Fenwick Tree 2D (Offline Query)
(Src/DataStructure/FenwickTree/OfflineFenwickTree2D.hpp)

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• Last update: 2025-06-24 20:48:55+09:00
• Include: #include "Src/DataStructure/FenwickTree/OfflineFenwickTree2D.hpp"

概要

二次元FenwickTree!!

一点加算が行われる点がどこか、先に要求される。(オフライン)

ライブラリの使い方

テンプレート引数

template <class T, class G>

T: 座標の型

G: データ構造に乗せる代数的構造。雛形は下から

class G {
public:
    using Element = ;
    static T identity() noexcept {
    }
    static T operation(const T& l, const T& r) noexcept {
    }
    static T inverse(const T& v) noexcept {
    }
};

operation

void operation(T x, T y)

$(x, y)$ に点が加算されることを知らせる。

計算量: $O(1)$

以下、登録された点の数が $N$ 個であるとする。

build

[[nodiscard]] internal::FenwickTree2D<T, G> build()

二次元FenwickTreeを構築する。構築されたFenwickTreeを返す。

計算量: $O(N\log N)$

以下で説明するものは、buildの返り値が持つメンバである。

operation

void operation(const T& x, const T& y, const V& v);

$(x, y)$ に $v$ を加算する。x, yはOfflineFenwickTree2Dのoperationで登録された点である必要がある。

計算量: $O(\log^2 N)$

product

[[nodiscard]] V product(const T& lx, const T& ly, const T& rx, const T& ry)

$\displaystyle \sum_{i=lx}^{rx-1}\sum_{j=ly}^{ry-1} A_{ij}$ を計算する。

計算量: $O(\log^2 N)$

Depends on

• Src/Algebra/Group/GroupConcept.hpp
• Src/Algebra/Monoid/MonoidConcept.hpp
• Src/Algebra/Semigroup/SemigroupConcept.hpp
• Fenwick Tree (Src/DataStructure/FenwickTree/FenwickTree.hpp)
• 座標圧縮 (Src/Sequence/CompressedSequence.hpp)
• 標準データ型のエイリアス (Src/Template/TypeAlias.hpp)

Verified with

• Test/LC/point_add_rectangle_sum/OfflineFenwickTree2D.test.cpp

Code

#pragma once

#include "../../Template/TypeAlias.hpp"
#include "../../Algebra/Group/GroupConcept.hpp"
#include "./FenwickTree.hpp"
#include "../../Sequence/CompressedSequence.hpp"

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

namespace zawa {

namespace internal {

template <class T, concepts::Group G>
class FenwickTree2D {
public:

    using V = G::Element;

    FenwickTree2D() = default;

    FenwickTree2D(const std::vector<T>& px, const std::vector<T>& py) 
        : xs_{px}, ys_(xs_.size() + 1), fen_(xs_.size() + 1) {
        assert(px.size());
        assert(px.size() == py.size());
        std::vector<std::vector<T>> appy(xs_.size() + 1);
        for (usize i = 0 ; i < px.size() ; i++) {
            auto x = xs_[px[i]];
            for (x++ ; x < appy.size() ; x += lsb(x)) {
                appy[x].push_back(py[i]);
            }
        }
        for (usize i = 1 ; i < fen_.size() ; i++) {
            ys_[i] = CompressedSequence{appy[i]};
            fen_[i] = FenwickTree<G>{ys_[i].size()};
        }
    }

    void operation(const T& x, const T& y, const V& v) {
        auto i = xs_.find(x);
        assert(i != CompressedSequence<T>::NotFound);
        for ( i++ ; i < fen_.size() ; i += lsb(i)) {
            auto j = ys_[i].find(y);
            assert(j != CompressedSequence<T>::NotFound);
            fen_[i].operation(j, v);
        }
    }

    [[nodiscard]] V prefixProduct(const T& x, const T& y) const {
        V res = G::identity();
        for (u32 i = xs_[x] ; i ; i &= i - 1) {
            res = G::operation(res, fen_[i].prefixProduct(ys_[i][y])); 
        }
        return res;
    }

    [[nodiscard]] V product(const T& lx, const T& ly, const T& rx, const T& ry) const {
        assert(lx <= rx);
        assert(ly <= ry);
        V add = G::operation(prefixProduct(rx, ry), prefixProduct(lx, ly));
        V sub = G::operation(prefixProduct(rx, ly), prefixProduct(lx, ry));
        return G::operation(add, G::inverse(sub));
    }

private:

    CompressedSequence<T> xs_;

    std::vector<CompressedSequence<T>> ys_;

    std::vector<FenwickTree<G>> fen_;

    static constexpr i32 lsb(i32 v) {
        return v & -v;
    }
};

} // namespace internal

template <class T, concepts::Group G>
class OfflineFenwickTree2D {
public:

    OfflineFenwickTree2D(usize q = 0) {
        xs_.reserve(q);
        ys_.reserve(q);
    }

    void operation(const T& x, const T& y) {
        xs_.push_back(x);
        ys_.push_back(y);
    }

    void operation(T&& x, T&& y) {
        xs_.push_back(std::move(x));
        ys_.push_back(std::move(y));
    }

    [[nodiscard]] internal::FenwickTree2D<T, G> build() const {
        return internal::FenwickTree2D<T, G>{xs_, ys_};
    }

private:

    std::vector<T> xs_{}, ys_{};
};

} // namespace zawa

#line 2 "Src/DataStructure/FenwickTree/OfflineFenwickTree2D.hpp"

#line 2 "Src/Template/TypeAlias.hpp"

#include <cstdint>
#include <cstddef>

namespace zawa {

using i16 = std::int16_t;
using i32 = std::int32_t;
using i64 = std::int64_t;
using i128 = __int128_t;

using u8 = std::uint8_t;
using u16 = std::uint16_t;
using u32 = std::uint32_t;
using u64 = std::uint64_t;

using usize = std::size_t;

} // namespace zawa
#line 2 "Src/Algebra/Group/GroupConcept.hpp"

#line 2 "Src/Algebra/Monoid/MonoidConcept.hpp"

#line 2 "Src/Algebra/Semigroup/SemigroupConcept.hpp"

#include <concepts>

namespace zawa {

namespace concepts {

template <class T>
concept Semigroup = requires {
    typename T::Element;
    { T::operation(std::declval<typename T::Element>(), std::declval<typename T::Element>()) } -> std::same_as<typename T::Element>;
};

} // namespace concepts

} // namespace zawa
#line 4 "Src/Algebra/Monoid/MonoidConcept.hpp"

#line 6 "Src/Algebra/Monoid/MonoidConcept.hpp"

namespace zawa {

namespace concepts {

template <class T>
concept Identitiable = requires {
    typename T::Element;
    { T::identity() } -> std::same_as<typename T::Element>;
};

template <class T>
concept Monoid = Semigroup<T> and Identitiable<T>;

} // namespace

} // namespace zawa
#line 4 "Src/Algebra/Group/GroupConcept.hpp"

namespace zawa {

namespace concepts {

template <class T>
concept Inversible = requires {
    typename T::Element;
    { T::inverse(std::declval<typename T::Element>()) } -> std::same_as<typename T::Element>;
};

template <class T>
concept Group = Monoid<T> and Inversible<T>;

} // namespace Concept

} // namespace zawa
#line 2 "Src/DataStructure/FenwickTree/FenwickTree.hpp"

#line 5 "Src/DataStructure/FenwickTree/FenwickTree.hpp"

#include <vector>
#include <cassert>
#include <ostream>
#include <functional>
#include <type_traits>

namespace zawa {

template <concepts::Group Group>
class FenwickTree {
public:

    using VM = Group;
    
    using V = typename VM::Element;

    FenwickTree() = default;

    explicit FenwickTree(usize n) : m_n{ n }, m_bitwidth{ std::__lg(n) + 1 }, m_a(n), m_dat(n + 1, VM::identity()) {
        m_dat.shrink_to_fit();
        m_a.shrink_to_fit();
    }

    explicit FenwickTree(const std::vector<V>& a) : m_n{ a.size() }, m_bitwidth{ std::__lg(a.size()) + 1 }, m_a(a), m_dat(a.size() + 1, VM::identity()) {
        m_dat.shrink_to_fit();  
        m_a.shrink_to_fit();
        for (i32 i{} ; i < static_cast<i32>(m_n) ; i++) {
            addDat(i, a[i]);
        }
    }

    inline usize size() const noexcept {
        return m_n;
    }

    // return a[i]
    const V& get(usize i) const noexcept {
        assert(i < size());
        return m_a[i];
    }

    // return a[i]
    const V& operator[](usize i) const noexcept {
        assert(i < size());
        return m_a[i];
    }

    // a[i] <- a[i] + v
    void operation(usize i, const V& v) {
        assert(i < size());
        addDat(i, v);
        m_a[i] = VM::operation(m_a[i], v);
    }

    // a[i] <- v
    void assign(usize i, const V& v) {
        assert(i < size());
        addDat(i, VM::operation(VM::inverse(m_a[i]), v));
        m_a[i] = v;
    }

    // return a[0] + a[1] + ... + a[r - 1]
    V prefixProduct(usize r) const {
        assert(r <= size());
        return product(r);
    }

    // return a[l] + a[l + 1] ... + a[r - 1]
    V product(usize l, usize r) const {
        assert(l <= r and r <= size());
        return VM::operation(VM::inverse(product(l)), product(r));
    }

    template <class Function>
    usize maxRight(usize l, const Function& f) const {
        static_assert(std::is_convertible_v<decltype(f), std::function<bool(V)>>, "maxRight's argument f must be function bool(T)");
        assert(l < size());
        V sum{ VM::inverse(product(l)) }; 
        usize r{};
        for (usize bit{ m_bitwidth } ; bit ; ) {
            bit--;
            usize nxt{ r | (1u << bit) };
            if (nxt < m_dat.size() and f(VM::operation(sum, m_dat[nxt]))) {
                sum = VM::operation(sum, m_dat[nxt]);
                r = std::move(nxt);
            }
        }
        assert(l <= r);
        return r;
    }

    template <class Function>
    usize minLeft(usize r, const Function& f) const {
        static_assert(std::is_convertible_v<decltype(f), std::function<bool(V)>>, "minLeft's argument f must be function bool(T)");
        assert(r <= size());
        V sum{ product(r) };
        usize l{};
        for (usize bit{ m_bitwidth } ; bit ; ) {
            bit--;
            usize nxt{ l | (1u << bit) };
            if (nxt <= r and not f(VM::operation(VM::inverse(m_dat[nxt]), sum))) {
                sum = VM::operation(VM::inverse(m_dat[nxt]), sum);
                l = std::move(nxt);
            }
        }
        assert(l <= r);
        return l;
    }

    // debug print
    friend std::ostream& operator<<(std::ostream& os, const FenwickTree& ft) {
        for (usize i{} ; i <= ft.size() ; i++) {
            os << ft.prefixProduct(i) << (i == ft.size() ? "" : " ");
        }
        return os;
    }

private:

    usize m_n{};

    usize m_bitwidth{};

    std::vector<V> m_a, m_dat;

    constexpr i32 lsb(i32 x) const noexcept {
        return x & -x;
    }
    
    // a[i] <- a[i] + v
    void addDat(i32 i, const V& v) {
        assert(0 <= i and i < static_cast<i32>(m_n));
        for ( i++ ; i < static_cast<i32>(m_dat.size()) ; i += lsb(i)) {
            m_dat[i] = VM::operation(m_dat[i], v);
        }
    }

    // return a[0] + a[1] + .. + a[i - 1]
    V product(i32 i) const {
        assert(0 <= i and i <= static_cast<i32>(m_n));
        V res{ VM::identity() };
        for ( ; i > 0 ; i -= lsb(i)) {
            res = VM::operation(res, m_dat[i]);
        }
        return res;
    }

};

} // namespace zawa
#line 2 "Src/Sequence/CompressedSequence.hpp"

#line 4 "Src/Sequence/CompressedSequence.hpp"

#line 6 "Src/Sequence/CompressedSequence.hpp"
#include <algorithm>
#line 8 "Src/Sequence/CompressedSequence.hpp"
#include <iterator>
#include <limits>

namespace zawa {

template <class T>
class CompressedSequence {
public:

    static constexpr u32 NotFound = std::numeric_limits<u32>::max();

    CompressedSequence() = default;

    template <class InputIterator>
    CompressedSequence(InputIterator first, InputIterator last) : comped_(first, last), f_{} {
        std::sort(comped_.begin(), comped_.end());
        comped_.erase(std::unique(comped_.begin(), comped_.end()), comped_.end());
        comped_.shrink_to_fit();
        f_.reserve(std::distance(first, last));
        for (auto it{first} ; it != last ; it++) {
            f_.emplace_back(std::distance(comped_.begin(), std::lower_bound(comped_.begin(), comped_.end(), *it)));
        }
    }

    CompressedSequence(const std::vector<T>& A) : CompressedSequence(A.begin(), A.end()) {}

    inline usize size() const noexcept {
        return comped_.size();
    }

    u32 operator[](const T& v) const {
        return std::distance(comped_.begin(), std::lower_bound(comped_.begin(), comped_.end(), v));
    }

    u32 upper_bound(const T& v) const {
        return std::distance(comped_.begin(), std::upper_bound(comped_.begin(), comped_.end(), v));
    }

    u32 find(const T& v) const {
        u32 i = std::distance(comped_.begin(), std::lower_bound(comped_.begin(), comped_.end(), v));
        return i == comped_.size() or comped_[i] != v ? NotFound : i;
    }

    bool contains(const T& v) const {
        u32 i = std::distance(comped_.begin(), std::lower_bound(comped_.begin(), comped_.end(), v));
        return i < comped_.size() and comped_[i] == v;
    }

    u32 at(const T& v) const {
        u32 res = find(v);
        assert(res != NotFound);
        return res;
    }

    inline u32 map(u32 i) const noexcept {
        assert(i < f_.size());
        return f_[i];
    }

    inline T inverse(u32 i) const noexcept {
        assert(i < size());
        return comped_[i];
    }

    inline std::vector<T> comped() const noexcept {
        return comped_;
    }

private:

    std::vector<T> comped_;

    std::vector<u32> f_;

};

} // namespace zawa
#line 7 "Src/DataStructure/FenwickTree/OfflineFenwickTree2D.hpp"

#line 9 "Src/DataStructure/FenwickTree/OfflineFenwickTree2D.hpp"
#include <utility>
#line 11 "Src/DataStructure/FenwickTree/OfflineFenwickTree2D.hpp"
#include <tuple>

namespace zawa {

namespace internal {

template <class T, concepts::Group G>
class FenwickTree2D {
public:

    using V = G::Element;

    FenwickTree2D() = default;

    FenwickTree2D(const std::vector<T>& px, const std::vector<T>& py) 
        : xs_{px}, ys_(xs_.size() + 1), fen_(xs_.size() + 1) {
        assert(px.size());
        assert(px.size() == py.size());
        std::vector<std::vector<T>> appy(xs_.size() + 1);
        for (usize i = 0 ; i < px.size() ; i++) {
            auto x = xs_[px[i]];
            for (x++ ; x < appy.size() ; x += lsb(x)) {
                appy[x].push_back(py[i]);
            }
        }
        for (usize i = 1 ; i < fen_.size() ; i++) {
            ys_[i] = CompressedSequence{appy[i]};
            fen_[i] = FenwickTree<G>{ys_[i].size()};
        }
    }

    void operation(const T& x, const T& y, const V& v) {
        auto i = xs_.find(x);
        assert(i != CompressedSequence<T>::NotFound);
        for ( i++ ; i < fen_.size() ; i += lsb(i)) {
            auto j = ys_[i].find(y);
            assert(j != CompressedSequence<T>::NotFound);
            fen_[i].operation(j, v);
        }
    }

    [[nodiscard]] V prefixProduct(const T& x, const T& y) const {
        V res = G::identity();
        for (u32 i = xs_[x] ; i ; i &= i - 1) {
            res = G::operation(res, fen_[i].prefixProduct(ys_[i][y])); 
        }
        return res;
    }

    [[nodiscard]] V product(const T& lx, const T& ly, const T& rx, const T& ry) const {
        assert(lx <= rx);
        assert(ly <= ry);
        V add = G::operation(prefixProduct(rx, ry), prefixProduct(lx, ly));
        V sub = G::operation(prefixProduct(rx, ly), prefixProduct(lx, ry));
        return G::operation(add, G::inverse(sub));
    }

private:

    CompressedSequence<T> xs_;

    std::vector<CompressedSequence<T>> ys_;

    std::vector<FenwickTree<G>> fen_;

    static constexpr i32 lsb(i32 v) {
        return v & -v;
    }
};

} // namespace internal

template <class T, concepts::Group G>
class OfflineFenwickTree2D {
public:

    OfflineFenwickTree2D(usize q = 0) {
        xs_.reserve(q);
        ys_.reserve(q);
    }

    void operation(const T& x, const T& y) {
        xs_.push_back(x);
        ys_.push_back(y);
    }

    void operation(T&& x, T&& y) {
        xs_.push_back(std::move(x));
        ys_.push_back(std::move(y));
    }

    [[nodiscard]] internal::FenwickTree2D<T, G> build() const {
        return internal::FenwickTree2D<T, G>{xs_, ys_};
    }

private:

    std::vector<T> xs_{}, ys_{};
};

} // namespace zawa

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