Lazy Propagation Segment Tree / 遅延伝播セグメント木 (monoids, 赤黒木)
(data_structure/lazy_propagation_red_black_tree.hpp)

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• Last update: 2020-03-04 20:56:55+09:00
• Include: #include "data_structure/lazy_propagation_red_black_tree.hpp"

Operations

完全二分木上の遅延伝播セグメント木でできることに加え、$O(\log N)$ で次ができる:

• $\mathtt{insert}(i, b)$: 列の $i$ 番目の要素の前に要素 $b$ を挿入する。
• $\mathtt{erase}(i)$: 列の $i$ 番目の要素を削除する。
• $\mathtt{reverse}(l, r)$: 列の $l, l+1, \dots, r-1$ 番目の要素を反転する。つまり $(a_0, a_1, \dots, a _ {l-1}, a_l, a _ {l+1}, \dots, a _ {r-1}, a_r, a _ {r + 1}, \dots, a _ {N - 1}) \gets (a_0, a_1, \dots, a _ {l-1}, a _ {r-1}, a _ {r-2}, \dots, a_l, a_r, a _ {r + 1}, \dots, a _ {N - 1})$ と更新する。

Verified with

• data_structure/lazy_propagation_red_black_tree.range_affine_range_sum.test.cpp

Code

#pragma once
#include <algorithm>
#include <cassert>
#include <memory>
#include <type_traits>
#include <vector>

/**
 * @brief Lazy Propagation Segment Tree / 遅延伝播セグメント木 (monoids, 赤黒木)
 * @docs data_structure/lazy_propagation_red_black_tree.md
 * @tparam MonoidX is a monoid
 * @tparam MonoidF is a monoid
 * @tparam Action is a function phi : F * X -> X where the partial applied phi(f, -) : X -> X is a homomorphism on X
 */
template <class MonoidX, class MonoidF, class Action>
class lazy_propagation_red_black_tree {
    static_assert (std::is_invocable_r<typename MonoidX::value_type, Action, typename MonoidF::value_type, typename MonoidX::value_type>::value, "");
    typedef typename MonoidX::value_type value_type;
    typedef typename MonoidF::value_type operator_type;

    enum color_t { BLACK, RED };
    struct node_t {
        bool is_leaf;
        value_type data;
        operator_type lazy;  // NOTE: this->lazy is already applied to this->data
        bool reversed;
        color_t color;
        int rank;
        int size;
        node_t *left;
        node_t *right;
        node_t() = default;
        node_t(value_type const & a_data)
                : is_leaf(true)
                , data(a_data)
                , color(BLACK)
                , rank(0)
                , size(1) {
        }
        node_t(node_t *l, node_t *r, color_t c)  // non-leaf node
                : is_leaf(false)
                , data(MonoidX().mult(l->data, r->data))
                , lazy(MonoidF().unit())
                , reversed(false)
                , color(c)
                , rank(std::max(l->rank + (l->color == BLACK),
                                r->rank + (r->color == BLACK)))
                , size(l->size + r->size)
                , left(l)
                , right(r) {
        }
    };
    struct node_deleter {
        void operator () (node_t *t) const {
            assert (t != nullptr);
            if (not t->is_leaf) {
                (*this)(t->right);
                (*this)(t->left);
            }
            delete t;
        }
    };

    static void propagate_only_operator(node_t *a) {
        MonoidF mon_f;
        Action act;
        if (not a->is_leaf) {
            if (a->lazy != mon_f.unit()) {
                auto const & l = a->left;
                auto const & r = a->right;
                l->data = act(a->lazy, l->data);
                r->data = act(a->lazy, r->data);
                if (not l->is_leaf) l->lazy = mon_f.mult(a->lazy, l->lazy);
                if (not r->is_leaf) r->lazy = mon_f.mult(a->lazy, r->lazy);
                a->lazy = mon_f.unit();
            }
        }
    }
    static void propagate_only_reverse(node_t *a) {
        if (not a->is_leaf) {
            if (a->reversed) {
                auto const & l = a->left;
                auto const & r = a->right;
                if (not l->is_leaf) l->reversed = not l->reversed;
                if (not r->is_leaf) r->reversed = not r->reversed;
                std::swap(a->left, a->right);  // CAUTION: auto const & l, r are destroyed
                a->reversed = false;
            }
        }
    }
    static void propagate(node_t *a) {
        propagate_only_operator(a);
        propagate_only_reverse(a);
    }

    /**
     * @note trees a, b are consumed  (at set_left()/set_right())
     */
    static node_t *merge(node_t *a, node_t *b) {
        if (a == nullptr) return b;
        if (b == nullptr) return a;
        node_t *c = merge_relax(a, b);
        c->color = BLACK;
        return c;
    }
    /*
     * @note the root of returned tree may violates the color constraint
     */
    static node_t *merge_relax(node_t *a, node_t *b) {
        if ((a->rank) < b->rank) {
            assert (not b->is_leaf);
            propagate(b);
            return set_left(b, merge_relax(a, b->left));
        } else if (a->rank > b->rank) {
            assert (not a->is_leaf);
            propagate(a);
            return set_right(a, merge_relax(a->right, b));
        } else {
            a->color = BLACK;
            b->color = BLACK;
            return new node_t(a, b, RED);
        }
    }
    static node_t *set_left(node_t *b, node_t *c) {
        if (b->color == BLACK and c->color == RED and c->left->color == RED) {
            if (b->right->color == BLACK) {
                *b = node_t(c->right, b->right, RED);
                *c = node_t(c->left, b, BLACK);
                std::swap(b, c);
            } else {
                b->right->color = BLACK;
                c->color = BLACK;
                *b = node_t(c, b->right, RED);
            }
        } else {
            *b = node_t(c, b->right, b->color);
        }
        return b;
    }
    static node_t *set_right(node_t *a, node_t *c) {
        if (a->color == BLACK and c->color == RED and c->right->color == RED) {
            if (a->left->color == BLACK) {
                *a = node_t(a->left, c->left, RED);
                *c = node_t(a, c->right, BLACK);
                std::swap(a, c);
            } else {
                a->left->color = BLACK;
                c->color = BLACK;
                *a = node_t(a->left, c, RED);
            }
        } else {
            *a = node_t(a->left, c, a->color);
        }
        return a;
    }

    /**
     * @note tree a is consumed  (at explicit delete and merge())
     */
    static std::pair<node_t *, node_t *> split(node_t *a, int k) {
        if (k == 0) {
            return std::make_pair( nullptr, a );
        }
        assert (a != nullptr);
        if (k == a->size) {
            return std::make_pair( a, nullptr );
        }
        assert (not a->is_leaf);
        propagate(a);
        node_t *a_left  = a->left;
        node_t *a_right = a->right;
        delete a;
        if (k < a_left->size) {
            node_t *l, *r; tie(l, r) = split(a_left, k);
            return std::make_pair( l, merge(r, a_right) );
        } else if (k > a_left->size) {
            node_t *l, *r; tie(l, r) = split(a_right, k - a_left->size);
            return std::make_pair( merge(a_left, l), r );
        } else {
            return std::make_pair( a_left, a_right );
        }
    }

    static void range_apply(node_t *a, int l, int r, const operator_type & func) {
        MonoidX mon_x;
        MonoidF mon_f;
        Action act;
        if (l == r) return;
        if (l == 0 and r == a->size) {
            a->data = act(func, a->data);
            if (not a->is_leaf) a->lazy = mon_f.mult(func, a->lazy);
            return;
        }
        assert (not a->is_leaf);
        propagate(a);
        int k = a->left->size;
        if (r <= k) {
            range_apply(a->left, l, r, func);
        } else if (k <= l) {
            range_apply(a->right, l - k, r - k, func);
        } else {
            range_apply(a->left, l, k, func);
            range_apply(a->right, 0, r - k, func);
        }
        a->data = act(a->lazy, mon_x.mult(a->left->data, a->right->data));
    }

    static value_type range_get(node_t *a, int l, int r) {
        MonoidX mon_x;
        assert (l < r);
        if (l == 0 and r == a->size) return a->data;
        assert (not a->is_leaf);
        propagate(a);
        int k = a->left->size;
        if (r <= k) {
            return range_get(a->left, l, r);
        } else if (k <= l) {
            return range_get(a->right, l - k, r - k);
        } else {
            return mon_x.mult(
                    range_get(a->left, l, k),
                    range_get(a->right, 0, r - k));
        }
    }

    static node_t *reverse(node_t *a, int l, int r) {
        // TODO: find ways to do without split. there may be clever ways using recursion
        if (l == r) return a;
        node_t *bl, *br; tie(bl, br) = split(a, r);
        node_t *bm; tie(bl, bm) = split(bl, l);
        if (not bm->is_leaf) bm->reversed = not bm->reversed;
        return merge(merge(bl, bm), br);
    }

    static void point_set(node_t *a, int i, const value_type & value) {
        MonoidX mon_x;
        Action act;
        if (a->is_leaf) {
            assert (i == 0);
            a->data = value;
        } else {
            propagate_only_reverse(a);  // should we do full propagation?
            if (i < a->left->size) {
                point_set(a->left, i, value);
            } else {
                point_set(a->right, i - a->left->size, value);
            }
            a->data = act(a->lazy,
                    mon_x.mult(a->left->data, a->right->data));
        }
    }

    static value_type & point_get(node_t *a, int i) {
        if (a->is_leaf) {
            assert (i == 0);
            return a->data;
        } else {
            propagate(a);
            if (i < a->left->size) {
                return point_get(a->left, i);
            } else {
                return point_get(a->right, i - a->left->size);
            }
        }
    }

private:
    std::unique_ptr<node_t, node_deleter> root;

public:
    lazy_propagation_red_black_tree() = default;
    lazy_propagation_red_black_tree(node_t *a_root)
            : root(a_root) {
    }
    template <class InputIterator>
    lazy_propagation_red_black_tree(InputIterator first, InputIterator last)
            : root(nullptr) {
        for (; first != last; ++ first) {
            this->push_back(*first);
        }
    }

    static lazy_propagation_red_black_tree merge(lazy_propagation_red_black_tree & l, lazy_propagation_red_black_tree & r) {
        node_t *a = l.root.release();
        node_t *b = r.root.release();
        if (a == nullptr) return lazy_propagation_red_black_tree(b);
        if (b == nullptr) return lazy_propagation_red_black_tree(a);
        return lazy_propagation_red_black_tree(merge(a, b));
    }
    std::pair<lazy_propagation_red_black_tree, lazy_propagation_red_black_tree> split(int k) {
        assert (0 <= k and k <= size());
        node_t *l, *r; tie(l, r) = split(root.release(), k);
        return std::make_pair( lazy_propagation_red_black_tree(l), lazy_propagation_red_black_tree(r) );
    }

    void insert(int i, const value_type & data) {
        assert (0 <= i and i <= size());
        if (empty()) {
            root.reset(new node_t(data));
            return;
        } else {
            node_t *l, *r; tie(l, r) = split(root.release(), i);
            root.reset( merge(merge(l, new node_t(data)), r) );
        }
    }
    void erase(int i) {
        assert (0 <= i and i < size());
        node_t *l, *r; tie(l, r) = split(root.release(), i + 1);
        node_t *m; tie(l, m) = split(l, i);
        node_deleter()(m);
        root.reset( merge(l, r) );
    }

    void point_set(int i, const value_type & value) {
        assert (0 <= i and i < size());
        point_set(root.get(), i, value);
    }
    value_type const & point_get(int i) const {
        assert (0 <= i and i < size());
        return point_get(const_cast<node_t *>(root.get()), i);
    }

    void range_apply(int l, int r, const operator_type & func) {
        assert (0 <= l and l <= r and r <= size());
        if (l == r) return;
        range_apply(root.get(), l, r, func);
    }
    value_type const range_get(int l, int r) const {
        assert (0 <= l and l <= r and r <= size());
        if (l == r) return MonoidX().unit();
        return range_get(const_cast<node_t *>(root.get()), l, r);
    }
    void reverse(int l, int r) {
        assert (0 <= l and l <= r and r <= size());
        if (not root) return;
        root.reset( reverse(root.release(), l, r) );
    }

    void push_back(const value_type & data) {
        root.reset( merge(root.release(), new node_t(data)) );
    }
    void push_front(const value_type & data) {
        root.reset( merge(new node_t(data), root.release()) );
    }
    void pop_back() {
        int k = size() - 1;
        auto lr = split(root.release(), k);
        root.reset(lr.first);
        node_deleter()(lr.second);
    }
    void pop_front() {
        auto lr = split(root.release(), 1);
        node_deleter()(lr.first);
        root.reset(lr.second);
    }
    int size() const {
        return root ? root.get()->size : 0;
    }
    bool empty() const {
        return not root;
    }
    void clear() {
        root = nullptr;
    }
};

#line 2 "data_structure/lazy_propagation_red_black_tree.hpp"
#include <algorithm>
#include <cassert>
#include <memory>
#include <type_traits>
#include <vector>

/**
 * @brief Lazy Propagation Segment Tree / 遅延伝播セグメント木 (monoids, 赤黒木)
 * @docs data_structure/lazy_propagation_red_black_tree.md
 * @tparam MonoidX is a monoid
 * @tparam MonoidF is a monoid
 * @tparam Action is a function phi : F * X -> X where the partial applied phi(f, -) : X -> X is a homomorphism on X
 */
template <class MonoidX, class MonoidF, class Action>
class lazy_propagation_red_black_tree {
    static_assert (std::is_invocable_r<typename MonoidX::value_type, Action, typename MonoidF::value_type, typename MonoidX::value_type>::value, "");
    typedef typename MonoidX::value_type value_type;
    typedef typename MonoidF::value_type operator_type;

    enum color_t { BLACK, RED };
    struct node_t {
        bool is_leaf;
        value_type data;
        operator_type lazy;  // NOTE: this->lazy is already applied to this->data
        bool reversed;
        color_t color;
        int rank;
        int size;
        node_t *left;
        node_t *right;
        node_t() = default;
        node_t(value_type const & a_data)
                : is_leaf(true)
                , data(a_data)
                , color(BLACK)
                , rank(0)
                , size(1) {
        }
        node_t(node_t *l, node_t *r, color_t c)  // non-leaf node
                : is_leaf(false)
                , data(MonoidX().mult(l->data, r->data))
                , lazy(MonoidF().unit())
                , reversed(false)
                , color(c)
                , rank(std::max(l->rank + (l->color == BLACK),
                                r->rank + (r->color == BLACK)))
                , size(l->size + r->size)
                , left(l)
                , right(r) {
        }
    };
    struct node_deleter {
        void operator () (node_t *t) const {
            assert (t != nullptr);
            if (not t->is_leaf) {
                (*this)(t->right);
                (*this)(t->left);
            }
            delete t;
        }
    };

    static void propagate_only_operator(node_t *a) {
        MonoidF mon_f;
        Action act;
        if (not a->is_leaf) {
            if (a->lazy != mon_f.unit()) {
                auto const & l = a->left;
                auto const & r = a->right;
                l->data = act(a->lazy, l->data);
                r->data = act(a->lazy, r->data);
                if (not l->is_leaf) l->lazy = mon_f.mult(a->lazy, l->lazy);
                if (not r->is_leaf) r->lazy = mon_f.mult(a->lazy, r->lazy);
                a->lazy = mon_f.unit();
            }
        }
    }
    static void propagate_only_reverse(node_t *a) {
        if (not a->is_leaf) {
            if (a->reversed) {
                auto const & l = a->left;
                auto const & r = a->right;
                if (not l->is_leaf) l->reversed = not l->reversed;
                if (not r->is_leaf) r->reversed = not r->reversed;
                std::swap(a->left, a->right);  // CAUTION: auto const & l, r are destroyed
                a->reversed = false;
            }
        }
    }
    static void propagate(node_t *a) {
        propagate_only_operator(a);
        propagate_only_reverse(a);
    }

    /**
     * @note trees a, b are consumed  (at set_left()/set_right())
     */
    static node_t *merge(node_t *a, node_t *b) {
        if (a == nullptr) return b;
        if (b == nullptr) return a;
        node_t *c = merge_relax(a, b);
        c->color = BLACK;
        return c;
    }
    /*
     * @note the root of returned tree may violates the color constraint
     */
    static node_t *merge_relax(node_t *a, node_t *b) {
        if ((a->rank) < b->rank) {
            assert (not b->is_leaf);
            propagate(b);
            return set_left(b, merge_relax(a, b->left));
        } else if (a->rank > b->rank) {
            assert (not a->is_leaf);
            propagate(a);
            return set_right(a, merge_relax(a->right, b));
        } else {
            a->color = BLACK;
            b->color = BLACK;
            return new node_t(a, b, RED);
        }
    }
    static node_t *set_left(node_t *b, node_t *c) {
        if (b->color == BLACK and c->color == RED and c->left->color == RED) {
            if (b->right->color == BLACK) {
                *b = node_t(c->right, b->right, RED);
                *c = node_t(c->left, b, BLACK);
                std::swap(b, c);
            } else {
                b->right->color = BLACK;
                c->color = BLACK;
                *b = node_t(c, b->right, RED);
            }
        } else {
            *b = node_t(c, b->right, b->color);
        }
        return b;
    }
    static node_t *set_right(node_t *a, node_t *c) {
        if (a->color == BLACK and c->color == RED and c->right->color == RED) {
            if (a->left->color == BLACK) {
                *a = node_t(a->left, c->left, RED);
                *c = node_t(a, c->right, BLACK);
                std::swap(a, c);
            } else {
                a->left->color = BLACK;
                c->color = BLACK;
                *a = node_t(a->left, c, RED);
            }
        } else {
            *a = node_t(a->left, c, a->color);
        }
        return a;
    }

    /**
     * @note tree a is consumed  (at explicit delete and merge())
     */
    static std::pair<node_t *, node_t *> split(node_t *a, int k) {
        if (k == 0) {
            return std::make_pair( nullptr, a );
        }
        assert (a != nullptr);
        if (k == a->size) {
            return std::make_pair( a, nullptr );
        }
        assert (not a->is_leaf);
        propagate(a);
        node_t *a_left  = a->left;
        node_t *a_right = a->right;
        delete a;
        if (k < a_left->size) {
            node_t *l, *r; tie(l, r) = split(a_left, k);
            return std::make_pair( l, merge(r, a_right) );
        } else if (k > a_left->size) {
            node_t *l, *r; tie(l, r) = split(a_right, k - a_left->size);
            return std::make_pair( merge(a_left, l), r );
        } else {
            return std::make_pair( a_left, a_right );
        }
    }

    static void range_apply(node_t *a, int l, int r, const operator_type & func) {
        MonoidX mon_x;
        MonoidF mon_f;
        Action act;
        if (l == r) return;
        if (l == 0 and r == a->size) {
            a->data = act(func, a->data);
            if (not a->is_leaf) a->lazy = mon_f.mult(func, a->lazy);
            return;
        }
        assert (not a->is_leaf);
        propagate(a);
        int k = a->left->size;
        if (r <= k) {
            range_apply(a->left, l, r, func);
        } else if (k <= l) {
            range_apply(a->right, l - k, r - k, func);
        } else {
            range_apply(a->left, l, k, func);
            range_apply(a->right, 0, r - k, func);
        }
        a->data = act(a->lazy, mon_x.mult(a->left->data, a->right->data));
    }

    static value_type range_get(node_t *a, int l, int r) {
        MonoidX mon_x;
        assert (l < r);
        if (l == 0 and r == a->size) return a->data;
        assert (not a->is_leaf);
        propagate(a);
        int k = a->left->size;
        if (r <= k) {
            return range_get(a->left, l, r);
        } else if (k <= l) {
            return range_get(a->right, l - k, r - k);
        } else {
            return mon_x.mult(
                    range_get(a->left, l, k),
                    range_get(a->right, 0, r - k));
        }
    }

    static node_t *reverse(node_t *a, int l, int r) {
        // TODO: find ways to do without split. there may be clever ways using recursion
        if (l == r) return a;
        node_t *bl, *br; tie(bl, br) = split(a, r);
        node_t *bm; tie(bl, bm) = split(bl, l);
        if (not bm->is_leaf) bm->reversed = not bm->reversed;
        return merge(merge(bl, bm), br);
    }

    static void point_set(node_t *a, int i, const value_type & value) {
        MonoidX mon_x;
        Action act;
        if (a->is_leaf) {
            assert (i == 0);
            a->data = value;
        } else {
            propagate_only_reverse(a);  // should we do full propagation?
            if (i < a->left->size) {
                point_set(a->left, i, value);
            } else {
                point_set(a->right, i - a->left->size, value);
            }
            a->data = act(a->lazy,
                    mon_x.mult(a->left->data, a->right->data));
        }
    }

    static value_type & point_get(node_t *a, int i) {
        if (a->is_leaf) {
            assert (i == 0);
            return a->data;
        } else {
            propagate(a);
            if (i < a->left->size) {
                return point_get(a->left, i);
            } else {
                return point_get(a->right, i - a->left->size);
            }
        }
    }

private:
    std::unique_ptr<node_t, node_deleter> root;

public:
    lazy_propagation_red_black_tree() = default;
    lazy_propagation_red_black_tree(node_t *a_root)
            : root(a_root) {
    }
    template <class InputIterator>
    lazy_propagation_red_black_tree(InputIterator first, InputIterator last)
            : root(nullptr) {
        for (; first != last; ++ first) {
            this->push_back(*first);
        }
    }

    static lazy_propagation_red_black_tree merge(lazy_propagation_red_black_tree & l, lazy_propagation_red_black_tree & r) {
        node_t *a = l.root.release();
        node_t *b = r.root.release();
        if (a == nullptr) return lazy_propagation_red_black_tree(b);
        if (b == nullptr) return lazy_propagation_red_black_tree(a);
        return lazy_propagation_red_black_tree(merge(a, b));
    }
    std::pair<lazy_propagation_red_black_tree, lazy_propagation_red_black_tree> split(int k) {
        assert (0 <= k and k <= size());
        node_t *l, *r; tie(l, r) = split(root.release(), k);
        return std::make_pair( lazy_propagation_red_black_tree(l), lazy_propagation_red_black_tree(r) );
    }

    void insert(int i, const value_type & data) {
        assert (0 <= i and i <= size());
        if (empty()) {
            root.reset(new node_t(data));
            return;
        } else {
            node_t *l, *r; tie(l, r) = split(root.release(), i);
            root.reset( merge(merge(l, new node_t(data)), r) );
        }
    }
    void erase(int i) {
        assert (0 <= i and i < size());
        node_t *l, *r; tie(l, r) = split(root.release(), i + 1);
        node_t *m; tie(l, m) = split(l, i);
        node_deleter()(m);
        root.reset( merge(l, r) );
    }

    void point_set(int i, const value_type & value) {
        assert (0 <= i and i < size());
        point_set(root.get(), i, value);
    }
    value_type const & point_get(int i) const {
        assert (0 <= i and i < size());
        return point_get(const_cast<node_t *>(root.get()), i);
    }

    void range_apply(int l, int r, const operator_type & func) {
        assert (0 <= l and l <= r and r <= size());
        if (l == r) return;
        range_apply(root.get(), l, r, func);
    }
    value_type const range_get(int l, int r) const {
        assert (0 <= l and l <= r and r <= size());
        if (l == r) return MonoidX().unit();
        return range_get(const_cast<node_t *>(root.get()), l, r);
    }
    void reverse(int l, int r) {
        assert (0 <= l and l <= r and r <= size());
        if (not root) return;
        root.reset( reverse(root.release(), l, r) );
    }

    void push_back(const value_type & data) {
        root.reset( merge(root.release(), new node_t(data)) );
    }
    void push_front(const value_type & data) {
        root.reset( merge(new node_t(data), root.release()) );
    }
    void pop_back() {
        int k = size() - 1;
        auto lr = split(root.release(), k);
        root.reset(lr.first);
        node_deleter()(lr.second);
    }
    void pop_front() {
        auto lr = split(root.release(), 1);
        node_deleter()(lr.first);
        root.reset(lr.second);
    }
    int size() const {
        return root ? root.get()->size : 0;
    }
    bool empty() const {
        return not root;
    }
    void clear() {
        root = nullptr;
    }
};

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