icpc_library

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:heavy_check_mark: Li Chao Tree
(data_structure/li_chao_tree.hpp)

説明

直線追加、線分追加、 $x$ における最小値を処理するデータ構造。 以下、 $N$ は処理する $x$ 座標の個数。

add(a, b)

直線 $y = ax + b$ を追加。 $O(\log N)$

add_segment(a, b, l, r)

$x$ 座標の区間 $[l, r)$ に、線分 $y = ax + b$ を追加する。 $O((\log N)^2)$

min(x)

$x$ における最小値を返す。 $O(\log N)$

Depends on

Verified with

Code

#pragma once

#include "../template/template.hpp"

namespace lib {

template <class T> struct li_chao_tree {
  private:
    T f(std::pair<T, T> y, T x) const {
        return y.first * x + y.second;
    }

    void add(std::pair<T, T> y, int l, int r, int index) {
        while (l < r) {
            bool left = f(y, xs[l]) < f(data[index], xs[l]);
            bool mid = f(y, xs[(l + r) / 2]) < f(data[index], xs[(l + r) / 2]);
            bool right = f(y, xs[r - 1]) < f(data[index], xs[r - 1]);
            if (left && right) {
                data[index] = y;
                return;
            }
            if (!(left || right)) {
                return;
            }
            if (mid) {
                std::swap(y, data[index]);
                left = !left;
                right = !right;
            }
            if (left) {
                r = (l + r) / 2;
                index *= 2;
            } else {
                l = (l + r) / 2;
                index = 2 * index + 1;
            }
        }
    }

    int get_index(T x) const {
        auto itr = std::lower_bound(xs.begin(), xs.end(), x);
        return itr - xs.begin();
    }

  public:
    li_chao_tree(std::vector<T> &_x) : xs(_x), sz(1) {
        std::sort(xs.begin(), xs.end());
        xs.erase(std::unique(xs.begin(), xs.end()), xs.end());
        while (sz < int(xs.size())) sz <<= 1;
        while (int(xs.size()) < sz) xs.emplace_back(xs.back() + 1);
        data.assign(2 * sz, {0, std::numeric_limits<T>::max()});
    }

    void add_line(T a, T b) {
        add({a, b}, 0, sz, 1);
    }

    void add_segment(T a, T b, T lx, T rx) {
        int l = get_index(lx);
        int r = get_index(rx);
        assert(0 <= l && l <= r && r <= sz);
        int il = l + sz;
        int ir = r + sz;
        int rank = 0;
        while (il < ir) {
            if (il & 1) {
                add({a, b}, l, l + (1 << rank), il++);
                l += (1 << rank);
            }
            if (ir & 1) {
                add({a, b}, r - (1 << rank), r, --ir);
                r -= (1 << rank);
            }
            rank++;
            il >>= 1;
            ir >>= 1;
        }
    }

    T min(T x) {
        int k = get_index(x) + sz;
        T val = std::numeric_limits<T>::max();
        while (k > 0) {
            val = std::min(val, f(data[k], x));
            k >>= 1;
        }
        return val;
    }

  private:
    std::vector<std::pair<T, T>> data;
    std::vector<T> xs;
    int sz;
};

}
#line 2 "data_structure/li_chao_tree.hpp"

#line 2 "template/template.hpp"

#include <bits/stdc++.h>

#define rep(i, s, n) for (int i = (int)(s); i < (int)(n); i++)
#define rrep(i, s, n) for (int i = (int)(n)-1; i >= (int)(s); i--)
#define all(v) v.begin(), v.end()

using ll = long long;
using ld = long double;
using ull = unsigned long long;

template <typename T> bool chmin(T &a, const T &b) {
    if (a <= b) return false;
    a = b;
    return true;
}
template <typename T> bool chmax(T &a, const T &b) {
    if (a >= b) return false;
    a = b;
    return true;
}

namespace lib {

using namespace std;

}  // namespace lib

// using namespace lib;
#line 4 "data_structure/li_chao_tree.hpp"

namespace lib {

template <class T> struct li_chao_tree {
  private:
    T f(std::pair<T, T> y, T x) const {
        return y.first * x + y.second;
    }

    void add(std::pair<T, T> y, int l, int r, int index) {
        while (l < r) {
            bool left = f(y, xs[l]) < f(data[index], xs[l]);
            bool mid = f(y, xs[(l + r) / 2]) < f(data[index], xs[(l + r) / 2]);
            bool right = f(y, xs[r - 1]) < f(data[index], xs[r - 1]);
            if (left && right) {
                data[index] = y;
                return;
            }
            if (!(left || right)) {
                return;
            }
            if (mid) {
                std::swap(y, data[index]);
                left = !left;
                right = !right;
            }
            if (left) {
                r = (l + r) / 2;
                index *= 2;
            } else {
                l = (l + r) / 2;
                index = 2 * index + 1;
            }
        }
    }

    int get_index(T x) const {
        auto itr = std::lower_bound(xs.begin(), xs.end(), x);
        return itr - xs.begin();
    }

  public:
    li_chao_tree(std::vector<T> &_x) : xs(_x), sz(1) {
        std::sort(xs.begin(), xs.end());
        xs.erase(std::unique(xs.begin(), xs.end()), xs.end());
        while (sz < int(xs.size())) sz <<= 1;
        while (int(xs.size()) < sz) xs.emplace_back(xs.back() + 1);
        data.assign(2 * sz, {0, std::numeric_limits<T>::max()});
    }

    void add_line(T a, T b) {
        add({a, b}, 0, sz, 1);
    }

    void add_segment(T a, T b, T lx, T rx) {
        int l = get_index(lx);
        int r = get_index(rx);
        assert(0 <= l && l <= r && r <= sz);
        int il = l + sz;
        int ir = r + sz;
        int rank = 0;
        while (il < ir) {
            if (il & 1) {
                add({a, b}, l, l + (1 << rank), il++);
                l += (1 << rank);
            }
            if (ir & 1) {
                add({a, b}, r - (1 << rank), r, --ir);
                r -= (1 << rank);
            }
            rank++;
            il >>= 1;
            ir >>= 1;
        }
    }

    T min(T x) {
        int k = get_index(x) + sz;
        T val = std::numeric_limits<T>::max();
        while (k > 0) {
            val = std::min(val, f(data[k], x));
            k >>= 1;
        }
        return val;
    }

  private:
    std::vector<std::pair<T, T>> data;
    std::vector<T> xs;
    int sz;
};

}
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