DynamicLiChaoTree
(src/data_structure/dynamic_li_chao_tree.hpp)

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• Last update: 2024-11-09 01:34:39+09:00
• Include: #include "src/data_structure/dynamic_li_chao_tree.hpp"

DynamicLiChaoTree

• 直線 $y = ax + b$ の追加
• 線分 $y = ax + b ~ (l \leq x < r)$ の追加
• 与えられた $x$ 座標における直線・線分群の $y$ 座標の最小値の取得

を処理するデータ構造です．

ConvexHullTrick よりやや計算量は重いですが，単調性が必要ありません．

コンストラクタ

DynamicLiChaoTree<T, T x_low, T x_high> cht

空の直線・線分群 cht を構築します．

$x_{low}$ には query で与える $x$ の最小値を， $x_{high}$ には query で与える $x$ の最大値を与えてください．
以下， $V = x_{high} - x_{low}$ とおきます．

計算量

• $O(1)$

add_line

void cht.add_line(T a, T b)

cht に直線 $y = ax + b$ を追加します．

計算量

• $O(\log V)$

add_segment

void cht.add_segment(T l, T r, T a, T b)

cht に線分 $y = ax + b ~ (l \leq x < r)$ を追加します．

計算量

• $O(\log V^2)$

operator ()

T cht(T x)

与えられた $x$ 座標における，直線・線分群 cht の $y$ 座標の最小値を取得します．

制約

• $x_{low} \leq x \leq x_{high}$

計算量

• $O(\log V)$

Depends on

• template (src/template/template.hpp)

Verified with

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

Code

#pragma once
#include "../template/template.hpp"
template <typename T, T x_low, T x_high>
struct DynamicLiChaoTree {
    DynamicLiChaoTree()
        : root{nullptr} {}
    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 operator()(const T& x) const {
        assert(x_low <= x and x <= x_high);
        return query(root, x_low, x_high, x);
    }

   private:
    struct Line {
        T a, b;
        Line(const T& a, const T& b)
            : a(a), b(b) {}
        inline T get(const T& x) const {
            return a * x + b;
        }
    };
    struct Node {
        Line x;
        Node *l, *r;
        Node(const Line& x)
            : x{x}, l{nullptr}, r{nullptr} {}
    };
    const T id = numeric_limits<T>::max();
    Node* root;
    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 and t_r <= x_r) {
            return t;
        } else if(t_l >= x_l and 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(x_l >= t_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(t_l >= x_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 or b < l) return t;
        if(a <= l and 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 and 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) const {
        if(!t) return id;
        if(l == r) return t->x.get(x);
        T m = (l + r) / 2;
        if(m == r) --m;
        if(x <= m) return min(t->x.get(x), query(t->l, l, m, x));
        else return min(t->x.get(x), query(t->r, m + 1, r, x));
    }
};

#line 2 "src/template/template.hpp"
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using P = pair<long long, long long>;
#define rep(i, a, b) for(long long i = (a); i < (b); ++i)
#define rrep(i, a, b) for(long long i = (a); i >= (b); --i)
constexpr long long inf = 4e18;
struct SetupIO {
    SetupIO() {
        ios::sync_with_stdio(0);
        cin.tie(0);
        cout << fixed << setprecision(30);
    }
} setup_io;
#line 3 "src/data_structure/dynamic_li_chao_tree.hpp"
template <typename T, T x_low, T x_high>
struct DynamicLiChaoTree {
    DynamicLiChaoTree()
        : root{nullptr} {}
    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 operator()(const T& x) const {
        assert(x_low <= x and x <= x_high);
        return query(root, x_low, x_high, x);
    }

   private:
    struct Line {
        T a, b;
        Line(const T& a, const T& b)
            : a(a), b(b) {}
        inline T get(const T& x) const {
            return a * x + b;
        }
    };
    struct Node {
        Line x;
        Node *l, *r;
        Node(const Line& x)
            : x{x}, l{nullptr}, r{nullptr} {}
    };
    const T id = numeric_limits<T>::max();
    Node* root;
    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 and t_r <= x_r) {
            return t;
        } else if(t_l >= x_l and 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(x_l >= t_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(t_l >= x_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 or b < l) return t;
        if(a <= l and 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 and 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) const {
        if(!t) return id;
        if(l == r) return t->x.get(x);
        T m = (l + r) / 2;
        if(m == r) --m;
        if(x <= m) return min(t->x.get(x), query(t->l, l, m, x));
        else return min(t->x.get(x), query(t->r, m + 1, r, x));
    }
};

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