Dual slope trick
(other_algorithms/dual_slope_trick.hpp)

• View this file on GitHub
• Last update: 2023-08-05 14:49:43+09:00
• Include: #include "other_algorithms/dual_slope_trick.hpp"
• Link: https://maspypy.com/slope-trick-3-slope-trick-%E3%81%AE%E5%87%B8%E5%85%B1%E5%BD%B9
• Link: https://yukicoder.me/problems/no/2114

凸関数 $f(x)$ に関して $f(0)$ の値や $f(i + 1) - f(i)$ の値の集合を保持し，各種クエリを高速に処理する．Slope trick が保持する関数の凸共役にあたる．

問題例

• No.2114 01 Matching - yukicoder

Links

• slope trick (3) slope trick の凸共役 | maspyのHP

Depends on

• 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)

Verified with

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

Code

#pragma once
#include "slope_trick.hpp"

// https://maspypy.com/slope-trick-3-slope-trick-%E3%81%AE%E5%87%B8%E5%85%B1%E5%BD%B9
// Verified: https://yukicoder.me/problems/no/2114
template <class T, T INF = std::numeric_limits<T>::max() / 2>
class dual_slope_trick : private slope_trick<T, INF> {
public:
    using Base = slope_trick<T, INF>;

    // Initialize: f(x) = 0 (x == 0), inf (otherwise)
    // Complexity: O(1)
    dual_slope_trick() : Base() {}

    // Get f(0)
    // Complexity: O(1)
    T get_at_zero() const { return -Base::get_min().min; }

    // f(x) <- f(x - d) (Move graph to right by d)
    // Complexity: O(log n)
    dual_slope_trick &shift(int d) {
        while (d > 0) --d, Base::add_relu(-INF).add_const(-INF);
        while (d < 0) ++d, Base::add_irelu(INF).add_const(-INF);
        return *this;
    }

    // f(x) += ax + b
    // Complexity: O(log n)
    dual_slope_trick &add_linear(T a, T b) { return Base::translate(a).add_const(b), *this; }

    // f(x) += max(c(x - a), 0)
    // Complexity: O(|a| log n)
    dual_slope_trick &add_linear_or_zero(T c, int a) {
        shift(-a);
        if (c > T()) Base::move_right_curve(c);
        if (c < T()) Base::move_left_curve(-c);
        return shift(a);
    }

    // f(x) <- min f(x - d), a <= d <= b
    // Complexity: O((|a| + |b|) log n)
    dual_slope_trick &slide_min(int a, int b) {
        assert(a <= b);
        shift(a);
        for (int t = 0; t < b - a; ++t) Base::add_relu(T());
        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;
    }
};
#line 3 "other_algorithms/dual_slope_trick.hpp"

// https://maspypy.com/slope-trick-3-slope-trick-%E3%81%AE%E5%87%B8%E5%85%B1%E5%BD%B9
// Verified: https://yukicoder.me/problems/no/2114
template <class T, T INF = std::numeric_limits<T>::max() / 2>
class dual_slope_trick : private slope_trick<T, INF> {
public:
    using Base = slope_trick<T, INF>;

    // Initialize: f(x) = 0 (x == 0), inf (otherwise)
    // Complexity: O(1)
    dual_slope_trick() : Base() {}

    // Get f(0)
    // Complexity: O(1)
    T get_at_zero() const { return -Base::get_min().min; }

    // f(x) <- f(x - d) (Move graph to right by d)
    // Complexity: O(log n)
    dual_slope_trick &shift(int d) {
        while (d > 0) --d, Base::add_relu(-INF).add_const(-INF);
        while (d < 0) ++d, Base::add_irelu(INF).add_const(-INF);
        return *this;
    }

    // f(x) += ax + b
    // Complexity: O(log n)
    dual_slope_trick &add_linear(T a, T b) { return Base::translate(a).add_const(b), *this; }

    // f(x) += max(c(x - a), 0)
    // Complexity: O(|a| log n)
    dual_slope_trick &add_linear_or_zero(T c, int a) {
        shift(-a);
        if (c > T()) Base::move_right_curve(c);
        if (c < T()) Base::move_left_curve(-c);
        return shift(a);
    }

    // f(x) <- min f(x - d), a <= d <= b
    // Complexity: O((|a| + |b|) log n)
    dual_slope_trick &slide_min(int a, int b) {
        assert(a <= b);
        shift(a);
        for (int t = 0; t < b - a; ++t) Base::add_relu(T());
        return *this;
    }
};

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