Slope trick （区分線形凸関数に関する特定の更新を高速に扱う手法）
(other_algorithms/slope_trick.hpp)

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• Last update: 2023-08-05 14:49:43+09:00
• Include: #include "other_algorithms/slope_trick.hpp"
• Link: https://ei1333.github.io/library/structure/others/slope-trick.cpp
• Link: https://maspypy.com/slope-trick-1-%E8%A7%A3%E8%AA%AC%E7%B7%A8

区分線形凸関数 $f(x)$ に関して傾きが 1 変化する $x$ 座標の多重集合と，$f(x)$ が最小値をとる区間を常に保持する．

slope_trick<long long> f; // f(x) := 0                                    O(1)
f.add_const(b);           // f(x) += b                                    O(1)
f.add_relu(a);            // f(x) += max(x - a, 0)                        O(log N)
f.add_irelu(a);           // f(x) += max(a - x, 0)                        O(log N)
f.add_abs(a);             // f(x) += |x - a|                              O(log N)
f.move_left_curve(w);     // f(x) <- min_{0 <= y <= w} f(x + y),  w >= 0  O(1)
f.move_right_curve(w);    // f(x) <- min_{0 <= y <= w} f(x - y),  w >= 0  O(1)
f.translate(dx);          // f(x) <- f(x - dx)                            O(1)
f.merge_destructive(g);   // f(x) += g(x), g(x) broken                    O(min(N_f, N_g) log (N_f + N_g))
auto v = f.get_min().min; // v = min_x f(x)                               O(1)
y = f.get_destructive(x); // y = f(x), f(x) broken                        O(log N)

問題例

• Codeforces LATOKEN Round 1 (Div. 1 + Div. 2) G. A New Beginning
• AtCoder Beginner Contest 217 H - Snuketoon

Links

• slope trick (1) 解説編 | maspyのHP
• Slope-Trick | Luzhiled’s Library

Required by

• Dual slope trick (other_algorithms/dual_slope_trick.hpp)

Verified with

• other_algorithms/test/dual_slope_trick.yuki2114.test.cpp
• other_algorithms/test/slope_trick_stress.test.cpp

Code

#pragma once
#include <algorithm>
#include <cassert>
#include <limits>
#include <queue>
#include <utility>

// Slope trick: fast operations for convex piecewise-linear functions
// Implementation idea:
// - https://maspypy.com/slope-trick-1-%E8%A7%A3%E8%AA%AC%E7%B7%A8
// - https://ei1333.github.io/library/structure/others/slope-trick.cpp
template <class T, T INF = std::numeric_limits<T>::max() / 2> class slope_trick {
    T min_f;
    T displacement_l, displacement_r;
    std::priority_queue<T, std::vector<T>, std::less<T>> L;
    std::priority_queue<T, std::vector<T>, std::greater<T>> R;
    void pushR(const T &a) { R.push(a - displacement_r); }
    T topR() const { return R.empty() ? INF : R.top() + displacement_r; }
    T popR() {
        auto ret = topR();
        if (R.size()) R.pop();
        return ret;
    }
    void pushL(const T &a) { L.push(a + displacement_l); }
    T topL() const { return L.empty() ? -INF : L.top() - displacement_l; }
    T popL() {
        auto ret = topL();
        if (L.size()) L.pop();
        return ret;
    }

public:
    // Initialize, f(x) = 0 everywhere
    // Complexity: O(1)
    slope_trick() : min_f(0), displacement_l(0), displacement_r(0) {
        static_assert(INF > 0, "INF must be greater than 0");
    }
    inline int sizeL() const noexcept { return L.size(); }
    inline int sizeR() const noexcept { return R.size(); }

    // argmin f(x), min f(x)
    // Complexity: O(1)
    using Q = struct {
        T min, lo, hi;
    };
    Q get_min() const { return {min_f, topL(), topR()}; }

    // f(x) += b
    // Complexity: O(1)
    slope_trick &add_const(const T &b) { return min_f += b, *this; }

    // f(x) += max(x - a, 0)  _/
    // Complexity: O(log n)
    slope_trick &add_relu(const T &a) {
        return min_f += std::max(T(0), topL() - a), pushL(a), pushR(popL()), *this;
    }

    // f(x) += max(a - x, 0)  \_
    // Complexity: O(log n)
    slope_trick &add_irelu(const T &a) {
        return min_f += std::max(T(0), a - topR()), pushR(a), pushL(popR()), *this;
    }

    // f(x) += |x - a|  \/
    // Complexity: O(log n)
    slope_trick &add_abs(const T &a) { return add_relu(a).add_irelu(a); }

    // f(x) <- min_{0 <= y <= w} f(x + y)  .\ -> \_
    // Complexity: O(1)
    slope_trick &move_left_curve(const T &w) { return assert(w >= 0), displacement_l += w, *this; }

    // f(x) <- min_{0 <= y <= w} f(x - y)  /. -> _/
    // Complexity: O(1)
    slope_trick &move_right_curve(const T &w) {
        return assert(w >= 0), displacement_r += w, *this;
    }

    // f(x) <- f(x - dx) \/. -> .\/
    // Complexity: O(1)
    slope_trick &translate(const T &dx) {
        return displacement_l -= dx, displacement_r += dx, *this;
    }

    // return f(x), f destructive
    T get_destructive(const T &x) {
        T ret = get_min().min;
        while (L.size()) ret += std::max(T(0), popL() - x);
        while (R.size()) ret += std::max(T(0), x - popR());
        return ret;
    }

    // f(x) += g(x), g destructive
    slope_trick &merge_destructive(slope_trick<T, INF> &g) {
        if (sizeL() + sizeR() > g.sizeL() + g.sizeR()) {
            std::swap(min_f, g.min_f);
            std::swap(displacement_l, g.displacement_l);
            std::swap(displacement_r, g.displacement_r);
            std::swap(L, g.L);
            std::swap(R, g.R);
        }
        min_f += g.get_min().min;
        while (g.L.size()) add_irelu(g.popL());
        while (g.R.size()) add_relu(g.popR());
        return *this;
    }
};

#line 2 "other_algorithms/slope_trick.hpp"
#include <algorithm>
#include <cassert>
#include <limits>
#include <queue>
#include <utility>

// Slope trick: fast operations for convex piecewise-linear functions
// Implementation idea:
// - https://maspypy.com/slope-trick-1-%E8%A7%A3%E8%AA%AC%E7%B7%A8
// - https://ei1333.github.io/library/structure/others/slope-trick.cpp
template <class T, T INF = std::numeric_limits<T>::max() / 2> class slope_trick {
    T min_f;
    T displacement_l, displacement_r;
    std::priority_queue<T, std::vector<T>, std::less<T>> L;
    std::priority_queue<T, std::vector<T>, std::greater<T>> R;
    void pushR(const T &a) { R.push(a - displacement_r); }
    T topR() const { return R.empty() ? INF : R.top() + displacement_r; }
    T popR() {
        auto ret = topR();
        if (R.size()) R.pop();
        return ret;
    }
    void pushL(const T &a) { L.push(a + displacement_l); }
    T topL() const { return L.empty() ? -INF : L.top() - displacement_l; }
    T popL() {
        auto ret = topL();
        if (L.size()) L.pop();
        return ret;
    }

public:
    // Initialize, f(x) = 0 everywhere
    // Complexity: O(1)
    slope_trick() : min_f(0), displacement_l(0), displacement_r(0) {
        static_assert(INF > 0, "INF must be greater than 0");
    }
    inline int sizeL() const noexcept { return L.size(); }
    inline int sizeR() const noexcept { return R.size(); }

    // argmin f(x), min f(x)
    // Complexity: O(1)
    using Q = struct {
        T min, lo, hi;
    };
    Q get_min() const { return {min_f, topL(), topR()}; }

    // f(x) += b
    // Complexity: O(1)
    slope_trick &add_const(const T &b) { return min_f += b, *this; }

    // f(x) += max(x - a, 0)  _/
    // Complexity: O(log n)
    slope_trick &add_relu(const T &a) {
        return min_f += std::max(T(0), topL() - a), pushL(a), pushR(popL()), *this;
    }

    // f(x) += max(a - x, 0)  \_
    // Complexity: O(log n)
    slope_trick &add_irelu(const T &a) {
        return min_f += std::max(T(0), a - topR()), pushR(a), pushL(popR()), *this;
    }

    // f(x) += |x - a|  \/
    // Complexity: O(log n)
    slope_trick &add_abs(const T &a) { return add_relu(a).add_irelu(a); }

    // f(x) <- min_{0 <= y <= w} f(x + y)  .\ -> \_
    // Complexity: O(1)
    slope_trick &move_left_curve(const T &w) { return assert(w >= 0), displacement_l += w, *this; }

    // f(x) <- min_{0 <= y <= w} f(x - y)  /. -> _/
    // Complexity: O(1)
    slope_trick &move_right_curve(const T &w) {
        return assert(w >= 0), displacement_r += w, *this;
    }

    // f(x) <- f(x - dx) \/. -> .\/
    // Complexity: O(1)
    slope_trick &translate(const T &dx) {
        return displacement_l -= dx, displacement_r += dx, *this;
    }

    // return f(x), f destructive
    T get_destructive(const T &x) {
        T ret = get_min().min;
        while (L.size()) ret += std::max(T(0), popL() - x);
        while (R.size()) ret += std::max(T(0), x - popR());
        return ret;
    }

    // f(x) += g(x), g destructive
    slope_trick &merge_destructive(slope_trick<T, INF> &g) {
        if (sizeL() + sizeR() > g.sizeL() + g.sizeR()) {
            std::swap(min_f, g.min_f);
            std::swap(displacement_l, g.displacement_l);
            std::swap(displacement_r, g.displacement_r);
            std::swap(L, g.L);
            std::swap(R, g.R);
        }
        min_f += g.get_min().min;
        while (g.L.size()) add_irelu(g.popL());
        while (g.R.size()) add_relu(g.popR());
        return *this;
    }
};

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