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:heavy_check_mark: Monge All
(convex/monge/all.hpp)

Overview

convex/monge/all.hpp includes the repository’s Monge and monotone-matrix algorithms.

Included Headers

Header Contents
convex/monge/smawk.hpp Linear-time row optima for implicit totally monotone matrices.
convex/monge/monotone_minima.hpp Divide-and-conquer row optima under monotone argmins.
convex/monge/divide_and_conquer_optimization.hpp DP-facing divide-and-conquer optimization returning values and choices.
convex/monge/knuth_optimization.hpp Quadratic Knuth optimization for interval DP.
convex/monge/larsch.hpp Linear-time online minima and shortest paths for triangular totally monotone matrices.
convex/monge/check.hpp Monge and anti-Monge quadrangle-inequality checks.
convex/monge/min_plus_convolution.hpp Linear-time structured min-plus/max-plus convolution, including infinity-aware and low-constant variants.

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Code

#ifndef M1UNE_CONVEX_MONGE_ALL_HPP
#define M1UNE_CONVEX_MONGE_ALL_HPP 1

#include "check.hpp"
#include "divide_and_conquer_optimization.hpp"
#include "knuth_optimization.hpp"
#include "larsch.hpp"
#include "min_plus_convolution.hpp"
#include "monotone_minima.hpp"
#include "smawk.hpp"

#endif  // M1UNE_CONVEX_MONGE_ALL_HPP
#line 1 "convex/monge/all.hpp"



#line 1 "convex/monge/check.hpp"



#include <cassert>
#include <vector>

namespace m1une {
namespace convex {

template <class Value>
bool is_monge(int row_count, int column_count, Value value) {
    assert(row_count >= 0);
    assert(column_count >= 0);
    for (int row = 0; row + 1 < row_count; row++) {
        for (int column = 0; column + 1 < column_count; column++) {
            if (value(row, column) + value(row + 1, column + 1) >
                value(row, column + 1) + value(row + 1, column)) {
                return false;
            }
        }
    }
    return true;
}

template <class Value>
bool is_anti_monge(int row_count, int column_count, Value value) {
    assert(row_count >= 0);
    assert(column_count >= 0);
    for (int row = 0; row + 1 < row_count; row++) {
        for (int column = 0; column + 1 < column_count; column++) {
            if (value(row, column) + value(row + 1, column + 1) <
                value(row, column + 1) + value(row + 1, column)) {
                return false;
            }
        }
    }
    return true;
}

template <class T>
bool is_monge(const std::vector<std::vector<T>>& matrix) {
    int row_count = int(matrix.size());
    int column_count = row_count == 0 ? 0 : int(matrix[0].size());
    for (const auto& row : matrix) assert(int(row.size()) == column_count);
    return is_monge(row_count, column_count,
                    [&](int row, int column) -> const T& { return matrix[row][column]; });
}

template <class T>
bool is_anti_monge(const std::vector<std::vector<T>>& matrix) {
    int row_count = int(matrix.size());
    int column_count = row_count == 0 ? 0 : int(matrix[0].size());
    for (const auto& row : matrix) assert(int(row.size()) == column_count);
    return is_anti_monge(
        row_count, column_count,
        [&](int row, int column) -> const T& { return matrix[row][column]; });
}

}  // namespace convex
}  // namespace m1une


#line 1 "convex/monge/divide_and_conquer_optimization.hpp"



#include <type_traits>
#include <utility>
#line 7 "convex/monge/divide_and_conquer_optimization.hpp"

#line 1 "convex/monge/monotone_minima.hpp"



#line 5 "convex/monge/monotone_minima.hpp"
#include <functional>
#line 7 "convex/monge/monotone_minima.hpp"

namespace m1une {
namespace convex {

namespace monotone_minima_detail {

template <class Value, class Compare>
void solve(int row_left, int row_right, int column_left, int column_right,
           const Value& value, const Compare& compare, std::vector<int>& answer) {
    if (row_left == row_right) return;
    int row = (row_left + row_right) / 2;
    int best = column_left;
    for (int column = column_left + 1; column < column_right; column++) {
        if (compare(value(row, column), value(row, best))) best = column;
    }
    answer[row] = best;
    solve(row_left, row, column_left, best + 1, value, compare, answer);
    solve(row + 1, row_right, best, column_right, value, compare, answer);
}

}  // namespace monotone_minima_detail

template <class Value, class Compare = std::less<>>
std::vector<int> monotone_row_optima(int row_count, int column_count, Value value,
                                     Compare compare = Compare()) {
    assert(row_count >= 0);
    assert(column_count >= 0);
    std::vector<int> answer(row_count, -1);
    if (row_count == 0 || column_count == 0) return answer;
    monotone_minima_detail::solve(0, row_count, 0, column_count, value, compare, answer);
    return answer;
}

template <class Value>
std::vector<int> monotone_row_argmin(int row_count, int column_count, Value value) {
    return monotone_row_optima(row_count, column_count, value, std::less<>());
}

template <class Value>
std::vector<int> monotone_row_argmax(int row_count, int column_count, Value value) {
    return monotone_row_optima(row_count, column_count, value, std::greater<>());
}

template <class T>
std::vector<int> monotone_row_argmin(const std::vector<std::vector<T>>& matrix) {
    int row_count = int(matrix.size());
    int column_count = row_count == 0 ? 0 : int(matrix[0].size());
    for (const auto& row : matrix) assert(int(row.size()) == column_count);
    return monotone_row_argmin(
        row_count, column_count,
        [&](int row, int column) -> const T& { return matrix[row][column]; });
}

template <class T>
std::vector<int> monotone_row_argmax(const std::vector<std::vector<T>>& matrix) {
    int row_count = int(matrix.size());
    int column_count = row_count == 0 ? 0 : int(matrix[0].size());
    for (const auto& row : matrix) assert(int(row.size()) == column_count);
    return monotone_row_argmax(
        row_count, column_count,
        [&](int row, int column) -> const T& { return matrix[row][column]; });
}

}  // namespace convex
}  // namespace m1une


#line 9 "convex/monge/divide_and_conquer_optimization.hpp"

namespace m1une {
namespace convex {

template <class T>
struct DivideAndConquerDpResult {
    std::vector<T> value;
    std::vector<int> argmin;
};

template <class Value>
auto divide_and_conquer_dp(int state_count, int candidate_count, Value value)
    -> DivideAndConquerDpResult<
        std::decay_t<std::invoke_result_t<Value, int, int>>> {
    using T = std::decay_t<std::invoke_result_t<Value, int, int>>;
    DivideAndConquerDpResult<T> result;
    result.argmin = monotone_row_argmin(state_count, candidate_count, value);
    result.value.resize(state_count);
    for (int state = 0; state < state_count; state++) {
        if (result.argmin[state] != -1) {
            result.value[state] = value(state, result.argmin[state]);
        }
    }
    return result;
}

template <class T, class Cost>
auto divide_and_conquer_transition(const std::vector<T>& previous, int state_count,
                                   Cost cost)
    -> DivideAndConquerDpResult<
        std::decay_t<decltype(std::declval<T>() + cost(0, 0))>> {
    using Result = std::decay_t<decltype(std::declval<T>() + cost(0, 0))>;
    return divide_and_conquer_dp(
        state_count, int(previous.size()),
        [&](int state, int candidate) -> Result {
            return previous[candidate] + cost(candidate, state);
        });
}

}  // namespace convex
}  // namespace m1une


#line 1 "convex/monge/knuth_optimization.hpp"



#include <algorithm>
#line 8 "convex/monge/knuth_optimization.hpp"

namespace m1une {
namespace convex {

template <class T>
struct KnuthOptimizationResult {
    std::vector<std::vector<T>> value;
    std::vector<std::vector<int>> split;

    T optimum() const {
        return value[0].back();
    }
};

template <class IntervalCost>
auto knuth_optimization(int element_count, IntervalCost interval_cost)
    -> KnuthOptimizationResult<
        std::decay_t<std::invoke_result_t<IntervalCost, int, int>>> {
    assert(element_count >= 0);
    using T = std::decay_t<std::invoke_result_t<IntervalCost, int, int>>;

    KnuthOptimizationResult<T> result;
    result.value.assign(element_count + 1, std::vector<T>(element_count + 1, T()));
    result.split.assign(element_count + 1, std::vector<int>(element_count + 1, -1));

    for (int left = 0; left <= element_count; left++) result.split[left][left] = left;
    for (int left = 0; left < element_count; left++) result.split[left][left + 1] = left + 1;

    for (int length = 2; length <= element_count; length++) {
        for (int left = 0; left + length <= element_count; left++) {
            int right = left + length;
            int first = std::max(left + 1, result.split[left][right - 1]);
            int last = std::min(right - 1, result.split[left + 1][right]);
            assert(first <= last);

            int best = first;
            T best_value = result.value[left][best] + result.value[best][right];
            for (int split = first + 1; split <= last; split++) {
                T candidate = result.value[left][split] + result.value[split][right];
                if (candidate < best_value) {
                    best = split;
                    best_value = candidate;
                }
            }
            result.value[left][right] = best_value + interval_cost(left, right);
            result.split[left][right] = best;
        }
    }
    return result;
}

}  // namespace convex
}  // namespace m1une


#line 1 "convex/monge/larsch.hpp"



#line 6 "convex/monge/larsch.hpp"
#include <memory>
#line 10 "convex/monge/larsch.hpp"

namespace m1une {
namespace convex {

template <class T>
class Larsch {
    struct ReduceColumn;

    struct ReduceRow {
        int size;
        std::function<T(int, int)> value;
        int current_row = 0;
        int boundary = 0;
        std::unique_ptr<ReduceColumn> recursive;

        explicit ReduceRow(int size_) : size(size_) {
            if (size / 2 != 0) recursive = std::make_unique<ReduceColumn>(size / 2);
        }

        void set_value(std::function<T(int, int)> value_) {
            value = std::move(value_);
            if (recursive) {
                recursive->set_value(
                    [&](int row, int column) { return value(row * 2 + 1, column); });
            }
        }

        int next_argmin() {
            int row = current_row++;
            if (row % 2 == 0) {
                int previous = boundary;
                int next = row + 1 == size ? size - 1 : recursive->next_argmin();
                boundary = next;
                int best = previous;
                for (int column = previous + 1; column <= next; column++) {
                    if (value(row, column) < value(row, best)) best = column;
                }
                return best;
            }
            return value(row, boundary) <= value(row, row) ? boundary : row;
        }
    };

    struct ReduceColumn {
        int size;
        std::function<T(int, int)> value;
        int current_row = 0;
        std::vector<int> columns;
        ReduceRow recursive;

        explicit ReduceColumn(int size_) : size(size_), recursive(size_) {}

        void set_value(std::function<T(int, int)> value_) {
            value = std::move(value_);
            recursive.set_value(
                [&](int row, int column) { return value(row, columns[column]); });
        }

        int next_argmin() {
            int row = current_row++;
            int first = row == 0 ? 0 : row * 2 - 1;
            int last = row * 2;
            for (int column = first; column <= last; column++) {
                while (int(columns.size()) != row &&
                       value(int(columns.size()) - 1, columns.back()) >
                           value(int(columns.size()) - 1, column)) {
                    columns.pop_back();
                }
                if (int(columns.size()) != size) columns.push_back(column);
            }
            return columns[recursive.next_argmin()];
        }
    };

    int _size;
    int _processed = 0;
    std::unique_ptr<ReduceRow> _base;

   public:
    template <class Value>
    explicit Larsch(int size, Value value)
        : _size(size), _base(std::make_unique<ReduceRow>(size)) {
        assert(size >= 0);
        _base->set_value(std::function<T(int, int)>(std::move(value)));
    }

    int size() const {
        return _size;
    }

    int processed_rows() const {
        return _processed;
    }

    bool finished() const {
        return _processed == _size;
    }

    int next_argmin() {
        assert(!finished());
        _processed++;
        return _base->next_argmin();
    }
};

template <class T>
struct LarschShortestPathResult {
    std::vector<T> distance;
    std::vector<int> parent;
};

template <class Cost>
auto larsch_shortest_path(int vertex_count, Cost cost)
    -> LarschShortestPathResult<
        std::decay_t<std::invoke_result_t<Cost, int, int>>> {
    using T = std::decay_t<std::invoke_result_t<Cost, int, int>>;
    assert(vertex_count >= 0);

    LarschShortestPathResult<T> result;
    result.distance.assign(vertex_count, T());
    result.parent.assign(vertex_count, -1);
    if (vertex_count <= 1) return result;

    Larsch<T> optimizer(vertex_count - 1, [&](int row, int column) {
        return result.distance[column] + cost(column, row + 1);
    });
    for (int vertex = 1; vertex < vertex_count; vertex++) {
        int parent = optimizer.next_argmin();
        result.parent[vertex] = parent;
        result.distance[vertex] = result.distance[parent] + cost(parent, vertex);
    }
    return result;
}

}  // namespace convex
}  // namespace m1une


#line 1 "convex/monge/min_plus_convolution.hpp"



#line 7 "convex/monge/min_plus_convolution.hpp"

#line 1 "convex/monge/smawk.hpp"



#line 6 "convex/monge/smawk.hpp"
#include <numeric>
#line 8 "convex/monge/smawk.hpp"

namespace m1une {
namespace convex {

namespace smawk_detail {

template <class Select>
void solve(const std::vector<int>& rows, const std::vector<int>& columns,
           const Select& select, std::vector<int>& answer) {
    if (rows.empty()) return;

    std::vector<int> reduced;
    reduced.reserve(columns.size());
    for (int column : columns) {
        while (!reduced.empty()) {
            int row = rows[int(reduced.size()) - 1];
            if (!select(row, reduced.back(), column)) break;
            reduced.pop_back();
        }
        if (reduced.size() < rows.size()) reduced.push_back(column);
    }

    std::vector<int> odd_rows;
    odd_rows.reserve(rows.size() / 2);
    for (int i = 1; i < int(rows.size()); i += 2) odd_rows.push_back(rows[i]);
    solve(odd_rows, reduced, select, answer);

    int left = 0;
    int right = 0;
    for (int i = 0; i < int(rows.size()); i += 2) {
        if (i + 1 < int(rows.size())) {
            while (reduced[right] != answer[rows[i + 1]]) right++;
        } else {
            right = int(reduced.size()) - 1;
        }

        int best = left;
        for (int j = left + 1; j <= right; j++) {
            if (select(rows[i], reduced[best], reduced[j])) {
                best = j;
            }
        }
        answer[rows[i]] = reduced[best];
        left = right;
    }
}

template <class Select>
std::vector<int> row_optima(int row_count, int column_count, const Select& select) {
    std::vector<int> answer(row_count, -1);
    if (row_count == 0 || column_count == 0) return answer;

    std::vector<int> rows(row_count), columns(column_count);
    std::iota(rows.begin(), rows.end(), 0);
    std::iota(columns.begin(), columns.end(), 0);
    solve(rows, columns, select, answer);
    return answer;
}

}  // namespace smawk_detail

template <class Value, class Compare = std::less<>>
std::vector<int> smawk_row_optima(int row_count, int column_count, Value value,
                                  Compare compare = Compare()) {
    assert(row_count >= 0);
    assert(column_count >= 0);
    return smawk_detail::row_optima(
        row_count, column_count,
        [&](int row, int current, int candidate) {
            return compare(value(row, candidate), value(row, current));
        });
}

template <class Value>
std::vector<int> smawk_row_argmin(int row_count, int column_count, Value value) {
    return smawk_row_optima(row_count, column_count, value, std::less<>());
}

template <class Value>
std::vector<int> smawk_row_argmax(int row_count, int column_count, Value value) {
    return smawk_row_optima(row_count, column_count, value, std::greater<>());
}

template <class T>
std::vector<int> smawk_row_argmin(const std::vector<std::vector<T>>& matrix) {
    int row_count = int(matrix.size());
    int column_count = row_count == 0 ? 0 : int(matrix[0].size());
    for (const auto& row : matrix) assert(int(row.size()) == column_count);
    return smawk_row_argmin(
        row_count, column_count,
        [&](int row, int column) -> const T& { return matrix[row][column]; });
}

template <class T>
std::vector<int> smawk_row_argmax(const std::vector<std::vector<T>>& matrix) {
    int row_count = int(matrix.size());
    int column_count = row_count == 0 ? 0 : int(matrix[0].size());
    for (const auto& row : matrix) assert(int(row.size()) == column_count);
    return smawk_row_argmax(
        row_count, column_count,
        [&](int row, int column) -> const T& { return matrix[row][column]; });
}

}  // namespace convex
}  // namespace m1une


#line 9 "convex/monge/min_plus_convolution.hpp"

namespace m1une {
namespace convex {

namespace convolution_detail {

template <class T, class Compare, class Add>
std::vector<T> structured_convolution(const std::vector<T>& arbitrary,
                                      const std::vector<T>& structured,
                                      Compare compare, Add add) {
    if (arbitrary.empty() || structured.empty()) return {};

    int first_size = int(arbitrary.size());
    int second_size = int(structured.size());
    int result_size = first_size + second_size - 1;
    auto select = [&](int index, int current, int candidate) {
        if (index < candidate) return false;
        if (index - current >= second_size) return true;
        T current_value = add(arbitrary[current], structured[index - current]);
        T candidate_value = add(arbitrary[candidate], structured[index - candidate]);
        return !compare(current_value, candidate_value);
    };

    std::vector<int> optima =
        smawk_detail::row_optima(result_size, first_size, select);
    std::vector<T> result;
    result.reserve(result_size);
    for (int index = 0; index < result_size; index++) {
        int first_index = optima[index];
        result.emplace_back(add(arbitrary[first_index],
                                structured[index - first_index]));
    }
    return result;
}

template <class T>
std::pair<int, int> finite_interval(const std::vector<T>& sequence,
                                    const T& infinity) {
    int left = 0;
    while (left < int(sequence.size()) && sequence[left] == infinity) left++;
    int right = int(sequence.size());
    while (right > left && sequence[right - 1] == infinity) right--;
    return {left, right};
}

template <class T, class Compare>
std::vector<T> structured_convolution_with_infinity(
    const std::vector<T>& arbitrary, const std::vector<T>& structured,
    const T& infinity, Compare compare) {
    if (arbitrary.empty() || structured.empty()) return {};

    auto [left, right] = finite_interval(structured, infinity);
    int result_size = int(arbitrary.size() + structured.size() - 1);
    std::vector<T> result(result_size, infinity);
    if (left == right) return result;

    std::vector<int> columns;
    columns.reserve(arbitrary.size());
    for (int i = 0; i < int(arbitrary.size()); i++) {
        if (arbitrary[i] != infinity) columns.push_back(i);
    }
    if (columns.empty()) return result;

    int finite_size = right - left;
    int middle_size = int(arbitrary.size()) + finite_size - 1;
    std::vector<int> rows;
    rows.reserve(middle_size);
    int active = 0;
    for (int row = 0; row < middle_size; row++) {
        if (row < int(arbitrary.size()) && arbitrary[row] != infinity) active++;
        if (row >= finite_size && arbitrary[row - finite_size] != infinity) active--;
        if (active > 0) rows.push_back(row);
    }

    auto select = [&](int index, int current, int candidate) {
        if (index < candidate) return false;
        if (index - current >= finite_size) return true;
        T current_value =
            arbitrary[current] + structured[left + index - current];
        T candidate_value =
            arbitrary[candidate] + structured[left + index - candidate];
        return !compare(current_value, candidate_value);
    };
    std::vector<int> optima(middle_size, -1);
    smawk_detail::solve(rows, columns, select, optima);
    for (int row : rows) {
        int first_index = optima[row];
        result[left + row] =
            arbitrary[first_index] + structured[left + row - first_index];
    }
    return result;
}

template <class T, class Compare>
std::vector<T> linear_structured_convolution(const std::vector<T>& first,
                                             const std::vector<T>& second,
                                             Compare compare) {
    if (first.empty() || second.empty()) return {};

    int first_size = int(first.size());
    int second_size = int(second.size());
    std::vector<T> result(first_size + second_size - 1);
    result[0] = first[0] + second[0];

    int first_index = 1;
    int second_index = 1;
    int result_index = 1;
    while (first_index < first_size && second_index < second_size) {
        T first_difference = first[first_index] - first[first_index - 1];
        T second_difference = second[second_index] - second[second_index - 1];
        if (compare(second_difference, first_difference)) {
            result[result_index] = result[result_index - 1] + second_difference;
            second_index++;
        } else {
            result[result_index] = result[result_index - 1] + first_difference;
            first_index++;
        }
        result_index++;
    }
    while (first_index < first_size) {
        T difference = first[first_index] - first[first_index - 1];
        result[result_index] = result[result_index - 1] + difference;
        first_index++;
        result_index++;
    }
    while (second_index < second_size) {
        T difference = second[second_index] - second[second_index - 1];
        result[result_index] = result[result_index - 1] + difference;
        second_index++;
        result_index++;
    }
    return result;
}

template <class T, class Compare>
std::vector<T> linear_structured_convolution_with_infinity(
    const std::vector<T>& first, const std::vector<T>& second,
    const T& infinity, Compare compare) {
    if (first.empty() || second.empty()) return {};

    auto [first_left, first_right] = finite_interval(first, infinity);
    auto [second_left, second_right] = finite_interval(second, infinity);
    int result_size = int(first.size() + second.size() - 1);
    std::vector<T> result(result_size, infinity);
    if (first_left == first_right || second_left == second_right) return result;

    int offset = first_left + second_left;
    result[offset] = first[first_left] + second[second_left];

    int first_index = first_left + 1;
    int second_index = second_left + 1;
    int result_index = offset + 1;
    while (first_index < first_right && second_index < second_right) {
        T first_difference = first[first_index] - first[first_index - 1];
        T second_difference = second[second_index] - second[second_index - 1];
        if (compare(second_difference, first_difference)) {
            result[result_index] = result[result_index - 1] + second_difference;
            second_index++;
        } else {
            result[result_index] = result[result_index - 1] + first_difference;
            first_index++;
        }
        result_index++;
    }
    while (first_index < first_right) {
        T difference = first[first_index] - first[first_index - 1];
        result[result_index] = result[result_index - 1] + difference;
        first_index++;
        result_index++;
    }
    while (second_index < second_right) {
        T difference = second[second_index] - second[second_index - 1];
        result[result_index] = result[result_index - 1] + difference;
        second_index++;
        result_index++;
    }
    return result;
}

template <class T, class Compare>
bool is_structured_sequence_with_infinity(const std::vector<T>& sequence,
                                          const T& infinity, Compare violation) {
    auto [left, right] = finite_interval(sequence, infinity);
    for (int i = left; i < right; i++) {
        if (sequence[i] == infinity) return false;
    }
    for (int i = left + 1; i + 1 < right; i++) {
        T first_difference = sequence[i] - sequence[i - 1];
        T second_difference = sequence[i + 1] - sequence[i];
        if (violation(first_difference, second_difference)) return false;
    }
    return true;
}

}  // namespace convolution_detail

template <class T>
bool is_convex_sequence(const std::vector<T>& sequence) {
    for (int i = 1; i + 1 < int(sequence.size()); i++) {
        if (sequence[i] - sequence[i - 1] > sequence[i + 1] - sequence[i]) {
            return false;
        }
    }
    return true;
}

template <class T>
bool is_convex_sequence(const std::vector<T>& sequence, const T& infinity) {
    return convolution_detail::is_structured_sequence_with_infinity(
        sequence, infinity, std::greater<>());
}

template <class T>
bool is_concave_sequence(const std::vector<T>& sequence) {
    for (int i = 1; i + 1 < int(sequence.size()); i++) {
        if (sequence[i] - sequence[i - 1] < sequence[i + 1] - sequence[i]) {
            return false;
        }
    }
    return true;
}

template <class T>
bool is_concave_sequence(const std::vector<T>& sequence,
                         const T& negative_infinity) {
    return convolution_detail::is_structured_sequence_with_infinity(
        sequence, negative_infinity, std::less<>());
}

template <class T>
std::vector<T> min_plus_convolution_convex(const std::vector<T>& arbitrary,
                                           const std::vector<T>& convex) {
    auto add = [](const T& first, const T& second) { return first + second; };
    return convolution_detail::structured_convolution(arbitrary, convex,
                                                      std::less<>(), add);
}

template <class T>
std::vector<T> min_plus_convolution_convex(const std::vector<T>& arbitrary,
                                           const std::vector<T>& convex,
                                           const T& infinity) {
    return convolution_detail::structured_convolution_with_infinity(
        arbitrary, convex, infinity, std::less<>());
}

template <class T>
std::vector<T> min_plus_convolution_convex_convex(const std::vector<T>& first,
                                                  const std::vector<T>& second) {
    return convolution_detail::linear_structured_convolution(first, second, std::less<>());
}

template <class T>
std::vector<T> min_plus_convolution_convex_convex(
    const std::vector<T>& first, const std::vector<T>& second,
    const T& infinity) {
    return convolution_detail::linear_structured_convolution_with_infinity(
        first, second, infinity, std::less<>());
}

template <class T>
std::vector<T> max_plus_convolution_concave(const std::vector<T>& arbitrary,
                                            const std::vector<T>& concave) {
    auto add = [](const T& first, const T& second) { return first + second; };
    return convolution_detail::structured_convolution(arbitrary, concave,
                                                      std::greater<>(), add);
}

template <class T>
std::vector<T> max_plus_convolution_concave(const std::vector<T>& arbitrary,
                                            const std::vector<T>& concave,
                                            const T& negative_infinity) {
    return convolution_detail::structured_convolution_with_infinity(
        arbitrary, concave, negative_infinity, std::greater<>());
}

template <class T>
std::vector<T> max_plus_convolution_concave_concave(const std::vector<T>& first,
                                                    const std::vector<T>& second) {
    return convolution_detail::linear_structured_convolution(first, second, std::greater<>());
}

template <class T>
std::vector<T> max_plus_convolution_concave_concave(
    const std::vector<T>& first, const std::vector<T>& second,
    const T& negative_infinity) {
    return convolution_detail::linear_structured_convolution_with_infinity(
        first, second, negative_infinity, std::greater<>());
}

}  // namespace convex
}  // namespace m1une


#line 11 "convex/monge/all.hpp"
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