COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/network_simplex.h @ 887:77704bf2a4ae

Last change on this file since 887:77704bf2a4ae was 877:141f9c0db4a3, checked in by Alpar Juttner <alpar@…>, 10 years ago

Unify the sources (#339)

File size: 49.9 KB
RevLine 
[601]1/* -*- mode: C++; indent-tabs-mode: nil; -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library.
4 *
[877]5 * Copyright (C) 2003-2010
[601]6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
8 *
9 * Permission to use, modify and distribute this software is granted
10 * provided that this copyright notice appears in all copies. For
11 * precise terms see the accompanying LICENSE file.
12 *
13 * This software is provided "AS IS" with no warranty of any kind,
14 * express or implied, and with no claim as to its suitability for any
15 * purpose.
16 *
17 */
18
19#ifndef LEMON_NETWORK_SIMPLEX_H
20#define LEMON_NETWORK_SIMPLEX_H
21
[663]22/// \ingroup min_cost_flow_algs
[601]23///
24/// \file
[605]25/// \brief Network Simplex algorithm for finding a minimum cost flow.
[601]26
27#include <vector>
28#include <limits>
29#include <algorithm>
30
[603]31#include <lemon/core.h>
[601]32#include <lemon/math.h>
33
34namespace lemon {
35
[663]36  /// \addtogroup min_cost_flow_algs
[601]37  /// @{
38
[605]39  /// \brief Implementation of the primal Network Simplex algorithm
[601]40  /// for finding a \ref min_cost_flow "minimum cost flow".
41  ///
[605]42  /// \ref NetworkSimplex implements the primal Network Simplex algorithm
[755]43  /// for finding a \ref min_cost_flow "minimum cost flow"
44  /// \ref amo93networkflows, \ref dantzig63linearprog,
45  /// \ref kellyoneill91netsimplex.
[812]46  /// This algorithm is a highly efficient specialized version of the
47  /// linear programming simplex method directly for the minimum cost
48  /// flow problem.
[606]49  ///
[812]50  /// In general, %NetworkSimplex is the fastest implementation available
51  /// in LEMON for this problem.
52  /// Moreover, it supports both directions of the supply/demand inequality
[786]53  /// constraints. For more information, see \ref SupplyType.
[640]54  ///
55  /// Most of the parameters of the problem (except for the digraph)
56  /// can be given using separate functions, and the algorithm can be
57  /// executed using the \ref run() function. If some parameters are not
58  /// specified, then default values will be used.
[601]59  ///
[605]60  /// \tparam GR The digraph type the algorithm runs on.
[812]61  /// \tparam V The number type used for flow amounts, capacity bounds
[786]62  /// and supply values in the algorithm. By default, it is \c int.
[812]63  /// \tparam C The number type used for costs and potentials in the
[786]64  /// algorithm. By default, it is the same as \c V.
[601]65  ///
[812]66  /// \warning Both number types must be signed and all input data must
[608]67  /// be integer.
[601]68  ///
[605]69  /// \note %NetworkSimplex provides five different pivot rule
[609]70  /// implementations, from which the most efficient one is used
[786]71  /// by default. For more information, see \ref PivotRule.
[641]72  template <typename GR, typename V = int, typename C = V>
[601]73  class NetworkSimplex
74  {
[605]75  public:
[601]76
[642]77    /// The type of the flow amounts, capacity bounds and supply values
[641]78    typedef V Value;
[642]79    /// The type of the arc costs
[607]80    typedef C Cost;
[605]81
82  public:
83
[640]84    /// \brief Problem type constants for the \c run() function.
[605]85    ///
[640]86    /// Enum type containing the problem type constants that can be
87    /// returned by the \ref run() function of the algorithm.
88    enum ProblemType {
89      /// The problem has no feasible solution (flow).
90      INFEASIBLE,
91      /// The problem has optimal solution (i.e. it is feasible and
92      /// bounded), and the algorithm has found optimal flow and node
93      /// potentials (primal and dual solutions).
94      OPTIMAL,
95      /// The objective function of the problem is unbounded, i.e.
96      /// there is a directed cycle having negative total cost and
97      /// infinite upper bound.
98      UNBOUNDED
99    };
[877]100
[640]101    /// \brief Constants for selecting the type of the supply constraints.
102    ///
103    /// Enum type containing constants for selecting the supply type,
104    /// i.e. the direction of the inequalities in the supply/demand
105    /// constraints of the \ref min_cost_flow "minimum cost flow problem".
106    ///
[663]107    /// The default supply type is \c GEQ, the \c LEQ type can be
108    /// selected using \ref supplyType().
109    /// The equality form is a special case of both supply types.
[640]110    enum SupplyType {
111      /// This option means that there are <em>"greater or equal"</em>
[663]112      /// supply/demand constraints in the definition of the problem.
[640]113      GEQ,
114      /// This option means that there are <em>"less or equal"</em>
[663]115      /// supply/demand constraints in the definition of the problem.
116      LEQ
[640]117    };
[877]118
[640]119    /// \brief Constants for selecting the pivot rule.
120    ///
121    /// Enum type containing constants for selecting the pivot rule for
122    /// the \ref run() function.
123    ///
[605]124    /// \ref NetworkSimplex provides five different pivot rule
125    /// implementations that significantly affect the running time
126    /// of the algorithm.
[786]127    /// By default, \ref BLOCK_SEARCH "Block Search" is used, which
[605]128    /// proved to be the most efficient and the most robust on various
[812]129    /// test inputs.
[786]130    /// However, another pivot rule can be selected using the \ref run()
[605]131    /// function with the proper parameter.
132    enum PivotRule {
133
[786]134      /// The \e First \e Eligible pivot rule.
[605]135      /// The next eligible arc is selected in a wraparound fashion
136      /// in every iteration.
137      FIRST_ELIGIBLE,
138
[786]139      /// The \e Best \e Eligible pivot rule.
[605]140      /// The best eligible arc is selected in every iteration.
141      BEST_ELIGIBLE,
142
[786]143      /// The \e Block \e Search pivot rule.
[605]144      /// A specified number of arcs are examined in every iteration
145      /// in a wraparound fashion and the best eligible arc is selected
146      /// from this block.
147      BLOCK_SEARCH,
148
[786]149      /// The \e Candidate \e List pivot rule.
[605]150      /// In a major iteration a candidate list is built from eligible arcs
151      /// in a wraparound fashion and in the following minor iterations
152      /// the best eligible arc is selected from this list.
153      CANDIDATE_LIST,
154
[786]155      /// The \e Altering \e Candidate \e List pivot rule.
[605]156      /// It is a modified version of the Candidate List method.
157      /// It keeps only the several best eligible arcs from the former
158      /// candidate list and extends this list in every iteration.
159      ALTERING_LIST
160    };
[877]161
[605]162  private:
163
164    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
165
[601]166    typedef std::vector<int> IntVector;
[642]167    typedef std::vector<Value> ValueVector;
[607]168    typedef std::vector<Cost> CostVector;
[839]169    typedef std::vector<char> BoolVector;
170    // Note: vector<char> is used instead of vector<bool> for efficiency reasons
[601]171
172    // State constants for arcs
[862]173    enum ArcState {
[601]174      STATE_UPPER = -1,
175      STATE_TREE  =  0,
176      STATE_LOWER =  1
177    };
178
[862]179    typedef std::vector<signed char> StateVector;
180    // Note: vector<signed char> is used instead of vector<ArcState> for
181    // efficiency reasons
182
[601]183  private:
184
[605]185    // Data related to the underlying digraph
186    const GR &_graph;
187    int _node_num;
188    int _arc_num;
[663]189    int _all_arc_num;
190    int _search_arc_num;
[605]191
192    // Parameters of the problem
[642]193    bool _have_lower;
[640]194    SupplyType _stype;
[641]195    Value _sum_supply;
[601]196
[605]197    // Data structures for storing the digraph
[603]198    IntNodeMap _node_id;
[642]199    IntArcMap _arc_id;
[603]200    IntVector _source;
201    IntVector _target;
[830]202    bool _arc_mixing;
[603]203
[605]204    // Node and arc data
[642]205    ValueVector _lower;
206    ValueVector _upper;
207    ValueVector _cap;
[607]208    CostVector _cost;
[642]209    ValueVector _supply;
210    ValueVector _flow;
[607]211    CostVector _pi;
[601]212
[603]213    // Data for storing the spanning tree structure
[601]214    IntVector _parent;
215    IntVector _pred;
216    IntVector _thread;
[604]217    IntVector _rev_thread;
218    IntVector _succ_num;
219    IntVector _last_succ;
220    IntVector _dirty_revs;
[839]221    BoolVector _forward;
[862]222    StateVector _state;
[601]223    int _root;
224
225    // Temporary data used in the current pivot iteration
[603]226    int in_arc, join, u_in, v_in, u_out, v_out;
227    int first, second, right, last;
[601]228    int stem, par_stem, new_stem;
[641]229    Value delta;
[877]230
[811]231    const Value MAX;
[601]232
[640]233  public:
[877]234
[640]235    /// \brief Constant for infinite upper bounds (capacities).
236    ///
237    /// Constant for infinite upper bounds (capacities).
[641]238    /// It is \c std::numeric_limits<Value>::infinity() if available,
239    /// \c std::numeric_limits<Value>::max() otherwise.
240    const Value INF;
[640]241
[601]242  private:
243
[605]244    // Implementation of the First Eligible pivot rule
[601]245    class FirstEligiblePivotRule
246    {
247    private:
248
249      // References to the NetworkSimplex class
250      const IntVector  &_source;
251      const IntVector  &_target;
[607]252      const CostVector &_cost;
[862]253      const StateVector &_state;
[607]254      const CostVector &_pi;
[601]255      int &_in_arc;
[663]256      int _search_arc_num;
[601]257
258      // Pivot rule data
259      int _next_arc;
260
261    public:
262
[605]263      // Constructor
[601]264      FirstEligiblePivotRule(NetworkSimplex &ns) :
[603]265        _source(ns._source), _target(ns._target),
[601]266        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
[663]267        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
268        _next_arc(0)
[601]269      {}
270
[605]271      // Find next entering arc
[601]272      bool findEnteringArc() {
[607]273        Cost c;
[839]274        for (int e = _next_arc; e != _search_arc_num; ++e) {
[601]275          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
276          if (c < 0) {
277            _in_arc = e;
278            _next_arc = e + 1;
279            return true;
280          }
281        }
[839]282        for (int e = 0; e != _next_arc; ++e) {
[601]283          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
284          if (c < 0) {
285            _in_arc = e;
286            _next_arc = e + 1;
287            return true;
288          }
289        }
290        return false;
291      }
292
293    }; //class FirstEligiblePivotRule
294
295
[605]296    // Implementation of the Best Eligible pivot rule
[601]297    class BestEligiblePivotRule
298    {
299    private:
300
301      // References to the NetworkSimplex class
302      const IntVector  &_source;
303      const IntVector  &_target;
[607]304      const CostVector &_cost;
[862]305      const StateVector &_state;
[607]306      const CostVector &_pi;
[601]307      int &_in_arc;
[663]308      int _search_arc_num;
[601]309
310    public:
311
[605]312      // Constructor
[601]313      BestEligiblePivotRule(NetworkSimplex &ns) :
[603]314        _source(ns._source), _target(ns._target),
[601]315        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
[663]316        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
[601]317      {}
318
[605]319      // Find next entering arc
[601]320      bool findEnteringArc() {
[607]321        Cost c, min = 0;
[839]322        for (int e = 0; e != _search_arc_num; ++e) {
[601]323          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
324          if (c < min) {
325            min = c;
326            _in_arc = e;
327          }
328        }
329        return min < 0;
330      }
331
332    }; //class BestEligiblePivotRule
333
334
[605]335    // Implementation of the Block Search pivot rule
[601]336    class BlockSearchPivotRule
337    {
338    private:
339
340      // References to the NetworkSimplex class
341      const IntVector  &_source;
342      const IntVector  &_target;
[607]343      const CostVector &_cost;
[862]344      const StateVector &_state;
[607]345      const CostVector &_pi;
[601]346      int &_in_arc;
[663]347      int _search_arc_num;
[601]348
349      // Pivot rule data
350      int _block_size;
351      int _next_arc;
352
353    public:
354
[605]355      // Constructor
[601]356      BlockSearchPivotRule(NetworkSimplex &ns) :
[603]357        _source(ns._source), _target(ns._target),
[601]358        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
[663]359        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
360        _next_arc(0)
[601]361      {
362        // The main parameters of the pivot rule
[839]363        const double BLOCK_SIZE_FACTOR = 1.0;
[601]364        const int MIN_BLOCK_SIZE = 10;
365
[612]366        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
[663]367                                    std::sqrt(double(_search_arc_num))),
[601]368                                MIN_BLOCK_SIZE );
369      }
370
[605]371      // Find next entering arc
[601]372      bool findEnteringArc() {
[607]373        Cost c, min = 0;
[601]374        int cnt = _block_size;
[727]375        int e;
[839]376        for (e = _next_arc; e != _search_arc_num; ++e) {
[601]377          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
378          if (c < min) {
379            min = c;
[727]380            _in_arc = e;
[601]381          }
382          if (--cnt == 0) {
[727]383            if (min < 0) goto search_end;
[601]384            cnt = _block_size;
385          }
386        }
[839]387        for (e = 0; e != _next_arc; ++e) {
[727]388          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
389          if (c < min) {
390            min = c;
391            _in_arc = e;
392          }
393          if (--cnt == 0) {
394            if (min < 0) goto search_end;
395            cnt = _block_size;
[601]396          }
397        }
398        if (min >= 0) return false;
[727]399
400      search_end:
[601]401        _next_arc = e;
402        return true;
403      }
404
405    }; //class BlockSearchPivotRule
406
407
[605]408    // Implementation of the Candidate List pivot rule
[601]409    class CandidateListPivotRule
410    {
411    private:
412
413      // References to the NetworkSimplex class
414      const IntVector  &_source;
415      const IntVector  &_target;
[607]416      const CostVector &_cost;
[862]417      const StateVector &_state;
[607]418      const CostVector &_pi;
[601]419      int &_in_arc;
[663]420      int _search_arc_num;
[601]421
422      // Pivot rule data
423      IntVector _candidates;
424      int _list_length, _minor_limit;
425      int _curr_length, _minor_count;
426      int _next_arc;
427
428    public:
429
430      /// Constructor
431      CandidateListPivotRule(NetworkSimplex &ns) :
[603]432        _source(ns._source), _target(ns._target),
[601]433        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
[663]434        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
435        _next_arc(0)
[601]436      {
437        // The main parameters of the pivot rule
[727]438        const double LIST_LENGTH_FACTOR = 0.25;
[601]439        const int MIN_LIST_LENGTH = 10;
440        const double MINOR_LIMIT_FACTOR = 0.1;
441        const int MIN_MINOR_LIMIT = 3;
442
[612]443        _list_length = std::max( int(LIST_LENGTH_FACTOR *
[663]444                                     std::sqrt(double(_search_arc_num))),
[601]445                                 MIN_LIST_LENGTH );
446        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
447                                 MIN_MINOR_LIMIT );
448        _curr_length = _minor_count = 0;
449        _candidates.resize(_list_length);
450      }
451
452      /// Find next entering arc
453      bool findEnteringArc() {
[607]454        Cost min, c;
[727]455        int e;
[601]456        if (_curr_length > 0 && _minor_count < _minor_limit) {
457          // Minor iteration: select the best eligible arc from the
458          // current candidate list
459          ++_minor_count;
460          min = 0;
461          for (int i = 0; i < _curr_length; ++i) {
462            e = _candidates[i];
463            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
464            if (c < min) {
465              min = c;
[727]466              _in_arc = e;
[601]467            }
[727]468            else if (c >= 0) {
[601]469              _candidates[i--] = _candidates[--_curr_length];
470            }
471          }
[727]472          if (min < 0) return true;
[601]473        }
474
475        // Major iteration: build a new candidate list
476        min = 0;
477        _curr_length = 0;
[839]478        for (e = _next_arc; e != _search_arc_num; ++e) {
[601]479          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
480          if (c < 0) {
481            _candidates[_curr_length++] = e;
482            if (c < min) {
483              min = c;
[727]484              _in_arc = e;
[601]485            }
[727]486            if (_curr_length == _list_length) goto search_end;
[601]487          }
488        }
[839]489        for (e = 0; e != _next_arc; ++e) {
[727]490          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
491          if (c < 0) {
492            _candidates[_curr_length++] = e;
493            if (c < min) {
494              min = c;
495              _in_arc = e;
[601]496            }
[727]497            if (_curr_length == _list_length) goto search_end;
[601]498          }
499        }
500        if (_curr_length == 0) return false;
[877]501
502      search_end:
[601]503        _minor_count = 1;
504        _next_arc = e;
505        return true;
506      }
507
508    }; //class CandidateListPivotRule
509
510
[605]511    // Implementation of the Altering Candidate List pivot rule
[601]512    class AlteringListPivotRule
513    {
514    private:
515
516      // References to the NetworkSimplex class
517      const IntVector  &_source;
518      const IntVector  &_target;
[607]519      const CostVector &_cost;
[862]520      const StateVector &_state;
[607]521      const CostVector &_pi;
[601]522      int &_in_arc;
[663]523      int _search_arc_num;
[601]524
525      // Pivot rule data
526      int _block_size, _head_length, _curr_length;
527      int _next_arc;
528      IntVector _candidates;
[607]529      CostVector _cand_cost;
[601]530
531      // Functor class to compare arcs during sort of the candidate list
532      class SortFunc
533      {
534      private:
[607]535        const CostVector &_map;
[601]536      public:
[607]537        SortFunc(const CostVector &map) : _map(map) {}
[601]538        bool operator()(int left, int right) {
539          return _map[left] > _map[right];
540        }
541      };
542
543      SortFunc _sort_func;
544
545    public:
546
[605]547      // Constructor
[601]548      AlteringListPivotRule(NetworkSimplex &ns) :
[603]549        _source(ns._source), _target(ns._target),
[601]550        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
[663]551        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
552        _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
[601]553      {
554        // The main parameters of the pivot rule
[727]555        const double BLOCK_SIZE_FACTOR = 1.0;
[601]556        const int MIN_BLOCK_SIZE = 10;
557        const double HEAD_LENGTH_FACTOR = 0.1;
558        const int MIN_HEAD_LENGTH = 3;
559
[612]560        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
[663]561                                    std::sqrt(double(_search_arc_num))),
[601]562                                MIN_BLOCK_SIZE );
563        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
564                                 MIN_HEAD_LENGTH );
565        _candidates.resize(_head_length + _block_size);
566        _curr_length = 0;
567      }
568
[605]569      // Find next entering arc
[601]570      bool findEnteringArc() {
571        // Check the current candidate list
572        int e;
[839]573        for (int i = 0; i != _curr_length; ++i) {
[601]574          e = _candidates[i];
575          _cand_cost[e] = _state[e] *
576            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
577          if (_cand_cost[e] >= 0) {
578            _candidates[i--] = _candidates[--_curr_length];
579          }
580        }
581
582        // Extend the list
583        int cnt = _block_size;
584        int limit = _head_length;
585
[839]586        for (e = _next_arc; e != _search_arc_num; ++e) {
[601]587          _cand_cost[e] = _state[e] *
588            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
589          if (_cand_cost[e] < 0) {
590            _candidates[_curr_length++] = e;
591          }
592          if (--cnt == 0) {
[727]593            if (_curr_length > limit) goto search_end;
[601]594            limit = 0;
595            cnt = _block_size;
596          }
597        }
[839]598        for (e = 0; e != _next_arc; ++e) {
[727]599          _cand_cost[e] = _state[e] *
600            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
601          if (_cand_cost[e] < 0) {
602            _candidates[_curr_length++] = e;
603          }
604          if (--cnt == 0) {
605            if (_curr_length > limit) goto search_end;
606            limit = 0;
607            cnt = _block_size;
[601]608          }
609        }
610        if (_curr_length == 0) return false;
[877]611
[727]612      search_end:
[601]613
614        // Make heap of the candidate list (approximating a partial sort)
615        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
616                   _sort_func );
617
618        // Pop the first element of the heap
619        _in_arc = _candidates[0];
[727]620        _next_arc = e;
[601]621        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
622                  _sort_func );
623        _curr_length = std::min(_head_length, _curr_length - 1);
624        return true;
625      }
626
627    }; //class AlteringListPivotRule
628
629  public:
630
[605]631    /// \brief Constructor.
[601]632    ///
[609]633    /// The constructor of the class.
[601]634    ///
[603]635    /// \param graph The digraph the algorithm runs on.
[728]636    /// \param arc_mixing Indicate if the arcs have to be stored in a
[877]637    /// mixed order in the internal data structure.
[728]638    /// In special cases, it could lead to better overall performance,
639    /// but it is usually slower. Therefore it is disabled by default.
640    NetworkSimplex(const GR& graph, bool arc_mixing = false) :
[642]641      _graph(graph), _node_id(graph), _arc_id(graph),
[830]642      _arc_mixing(arc_mixing),
[811]643      MAX(std::numeric_limits<Value>::max()),
[641]644      INF(std::numeric_limits<Value>::has_infinity ?
[811]645          std::numeric_limits<Value>::infinity() : MAX)
[605]646    {
[812]647      // Check the number types
[641]648      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
[640]649        "The flow type of NetworkSimplex must be signed");
650      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
651        "The cost type of NetworkSimplex must be signed");
[877]652
[830]653      // Reset data structures
[729]654      reset();
[601]655    }
656
[609]657    /// \name Parameters
658    /// The parameters of the algorithm can be specified using these
659    /// functions.
660
661    /// @{
662
[605]663    /// \brief Set the lower bounds on the arcs.
664    ///
665    /// This function sets the lower bounds on the arcs.
[640]666    /// If it is not used before calling \ref run(), the lower bounds
667    /// will be set to zero on all arcs.
[605]668    ///
669    /// \param map An arc map storing the lower bounds.
[641]670    /// Its \c Value type must be convertible to the \c Value type
[605]671    /// of the algorithm.
672    ///
673    /// \return <tt>(*this)</tt>
[640]674    template <typename LowerMap>
675    NetworkSimplex& lowerMap(const LowerMap& map) {
[642]676      _have_lower = true;
[605]677      for (ArcIt a(_graph); a != INVALID; ++a) {
[642]678        _lower[_arc_id[a]] = map[a];
[605]679      }
680      return *this;
681    }
682
683    /// \brief Set the upper bounds (capacities) on the arcs.
684    ///
685    /// This function sets the upper bounds (capacities) on the arcs.
[640]686    /// If it is not used before calling \ref run(), the upper bounds
687    /// will be set to \ref INF on all arcs (i.e. the flow value will be
[812]688    /// unbounded from above).
[605]689    ///
690    /// \param map An arc map storing the upper bounds.
[641]691    /// Its \c Value type must be convertible to the \c Value type
[605]692    /// of the algorithm.
693    ///
694    /// \return <tt>(*this)</tt>
[640]695    template<typename UpperMap>
696    NetworkSimplex& upperMap(const UpperMap& map) {
[605]697      for (ArcIt a(_graph); a != INVALID; ++a) {
[642]698        _upper[_arc_id[a]] = map[a];
[605]699      }
700      return *this;
701    }
702
703    /// \brief Set the costs of the arcs.
704    ///
705    /// This function sets the costs of the arcs.
706    /// If it is not used before calling \ref run(), the costs
707    /// will be set to \c 1 on all arcs.
708    ///
709    /// \param map An arc map storing the costs.
[607]710    /// Its \c Value type must be convertible to the \c Cost type
[605]711    /// of the algorithm.
712    ///
713    /// \return <tt>(*this)</tt>
[640]714    template<typename CostMap>
715    NetworkSimplex& costMap(const CostMap& map) {
[605]716      for (ArcIt a(_graph); a != INVALID; ++a) {
[642]717        _cost[_arc_id[a]] = map[a];
[605]718      }
719      return *this;
720    }
721
722    /// \brief Set the supply values of the nodes.
723    ///
724    /// This function sets the supply values of the nodes.
725    /// If neither this function nor \ref stSupply() is used before
726    /// calling \ref run(), the supply of each node will be set to zero.
727    ///
728    /// \param map A node map storing the supply values.
[641]729    /// Its \c Value type must be convertible to the \c Value type
[605]730    /// of the algorithm.
731    ///
732    /// \return <tt>(*this)</tt>
[640]733    template<typename SupplyMap>
734    NetworkSimplex& supplyMap(const SupplyMap& map) {
[605]735      for (NodeIt n(_graph); n != INVALID; ++n) {
[642]736        _supply[_node_id[n]] = map[n];
[605]737      }
738      return *this;
739    }
740
741    /// \brief Set single source and target nodes and a supply value.
742    ///
743    /// This function sets a single source node and a single target node
744    /// and the required flow value.
745    /// If neither this function nor \ref supplyMap() is used before
746    /// calling \ref run(), the supply of each node will be set to zero.
747    ///
[640]748    /// Using this function has the same effect as using \ref supplyMap()
749    /// with such a map in which \c k is assigned to \c s, \c -k is
750    /// assigned to \c t and all other nodes have zero supply value.
751    ///
[605]752    /// \param s The source node.
753    /// \param t The target node.
754    /// \param k The required amount of flow from node \c s to node \c t
755    /// (i.e. the supply of \c s and the demand of \c t).
756    ///
757    /// \return <tt>(*this)</tt>
[641]758    NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
[642]759      for (int i = 0; i != _node_num; ++i) {
760        _supply[i] = 0;
761      }
762      _supply[_node_id[s]] =  k;
763      _supply[_node_id[t]] = -k;
[605]764      return *this;
765    }
[877]766
[640]767    /// \brief Set the type of the supply constraints.
[609]768    ///
[640]769    /// This function sets the type of the supply/demand constraints.
770    /// If it is not used before calling \ref run(), the \ref GEQ supply
[609]771    /// type will be used.
772    ///
[786]773    /// For more information, see \ref SupplyType.
[609]774    ///
775    /// \return <tt>(*this)</tt>
[640]776    NetworkSimplex& supplyType(SupplyType supply_type) {
777      _stype = supply_type;
[609]778      return *this;
779    }
[605]780
[609]781    /// @}
[601]782
[605]783    /// \name Execution Control
784    /// The algorithm can be executed using \ref run().
785
[601]786    /// @{
787
788    /// \brief Run the algorithm.
789    ///
790    /// This function runs the algorithm.
[609]791    /// The paramters can be specified using functions \ref lowerMap(),
[877]792    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
[642]793    /// \ref supplyType().
[609]794    /// For example,
[605]795    /// \code
796    ///   NetworkSimplex<ListDigraph> ns(graph);
[640]797    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
[605]798    ///     .supplyMap(sup).run();
799    /// \endcode
[601]800    ///
[830]801    /// This function can be called more than once. All the given parameters
802    /// are kept for the next call, unless \ref resetParams() or \ref reset()
803    /// is used, thus only the modified parameters have to be set again.
804    /// If the underlying digraph was also modified after the construction
805    /// of the class (or the last \ref reset() call), then the \ref reset()
806    /// function must be called.
[606]807    ///
[605]808    /// \param pivot_rule The pivot rule that will be used during the
[786]809    /// algorithm. For more information, see \ref PivotRule.
[601]810    ///
[640]811    /// \return \c INFEASIBLE if no feasible flow exists,
812    /// \n \c OPTIMAL if the problem has optimal solution
813    /// (i.e. it is feasible and bounded), and the algorithm has found
814    /// optimal flow and node potentials (primal and dual solutions),
815    /// \n \c UNBOUNDED if the objective function of the problem is
816    /// unbounded, i.e. there is a directed cycle having negative total
817    /// cost and infinite upper bound.
818    ///
819    /// \see ProblemType, PivotRule
[830]820    /// \see resetParams(), reset()
[640]821    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
822      if (!init()) return INFEASIBLE;
823      return start(pivot_rule);
[601]824    }
825
[606]826    /// \brief Reset all the parameters that have been given before.
827    ///
828    /// This function resets all the paramaters that have been given
[609]829    /// before using functions \ref lowerMap(), \ref upperMap(),
[642]830    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
[606]831    ///
[830]832    /// It is useful for multiple \ref run() calls. Basically, all the given
833    /// parameters are kept for the next \ref run() call, unless
834    /// \ref resetParams() or \ref reset() is used.
835    /// If the underlying digraph was also modified after the construction
836    /// of the class or the last \ref reset() call, then the \ref reset()
837    /// function must be used, otherwise \ref resetParams() is sufficient.
[606]838    ///
839    /// For example,
840    /// \code
841    ///   NetworkSimplex<ListDigraph> ns(graph);
842    ///
843    ///   // First run
[640]844    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
[606]845    ///     .supplyMap(sup).run();
846    ///
[830]847    ///   // Run again with modified cost map (resetParams() is not called,
[606]848    ///   // so only the cost map have to be set again)
849    ///   cost[e] += 100;
850    ///   ns.costMap(cost).run();
851    ///
[830]852    ///   // Run again from scratch using resetParams()
[606]853    ///   // (the lower bounds will be set to zero on all arcs)
[830]854    ///   ns.resetParams();
[640]855    ///   ns.upperMap(capacity).costMap(cost)
[606]856    ///     .supplyMap(sup).run();
857    /// \endcode
858    ///
859    /// \return <tt>(*this)</tt>
[830]860    ///
861    /// \see reset(), run()
862    NetworkSimplex& resetParams() {
[642]863      for (int i = 0; i != _node_num; ++i) {
864        _supply[i] = 0;
865      }
866      for (int i = 0; i != _arc_num; ++i) {
867        _lower[i] = 0;
868        _upper[i] = INF;
869        _cost[i] = 1;
870      }
871      _have_lower = false;
[640]872      _stype = GEQ;
[606]873      return *this;
874    }
875
[830]876    /// \brief Reset the internal data structures and all the parameters
877    /// that have been given before.
878    ///
879    /// This function resets the internal data structures and all the
880    /// paramaters that have been given before using functions \ref lowerMap(),
881    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
882    /// \ref supplyType().
883    ///
884    /// It is useful for multiple \ref run() calls. Basically, all the given
885    /// parameters are kept for the next \ref run() call, unless
886    /// \ref resetParams() or \ref reset() is used.
887    /// If the underlying digraph was also modified after the construction
888    /// of the class or the last \ref reset() call, then the \ref reset()
889    /// function must be used, otherwise \ref resetParams() is sufficient.
890    ///
891    /// See \ref resetParams() for examples.
892    ///
893    /// \return <tt>(*this)</tt>
894    ///
895    /// \see resetParams(), run()
896    NetworkSimplex& reset() {
897      // Resize vectors
898      _node_num = countNodes(_graph);
899      _arc_num = countArcs(_graph);
900      int all_node_num = _node_num + 1;
901      int max_arc_num = _arc_num + 2 * _node_num;
902
903      _source.resize(max_arc_num);
904      _target.resize(max_arc_num);
905
906      _lower.resize(_arc_num);
907      _upper.resize(_arc_num);
908      _cap.resize(max_arc_num);
909      _cost.resize(max_arc_num);
910      _supply.resize(all_node_num);
911      _flow.resize(max_arc_num);
912      _pi.resize(all_node_num);
913
914      _parent.resize(all_node_num);
915      _pred.resize(all_node_num);
916      _forward.resize(all_node_num);
917      _thread.resize(all_node_num);
918      _rev_thread.resize(all_node_num);
919      _succ_num.resize(all_node_num);
920      _last_succ.resize(all_node_num);
921      _state.resize(max_arc_num);
922
923      // Copy the graph
924      int i = 0;
925      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
926        _node_id[n] = i;
927      }
928      if (_arc_mixing) {
929        // Store the arcs in a mixed order
930        int k = std::max(int(std::sqrt(double(_arc_num))), 10);
931        int i = 0, j = 0;
932        for (ArcIt a(_graph); a != INVALID; ++a) {
933          _arc_id[a] = i;
934          _source[i] = _node_id[_graph.source(a)];
935          _target[i] = _node_id[_graph.target(a)];
936          if ((i += k) >= _arc_num) i = ++j;
937        }
938      } else {
939        // Store the arcs in the original order
940        int i = 0;
941        for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
942          _arc_id[a] = i;
943          _source[i] = _node_id[_graph.source(a)];
944          _target[i] = _node_id[_graph.target(a)];
945        }
946      }
[877]947
[830]948      // Reset parameters
949      resetParams();
950      return *this;
951    }
[877]952
[601]953    /// @}
954
955    /// \name Query Functions
956    /// The results of the algorithm can be obtained using these
957    /// functions.\n
[605]958    /// The \ref run() function must be called before using them.
959
[601]960    /// @{
961
[605]962    /// \brief Return the total cost of the found flow.
963    ///
964    /// This function returns the total cost of the found flow.
[640]965    /// Its complexity is O(e).
[605]966    ///
967    /// \note The return type of the function can be specified as a
968    /// template parameter. For example,
969    /// \code
970    ///   ns.totalCost<double>();
971    /// \endcode
[607]972    /// It is useful if the total cost cannot be stored in the \c Cost
[605]973    /// type of the algorithm, which is the default return type of the
974    /// function.
975    ///
976    /// \pre \ref run() must be called before using this function.
[642]977    template <typename Number>
978    Number totalCost() const {
979      Number c = 0;
980      for (ArcIt a(_graph); a != INVALID; ++a) {
981        int i = _arc_id[a];
982        c += Number(_flow[i]) * Number(_cost[i]);
[605]983      }
984      return c;
985    }
986
987#ifndef DOXYGEN
[607]988    Cost totalCost() const {
989      return totalCost<Cost>();
[605]990    }
991#endif
992
993    /// \brief Return the flow on the given arc.
994    ///
995    /// This function returns the flow on the given arc.
996    ///
997    /// \pre \ref run() must be called before using this function.
[641]998    Value flow(const Arc& a) const {
[642]999      return _flow[_arc_id[a]];
[605]1000    }
1001
[642]1002    /// \brief Return the flow map (the primal solution).
[601]1003    ///
[642]1004    /// This function copies the flow value on each arc into the given
1005    /// map. The \c Value type of the algorithm must be convertible to
1006    /// the \c Value type of the map.
[601]1007    ///
1008    /// \pre \ref run() must be called before using this function.
[642]1009    template <typename FlowMap>
1010    void flowMap(FlowMap &map) const {
1011      for (ArcIt a(_graph); a != INVALID; ++a) {
1012        map.set(a, _flow[_arc_id[a]]);
1013      }
[601]1014    }
1015
[605]1016    /// \brief Return the potential (dual value) of the given node.
1017    ///
1018    /// This function returns the potential (dual value) of the
1019    /// given node.
1020    ///
1021    /// \pre \ref run() must be called before using this function.
[607]1022    Cost potential(const Node& n) const {
[642]1023      return _pi[_node_id[n]];
[605]1024    }
1025
[642]1026    /// \brief Return the potential map (the dual solution).
[601]1027    ///
[642]1028    /// This function copies the potential (dual value) of each node
1029    /// into the given map.
1030    /// The \c Cost type of the algorithm must be convertible to the
1031    /// \c Value type of the map.
[601]1032    ///
1033    /// \pre \ref run() must be called before using this function.
[642]1034    template <typename PotentialMap>
1035    void potentialMap(PotentialMap &map) const {
1036      for (NodeIt n(_graph); n != INVALID; ++n) {
1037        map.set(n, _pi[_node_id[n]]);
1038      }
[601]1039    }
1040
1041    /// @}
1042
1043  private:
1044
1045    // Initialize internal data structures
1046    bool init() {
[605]1047      if (_node_num == 0) return false;
[601]1048
[642]1049      // Check the sum of supply values
1050      _sum_supply = 0;
1051      for (int i = 0; i != _node_num; ++i) {
1052        _sum_supply += _supply[i];
1053      }
[643]1054      if ( !((_stype == GEQ && _sum_supply <= 0) ||
1055             (_stype == LEQ && _sum_supply >= 0)) ) return false;
[601]1056
[642]1057      // Remove non-zero lower bounds
1058      if (_have_lower) {
1059        for (int i = 0; i != _arc_num; ++i) {
1060          Value c = _lower[i];
1061          if (c >= 0) {
[811]1062            _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;
[642]1063          } else {
[811]1064            _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;
[642]1065          }
1066          _supply[_source[i]] -= c;
1067          _supply[_target[i]] += c;
1068        }
1069      } else {
1070        for (int i = 0; i != _arc_num; ++i) {
1071          _cap[i] = _upper[i];
1072        }
[605]1073      }
[601]1074
[609]1075      // Initialize artifical cost
[640]1076      Cost ART_COST;
[609]1077      if (std::numeric_limits<Cost>::is_exact) {
[663]1078        ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
[609]1079      } else {
[640]1080        ART_COST = std::numeric_limits<Cost>::min();
[609]1081        for (int i = 0; i != _arc_num; ++i) {
[640]1082          if (_cost[i] > ART_COST) ART_COST = _cost[i];
[609]1083        }
[640]1084        ART_COST = (ART_COST + 1) * _node_num;
[609]1085      }
1086
[642]1087      // Initialize arc maps
1088      for (int i = 0; i != _arc_num; ++i) {
1089        _flow[i] = 0;
1090        _state[i] = STATE_LOWER;
1091      }
[877]1092
[601]1093      // Set data for the artificial root node
1094      _root = _node_num;
1095      _parent[_root] = -1;
1096      _pred[_root] = -1;
1097      _thread[_root] = 0;
[604]1098      _rev_thread[0] = _root;
[642]1099      _succ_num[_root] = _node_num + 1;
[604]1100      _last_succ[_root] = _root - 1;
[640]1101      _supply[_root] = -_sum_supply;
[663]1102      _pi[_root] = 0;
[601]1103
1104      // Add artificial arcs and initialize the spanning tree data structure
[663]1105      if (_sum_supply == 0) {
1106        // EQ supply constraints
1107        _search_arc_num = _arc_num;
1108        _all_arc_num = _arc_num + _node_num;
1109        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1110          _parent[u] = _root;
1111          _pred[u] = e;
1112          _thread[u] = u + 1;
1113          _rev_thread[u + 1] = u;
1114          _succ_num[u] = 1;
1115          _last_succ[u] = u;
1116          _cap[e] = INF;
1117          _state[e] = STATE_TREE;
1118          if (_supply[u] >= 0) {
1119            _forward[u] = true;
1120            _pi[u] = 0;
1121            _source[e] = u;
1122            _target[e] = _root;
1123            _flow[e] = _supply[u];
1124            _cost[e] = 0;
1125          } else {
1126            _forward[u] = false;
1127            _pi[u] = ART_COST;
1128            _source[e] = _root;
1129            _target[e] = u;
1130            _flow[e] = -_supply[u];
1131            _cost[e] = ART_COST;
1132          }
[601]1133        }
1134      }
[663]1135      else if (_sum_supply > 0) {
1136        // LEQ supply constraints
1137        _search_arc_num = _arc_num + _node_num;
1138        int f = _arc_num + _node_num;
1139        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1140          _parent[u] = _root;
1141          _thread[u] = u + 1;
1142          _rev_thread[u + 1] = u;
1143          _succ_num[u] = 1;
1144          _last_succ[u] = u;
1145          if (_supply[u] >= 0) {
1146            _forward[u] = true;
1147            _pi[u] = 0;
1148            _pred[u] = e;
1149            _source[e] = u;
1150            _target[e] = _root;
1151            _cap[e] = INF;
1152            _flow[e] = _supply[u];
1153            _cost[e] = 0;
1154            _state[e] = STATE_TREE;
1155          } else {
1156            _forward[u] = false;
1157            _pi[u] = ART_COST;
1158            _pred[u] = f;
1159            _source[f] = _root;
1160            _target[f] = u;
1161            _cap[f] = INF;
1162            _flow[f] = -_supply[u];
1163            _cost[f] = ART_COST;
1164            _state[f] = STATE_TREE;
1165            _source[e] = u;
1166            _target[e] = _root;
1167            _cap[e] = INF;
1168            _flow[e] = 0;
1169            _cost[e] = 0;
1170            _state[e] = STATE_LOWER;
1171            ++f;
1172          }
1173        }
1174        _all_arc_num = f;
1175      }
1176      else {
1177        // GEQ supply constraints
1178        _search_arc_num = _arc_num + _node_num;
1179        int f = _arc_num + _node_num;
1180        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1181          _parent[u] = _root;
1182          _thread[u] = u + 1;
1183          _rev_thread[u + 1] = u;
1184          _succ_num[u] = 1;
1185          _last_succ[u] = u;
1186          if (_supply[u] <= 0) {
1187            _forward[u] = false;
1188            _pi[u] = 0;
1189            _pred[u] = e;
1190            _source[e] = _root;
1191            _target[e] = u;
1192            _cap[e] = INF;
1193            _flow[e] = -_supply[u];
1194            _cost[e] = 0;
1195            _state[e] = STATE_TREE;
1196          } else {
1197            _forward[u] = true;
1198            _pi[u] = -ART_COST;
1199            _pred[u] = f;
1200            _source[f] = u;
1201            _target[f] = _root;
1202            _cap[f] = INF;
1203            _flow[f] = _supply[u];
1204            _state[f] = STATE_TREE;
1205            _cost[f] = ART_COST;
1206            _source[e] = _root;
1207            _target[e] = u;
1208            _cap[e] = INF;
1209            _flow[e] = 0;
1210            _cost[e] = 0;
1211            _state[e] = STATE_LOWER;
1212            ++f;
1213          }
1214        }
1215        _all_arc_num = f;
1216      }
[601]1217
1218      return true;
1219    }
1220
1221    // Find the join node
1222    void findJoinNode() {
[603]1223      int u = _source[in_arc];
1224      int v = _target[in_arc];
[601]1225      while (u != v) {
[604]1226        if (_succ_num[u] < _succ_num[v]) {
1227          u = _parent[u];
1228        } else {
1229          v = _parent[v];
1230        }
[601]1231      }
1232      join = u;
1233    }
1234
1235    // Find the leaving arc of the cycle and returns true if the
1236    // leaving arc is not the same as the entering arc
1237    bool findLeavingArc() {
1238      // Initialize first and second nodes according to the direction
1239      // of the cycle
[603]1240      if (_state[in_arc] == STATE_LOWER) {
1241        first  = _source[in_arc];
1242        second = _target[in_arc];
[601]1243      } else {
[603]1244        first  = _target[in_arc];
1245        second = _source[in_arc];
[601]1246      }
[603]1247      delta = _cap[in_arc];
[601]1248      int result = 0;
[641]1249      Value d;
[601]1250      int e;
1251
1252      // Search the cycle along the path form the first node to the root
1253      for (int u = first; u != join; u = _parent[u]) {
1254        e = _pred[u];
[640]1255        d = _forward[u] ?
[811]1256          _flow[e] : (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]);
[601]1257        if (d < delta) {
1258          delta = d;
1259          u_out = u;
1260          result = 1;
1261        }
1262      }
1263      // Search the cycle along the path form the second node to the root
1264      for (int u = second; u != join; u = _parent[u]) {
1265        e = _pred[u];
[877]1266        d = _forward[u] ?
[811]1267          (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]) : _flow[e];
[601]1268        if (d <= delta) {
1269          delta = d;
1270          u_out = u;
1271          result = 2;
1272        }
1273      }
1274
1275      if (result == 1) {
1276        u_in = first;
1277        v_in = second;
1278      } else {
1279        u_in = second;
1280        v_in = first;
1281      }
1282      return result != 0;
1283    }
1284
1285    // Change _flow and _state vectors
1286    void changeFlow(bool change) {
1287      // Augment along the cycle
1288      if (delta > 0) {
[641]1289        Value val = _state[in_arc] * delta;
[603]1290        _flow[in_arc] += val;
1291        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
[601]1292          _flow[_pred[u]] += _forward[u] ? -val : val;
1293        }
[603]1294        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
[601]1295          _flow[_pred[u]] += _forward[u] ? val : -val;
1296        }
1297      }
1298      // Update the state of the entering and leaving arcs
1299      if (change) {
[603]1300        _state[in_arc] = STATE_TREE;
[601]1301        _state[_pred[u_out]] =
1302          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1303      } else {
[603]1304        _state[in_arc] = -_state[in_arc];
[601]1305      }
1306    }
1307
[604]1308    // Update the tree structure
1309    void updateTreeStructure() {
1310      int u, w;
1311      int old_rev_thread = _rev_thread[u_out];
1312      int old_succ_num = _succ_num[u_out];
1313      int old_last_succ = _last_succ[u_out];
[601]1314      v_out = _parent[u_out];
1315
[604]1316      u = _last_succ[u_in];  // the last successor of u_in
1317      right = _thread[u];    // the node after it
1318
1319      // Handle the case when old_rev_thread equals to v_in
1320      // (it also means that join and v_out coincide)
1321      if (old_rev_thread == v_in) {
1322        last = _thread[_last_succ[u_out]];
1323      } else {
1324        last = _thread[v_in];
[601]1325      }
1326
[604]1327      // Update _thread and _parent along the stem nodes (i.e. the nodes
1328      // between u_in and u_out, whose parent have to be changed)
[601]1329      _thread[v_in] = stem = u_in;
[604]1330      _dirty_revs.clear();
1331      _dirty_revs.push_back(v_in);
[601]1332      par_stem = v_in;
1333      while (stem != u_out) {
[604]1334        // Insert the next stem node into the thread list
1335        new_stem = _parent[stem];
1336        _thread[u] = new_stem;
1337        _dirty_revs.push_back(u);
[601]1338
[604]1339        // Remove the subtree of stem from the thread list
1340        w = _rev_thread[stem];
1341        _thread[w] = right;
1342        _rev_thread[right] = w;
[601]1343
[604]1344        // Change the parent node and shift stem nodes
[601]1345        _parent[stem] = par_stem;
1346        par_stem = stem;
1347        stem = new_stem;
1348
[604]1349        // Update u and right
1350        u = _last_succ[stem] == _last_succ[par_stem] ?
1351          _rev_thread[par_stem] : _last_succ[stem];
[601]1352        right = _thread[u];
1353      }
1354      _parent[u_out] = par_stem;
1355      _thread[u] = last;
[604]1356      _rev_thread[last] = u;
1357      _last_succ[u_out] = u;
[601]1358
[604]1359      // Remove the subtree of u_out from the thread list except for
1360      // the case when old_rev_thread equals to v_in
1361      // (it also means that join and v_out coincide)
1362      if (old_rev_thread != v_in) {
1363        _thread[old_rev_thread] = right;
1364        _rev_thread[right] = old_rev_thread;
1365      }
1366
1367      // Update _rev_thread using the new _thread values
[839]1368      for (int i = 0; i != int(_dirty_revs.size()); ++i) {
[604]1369        u = _dirty_revs[i];
1370        _rev_thread[_thread[u]] = u;
1371      }
1372
1373      // Update _pred, _forward, _last_succ and _succ_num for the
1374      // stem nodes from u_out to u_in
1375      int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1376      u = u_out;
1377      while (u != u_in) {
1378        w = _parent[u];
1379        _pred[u] = _pred[w];
1380        _forward[u] = !_forward[w];
1381        tmp_sc += _succ_num[u] - _succ_num[w];
1382        _succ_num[u] = tmp_sc;
1383        _last_succ[w] = tmp_ls;
1384        u = w;
1385      }
1386      _pred[u_in] = in_arc;
1387      _forward[u_in] = (u_in == _source[in_arc]);
1388      _succ_num[u_in] = old_succ_num;
1389
1390      // Set limits for updating _last_succ form v_in and v_out
1391      // towards the root
1392      int up_limit_in = -1;
1393      int up_limit_out = -1;
1394      if (_last_succ[join] == v_in) {
1395        up_limit_out = join;
[601]1396      } else {
[604]1397        up_limit_in = join;
1398      }
1399
1400      // Update _last_succ from v_in towards the root
1401      for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
1402           u = _parent[u]) {
1403        _last_succ[u] = _last_succ[u_out];
1404      }
1405      // Update _last_succ from v_out towards the root
1406      if (join != old_rev_thread && v_in != old_rev_thread) {
1407        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1408             u = _parent[u]) {
1409          _last_succ[u] = old_rev_thread;
1410        }
1411      } else {
1412        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1413             u = _parent[u]) {
1414          _last_succ[u] = _last_succ[u_out];
1415        }
1416      }
1417
1418      // Update _succ_num from v_in to join
1419      for (u = v_in; u != join; u = _parent[u]) {
1420        _succ_num[u] += old_succ_num;
1421      }
1422      // Update _succ_num from v_out to join
1423      for (u = v_out; u != join; u = _parent[u]) {
1424        _succ_num[u] -= old_succ_num;
[601]1425      }
1426    }
1427
[604]1428    // Update potentials
1429    void updatePotential() {
[607]1430      Cost sigma = _forward[u_in] ?
[601]1431        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
1432        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
[608]1433      // Update potentials in the subtree, which has been moved
1434      int end = _thread[_last_succ[u_in]];
1435      for (int u = u_in; u != end; u = _thread[u]) {
1436        _pi[u] += sigma;
[601]1437      }
1438    }
1439
[839]1440    // Heuristic initial pivots
1441    bool initialPivots() {
1442      Value curr, total = 0;
1443      std::vector<Node> supply_nodes, demand_nodes;
1444      for (NodeIt u(_graph); u != INVALID; ++u) {
1445        curr = _supply[_node_id[u]];
1446        if (curr > 0) {
1447          total += curr;
1448          supply_nodes.push_back(u);
1449        }
1450        else if (curr < 0) {
1451          demand_nodes.push_back(u);
1452        }
1453      }
1454      if (_sum_supply > 0) total -= _sum_supply;
1455      if (total <= 0) return true;
1456
1457      IntVector arc_vector;
1458      if (_sum_supply >= 0) {
1459        if (supply_nodes.size() == 1 && demand_nodes.size() == 1) {
1460          // Perform a reverse graph search from the sink to the source
1461          typename GR::template NodeMap<bool> reached(_graph, false);
1462          Node s = supply_nodes[0], t = demand_nodes[0];
1463          std::vector<Node> stack;
1464          reached[t] = true;
1465          stack.push_back(t);
1466          while (!stack.empty()) {
1467            Node u, v = stack.back();
1468            stack.pop_back();
1469            if (v == s) break;
1470            for (InArcIt a(_graph, v); a != INVALID; ++a) {
1471              if (reached[u = _graph.source(a)]) continue;
1472              int j = _arc_id[a];
1473              if (_cap[j] >= total) {
1474                arc_vector.push_back(j);
1475                reached[u] = true;
1476                stack.push_back(u);
1477              }
1478            }
1479          }
1480        } else {
1481          // Find the min. cost incomming arc for each demand node
1482          for (int i = 0; i != int(demand_nodes.size()); ++i) {
1483            Node v = demand_nodes[i];
1484            Cost c, min_cost = std::numeric_limits<Cost>::max();
1485            Arc min_arc = INVALID;
1486            for (InArcIt a(_graph, v); a != INVALID; ++a) {
1487              c = _cost[_arc_id[a]];
1488              if (c < min_cost) {
1489                min_cost = c;
1490                min_arc = a;
1491              }
1492            }
1493            if (min_arc != INVALID) {
1494              arc_vector.push_back(_arc_id[min_arc]);
1495            }
1496          }
1497        }
1498      } else {
1499        // Find the min. cost outgoing arc for each supply node
1500        for (int i = 0; i != int(supply_nodes.size()); ++i) {
1501          Node u = supply_nodes[i];
1502          Cost c, min_cost = std::numeric_limits<Cost>::max();
1503          Arc min_arc = INVALID;
1504          for (OutArcIt a(_graph, u); a != INVALID; ++a) {
1505            c = _cost[_arc_id[a]];
1506            if (c < min_cost) {
1507              min_cost = c;
1508              min_arc = a;
1509            }
1510          }
1511          if (min_arc != INVALID) {
1512            arc_vector.push_back(_arc_id[min_arc]);
1513          }
1514        }
1515      }
1516
1517      // Perform heuristic initial pivots
1518      for (int i = 0; i != int(arc_vector.size()); ++i) {
1519        in_arc = arc_vector[i];
1520        if (_state[in_arc] * (_cost[in_arc] + _pi[_source[in_arc]] -
1521            _pi[_target[in_arc]]) >= 0) continue;
1522        findJoinNode();
1523        bool change = findLeavingArc();
1524        if (delta >= MAX) return false;
1525        changeFlow(change);
1526        if (change) {
1527          updateTreeStructure();
1528          updatePotential();
1529        }
1530      }
1531      return true;
1532    }
1533
[601]1534    // Execute the algorithm
[640]1535    ProblemType start(PivotRule pivot_rule) {
[601]1536      // Select the pivot rule implementation
1537      switch (pivot_rule) {
[605]1538        case FIRST_ELIGIBLE:
[601]1539          return start<FirstEligiblePivotRule>();
[605]1540        case BEST_ELIGIBLE:
[601]1541          return start<BestEligiblePivotRule>();
[605]1542        case BLOCK_SEARCH:
[601]1543          return start<BlockSearchPivotRule>();
[605]1544        case CANDIDATE_LIST:
[601]1545          return start<CandidateListPivotRule>();
[605]1546        case ALTERING_LIST:
[601]1547          return start<AlteringListPivotRule>();
1548      }
[640]1549      return INFEASIBLE; // avoid warning
[601]1550    }
1551
[605]1552    template <typename PivotRuleImpl>
[640]1553    ProblemType start() {
[605]1554      PivotRuleImpl pivot(*this);
[601]1555
[839]1556      // Perform heuristic initial pivots
1557      if (!initialPivots()) return UNBOUNDED;
1558
[605]1559      // Execute the Network Simplex algorithm
[601]1560      while (pivot.findEnteringArc()) {
1561        findJoinNode();
1562        bool change = findLeavingArc();
[811]1563        if (delta >= MAX) return UNBOUNDED;
[601]1564        changeFlow(change);
1565        if (change) {
[604]1566          updateTreeStructure();
1567          updatePotential();
[601]1568        }
1569      }
[877]1570
[640]1571      // Check feasibility
[663]1572      for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1573        if (_flow[e] != 0) return INFEASIBLE;
[640]1574      }
[601]1575
[642]1576      // Transform the solution and the supply map to the original form
1577      if (_have_lower) {
[601]1578        for (int i = 0; i != _arc_num; ++i) {
[642]1579          Value c = _lower[i];
1580          if (c != 0) {
1581            _flow[i] += c;
1582            _supply[_source[i]] += c;
1583            _supply[_target[i]] -= c;
1584          }
[601]1585        }
1586      }
[877]1587
[663]1588      // Shift potentials to meet the requirements of the GEQ/LEQ type
1589      // optimality conditions
1590      if (_sum_supply == 0) {
1591        if (_stype == GEQ) {
1592          Cost max_pot = std::numeric_limits<Cost>::min();
1593          for (int i = 0; i != _node_num; ++i) {
1594            if (_pi[i] > max_pot) max_pot = _pi[i];
1595          }
1596          if (max_pot > 0) {
1597            for (int i = 0; i != _node_num; ++i)
1598              _pi[i] -= max_pot;
1599          }
1600        } else {
1601          Cost min_pot = std::numeric_limits<Cost>::max();
1602          for (int i = 0; i != _node_num; ++i) {
1603            if (_pi[i] < min_pot) min_pot = _pi[i];
1604          }
1605          if (min_pot < 0) {
1606            for (int i = 0; i != _node_num; ++i)
1607              _pi[i] -= min_pot;
1608          }
1609        }
1610      }
[601]1611
[640]1612      return OPTIMAL;
[601]1613    }
1614
1615  }; //class NetworkSimplex
1616
1617  ///@}
1618
1619} //namespace lemon
1620
1621#endif //LEMON_NETWORK_SIMPLEX_H
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