verify/geometry/convex_layers.test.cpp

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• Last update: 2026-10-06 01:57:24+09:00
• Problem: https://judge.yosupo.jp/problem/convex_layers

Depends on

• Convex Hull (geometry/convex_hull.hpp)
• Convex Layers (geometry/convex_layers.hpp)
• geometry/detail/floating_predicate.hpp
• 2D Point and Predicates (geometry/point.hpp)
• 2D Point and Predicates (geometry/point.hpp)
• Fast IO (utilities/fast_io.hpp)

Code

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

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

#include "../../geometry/convex_hull.hpp"
#include "../../geometry/convex_layers.hpp"
#include "../../geometry/point.hpp"

namespace {

using Point = m1une::geometry::Point<long long>;

std::vector<int> naive_convex_layers(const std::vector<Point>& points) {
    const int n = int(points.size());
    std::vector<int> result(n, 0);
    int remaining = n;
    for (int layer = 1; remaining > 0; layer++) {
        std::vector<Point> active;
        for (int index = 0; index < n; index++) {
            if (result[index] == 0) active.push_back(points[index]);
        }
        std::vector<Point> hull = m1une::geometry::convex_hull(active, true);
        for (int index = 0; index < n; index++) {
            if (result[index] != 0) continue;
            if (std::find(hull.begin(), hull.end(), points[index]) == hull.end()) {
                continue;
            }
            result[index] = layer;
            remaining--;
        }
    }
    return result;
}

void test_randomized() {
    assert(m1une::geometry::convex_layers(std::vector<Point>()).empty());

    std::uint64_t state = 0x3141592653589793ULL;
    auto random = [&]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };
    for (int test = 0; test < 500; test++) {
        const int n = int(random() % 45);
        std::vector<Point> points;
        points.reserve(n);
        for (int index = 0; index < n; index++) {
            points.emplace_back(
                static_cast<long long>(random() % 17) - 8,
                static_cast<long long>(random() % 17) - 8
            );
        }
        assert(
            m1une::geometry::convex_layers(points) ==
            naive_convex_layers(points)
        );
    }
}

void test_coordinate_types() {
    std::vector<m1une::geometry::Point<long double>> floating;
    floating.emplace_back(0.5L, 0.5L);
    floating.emplace_back(4.5L, 0.5L);
    floating.emplace_back(4.5L, 4.5L);
    floating.emplace_back(0.5L, 4.5L);
    floating.emplace_back(2.5L, 2.5L);
    std::vector<int> expected_floating = {1, 1, 1, 1, 2};
    assert(m1une::geometry::convex_layers(floating) == expected_floating);

    const long long base = std::numeric_limits<long long>::lowest();
    std::vector<Point> extreme;
    extreme.emplace_back(base, base);
    extreme.emplace_back(base + 4, base);
    extreme.emplace_back(base + 4, base + 4);
    extreme.emplace_back(base, base + 4);
    extreme.emplace_back(base + 2, base + 2);
    std::vector<int> expected_extreme = {1, 1, 1, 1, 2};
    assert(m1une::geometry::convex_layers(extreme) == expected_extreme);
}

}  // namespace

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

    test_coordinate_types();
    test_randomized();
    int point_count;
    fast_input >> point_count;
    std::vector<Point> points(point_count);
    for (Point& point : points) fast_input >> point.x >> point.y;

    for (int layer : m1une::geometry::convex_layers(points)) {
        fast_output << layer << '\n';
    }
}

#line 1 "verify/geometry/convex_layers.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/convex_layers"

#include <algorithm>
#include <cassert>
#include <cstdint>
#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>
#include <utility>
#include <unistd.h>
#include <vector>

namespace m1une {
namespace utilities {

struct FastOutput;

namespace internal {

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

}  // namespace internal

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

}  // namespace utilities
}  // namespace m1une


#line 7 "verify/geometry/convex_layers.test.cpp"
#include <limits>
#line 9 "verify/geometry/convex_layers.test.cpp"

#line 1 "geometry/convex_hull.hpp"



#line 8 "geometry/convex_hull.hpp"

#line 1 "geometry/point.hpp"



#include <cmath>
#include <concepts>
#line 8 "geometry/point.hpp"

#line 1 "geometry/detail/floating_predicate.hpp"



namespace m1une {
namespace geometry {
namespace predicate_detail {

template <typename T>
constexpr T absolute(T value) {
    return value < T(0) ? -value : value;
}

template <typename T>
constexpr T max_value(T first, T second) {
    return first < second ? second : first;
}

template <typename T>
constexpr T vector_scale(T x, T y) {
    return max_value(absolute(x), absolute(y));
}

template <bool Exact, typename T>
constexpr int scaled_sign(T value, T scale, long double eps) {
    if constexpr (Exact) {
        return (value > T(0)) - (value < T(0));
    } else {
        const T tolerance = T(eps) * scale;
        return (value > tolerance) - (value < -tolerance);
    }
}

template <bool Exact, typename T>
constexpr T determinant_scale(T ax, T ay, T bx, T by) {
    if constexpr (Exact) {
        return T(0);
    } else {
        return vector_scale(ax, ay) * vector_scale(bx, by);
    }
}

template <bool Exact, typename T>
constexpr int determinant_sign(
    T ax,
    T ay,
    T bx,
    T by,
    long double eps
) {
    const T determinant = ax * by - ay * bx;
    return scaled_sign<Exact>(
        determinant,
        determinant_scale<Exact>(ax, ay, bx, by),
        eps
    );
}

template <bool Exact, typename T>
constexpr int orientation_sign(
    T direction_x,
    T direction_y,
    T offset_x,
    T offset_y,
    long double eps
) {
    const T determinant =
        direction_x * offset_y - direction_y * offset_x;
    T scale = T(0);
    if constexpr (!Exact) {
        const T direction_scale =
            vector_scale(direction_x, direction_y);
        scale = direction_scale * max_value(
            direction_scale,
            vector_scale(offset_x, offset_y)
        );
    }
    return scaled_sign<Exact>(determinant, scale, eps);
}

template <bool Exact, typename T>
constexpr int dot_sign(
    T ax,
    T ay,
    T bx,
    T by,
    long double eps
) {
    const T value = ax * bx + ay * by;
    T scale = T(0);
    if constexpr (!Exact) {
        scale = vector_scale(ax, ay) * vector_scale(bx, by);
    }
    return scaled_sign<Exact>(value, scale, eps);
}

}  // namespace predicate_detail
}  // namespace geometry
}  // namespace m1une


#line 10 "geometry/point.hpp"

namespace m1une {
namespace geometry {

template <typename T>
concept Coordinate = !std::same_as<std::remove_cv_t<T>, bool> &&
    (std::is_arithmetic_v<T> ||
     (std::copyable<T> && std::totally_ordered<T> && requires(T a, T b) {
         T(0);
         T(1);
         static_cast<long double>(a);
         { +a } -> std::same_as<T>;
         { -a } -> std::same_as<T>;
         { a + b } -> std::same_as<T>;
         { a - b } -> std::same_as<T>;
         { a * b } -> std::same_as<T>;
         { a / b } -> std::same_as<T>;
         { a += b } -> std::same_as<T&>;
         { a -= b } -> std::same_as<T&>;
     }));

// Custom coordinate types keep their own exact arithmetic.
template <typename T>
concept ExactCoordinate = Coordinate<T> && !std::floating_point<T>;

template <Coordinate T>
using wide_type = std::conditional_t<std::integral<T>, __int128_t,
    std::conditional_t<std::floating_point<T>, long double, T>>;

template <Coordinate T>
struct Point {
    T x;
    T y;

    constexpr Point() : x(0), y(0) {}
    constexpr Point(T x_value, T y_value) : x(x_value), y(y_value) {}

    template <Coordinate U>
    explicit constexpr Point(const Point<U>& other)
        : x(static_cast<T>(other.x)), y(static_cast<T>(other.y)) {}

    constexpr Point& operator+=(const Point& other) {
        x += other.x;
        y += other.y;
        return *this;
    }

    constexpr Point& operator-=(const Point& other) {
        x -= other.x;
        y -= other.y;
        return *this;
    }

    constexpr Point operator+() const {
        return *this;
    }

    constexpr Point operator-() const {
        return Point(-x, -y);
    }

    friend constexpr Point operator+(Point left, const Point& right) {
        return left += right;
    }

    friend constexpr Point operator-(Point left, const Point& right) {
        return left -= right;
    }

    friend constexpr bool operator==(const Point&, const Point&) = default;

    friend constexpr bool operator<(const Point& left, const Point& right) {
        if (left.x != right.x) return left.x < right.x;
        return left.y < right.y;
    }
};

template <Coordinate T>
constexpr Point<long double> centroid(const Point<T>& point) {
    return Point<long double>(point);
}

template <Coordinate T, typename Scalar>
requires (std::is_arithmetic_v<Scalar> || Coordinate<Scalar>)
constexpr auto operator*(const Point<T>& point, Scalar scalar) {
    using Result = std::common_type_t<T, Scalar>;
    return Point<Result>(
        Result(point.x) * Result(scalar),
        Result(point.y) * Result(scalar)
    );
}

template <typename Scalar, Coordinate T>
requires (std::is_arithmetic_v<Scalar> || Coordinate<Scalar>)
constexpr auto operator*(Scalar scalar, const Point<T>& point) {
    return point * scalar;
}

template <Coordinate T, typename Scalar>
requires (std::is_arithmetic_v<Scalar> || Coordinate<Scalar>)
constexpr auto operator/(const Point<T>& point, Scalar scalar) {
    using Result = std::common_type_t<T, Scalar>;
    return Point<Result>(
        Result(point.x) / Result(scalar),
        Result(point.y) / Result(scalar)
    );
}

template <Coordinate T>
constexpr wide_type<T> dot(const Point<T>& a, const Point<T>& b) {
    using W = wide_type<T>;
    return W(a.x) * W(b.x) + W(a.y) * W(b.y);
}

template <Coordinate T>
constexpr wide_type<T> cross(const Point<T>& a, const Point<T>& b) {
    using W = wide_type<T>;
    return W(a.x) * W(b.y) - W(a.y) * W(b.x);
}

template <Coordinate T>
constexpr wide_type<T> cross(
    const Point<T>& origin,
    const Point<T>& a,
    const Point<T>& b
) {
    using W = wide_type<T>;
    W ax = W(a.x) - W(origin.x);
    W ay = W(a.y) - W(origin.y);
    W bx = W(b.x) - W(origin.x);
    W by = W(b.y) - W(origin.y);
    return ax * by - ay * bx;
}

template <Coordinate T>
constexpr wide_type<T> norm2(const Point<T>& point) {
    return dot(point, point);
}

template <Coordinate T>
constexpr wide_type<T> distance2(const Point<T>& a, const Point<T>& b) {
    using W = wide_type<T>;
    W dx = W(a.x) - W(b.x);
    W dy = W(a.y) - W(b.y);
    return dx * dx + dy * dy;
}

template <Coordinate T>
long double norm(const Point<T>& point) {
    return std::hypot(
        static_cast<long double>(point.x),
        static_cast<long double>(point.y)
    );
}

template <Coordinate T>
long double distance(const Point<T>& a, const Point<T>& b) {
    return std::hypot(
        static_cast<long double>(a.x) - static_cast<long double>(b.x),
        static_cast<long double>(a.y) - static_cast<long double>(b.y)
    );
}

template <Coordinate T, typename M, typename N>
requires (std::is_arithmetic_v<M> || Coordinate<M>) &&
         (std::is_arithmetic_v<N> || Coordinate<N>)
constexpr Point<long double> internal_division_point(
    const Point<T>& a,
    const Point<T>& b,
    M m,
    N n
) {
    long double first_ratio = static_cast<long double>(m);
    long double second_ratio = static_cast<long double>(n);
    long double denominator = first_ratio + second_ratio;
    assert(denominator != 0);
    Point<long double> first(a);
    Point<long double> direction = Point<long double>(b) - first;
    return first + direction * (first_ratio / denominator);
}

template <Coordinate T, typename M, typename N>
requires (std::is_arithmetic_v<M> || Coordinate<M>) &&
         (std::is_arithmetic_v<N> || Coordinate<N>)
constexpr Point<long double> external_division_point(
    const Point<T>& a,
    const Point<T>& b,
    M m,
    N n
) {
    long double first_ratio = static_cast<long double>(m);
    long double second_ratio = static_cast<long double>(n);
    long double denominator = first_ratio - second_ratio;
    assert(denominator != 0);
    Point<long double> first(a);
    Point<long double> direction = Point<long double>(b) - first;
    return first + direction * (first_ratio / denominator);
}

template <Coordinate T>
constexpr int sign(wide_type<T> value, long double eps = 1e-12L) {
    return predicate_detail::scaled_sign<ExactCoordinate<T>>(
        value,
        wide_type<T>(1),
        eps
    );
}

template <Coordinate T>
constexpr int orientation(
    const Point<T>& a,
    const Point<T>& b,
    const Point<T>& c,
    long double eps = 1e-12L
) {
    using W = wide_type<T>;
    const W first_x = W(b.x) - W(a.x);
    const W first_y = W(b.y) - W(a.y);
    const W second_x = W(c.x) - W(a.x);
    const W second_y = W(c.y) - W(a.y);
    return predicate_detail::orientation_sign<ExactCoordinate<T>>(
        first_x,
        first_y,
        second_x,
        second_y,
        eps
    );
}

template <Coordinate T>
constexpr bool collinear(
    const Point<T>& a,
    const Point<T>& b,
    const Point<T>& c,
    long double eps = 1e-12L
) {
    return orientation(a, b, c, eps) == 0;
}

template <Coordinate T>
Point<long double> rotate(const Point<T>& point, long double angle) {
    long double cosine = std::cos(angle);
    long double sine = std::sin(angle);
    return Point<long double>(
        static_cast<long double>(point.x) * cosine -
            static_cast<long double>(point.y) * sine,
        static_cast<long double>(point.x) * sine +
            static_cast<long double>(point.y) * cosine
    );
}

template <Coordinate T>
Point<long double> normalized(const Point<T>& point) {
    long double length = norm(point);
    assert(length != 0);
    return Point<long double>(
        static_cast<long double>(point.x) / length,
        static_cast<long double>(point.y) / length
    );
}

}  // namespace geometry
}  // namespace m1une


#line 10 "geometry/convex_hull.hpp"

namespace m1une {
namespace geometry {

// Returns the convex hull counterclockwise from its lexicographically smallest
// point. The first point is not repeated at the end.
template <Coordinate T>
std::vector<Point<T>> convex_hull(
    std::vector<Point<T>> points,
    bool include_collinear = false
) {
    std::sort(points.begin(), points.end());
    points.erase(std::unique(points.begin(), points.end()), points.end());
    std::size_t size = points.size();
    if (size <= 1) return points;

    std::vector<Point<T>> hull;
    hull.reserve(2 * size);
    auto should_pop = [include_collinear](
        const Point<T>& first,
        const Point<T>& second,
        const Point<T>& third
    ) {
        int turn = orientation(first, second, third);
        return include_collinear ? turn < 0 : turn <= 0;
    };

    for (const Point<T>& point : points) {
        while (
            hull.size() >= 2 &&
            should_pop(hull[hull.size() - 2], hull.back(), point)
        ) {
            hull.pop_back();
        }
        hull.push_back(point);
    }

    std::size_t lower_size = hull.size();
    for (std::size_t index = size - 1; index-- > 0;) {
        const Point<T>& point = points[index];
        while (
            hull.size() > lower_size &&
            should_pop(hull[hull.size() - 2], hull.back(), point)
        ) {
            hull.pop_back();
        }
        hull.push_back(point);
    }
    hull.pop_back();

    if (include_collinear && hull.size() == 2 * points.size() - 2) {
        hull = std::move(points);
    }
    return hull;
}

}  // namespace geometry
}  // namespace m1une


#line 1 "geometry/convex_layers.hpp"



#line 9 "geometry/convex_layers.hpp"

#line 11 "geometry/convex_layers.hpp"

namespace m1une {
namespace geometry {

namespace convex_layers_detail {

template <Coordinate T>
struct LayerPoint {
    wide_type<T> x;
    wide_type<T> y;
};

template <Coordinate T>
wide_type<T> layer_cross(
    const LayerPoint<T>& first,
    const LayerPoint<T>& second,
    const LayerPoint<T>& third
) {
    return
        (second.x - first.x) * (third.y - first.y) -
        (second.y - first.y) * (third.x - first.x);
}

template <Coordinate T>
class DecrementalHull {
   private:
    struct Node {
        int left_bound;
        int right_bound;
        int bridge_left;
        int bridge_right;
        int left_child;
        int right_child;
    };

    std::vector<LayerPoint<T>> points;
    std::vector<Node> nodes;
    int root;

    bool is_leaf(int node) const {
        return nodes[node].left_child == -1 && nodes[node].right_child == -1;
    }

    void pull(int node) {
        int left = nodes[node].left_child;
        int right = nodes[node].right_child;
        assert(left != -1 && right != -1);
        using Wide = wide_type<T>;
        const Wide split_y = points[nodes[right].left_bound].y;

        while (!is_leaf(left) || !is_leaf(right)) {
            const int a = nodes[left].bridge_left;
            const int b = nodes[left].bridge_right;
            const int c = nodes[right].bridge_left;
            const int d = nodes[right].bridge_right;

            if (
                a != b &&
                sign<T>(layer_cross<T>(points[a], points[b], points[c])) > 0
            ) {
                left = nodes[left].left_child;
            } else if (
                c != d &&
                sign<T>(layer_cross<T>(points[b], points[c], points[d])) > 0
            ) {
                right = nodes[right].right_child;
            } else if (a == b) {
                right = nodes[right].left_child;
            } else if (c == d) {
                left = nodes[left].right_child;
            } else {
                const Wide first =
                    layer_cross<T>(points[a], points[b], points[c]);
                const Wide second =
                    layer_cross<T>(points[b], points[a], points[d]);
                const Wide sum = first + second;
                assert(sign<T>(sum) >= 0);
                const Wide comparison =
                    first * points[d].y + second * points[c].y - split_y * sum;
                if (sign<T>(sum) == 0 || sign<T>(comparison) < 0) {
                    left = nodes[left].right_child;
                } else {
                    right = nodes[right].left_child;
                }
            }
        }
        nodes[node].bridge_left = nodes[left].left_bound;
        nodes[node].bridge_right = nodes[right].left_bound;
    }

    void build(int node, int left, int right) {
        nodes[node].left_bound = left;
        nodes[node].right_bound = right;
        if (right - left == 1) {
            nodes[node].bridge_left = left;
            nodes[node].bridge_right = left;
            nodes[node].left_child = -1;
            nodes[node].right_child = -1;
            return;
        }

        const int middle = (left + right) / 2;
        nodes[node].left_child = node + 1;
        nodes[node].right_child = node + 2 * (middle - left);
        build(nodes[node].left_child, left, middle);
        build(nodes[node].right_child, middle, right);
        pull(node);
    }

    int erase(int node, int position) {
        if (
            position < nodes[node].left_bound ||
            nodes[node].right_bound <= position
        ) {
            return node;
        }
        if (nodes[node].right_bound - nodes[node].left_bound == 1) return -1;

        nodes[node].left_child = erase(nodes[node].left_child, position);
        nodes[node].right_child = erase(nodes[node].right_child, position);
        if (nodes[node].left_child == -1) return nodes[node].right_child;
        if (nodes[node].right_child == -1) return nodes[node].left_child;
        pull(node);
        return node;
    }

    void collect(
        int node,
        int left,
        int right,
        std::vector<int>& result
    ) const {
        if (is_leaf(node)) {
            result.push_back(nodes[node].left_bound);
        } else if (right <= nodes[node].bridge_left) {
            collect(nodes[node].left_child, left, right, result);
        } else if (nodes[node].bridge_right <= left) {
            collect(nodes[node].right_child, left, right, result);
        } else {
            assert(
                left <= nodes[node].bridge_left &&
                nodes[node].bridge_right <= right
            );
            collect(
                nodes[node].left_child,
                left,
                nodes[node].bridge_left,
                result
            );
            collect(
                nodes[node].right_child,
                nodes[node].bridge_right,
                right,
                result
            );
        }
    }

   public:
    explicit DecrementalHull(std::vector<LayerPoint<T>> ordered_points)
        : points(std::move(ordered_points)),
          nodes(2 * points.size()),
          root(points.empty() ? -1 : 0) {
        if (!points.empty()) build(0, 0, int(points.size()));
    }

    std::vector<int> hull() const {
        std::vector<int> result;
        if (root != -1) collect(root, 0, int(points.size()) - 1, result);
        return result;
    }

    void erase(int position) {
        assert(root != -1);
        assert(0 <= position && position < int(points.size()));
        root = erase(root, position);
    }
};

}  // namespace convex_layers_detail

template <Coordinate T>
std::vector<int> convex_layers(const std::vector<Point<T>>& points) {
    const int n = int(points.size());
    if (n == 0) return {};

    struct IndexedPoint {
        Point<T> point;
        int original_index;
    };
    std::vector<IndexedPoint> indexed;
    indexed.reserve(n);
    for (int index = 0; index < n; index++) {
        indexed.push_back(IndexedPoint{points[index], index});
    }
    std::sort(
        indexed.begin(),
        indexed.end(),
        [](const IndexedPoint& first, const IndexedPoint& second) {
            if (first.point.y != second.point.y) {
                return first.point.y < second.point.y;
            }
            if (first.point.x != second.point.x) {
                return first.point.x < second.point.x;
            }
            return first.original_index < second.original_index;
        }
    );

    std::vector<Point<T>> ordered;
    std::vector<int> position(n);
    ordered.reserve(n);
    for (const IndexedPoint& item : indexed) {
        if (ordered.empty() || !(ordered.back() == item.point)) {
            ordered.push_back(item.point);
        }
        position[item.original_index] = int(ordered.size()) - 1;
    }

    using LayerPoint = convex_layers_detail::LayerPoint<T>;
    using Wide = wide_type<T>;
    std::vector<LayerPoint> left_points;
    left_points.reserve(ordered.size());
    for (const Point<T>& point : ordered) {
        left_points.push_back(LayerPoint{Wide(point.x), Wide(point.y)});
    }
    convex_layers_detail::DecrementalHull<T> left_hull(
        std::move(left_points)
    );

    std::vector<LayerPoint> reversed;
    reversed.reserve(ordered.size());
    for (auto iterator = ordered.rbegin(); iterator != ordered.rend(); ++iterator) {
        reversed.push_back(LayerPoint{-Wide(iterator->x), -Wide(iterator->y)});
    }
    convex_layers_detail::DecrementalHull<T> right_hull(std::move(reversed));

    const int distinct_count = int(ordered.size());
    std::vector<int> layer_by_position(distinct_count, 0);
    std::vector<int> selected_in_layer(distinct_count, 0);
    int remaining = distinct_count;
    for (int layer = 1; remaining > 0; layer++) {
        std::vector<int> boundary;
        auto add_boundary = [&](int ordered_position) {
            if (selected_in_layer[ordered_position] == layer) return;
            selected_in_layer[ordered_position] = layer;
            boundary.push_back(ordered_position);
        };
        for (int ordered_position : left_hull.hull()) {
            add_boundary(ordered_position);
        }
        for (int reversed_position : right_hull.hull()) {
            add_boundary(distinct_count - 1 - reversed_position);
        }

        assert(!boundary.empty());
        for (int ordered_position : boundary) {
            layer_by_position[ordered_position] = layer;
            left_hull.erase(ordered_position);
            right_hull.erase(distinct_count - 1 - ordered_position);
            remaining--;
        }
    }

    std::vector<int> result(n);
    for (int index = 0; index < n; index++) {
        result[index] = layer_by_position[position[index]];
    }
    return result;
}

}  // namespace geometry
}  // namespace m1une


#line 13 "verify/geometry/convex_layers.test.cpp"

namespace {

using Point = m1une::geometry::Point<long long>;

std::vector<int> naive_convex_layers(const std::vector<Point>& points) {
    const int n = int(points.size());
    std::vector<int> result(n, 0);
    int remaining = n;
    for (int layer = 1; remaining > 0; layer++) {
        std::vector<Point> active;
        for (int index = 0; index < n; index++) {
            if (result[index] == 0) active.push_back(points[index]);
        }
        std::vector<Point> hull = m1une::geometry::convex_hull(active, true);
        for (int index = 0; index < n; index++) {
            if (result[index] != 0) continue;
            if (std::find(hull.begin(), hull.end(), points[index]) == hull.end()) {
                continue;
            }
            result[index] = layer;
            remaining--;
        }
    }
    return result;
}

void test_randomized() {
    assert(m1une::geometry::convex_layers(std::vector<Point>()).empty());

    std::uint64_t state = 0x3141592653589793ULL;
    auto random = [&]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };
    for (int test = 0; test < 500; test++) {
        const int n = int(random() % 45);
        std::vector<Point> points;
        points.reserve(n);
        for (int index = 0; index < n; index++) {
            points.emplace_back(
                static_cast<long long>(random() % 17) - 8,
                static_cast<long long>(random() % 17) - 8
            );
        }
        assert(
            m1une::geometry::convex_layers(points) ==
            naive_convex_layers(points)
        );
    }
}

void test_coordinate_types() {
    std::vector<m1une::geometry::Point<long double>> floating;
    floating.emplace_back(0.5L, 0.5L);
    floating.emplace_back(4.5L, 0.5L);
    floating.emplace_back(4.5L, 4.5L);
    floating.emplace_back(0.5L, 4.5L);
    floating.emplace_back(2.5L, 2.5L);
    std::vector<int> expected_floating = {1, 1, 1, 1, 2};
    assert(m1une::geometry::convex_layers(floating) == expected_floating);

    const long long base = std::numeric_limits<long long>::lowest();
    std::vector<Point> extreme;
    extreme.emplace_back(base, base);
    extreme.emplace_back(base + 4, base);
    extreme.emplace_back(base + 4, base + 4);
    extreme.emplace_back(base, base + 4);
    extreme.emplace_back(base + 2, base + 2);
    std::vector<int> expected_extreme = {1, 1, 1, 1, 2};
    assert(m1une::geometry::convex_layers(extreme) == expected_extreme);
}

}  // namespace

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

    test_coordinate_types();
    test_randomized();
    int point_count;
    fast_input >> point_count;
    std::vector<Point> points(point_count);
    for (Point& point : points) fast_input >> point.x >> point.y;

    for (int layer : m1une::geometry::convex_layers(points)) {
        fast_output << layer << '\n';
    }
}

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