#include <bits/stdc++.h>
using namespace std;
using lint = long long;
using pint = pair<int, int>;
using plint = pair<lint, lint>;
struct fast_ios { fast_ios(){ cin.tie(nullptr), ios::sync_with_stdio(false), cout << fixed << setprecision(20); }; } fast_ios_;
#define ALL(x) (x).begin(), (x).end()
#define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i<i##_end_;i++)
#define IFOR(i, begin, end) for(int i=(end)-1,i##_begin_=(begin);i>=i##_begin_;i--)
#define REP(i, n) FOR(i,0,n)
#define IREP(i, n) IFOR(i,0,n)
template <typename T, typename V>
void ndarray(vector<T>& vec, const V& val, int len) { vec.assign(len, val); }
template <typename T, typename V, typename... Args> void ndarray(vector<T>& vec, const V& val, int len, Args... args) { vec.resize(len), for_each(begin(vec), end(vec), [&](T& v) { ndarray(v, val, args...); }); }
template <typename T> bool chmax(T &m, const T q) { if (m < q) {m = q; return true;} else return false; }
template <typename T> bool chmin(T &m, const T q) { if (m > q) {m = q; return true;} else return false; }
int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); }
template <typename T1, typename T2> pair<T1, T2> operator+(const pair<T1, T2> &l, const pair<T1, T2> &r) { return make_pair(l.first + r.first, l.second + r.second); }
template <typename T1, typename T2> pair<T1, T2> operator-(const pair<T1, T2> &l, const pair<T1, T2> &r) { return make_pair(l.first - r.first, l.second - r.second); }
template <typename T> vector<T> sort_unique(vector<T> vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; }
template <typename T> istream &operator>>(istream &is, vector<T> &vec) { for (auto &v : vec) is >> v; return is; }
template <typename T> ostream &operator<<(ostream &os, const vector<T> &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; }
template <typename T, size_t sz> ostream &operator<<(ostream &os, const array<T, sz> &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; }
#if __cplusplus >= 201703L
template <typename... T> istream &operator>>(istream &is, tuple<T...> &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; }
template <typename... T> ostream &operator<<(ostream &os, const tuple<T...> &tpl) { std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os; }
#endif
template <typename T> ostream &operator<<(ostream &os, const deque<T> &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; }
template <typename T> ostream &operator<<(ostream &os, const set<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <typename T, typename TH> ostream &operator<<(ostream &os, const unordered_set<T, TH> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <typename T> ostream &operator<<(ostream &os, const multiset<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <typename T> ostream &operator<<(ostream &os, const unordered_multiset<T> &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; }
template <typename T1, typename T2> ostream &operator<<(ostream &os, const pair<T1, T2> &pa) { os << '(' << pa.first << ',' << pa.second << ')'; return os; }
template <typename TK, typename TV> ostream &operator<<(ostream &os, const map<TK, TV> &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; }
template <typename TK, typename TV, typename TH> ostream &operator<<(ostream &os, const unordered_map<TK, TV, TH> &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; }
#ifdef HITONANODE_LOCAL
const string COLOR_RESET = "\033[0m", BRIGHT_GREEN = "\033[1;32m", BRIGHT_RED = "\033[1;31m", BRIGHT_CYAN = "\033[1;36m", NORMAL_CROSSED = "\033[0;9;37m", RED_BACKGROUND = "\033[1;41m", NORMAL_FAINT = "\033[0;2m";
#define dbg(x) cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << endl
#else
#define dbg(x) (x)
#endif

// This program is the modificatiosn of the [lemon::NetworkSimplex](http://lemon.cs.elte.hu/pub/doc/latest-svn/a00404.html)
//
/* -*- mode: C++; indent-tabs-mode: nil; -*-
 *
 * This file is a part of LEMON, a generic C++ optimization library.
 *
 * Copyright (C) 2003-2013
 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
 * (Egervary Research Group on Combinatorial Optimization, EGRES).
 *
 * Permission to use, modify and distribute this software is granted
 * provided that this copyright notice appears in all copies. For
 * precise terms see the accompanying LICENSE file.
 *
 * This software is provided "AS IS" with no warranty of any kind,
 * express or implied, and with no claim as to its suitability for any
 * purpose.
 *
 */

template <class Digraph, typename V = int, typename C = V> class NetworkSimplex {
public:
    using Node = int;
    using Arc = int;
    static const int INVALID = -1;
    typedef V Value; /// The type of the flow amounts, capacity bounds and supply values
    typedef C Cost;  /// The type of the arc costs

public:
    enum ProblemType { INFEASIBLE, OPTIMAL, UNBOUNDED };

    /// \brief Constants for selecting the type of the supply constraints.
    ///
    /// Enum type containing constants for selecting the supply type,
    /// i.e. the direction of the inequalities in the supply/demand
    /// constraints of the \ref min_cost_flow "minimum cost flow problem".
    ///
    /// The default supply type is \c GEQ, the \c LEQ type can be
    /// selected using \ref supplyType().
    /// The equality form is a special case of both supply types.
    enum SupplyType {
        /// This option means that there are <em>"greater or equal"</em>
        /// supply/demand constraints in the definition of the problem.
        GEQ,
        /// This option means that there are <em>"less or equal"</em>
        /// supply/demand constraints in the definition of the problem.
        LEQ
    };

    /// \brief Constants for selecting the pivot rule.
    ///
    /// Enum type containing constants for selecting the pivot rule for
    /// the \ref run() function.
    ///
    /// \ref NetworkSimplex provides five different implementations for
    /// the pivot strategy that significantly affects the running time
    /// of the algorithm.
    /// According to experimental tests conducted on various problem
    /// instances, \ref BLOCK_SEARCH "Block Search" and
    /// \ref ALTERING_LIST "Altering Candidate List" rules turned out
    /// to be the most efficient.
    /// Since \ref BLOCK_SEARCH "Block Search" is a simpler strategy that
    /// seemed to be slightly more robust, it is used by default.
    /// However, another pivot rule can easily be selected using the
    /// \ref run() function with the proper parameter.
    enum PivotRule {

        /// The \e First \e Eligible pivot rule.
        /// The next eligible arc is selected in a wraparound fashion
        /// in every iteration.
        FIRST_ELIGIBLE,

        /// The \e Best \e Eligible pivot rule.
        /// The best eligible arc is selected in every iteration.
        BEST_ELIGIBLE,

        /// The \e Block \e Search pivot rule.
        /// A specified number of arcs are examined in every iteration
        /// in a wraparound fashion and the best eligible arc is selected
        /// from this block.
        BLOCK_SEARCH,

        /// The \e Candidate \e List pivot rule.
        /// In a major iteration a candidate list is built from eligible arcs
        /// in a wraparound fashion and in the following minor iterations
        /// the best eligible arc is selected from this list.
        CANDIDATE_LIST,

        /// The \e Altering \e Candidate \e List pivot rule.
        /// It is a modified version of the Candidate List method.
        /// It keeps only a few of the best eligible arcs from the former
        /// candidate list and extends this list in every iteration.
        ALTERING_LIST
    };

private:
    using IntVector = std::vector<int>;
    using ValueVector = std::vector<Value>;
    using CostVector = std::vector<Cost>;
    using CharVector = std::vector<signed char>;
    enum ArcState { STATE_UPPER = -1, STATE_TREE = 0, STATE_LOWER = 1 };
    enum ArcDirection { DIR_DOWN = -1, DIR_UP = 1 };

private:
    // Data related to the underlying digraph
    const Digraph &_graph;
    int _node_num;
    int _arc_num;
    int _all_arc_num;
    int _search_arc_num;

    // Parameters of the problem
    bool _has_lower;
    SupplyType _stype;
    Value _sum_supply;

    // Data structures for storing the digraph
    IntVector _source;
    IntVector _target;

    // Node and arc data
    ValueVector _lower;
    ValueVector _upper;
    ValueVector _cap;
    CostVector _cost;
    ValueVector _supply;
    ValueVector _flow;
    CostVector _pi;

    // Data for storing the spanning tree structure
    IntVector _parent;
    IntVector _pred;
    IntVector _thread;
    IntVector _rev_thread;
    IntVector _succ_num;
    IntVector _last_succ;
    CharVector _pred_dir;
    CharVector _state;
    IntVector _dirty_revs;
    int _root;

    // Temporary data used in the current pivot iteration
    int in_arc, join, u_in, v_in, u_out, v_out;
    Value delta;

    const Value MAX;

public:
    /// \brief Constant for infinite upper bounds (capacities).
    ///
    /// Constant for infinite upper bounds (capacities).
    /// It is \c std::numeric_limits<Value>::infinity() if available,
    /// \c std::numeric_limits<Value>::max() otherwise.
    const Value INF;

private:
    // Implementation of the First Eligible pivot rule
    class FirstEligiblePivotRule {
    private:
        // References to the NetworkSimplex class
        const IntVector &_source;
        const IntVector &_target;
        const CostVector &_cost;
        const CharVector &_state;
        const CostVector &_pi;
        int &_in_arc;
        int _search_arc_num;

        // Pivot rule data
        int _next_arc;

    public:
        // Constructor
        FirstEligiblePivotRule(NetworkSimplex &ns)
            : _source(ns._source), _target(ns._target), _cost(ns._cost), _state(ns._state), _pi(ns._pi), _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num), _next_arc(0) {}

        // Find next entering arc
        bool findEnteringArc() {
            Cost c;
            for (int e = _next_arc; e != _search_arc_num; ++e) {
                c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
                if (c < 0) {
                    _in_arc = e;
                    _next_arc = e + 1;
                    return true;
                }
            }
            for (int e = 0; e != _next_arc; ++e) {
                c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
                if (c < 0) {
                    _in_arc = e;
                    _next_arc = e + 1;
                    return true;
                }
            }
            return false;
        }

    }; // class FirstEligiblePivotRule

    // Implementation of the Best Eligible pivot rule
    class BestEligiblePivotRule {
    private:
        // References to the NetworkSimplex class
        const IntVector &_source;
        const IntVector &_target;
        const CostVector &_cost;
        const CharVector &_state;
        const CostVector &_pi;
        int &_in_arc;
        int _search_arc_num;

    public:
        // Constructor
        BestEligiblePivotRule(NetworkSimplex &ns)
            : _source(ns._source), _target(ns._target), _cost(ns._cost), _state(ns._state), _pi(ns._pi), _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num) {}

        // Find next entering arc
        bool findEnteringArc() {
            Cost c, min = 0;
            for (int e = 0; e != _search_arc_num; ++e) {
                c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
                if (c < min) {
                    min = c;
                    _in_arc = e;
                }
            }
            return min < 0;
        }

    }; // class BestEligiblePivotRule

    // Implementation of the Block Search pivot rule
    class BlockSearchPivotRule {
    private:
        // References to the NetworkSimplex class
        const IntVector &_source;
        const IntVector &_target;
        const CostVector &_cost;
        const CharVector &_state;
        const CostVector &_pi;
        int &_in_arc;
        int _search_arc_num;

        // Pivot rule data
        int _block_size;
        int _next_arc;

    public:
        // Constructor
        BlockSearchPivotRule(NetworkSimplex &ns)
            : _source(ns._source), _target(ns._target), _cost(ns._cost), _state(ns._state), _pi(ns._pi), _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num), _next_arc(0) {
            // The main parameters of the pivot rule
            const double BLOCK_SIZE_FACTOR = 1.0;
            const int MIN_BLOCK_SIZE = 10;

            _block_size = std::max(int(BLOCK_SIZE_FACTOR * std::sqrt(double(_search_arc_num))), MIN_BLOCK_SIZE);
        }

        // Find next entering arc
        bool findEnteringArc() {
            Cost c, min = 0;
            int cnt = _block_size;
            int e;
            for (e = _next_arc; e != _search_arc_num; ++e) {
                c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
                if (c < min) {
                    min = c;
                    _in_arc = e;
                }
                if (--cnt == 0) {
                    if (min < 0) goto search_end;
                    cnt = _block_size;
                }
            }
            for (e = 0; e != _next_arc; ++e) {
                c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
                if (c < min) {
                    min = c;
                    _in_arc = e;
                }
                if (--cnt == 0) {
                    if (min < 0) goto search_end;
                    cnt = _block_size;
                }
            }
            if (min >= 0) return false;

        search_end:
            _next_arc = e;
            return true;
        }

    }; // class BlockSearchPivotRule

    // Implementation of the Candidate List pivot rule
    class CandidateListPivotRule {
    private:
        // References to the NetworkSimplex class
        const IntVector &_source;
        const IntVector &_target;
        const CostVector &_cost;
        const CharVector &_state;
        const CostVector &_pi;
        int &_in_arc;
        int _search_arc_num;

        // Pivot rule data
        IntVector _candidates;
        int _list_length, _minor_limit;
        int _curr_length, _minor_count;
        int _next_arc;

    public:
        /// Constructor
        CandidateListPivotRule(NetworkSimplex &ns)
            : _source(ns._source), _target(ns._target), _cost(ns._cost), _state(ns._state), _pi(ns._pi), _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num), _next_arc(0) {
            // The main parameters of the pivot rule
            const double LIST_LENGTH_FACTOR = 0.25;
            const int MIN_LIST_LENGTH = 10;
            const double MINOR_LIMIT_FACTOR = 0.1;
            const int MIN_MINOR_LIMIT = 3;

            _list_length = std::max(int(LIST_LENGTH_FACTOR * std::sqrt(double(_search_arc_num))), MIN_LIST_LENGTH);
            _minor_limit = std::max(int(MINOR_LIMIT_FACTOR * _list_length), MIN_MINOR_LIMIT);
            _curr_length = _minor_count = 0;
            _candidates.resize(_list_length);
        }

        /// Find next entering arc
        bool findEnteringArc() {
            Cost min, c;
            int e;
            if (_curr_length > 0 && _minor_count < _minor_limit) {
                // Minor iteration: select the best eligible arc from the
                // current candidate list
                ++_minor_count;
                min = 0;
                for (int i = 0; i < _curr_length; ++i) {
                    e = _candidates[i];
                    c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
                    if (c < min) {
                        min = c;
                        _in_arc = e;
                    } else if (c >= 0) {
                        _candidates[i--] = _candidates[--_curr_length];
                    }
                }
                if (min < 0) return true;
            }

            // Major iteration: build a new candidate list
            min = 0;
            _curr_length = 0;
            for (e = _next_arc; e != _search_arc_num; ++e) {
                c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
                if (c < 0) {
                    _candidates[_curr_length++] = e;
                    if (c < min) {
                        min = c;
                        _in_arc = e;
                    }
                    if (_curr_length == _list_length) goto search_end;
                }
            }
            for (e = 0; e != _next_arc; ++e) {
                c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
                if (c < 0) {
                    _candidates[_curr_length++] = e;
                    if (c < min) {
                        min = c;
                        _in_arc = e;
                    }
                    if (_curr_length == _list_length) goto search_end;
                }
            }
            if (_curr_length == 0) return false;

        search_end:
            _minor_count = 1;
            _next_arc = e;
            return true;
        }

    }; // class CandidateListPivotRule

    // Implementation of the Altering Candidate List pivot rule
    class AlteringListPivotRule {
    private:
        // References to the NetworkSimplex class
        const IntVector &_source;
        const IntVector &_target;
        const CostVector &_cost;
        const CharVector &_state;
        const CostVector &_pi;
        int &_in_arc;
        int _search_arc_num;

        // Pivot rule data
        int _block_size, _head_length, _curr_length;
        int _next_arc;
        IntVector _candidates;
        CostVector _cand_cost;

        // Functor class to compare arcs during sort of the candidate list
        class SortFunc {
        private:
            const CostVector &_map;

        public:
            SortFunc(const CostVector &map) : _map(map) {}
            bool operator()(int left, int right) { return _map[left] < _map[right]; }
        };

        SortFunc _sort_func;

    public:
        // Constructor
        AlteringListPivotRule(NetworkSimplex &ns)
            : _source(ns._source), _target(ns._target), _cost(ns._cost), _state(ns._state), _pi(ns._pi), _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num), _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost) {
            // The main parameters of the pivot rule
            const double BLOCK_SIZE_FACTOR = 1.0;
            const int MIN_BLOCK_SIZE = 10;
            const double HEAD_LENGTH_FACTOR = 0.01;
            const int MIN_HEAD_LENGTH = 3;

            _block_size = std::max(int(BLOCK_SIZE_FACTOR * std::sqrt(double(_search_arc_num))), MIN_BLOCK_SIZE);
            _head_length = std::max(int(HEAD_LENGTH_FACTOR * _block_size), MIN_HEAD_LENGTH);
            _candidates.resize(_head_length + _block_size);
            _curr_length = 0;
        }

        // Find next entering arc
        bool findEnteringArc() {
            // Check the current candidate list
            int e;
            Cost c;
            for (int i = 0; i != _curr_length; ++i) {
                e = _candidates[i];
                c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
                if (c < 0) {
                    _cand_cost[e] = c;
                } else {
                    _candidates[i--] = _candidates[--_curr_length];
                }
            }

            // Extend the list
            int cnt = _block_size;
            int limit = _head_length;

            for (e = _next_arc; e != _search_arc_num; ++e) {
                c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
                if (c < 0) {
                    _cand_cost[e] = c;
                    _candidates[_curr_length++] = e;
                }
                if (--cnt == 0) {
                    if (_curr_length > limit) goto search_end;
                    limit = 0;
                    cnt = _block_size;
                }
            }
            for (e = 0; e != _next_arc; ++e) {
                c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
                if (c < 0) {
                    _cand_cost[e] = c;
                    _candidates[_curr_length++] = e;
                }
                if (--cnt == 0) {
                    if (_curr_length > limit) goto search_end;
                    limit = 0;
                    cnt = _block_size;
                }
            }
            if (_curr_length == 0) return false;

        search_end:

            // Perform partial sort operation on the candidate list
            int new_length = std::min(_head_length + 1, _curr_length);
            std::partial_sort(_candidates.begin(), _candidates.begin() + new_length, _candidates.begin() + _curr_length, _sort_func);

            // Select the entering arc and remove it from the list
            _in_arc = _candidates[0];
            _next_arc = e;
            _candidates[0] = _candidates[new_length - 1];
            _curr_length = new_length - 1;
            return true;
        }

    }; // class AlteringListPivotRule

public:
    NetworkSimplex(const Digraph &graph)
        : _graph(graph), MAX(std::numeric_limits<Value>::max()), INF(std::numeric_limits<Value>::has_infinity ? std::numeric_limits<Value>::infinity() : MAX) {
        // Check the number types
        static_assert(std::numeric_limits<Value>::is_signed);
        static_assert(std::numeric_limits<Cost>::is_signed);
        static_assert(std::numeric_limits<Value>::max() > 0);

        // Reset data structures
        reset();
    }

    template <typename LowerMap> NetworkSimplex &lowerMap(const LowerMap &map) {
        _has_lower = true;
        for (Arc a = 0; a < _arc_num; a++) _lower[a] = map[a];
        return *this;
    }
    template <typename UpperMap> NetworkSimplex &upperMap(const UpperMap &map) {
        for (Arc a = 0; a < _arc_num; a++) _upper[a] = map[a];
        return *this;
    }
    // Set costs of arcs (default value: 1)
    template <typename CostMap> NetworkSimplex &costMap(const CostMap &map) {
        for (Arc a = 0; a < _arc_num; a++) _cost[a] = map[a];
        return *this;
    }
    template <typename SupplyMap> NetworkSimplex &supplyMap(const SupplyMap &map) {
        for (Node n = 0; n < _node_num; n++) _supply[n] = map[n];
        return *this;
    }
    NetworkSimplex &stSupply(const Node &s, const Node &t, Value k) { // set s-t flow
        for (int i = 0; i != _node_num; ++i) _supply[i] = 0;
        _supply[s] = k, _supply[t] = -k;
        return *this;
    }

    /// \brief Set the type of the supply constraints.
    ///
    /// This function sets the type of the supply/demand constraints.
    /// If it is not used before calling \ref run(), the \ref GEQ supply
    /// type will be used.
    NetworkSimplex &supplyType(SupplyType supply_type) {
        _stype = supply_type;
        return *this;
    }

    /// @}

    /// This function can be called more than once. All the given parameters
    /// are kept for the next call, unless \ref resetParams() or \ref reset()
    /// is used, thus only the modified parameters have to be set again.
    /// If the underlying digraph was also modified after the construction
    /// of the class (or the last \ref reset() call), then the \ref reset()
    /// function must be called.
    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
        if (!init()) return INFEASIBLE;
        return start(pivot_rule);
    }

    /// \brief Reset all the parameters that have been given before.
    ///
    /// This function resets all the paramaters that have been given
    /// before using functions \ref lowerMap(), \ref upperMap(),
    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
    ///
    /// It is useful for multiple \ref run() calls. Basically, all the given
    /// parameters are kept for the next \ref run() call, unless
    /// \ref resetParams() or \ref reset() is used.
    /// If the underlying digraph was also modified after the construction
    /// of the class or the last \ref reset() call, then the \ref reset()
    /// function must be used, otherwise \ref resetParams() is sufficient.
    ///
    /// For example,
    /// \code
    ///   NetworkSimplex<ListDigraph> ns(graph);
    ///
    ///   // First run
    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
    ///     .supplyMap(sup).run();
    ///
    ///   // Run again with modified cost map (resetParams() is not called,
    ///   // so only the cost map have to be set again)
    ///   cost[e] += 100;
    ///   ns.costMap(cost).run();
    ///
    ///   // Run again from scratch using resetParams()
    ///   // (the lower bounds will be set to zero on all arcs)
    ///   ns.resetParams();
    ///   ns.upperMap(capacity).costMap(cost)
    ///     .supplyMap(sup).run();
    /// \endcode
    ///
    /// \return <tt>(*this)</tt>
    ///
    /// \see reset(), run()
    NetworkSimplex &resetParams() {
        for (int i = 0; i != _node_num; ++i) { _supply[i] = 0; }
        for (int i = 0; i != _arc_num; ++i) {
            _lower[i] = 0;
            _upper[i] = INF;
            _cost[i] = 1;
        }
        _has_lower = false;
        _stype = GEQ;
        return *this;
    }

    /// \brief Reset the internal data structures and all the parameters
    /// that have been given before.
    ///
    /// This function resets the internal data structures and all the
    /// paramaters that have been given before using functions \ref lowerMap(),
    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
    /// \ref supplyType().
    ///
    /// It is useful for multiple \ref run() calls. Basically, all the given
    /// parameters are kept for the next \ref run() call, unless
    /// \ref resetParams() or \ref reset() is used.
    /// If the underlying digraph was also modified after the construction
    /// of the class or the last \ref reset() call, then the \ref reset()
    /// function must be used, otherwise \ref resetParams() is sufficient.
    ///
    /// See \ref resetParams() for examples.
    ///
    /// \return <tt>(*this)</tt>
    ///
    /// \see resetParams(), run()
    NetworkSimplex &reset() {
        // Resize vectors
        _node_num = _graph.countNodes();
        _arc_num = _graph.countArcs();
        int all_node_num = _node_num + 1;
        int max_arc_num = _arc_num + 2 * _node_num;

        _source.resize(max_arc_num);
        _target.resize(max_arc_num);

        _lower.resize(_arc_num);
        _upper.resize(_arc_num);
        _cap.resize(max_arc_num);
        _cost.resize(max_arc_num);
        _supply.resize(all_node_num);
        _flow.resize(max_arc_num);
        _pi.resize(all_node_num);

        _parent.resize(all_node_num);
        _pred.resize(all_node_num);
        _pred_dir.resize(all_node_num);
        _thread.resize(all_node_num);
        _rev_thread.resize(all_node_num);
        _succ_num.resize(all_node_num);
        _last_succ.resize(all_node_num);
        _state.resize(max_arc_num);

        for (int a = 0; a < _arc_num; ++a) {
            _source[a] = _graph.source(a);
            _target[a] = _graph.target(a);
        }
        // Reset parameters
        resetParams();
        return *this;
    }

    /// @}

    template <typename Number = Cost> Number totalCost() const {
        Number c = 0;
        for (Arc a = 0; a < _arc_num; a++) c += Number(_flow[a]) * Number(_cost[a]);
        return c;
    }

    Value flow(const Arc &a) const { return _flow[a]; }
    template <typename FlowMap> void flowMap(FlowMap &map) const {
        for (Arc a = 0; a < _arc_num; a++) { map.set(a, _flow[a]); }
    }
    ValueVector flowMap() const { return _flow; }

    Cost potential(const Node &n) const { return _pi[n]; }
    template <typename PotentialMap> void potentialMap(PotentialMap &map) const {
        for (int n = 0; n < _graph.V; n++) { map.set(n, _pi[n]); }
    }
    CostVector potentialMap() const { return _pi; }

private:
    // Initialize internal data structures
    bool init() {
        if (_node_num == 0) return false;

        // Check the sum of supply values
        _sum_supply = 0;
        for (int i = 0; i != _node_num; ++i) { _sum_supply += _supply[i]; }
        if (!((_stype == GEQ && _sum_supply <= 0) || (_stype == LEQ && _sum_supply >= 0))) return false;

        // Check lower and upper bounds
        // LEMON_DEBUG(checkBoundMaps(), "Upper bounds must be greater or equal to the lower bounds");

        // Remove non-zero lower bounds
        if (_has_lower) {
            for (int i = 0; i != _arc_num; ++i) {
                Value c = _lower[i];
                if (c >= 0) {
                    _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;
                } else {
                    _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;
                }
                _supply[_source[i]] -= c;
                _supply[_target[i]] += c;
            }
        } else {
            for (int i = 0; i != _arc_num; ++i) { _cap[i] = _upper[i]; }
        }

        // Initialize artifical cost
        Cost ART_COST;
        if (std::numeric_limits<Cost>::is_exact) {
            ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
        } else {
            ART_COST = 0;
            for (int i = 0; i != _arc_num; ++i) {
                if (_cost[i] > ART_COST) ART_COST = _cost[i];
            }
            ART_COST = (ART_COST + 1) * _node_num;
        }

        // Initialize arc maps
        for (int i = 0; i != _arc_num; ++i) {
            _flow[i] = 0;
            _state[i] = STATE_LOWER;
        }

        // Set data for the artificial root node
        _root = _node_num;
        _parent[_root] = -1;
        _pred[_root] = -1;
        _thread[_root] = 0;
        _rev_thread[0] = _root;
        _succ_num[_root] = _node_num + 1;
        _last_succ[_root] = _root - 1;
        _supply[_root] = -_sum_supply;
        _pi[_root] = 0;

        // Add artificial arcs and initialize the spanning tree data structure
        if (_sum_supply == 0) {
            // EQ supply constraints
            _search_arc_num = _arc_num;
            _all_arc_num = _arc_num + _node_num;
            for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
                _parent[u] = _root;
                _pred[u] = e;
                _thread[u] = u + 1;
                _rev_thread[u + 1] = u;
                _succ_num[u] = 1;
                _last_succ[u] = u;
                _cap[e] = INF;
                _state[e] = STATE_TREE;
                if (_supply[u] >= 0) {
                    _pred_dir[u] = DIR_UP;
                    _pi[u] = 0;
                    _source[e] = u;
                    _target[e] = _root;
                    _flow[e] = _supply[u];
                    _cost[e] = 0;
                } else {
                    _pred_dir[u] = DIR_DOWN;
                    _pi[u] = ART_COST;
                    _source[e] = _root;
                    _target[e] = u;
                    _flow[e] = -_supply[u];
                    _cost[e] = ART_COST;
                }
            }
        } else if (_sum_supply > 0) {
            // LEQ supply constraints
            _search_arc_num = _arc_num + _node_num;
            int f = _arc_num + _node_num;
            for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
                _parent[u] = _root;
                _thread[u] = u + 1;
                _rev_thread[u + 1] = u;
                _succ_num[u] = 1;
                _last_succ[u] = u;
                if (_supply[u] >= 0) {
                    _pred_dir[u] = DIR_UP;
                    _pi[u] = 0;
                    _pred[u] = e;
                    _source[e] = u;
                    _target[e] = _root;
                    _cap[e] = INF;
                    _flow[e] = _supply[u];
                    _cost[e] = 0;
                    _state[e] = STATE_TREE;
                } else {
                    _pred_dir[u] = DIR_DOWN;
                    _pi[u] = ART_COST;
                    _pred[u] = f;
                    _source[f] = _root;
                    _target[f] = u;
                    _cap[f] = INF;
                    _flow[f] = -_supply[u];
                    _cost[f] = ART_COST;
                    _state[f] = STATE_TREE;
                    _source[e] = u;
                    _target[e] = _root;
                    _cap[e] = INF;
                    _flow[e] = 0;
                    _cost[e] = 0;
                    _state[e] = STATE_LOWER;
                    ++f;
                }
            }
            _all_arc_num = f;
        } else {
            // GEQ supply constraints
            _search_arc_num = _arc_num + _node_num;
            int f = _arc_num + _node_num;
            for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
                _parent[u] = _root;
                _thread[u] = u + 1;
                _rev_thread[u + 1] = u;
                _succ_num[u] = 1;
                _last_succ[u] = u;
                if (_supply[u] <= 0) {
                    _pred_dir[u] = DIR_DOWN;
                    _pi[u] = 0;
                    _pred[u] = e;
                    _source[e] = _root;
                    _target[e] = u;
                    _cap[e] = INF;
                    _flow[e] = -_supply[u];
                    _cost[e] = 0;
                    _state[e] = STATE_TREE;
                } else {
                    _pred_dir[u] = DIR_UP;
                    _pi[u] = -ART_COST;
                    _pred[u] = f;
                    _source[f] = u;
                    _target[f] = _root;
                    _cap[f] = INF;
                    _flow[f] = _supply[u];
                    _state[f] = STATE_TREE;
                    _cost[f] = ART_COST;
                    _source[e] = _root;
                    _target[e] = u;
                    _cap[e] = INF;
                    _flow[e] = 0;
                    _cost[e] = 0;
                    _state[e] = STATE_LOWER;
                    ++f;
                }
            }
            _all_arc_num = f;
        }

        return true;
    }

    // Check if the upper bound is greater than or equal to the lower bound
    // on each arc.
    bool checkBoundMaps() {
        for (int j = 0; j != _arc_num; ++j) {
            if (_upper[j] < _lower[j]) return false;
        }
        return true;
    }

    // Find the join node
    void findJoinNode() {
        int u = _source[in_arc];
        int v = _target[in_arc];
        while (u != v) {
            if (_succ_num[u] < _succ_num[v]) {
                u = _parent[u];
            } else {
                v = _parent[v];
            }
        }
        join = u;
    }

    // Find the leaving arc of the cycle and returns true if the
    // leaving arc is not the same as the entering arc
    bool findLeavingArc() {
        // Initialize first and second nodes according to the direction
        // of the cycle
        int first, second;
        if (_state[in_arc] == STATE_LOWER) {
            first = _source[in_arc];
            second = _target[in_arc];
        } else {
            first = _target[in_arc];
            second = _source[in_arc];
        }
        delta = _cap[in_arc];
        int result = 0;
        Value c, d;
        int e;

        // Search the cycle form the first node to the join node
        for (int u = first; u != join; u = _parent[u]) {
            e = _pred[u];
            d = _flow[e];
            if (_pred_dir[u] == DIR_DOWN) {
                c = _cap[e];
                d = c >= MAX ? INF : c - d;
            }
            if (d < delta) {
                delta = d;
                u_out = u;
                result = 1;
            }
        }

        // Search the cycle form the second node to the join node
        for (int u = second; u != join; u = _parent[u]) {
            e = _pred[u];
            d = _flow[e];
            if (_pred_dir[u] == DIR_UP) {
                c = _cap[e];
                d = c >= MAX ? INF : c - d;
            }
            if (d <= delta) {
                delta = d;
                u_out = u;
                result = 2;
            }
        }

        if (result == 1) {
            u_in = first;
            v_in = second;
        } else {
            u_in = second;
            v_in = first;
        }
        return result != 0;
    }

    // Change _flow and _state vectors
    void changeFlow(bool change) {
        // Augment along the cycle
        if (delta > 0) {
            Value val = _state[in_arc] * delta;
            _flow[in_arc] += val;
            for (int u = _source[in_arc]; u != join; u = _parent[u]) { _flow[_pred[u]] -= _pred_dir[u] * val; }
            for (int u = _target[in_arc]; u != join; u = _parent[u]) { _flow[_pred[u]] += _pred_dir[u] * val; }
        }
        // Update the state of the entering and leaving arcs
        if (change) {
            _state[in_arc] = STATE_TREE;
            _state[_pred[u_out]] = (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
        } else {
            _state[in_arc] = -_state[in_arc];
        }
    }

    // Update the tree structure
    void updateTreeStructure() {
        int old_rev_thread = _rev_thread[u_out];
        int old_succ_num = _succ_num[u_out];
        int old_last_succ = _last_succ[u_out];
        v_out = _parent[u_out];

        // Check if u_in and u_out coincide
        if (u_in == u_out) {
            // Update _parent, _pred, _pred_dir
            _parent[u_in] = v_in;
            _pred[u_in] = in_arc;
            _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;

            // Update _thread and _rev_thread
            if (_thread[v_in] != u_out) {
                int after = _thread[old_last_succ];
                _thread[old_rev_thread] = after;
                _rev_thread[after] = old_rev_thread;
                after = _thread[v_in];
                _thread[v_in] = u_out;
                _rev_thread[u_out] = v_in;
                _thread[old_last_succ] = after;
                _rev_thread[after] = old_last_succ;
            }
        } else {
            // Handle the case when old_rev_thread equals to v_in
            // (it also means that join and v_out coincide)
            int thread_continue = old_rev_thread == v_in ? _thread[old_last_succ] : _thread[v_in];

            // Update _thread and _parent along the stem nodes (i.e. the nodes
            // between u_in and u_out, whose parent have to be changed)
            int stem = u_in;             // the current stem node
            int par_stem = v_in;         // the new parent of stem
            int next_stem;               // the next stem node
            int last = _last_succ[u_in]; // the last successor of stem
            int before, after = _thread[last];
            _thread[v_in] = u_in;
            _dirty_revs.clear();
            _dirty_revs.push_back(v_in);
            while (stem != u_out) {
                // Insert the next stem node into the thread list
                next_stem = _parent[stem];
                _thread[last] = next_stem;
                _dirty_revs.push_back(last);

                // Remove the subtree of stem from the thread list
                before = _rev_thread[stem];
                _thread[before] = after;
                _rev_thread[after] = before;

                // Change the parent node and shift stem nodes
                _parent[stem] = par_stem;
                par_stem = stem;
                stem = next_stem;

                // Update last and after
                last = _last_succ[stem] == _last_succ[par_stem] ? _rev_thread[par_stem] : _last_succ[stem];
                after = _thread[last];
            }
            _parent[u_out] = par_stem;
            _thread[last] = thread_continue;
            _rev_thread[thread_continue] = last;
            _last_succ[u_out] = last;

            // Remove the subtree of u_out from the thread list except for
            // the case when old_rev_thread equals to v_in
            if (old_rev_thread != v_in) {
                _thread[old_rev_thread] = after;
                _rev_thread[after] = old_rev_thread;
            }

            // Update _rev_thread using the new _thread values
            for (int i = 0; i != int(_dirty_revs.size()); ++i) {
                int u = _dirty_revs[i];
                _rev_thread[_thread[u]] = u;
            }

            // Update _pred, _pred_dir, _last_succ and _succ_num for the
            // stem nodes from u_out to u_in
            int tmp_sc = 0, tmp_ls = _last_succ[u_out];
            for (int u = u_out, p = _parent[u]; u != u_in; u = p, p = _parent[u]) {
                _pred[u] = _pred[p];
                _pred_dir[u] = -_pred_dir[p];
                tmp_sc += _succ_num[u] - _succ_num[p];
                _succ_num[u] = tmp_sc;
                _last_succ[p] = tmp_ls;
            }
            _pred[u_in] = in_arc;
            _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;
            _succ_num[u_in] = old_succ_num;
        }

        // Update _last_succ from v_in towards the root
        int up_limit_out = _last_succ[join] == v_in ? join : -1;
        int last_succ_out = _last_succ[u_out];
        for (int u = v_in; u != -1 && _last_succ[u] == v_in; u = _parent[u]) { _last_succ[u] = last_succ_out; }

        // Update _last_succ from v_out towards the root
        if (join != old_rev_thread && v_in != old_rev_thread) {
            for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ; u = _parent[u]) {
                _last_succ[u] = old_rev_thread;
            }
        } else if (last_succ_out != old_last_succ) {
            for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ; u = _parent[u]) {
                _last_succ[u] = last_succ_out;
            }
        }

        // Update _succ_num from v_in to join
        for (int u = v_in; u != join; u = _parent[u]) { _succ_num[u] += old_succ_num; }
        // Update _succ_num from v_out to join
        for (int u = v_out; u != join; u = _parent[u]) { _succ_num[u] -= old_succ_num; }
    }

    // Update potentials in the subtree that has been moved
    void updatePotential() {
        Cost sigma = _pi[v_in] - _pi[u_in] - _pred_dir[u_in] * _cost[in_arc];
        int end = _thread[_last_succ[u_in]];
        for (int u = u_in; u != end; u = _thread[u]) { _pi[u] += sigma; }
    }

    // Heuristic initial pivots
    bool initialPivots() {
        Value curr, total = 0;
        std::vector<Node> supply_nodes, demand_nodes;
        for (int u = 0; u < _node_num; ++u) {
            curr = _supply[u];
            if (curr > 0) {
                total += curr;
                supply_nodes.push_back(u);
            } else if (curr < 0) {
                demand_nodes.push_back(u);
            }
        }
        if (_sum_supply > 0) total -= _sum_supply;
        if (total <= 0) return true;

        IntVector arc_vector;
        if (_sum_supply >= 0) {
            if (supply_nodes.size() == 1 && demand_nodes.size() == 1) {
                // Perform a reverse graph search from the sink to the source
                std::vector<char> reached(_node_num, false);
                Node s = supply_nodes[0], t = demand_nodes[0];
                std::vector<Node> stack;
                reached[t] = true;
                stack.push_back(t);
                while (!stack.empty()) {
                    Node u, v = stack.back();
                    stack.pop_back();
                    if (v == s) break;
                    // for (InArcIt a(_graph, v); a != INVALID; ++a) {
                    for (auto a : _graph.in_eids[v]) {
                        if (reached[u = _graph.source(a)]) continue;
                        int j = a;
                        if (_cap[j] >= total) {
                            arc_vector.push_back(j);
                            reached[u] = true;
                            stack.push_back(u);
                        }
                    }
                }
            } else {
                // Find the min. cost incoming arc for each demand node
                for (int i = 0; i != int(demand_nodes.size()); ++i) {
                    Node v = demand_nodes[i];
                    Cost c, min_cost = std::numeric_limits<Cost>::max();
                    Arc min_arc = INVALID;
                    for (auto a : _graph.in_eids[v]) {
                        // for (InArcIt a(_graph, v); a != INVALID; ++a) {
                        c = _cost[a];
                        if (c < min_cost) {
                            min_cost = c;
                            min_arc = a;
                        }
                    }
                    if (min_arc != INVALID) { arc_vector.push_back(min_arc); }
                }
            }
        } else {
            // Find the min. cost outgoing arc for each supply node
            for (Node u : supply_nodes) {
                Cost c, min_cost = std::numeric_limits<Cost>::max();
                Arc min_arc = INVALID;
                for (auto a : _graph.out_eids[u]) {
                    c = _cost[a];
                    if (c < min_cost) {
                        min_cost = c;
                        min_arc = a;
                    }
                }
                if (min_arc != INVALID) { arc_vector.push_back(min_arc); }
            }
        }

        // Perform heuristic initial pivots
        for (int i = 0; i != int(arc_vector.size()); ++i) {
            in_arc = arc_vector[i];
            if (_state[in_arc] * (_cost[in_arc] + _pi[_source[in_arc]] - _pi[_target[in_arc]]) >= 0) continue;
            findJoinNode();
            bool change = findLeavingArc();
            if (delta >= MAX) return false;
            changeFlow(change);
            if (change) {
                updateTreeStructure();
                updatePotential();
            }
        }
        return true;
    }

    // Execute the algorithm
    ProblemType start(PivotRule pivot_rule) {
        // Select the pivot rule implementation
        switch (pivot_rule) {
        case FIRST_ELIGIBLE: return start<FirstEligiblePivotRule>();
        case BEST_ELIGIBLE: return start<BestEligiblePivotRule>();
        case BLOCK_SEARCH: return start<BlockSearchPivotRule>();
        case CANDIDATE_LIST: return start<CandidateListPivotRule>();
        case ALTERING_LIST: return start<AlteringListPivotRule>();
        }
        return INFEASIBLE; // avoid warning
    }

    template <typename PivotRuleImpl> ProblemType start() {
        PivotRuleImpl pivot(*this);

        // Perform heuristic initial pivots
        if (!initialPivots()) return UNBOUNDED;

        // Execute the Network Simplex algorithm
        while (pivot.findEnteringArc()) {
            findJoinNode();
            bool change = findLeavingArc();
            if (delta >= MAX) return UNBOUNDED;
            changeFlow(change);
            if (change) {
                updateTreeStructure();
                updatePotential();
            }
        }

        // Check feasibility
        for (int e = _search_arc_num; e != _all_arc_num; ++e) {
            if (_flow[e] != 0) return INFEASIBLE;
        }

        // Transform the solution and the supply map to the original form
        if (_has_lower) {
            for (int i = 0; i != _arc_num; ++i) {
                Value c = _lower[i];
                if (c != 0) {
                    _flow[i] += c;
                    _supply[_source[i]] += c;
                    _supply[_target[i]] -= c;
                }
            }
        }

        // Shift potentials to meet the requirements of the GEQ/LEQ type
        // optimality conditions
        if (_sum_supply == 0) {
            if (_stype == GEQ) {
                Cost max_pot = -std::numeric_limits<Cost>::max();
                for (int i = 0; i != _node_num; ++i) {
                    if (_pi[i] > max_pot) max_pot = _pi[i];
                }
                if (max_pot > 0) {
                    for (int i = 0; i != _node_num; ++i) _pi[i] -= max_pot;
                }
            } else {
                Cost min_pot = std::numeric_limits<Cost>::max();
                for (int i = 0; i != _node_num; ++i) {
                    if (_pi[i] < min_pot) min_pot = _pi[i];
                }
                if (min_pot < 0) {
                    for (int i = 0; i != _node_num; ++i) _pi[i] -= min_pot;
                }
            }
        }

        return OPTIMAL;
    }
}; // class NetworkSimplex

template <typename Capacity = long long, typename Weight = long long> struct mcf_graph_ns {

    struct Digraph {
        const int V;
        int E;
        std::vector<std::vector<int>> in_eids, out_eids;
        std::vector<std::pair<int, int>> arcs;
        Digraph(int V = 0) : V(V), E(0), in_eids(V), out_eids(V){};
        int add_edge(int s, int t) {
            assert(0 <= s and s < V);
            assert(0 <= t and t < V);
            in_eids[t].push_back(E), out_eids[s].push_back(E), arcs.emplace_back(s, t), E++;
            return E - 1;
        }
        int countNodes() const noexcept { return V; }
        int countArcs() const noexcept { return E; }
        int source(int arcid) const { return arcs[arcid].first; }
        int target(int arcid) const { return arcs[arcid].second; }
    };

    struct edge {
        int eid;
        int from, to;
        Capacity lo, hi;
        Weight weight;
    };

    int n;
    std::vector<Capacity> bs;
    bool infeasible;
    std::vector<edge> Edges;

    mcf_graph_ns(int V = 0) : n(V), bs(V), infeasible(false) {}

    int add_edge(int from, int to, Capacity lower, Capacity upper, Weight weight) {
        assert(from >= 0 and from < n);
        assert(to >= 0 and to < n);
        int idnow = Edges.size();
        Edges.push_back({idnow, from, to, lower, upper, weight});
        return idnow;
    }
    void set_supply(int v, Capacity b) {
        assert(v >= 0 and v < n);
        bs[v] = b;
    }
    std::vector<Capacity> flow;
    std::vector<Capacity> potential;

    template <typename RetVal = __int128> [[nodiscard]] RetVal solve() {
        std::mt19937 rng(std::chrono::steady_clock::now().time_since_epoch().count());

        std::vector<int> vid(n), eid(Edges.size());
        std::iota(vid.begin(), vid.end(), 0);
        std::shuffle(vid.begin(), vid.end(), rng);
        std::iota(eid.begin(), eid.end(), 0);
        std::shuffle(eid.begin(), eid.end(), rng);

        flow.clear();
        potential.clear();
        Digraph graph(n + 1);

        std::vector<Capacity> supplies(graph.countNodes());
        std::vector<Capacity> lowers(Edges.size());
        std::vector<Capacity> uppers(Edges.size());
        std::vector<Weight> weights(Edges.size());

        for (int i = 0; i < n; i++) supplies[vid[i]] = bs[i];
        for (auto i : eid) {
            const auto &e = Edges[i];
            int arc = graph.add_edge(vid[e.from], vid[e.to]);
            lowers[arc] = e.lo;
            uppers[arc] = e.hi;
            weights[arc] = e.weight;
        }

        NetworkSimplex<Digraph, Capacity, Weight> ns(graph);
        auto status = ns.supplyMap(supplies).costMap(weights).lowerMap(lowers).upperMap(uppers).run(decltype(ns)::BLOCK_SEARCH);

        if (status == decltype(ns)::INFEASIBLE) {
            return infeasible = true, 0;
        } else {
            flow.resize(Edges.size());
            potential.resize(n);
            for (int i = 0; i < int(Edges.size()); i++) flow[eid[i]] = ns.flow(i);
            for (int i = 0; i < n; i++) potential[i] = ns.potential(vid[i]);
            return ns.template totalCost<RetVal>();
        }
    }
};


int main() {
    int M, N;
    cin >> M >> N;
    vector<int> A(M), B(N);
    cin >> A >> B;
    sort(A.begin(), A.end());
    sort(B.begin(), B.end());
    // vector<int> cnt(N);
    // vector<lint> ans(M + 1);

    // vector<int> idx(M);
    // REP(i, M) {
    //     int j = lower_bound(B.begin(), B.end(), A[i]) - B.begin();
    //     if (j >= N) j--;
    //     if (j and abs(B[j - 1] - A[i]) < abs(B[j] - A[i])) j--;
    //     ans[M] += abs(B[j] - A[i]);
    //     cnt[j]++;
    // }
    // int r = *max(ALL(cnt));
    // FOR(i, r, M) ans[i] = ans[M];
    // while (r > 0) {
    //     vector<int> muri;
    //     vector<int> free_js;
    // }

    // FOR(k, 1, M + 1) cout << ans[k] << '\n';
    vector<int> xs = A;
    xs.insert(xs.end(), B.begin(), B.end());
    xs = sort_unique(xs);

    for (auto &x : A) x = lower_bound(xs.begin(), xs.end(), x) - xs.begin();
    for (auto &x : B) x = lower_bound(xs.begin(), xs.end(), x) - xs.begin();

    FOR(k, 1, M + 1) {
        mcf_graph_ns<int, lint> graph(xs.size() + 1);
        for (auto i : A) graph.set_supply(i, 1);
        for (auto i : B) graph.add_edge(i, xs.size(), 0, k, 0);
        REP(j, xs.size() - 1) {
            graph.add_edge(j, j + 1, 0, M, xs[j + 1] - xs[j]);
            graph.add_edge(j + 1, j, 0, M, xs[j + 1] - xs[j]);
        }
        graph.set_supply(xs.size(), -M);
        cout << graph.solve<lint>() << '\n';
    }
}