Dual Segtree 2D
(ds/segtree/dual_segtree_2d.hpp)

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• Last update: 2026-07-16 20:44:42+09:00
• Include: #include "ds/segtree/dual_segtree_2d.hpp"

Overview

DualSegtree2D is a static compressed two-dimensional dual segment tree. It applies a monoid value to every registered point in a half-open rectangle and answers the current value of one registered point.

Use it when all point-query coordinates are known before processing updates. Rectangle boundaries do not need to be registered.

Requirements

template <class Monoid, class X = int, class Y = X>
struct DualSegtree2D;

Monoid must satisfy m1une::monoid::IsMonoid, and its operation must be commutative. Commutativity is necessary because one rectangle update is stored in independently compressed x- and y-segment nodes whose values are combined later at a point query.

X and Y must be ordered coordinate types. Rectangles use [x_lower, x_upper) x [y_lower, y_upper).

Construction

DualSegtree2D();
explicit DualSegtree2D(const std::vector<std::pair<X, Y>>& points);
explicit DualSegtree2D(std::vector<std::pair<X, Y>>&& points);
explicit DualSegtree2D(const std::vector<std::tuple<X, Y, T>>& points);
void build(std::vector<std::pair<X, Y>> points);

Coordinate constructors initialize every distinct point with Monoid::id(). The weighted constructor combines values at duplicate coordinates with Monoid::op. build discards all previous points, values, and updates.

Construction takes $O(N\log^2 N)$ time and $O(N\log N)$ memory.

Methods

Method	Description	Complexity
int size() const	Returns the number of distinct registered points.	$O(1)$
bool empty() const	Returns whether no point is registered.	$O(1)$
int x_size() const	Returns the number of distinct registered x-coordinates.	$O(1)$
const std::vector<X>& xs() const	Returns the sorted compressed x-coordinates.	$O(1)$
bool contains_point(const X& x, const Y& y) const	Tests whether a point is registered.	$O(\log N)$
void apply(const X& x, const Y& y, const T& value)	Applies value to one registered point.	$O(\log N)$
void apply(const X& xl, const X& xr, const Y& yl, const Y& yr, const T& value)	Applies value to every registered point in the rectangle.	$O(\log^2 N)$
T get(const X& x, const Y& y) const	Returns a point value, or Monoid::id() when unregistered.	$O(\log^2 N)$
T operator()(const X& x, const Y& y) const	Alias of get(x, y).	$O(\log^2 N)$
std::vector<std::tuple<X, Y, T>> to_vector() const	Returns all registered points and values in coordinate order.	$O(N\log^2 N)$

For both apply overloads, a stored value current becomes Monoid::op(value, current).

Example

#include "ds/segtree/dual_segtree_2d.hpp"
#include "monoid/add.hpp"

#include <iostream>
#include <utility>
#include <vector>

int main() {
    using Add = m1une::monoid::Add<long long>;
    std::vector<std::pair<int, int>> points = {
        {1, 2},
        {3, 4},
        {5, 1},
    };
    m1une::ds::DualSegtree2D<Add> seg(points);

    seg.apply(0, 4, 0, 5, 10);
    seg.apply(3, 4, 7);

    std::cout << seg.get(1, 2) << "\n"; // 10
    std::cout << seg.get(3, 4) << "\n"; // 17
    std::cout << seg.get(5, 1) << "\n"; // 0
}

Depends on

• Bit Ceil (math/bit_ceil.hpp)
• Monoid Concept (monoid/concept.hpp)

Verified with

• verify/ds/segtree/dual_segtree_2d.test.cpp

Code

#ifndef M1UNE_DUAL_SEGTREE_2D_HPP
#define M1UNE_DUAL_SEGTREE_2D_HPP 1

#include <algorithm>
#include <cassert>
#include <tuple>
#include <utility>
#include <vector>

#include "../../math/bit_ceil.hpp"
#include "../../monoid/concept.hpp"

namespace m1une {
namespace ds {

// A static compressed 2D dual segment tree.
// It supports rectangle monoid updates and point queries on registered points.
// Monoid::op must be commutative.
template <class Monoid, class X = int, class Y = X>
requires m1une::monoid::IsMonoid<Monoid>
struct DualSegtree2D {
    using T = typename Monoid::value_type;
    using point_type = std::pair<X, Y>;
    using weighted_point_type = std::tuple<X, Y, T>;

private:
    int _n;
    int _size;
    int _point_count;
    std::vector<X> _xs;
    std::vector<std::vector<Y>> _ys;
    std::vector<std::vector<T>> _lazy;
    std::vector<std::vector<T>> _values;

    static std::vector<point_type> normalize_points(std::vector<point_type> points) {
        std::sort(points.begin(), points.end());
        points.erase(std::unique(points.begin(), points.end()), points.end());
        return points;
    }

    int x_index(const X& x) const {
        auto it = std::lower_bound(_xs.begin(), _xs.end(), x);
        if (it == _xs.end() || *it != x) return -1;
        return int(it - _xs.begin());
    }

    int y_index(int node, const Y& y) const {
        const auto& ys = _ys[node];
        auto it = std::lower_bound(ys.begin(), ys.end(), y);
        if (it == ys.end() || *it != y) return -1;
        return int(it - ys.begin());
    }

    void apply_y(int node, const Y& lower, const Y& upper, const T& value) {
        const auto& ys = _ys[node];
        if (ys.empty()) return;
        int left = int(std::lower_bound(ys.begin(), ys.end(), lower) - ys.begin());
        int right = int(std::lower_bound(ys.begin(), ys.end(), upper) - ys.begin());
        int size = int(ys.size());
        left += size;
        right += size;
        while (left < right) {
            if (left & 1) {
                _lazy[node][left] = Monoid::op(value, _lazy[node][left]);
                left++;
            }
            if (right & 1) {
                --right;
                _lazy[node][right] = Monoid::op(value, _lazy[node][right]);
            }
            left >>= 1;
            right >>= 1;
        }
    }

    T get_y(int node, const Y& y) const {
        int position = y_index(node, y);
        assert(position != -1);
        int size = int(_ys[node].size());
        T result = Monoid::id();
        for (int index = size + position; index; index >>= 1) {
            result = Monoid::op(_lazy[node][index], result);
        }
        return result;
    }

public:
    DualSegtree2D()
        : _n(0), _size(1), _point_count(0), _ys(2), _lazy(2) {}

    explicit DualSegtree2D(const std::vector<point_type>& points) {
        build(points);
    }

    explicit DualSegtree2D(std::vector<point_type>&& points) {
        build(std::move(points));
    }

    explicit DualSegtree2D(const std::vector<weighted_point_type>& points) {
        std::vector<point_type> coordinates;
        coordinates.reserve(points.size());
        for (const auto& [x, y, value] : points) {
            (void)value;
            coordinates.emplace_back(x, y);
        }
        build(std::move(coordinates));
        for (const auto& [x, y, value] : points) apply(x, y, value);
    }

    void build(std::vector<point_type> points) {
        points = normalize_points(std::move(points));
        _point_count = int(points.size());

        _xs.clear();
        _xs.reserve(points.size());
        for (const auto& [x, y] : points) {
            (void)y;
            if (_xs.empty() || _xs.back() != x) _xs.push_back(x);
        }

        _n = int(_xs.size());
        _size = int(m1une::math::bit_ceil((unsigned int)std::max(1, _n)));
        _ys.assign(2 * _size, {});
        _lazy.assign(2 * _size, {});

        for (const auto& [x, y] : points) {
            int x_position =
                int(std::lower_bound(_xs.begin(), _xs.end(), x) - _xs.begin());
            for (int node = x_position + _size; node; node >>= 1) {
                _ys[node].push_back(y);
            }
        }

        for (int node = 1; node < 2 * _size; node++) {
            auto& ys = _ys[node];
            std::sort(ys.begin(), ys.end());
            ys.erase(std::unique(ys.begin(), ys.end()), ys.end());
            _lazy[node].assign(2 * ys.size(), Monoid::id());
        }

        _values.resize(_n);
        for (int x_position = 0; x_position < _n; x_position++) {
            _values[x_position].assign(_ys[x_position + _size].size(), Monoid::id());
        }
    }

    int size() const {
        return _point_count;
    }

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

    int x_size() const {
        return _n;
    }

    const std::vector<X>& xs() const {
        return _xs;
    }

    bool contains_point(const X& x, const Y& y) const {
        int x_position = x_index(x);
        if (x_position == -1) return false;
        return y_index(x_position + _size, y) != -1;
    }

    void apply(const X& x, const Y& y, const T& value) {
        int x_position = x_index(x);
        assert(x_position != -1);
        int y_position = y_index(x_position + _size, y);
        assert(y_position != -1);
        _values[x_position][y_position] =
            Monoid::op(value, _values[x_position][y_position]);
    }

    void apply(const X& x_lower, const X& x_upper, const Y& y_lower,
               const Y& y_upper, const T& value) {
        assert(x_lower <= x_upper);
        assert(y_lower <= y_upper);
        if (x_lower == x_upper || y_lower == y_upper || empty()) return;

        int left = int(std::lower_bound(_xs.begin(), _xs.end(), x_lower) - _xs.begin());
        int right = int(std::lower_bound(_xs.begin(), _xs.end(), x_upper) - _xs.begin());
        left += _size;
        right += _size;
        while (left < right) {
            if (left & 1) apply_y(left++, y_lower, y_upper, value);
            if (right & 1) apply_y(--right, y_lower, y_upper, value);
            left >>= 1;
            right >>= 1;
        }
    }

    T get(const X& x, const Y& y) const {
        int x_position = x_index(x);
        if (x_position == -1) return Monoid::id();
        int leaf = x_position + _size;
        int y_position = y_index(leaf, y);
        if (y_position == -1) return Monoid::id();

        T result = _values[x_position][y_position];
        for (int node = leaf; node; node >>= 1) {
            result = Monoid::op(get_y(node, y), result);
        }
        return result;
    }

    T operator()(const X& x, const Y& y) const {
        return get(x, y);
    }

    std::vector<weighted_point_type> to_vector() const {
        std::vector<weighted_point_type> result;
        result.reserve(_point_count);
        for (int x_position = 0; x_position < _n; x_position++) {
            int leaf = x_position + _size;
            for (const Y& y : _ys[leaf]) {
                result.emplace_back(_xs[x_position], y, get(_xs[x_position], y));
            }
        }
        return result;
    }
};

}  // namespace ds
}  // namespace m1une

#endif  // M1UNE_DUAL_SEGTREE_2D_HPP

#line 1 "ds/segtree/dual_segtree_2d.hpp"



#include <algorithm>
#include <cassert>
#include <tuple>
#include <utility>
#include <vector>

#line 1 "math/bit_ceil.hpp"



namespace m1une {
namespace math {

template <typename T>
constexpr T bit_ceil(T n) {
    if (n <= 1) return 1;
    T x = 1;
    while (x < n) x <<= 1;
    return x;
}

}  // namespace math
}  // namespace m1une


#line 1 "monoid/concept.hpp"



#include <concepts>

namespace m1une {
namespace monoid {

// Concept to check if a type satisfies the requirements of a Monoid.
// A Monoid must have a `value_type`, an identity element `id()`, and an associative binary operation `op()`.
template <typename M>
concept IsMonoid = requires(typename M::value_type a, typename M::value_type b) {
    // 1. Must define `value_type`
    typename M::value_type;

    // 2. Must have a static method `id()` returning `value_type`
    { M::id() } -> std::same_as<typename M::value_type>;

    // 3. Must have a static method `op(a, b)` returning `value_type`
    { M::op(a, b) } -> std::same_as<typename M::value_type>;
};

// Concept for groups. A type satisfying this concept must also obey the group
// laws; concepts can check the interface but not the algebraic properties.
template <typename M>
concept IsGroup = IsMonoid<M> && requires(typename M::value_type a) {
    { M::inv(a) } -> std::same_as<typename M::value_type>;
};

// Concept for commutative groups. Commutativity is a semantic requirement and
// cannot be checked by a C++ concept.
template <typename M>
concept IsCommutativeGroup = IsGroup<M>;

}  // namespace monoid
}  // namespace m1une


#line 12 "ds/segtree/dual_segtree_2d.hpp"

namespace m1une {
namespace ds {

// A static compressed 2D dual segment tree.
// It supports rectangle monoid updates and point queries on registered points.
// Monoid::op must be commutative.
template <class Monoid, class X = int, class Y = X>
requires m1une::monoid::IsMonoid<Monoid>
struct DualSegtree2D {
    using T = typename Monoid::value_type;
    using point_type = std::pair<X, Y>;
    using weighted_point_type = std::tuple<X, Y, T>;

private:
    int _n;
    int _size;
    int _point_count;
    std::vector<X> _xs;
    std::vector<std::vector<Y>> _ys;
    std::vector<std::vector<T>> _lazy;
    std::vector<std::vector<T>> _values;

    static std::vector<point_type> normalize_points(std::vector<point_type> points) {
        std::sort(points.begin(), points.end());
        points.erase(std::unique(points.begin(), points.end()), points.end());
        return points;
    }

    int x_index(const X& x) const {
        auto it = std::lower_bound(_xs.begin(), _xs.end(), x);
        if (it == _xs.end() || *it != x) return -1;
        return int(it - _xs.begin());
    }

    int y_index(int node, const Y& y) const {
        const auto& ys = _ys[node];
        auto it = std::lower_bound(ys.begin(), ys.end(), y);
        if (it == ys.end() || *it != y) return -1;
        return int(it - ys.begin());
    }

    void apply_y(int node, const Y& lower, const Y& upper, const T& value) {
        const auto& ys = _ys[node];
        if (ys.empty()) return;
        int left = int(std::lower_bound(ys.begin(), ys.end(), lower) - ys.begin());
        int right = int(std::lower_bound(ys.begin(), ys.end(), upper) - ys.begin());
        int size = int(ys.size());
        left += size;
        right += size;
        while (left < right) {
            if (left & 1) {
                _lazy[node][left] = Monoid::op(value, _lazy[node][left]);
                left++;
            }
            if (right & 1) {
                --right;
                _lazy[node][right] = Monoid::op(value, _lazy[node][right]);
            }
            left >>= 1;
            right >>= 1;
        }
    }

    T get_y(int node, const Y& y) const {
        int position = y_index(node, y);
        assert(position != -1);
        int size = int(_ys[node].size());
        T result = Monoid::id();
        for (int index = size + position; index; index >>= 1) {
            result = Monoid::op(_lazy[node][index], result);
        }
        return result;
    }

public:
    DualSegtree2D()
        : _n(0), _size(1), _point_count(0), _ys(2), _lazy(2) {}

    explicit DualSegtree2D(const std::vector<point_type>& points) {
        build(points);
    }

    explicit DualSegtree2D(std::vector<point_type>&& points) {
        build(std::move(points));
    }

    explicit DualSegtree2D(const std::vector<weighted_point_type>& points) {
        std::vector<point_type> coordinates;
        coordinates.reserve(points.size());
        for (const auto& [x, y, value] : points) {
            (void)value;
            coordinates.emplace_back(x, y);
        }
        build(std::move(coordinates));
        for (const auto& [x, y, value] : points) apply(x, y, value);
    }

    void build(std::vector<point_type> points) {
        points = normalize_points(std::move(points));
        _point_count = int(points.size());

        _xs.clear();
        _xs.reserve(points.size());
        for (const auto& [x, y] : points) {
            (void)y;
            if (_xs.empty() || _xs.back() != x) _xs.push_back(x);
        }

        _n = int(_xs.size());
        _size = int(m1une::math::bit_ceil((unsigned int)std::max(1, _n)));
        _ys.assign(2 * _size, {});
        _lazy.assign(2 * _size, {});

        for (const auto& [x, y] : points) {
            int x_position =
                int(std::lower_bound(_xs.begin(), _xs.end(), x) - _xs.begin());
            for (int node = x_position + _size; node; node >>= 1) {
                _ys[node].push_back(y);
            }
        }

        for (int node = 1; node < 2 * _size; node++) {
            auto& ys = _ys[node];
            std::sort(ys.begin(), ys.end());
            ys.erase(std::unique(ys.begin(), ys.end()), ys.end());
            _lazy[node].assign(2 * ys.size(), Monoid::id());
        }

        _values.resize(_n);
        for (int x_position = 0; x_position < _n; x_position++) {
            _values[x_position].assign(_ys[x_position + _size].size(), Monoid::id());
        }
    }

    int size() const {
        return _point_count;
    }

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

    int x_size() const {
        return _n;
    }

    const std::vector<X>& xs() const {
        return _xs;
    }

    bool contains_point(const X& x, const Y& y) const {
        int x_position = x_index(x);
        if (x_position == -1) return false;
        return y_index(x_position + _size, y) != -1;
    }

    void apply(const X& x, const Y& y, const T& value) {
        int x_position = x_index(x);
        assert(x_position != -1);
        int y_position = y_index(x_position + _size, y);
        assert(y_position != -1);
        _values[x_position][y_position] =
            Monoid::op(value, _values[x_position][y_position]);
    }

    void apply(const X& x_lower, const X& x_upper, const Y& y_lower,
               const Y& y_upper, const T& value) {
        assert(x_lower <= x_upper);
        assert(y_lower <= y_upper);
        if (x_lower == x_upper || y_lower == y_upper || empty()) return;

        int left = int(std::lower_bound(_xs.begin(), _xs.end(), x_lower) - _xs.begin());
        int right = int(std::lower_bound(_xs.begin(), _xs.end(), x_upper) - _xs.begin());
        left += _size;
        right += _size;
        while (left < right) {
            if (left & 1) apply_y(left++, y_lower, y_upper, value);
            if (right & 1) apply_y(--right, y_lower, y_upper, value);
            left >>= 1;
            right >>= 1;
        }
    }

    T get(const X& x, const Y& y) const {
        int x_position = x_index(x);
        if (x_position == -1) return Monoid::id();
        int leaf = x_position + _size;
        int y_position = y_index(leaf, y);
        if (y_position == -1) return Monoid::id();

        T result = _values[x_position][y_position];
        for (int node = leaf; node; node >>= 1) {
            result = Monoid::op(get_y(node, y), result);
        }
        return result;
    }

    T operator()(const X& x, const Y& y) const {
        return get(x, y);
    }

    std::vector<weighted_point_type> to_vector() const {
        std::vector<weighted_point_type> result;
        result.reserve(_point_count);
        for (int x_position = 0; x_position < _n; x_position++) {
            int leaf = x_position + _size;
            for (const Y& y : _ys[leaf]) {
                result.emplace_back(_xs[x_position], y, get(_xs[x_position], y));
            }
        }
        return result;
    }
};

}  // namespace ds
}  // namespace m1une

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