buffer_size) flush();
            const std::size_t copied =
                std::min<std::size_t>(buffer_size - _position, s.size() - position);
            std::memcpy(_buffer + _position, s.data() + position, copied);
            _position += int(copied);
            position += copied;
        }
    }

    void write(char c) {
        write_char(c);
    }

    void write(bool value) {
        write_char(value ? '1' : '0');
    }

    template <class T>
    std::enable_if_t<std::is_floating_point_v<T>>
    write(T value) {
        char digits[128];
        auto [end, error] = std::to_chars(
            digits,
            digits + sizeof(digits),
            value,
            _float_format,
            _precision
        );
        if (error != std::errc()) std::abort();
        for (const char* pointer = digits; pointer != end; pointer++) {
            write_char(*pointer);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::is_integral_v<T>
            && !std::is_same_v<std::remove_cv_t<T>, bool>
            && !std::is_same_v<std::remove_cv_t<T>, char>
    >
    write(T value) {
        using Raw = std::remove_cv_t<T>;
        using Unsigned = internal::make_unsigned_t<Raw>;

        Unsigned magnitude;
        if constexpr (internal::is_signed_v<Raw>) {
            if (value < 0) {
                write_char('-');
                magnitude = Unsigned(0) - Unsigned(value);
            } else {
                magnitude = Unsigned(value);
            }
        } else {
            magnitude = value;
        }

        if (magnitude == 0) {
            write_char('0');
            return;
        }

        unsigned chunks[16];
        int count = 0;
        while (magnitude >= 10000) {
            const Unsigned quotient = magnitude / 10000;
            chunks[count++] = unsigned(magnitude - quotient * 10000);
            magnitude = quotient;
        }
        if (_capture == nullptr && _position > buffer_size - 64) flush();
        char captured[64];
        char* const begin = _capture != nullptr ? captured : _buffer + _position;
        char* destination = begin;
        const unsigned leading = unsigned(magnitude);
        const char* first = digit_quads.data() + 4 * leading;
        int skip = leading < 10 ? 3 : leading < 100 ? 2 : leading < 1000 ? 1 : 0;
        for (; skip < 4; skip++) *destination++ = first[skip];
        while (count--) {
            const char* digits = digit_quads.data() + 4 * chunks[count];
            std::memcpy(destination, digits, 4);
            destination += 4;
        }
        if (_capture != nullptr) {
            _capture->append(begin, destination - begin);
        } else {
            _position += int(destination - begin);
        }
    }

    template <class T>
    std::enable_if_t<
        internal::has_val_method_v<T>
            && !internal::is_integral_v<T>
            && !internal::is_range_v<T>
    >
    write(const T& value) {
        write(value.val());
    }

    template <class First, class Second>
    void write(const std::pair<First, Second>& value) {
        write(value.first);
        write_char(' ');
        write(value.second);
    }

    template <class Range>
    std::enable_if_t<
        internal::is_range_v<Range>
            && !internal::is_string_like_v<Range>
    >
    write(const Range& range) {
        using StoredValue = internal::range_stored_value_t<const Range>;
        constexpr bool nested = internal::is_range_v<StoredValue>
                                && !internal::is_string_like_v<StoredValue>;

        bool first = true;
        for (const auto& value : range) {
            if (!first) write_char(nested ? '\n' : _range_separator);
            first = false;
            if constexpr (std::is_same_v<StoredValue, bool> && !nested) {
                write(static_cast<bool>(value));
            } else {
                write(value);
            }
        }
    }

    template <class First, class... Rest>
    void print(const First& first, const Rest&... rest) {
        write(first);
        ((write_char(' '), write(rest)), ...);
    }

    void println() {
        write_char('\n');
    }

    void set_precision(int precision) {
        _precision = precision;
    }

    void set_fixed(int precision = 6) {
        _float_format = std::chars_format::fixed;
        _precision = precision;
    }

    void set_general(int precision = 6) {
        _float_format = std::chars_format::general;
        _precision = precision;
    }

    void set_range_separator(char separator) {
        _range_separator = separator;
    }

    template <class Matrix>
    void write_aligned(const Matrix& matrix) {
        using Row = internal::range_stored_value_t<const Matrix>;
        using Cell = internal::range_stored_value_t<const Row>;
        static_assert(internal::is_range_v<Row> && !internal::is_string_like_v<Row>,
                      "write_aligned requires a two-dimensional range");
        static_assert(!internal::is_range_v<Cell> || internal::is_string_like_v<Cell>,
                      "write_aligned requires scalar cells");
        write_aligned_matrix(matrix);
    }

    template <class Matrix>
    void println_aligned(const Matrix& matrix) {
        write_aligned(matrix);
        write_char('\n');
    }

    template <class... Args>
    void println(const Args&... args) {
        print(args...);
        write_char('\n');
    }

    template <class T>
    FastOutput& operator<<(const T& value) {
        write(value);
        return *this;
    }
};

}  // namespace utilities
}  // namespace m1une


#line 6 "verify/ds/bst/mex_multiset.test.cpp"

#line 11 "verify/ds/bst/mex_multiset.test.cpp"
#include <numeric>
#include <random>
#include <set>
#line 15 "verify/ds/bst/mex_multiset.test.cpp"

namespace {

template <class T>
int naive_mex(const std::vector<T>& values) {
    std::set<T> present(values.begin(), values.end());
    int answer = 0;
    while (present.contains(T(answer))) ++answer;
    return answer;
}

void test_static() {
    using m1une::algo::mex;
    assert(mex(std::vector<int>()) == 0);
    assert(mex(std::vector<int>{0, 1, 1, 3}) == 2);
    assert(mex(std::vector<int>{1, 2, 3}) == 0);
    assert(mex(std::vector<long long>{
        std::numeric_limits<long long>::min(), 0, 1,
        std::numeric_limits<long long>::max()
    }) == 2);
    assert(mex(std::vector<std::uint64_t>{
        0, 1, std::numeric_limits<std::uint64_t>::max()
    }) == 2);
    assert(mex(std::vector<bool>{false, true}) == 2);

    // Exhaust all short vectors over an alphabet containing a negative value.
    int cases = 1;
    for (int n = 0; n <= 6; ++n, cases *= 6) {
        for (int code = 0; code < cases; ++code) {
            int remaining = code;
            std::vector<int> values(n);
            for (int& value : values) {
                value = remaining % 6 - 1;
                remaining /= 6;
            }
            const auto original = values;
            assert(mex(values) == naive_mex(values));
            assert(values == original);
        }
    }

    std::mt19937_64 random(1602176623);
    for (int iteration = 0; iteration < 5000; ++iteration) {
        const int n = int(random() % 200);
        std::vector<long long> values(n);
        for (long long& value : values) {
            value = static_cast<long long>(random() % (2 * n + 3)) - n / 2;
        }
        assert(mex(values) == naive_mex(values));
        std::vector<std::uint64_t> unsigned_values(n);
        for (auto& value : unsigned_values) {
            value = random() % 3 == 0 ? random() : random() % (n + 1);
        }
        assert(mex(unsigned_values) == naive_mex(unsigned_values));
    }
}

void test_boundaries() {
    using m1une::ds::MexMultiset;
    MexMultiset empty;
    assert(empty.universe_size() == 0);
    empty.insert(0);
    empty.insert(-1);
    assert(!empty.erase(0));
    assert(empty.count(0) == 0);
    assert(empty.mex() == 0);

    for (int n : {0, 1, 2, 63, 64, 65, 127, 128, 129, 4095, 4096, 4097, 262145}) {
        MexMultiset values(n);
        const MexMultiset& query = values;
        for (int value = 0; value < n; ++value) {
            assert(query.mex() == value);
            values.insert(value);
            values.insert(value);
            assert(query.count(value) == 2);
            assert(query.mex() == value + 1);
            assert(values.erase(value));
            assert(query.mex() == value + 1);
        }
        assert(query.mex() == n);
        values.insert(std::numeric_limits<long long>::min());
        values.insert(std::numeric_limits<long long>::max());
        values.insert(n);
        assert(!values.erase(-1));
        assert(!values.erase(n));
        assert(query.count(-1) == 0);
        assert(query.count(n) == 0);
        assert(query.mex() == n);

        const MexMultiset original = values;
        for (int hole : {0, 63, 64, 127, 128, 4095, 4096, 262143, 262144}) {
            if (hole >= n) continue;
            assert(values.erase(hole));
            assert(!values.erase(hole));
            assert(query.mex() == hole);
            assert(original.mex() == n);
            values.insert(hole);
            assert(query.mex() == n);
        }
        for (int value = n - 1; value >= 0; --value) {
            assert(values.erase(value));
            assert(query.mex() == value);
        }

        std::vector<int> permutation(n);
        std::iota(permutation.begin(), permutation.end(), 0);
        const MexMultiset built(permutation);
        assert(built.universe_size() == n);
        assert(built.mex() == n);
    }

    const MexMultiset wide(std::vector<std::uint64_t>{
        0, 1, std::numeric_limits<std::uint64_t>::max()
    });
    assert(wide.mex() == 2);
    assert(wide.count(0) == 1);
    assert(wide.count(1) == 1);
    assert(wide.count(2) == 0);
    assert(MexMultiset(std::vector<bool>{false, true}).mex() == 2);
}

void test_randomized_updates() {
    std::mt19937 random(314159265);
    for (int iteration = 0; iteration < 100; ++iteration) {
        const int n = int(random() % 257);
        std::vector<long long> initial(n);
        std::vector<int> counts(n, 0);
        for (long long& value : initial) {
            value = int(random() % (n + 7)) - 3;
            if (0 <= value && value < n) ++counts[int(value)];
        }
        m1une::ds::MexMultiset values(initial);
        assert(values.mex() == naive_mex(initial));
        for (int step = 0; step < 2000; ++step) {
            const int key = int(random() % (n + 7)) - 3;
            if (random() % 2 == 0) {
                values.insert(key);
                if (0 <= key && key < n) ++counts[key];
            } else {
                const bool exists = 0 <= key && key < n && counts[key] != 0;
                assert(values.erase(key) == exists);
                if (exists) --counts[key];
            }
            int expected = 0;
            while (expected < n && counts[expected] != 0) ++expected;
            assert(values.mex() == expected);
            assert(values.count(key) == (0 <= key && key < n ? counts[key] : 0));
        }
    }
}

long long recurrence_term(std::vector<long long> window, long long index) {
    const int n = int(window.size());
    if (index <= n) return window[int(index - 1)];
    m1une::ds::MexMultiset values(window);
    assert(values.mex() == m1une::algo::mex(window));
    std::vector<int> period(n + 1);
    for (int step = 0; step <= n; ++step) {
        const int next = values.mex();
        period[step] = next;
        values.erase(window[step % n]);
        values.insert(next);
        window[step % n] = next;
    }
    // Consecutive generated terms cannot repeat within N positions.
    // These N+1 terms are therefore a permutation of [0, N], which repeats.
    return period[static_cast<std::size_t>((index - n - 1) % (n + 1))];
}

void test_recurrence() {
    std::mt19937 random(271828182);
    for (int iteration = 0; iteration < 300; ++iteration) {
        const int n = 1 + int(random() % 30);
        std::vector<long long> initial(n);
        for (auto& value : initial) value = random() % (2 * n + 5);
        std::vector<long long> sequence = initial;
        for (int index = n; index < 5 * n + 10; ++index) {
            std::vector<long long> window(sequence.end() - n, sequence.end());
            sequence.push_back(naive_mex(window));
        }
        for (int index = 1; index <= int(sequence.size()); ++index) {
            assert(recurrence_term(initial, index) == sequence[index - 1]);
        }
        const long long distant = 1000000000000000000LL;
        const int equivalent = n + 1 + int((distant - n - 1) % (n + 1));
        assert(recurrence_term(initial, distant) == sequence[equivalent - 1]);
    }
}

}  // namespace

int main() {
    test_static();
    test_boundaries();
    test_randomized_updates();
    test_recurrence();

    m1une::utilities::FastInput input;
    m1une::utilities::FastOutput output;
    int n = 0;
    long long index = 0;
    input.read(n, index);
    std::vector<long long> initial(n);
    for (auto& value : initial) input.read(value);
    output.println(recurrence_term(initial, index));
}
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