lemon/network_simplex.h
author deba
Sun, 05 Oct 2008 20:08:13 +0000
changeset 2622 fa2877651022
parent 2593 8eed667ea23c
child 2623 90defb96ee61
permissions -rw-r--r--
Fix _setCoeff
     1 /* -*- C++ -*-
     2  *
     3  * This file is a part of LEMON, a generic C++ optimization library
     4  *
     5  * Copyright (C) 2003-2008
     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 
    22 /// \ingroup min_cost_flow
    23 ///
    24 /// \file
    25 /// \brief Network simplex algorithm for finding a minimum cost flow.
    26 
    27 #include <vector>
    28 #include <limits>
    29 
    30 #include <lemon/graph_adaptor.h>
    31 #include <lemon/graph_utils.h>
    32 #include <lemon/smart_graph.h>
    33 #include <lemon/math.h>
    34 
    35 #define _DEBUG_
    36 
    37 namespace lemon {
    38 
    39   /// \addtogroup min_cost_flow
    40   /// @{
    41 
    42   /// \brief Implementation of the primal network simplex algorithm
    43   /// for finding a minimum cost flow.
    44   ///
    45   /// \ref NetworkSimplex implements the primal network simplex algorithm
    46   /// for finding a minimum cost flow.
    47   ///
    48   /// \tparam Graph The directed graph type the algorithm runs on.
    49   /// \tparam LowerMap The type of the lower bound map.
    50   /// \tparam CapacityMap The type of the capacity (upper bound) map.
    51   /// \tparam CostMap The type of the cost (length) map.
    52   /// \tparam SupplyMap The type of the supply map.
    53   ///
    54   /// \warning
    55   /// - Edge capacities and costs should be \e non-negative \e integers.
    56   /// - Supply values should be \e signed \e integers.
    57   /// - The value types of the maps should be convertible to each other.
    58   /// - \c CostMap::Value must be signed type.
    59   ///
    60   /// \note \ref NetworkSimplex provides five different pivot rule
    61   /// implementations that significantly affect the efficiency of the
    62   /// algorithm.
    63   /// By default "Block Search" pivot rule is used, which proved to be
    64   /// by far the most efficient according to our benchmark tests.
    65   /// However another pivot rule can be selected using \ref run()
    66   /// function with the proper parameter.
    67   ///
    68   /// \author Peter Kovacs
    69   template < typename Graph,
    70              typename LowerMap = typename Graph::template EdgeMap<int>,
    71              typename CapacityMap = typename Graph::template EdgeMap<int>,
    72              typename CostMap = typename Graph::template EdgeMap<int>,
    73              typename SupplyMap = typename Graph::template NodeMap<int> >
    74   class NetworkSimplex
    75   {
    76     typedef typename CapacityMap::Value Capacity;
    77     typedef typename CostMap::Value Cost;
    78     typedef typename SupplyMap::Value Supply;
    79 
    80     typedef SmartGraph SGraph;
    81     GRAPH_TYPEDEFS(typename SGraph);
    82 
    83     typedef typename SGraph::template EdgeMap<Capacity> SCapacityMap;
    84     typedef typename SGraph::template EdgeMap<Cost> SCostMap;
    85     typedef typename SGraph::template NodeMap<Supply> SSupplyMap;
    86     typedef typename SGraph::template NodeMap<Cost> SPotentialMap;
    87 
    88     typedef typename SGraph::template NodeMap<int> IntNodeMap;
    89     typedef typename SGraph::template NodeMap<bool> BoolNodeMap;
    90     typedef typename SGraph::template NodeMap<Node> NodeNodeMap;
    91     typedef typename SGraph::template NodeMap<Edge> EdgeNodeMap;
    92     typedef typename SGraph::template EdgeMap<int> IntEdgeMap;
    93     typedef typename SGraph::template EdgeMap<bool> BoolEdgeMap;
    94 
    95     typedef typename Graph::template NodeMap<Node> NodeRefMap;
    96     typedef typename Graph::template EdgeMap<Edge> EdgeRefMap;
    97 
    98     typedef std::vector<Edge> EdgeVector;
    99 
   100   public:
   101 
   102     /// The type of the flow map.
   103     typedef typename Graph::template EdgeMap<Capacity> FlowMap;
   104     /// The type of the potential map.
   105     typedef typename Graph::template NodeMap<Cost> PotentialMap;
   106 
   107   public:
   108 
   109     /// Enum type to select the pivot rule used by \ref run().
   110     enum PivotRuleEnum {
   111       FIRST_ELIGIBLE_PIVOT,
   112       BEST_ELIGIBLE_PIVOT,
   113       BLOCK_SEARCH_PIVOT,
   114       CANDIDATE_LIST_PIVOT,
   115       ALTERING_LIST_PIVOT
   116     };
   117 
   118   private:
   119 
   120     /// \brief Map adaptor class for handling reduced edge costs.
   121     ///
   122     /// Map adaptor class for handling reduced edge costs.
   123     class ReducedCostMap : public MapBase<Edge, Cost>
   124     {
   125     private:
   126 
   127       const SGraph &_gr;
   128       const SCostMap &_cost_map;
   129       const SPotentialMap &_pot_map;
   130 
   131     public:
   132 
   133       ///\e
   134       ReducedCostMap( const SGraph &gr,
   135                       const SCostMap &cost_map,
   136                       const SPotentialMap &pot_map ) :
   137         _gr(gr), _cost_map(cost_map), _pot_map(pot_map) {}
   138 
   139       ///\e
   140       Cost operator[](const Edge &e) const {
   141         return _cost_map[e] + _pot_map[_gr.source(e)]
   142                            - _pot_map[_gr.target(e)];
   143       }
   144 
   145     }; //class ReducedCostMap
   146 
   147   private:
   148 
   149     /// \brief Implementation of the "First Eligible" pivot rule for the
   150     /// \ref NetworkSimplex "network simplex" algorithm.
   151     ///
   152     /// This class implements the "First Eligible" pivot rule
   153     /// for the \ref NetworkSimplex "network simplex" algorithm.
   154     ///
   155     /// For more information see \ref NetworkSimplex::run().
   156     class FirstEligiblePivotRule
   157     {
   158     private:
   159 
   160       // References to the NetworkSimplex class
   161       NetworkSimplex &_ns;
   162       EdgeVector &_edges;
   163 
   164       int _next_edge;
   165 
   166     public:
   167 
   168       /// Constructor
   169       FirstEligiblePivotRule(NetworkSimplex &ns, EdgeVector &edges) :
   170         _ns(ns), _edges(edges), _next_edge(0) {}
   171 
   172       /// Find next entering edge
   173       inline bool findEnteringEdge() {
   174         Edge e;
   175         for (int i = _next_edge; i < int(_edges.size()); ++i) {
   176           e = _edges[i];
   177           if (_ns._state[e] * _ns._red_cost[e] < 0) {
   178             _ns._in_edge = e;
   179             _next_edge = i + 1;
   180             return true;
   181           }
   182         }
   183         for (int i = 0; i < _next_edge; ++i) {
   184           e = _edges[i];
   185           if (_ns._state[e] * _ns._red_cost[e] < 0) {
   186             _ns._in_edge = e;
   187             _next_edge = i + 1;
   188             return true;
   189           }
   190         }
   191         return false;
   192       }
   193     }; //class FirstEligiblePivotRule
   194 
   195     /// \brief Implementation of the "Best Eligible" pivot rule for the
   196     /// \ref NetworkSimplex "network simplex" algorithm.
   197     ///
   198     /// This class implements the "Best Eligible" pivot rule
   199     /// for the \ref NetworkSimplex "network simplex" algorithm.
   200     ///
   201     /// For more information see \ref NetworkSimplex::run().
   202     class BestEligiblePivotRule
   203     {
   204     private:
   205 
   206       // References to the NetworkSimplex class
   207       NetworkSimplex &_ns;
   208       EdgeVector &_edges;
   209 
   210     public:
   211 
   212       /// Constructor
   213       BestEligiblePivotRule(NetworkSimplex &ns, EdgeVector &edges) :
   214         _ns(ns), _edges(edges) {}
   215 
   216       /// Find next entering edge
   217       inline bool findEnteringEdge() {
   218         Cost min = 0;
   219         Edge e;
   220         for (int i = 0; i < int(_edges.size()); ++i) {
   221           e = _edges[i];
   222           if (_ns._state[e] * _ns._red_cost[e] < min) {
   223             min = _ns._state[e] * _ns._red_cost[e];
   224             _ns._in_edge = e;
   225           }
   226         }
   227         return min < 0;
   228       }
   229     }; //class BestEligiblePivotRule
   230 
   231     /// \brief Implementation of the "Block Search" pivot rule for the
   232     /// \ref NetworkSimplex "network simplex" algorithm.
   233     ///
   234     /// This class implements the "Block Search" pivot rule
   235     /// for the \ref NetworkSimplex "network simplex" algorithm.
   236     ///
   237     /// For more information see \ref NetworkSimplex::run().
   238     class BlockSearchPivotRule
   239     {
   240     private:
   241 
   242       // References to the NetworkSimplex class
   243       NetworkSimplex &_ns;
   244       EdgeVector &_edges;
   245 
   246       int _block_size;
   247       int _next_edge, _min_edge;
   248 
   249     public:
   250 
   251       /// Constructor
   252       BlockSearchPivotRule(NetworkSimplex &ns, EdgeVector &edges) :
   253         _ns(ns), _edges(edges), _next_edge(0), _min_edge(0)
   254       {
   255         // The main parameters of the pivot rule
   256         const double BLOCK_SIZE_FACTOR = 2.0;
   257         const int MIN_BLOCK_SIZE = 10;
   258 
   259         _block_size = std::max( int(BLOCK_SIZE_FACTOR * sqrt(_edges.size())),
   260                                 MIN_BLOCK_SIZE );
   261       }
   262 
   263       /// Find next entering edge
   264       inline bool findEnteringEdge() {
   265         Cost curr, min = 0;
   266         Edge e;
   267         int cnt = _block_size;
   268         int i;
   269         for (i = _next_edge; i < int(_edges.size()); ++i) {
   270           e = _edges[i];
   271           if ((curr = _ns._state[e] * _ns._red_cost[e]) < min) {
   272             min = curr;
   273             _min_edge = i;
   274           }
   275           if (--cnt == 0) {
   276             if (min < 0) break;
   277             cnt = _block_size;
   278           }
   279         }
   280         if (min == 0 || cnt > 0) {
   281           for (i = 0; i < _next_edge; ++i) {
   282             e = _edges[i];
   283             if ((curr = _ns._state[e] * _ns._red_cost[e]) < min) {
   284               min = curr;
   285               _min_edge = i;
   286             }
   287             if (--cnt == 0) {
   288               if (min < 0) break;
   289               cnt = _block_size;
   290             }
   291           }
   292         }
   293         if (min >= 0) return false;
   294         _ns._in_edge = _edges[_min_edge];
   295         _next_edge = i;
   296         return true;
   297       }
   298     }; //class BlockSearchPivotRule
   299 
   300     /// \brief Implementation of the "Candidate List" pivot rule for the
   301     /// \ref NetworkSimplex "network simplex" algorithm.
   302     ///
   303     /// This class implements the "Candidate List" pivot rule
   304     /// for the \ref NetworkSimplex "network simplex" algorithm.
   305     ///
   306     /// For more information see \ref NetworkSimplex::run().
   307     class CandidateListPivotRule
   308     {
   309     private:
   310 
   311       // References to the NetworkSimplex class
   312       NetworkSimplex &_ns;
   313       EdgeVector &_edges;
   314 
   315       EdgeVector _candidates;
   316       int _list_length, _minor_limit;
   317       int _curr_length, _minor_count;
   318       int _next_edge, _min_edge;
   319 
   320     public:
   321 
   322       /// Constructor
   323       CandidateListPivotRule(NetworkSimplex &ns, EdgeVector &edges) :
   324         _ns(ns), _edges(edges), _next_edge(0), _min_edge(0)
   325       {
   326         // The main parameters of the pivot rule
   327         const double LIST_LENGTH_FACTOR = 1.0;
   328         const int MIN_LIST_LENGTH = 10;
   329         const double MINOR_LIMIT_FACTOR = 0.1;
   330         const int MIN_MINOR_LIMIT = 3;
   331 
   332         _list_length = std::max( int(LIST_LENGTH_FACTOR * sqrt(_edges.size())),
   333                                  MIN_LIST_LENGTH );
   334         _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
   335                                  MIN_MINOR_LIMIT );
   336         _curr_length = _minor_count = 0;
   337         _candidates.resize(_list_length);
   338       }
   339 
   340       /// Find next entering edge
   341       inline bool findEnteringEdge() {
   342         Cost min, curr;
   343         if (_curr_length > 0 && _minor_count < _minor_limit) {
   344           // Minor iteration: selecting the best eligible edge from
   345           // the current candidate list
   346           ++_minor_count;
   347           Edge e;
   348           min = 0;
   349           for (int i = 0; i < _curr_length; ++i) {
   350             e = _candidates[i];
   351             curr = _ns._state[e] * _ns._red_cost[e];
   352             if (curr < min) {
   353               min = curr;
   354               _ns._in_edge = e;
   355             }
   356             if (curr >= 0) {
   357               _candidates[i--] = _candidates[--_curr_length];
   358             }
   359           }
   360           if (min < 0) return true;
   361         }
   362 
   363         // Major iteration: building a new candidate list
   364         Edge e;
   365         min = 0;
   366         _curr_length = 0;
   367         int i;
   368         for (i = _next_edge; i < int(_edges.size()); ++i) {
   369           e = _edges[i];
   370           if ((curr = _ns._state[e] * _ns._red_cost[e]) < 0) {
   371             _candidates[_curr_length++] = e;
   372             if (curr < min) {
   373               min = curr;
   374               _min_edge = i;
   375             }
   376             if (_curr_length == _list_length) break;
   377           }
   378         }
   379         if (_curr_length < _list_length) {
   380           for (i = 0; i < _next_edge; ++i) {
   381             e = _edges[i];
   382             if ((curr = _ns._state[e] * _ns._red_cost[e]) < 0) {
   383               _candidates[_curr_length++] = e;
   384               if (curr < min) {
   385                 min = curr;
   386                 _min_edge = i;
   387               }
   388               if (_curr_length == _list_length) break;
   389             }
   390           }
   391         }
   392         if (_curr_length == 0) return false;
   393         _minor_count = 1;
   394         _ns._in_edge = _edges[_min_edge];
   395         _next_edge = i;
   396         return true;
   397       }
   398     }; //class CandidateListPivotRule
   399 
   400     /// \brief Implementation of the "Altering Candidate List" pivot rule
   401     /// for the \ref NetworkSimplex "network simplex" algorithm.
   402     ///
   403     /// This class implements the "Altering Candidate List" pivot rule
   404     /// for the \ref NetworkSimplex "network simplex" algorithm.
   405     ///
   406     /// For more information see \ref NetworkSimplex::run().
   407     class AlteringListPivotRule
   408     {
   409     private:
   410 
   411       // References to the NetworkSimplex class
   412       NetworkSimplex &_ns;
   413       EdgeVector &_edges;
   414 
   415       EdgeVector _candidates;
   416       SCostMap _cand_cost;
   417       int _block_size, _head_length, _curr_length;
   418       int _next_edge;
   419 
   420       // Functor class to compare edges during sort of the candidate list
   421       class SortFunc
   422       {
   423       private:
   424         const SCostMap &_map;
   425       public:
   426         SortFunc(const SCostMap &map) : _map(map) {}
   427         bool operator()(const Edge &e1, const Edge &e2) {
   428 	  return _map[e1] < _map[e2];
   429         }
   430       };
   431 
   432       SortFunc _sort_func;
   433 
   434     public:
   435 
   436       /// Constructor
   437       AlteringListPivotRule(NetworkSimplex &ns, EdgeVector &edges) :
   438         _ns(ns), _edges(edges), _cand_cost(_ns._graph),
   439         _next_edge(0), _sort_func(_cand_cost)
   440       {
   441         // The main parameters of the pivot rule
   442         const double BLOCK_SIZE_FACTOR = 1.0;
   443         const int MIN_BLOCK_SIZE = 10;
   444         const double HEAD_LENGTH_FACTOR = 0.1;
   445         const int MIN_HEAD_LENGTH = 5;
   446 
   447         _block_size = std::max( int(BLOCK_SIZE_FACTOR * sqrt(_edges.size())),
   448                                 MIN_BLOCK_SIZE );
   449         _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
   450                                  MIN_HEAD_LENGTH );
   451         _candidates.resize(_head_length + _block_size);
   452         _curr_length = 0;
   453       }
   454 
   455       /// Find next entering edge
   456       inline bool findEnteringEdge() {
   457         // Checking the current candidate list
   458         Edge e;
   459         for (int idx = 0; idx < _curr_length; ++idx) {
   460           e = _candidates[idx];
   461           if ((_cand_cost[e] = _ns._state[e] * _ns._red_cost[e]) >= 0) {
   462             _candidates[idx--] = _candidates[--_curr_length];
   463           }
   464         }
   465 
   466         // Extending the list
   467         int cnt = _block_size;
   468         int last_edge = 0;
   469         int limit = _head_length;
   470         for (int i = _next_edge; i < int(_edges.size()); ++i) {
   471           e = _edges[i];
   472           if ((_cand_cost[e] = _ns._state[e] * _ns._red_cost[e]) < 0) {
   473             _candidates[_curr_length++] = e;
   474             last_edge = i;
   475           }
   476           if (--cnt == 0) {
   477             if (_curr_length > limit) break;
   478             limit = 0;
   479             cnt = _block_size;
   480           }
   481         }
   482         if (_curr_length <= limit) {
   483           for (int i = 0; i < _next_edge; ++i) {
   484             e = _edges[i];
   485             if ((_cand_cost[e] = _ns._state[e] * _ns._red_cost[e]) < 0) {
   486               _candidates[_curr_length++] = e;
   487               last_edge = i;
   488             }
   489             if (--cnt == 0) {
   490               if (_curr_length > limit) break;
   491               limit = 0;
   492               cnt = _block_size;
   493             }
   494           }
   495         }
   496         if (_curr_length == 0) return false;
   497         _next_edge = last_edge + 1;
   498 
   499         // Sorting the list partially
   500         EdgeVector::iterator sort_end = _candidates.begin();
   501         EdgeVector::iterator vector_end = _candidates.begin();
   502         for (int idx = 0; idx < _curr_length; ++idx) {
   503           ++vector_end;
   504           if (idx <= _head_length) ++sort_end;
   505         }
   506         partial_sort(_candidates.begin(), sort_end, vector_end, _sort_func);
   507 
   508         _ns._in_edge = _candidates[0];
   509         if (_curr_length > _head_length) {
   510           _candidates[0] = _candidates[_head_length - 1];
   511           _curr_length = _head_length - 1;
   512         } else {
   513           _candidates[0] = _candidates[_curr_length - 1];
   514           --_curr_length;
   515         }
   516 
   517         return true;
   518       }
   519     }; //class AlteringListPivotRule
   520 
   521   private:
   522 
   523     // State constants for edges
   524     enum EdgeStateEnum {
   525       STATE_UPPER = -1,
   526       STATE_TREE  =  0,
   527       STATE_LOWER =  1
   528     };
   529 
   530   private:
   531 
   532     // The directed graph the algorithm runs on
   533     SGraph _graph;
   534     // The original graph
   535     const Graph &_graph_ref;
   536     // The original lower bound map
   537     const LowerMap *_lower;
   538     // The capacity map
   539     SCapacityMap _capacity;
   540     // The cost map
   541     SCostMap _cost;
   542     // The supply map
   543     SSupplyMap _supply;
   544     bool _valid_supply;
   545 
   546     // Edge map of the current flow
   547     SCapacityMap _flow;
   548     // Node map of the current potentials
   549     SPotentialMap _potential;
   550 
   551     // The depth node map of the spanning tree structure
   552     IntNodeMap _depth;
   553     // The parent node map of the spanning tree structure
   554     NodeNodeMap _parent;
   555     // The pred_edge node map of the spanning tree structure
   556     EdgeNodeMap _pred_edge;
   557     // The thread node map of the spanning tree structure
   558     NodeNodeMap _thread;
   559     // The forward node map of the spanning tree structure
   560     BoolNodeMap _forward;
   561     // The state edge map
   562     IntEdgeMap _state;
   563     // The root node of the starting spanning tree
   564     Node _root;
   565 
   566     // The reduced cost map
   567     ReducedCostMap _red_cost;
   568 
   569     // The non-artifical edges
   570     EdgeVector _edges;
   571 
   572     // Members for handling the original graph
   573     FlowMap *_flow_result;
   574     PotentialMap *_potential_result;
   575     bool _local_flow;
   576     bool _local_potential;
   577     NodeRefMap _node_ref;
   578     EdgeRefMap _edge_ref;
   579 
   580     // The entering edge of the current pivot iteration.
   581     Edge _in_edge;
   582 
   583     // Temporary nodes used in the current pivot iteration.
   584     Node join, u_in, v_in, u_out, v_out;
   585     Node right, first, second, last;
   586     Node stem, par_stem, new_stem;
   587     // The maximum augment amount along the found cycle in the current
   588     // pivot iteration.
   589     Capacity delta;
   590 
   591   public :
   592 
   593     /// \brief General constructor (with lower bounds).
   594     ///
   595     /// General constructor (with lower bounds).
   596     ///
   597     /// \param graph The directed graph the algorithm runs on.
   598     /// \param lower The lower bounds of the edges.
   599     /// \param capacity The capacities (upper bounds) of the edges.
   600     /// \param cost The cost (length) values of the edges.
   601     /// \param supply The supply values of the nodes (signed).
   602     NetworkSimplex( const Graph &graph,
   603                     const LowerMap &lower,
   604                     const CapacityMap &capacity,
   605                     const CostMap &cost,
   606                     const SupplyMap &supply ) :
   607       _graph(), _graph_ref(graph), _lower(&lower), _capacity(_graph),
   608       _cost(_graph), _supply(_graph), _flow(_graph),
   609       _potential(_graph), _depth(_graph), _parent(_graph),
   610       _pred_edge(_graph), _thread(_graph), _forward(_graph),
   611       _state(_graph), _red_cost(_graph, _cost, _potential),
   612       _flow_result(0), _potential_result(0),
   613       _local_flow(false), _local_potential(false),
   614       _node_ref(graph), _edge_ref(graph)
   615     {
   616       // Checking the sum of supply values
   617       Supply sum = 0;
   618       for (typename Graph::NodeIt n(_graph_ref); n != INVALID; ++n)
   619         sum += supply[n];
   620       if (!(_valid_supply = sum == 0)) return;
   621 
   622       // Copying _graph_ref to _graph
   623       _graph.reserveNode(countNodes(_graph_ref) + 1);
   624       _graph.reserveEdge(countEdges(_graph_ref) + countNodes(_graph_ref));
   625       copyGraph(_graph, _graph_ref)
   626         .edgeMap(_cost, cost)
   627         .nodeRef(_node_ref)
   628         .edgeRef(_edge_ref)
   629         .run();
   630 
   631       // Removing non-zero lower bounds
   632       for (typename Graph::EdgeIt e(_graph_ref); e != INVALID; ++e) {
   633         _capacity[_edge_ref[e]] = capacity[e] - lower[e];
   634       }
   635       for (typename Graph::NodeIt n(_graph_ref); n != INVALID; ++n) {
   636         Supply s = supply[n];
   637         for (typename Graph::InEdgeIt e(_graph_ref, n); e != INVALID; ++e)
   638           s += lower[e];
   639         for (typename Graph::OutEdgeIt e(_graph_ref, n); e != INVALID; ++e)
   640           s -= lower[e];
   641         _supply[_node_ref[n]] = s;
   642       }
   643     }
   644 
   645     /// \brief General constructor (without lower bounds).
   646     ///
   647     /// General constructor (without lower bounds).
   648     ///
   649     /// \param graph The directed graph the algorithm runs on.
   650     /// \param capacity The capacities (upper bounds) of the edges.
   651     /// \param cost The cost (length) values of the edges.
   652     /// \param supply The supply values of the nodes (signed).
   653     NetworkSimplex( const Graph &graph,
   654                     const CapacityMap &capacity,
   655                     const CostMap &cost,
   656                     const SupplyMap &supply ) :
   657       _graph(), _graph_ref(graph), _lower(NULL), _capacity(_graph),
   658       _cost(_graph), _supply(_graph), _flow(_graph),
   659       _potential(_graph), _depth(_graph), _parent(_graph),
   660       _pred_edge(_graph), _thread(_graph), _forward(_graph),
   661       _state(_graph), _red_cost(_graph, _cost, _potential),
   662       _flow_result(0), _potential_result(0),
   663       _local_flow(false), _local_potential(false),
   664       _node_ref(graph), _edge_ref(graph)
   665     {
   666       // Checking the sum of supply values
   667       Supply sum = 0;
   668       for (typename Graph::NodeIt n(_graph_ref); n != INVALID; ++n)
   669         sum += supply[n];
   670       if (!(_valid_supply = sum == 0)) return;
   671 
   672       // Copying _graph_ref to graph
   673       copyGraph(_graph, _graph_ref)
   674         .edgeMap(_capacity, capacity)
   675         .edgeMap(_cost, cost)
   676         .nodeMap(_supply, supply)
   677         .nodeRef(_node_ref)
   678         .edgeRef(_edge_ref)
   679         .run();
   680     }
   681 
   682     /// \brief Simple constructor (with lower bounds).
   683     ///
   684     /// Simple constructor (with lower bounds).
   685     ///
   686     /// \param graph The directed graph the algorithm runs on.
   687     /// \param lower The lower bounds of the edges.
   688     /// \param capacity The capacities (upper bounds) of the edges.
   689     /// \param cost The cost (length) values of the edges.
   690     /// \param s The source node.
   691     /// \param t The target node.
   692     /// \param flow_value The required amount of flow from node \c s
   693     /// to node \c t (i.e. the supply of \c s and the demand of \c t).
   694     NetworkSimplex( const Graph &graph,
   695                     const LowerMap &lower,
   696                     const CapacityMap &capacity,
   697                     const CostMap &cost,
   698                     typename Graph::Node s,
   699                     typename Graph::Node t,
   700                     typename SupplyMap::Value flow_value ) :
   701       _graph(), _graph_ref(graph), _lower(&lower), _capacity(_graph),
   702       _cost(_graph), _supply(_graph), _flow(_graph),
   703       _potential(_graph), _depth(_graph), _parent(_graph),
   704       _pred_edge(_graph), _thread(_graph), _forward(_graph),
   705       _state(_graph), _red_cost(_graph, _cost, _potential),
   706       _flow_result(0), _potential_result(0),
   707       _local_flow(false), _local_potential(false),
   708       _node_ref(graph), _edge_ref(graph)
   709     {
   710       // Copying _graph_ref to graph
   711       copyGraph(_graph, _graph_ref)
   712         .edgeMap(_cost, cost)
   713         .nodeRef(_node_ref)
   714         .edgeRef(_edge_ref)
   715         .run();
   716 
   717       // Removing non-zero lower bounds
   718       for (typename Graph::EdgeIt e(_graph_ref); e != INVALID; ++e) {
   719         _capacity[_edge_ref[e]] = capacity[e] - lower[e];
   720       }
   721       for (typename Graph::NodeIt n(_graph_ref); n != INVALID; ++n) {
   722         Supply sum = 0;
   723         if (n == s) sum =  flow_value;
   724         if (n == t) sum = -flow_value;
   725         for (typename Graph::InEdgeIt e(_graph_ref, n); e != INVALID; ++e)
   726           sum += lower[e];
   727         for (typename Graph::OutEdgeIt e(_graph_ref, n); e != INVALID; ++e)
   728           sum -= lower[e];
   729         _supply[_node_ref[n]] = sum;
   730       }
   731       _valid_supply = true;
   732     }
   733 
   734     /// \brief Simple constructor (without lower bounds).
   735     ///
   736     /// Simple constructor (without lower bounds).
   737     ///
   738     /// \param graph The directed graph the algorithm runs on.
   739     /// \param capacity The capacities (upper bounds) of the edges.
   740     /// \param cost The cost (length) values of the edges.
   741     /// \param s The source node.
   742     /// \param t The target node.
   743     /// \param flow_value The required amount of flow from node \c s
   744     /// to node \c t (i.e. the supply of \c s and the demand of \c t).
   745     NetworkSimplex( const Graph &graph,
   746                     const CapacityMap &capacity,
   747                     const CostMap &cost,
   748                     typename Graph::Node s,
   749                     typename Graph::Node t,
   750                     typename SupplyMap::Value flow_value ) :
   751       _graph(), _graph_ref(graph), _lower(NULL), _capacity(_graph),
   752       _cost(_graph), _supply(_graph, 0), _flow(_graph),
   753       _potential(_graph), _depth(_graph), _parent(_graph),
   754       _pred_edge(_graph), _thread(_graph), _forward(_graph),
   755       _state(_graph), _red_cost(_graph, _cost, _potential),
   756       _flow_result(0), _potential_result(0),
   757       _local_flow(false), _local_potential(false),
   758       _node_ref(graph), _edge_ref(graph)
   759     {
   760       // Copying _graph_ref to graph
   761       copyGraph(_graph, _graph_ref)
   762         .edgeMap(_capacity, capacity)
   763         .edgeMap(_cost, cost)
   764         .nodeRef(_node_ref)
   765         .edgeRef(_edge_ref)
   766         .run();
   767       _supply[_node_ref[s]] =  flow_value;
   768       _supply[_node_ref[t]] = -flow_value;
   769       _valid_supply = true;
   770     }
   771 
   772     /// Destructor.
   773     ~NetworkSimplex() {
   774       if (_local_flow) delete _flow_result;
   775       if (_local_potential) delete _potential_result;
   776     }
   777 
   778     /// \brief Set the flow map.
   779     ///
   780     /// Set the flow map.
   781     ///
   782     /// \return \c (*this)
   783     NetworkSimplex& flowMap(FlowMap &map) {
   784       if (_local_flow) {
   785         delete _flow_result;
   786         _local_flow = false;
   787       }
   788       _flow_result = &map;
   789       return *this;
   790     }
   791 
   792     /// \brief Set the potential map.
   793     ///
   794     /// Set the potential map.
   795     ///
   796     /// \return \c (*this)
   797     NetworkSimplex& potentialMap(PotentialMap &map) {
   798       if (_local_potential) {
   799         delete _potential_result;
   800         _local_potential = false;
   801       }
   802       _potential_result = &map;
   803       return *this;
   804     }
   805 
   806     /// \name Execution control
   807 
   808     /// @{
   809 
   810     /// \brief Runs the algorithm.
   811     ///
   812     /// Runs the algorithm.
   813     ///
   814     /// \param pivot_rule The pivot rule that is used during the
   815     /// algorithm.
   816     ///
   817     /// The available pivot rules:
   818     ///
   819     /// - FIRST_ELIGIBLE_PIVOT The next eligible edge is selected in
   820     /// a wraparound fashion in every iteration
   821     /// (\ref FirstEligiblePivotRule).
   822     ///
   823     /// - BEST_ELIGIBLE_PIVOT The best eligible edge is selected in
   824     /// every iteration (\ref BestEligiblePivotRule).
   825     ///
   826     /// - BLOCK_SEARCH_PIVOT A specified number of edges are examined in
   827     /// every iteration in a wraparound fashion and the best eligible
   828     /// edge is selected from this block (\ref BlockSearchPivotRule).
   829     ///
   830     /// - CANDIDATE_LIST_PIVOT In a major iteration a candidate list is
   831     /// built from eligible edges in a wraparound fashion and in the
   832     /// following minor iterations the best eligible edge is selected
   833     /// from this list (\ref CandidateListPivotRule).
   834     ///
   835     /// - ALTERING_LIST_PIVOT It is a modified version of the
   836     /// "Candidate List" pivot rule. It keeps only the several best
   837     /// eligible edges from the former candidate list and extends this
   838     /// list in every iteration (\ref AlteringListPivotRule).
   839     ///
   840     /// According to our comprehensive benchmark tests the "Block Search"
   841     /// pivot rule proved to be by far the fastest and the most robust
   842     /// on various test inputs. Thus it is the default option.
   843     ///
   844     /// \return \c true if a feasible flow can be found.
   845     bool run(PivotRuleEnum pivot_rule = BLOCK_SEARCH_PIVOT) {
   846       return init() && start(pivot_rule);
   847     }
   848 
   849     /// @}
   850 
   851     /// \name Query Functions
   852     /// The results of the algorithm can be obtained using these
   853     /// functions.\n
   854     /// \ref lemon::NetworkSimplex::run() "run()" must be called before
   855     /// using them.
   856 
   857     /// @{
   858 
   859     /// \brief Return a const reference to the edge map storing the
   860     /// found flow.
   861     ///
   862     /// Return a const reference to the edge map storing the found flow.
   863     ///
   864     /// \pre \ref run() must be called before using this function.
   865     const FlowMap& flowMap() const {
   866       return *_flow_result;
   867     }
   868 
   869     /// \brief Return a const reference to the node map storing the
   870     /// found potentials (the dual solution).
   871     ///
   872     /// Return a const reference to the node map storing the found
   873     /// potentials (the dual solution).
   874     ///
   875     /// \pre \ref run() must be called before using this function.
   876     const PotentialMap& potentialMap() const {
   877       return *_potential_result;
   878     }
   879 
   880     /// \brief Return the flow on the given edge.
   881     ///
   882     /// Return the flow on the given edge.
   883     ///
   884     /// \pre \ref run() must be called before using this function.
   885     Capacity flow(const typename Graph::Edge& edge) const {
   886       return (*_flow_result)[edge];
   887     }
   888 
   889     /// \brief Return the potential of the given node.
   890     ///
   891     /// Return the potential of the given node.
   892     ///
   893     /// \pre \ref run() must be called before using this function.
   894     Cost potential(const typename Graph::Node& node) const {
   895       return (*_potential_result)[node];
   896     }
   897 
   898     /// \brief Return the total cost of the found flow.
   899     ///
   900     /// Return the total cost of the found flow. The complexity of the
   901     /// function is \f$ O(e) \f$.
   902     ///
   903     /// \pre \ref run() must be called before using this function.
   904     Cost totalCost() const {
   905       Cost c = 0;
   906       for (typename Graph::EdgeIt e(_graph_ref); e != INVALID; ++e)
   907         c += (*_flow_result)[e] * _cost[_edge_ref[e]];
   908       return c;
   909     }
   910 
   911     /// @}
   912 
   913   private:
   914 
   915     /// \brief Extend the underlying graph and initialize all the
   916     /// node and edge maps.
   917     bool init() {
   918       if (!_valid_supply) return false;
   919 
   920       // Initializing result maps
   921       if (!_flow_result) {
   922         _flow_result = new FlowMap(_graph_ref);
   923         _local_flow = true;
   924       }
   925       if (!_potential_result) {
   926         _potential_result = new PotentialMap(_graph_ref);
   927         _local_potential = true;
   928       }
   929 
   930       // Initializing the edge vector and the edge maps
   931       _edges.reserve(countEdges(_graph));
   932       for (EdgeIt e(_graph); e != INVALID; ++e) {
   933         _edges.push_back(e);
   934         _flow[e] = 0;
   935         _state[e] = STATE_LOWER;
   936       }
   937 
   938       // Adding an artificial root node to the graph
   939       _root = _graph.addNode();
   940       _parent[_root] = INVALID;
   941       _pred_edge[_root] = INVALID;
   942       _depth[_root] = 0;
   943       _supply[_root] = 0;
   944       _potential[_root] = 0;
   945 
   946       // Adding artificial edges to the graph and initializing the node
   947       // maps of the spanning tree data structure
   948       Node last = _root;
   949       Edge e;
   950       Cost max_cost = std::numeric_limits<Cost>::max() / 4;
   951       for (NodeIt u(_graph); u != INVALID; ++u) {
   952         if (u == _root) continue;
   953         _thread[last] = u;
   954         last = u;
   955         _parent[u] = _root;
   956         _depth[u] = 1;
   957         if (_supply[u] >= 0) {
   958           e = _graph.addEdge(u, _root);
   959           _flow[e] = _supply[u];
   960           _forward[u] = true;
   961           _potential[u] = -max_cost;
   962         } else {
   963           e = _graph.addEdge(_root, u);
   964           _flow[e] = -_supply[u];
   965           _forward[u] = false;
   966           _potential[u] = max_cost;
   967         }
   968         _cost[e] = max_cost;
   969         _capacity[e] = std::numeric_limits<Capacity>::max();
   970         _state[e] = STATE_TREE;
   971         _pred_edge[u] = e;
   972       }
   973       _thread[last] = _root;
   974 
   975       return true;
   976     }
   977 
   978     /// Find the join node.
   979     inline Node findJoinNode() {
   980       Node u = _graph.source(_in_edge);
   981       Node v = _graph.target(_in_edge);
   982       while (u != v) {
   983         if (_depth[u] == _depth[v]) {
   984           u = _parent[u];
   985           v = _parent[v];
   986         }
   987         else if (_depth[u] > _depth[v]) u = _parent[u];
   988         else v = _parent[v];
   989       }
   990       return u;
   991     }
   992 
   993     /// \brief Find the leaving edge of the cycle.
   994     /// \return \c true if the leaving edge is not the same as the
   995     /// entering edge.
   996     inline bool findLeavingEdge() {
   997       // Initializing first and second nodes according to the direction
   998       // of the cycle
   999       if (_state[_in_edge] == STATE_LOWER) {
  1000         first  = _graph.source(_in_edge);
  1001         second = _graph.target(_in_edge);
  1002       } else {
  1003         first  = _graph.target(_in_edge);
  1004         second = _graph.source(_in_edge);
  1005       }
  1006       delta = _capacity[_in_edge];
  1007       bool result = false;
  1008       Capacity d;
  1009       Edge e;
  1010 
  1011       // Searching the cycle along the path form the first node to the
  1012       // root node
  1013       for (Node u = first; u != join; u = _parent[u]) {
  1014         e = _pred_edge[u];
  1015         d = _forward[u] ? _flow[e] : _capacity[e] - _flow[e];
  1016         if (d < delta) {
  1017           delta = d;
  1018           u_out = u;
  1019           u_in = first;
  1020           v_in = second;
  1021           result = true;
  1022         }
  1023       }
  1024       // Searching the cycle along the path form the second node to the
  1025       // root node
  1026       for (Node u = second; u != join; u = _parent[u]) {
  1027         e = _pred_edge[u];
  1028         d = _forward[u] ? _capacity[e] - _flow[e] : _flow[e];
  1029         if (d <= delta) {
  1030           delta = d;
  1031           u_out = u;
  1032           u_in = second;
  1033           v_in = first;
  1034           result = true;
  1035         }
  1036       }
  1037       return result;
  1038     }
  1039 
  1040     /// Change \c flow and \c state edge maps.
  1041     inline void changeFlows(bool change) {
  1042       // Augmenting along the cycle
  1043       if (delta > 0) {
  1044         Capacity val = _state[_in_edge] * delta;
  1045         _flow[_in_edge] += val;
  1046         for (Node u = _graph.source(_in_edge); u != join; u = _parent[u]) {
  1047           _flow[_pred_edge[u]] += _forward[u] ? -val : val;
  1048         }
  1049         for (Node u = _graph.target(_in_edge); u != join; u = _parent[u]) {
  1050           _flow[_pred_edge[u]] += _forward[u] ? val : -val;
  1051         }
  1052       }
  1053       // Updating the state of the entering and leaving edges
  1054       if (change) {
  1055         _state[_in_edge] = STATE_TREE;
  1056         _state[_pred_edge[u_out]] =
  1057           (_flow[_pred_edge[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
  1058       } else {
  1059         _state[_in_edge] = -_state[_in_edge];
  1060       }
  1061     }
  1062 
  1063     /// Update \c thread and \c parent node maps.
  1064     inline void updateThreadParent() {
  1065       Node u;
  1066       v_out = _parent[u_out];
  1067 
  1068       // Handling the case when join and v_out coincide
  1069       bool par_first = false;
  1070       if (join == v_out) {
  1071         for (u = join; u != u_in && u != v_in; u = _thread[u]) ;
  1072         if (u == v_in) {
  1073           par_first = true;
  1074           while (_thread[u] != u_out) u = _thread[u];
  1075           first = u;
  1076         }
  1077       }
  1078 
  1079       // Finding the last successor of u_in (u) and the node after it
  1080       // (right) according to the thread index
  1081       for (u = u_in; _depth[_thread[u]] > _depth[u_in]; u = _thread[u]) ;
  1082       right = _thread[u];
  1083       if (_thread[v_in] == u_out) {
  1084         for (last = u; _depth[last] > _depth[u_out]; last = _thread[last]) ;
  1085         if (last == u_out) last = _thread[last];
  1086       }
  1087       else last = _thread[v_in];
  1088 
  1089       // Updating stem nodes
  1090       _thread[v_in] = stem = u_in;
  1091       par_stem = v_in;
  1092       while (stem != u_out) {
  1093         _thread[u] = new_stem = _parent[stem];
  1094 
  1095         // Finding the node just before the stem node (u) according to
  1096         // the original thread index
  1097         for (u = new_stem; _thread[u] != stem; u = _thread[u]) ;
  1098         _thread[u] = right;
  1099 
  1100         // Changing the parent node of stem and shifting stem and
  1101         // par_stem nodes
  1102         _parent[stem] = par_stem;
  1103         par_stem = stem;
  1104         stem = new_stem;
  1105 
  1106         // Finding the last successor of stem (u) and the node after it
  1107         // (right) according to the thread index
  1108         for (u = stem; _depth[_thread[u]] > _depth[stem]; u = _thread[u]) ;
  1109         right = _thread[u];
  1110       }
  1111       _parent[u_out] = par_stem;
  1112       _thread[u] = last;
  1113 
  1114       if (join == v_out && par_first) {
  1115         if (first != v_in) _thread[first] = right;
  1116       } else {
  1117         for (u = v_out; _thread[u] != u_out; u = _thread[u]) ;
  1118         _thread[u] = right;
  1119       }
  1120     }
  1121 
  1122     /// Update \c pred_edge and \c forward node maps.
  1123     inline void updatePredEdge() {
  1124       Node u = u_out, v;
  1125       while (u != u_in) {
  1126         v = _parent[u];
  1127         _pred_edge[u] = _pred_edge[v];
  1128         _forward[u] = !_forward[v];
  1129         u = v;
  1130       }
  1131       _pred_edge[u_in] = _in_edge;
  1132       _forward[u_in] = (u_in == _graph.source(_in_edge));
  1133     }
  1134 
  1135     /// Update \c depth and \c potential node maps.
  1136     inline void updateDepthPotential() {
  1137       _depth[u_in] = _depth[v_in] + 1;
  1138       _potential[u_in] = _forward[u_in] ?
  1139         _potential[v_in] - _cost[_pred_edge[u_in]] :
  1140         _potential[v_in] + _cost[_pred_edge[u_in]];
  1141 
  1142       Node u = _thread[u_in], v;
  1143       while (true) {
  1144         v = _parent[u];
  1145         if (v == INVALID) break;
  1146         _depth[u] = _depth[v] + 1;
  1147         _potential[u] = _forward[u] ?
  1148           _potential[v] - _cost[_pred_edge[u]] :
  1149           _potential[v] + _cost[_pred_edge[u]];
  1150         if (_depth[u] <= _depth[v_in]) break;
  1151         u = _thread[u];
  1152       }
  1153     }
  1154 
  1155     /// Execute the algorithm.
  1156     bool start(PivotRuleEnum pivot_rule) {
  1157       // Selecting the pivot rule implementation
  1158       switch (pivot_rule) {
  1159         case FIRST_ELIGIBLE_PIVOT:
  1160           return start<FirstEligiblePivotRule>();
  1161         case BEST_ELIGIBLE_PIVOT:
  1162           return start<BestEligiblePivotRule>();
  1163         case BLOCK_SEARCH_PIVOT:
  1164           return start<BlockSearchPivotRule>();
  1165         case CANDIDATE_LIST_PIVOT:
  1166           return start<CandidateListPivotRule>();
  1167         case ALTERING_LIST_PIVOT:
  1168           return start<AlteringListPivotRule>();
  1169       }
  1170       return false;
  1171     }
  1172 
  1173     template<class PivotRuleImplementation>
  1174     bool start() {
  1175       PivotRuleImplementation pivot(*this, _edges);
  1176 
  1177       // Executing the network simplex algorithm
  1178       while (pivot.findEnteringEdge()) {
  1179         join = findJoinNode();
  1180         bool change = findLeavingEdge();
  1181         changeFlows(change);
  1182         if (change) {
  1183           updateThreadParent();
  1184           updatePredEdge();
  1185           updateDepthPotential();
  1186         }
  1187       }
  1188 
  1189       // Checking if the flow amount equals zero on all the artificial
  1190       // edges
  1191       for (InEdgeIt e(_graph, _root); e != INVALID; ++e)
  1192         if (_flow[e] > 0) return false;
  1193       for (OutEdgeIt e(_graph, _root); e != INVALID; ++e)
  1194         if (_flow[e] > 0) return false;
  1195 
  1196       // Copying flow values to _flow_result
  1197       if (_lower) {
  1198         for (typename Graph::EdgeIt e(_graph_ref); e != INVALID; ++e)
  1199           (*_flow_result)[e] = (*_lower)[e] + _flow[_edge_ref[e]];
  1200       } else {
  1201         for (typename Graph::EdgeIt e(_graph_ref); e != INVALID; ++e)
  1202           (*_flow_result)[e] = _flow[_edge_ref[e]];
  1203       }
  1204       // Copying potential values to _potential_result
  1205       for (typename Graph::NodeIt n(_graph_ref); n != INVALID; ++n)
  1206         (*_potential_result)[n] = _potential[_node_ref[n]];
  1207 
  1208       return true;
  1209     }
  1210 
  1211   }; //class NetworkSimplex
  1212 
  1213   ///@}
  1214 
  1215 } //namespace lemon
  1216 
  1217 #endif //LEMON_NETWORK_SIMPLEX_H