monotone_minima
(src/dp/monotone_minima.hpp)

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• Last update: 2025-07-24 13:14:34+09:00
• Include: #include "src/dp/monotone_minima.hpp"

monotone_minima

vector<int> monotome_minima<F>(int n, int m, F f)

$n \times m$ のmonotone行列 $A$ に対して，各行における最小値を取る列を列挙します．
以下の要件を満たす関数 bool f(int i, int j, int k) を定義する必要があります．

• $A[i][j] < A[i][k]$ のとき true を返す ( $j < k$ が保証される．等号はどちらでもよい)．

計算量

• $O((n + m) \log n)$

Depends on

• template (src/template/template.hpp)

Required by

• min_plus_convolution (src/convolution/min_plus_convolution.hpp)

Verified with

• verify/library_checker/convolution/min_plus_convolution(convex_and_arbitrary).test.cpp
• verify/library_checker/convolution/min_plus_convolution(convex_and_convex).test.cpp

Code

#pragma once
#include "../template/template.hpp"
template <class F>
vector<int> monotone_minima(const int n, const int m, const F& f) {
    vector<int> res(n);
    auto dfs = [&](const auto& dfs, const int is, const int ie, const int l, const int r) -> void {
        if(is == ie) return;
        const int i = (is + ie) / 2;
        int m = l;
        for(int k = l + 1; k < r; ++k) {
            if(!f(i, m, k)) m = k;
        }
        res[i] = m;
        dfs(dfs, is, i, l, m + 1);
        dfs(dfs, i + 1, ie, m, r);
    };
    dfs(dfs, 0, n, 0, m);
    return res;
}

#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/dp/monotone_minima.hpp"
template <class F>
vector<int> monotone_minima(const int n, const int m, const F& f) {
    vector<int> res(n);
    auto dfs = [&](const auto& dfs, const int is, const int ie, const int l, const int r) -> void {
        if(is == ie) return;
        const int i = (is + ie) / 2;
        int m = l;
        for(int k = l + 1; k < r; ++k) {
            if(!f(i, m, k)) m = k;
        }
        res[i] = m;
        dfs(dfs, is, i, l, m + 1);
        dfs(dfs, i + 1, ie, m, r);
    };
    dfs(dfs, 0, n, 0, m);
    return res;
}

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