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:heavy_check_mark: verify/ds/wavelet_matrix/wavelet_matrix_2d.test.cpp

Depends on

Code

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

#include "../../../ds/wavelet_matrix/wavelet_matrix_2d.hpp"

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

namespace {

using Matrix = m1une::ds::WaveletMatrix2D<int, int>;

#ifndef NDEBUG
void randomized_test() {
    Matrix empty;
    assert(empty.empty());
    assert(empty.size() == 0);
    assert(empty.count(0, 0, -10, 10, -10, 10) == 0);

    std::uint64_t state = 20260713;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int trial = 0; trial < 80; trial++) {
        int n = int(random() % 40);
        std::vector<Matrix::value_type> values;
        std::vector<int> first;
        std::vector<int> second;
        values.reserve(n);
        first.reserve(n);
        second.reserve(n);
        for (int i = 0; i < n; i++) {
            int x = int(random() % 21) - 10;
            int y = int(random() % 21) - 10;
            values.emplace_back(x, y);
            first.push_back(x);
            second.push_back(y);
        }

        Matrix matrix(values);
        Matrix parallel(first, second);
        assert(matrix.size() == n);
        assert(matrix.empty() == (n == 0));
        for (int i = 0; i < n; i++) {
            assert(matrix[i] == values[i]);
            assert(parallel.access(i) == values[i]);
        }

        for (int query = 0; query < 150; query++) {
            int l = int(random() % std::uint64_t(n + 1));
            int r = int(random() % std::uint64_t(n + 1));
            if (r < l) std::swap(l, r);
            int first_lower = int(random() % 31) - 15;
            int first_upper = int(random() % 31) - 15;
            int second_lower = int(random() % 31) - 15;
            int second_upper = int(random() % 31) - 15;
            if (first_upper < first_lower) {
                std::swap(first_lower, first_upper);
            }
            if (second_upper < second_lower) {
                std::swap(second_lower, second_upper);
            }

            int expected_count = 0;
            std::vector<int> selected;
            for (int i = l; i < r; i++) {
                if (first_lower <= values[i].first &&
                    values[i].first < first_upper) {
                    selected.push_back(values[i].second);
                    if (second_lower <= values[i].second &&
                        values[i].second < second_upper) {
                        expected_count++;
                    }
                }
            }
            std::sort(selected.begin(), selected.end());

            assert(
                matrix.count(
                    l,
                    r,
                    first_lower,
                    first_upper,
                    second_lower,
                    second_upper
                ) == expected_count
            );
            assert(
                parallel.count(
                    l,
                    r,
                    first_lower,
                    first_upper,
                    second_lower,
                    second_upper
                ) == expected_count
            );
            for (int k = 0; k < int(selected.size()); k++) {
                assert(
                    matrix.quantile(
                        l,
                        r,
                        first_lower,
                        first_upper,
                        k
                    ) == selected[k]
                );
            }
        }
    }

    std::vector<Matrix::value_type> equal_values(20, Matrix::value_type(4, 7));
    Matrix equal_matrix(equal_values);
    assert(equal_matrix.count(3, 18, 4, 5, 7, 8) == 15);
    assert(equal_matrix.quantile(3, 18, 4, 5, 14) == 7);
}
#endif

}  // namespace

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

#ifndef NDEBUG
    randomized_test();
#endif
    int n, query_count;
    fast_input >> n >> query_count;
    std::vector<std::pair<int, long long>> values(n);
    for (auto& value : values) {
        value.first = 0;
        fast_input >> value.second;
    }

    m1une::ds::WaveletMatrix2D<int, long long> matrix(values);
    while (query_count--) {
        int l, r, k;
        fast_input >> l >> r >> k;
        fast_output << matrix.quantile(l, r, 0, 1, k) << '\n';
    }
}
#line 1 "verify/ds/wavelet_matrix/wavelet_matrix_2d.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/range_kth_smallest"

#line 1 "ds/wavelet_matrix/wavelet_matrix_2d.hpp"



#include <algorithm>
#include <bit>
#include <cassert>
#include <cstdint>
#include <utility>
#include <vector>

#if defined(__AVX2__) || defined(__BMI2__)
#include <immintrin.h>
#endif

namespace m1une {
namespace ds {

// A static wavelet matrix for a sequence of pairs.
//
// Besides the index range, queries can restrict the first component and count
// or select by the second component. This corresponds to orthogonal queries in
// the three dimensions (index, first, second).
template <class X, class Y = X>
class WaveletMatrix2D {
   public:
    using first_type = X;
    using second_type = Y;
    using value_type = std::pair<X, Y>;

   private:
    struct BitVector {
        std::vector<std::uint64_t> bits;
        std::vector<int> prefix;

        BitVector() = default;

        explicit BitVector(int n)
            : bits(((std::size_t(n) + 63) >> 6) + 1, 0),
              prefix(bits.size(), 0) {}

        void build() {
            for (std::size_t i = 0; i + 1 < bits.size(); i++) {
                prefix[i + 1] = prefix[i] + std::popcount(bits[i]);
            }
        }

        int rank1(int r) const {
            std::size_t word = std::size_t(r) >> 6;
            int offset = r & 63;
            int result = prefix[word];
#if defined(__BMI2__)
            result += std::popcount(
                _bzhi_u64(bits[word], static_cast<unsigned int>(offset))
            );
#else
            if (offset != 0) {
                result += std::popcount(
                    bits[word] & ((std::uint64_t(1) << offset) - 1)
                );
            }
#endif
            return result;
        }
    };

    static std::uint64_t extract_bits(
        const int* values,
        int count,
        int shift
    ) {
        std::uint64_t result = 0;
        int i = 0;
#if defined(__AVX2__)
        __m128i left = _mm_cvtsi32_si128(31 - shift);
        for (; i + 8 <= count; i += 8) {
            __m256i data = _mm256_loadu_si256(
                reinterpret_cast<const __m256i*>(values + i)
            );
            data = _mm256_sll_epi32(data, left);
            int mask = _mm256_movemask_ps(_mm256_castsi256_ps(data));
            result |= std::uint64_t(mask) << i;
        }
#endif
        for (; i < count; i++) {
            result |= std::uint64_t((unsigned(values[i]) >> shift) & 1) << i;
        }
        return result;
    }

    class RankWaveletMatrix {
       private:
        int _n = 0;
        int _alphabet_size = 0;
        int _log = 0;
        std::vector<BitVector> _matrix;
        std::vector<int> _zero_count;

        bool bit(int value, int level) const {
            return (value >> (_log - 1 - level)) & 1;
        }

        int count_less(int l, int r, int upper) const {
            if (upper <= 0) return 0;
            if (upper >= _alphabet_size) return r - l;

            int result = 0;
            for (int level = 0; level < _log; level++) {
                int l1 = _matrix[level].rank1(l);
                int r1 = _matrix[level].rank1(r);
                if (bit(upper, level)) {
                    result += (r - l) - (r1 - l1);
                    l = _zero_count[level] + l1;
                    r = _zero_count[level] + r1;
                } else {
                    l -= l1;
                    r -= r1;
                }
            }
            return result;
        }

        void build_owned(std::vector<int> current, int alphabet_size) {
            assert(alphabet_size >= 0);
            _n = int(current.size());
            _alphabet_size = alphabet_size;
            _log = alphabet_size == 0
                       ? 0
                       : std::max(
                             1,
                             int(std::bit_width(unsigned(alphabet_size - 1)))
                         );

            _matrix.clear();
            _matrix.reserve(_log);
            _zero_count.assign(_log, 0);
            std::vector<int> next(_n);
            for (int value : current) {
                assert(0 <= value && value < alphabet_size);
                (void)value;
            }

            for (int level = 0; level < _log; level++) {
                _matrix.emplace_back(_n);
                BitVector& bit_vector = _matrix.back();
                int shift = _log - 1 - level;
                int zeros = 0;
                for (int base = 0; base < _n; base += 64) {
                    int count = std::min(64, _n - base);
                    std::uint64_t word = WaveletMatrix2D::extract_bits(
                        current.data() + base,
                        count,
                        shift
                    );
                    bit_vector.bits[std::size_t(base) >> 6] = word;
                    zeros += count - std::popcount(word);
                }
                bit_vector.build();

                _zero_count[level] = zeros;
                int zero_position = 0;
                int one_position = zeros;
                for (int base = 0; base < _n; base += 64) {
                    int count = std::min(64, _n - base);
                    std::uint64_t ones =
                        bit_vector.bits[std::size_t(base) >> 6];
                    std::uint64_t valid = count == 64
                                              ? ~std::uint64_t(0)
                                              : (std::uint64_t(1) << count) - 1;
                    std::uint64_t zeroes = (~ones) & valid;
                    while (zeroes != 0) {
                        int offset = std::countr_zero(zeroes);
                        next[zero_position++] = current[base + offset];
                        zeroes &= zeroes - 1;
                    }
                    while (ones != 0) {
                        int offset = std::countr_zero(ones);
                        next[one_position++] = current[base + offset];
                        ones &= ones - 1;
                    }
                }
                current.swap(next);
            }
        }

       public:
        RankWaveletMatrix() = default;

        RankWaveletMatrix(
            const std::vector<int>& values,
            int alphabet_size
        ) {
            build(values, alphabet_size);
        }

        RankWaveletMatrix(
            std::vector<int>&& values,
            int alphabet_size
        ) {
            build(std::move(values), alphabet_size);
        }

        void build(const std::vector<int>& values, int alphabet_size) {
            build_owned(values, alphabet_size);
        }

        void build(std::vector<int>&& values, int alphabet_size) {
            build_owned(std::move(values), alphabet_size);
        }

        int range_freq(int l, int r, int lower, int upper) const {
            assert(0 <= l && l <= r && r <= _n);
            if (upper <= lower) return 0;
            return count_less(l, r, upper) - count_less(l, r, lower);
        }
    };

    int _n = 0;
    int _log = 0;
    std::vector<value_type> _values;
    std::vector<X> _first_coordinates;
    std::vector<Y> _second_coordinates;
    RankWaveletMatrix _first_matrix;
    std::vector<BitVector> _matrix;
    std::vector<int> _zero_count;
    std::vector<RankWaveletMatrix> _zero_first_matrix;

    template <class T>
    static bool equal(const T& first, const T& second) {
        return !(first < second) && !(second < first);
    }

    template <class T>
    static void sort_unique(std::vector<T>& values) {
        std::sort(values.begin(), values.end());
        values.erase(
            std::unique(
                values.begin(),
                values.end(),
                [](const T& first, const T& second) {
                    return equal(first, second);
                }
            ),
            values.end()
        );
    }

    bool bit(int value, int level) const {
        return (value >> (_log - 1 - level)) & 1;
    }

    int first_lower_bound(const X& value) const {
        return int(
            std::lower_bound(
                _first_coordinates.begin(),
                _first_coordinates.end(),
                value
            ) - _first_coordinates.begin()
        );
    }

    int second_lower_bound(const Y& value) const {
        return int(
            std::lower_bound(
                _second_coordinates.begin(),
                _second_coordinates.end(),
                value
            ) - _second_coordinates.begin()
        );
    }

    int count_first_rank(
        int l,
        int r,
        int first_lower,
        int first_upper
    ) const {
        return _first_matrix.range_freq(l, r, first_lower, first_upper);
    }

    int count_second_less(
        int l,
        int r,
        int first_lower,
        int first_upper,
        int second_upper
    ) const {
        if (second_upper <= 0) return 0;
        if (second_upper >= int(_second_coordinates.size())) {
            return count_first_rank(l, r, first_lower, first_upper);
        }

        int result = 0;
        for (int level = 0; level < _log; level++) {
            int l1 = _matrix[level].rank1(l);
            int r1 = _matrix[level].rank1(r);
            int l0 = l - l1;
            int r0 = r - r1;
            if (bit(second_upper, level)) {
                result += _zero_first_matrix[level].range_freq(
                    l0,
                    r0,
                    first_lower,
                    first_upper
                );
                l = _zero_count[level] + l1;
                r = _zero_count[level] + r1;
            } else {
                l = l0;
                r = r0;
            }
        }
        return result;
    }

   public:
    WaveletMatrix2D() = default;

    explicit WaveletMatrix2D(const std::vector<value_type>& values) {
        build(values);
    }

    explicit WaveletMatrix2D(std::vector<value_type>&& values) {
        build(std::move(values));
    }

    WaveletMatrix2D(
        const std::vector<X>& first,
        const std::vector<Y>& second
    ) {
        build(first, second);
    }

    void build(std::vector<value_type> values) {
        _values = std::move(values);
        _n = int(_values.size());

        _first_coordinates.clear();
        _second_coordinates.clear();
        _first_coordinates.reserve(_n);
        _second_coordinates.reserve(_n);
        for (const auto& value : _values) {
            _first_coordinates.push_back(value.first);
            _second_coordinates.push_back(value.second);
        }
        sort_unique(_first_coordinates);
        sort_unique(_second_coordinates);

        if (_n == 0) {
            _log = 0;
            _first_matrix = RankWaveletMatrix();
            _matrix.clear();
            _zero_count.clear();
            _zero_first_matrix.clear();
            return;
        }

        std::vector<int> current_first(_n);
        std::vector<int> current_second(_n);
        std::vector<int> next_first(_n);
        std::vector<int> next_second(_n);
        for (int i = 0; i < _n; i++) {
            current_first[i] = first_lower_bound(_values[i].first);
            current_second[i] = second_lower_bound(_values[i].second);
        }

        int first_size = int(_first_coordinates.size());
        int second_size = int(_second_coordinates.size());
        _first_matrix.build(current_first, first_size);
        _log = std::max(
            1,
            int(std::bit_width(unsigned(second_size - 1)))
        );
        _matrix.clear();
        _matrix.reserve(_log);
        _zero_count.assign(_log, 0);
        _zero_first_matrix.clear();
        _zero_first_matrix.reserve(_log);

        for (int level = 0; level < _log; level++) {
            _matrix.emplace_back(_n);
            BitVector& bit_vector = _matrix.back();
            int shift = _log - 1 - level;
            int zeros = 0;
            for (int base = 0; base < _n; base += 64) {
                int count = std::min(64, _n - base);
                std::uint64_t word = extract_bits(
                    current_second.data() + base,
                    count,
                    shift
                );
                bit_vector.bits[std::size_t(base) >> 6] = word;
                zeros += count - std::popcount(word);
            }
            bit_vector.build();

            _zero_count[level] = zeros;
            int zero_position = 0;
            int one_position = zeros;
            for (int base = 0; base < _n; base += 64) {
                int count = std::min(64, _n - base);
                std::uint64_t ones = bit_vector.bits[std::size_t(base) >> 6];
                std::uint64_t valid = count == 64
                                          ? ~std::uint64_t(0)
                                          : (std::uint64_t(1) << count) - 1;
                std::uint64_t zeroes = (~ones) & valid;
                while (zeroes != 0) {
                    int offset = std::countr_zero(zeroes);
                    next_first[zero_position] = current_first[base + offset];
                    next_second[zero_position] = current_second[base + offset];
                    zero_position++;
                    zeroes &= zeroes - 1;
                }
                while (ones != 0) {
                    int offset = std::countr_zero(ones);
                    next_first[one_position] = current_first[base + offset];
                    next_second[one_position] = current_second[base + offset];
                    one_position++;
                    ones &= ones - 1;
                }
            }

            std::vector<int> zero_first(
                next_first.begin(),
                next_first.begin() + zeros
            );
            _zero_first_matrix.emplace_back(std::move(zero_first), first_size);
            current_first.swap(next_first);
            current_second.swap(next_second);
        }
    }

    void build(
        const std::vector<X>& first,
        const std::vector<Y>& second
    ) {
        assert(first.size() == second.size());
        std::vector<value_type> values;
        values.reserve(first.size());
        for (int i = 0; i < int(first.size()); i++) {
            values.emplace_back(first[i], second[i]);
        }
        build(std::move(values));
    }

    int size() const {
        return _n;
    }

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

    const value_type& access(int p) const {
        assert(0 <= p && p < _n);
        return _values[p];
    }

    const value_type& operator[](int p) const {
        return access(p);
    }

    int count(
        int l,
        int r,
        const X& first_lower,
        const X& first_upper,
        const Y& second_lower,
        const Y& second_upper
    ) const {
        assert(0 <= l && l <= r && r <= _n);
        assert(!(first_upper < first_lower));
        assert(!(second_upper < second_lower));
        int first_l = first_lower_bound(first_lower);
        int first_r = first_lower_bound(first_upper);
        int second_l = second_lower_bound(second_lower);
        int second_r = second_lower_bound(second_upper);
        return count_second_less(l, r, first_l, first_r, second_r) -
               count_second_less(l, r, first_l, first_r, second_l);
    }

    Y quantile(
        int l,
        int r,
        const X& first_lower,
        const X& first_upper,
        int k
    ) const {
        assert(0 <= l && l <= r && r <= _n);
        assert(!(first_upper < first_lower));
        int first_l = first_lower_bound(first_lower);
        int first_r = first_lower_bound(first_upper);
        int candidates = count_first_rank(l, r, first_l, first_r);
        assert(0 <= k && k < candidates);
        (void)candidates;

        int second_rank = 0;
        for (int level = 0; level < _log; level++) {
            int l1 = _matrix[level].rank1(l);
            int r1 = _matrix[level].rank1(r);
            int l0 = l - l1;
            int r0 = r - r1;
            int zeros = _zero_first_matrix[level].range_freq(
                l0,
                r0,
                first_l,
                first_r
            );
            if (k < zeros) {
                l = l0;
                r = r0;
            } else {
                k -= zeros;
                second_rank |= 1 << (_log - 1 - level);
                l = _zero_count[level] + l1;
                r = _zero_count[level] + r1;
            }
        }
        return _second_coordinates[second_rank];
    }
};

}  // namespace ds
}  // namespace m1une


#line 4 "verify/ds/wavelet_matrix/wavelet_matrix_2d.test.cpp"

#line 1 "utilities/fast_io.hpp"



#line 5 "utilities/fast_io.hpp"
#include <array>
#include <cerrno>
#include <charconv>
#include <cstddef>
#include <cstdio>
#include <cstdlib>
#line 12 "utilities/fast_io.hpp"
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#include <type_traits>
#line 18 "utilities/fast_io.hpp"
#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 11 "verify/ds/wavelet_matrix/wavelet_matrix_2d.test.cpp"

namespace {

using Matrix = m1une::ds::WaveletMatrix2D<int, int>;

#ifndef NDEBUG
void randomized_test() {
    Matrix empty;
    assert(empty.empty());
    assert(empty.size() == 0);
    assert(empty.count(0, 0, -10, 10, -10, 10) == 0);

    std::uint64_t state = 20260713;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int trial = 0; trial < 80; trial++) {
        int n = int(random() % 40);
        std::vector<Matrix::value_type> values;
        std::vector<int> first;
        std::vector<int> second;
        values.reserve(n);
        first.reserve(n);
        second.reserve(n);
        for (int i = 0; i < n; i++) {
            int x = int(random() % 21) - 10;
            int y = int(random() % 21) - 10;
            values.emplace_back(x, y);
            first.push_back(x);
            second.push_back(y);
        }

        Matrix matrix(values);
        Matrix parallel(first, second);
        assert(matrix.size() == n);
        assert(matrix.empty() == (n == 0));
        for (int i = 0; i < n; i++) {
            assert(matrix[i] == values[i]);
            assert(parallel.access(i) == values[i]);
        }

        for (int query = 0; query < 150; query++) {
            int l = int(random() % std::uint64_t(n + 1));
            int r = int(random() % std::uint64_t(n + 1));
            if (r < l) std::swap(l, r);
            int first_lower = int(random() % 31) - 15;
            int first_upper = int(random() % 31) - 15;
            int second_lower = int(random() % 31) - 15;
            int second_upper = int(random() % 31) - 15;
            if (first_upper < first_lower) {
                std::swap(first_lower, first_upper);
            }
            if (second_upper < second_lower) {
                std::swap(second_lower, second_upper);
            }

            int expected_count = 0;
            std::vector<int> selected;
            for (int i = l; i < r; i++) {
                if (first_lower <= values[i].first &&
                    values[i].first < first_upper) {
                    selected.push_back(values[i].second);
                    if (second_lower <= values[i].second &&
                        values[i].second < second_upper) {
                        expected_count++;
                    }
                }
            }
            std::sort(selected.begin(), selected.end());

            assert(
                matrix.count(
                    l,
                    r,
                    first_lower,
                    first_upper,
                    second_lower,
                    second_upper
                ) == expected_count
            );
            assert(
                parallel.count(
                    l,
                    r,
                    first_lower,
                    first_upper,
                    second_lower,
                    second_upper
                ) == expected_count
            );
            for (int k = 0; k < int(selected.size()); k++) {
                assert(
                    matrix.quantile(
                        l,
                        r,
                        first_lower,
                        first_upper,
                        k
                    ) == selected[k]
                );
            }
        }
    }

    std::vector<Matrix::value_type> equal_values(20, Matrix::value_type(4, 7));
    Matrix equal_matrix(equal_values);
    assert(equal_matrix.count(3, 18, 4, 5, 7, 8) == 15);
    assert(equal_matrix.quantile(3, 18, 4, 5, 14) == 7);
}
#endif

}  // namespace

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

#ifndef NDEBUG
    randomized_test();
#endif
    int n, query_count;
    fast_input >> n >> query_count;
    std::vector<std::pair<int, long long>> values(n);
    for (auto& value : values) {
        value.first = 0;
        fast_input >> value.second;
    }

    m1une::ds::WaveletMatrix2D<int, long long> matrix(values);
    while (query_count--) {
        int l, r, k;
        fast_input >> l >> r >> k;
        fast_output << matrix.quantile(l, r, 0, 1, k) << '\n';
    }
}
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