verify/geometry/delaunay_triangulation.test.cpp

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

Depends on

• DSU (Disjoint Set Union) (ds/dsu/dsu.hpp)
• DSU (Disjoint Set Union) (ds/dsu/dsu.hpp)
• Convex Hull (geometry/convex_hull.hpp)
• Delaunay Triangulation (geometry/delaunay_triangulation.hpp)
• geometry/detail/floating_predicate.hpp
• Euclidean Minimum Spanning Tree (geometry/euclidean_mst.hpp)
• 2D Point and Predicates (geometry/point.hpp)
• Fast IO (utilities/fast_io.hpp)

Code

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

#include "../../geometry/convex_hull.hpp"
#include "../../geometry/delaunay_triangulation.hpp"

#include <algorithm>
#include <array>
#include <cassert>
#include <cstdint>
#include <numeric>
#include <random>
#include <set>
#include <utility>
#include <vector>

#include "../../ds/dsu/dsu.hpp"
#include "../../utilities/fast_io.hpp"

namespace {

using Point = m1une::geometry::Point<long long>;
using Wide = __int128_t;

bool inside_circumcircle(
    const Point& first,
    const Point& second,
    const Point& third,
    const Point& point
) {
    Wide ax = Wide(first.x) - Wide(point.x);
    Wide ay = Wide(first.y) - Wide(point.y);
    Wide bx = Wide(second.x) - Wide(point.x);
    Wide by = Wide(second.y) - Wide(point.y);
    Wide cx = Wide(third.x) - Wide(point.x);
    Wide cy = Wide(third.y) - Wide(point.y);
    Wide first_norm = ax * ax + ay * ay;
    Wide second_norm = bx * bx + by * by;
    Wide third_norm = cx * cx + cy * cy;
    Wide determinant =
        (bx * cy - by * cx) * first_norm +
        (cx * ay - cy * ax) * second_norm +
        (ax * by - ay * bx) * third_norm;
    return determinant > 0;
}

void check(const std::vector<Point>& points) {
    auto triangulation = m1une::geometry::delaunay_triangulation(points);
    assert(std::is_sorted(
        triangulation.edges.begin(),
        triangulation.edges.end()
    ));
    assert(std::adjacent_find(
        triangulation.edges.begin(),
        triangulation.edges.end()
    ) == triangulation.edges.end());
    assert(std::is_sorted(
        triangulation.triangles.begin(),
        triangulation.triangles.end()
    ));
    assert(std::adjacent_find(
        triangulation.triangles.begin(),
        triangulation.triangles.end()
    ) == triangulation.triangles.end());

    std::set<std::pair<int, int>> edge_set;
    for (auto [first, second] : triangulation.edges) {
        assert(0 <= first && first < int(points.size()));
        assert(0 <= second && second < int(points.size()));
        assert(first < second);
        edge_set.emplace(first, second);
    }

    std::vector<int> incidence(triangulation.edges.size(), 0);
    Wide triangle_area_twice = 0;
    for (const auto& triangle : triangulation.triangles) {
        int first = triangle[0];
        int second = triangle[1];
        int third = triangle[2];
        assert(first < second && first < third);
        Wide area = m1une::geometry::cross(
            points[first],
            points[second],
            points[third]
        );
        assert(area > 0);
        triangle_area_twice += area;

        for (int side = 0; side < 3; ++side) {
            int from = triangle[side];
            int to = triangle[(side + 1) % 3];
            if (from > to) std::swap(from, to);
            auto iterator = edge_set.find(std::pair(from, to));
            assert(iterator != edge_set.end());
            auto sorted_iterator = std::lower_bound(
                triangulation.edges.begin(),
                triangulation.edges.end(),
                *iterator
            );
            ++incidence[sorted_iterator - triangulation.edges.begin()];
        }
        for (const Point& point : points) {
            assert(!inside_circumcircle(
                points[first],
                points[second],
                points[third],
                point
            ));
        }
    }

    std::vector<Point> hull = m1une::geometry::convex_hull(points, true);
    bool non_collinear = false;
    for (int index = 2; index < int(points.size()); ++index) {
        if (m1une::geometry::orientation(
                points[0],
                points[1],
                points[index]
            ) != 0) {
            non_collinear = true;
        }
    }
    if (!non_collinear) {
        assert(triangulation.triangles.empty());
        assert(triangulation.edges.size() ==
               (points.empty() ? 0 : points.size() - 1));
        return;
    }

    int size = int(points.size());
    int hull_size = int(hull.size());
    assert(int(triangulation.edges.size()) == 3 * size - 3 - hull_size);
    assert(int(triangulation.triangles.size()) == 2 * size - 2 - hull_size);
    for (int count : incidence) assert(count == 1 || count == 2);

    Wide hull_area_twice = 0;
    for (int index = 0; index < hull_size; ++index) {
        hull_area_twice += m1une::geometry::cross(
            hull[index],
            hull[(index + 1) % hull_size]
        );
    }
    assert(triangle_area_twice == hull_area_twice);
}

void test_fixed() {
    check({});
    check({Point(3, -2)});
    check({Point(0, 0), Point(7, 1)});
    check({
        Point(-5, 0),
        Point(-2, 0),
        Point(1, 0),
        Point(7, 0),
    });
    check({Point(0, 0), Point(4, 0), Point(1, 3)});
    check({
        Point(0, 0),
        Point(2, 0),
        Point(2, 2),
        Point(0, 2),
    });
    check({
        Point(5, 0),
        Point(3, 4),
        Point(0, 5),
        Point(-3, 4),
        Point(-5, 0),
        Point(-3, -4),
        Point(0, -5),
        Point(3, -4),
    });
    check({
        Point(0, 0),
        Point(2, 0),
        Point(4, 0),
        Point(4, 3),
        Point(0, 3),
        Point(2, 1),
    });
    check({
        Point(0, 0),
        Point(4, 0),
        Point(5, 3),
        Point(2, 6),
        Point(-1, 3),
        Point(2, 3),
    });
}

void test_randomized() {
    std::uint64_t state = 0xd31a'2026'0722ULL;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int trial = 0; trial < 3000; ++trial) {
        int size = int(random() % 15);
        std::set<std::pair<long long, long long>> coordinates;
        while (int(coordinates.size()) < size) {
            coordinates.emplace(
                static_cast<long long>(random() % 21) - 10,
                static_cast<long long>(random() % 21) - 10
            );
        }
        std::vector<Point> points;
        for (auto [x, y] : coordinates) points.emplace_back(x, y);
        std::shuffle(
            points.begin(),
            points.end(),
            std::mt19937_64(random())
        );
        check(points);
    }
}

std::vector<std::pair<int, int>> delaunay_mst(
    const std::vector<Point>& points
) {
    struct Edge {
        int from;
        int to;
        Wide squared_distance;
    };

    std::vector<int> order(points.size());
    std::iota(order.begin(), order.end(), 0);
    std::sort(order.begin(), order.end(), [&](int first, int second) {
        return points[first] < points[second];
    });

    std::vector<Point> unique_points;
    std::vector<int> representatives;
    std::vector<Edge> candidates;
    unique_points.reserve(points.size());
    representatives.reserve(points.size());
    candidates.reserve(3 * points.size());
    for (int index : order) {
        if (unique_points.empty() || unique_points.back() != points[index]) {
            unique_points.push_back(points[index]);
            representatives.push_back(index);
        } else {
            candidates.push_back(Edge{index, representatives.back(), 0});
        }
    }

    auto triangulation =
        m1une::geometry::delaunay_triangulation(unique_points);
    for (auto [first, second] : triangulation.edges) {
        int from = representatives[first];
        int to = representatives[second];
        candidates.push_back(Edge{
            from,
            to,
            m1une::geometry::distance2(points[from], points[to]),
        });
    }
    std::sort(
        candidates.begin(),
        candidates.end(),
        [](const Edge& left, const Edge& right) {
            if (left.squared_distance != right.squared_distance) {
                return left.squared_distance < right.squared_distance;
            }
            if (left.from != right.from) return left.from < right.from;
            return left.to < right.to;
        }
    );

    m1une::ds::Dsu dsu(int(points.size()));
    std::vector<std::pair<int, int>> result;
    for (const Edge& edge : candidates) {
        if (dsu.same(edge.from, edge.to)) continue;
        dsu.merge(edge.from, edge.to);
        result.emplace_back(edge.from, edge.to);
    }
    assert(points.empty() || result.size() + 1 == points.size());
    return result;
}

}  // namespace

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

    test_fixed();
    test_randomized();

    int size;
    fast_input >> size;
    std::vector<Point> points(size);
    for (Point& point : points) fast_input >> point.x >> point.y;
    for (auto [from, to] : delaunay_mst(points)) {
        fast_output << from << ' ' << to << '\n';
    }
}

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

#line 1 "geometry/convex_hull.hpp"



#include <algorithm>
#include <cstddef>
#include <utility>
#include <vector>

#line 1 "geometry/point.hpp"



#include <cmath>
#include <concepts>
#include <cassert>
#include <type_traits>

#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/delaunay_triangulation.hpp"



#line 5 "geometry/delaunay_triangulation.hpp"
#include <array>
#line 10 "geometry/delaunay_triangulation.hpp"

#line 1 "geometry/euclidean_mst.hpp"



#line 9 "geometry/euclidean_mst.hpp"
#include <limits>
#include <tuple>
#line 13 "geometry/euclidean_mst.hpp"

#line 1 "ds/dsu/dsu.hpp"



#line 5 "ds/dsu/dsu.hpp"
#include <numeric>
#line 8 "ds/dsu/dsu.hpp"

namespace m1une {
namespace ds {

struct Dsu {
   private:
    int _n;
    // parent_or_size[i] is the parent of i if it's >= 0.
    // If it's < 0, then i is a root and -parent_or_size[i] is the size of the group.
    std::vector<int> parent_or_size;

    // Returns {new leader, absorbed leader}. The absorbed leader is -1 when
    // both vertices already belong to the same component.
    std::pair<int, int> merge_leaders(int a, int b) {
        int x = leader(a), y = leader(b);
        if (x == y) return {x, -1};
        if (-parent_or_size[x] < -parent_or_size[y]) std::swap(x, y);
        parent_or_size[x] += parent_or_size[y];
        parent_or_size[y] = x;
        return {x, y};
    }

   public:
    Dsu() : _n(0) {}
    explicit Dsu(int n) : _n(n), parent_or_size(n, -1) {}

    // Merges the group containing 'a' with the group containing 'b'.
    // Returns the leader of the merged group.
    int merge(int a, int b) {
        return merge_leaders(a, b).first;
    }

    // Invokes callback(new_leader, absorbed_leader) after an actual merge.
    // Returns the leader of the merged group.
    template <class Callback>
    int merge(int a, int b, Callback&& callback) {
        std::pair<int, int> merged = merge_leaders(a, b);
        if (merged.second != -1) callback(merged.first, merged.second);
        return merged.first;
    }

    // Returns true if 'a' and 'b' belong to the same group.
    bool same(int a, int b) {
        return leader(a) == leader(b);
    }

    // Returns the leader (representative) of the group containing 'a'.
    int leader(int a) {
        if (parent_or_size[a] < 0) return a;
        // Path compression
        return parent_or_size[a] = leader(parent_or_size[a]);
    }

    // Returns the size of the group containing 'a'.
    int size(int a) {
        return -parent_or_size[leader(a)];
    }

    // Returns a list of all groups, where each group is a vector of its elements.
    std::vector<std::vector<int>> groups() {
        std::vector<int> leader_buf(_n), group_size(_n);
        for (int i = 0; i < _n; i++) {
            leader_buf[i] = leader(i);
            group_size[leader_buf[i]]++;
        }
        std::vector<std::vector<int>> result(_n);
        for (int i = 0; i < _n; i++) {
            result[i].reserve(group_size[i]);
        }
        for (int i = 0; i < _n; i++) {
            result[leader_buf[i]].push_back(i);
        }
        result.erase(std::remove_if(result.begin(), result.end(), [&](const std::vector<int>& v) { return v.empty(); }),
                     result.end());
        return result;
    }
};

}  // namespace ds
}  // namespace m1une


#line 16 "geometry/euclidean_mst.hpp"

namespace m1une {
namespace geometry {

template <class T>
struct EuclideanMstEdge {
    int from;
    int to;
    T squared_distance;
};

template <class T>
struct EuclideanMst {
    long double cost;
    std::vector<EuclideanMstEdge<T>> edges;
};

namespace detail {

template <ExactCoordinate T>
class EuclideanDelaunay {
   private:
    using W = wide_type<T>;

    struct InternalPoint {
        W x;
        W y;

        friend bool operator==(const InternalPoint&, const InternalPoint&) = default;
    };

    struct Edge {
        int to;
        int ccw;
        int cw;
        int reverse;
        bool enabled = false;
    };

    std::vector<int> open_addresses;
    std::vector<InternalPoint> points;
    std::vector<Edge> edges;
    std::vector<int> duplicate_representative;

    static InternalPoint subtract(const InternalPoint& a, const InternalPoint& b) {
        return InternalPoint{a.x - b.x, a.y - b.y};
    }

    static W cross_product(const InternalPoint& a, const InternalPoint& b) {
        return a.x * b.y - a.y * b.x;
    }

    static W squared_norm(const InternalPoint& point) {
        return point.x * point.x + point.y * point.y;
    }

    static bool inside_circumcircle(
        InternalPoint a,
        InternalPoint b,
        InternalPoint c,
        const InternalPoint& d
    ) {
        a = subtract(a, d);
        b = subtract(b, d);
        c = subtract(c, d);
        W determinant = cross_product(b, c) * squared_norm(a)
                      + cross_product(c, a) * squared_norm(b)
                      + cross_product(a, b) * squared_norm(c);
        return determinant > 0;
    }

    int get_open_address() {
        if (open_addresses.empty()) {
            edges.push_back(Edge());
            return int(edges.size()) - 1;
        }
        int result = open_addresses.back();
        open_addresses.pop_back();
        return result;
    }

    std::pair<int, int> add_edge(int from, int to) {
        int forward = get_open_address();
        int backward = get_open_address();
        edges[forward].to = to;
        edges[forward].ccw = forward;
        edges[forward].cw = forward;
        edges[forward].reverse = backward;
        edges[forward].enabled = true;
        edges[backward].to = from;
        edges[backward].ccw = backward;
        edges[backward].cw = backward;
        edges[backward].reverse = forward;
        edges[backward].enabled = true;
        return {forward, backward};
    }

    void erase_directed_edge(int edge) {
        int ccw = edges[edge].ccw;
        int cw = edges[edge].cw;
        edges[ccw].cw = cw;
        edges[cw].ccw = ccw;
        edges[edge].enabled = false;
    }

    void erase_edge(int edge) {
        int reverse = edges[edge].reverse;
        erase_directed_edge(edge);
        erase_directed_edge(reverse);
        open_addresses.push_back(edge);
        open_addresses.push_back(reverse);
    }

    void insert_ccw_after(int edge, int position) {
        int next = edges[position].ccw;
        edges[edge].ccw = next;
        edges[next].cw = edge;
        edges[edge].cw = position;
        edges[position].ccw = edge;
    }

    void insert_cw_after(int edge, int position) {
        int next = edges[position].cw;
        edges[edge].cw = next;
        edges[next].ccw = edge;
        edges[edge].ccw = position;
        edges[position].cw = edge;
    }

    int orientation(int a, int b, int c) const {
        InternalPoint ab = subtract(points[b], points[a]);
        InternalPoint ac = subtract(points[c], points[a]);
        W value = cross_product(ab, ac);
        return (value > 0) - (value < 0);
    }

    std::pair<int, int> go_next(int edge) const {
        int vertex = edges[edge].to;
        int next_edge = edges[edges[edge].reverse].ccw;
        return {vertex, next_edge};
    }

    std::pair<int, int> go_previous(int edge) const {
        int vertex = edges[edges[edge].cw].to;
        int next_edge = edges[edges[edge].cw].reverse;
        return {vertex, next_edge};
    }

    std::tuple<int, int, int, int> lower_tangent(
        int left_vertex,
        int left_edge,
        int right_vertex,
        int right_edge
    ) const {
        while (true) {
            auto [next_left_vertex, next_left_edge] = go_previous(left_edge);
            if (orientation(right_vertex, left_vertex, next_left_vertex) > 0) {
                left_vertex = next_left_vertex;
                left_edge = next_left_edge;
                continue;
            }
            auto [next_right_vertex, next_right_edge] = go_next(right_edge);
            if (orientation(left_vertex, right_vertex, next_right_vertex) < 0) {
                right_vertex = next_right_vertex;
                right_edge = next_right_edge;
                continue;
            }
            break;
        }
        return {left_vertex, left_edge, right_vertex, right_edge};
    }

    std::pair<int, int> extreme_vertex(int vertex, int edge, bool minimum) const {
        std::pair<int, int> result = {vertex, edge};
        int current_vertex = vertex;
        int current_edge = edge;
        do {
            std::tie(current_vertex, current_edge) = go_next(current_edge);
            std::pair<int, int> candidate = {current_vertex, current_edge};
            if ((minimum && candidate < result) || (!minimum && result < candidate)) {
                result = candidate;
            }
        } while (current_edge != edge);
        return result;
    }

    bool inside_circumcircle(int a, int b, int c, int d) const {
        return inside_circumcircle(points[a], points[b], points[c], points[d]);
    }

    std::pair<int, int> merge_triangulations(
        int left_vertex,
        int left_edge,
        int right_vertex,
        int right_edge
    ) {
        std::tie(left_vertex, left_edge) = extreme_vertex(left_vertex, left_edge, false);
        std::tie(right_vertex, right_edge) = extreme_vertex(right_vertex, right_edge, true);

        auto [lower_left, lower_left_edge, lower_right, lower_right_edge]
            = lower_tangent(left_vertex, left_edge, right_vertex, right_edge);
        auto [upper_right, upper_right_edge, upper_left, upper_left_edge]
            = lower_tangent(right_vertex, right_edge, left_vertex, left_edge);
        lower_right_edge = edges[lower_right_edge].cw;
        upper_right_edge = edges[upper_right_edge].cw;

        auto [base, reverse_base] = add_edge(lower_left, lower_right);
        insert_cw_after(base, lower_left_edge);
        insert_ccw_after(reverse_base, lower_right_edge);
        if (lower_left == upper_left) upper_left_edge = base;
        if (lower_right == upper_right) upper_right_edge = reverse_base;

        int left = lower_left;
        int left_candidate = lower_left_edge;
        int right = lower_right;
        int right_candidate = lower_right_edge;
        while (left != upper_left || right != upper_right) {
            int next_left = edges[left_candidate].to;
            int next_right = edges[right_candidate].to;
            int next_left_candidate = edges[left_candidate].ccw;
            int next_right_candidate = edges[right_candidate].cw;

            if (left_candidate != upper_left_edge && next_left_candidate != base) {
                int second_left = edges[next_left_candidate].to;
                if (inside_circumcircle(left, right, next_left, second_left)) {
                    erase_edge(left_candidate);
                    left_candidate = next_left_candidate;
                    continue;
                }
            }

            if (right_candidate != upper_right_edge && next_right_candidate != reverse_base) {
                int second_right = edges[next_right_candidate].to;
                if (inside_circumcircle(next_right, left, right, second_right)) {
                    erase_edge(right_candidate);
                    right_candidate = next_right_candidate;
                    continue;
                }
            }

            bool choose_left = right_candidate == upper_right_edge;
            if (left_candidate != upper_left_edge && right_candidate != upper_right_edge) {
                if (orientation(left, right, next_right) < 0) {
                    choose_left = true;
                } else if (orientation(next_left, left, right) < 0) {
                    choose_left = false;
                } else {
                    choose_left = inside_circumcircle(left, right, next_right, next_left);
                }
            }

            if (choose_left) {
                next_left_candidate = edges[edges[left_candidate].reverse].ccw;
                auto [new_base, new_reverse_base] = add_edge(next_left, right);
                insert_cw_after(new_base, next_left_candidate);
                insert_ccw_after(new_reverse_base, right_candidate);
                left_candidate = next_left_candidate;
                left = next_left;
            } else {
                next_right_candidate = edges[edges[right_candidate].reverse].cw;
                auto [new_reverse_base, new_base] = add_edge(next_right, left);
                insert_ccw_after(new_reverse_base, next_right_candidate);
                insert_cw_after(new_base, left_candidate);
                right_candidate = next_right_candidate;
                right = next_right;
            }
        }
        return {lower_left, base};
    }

    std::pair<int, int> solve_range(int left, int right) {
        if (right - left == 2) {
            auto [forward, backward] = add_edge(left, left + 1);
            (void)backward;
            return {left, forward};
        }
        if (right - left == 3) {
            int middle = left + 1;
            int last = left + 2;
            auto [first_middle, middle_first] = add_edge(left, middle);
            auto [middle_last, last_middle] = add_edge(middle, last);
            int direction = orientation(left, middle, last);
            if (direction == 0) {
                insert_ccw_after(middle_first, middle_last);
                return {left, first_middle};
            }

            auto [first_last, last_first] = add_edge(left, last);
            if (direction > 0) {
                insert_cw_after(first_middle, first_last);
                insert_cw_after(middle_last, middle_first);
                insert_cw_after(last_first, last_middle);
                return {left, first_middle};
            }
            insert_ccw_after(first_middle, first_last);
            insert_ccw_after(middle_last, middle_first);
            insert_ccw_after(last_first, last_middle);
            return {middle, middle_first};
        }

        int middle = (left + right) / 2;
        auto [left_vertex, left_edge] = solve_range(left, middle);
        auto [right_vertex, right_edge] = solve_range(middle, right);
        return merge_triangulations(left_vertex, left_edge, right_vertex, right_edge);
    }

    void solve() {
        int size = int(points.size());
        if (size <= 1) return;

        std::vector<int> order(size);
        for (int i = 0; i < size; i++) order[i] = i;
        std::stable_sort(order.begin(), order.end(), [&](int left, int right) {
            if (points[left].x != points[right].x) {
                return points[left].x < points[right].x;
            }
            return points[left].y < points[right].y;
        });

        std::vector<InternalPoint> original_points = points;
        duplicate_representative.assign(size, 0);
        int unique_size = 0;
        for (int i = 0; i < size; i++) {
            int vertex = order[i];
            if (i == 0 || !(original_points[order[unique_size - 1]] == original_points[vertex])) {
                order[unique_size] = vertex;
                points[unique_size] = original_points[vertex];
                unique_size++;
                duplicate_representative[vertex] = vertex;
            } else {
                duplicate_representative[vertex] = order[unique_size - 1];
            }
        }

        if (unique_size >= 2) solve_range(0, unique_size);
        points.swap(original_points);
        for (auto& edge : edges) edge.to = order[edge.to];
    }

   public:
    explicit EuclideanDelaunay(const std::vector<Point<T>>& input_points) {
        assert(input_points.size() <= std::size_t(std::numeric_limits<int>::max()));
        points.reserve(input_points.size());
        edges.reserve(std::size_t(6) * input_points.size());
        for (const auto& point : input_points) {
            points.push_back(InternalPoint{W(point.x), W(point.y)});
        }
        solve();
    }

    bool has_duplicates() const {
        for (
            int vertex = 0;
            vertex < int(duplicate_representative.size());
            ++vertex
        ) {
            if (duplicate_representative[vertex] != vertex) return true;
        }
        return false;
    }

    std::vector<std::pair<int, int>> get_edges() const {
        std::vector<std::pair<int, int>> result;
        result.reserve(edges.size() / 2 + duplicate_representative.size());
        for (int edge = 0; edge < int(edges.size()); edge++) {
            if (!edges[edge].enabled) continue;
            int reverse = edges[edge].reverse;
            if (edge < reverse) continue;
            result.emplace_back(edges[edge].to, edges[reverse].to);
        }
        for (int vertex = 0; vertex < int(duplicate_representative.size()); vertex++) {
            if (duplicate_representative[vertex] != vertex) {
                result.emplace_back(vertex, duplicate_representative[vertex]);
            }
        }
        return result;
    }
};

}  // namespace detail

// Returns O(n) Delaunay edges containing a Euclidean minimum spanning tree.
template <ExactCoordinate T>
std::vector<EuclideanMstEdge<wide_type<T>>> euclidean_mst_edges(
    const std::vector<Point<T>>& points
) {
    using W = wide_type<T>;
    auto delaunay_edges = detail::EuclideanDelaunay<T>(points).get_edges();
    std::vector<EuclideanMstEdge<W>> result;
    result.reserve(delaunay_edges.size());
    for (auto [from, to] : delaunay_edges) {
        result.push_back(EuclideanMstEdge<W>{from, to, distance2(points[from], points[to])});
    }
    return result;
}

// Returns a Euclidean minimum spanning tree.
template <ExactCoordinate T>
EuclideanMst<wide_type<T>> euclidean_mst(const std::vector<Point<T>>& points) {
    using W = wide_type<T>;
    auto candidates = euclidean_mst_edges(points);
    std::sort(candidates.begin(), candidates.end(), [](const auto& left, const auto& right) {
        if (left.squared_distance != right.squared_distance) {
            return left.squared_distance < right.squared_distance;
        }
        if (left.from != right.from) return left.from < right.from;
        return left.to < right.to;
    });

    m1une::ds::Dsu dsu(int(points.size()));
    EuclideanMst<W> result;
    result.cost = 0;
    result.edges.reserve(points.empty() ? 0 : points.size() - 1);
    for (const auto& edge : candidates) {
        if (dsu.same(edge.from, edge.to)) continue;
        dsu.merge(edge.from, edge.to);
        result.cost += std::sqrt(static_cast<long double>(edge.squared_distance));
        result.edges.push_back(edge);
        if (result.edges.size() + 1 == points.size()) break;
    }
    assert(points.empty() || result.edges.size() + 1 == points.size());
    return result;
}

}  // namespace geometry
}  // namespace m1une


#line 12 "geometry/delaunay_triangulation.hpp"

namespace m1une {
namespace geometry {

struct DelaunayTriangulation {
    std::vector<std::pair<int, int>> edges;
    std::vector<std::array<int, 3>> triangles;
};

namespace delaunay_triangulation_detail {

template <ExactCoordinate T>
int direction_half(
    const Point<T>& origin,
    const Point<T>& destination
) {
    using W = wide_type<T>;
    W x = W(destination.x) - W(origin.x);
    W y = W(destination.y) - W(origin.y);
    return y > 0 || (y == 0 && x >= 0) ? 0 : 1;
}

template <ExactCoordinate T>
bool direction_less(
    const std::vector<Point<T>>& points,
    int origin,
    int first,
    int second
) {
    int first_half = direction_half(points[origin], points[first]);
    int second_half = direction_half(points[origin], points[second]);
    if (first_half != second_half) return first_half < second_half;

    using W = wide_type<T>;
    W first_x = W(points[first].x) - W(points[origin].x);
    W first_y = W(points[first].y) - W(points[origin].y);
    W second_x = W(points[second].x) - W(points[origin].x);
    W second_y = W(points[second].y) - W(points[origin].y);
    W product = first_x * second_y - first_y * second_x;
    if (product != 0) return product > 0;

    W first_norm = first_x * first_x + first_y * first_y;
    W second_norm = second_x * second_x + second_y * second_y;
    if (first_norm != second_norm) return first_norm < second_norm;
    return first < second;
}

inline void rotate_minimum_first(std::array<int, 3>& triangle) {
    int minimum = int(std::min_element(triangle.begin(), triangle.end()) -
                      triangle.begin());
    std::rotate(
        triangle.begin(),
        triangle.begin() + minimum,
        triangle.end()
    );
}

}  // namespace delaunay_triangulation_detail

// Constructs one Delaunay triangulation of distinct exact-coordinate points.
template <ExactCoordinate T>
DelaunayTriangulation delaunay_triangulation(
    const std::vector<Point<T>>& points
) {
    namespace detail = delaunay_triangulation_detail;

    DelaunayTriangulation result;
    geometry::detail::EuclideanDelaunay<T> builder(points);
    assert(!builder.has_duplicates());
    result.edges = builder.get_edges();
    for (auto& [first, second] : result.edges) {
        if (first > second) std::swap(first, second);
    }
    std::sort(result.edges.begin(), result.edges.end());
    result.edges.erase(
        std::unique(result.edges.begin(), result.edges.end()),
        result.edges.end()
    );

    std::vector<std::vector<int>> neighbors(points.size());
    for (auto [first, second] : result.edges) {
        neighbors[first].push_back(second);
        neighbors[second].push_back(first);
    }
    for (int point = 0; point < int(points.size()); ++point) {
        std::sort(
            neighbors[point].begin(),
            neighbors[point].end(),
            [&](int first, int second) {
                return detail::direction_less(points, point, first, second);
            }
        );
    }

    auto has_edge = [&](int first, int second) {
        if (first > second) std::swap(first, second);
        return std::binary_search(
            result.edges.begin(),
            result.edges.end(),
            std::pair(first, second)
        );
    };
    result.triangles.reserve(result.edges.size());
    for (int point = 0; point < int(points.size()); ++point) {
        int degree = int(neighbors[point].size());
        for (int index = 0; index < degree; ++index) {
            int first = neighbors[point][index];
            int second = neighbors[point][(index + 1) % degree];
            if (orientation(points[point], points[first], points[second]) <= 0) {
                continue;
            }
            if (!has_edge(first, second)) continue;
            std::array<int, 3> triangle{point, first, second};
            detail::rotate_minimum_first(triangle);
            result.triangles.push_back(triangle);
        }
    }
    std::sort(result.triangles.begin(), result.triangles.end());
    result.triangles.erase(
        std::unique(result.triangles.begin(), result.triangles.end()),
        result.triangles.end()
    );
    return result;
}

}  // namespace geometry
}  // namespace m1une


#line 5 "verify/geometry/delaunay_triangulation.test.cpp"

#line 9 "verify/geometry/delaunay_triangulation.test.cpp"
#include <cstdint>
#line 11 "verify/geometry/delaunay_triangulation.test.cpp"
#include <random>
#include <set>
#line 15 "verify/geometry/delaunay_triangulation.test.cpp"

#line 1 "utilities/fast_io.hpp"



#line 6 "utilities/fast_io.hpp"
#include <cerrno>
#include <charconv>
#line 9 "utilities/fast_io.hpp"
#include <cstdio>
#include <cstdlib>
#line 12 "utilities/fast_io.hpp"
#include <cstring>
#include <iterator>
#include <string>
#include <sys/stat.h>
#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 18 "verify/geometry/delaunay_triangulation.test.cpp"

namespace {

using Point = m1une::geometry::Point<long long>;
using Wide = __int128_t;

bool inside_circumcircle(
    const Point& first,
    const Point& second,
    const Point& third,
    const Point& point
) {
    Wide ax = Wide(first.x) - Wide(point.x);
    Wide ay = Wide(first.y) - Wide(point.y);
    Wide bx = Wide(second.x) - Wide(point.x);
    Wide by = Wide(second.y) - Wide(point.y);
    Wide cx = Wide(third.x) - Wide(point.x);
    Wide cy = Wide(third.y) - Wide(point.y);
    Wide first_norm = ax * ax + ay * ay;
    Wide second_norm = bx * bx + by * by;
    Wide third_norm = cx * cx + cy * cy;
    Wide determinant =
        (bx * cy - by * cx) * first_norm +
        (cx * ay - cy * ax) * second_norm +
        (ax * by - ay * bx) * third_norm;
    return determinant > 0;
}

void check(const std::vector<Point>& points) {
    auto triangulation = m1une::geometry::delaunay_triangulation(points);
    assert(std::is_sorted(
        triangulation.edges.begin(),
        triangulation.edges.end()
    ));
    assert(std::adjacent_find(
        triangulation.edges.begin(),
        triangulation.edges.end()
    ) == triangulation.edges.end());
    assert(std::is_sorted(
        triangulation.triangles.begin(),
        triangulation.triangles.end()
    ));
    assert(std::adjacent_find(
        triangulation.triangles.begin(),
        triangulation.triangles.end()
    ) == triangulation.triangles.end());

    std::set<std::pair<int, int>> edge_set;
    for (auto [first, second] : triangulation.edges) {
        assert(0 <= first && first < int(points.size()));
        assert(0 <= second && second < int(points.size()));
        assert(first < second);
        edge_set.emplace(first, second);
    }

    std::vector<int> incidence(triangulation.edges.size(), 0);
    Wide triangle_area_twice = 0;
    for (const auto& triangle : triangulation.triangles) {
        int first = triangle[0];
        int second = triangle[1];
        int third = triangle[2];
        assert(first < second && first < third);
        Wide area = m1une::geometry::cross(
            points[first],
            points[second],
            points[third]
        );
        assert(area > 0);
        triangle_area_twice += area;

        for (int side = 0; side < 3; ++side) {
            int from = triangle[side];
            int to = triangle[(side + 1) % 3];
            if (from > to) std::swap(from, to);
            auto iterator = edge_set.find(std::pair(from, to));
            assert(iterator != edge_set.end());
            auto sorted_iterator = std::lower_bound(
                triangulation.edges.begin(),
                triangulation.edges.end(),
                *iterator
            );
            ++incidence[sorted_iterator - triangulation.edges.begin()];
        }
        for (const Point& point : points) {
            assert(!inside_circumcircle(
                points[first],
                points[second],
                points[third],
                point
            ));
        }
    }

    std::vector<Point> hull = m1une::geometry::convex_hull(points, true);
    bool non_collinear = false;
    for (int index = 2; index < int(points.size()); ++index) {
        if (m1une::geometry::orientation(
                points[0],
                points[1],
                points[index]
            ) != 0) {
            non_collinear = true;
        }
    }
    if (!non_collinear) {
        assert(triangulation.triangles.empty());
        assert(triangulation.edges.size() ==
               (points.empty() ? 0 : points.size() - 1));
        return;
    }

    int size = int(points.size());
    int hull_size = int(hull.size());
    assert(int(triangulation.edges.size()) == 3 * size - 3 - hull_size);
    assert(int(triangulation.triangles.size()) == 2 * size - 2 - hull_size);
    for (int count : incidence) assert(count == 1 || count == 2);

    Wide hull_area_twice = 0;
    for (int index = 0; index < hull_size; ++index) {
        hull_area_twice += m1une::geometry::cross(
            hull[index],
            hull[(index + 1) % hull_size]
        );
    }
    assert(triangle_area_twice == hull_area_twice);
}

void test_fixed() {
    check({});
    check({Point(3, -2)});
    check({Point(0, 0), Point(7, 1)});
    check({
        Point(-5, 0),
        Point(-2, 0),
        Point(1, 0),
        Point(7, 0),
    });
    check({Point(0, 0), Point(4, 0), Point(1, 3)});
    check({
        Point(0, 0),
        Point(2, 0),
        Point(2, 2),
        Point(0, 2),
    });
    check({
        Point(5, 0),
        Point(3, 4),
        Point(0, 5),
        Point(-3, 4),
        Point(-5, 0),
        Point(-3, -4),
        Point(0, -5),
        Point(3, -4),
    });
    check({
        Point(0, 0),
        Point(2, 0),
        Point(4, 0),
        Point(4, 3),
        Point(0, 3),
        Point(2, 1),
    });
    check({
        Point(0, 0),
        Point(4, 0),
        Point(5, 3),
        Point(2, 6),
        Point(-1, 3),
        Point(2, 3),
    });
}

void test_randomized() {
    std::uint64_t state = 0xd31a'2026'0722ULL;
    auto random = [&state]() {
        state ^= state << 7;
        state ^= state >> 9;
        return state;
    };

    for (int trial = 0; trial < 3000; ++trial) {
        int size = int(random() % 15);
        std::set<std::pair<long long, long long>> coordinates;
        while (int(coordinates.size()) < size) {
            coordinates.emplace(
                static_cast<long long>(random() % 21) - 10,
                static_cast<long long>(random() % 21) - 10
            );
        }
        std::vector<Point> points;
        for (auto [x, y] : coordinates) points.emplace_back(x, y);
        std::shuffle(
            points.begin(),
            points.end(),
            std::mt19937_64(random())
        );
        check(points);
    }
}

std::vector<std::pair<int, int>> delaunay_mst(
    const std::vector<Point>& points
) {
    struct Edge {
        int from;
        int to;
        Wide squared_distance;
    };

    std::vector<int> order(points.size());
    std::iota(order.begin(), order.end(), 0);
    std::sort(order.begin(), order.end(), [&](int first, int second) {
        return points[first] < points[second];
    });

    std::vector<Point> unique_points;
    std::vector<int> representatives;
    std::vector<Edge> candidates;
    unique_points.reserve(points.size());
    representatives.reserve(points.size());
    candidates.reserve(3 * points.size());
    for (int index : order) {
        if (unique_points.empty() || unique_points.back() != points[index]) {
            unique_points.push_back(points[index]);
            representatives.push_back(index);
        } else {
            candidates.push_back(Edge{index, representatives.back(), 0});
        }
    }

    auto triangulation =
        m1une::geometry::delaunay_triangulation(unique_points);
    for (auto [first, second] : triangulation.edges) {
        int from = representatives[first];
        int to = representatives[second];
        candidates.push_back(Edge{
            from,
            to,
            m1une::geometry::distance2(points[from], points[to]),
        });
    }
    std::sort(
        candidates.begin(),
        candidates.end(),
        [](const Edge& left, const Edge& right) {
            if (left.squared_distance != right.squared_distance) {
                return left.squared_distance < right.squared_distance;
            }
            if (left.from != right.from) return left.from < right.from;
            return left.to < right.to;
        }
    );

    m1une::ds::Dsu dsu(int(points.size()));
    std::vector<std::pair<int, int>> result;
    for (const Edge& edge : candidates) {
        if (dsu.same(edge.from, edge.to)) continue;
        dsu.merge(edge.from, edge.to);
        result.emplace_back(edge.from, edge.to);
    }
    assert(points.empty() || result.size() + 1 == points.size());
    return result;
}

}  // namespace

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

    test_fixed();
    test_randomized();

    int size;
    fast_input >> size;
    std::vector<Point> points(size);
    for (Point& point : points) fast_input >> point.x >> point.y;
    for (auto [from, to] : delaunay_mst(points)) {
        fast_output << from << ' ' << to << '\n';
    }
}

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