Convex Hull Trick
(data_structure/cht.hpp)

• View this file on GitHub
• Last update: 2022-10-20 00:25:59+09:00
• Include: #include "data_structure/cht.hpp"

概要

• 以下の二種類のクエリを処理する。
  • 座標平面 $V \times V$ 上の直線 $y=ax+b$ を集合 $L$ に追加する。
  • $x=c$ において $L$ 内の直線がとる $y$ の最小値を答える。
• 単調性などは特に要求しない。

詳細

• <typename S, bool MIN> struct cht
  Convex Hull Trick を用いてクエリを処理するデータ構造本体。

  • typename S
    テンプレート引数として与える構造体。

    • using val_t
      $V$ を表す型。

    • static val_t div(val_t num, val_t den)
      $V$ 内で $den \times x \leq num$ となる最大の $x$ を返す。整数なら切り捨て除算、実数なら単なる除算に相当する。

    • static val_t inf()
      $V$ の最大元を返す。

  • using V
    $V$ を表す型。

  • void add(V a, V b)
    直線 $y=ax+b$ を集合 $L$ に追加する。 $O(\log |L|)$ 時間。

  • V get(V c)
    $x=c$ において $L$ 内の直線がとる $y$ の最小値を答える。 $O(\log |L|)$ 時間。

参考

• えびちゃんさんのスライド
• kactl

Depends on

• template.hpp

Verified with

• test/judge.yosupo.jp/Line_Add_Get_Min.0.test.cpp

Code

#pragma once

#include "../template.hpp"

template <typename P, bool MIN> struct cht {
    using V = typename P::V;
    struct line {
        mutable V a, b, l, r;
        bool operator<(const line &o) const { return a < o.a; };
        bool operator<(V x) const { return r < x; };
    };
    set<line, less<>> lines;
    void add(V a, V b) {
        if (MIN) a = -a, b = -b;
        line cur = {a, b, -P::inf(), P::inf()};
        auto p = lines.lower_bound(cur);
        if (p != lines.end() && p->a == a) {
            if (p->b > b) {
                return;
            } else {
                p = lines.erase(p);
            }
        }
        auto fi = [](const line &s, const line &t) -> V { return P::div(-s.b + t.b, s.a - t.a); };
        auto ci = [](const line &s, const line &t) -> V { return -P::div(s.b - t.b, s.a - t.a); };
        if (p != lines.begin() && ci(cur, *prev(p)) >= prev(p)->r && p != lines.end() && fi(cur, *p) <= p->l) return;
        while (p != lines.begin()) {
            auto pp = prev(p);
            if (ci(cur, *pp) <= pp->l) {
                lines.erase(pp);
                continue;
            }
            cur.l = ci(cur, *pp), pp->r = fi(cur, *pp);
            break;
        }
        while (p != lines.end()) {
            if (fi(cur, *p) >= p->r) {
                p = lines.erase(p);
                continue;
            }
            cur.r = fi(cur, *p), p->l = ci(cur, *p);
            break;
        }
        lines.insert(cur);
    }
    V get(V c) {
        line l = *lines.lower_bound(c);
        V ret = l.a * c + l.b;
        if (MIN) ret = -ret;
        return ret;
    }
};

struct ll_cht {
    using V = ll;
    static V div(V num, V den) { return fdiv(num, den); }
    static V inf() { return LLONG_MAX; }
};

struct double_cht {
    using V = double;
    static V div(V num, V den) { return num / den; }
    static V inf() { return numeric_limits<double>::infinity(); }
};

#line 2 "data_structure/cht.hpp"

#line 2 "template.hpp"

#include <bits/stdc++.h>
using namespace std;

#define all(a) begin(a), end(a)
#define rall(a) rbegin(a), rend(a)
#define uniq(a) (a).erase(unique(all(a)), (a).end())
#define t0 first
#define t1 second
using ll = long long;
using ull = unsigned long long;
using pll = pair<ll, ll>;
using vll = vector<ll>;
constexpr double pi = 3.14159265358979323846;
constexpr ll dy[9] = {0, 1, 0, -1, 1, 1, -1, -1, 0};
constexpr ll dx[9] = {1, 0, -1, 0, 1, -1, -1, 1, 0};
constexpr ll sign(ll a) { return (a > 0) - (a < 0); }
constexpr ll fdiv(ll a, ll b) { return a / b - ((a ^ b) < 0 && a % b); }
constexpr ll cdiv(ll a, ll b) { return -fdiv(-a, b); }
constexpr ll pw(ll n) { return 1ll << n; }
constexpr ll flg(ll n) { return 63 - __builtin_clzll(n); }
constexpr ll clg(ll n) { return n == 1 ? 0 : flg(n - 1) + 1; }
constexpr ll safemod(ll x, ll mod) { return (x % mod + mod) % mod; }
template <typename T> using priority_queue_rev = priority_queue<T, vector<T>, greater<T>>;
template <typename T> constexpr T sq(const T &a) { return a * a; }
template <typename T, typename U> constexpr bool chmax(T &a, const U &b) { return a < b ? a = b, true : false; }
template <typename T, typename U> constexpr bool chmin(T &a, const U &b) { return a > b ? a = b, true : false; }
template <typename T, typename U> ostream &operator<<(ostream &os, const pair<T, U> &a) {
    os << "(" << a.first << ", " << a.second << ")";
    return os;
}
template <typename T, typename U, typename V> ostream &operator<<(ostream &os, const tuple<T, U, V> &a) {
    os << "(" << get<0>(a) << ", " << get<1>(a) << ", " << get<2>(a) << ")";
    return os;
}
template <typename T> ostream &operator<<(ostream &os, const vector<T> &a) {
    os << "(";
    for (auto itr = a.begin(); itr != a.end(); ++itr) os << *itr << (next(itr) != a.end() ? ", " : "");
    os << ")";
    return os;
}
template <typename T> ostream &operator<<(ostream &os, const set<T> &a) {
    os << "(";
    for (auto itr = a.begin(); itr != a.end(); ++itr) os << *itr << (next(itr) != a.end() ? ", " : "");
    os << ")";
    return os;
}
template <typename T> ostream &operator<<(ostream &os, const multiset<T> &a) {
    os << "(";
    for (auto itr = a.begin(); itr != a.end(); ++itr) os << *itr << (next(itr) != a.end() ? ", " : "");
    os << ")";
    return os;
}
template <typename T, typename U> ostream &operator<<(ostream &os, const map<T, U> &a) {
    os << "(";
    for (auto itr = a.begin(); itr != a.end(); ++itr) os << *itr << (next(itr) != a.end() ? ", " : "");
    os << ")";
    return os;
}
#ifdef ONLINE_JUDGE
#define dump(...) (void(0))
#else
void debug() { cerr << endl; }
template <typename Head, typename... Tail> void debug(Head &&head, Tail &&... tail) {
    cerr << head;
    if (sizeof...(Tail)) cerr << ", ";
    debug(tail...);
}
#define dump(...) cerr << __LINE__ << ": " << #__VA_ARGS__ << " = ", debug(__VA_ARGS__)
#endif
struct rep {
    struct itr {
        ll v;
        itr(ll v) : v(v) {}
        void operator++() { ++v; }
        ll operator*() const { return v; }
        bool operator!=(itr i) const { return v < *i; }
    };
    ll l, r;
    rep(ll l, ll r) : l(l), r(r) {}
    rep(ll r) : rep(0, r) {}
    itr begin() const { return l; };
    itr end() const { return r; };
};
struct per {
    struct itr {
        ll v;
        itr(ll v) : v(v) {}
        void operator++() { --v; }
        ll operator*() const { return v; }
        bool operator!=(itr i) const { return v > *i; }
    };
    ll l, r;
    per(ll l, ll r) : l(l), r(r) {}
    per(ll r) : per(0, r) {}
    itr begin() const { return r - 1; };
    itr end() const { return l - 1; };
};
struct io_setup {
    static constexpr int PREC = 20;
    io_setup() {
        cout << fixed << setprecision(PREC);
        cerr << fixed << setprecision(PREC);
    };
} iOS;
#line 4 "data_structure/cht.hpp"

template <typename P, bool MIN> struct cht {
    using V = typename P::V;
    struct line {
        mutable V a, b, l, r;
        bool operator<(const line &o) const { return a < o.a; };
        bool operator<(V x) const { return r < x; };
    };
    set<line, less<>> lines;
    void add(V a, V b) {
        if (MIN) a = -a, b = -b;
        line cur = {a, b, -P::inf(), P::inf()};
        auto p = lines.lower_bound(cur);
        if (p != lines.end() && p->a == a) {
            if (p->b > b) {
                return;
            } else {
                p = lines.erase(p);
            }
        }
        auto fi = [](const line &s, const line &t) -> V { return P::div(-s.b + t.b, s.a - t.a); };
        auto ci = [](const line &s, const line &t) -> V { return -P::div(s.b - t.b, s.a - t.a); };
        if (p != lines.begin() && ci(cur, *prev(p)) >= prev(p)->r && p != lines.end() && fi(cur, *p) <= p->l) return;
        while (p != lines.begin()) {
            auto pp = prev(p);
            if (ci(cur, *pp) <= pp->l) {
                lines.erase(pp);
                continue;
            }
            cur.l = ci(cur, *pp), pp->r = fi(cur, *pp);
            break;
        }
        while (p != lines.end()) {
            if (fi(cur, *p) >= p->r) {
                p = lines.erase(p);
                continue;
            }
            cur.r = fi(cur, *p), p->l = ci(cur, *p);
            break;
        }
        lines.insert(cur);
    }
    V get(V c) {
        line l = *lines.lower_bound(c);
        V ret = l.a * c + l.b;
        if (MIN) ret = -ret;
        return ret;
    }
};

struct ll_cht {
    using V = ll;
    static V div(V num, V den) { return fdiv(num, den); }
    static V inf() { return LLONG_MAX; }
};

struct double_cht {
    using V = double;
    static V div(V num, V den) { return num / den; }
    static V inf() { return numeric_limits<double>::infinity(); }
};

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