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:warning: Convex All
(convex/all.hpp)

Overview

convex/all.hpp includes convex data structures and convex optimization helpers. The public namespace is m1une::convex.

Included Headers

Header Contents
convex/alien_trick.hpp Exact-count optimization through Lagrangian relaxation.
convex/convex_hull_trick.hpp Monotone-slope CHT for minimum or maximum line queries.
convex/li_chao_tree.hpp Dynamic Li Chao tree for arbitrary line and line-segment insertion.
convex/monge/all.hpp SMAWK, monotone minima, Knuth/D&C DP optimization, LARSCH, Monge checks, and structured min-plus/max-plus convolution.
convex/slope_trick.hpp Heap-based slope trick for convex piecewise-linear functions.

Depends on

Code

#ifndef M1UNE_CONVEX_ALL_HPP
#define M1UNE_CONVEX_ALL_HPP 1

#include "alien_trick.hpp"
#include "convex_hull_trick.hpp"
#include "li_chao_tree.hpp"
#include "monge/all.hpp"
#include "slope_trick.hpp"

#endif  // M1UNE_CONVEX_ALL_HPP
#line 1 "convex/all.hpp"



#line 1 "convex/alien_trick.hpp"



#include <cassert>
#include <concepts>
#include <numeric>
#include <type_traits>
#include <utility>

namespace m1une {
namespace convex {

namespace detail {

template <std::integral Penalty, std::integral Count, class Oracle>
Penalty alien_trick_penalty(Penalty lower, Penalty upper, Count target_count, Oracle& oracle) {
    assert(lower <= upper);
    assert(oracle(lower).second >= target_count);
    assert(oracle(upper).second <= target_count);

    while (lower < upper) {
        Penalty middle = std::midpoint(lower, upper);
        if (middle == lower) ++middle;
        if (oracle(middle).second >= target_count) {
            lower = middle;
        } else {
            upper = middle - 1;
        }
    }
    return lower;
}

}  // namespace detail

// Recovers the minimum value among solutions using exactly `target_count`
// items. The oracle minimizes value + penalty * count and breaks ties in favor
// of the larger count.
template <std::integral Penalty, std::integral Count, class Oracle>
auto alien_trick_minimize(Penalty lower, Penalty upper, Count target_count, Oracle oracle) {
    Penalty penalty = detail::alien_trick_penalty(lower, upper, target_count, oracle);
    auto result = oracle(penalty);
    using Value = std::remove_cvref_t<decltype(result.first)>;
    return result.first - static_cast<Value>(penalty) * static_cast<Value>(target_count);
}

// Recovers the maximum value among solutions using exactly `target_count`
// items. The oracle maximizes value - penalty * count and breaks ties in favor
// of the larger count.
template <std::integral Penalty, std::integral Count, class Oracle>
auto alien_trick_maximize(Penalty lower, Penalty upper, Count target_count, Oracle oracle) {
    Penalty penalty = detail::alien_trick_penalty(lower, upper, target_count, oracle);
    auto result = oracle(penalty);
    using Value = std::remove_cvref_t<decltype(result.first)>;
    return result.first + static_cast<Value>(penalty) * static_cast<Value>(target_count);
}

}  // namespace convex
}  // namespace m1une


#line 1 "convex/convex_hull_trick.hpp"



#line 6 "convex/convex_hull_trick.hpp"
#include <cstddef>
#include <optional>
#line 9 "convex/convex_hull_trick.hpp"
#include <vector>

namespace m1une {
namespace convex {

enum class LineOptimization {
    Minimize,
    Maximize,
};

template <std::signed_integral T>
using line_wide_type = __int128_t;

template <std::signed_integral T>
struct LinearFunction {
    using value_type = line_wide_type<T>;

    value_type slope;
    value_type intercept;

    constexpr LinearFunction() : slope(0), intercept(0) {}

    constexpr LinearFunction(T slope_value, T intercept_value) : slope(slope_value), intercept(intercept_value) {}

    constexpr value_type operator()(T x) const {
        return slope * value_type(x) + intercept;
    }
};

// Convex hull trick for lines inserted in nondecreasing slope order.
template <std::signed_integral T, LineOptimization Objective = LineOptimization::Minimize>
struct ConvexHullTrick {
    using Line = LinearFunction<T>;
    using value_type = typename Line::value_type;

   private:
    std::vector<Line> _lines;

    static bool better(value_type first, value_type second) {
        if constexpr (Objective == LineOptimization::Minimize) {
            return first < second;
        } else {
            return second < first;
        }
    }

    static bool redundant(const Line& first, const Line& middle, const Line& last) {
        value_type left = (first.intercept - middle.intercept) * (last.slope - middle.slope);
        value_type right = (middle.intercept - last.intercept) * (middle.slope - first.slope);
        if constexpr (Objective == LineOptimization::Minimize) {
            return left <= right;
        } else {
            return right <= left;
        }
    }

   public:
    ConvexHullTrick() = default;

    int size() const {
        return int(_lines.size());
    }

    bool empty() const {
        return _lines.empty();
    }

    const std::vector<Line>& lines() const {
        return _lines;
    }

    void reserve(std::size_t line_capacity) {
        _lines.reserve(line_capacity);
    }

    void clear() {
        _lines.clear();
    }

    // Slopes must be inserted in nondecreasing order.
    void add_line(T slope, T intercept) {
        Line line(slope, intercept);
        if (!_lines.empty()) {
            assert(_lines.back().slope <= line.slope);
        }

        if (!_lines.empty() && _lines.back().slope == line.slope) {
            if (!better(line.intercept, _lines.back().intercept)) return;
            _lines.pop_back();
        }

        while (_lines.size() >= 2 && redundant(_lines[_lines.size() - 2], _lines.back(), line)) {
            _lines.pop_back();
        }
        _lines.push_back(line);
    }

    std::optional<value_type> try_query(T x) const {
        if (_lines.empty()) return std::nullopt;
        int low = 0;
        int high = int(_lines.size()) - 1;
        while (low < high) {
            int middle = low + (high - low) / 2;
            value_type first = _lines[middle](x);
            value_type second = _lines[middle + 1](x);
            if (better(first, second) || first == second) {
                high = middle;
            } else {
                low = middle + 1;
            }
        }
        return _lines[low](x);
    }

    value_type query(T x) const {
        assert(!empty());
        return *try_query(x);
    }
};

template <std::signed_integral T>
using MinConvexHullTrick = ConvexHullTrick<T, LineOptimization::Minimize>;

template <std::signed_integral T>
using MaxConvexHullTrick = ConvexHullTrick<T, LineOptimization::Maximize>;

}  // namespace convex
}  // namespace m1une


#line 1 "convex/li_chao_tree.hpp"



#line 7 "convex/li_chao_tree.hpp"
#include <limits>
#line 13 "convex/li_chao_tree.hpp"

#line 15 "convex/li_chao_tree.hpp"

namespace m1une {
namespace convex {

// Dynamic Li Chao tree over an integral half-open coordinate domain.
template <std::signed_integral T, LineOptimization Objective = LineOptimization::Minimize>
struct LiChaoTree {
    using Line = LinearFunction<T>;
    using value_type = typename Line::value_type;

   private:
    struct Node {
        Line line;
        bool has_line;
        int left;
        int right;

        Node() : has_line(false), left(-1), right(-1) {}

        explicit Node(Line value) : line(std::move(value)), has_line(true), left(-1), right(-1) {}
    };

    T _left;
    T _right;
    int _root;
    std::vector<Node> _nodes;

    static bool better(value_type first, value_type second) {
        if constexpr (Objective == LineOptimization::Minimize) {
            return first < second;
        } else {
            return second < first;
        }
    }

    int new_node() {
        assert(_nodes.size() < std::size_t(std::numeric_limits<int>::max()));
        _nodes.emplace_back();
        return int(_nodes.size()) - 1;
    }

    int new_node(Line line) {
        assert(_nodes.size() < std::size_t(std::numeric_limits<int>::max()));
        _nodes.emplace_back(std::move(line));
        return int(_nodes.size()) - 1;
    }

    int add_line_node(int node, T left, T right, Line line) {
        if (node == -1) return new_node(std::move(line));
        if (!_nodes[node].has_line) {
            _nodes[node].line = std::move(line);
            _nodes[node].has_line = true;
            return node;
        }

        T middle = std::midpoint(left, right);
        bool left_better = better(line(left), _nodes[node].line(left));
        bool middle_better = better(line(middle), _nodes[node].line(middle));
        if (middle_better) std::swap(line, _nodes[node].line);
        if (middle == left) return node;

        if (left_better != middle_better) {
            int child = add_line_node(_nodes[node].left, left, middle, std::move(line));
            _nodes[node].left = child;
        } else {
            int child = add_line_node(_nodes[node].right, middle, right, std::move(line));
            _nodes[node].right = child;
        }
        return node;
    }

    int add_segment_node(int node, T left, T right, T query_left, T query_right, const Line& line) {
        if (query_right <= left || right <= query_left) return node;
        if (query_left <= left && right <= query_right) {
            return add_line_node(node, left, right, line);
        }
        if (node == -1) node = new_node();

        T middle = std::midpoint(left, right);
        if (middle == left) return add_line_node(node, left, right, line);
        int left_child = add_segment_node(_nodes[node].left, left, middle, query_left, query_right, line);
        int right_child = add_segment_node(_nodes[node].right, middle, right, query_left, query_right, line);
        _nodes[node].left = left_child;
        _nodes[node].right = right_child;
        return node;
    }

   public:
    LiChaoTree() : _left(0), _right(0), _root(-1) {}

    LiChaoTree(T left, T right) : _left(left), _right(right), _root(-1) {
        assert(left <= right);
    }

    T left_bound() const {
        return _left;
    }

    T right_bound() const {
        return _right;
    }

    bool empty() const {
        return _root == -1;
    }

    std::size_t node_count() const {
        return _nodes.size();
    }

    void reserve(std::size_t node_capacity) {
        _nodes.reserve(node_capacity);
    }

    void clear() {
        _root = -1;
        _nodes.clear();
    }

    void add_line(T slope, T intercept) {
        assert(_left < _right);
        _root = add_line_node(_root, _left, _right, Line(slope, intercept));
    }

    void add_segment(T segment_left, T segment_right, T slope, T intercept) {
        assert(_left <= segment_left && segment_left <= segment_right && segment_right <= _right);
        if (segment_left == segment_right) return;
        _root = add_segment_node(_root, _left, _right, segment_left, segment_right, Line(slope, intercept));
    }

    // Returns nullopt when no inserted line covers x.
    std::optional<value_type> query(T x) const {
        assert(_left <= x && x < _right);
        std::optional<value_type> result;
        int node = _root;
        T left = _left;
        T right = _right;
        while (node != -1) {
            if (_nodes[node].has_line) {
                value_type candidate = _nodes[node].line(x);
                if (!result || better(candidate, *result)) {
                    result = candidate;
                }
            }

            T middle = std::midpoint(left, right);
            if (middle == left) break;
            if (x < middle) {
                node = _nodes[node].left;
                right = middle;
            } else {
                node = _nodes[node].right;
                left = middle;
            }
        }
        return result;
    }

    value_type get(T x) const {
        std::optional<value_type> result = query(x);
        assert(result.has_value());
        return result.value_or(value_type());
    }
};

template <std::signed_integral T>
using MinLiChaoTree = LiChaoTree<T, LineOptimization::Minimize>;

template <std::signed_integral T>
using MaxLiChaoTree = LiChaoTree<T, LineOptimization::Maximize>;

}  // namespace convex
}  // namespace m1une


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



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



#line 6 "convex/monge/check.hpp"

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"



#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 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"


#line 1 "convex/slope_trick.hpp"



#line 7 "convex/slope_trick.hpp"
#include <queue>
#line 11 "convex/slope_trick.hpp"

namespace m1une {
namespace convex {

template <class T>
struct SlopeTrickArgmin {
    std::optional<T> left;
    std::optional<T> right;
};

template <class T>
class SlopeTrick {
    static_assert(std::is_arithmetic_v<T> && std::is_signed_v<T>);

    T _minimum = T();
    T _left_offset = T();
    T _right_offset = T();
    std::priority_queue<T> _left;
    std::priority_queue<T, std::vector<T>, std::greater<T>> _right;

    T left_top() const {
        return _left.top() + _left_offset;
    }

    T right_top() const {
        return _right.top() + _right_offset;
    }

    void push_left(T value) {
        _left.push(value - _left_offset);
    }

    void push_right(T value) {
        _right.push(value - _right_offset);
    }

   public:
    SlopeTrick() = default;

    T minimum() const {
        return _minimum;
    }

    int breakpoint_count() const {
        return int(_left.size() + _right.size());
    }

    SlopeTrickArgmin<T> argmin() const {
        SlopeTrickArgmin<T> result;
        if (!_left.empty()) result.left = left_top();
        if (!_right.empty()) result.right = right_top();
        return result;
    }

    void add_constant(T value) {
        _minimum += value;
    }

    void add_x_minus_a(T a) {
        if (!_left.empty() && left_top() > a) {
            T old = left_top();
            _minimum += old - a;
            _left.pop();
            push_left(a);
            push_right(old);
        } else {
            push_right(a);
        }
    }

    void add_a_minus_x(T a) {
        if (!_right.empty() && right_top() < a) {
            T old = right_top();
            _minimum += a - old;
            _right.pop();
            push_right(a);
            push_left(old);
        } else {
            push_left(a);
        }
    }

    void add_abs(T a) {
        add_a_minus_x(a);
        add_x_minus_a(a);
    }

    void clear_left() {
        _left = std::priority_queue<T>();
    }

    void clear_right() {
        _right = std::priority_queue<T, std::vector<T>, std::greater<T>>();
    }

    void prefix_minimum() {
        clear_right();
    }

    void suffix_minimum() {
        clear_left();
    }

    void shift(T delta) {
        _left_offset += delta;
        _right_offset += delta;
    }

    void shift(T left_delta, T right_delta) {
        assert(left_delta <= right_delta);
        _left_offset += left_delta;
        _right_offset += right_delta;
    }

    T evaluate(T x) const {
        T result = _minimum;
        auto left = _left;
        while (!left.empty()) {
            T breakpoint = left.top() + _left_offset;
            if (breakpoint > x) result += breakpoint - x;
            left.pop();
        }

        auto right = _right;
        while (!right.empty()) {
            T breakpoint = right.top() + _right_offset;
            if (x > breakpoint) result += x - breakpoint;
            right.pop();
        }
        return result;
    }

    void merge(SlopeTrick other) {
        add_constant(other._minimum);
        while (!other._left.empty()) {
            add_a_minus_x(other.left_top());
            other._left.pop();
        }
        while (!other._right.empty()) {
            add_x_minus_a(other.right_top());
            other._right.pop();
        }
    }

    void min_plus_convolve(SlopeTrick other) {
        SlopeTrick result;
        result._minimum = _minimum + other._minimum;

        while (!_left.empty() && !other._left.empty()) {
            result.push_left(left_top() + other.left_top());
            _left.pop();
            other._left.pop();
        }
        while (!_right.empty() && !other._right.empty()) {
            result.push_right(right_top() + other.right_top());
            _right.pop();
            other._right.pop();
        }
        *this = std::move(result);
    }
};

template <class T>
SlopeTrick<T> min_plus_convolution(SlopeTrick<T> first,
                                   SlopeTrick<T> second) {
    first.min_plus_convolve(std::move(second));
    return first;
}

}  // namespace convex
}  // namespace m1une


#line 9 "convex/all.hpp"
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