Dynamic Li Chao Tree
(lib/data_structure/dynamic_li_chao_tree.hpp)

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• Last update: 2024-05-04 18:06:16+09:00
• Include: #include "lib/data_structure/dynamic_li_chao_tree.hpp"

Dynamic Li Chao Tree

使い方

• DynamicLiChaoTree<T, x_low, x_high, id>(): Dynamic Li Chao Tree を構築します。x_low に $x$ の最小値、x_high に $x$ の最大値、id に十分大きい値 (単位元) を設定します。
• add_line(a, b): ax + b の直線を追加します。
• add_segment(l, r, a, b): [l, r) に ax + b の直線を追加します。
• query(x): x で最小値をとるような値を求めます。

計算量

$R$ を $x$ の範囲とすると

• add(a, b): $\mathrm{O}(\log R)$
• add_segment(l, r, a, b): $\mathrm{O}(\log^2 R)$
• query(x): $\mathrm{O}(\log R)$

Verified with

• test/library_checker/data_structure/line_add_get_min.test.cpp
• test/library_checker/data_structure/segment_add_get_min.test.cpp

Code

#pragma once

/**
 * @brief Dynamic Li Chao Tree
 * @docs docs/data_structure/dynamic_li_chao_tree.md
 */

template<typename T, T x_low, T x_high, T id>
struct DynamicLiChaoTree{
    struct Line{
        T a, b;

        Line(T _a, T _b) : a(_a), b(_b) {}

        inline T get(T x) const {
            return a * x + b;
        }
    };

    struct Node{
        Line x;
        Node *l, *r;

        Node(const Line &x) : x{x}, l{nullptr}, r{nullptr} {}
    };

    Node *root = nullptr;

    DynamicLiChaoTree() {}

private:
    Node *add_line_(Node *t, Line &x, const T &l, const T &r, const T &x_l, const T &x_r){
        if(!t){
            return new Node(x);
        }

        T t_l = t->x.get(l), t_r = t->x.get(r);

        if(t_l <= x_l && t_r <= x_r){
            return t;
        } else if(t_l >= x_l && t_r >= x_r){
            t->x = x;
            return t;
        } else{
            T m = (l + r) / 2;
            if(m == r) --m;
            T t_m = t->x.get(m), x_m = x.get(m);
            if(t_m > x_m){
                swap(t->x, x);
                if(t_l <= x_l){
                    t->l = add_line_(t->l, x, l, m, t_l, t_m);
                } else{
                    t->r = add_line_(t->r, x, m + 1, r, t_m + x.a, t_r);
                }
            } else{
                if(x_l <= t_l){
                    t->l = add_line_(t->l, x, l, m, x_l, x_m);
                } else{
                    t->r = add_line_(t->r, x, m + 1, r, x_m + x.a, x_r);
                }
            }
            return t;
        }
    }

    Node *add_segment_(Node *t, Line &x, const T &a, const T &b, const T &l, const T &r, const T &x_l, const T &x_r){
        if(r < a || b < l) return t;
        if(a <= l && r <= b){
            Line y{x};
            return add_line_(t, y, l, r, x_l, x_r);
        }
        if(t){
            T t_l = t->x.get(l), t_r = t->x.get(r);
            if(t_l <= x_l && t_r <= x_r) return t;
        } else{
            t = new Node(Line(0, id));
        }

        T m = (l + r) / 2;
        if(m == r) --m;
        T x_m = x.get(m);
        t->l = add_segment_(t->l, x, a, b, l, m, x_l, x_m);
        t->r = add_segment_(t->r, x, a, b, m + 1, r, x_m + x.a, x_r);
        return t;
    }

    T query_(const Node *t, const T &l, const T &r, const T &x_c) const {
        if(!t) return id;
        if(l == r) return t->x.get(x_c);
        T m = (l + r) / 2;
        if(m == r) --m;
        if(x_c <= m){
            return min(t->x.get(x_c), query_(t->l, l, m, x_c));
        } else{
            return min(t->x.get(x_c), query_(t->r, m + 1, r, x_c));
        }
    }

public:
    void add_line(const T &a, const T &b){
        Line x(a, b);
        root = add_line_(root, x, x_low, x_high, x.get(x_low), x.get(x_high));
    }

    void add_segment(const T &l, const T &r, const T &a, const T &b){
        Line x(a, b);
        root = add_segment_(root, x, l, r - 1, x_low, x_high, x.get(x_low), x.get(x_high));
    }

    T query(const T &x) const {
        return query_(root, x_low, x_high, x);
    }
};

#line 2 "lib/data_structure/dynamic_li_chao_tree.hpp"

/**
 * @brief Dynamic Li Chao Tree
 * @docs docs/data_structure/dynamic_li_chao_tree.md
 */

template<typename T, T x_low, T x_high, T id>
struct DynamicLiChaoTree{
    struct Line{
        T a, b;

        Line(T _a, T _b) : a(_a), b(_b) {}

        inline T get(T x) const {
            return a * x + b;
        }
    };

    struct Node{
        Line x;
        Node *l, *r;

        Node(const Line &x) : x{x}, l{nullptr}, r{nullptr} {}
    };

    Node *root = nullptr;

    DynamicLiChaoTree() {}

private:
    Node *add_line_(Node *t, Line &x, const T &l, const T &r, const T &x_l, const T &x_r){
        if(!t){
            return new Node(x);
        }

        T t_l = t->x.get(l), t_r = t->x.get(r);

        if(t_l <= x_l && t_r <= x_r){
            return t;
        } else if(t_l >= x_l && t_r >= x_r){
            t->x = x;
            return t;
        } else{
            T m = (l + r) / 2;
            if(m == r) --m;
            T t_m = t->x.get(m), x_m = x.get(m);
            if(t_m > x_m){
                swap(t->x, x);
                if(t_l <= x_l){
                    t->l = add_line_(t->l, x, l, m, t_l, t_m);
                } else{
                    t->r = add_line_(t->r, x, m + 1, r, t_m + x.a, t_r);
                }
            } else{
                if(x_l <= t_l){
                    t->l = add_line_(t->l, x, l, m, x_l, x_m);
                } else{
                    t->r = add_line_(t->r, x, m + 1, r, x_m + x.a, x_r);
                }
            }
            return t;
        }
    }

    Node *add_segment_(Node *t, Line &x, const T &a, const T &b, const T &l, const T &r, const T &x_l, const T &x_r){
        if(r < a || b < l) return t;
        if(a <= l && r <= b){
            Line y{x};
            return add_line_(t, y, l, r, x_l, x_r);
        }
        if(t){
            T t_l = t->x.get(l), t_r = t->x.get(r);
            if(t_l <= x_l && t_r <= x_r) return t;
        } else{
            t = new Node(Line(0, id));
        }

        T m = (l + r) / 2;
        if(m == r) --m;
        T x_m = x.get(m);
        t->l = add_segment_(t->l, x, a, b, l, m, x_l, x_m);
        t->r = add_segment_(t->r, x, a, b, m + 1, r, x_m + x.a, x_r);
        return t;
    }

    T query_(const Node *t, const T &l, const T &r, const T &x_c) const {
        if(!t) return id;
        if(l == r) return t->x.get(x_c);
        T m = (l + r) / 2;
        if(m == r) --m;
        if(x_c <= m){
            return min(t->x.get(x_c), query_(t->l, l, m, x_c));
        } else{
            return min(t->x.get(x_c), query_(t->r, m + 1, r, x_c));
        }
    }

public:
    void add_line(const T &a, const T &b){
        Line x(a, b);
        root = add_line_(root, x, x_low, x_high, x.get(x_low), x.get(x_high));
    }

    void add_segment(const T &l, const T &r, const T &a, const T &b){
        Line x(a, b);
        root = add_segment_(root, x, l, r - 1, x_low, x_high, x.get(x_low), x.get(x_high));
    }

    T query(const T &x) const {
        return query_(root, x_low, x_high, x);
    }
};

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