lemon/capacity_scaling.h
author Alpar Juttner <alpar@cs.elte.hu>
Fri, 22 Feb 2013 16:49:41 +0100
changeset 1016 97975184f4aa
parent 985 eb12ad2789fc
parent 1003 16f55008c863
child 1049 7bf489cf624e
child 1070 ee9bac10f58e
permissions -rw-r--r--
Merge bugfix #445
     1 /* -*- mode: C++; indent-tabs-mode: nil; -*-
     2  *
     3  * This file is a part of LEMON, a generic C++ optimization library.
     4  *
     5  * Copyright (C) 2003-2010
     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_CAPACITY_SCALING_H
    20 #define LEMON_CAPACITY_SCALING_H
    21 
    22 /// \ingroup min_cost_flow_algs
    23 ///
    24 /// \file
    25 /// \brief Capacity Scaling algorithm for finding a minimum cost flow.
    26 
    27 #include <vector>
    28 #include <limits>
    29 #include <lemon/core.h>
    30 #include <lemon/bin_heap.h>
    31 
    32 namespace lemon {
    33 
    34   /// \brief Default traits class of CapacityScaling algorithm.
    35   ///
    36   /// Default traits class of CapacityScaling algorithm.
    37   /// \tparam GR Digraph type.
    38   /// \tparam V The number type used for flow amounts, capacity bounds
    39   /// and supply values. By default it is \c int.
    40   /// \tparam C The number type used for costs and potentials.
    41   /// By default it is the same as \c V.
    42   template <typename GR, typename V = int, typename C = V>
    43   struct CapacityScalingDefaultTraits
    44   {
    45     /// The type of the digraph
    46     typedef GR Digraph;
    47     /// The type of the flow amounts, capacity bounds and supply values
    48     typedef V Value;
    49     /// The type of the arc costs
    50     typedef C Cost;
    51 
    52     /// \brief The type of the heap used for internal Dijkstra computations.
    53     ///
    54     /// The type of the heap used for internal Dijkstra computations.
    55     /// It must conform to the \ref lemon::concepts::Heap "Heap" concept,
    56     /// its priority type must be \c Cost and its cross reference type
    57     /// must be \ref RangeMap "RangeMap<int>".
    58     typedef BinHeap<Cost, RangeMap<int> > Heap;
    59   };
    60 
    61   /// \addtogroup min_cost_flow_algs
    62   /// @{
    63 
    64   /// \brief Implementation of the Capacity Scaling algorithm for
    65   /// finding a \ref min_cost_flow "minimum cost flow".
    66   ///
    67   /// \ref CapacityScaling implements the capacity scaling version
    68   /// of the successive shortest path algorithm for finding a
    69   /// \ref min_cost_flow "minimum cost flow" \ref amo93networkflows,
    70   /// \ref edmondskarp72theoretical. It is an efficient dual
    71   /// solution method.
    72   ///
    73   /// This algorithm is typically slower than \ref CostScaling and
    74   /// \ref NetworkSimplex, but in special cases, it can be more
    75   /// efficient than them.
    76   /// (For more information, see \ref min_cost_flow_algs "the module page".)
    77   ///
    78   /// Most of the parameters of the problem (except for the digraph)
    79   /// can be given using separate functions, and the algorithm can be
    80   /// executed using the \ref run() function. If some parameters are not
    81   /// specified, then default values will be used.
    82   ///
    83   /// \tparam GR The digraph type the algorithm runs on.
    84   /// \tparam V The number type used for flow amounts, capacity bounds
    85   /// and supply values in the algorithm. By default, it is \c int.
    86   /// \tparam C The number type used for costs and potentials in the
    87   /// algorithm. By default, it is the same as \c V.
    88   /// \tparam TR The traits class that defines various types used by the
    89   /// algorithm. By default, it is \ref CapacityScalingDefaultTraits
    90   /// "CapacityScalingDefaultTraits<GR, V, C>".
    91   /// In most cases, this parameter should not be set directly,
    92   /// consider to use the named template parameters instead.
    93   ///
    94   /// \warning Both \c V and \c C must be signed number types.
    95   /// \warning Capacity bounds and supply values must be integer, but
    96   /// arc costs can be arbitrary real numbers.
    97   /// \warning This algorithm does not support negative costs for
    98   /// arcs having infinite upper bound.
    99 #ifdef DOXYGEN
   100   template <typename GR, typename V, typename C, typename TR>
   101 #else
   102   template < typename GR, typename V = int, typename C = V,
   103              typename TR = CapacityScalingDefaultTraits<GR, V, C> >
   104 #endif
   105   class CapacityScaling
   106   {
   107   public:
   108 
   109     /// The type of the digraph
   110     typedef typename TR::Digraph Digraph;
   111     /// The type of the flow amounts, capacity bounds and supply values
   112     typedef typename TR::Value Value;
   113     /// The type of the arc costs
   114     typedef typename TR::Cost Cost;
   115 
   116     /// The type of the heap used for internal Dijkstra computations
   117     typedef typename TR::Heap Heap;
   118 
   119     /// The \ref CapacityScalingDefaultTraits "traits class" of the algorithm
   120     typedef TR Traits;
   121 
   122   public:
   123 
   124     /// \brief Problem type constants for the \c run() function.
   125     ///
   126     /// Enum type containing the problem type constants that can be
   127     /// returned by the \ref run() function of the algorithm.
   128     enum ProblemType {
   129       /// The problem has no feasible solution (flow).
   130       INFEASIBLE,
   131       /// The problem has optimal solution (i.e. it is feasible and
   132       /// bounded), and the algorithm has found optimal flow and node
   133       /// potentials (primal and dual solutions).
   134       OPTIMAL,
   135       /// The digraph contains an arc of negative cost and infinite
   136       /// upper bound. It means that the objective function is unbounded
   137       /// on that arc, however, note that it could actually be bounded
   138       /// over the feasible flows, but this algroithm cannot handle
   139       /// these cases.
   140       UNBOUNDED
   141     };
   142 
   143   private:
   144 
   145     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
   146 
   147     typedef std::vector<int> IntVector;
   148     typedef std::vector<Value> ValueVector;
   149     typedef std::vector<Cost> CostVector;
   150     typedef std::vector<char> BoolVector;
   151     // Note: vector<char> is used instead of vector<bool> for efficiency reasons
   152 
   153   private:
   154 
   155     // Data related to the underlying digraph
   156     const GR &_graph;
   157     int _node_num;
   158     int _arc_num;
   159     int _res_arc_num;
   160     int _root;
   161 
   162     // Parameters of the problem
   163     bool _have_lower;
   164     Value _sum_supply;
   165 
   166     // Data structures for storing the digraph
   167     IntNodeMap _node_id;
   168     IntArcMap _arc_idf;
   169     IntArcMap _arc_idb;
   170     IntVector _first_out;
   171     BoolVector _forward;
   172     IntVector _source;
   173     IntVector _target;
   174     IntVector _reverse;
   175 
   176     // Node and arc data
   177     ValueVector _lower;
   178     ValueVector _upper;
   179     CostVector _cost;
   180     ValueVector _supply;
   181 
   182     ValueVector _res_cap;
   183     CostVector _pi;
   184     ValueVector _excess;
   185     IntVector _excess_nodes;
   186     IntVector _deficit_nodes;
   187 
   188     Value _delta;
   189     int _factor;
   190     IntVector _pred;
   191 
   192   public:
   193 
   194     /// \brief Constant for infinite upper bounds (capacities).
   195     ///
   196     /// Constant for infinite upper bounds (capacities).
   197     /// It is \c std::numeric_limits<Value>::infinity() if available,
   198     /// \c std::numeric_limits<Value>::max() otherwise.
   199     const Value INF;
   200 
   201   private:
   202 
   203     // Special implementation of the Dijkstra algorithm for finding
   204     // shortest paths in the residual network of the digraph with
   205     // respect to the reduced arc costs and modifying the node
   206     // potentials according to the found distance labels.
   207     class ResidualDijkstra
   208     {
   209     private:
   210 
   211       int _node_num;
   212       bool _geq;
   213       const IntVector &_first_out;
   214       const IntVector &_target;
   215       const CostVector &_cost;
   216       const ValueVector &_res_cap;
   217       const ValueVector &_excess;
   218       CostVector &_pi;
   219       IntVector &_pred;
   220 
   221       IntVector _proc_nodes;
   222       CostVector _dist;
   223 
   224     public:
   225 
   226       ResidualDijkstra(CapacityScaling& cs) :
   227         _node_num(cs._node_num), _geq(cs._sum_supply < 0),
   228         _first_out(cs._first_out), _target(cs._target), _cost(cs._cost),
   229         _res_cap(cs._res_cap), _excess(cs._excess), _pi(cs._pi),
   230         _pred(cs._pred), _dist(cs._node_num)
   231       {}
   232 
   233       int run(int s, Value delta = 1) {
   234         RangeMap<int> heap_cross_ref(_node_num, Heap::PRE_HEAP);
   235         Heap heap(heap_cross_ref);
   236         heap.push(s, 0);
   237         _pred[s] = -1;
   238         _proc_nodes.clear();
   239 
   240         // Process nodes
   241         while (!heap.empty() && _excess[heap.top()] > -delta) {
   242           int u = heap.top(), v;
   243           Cost d = heap.prio() + _pi[u], dn;
   244           _dist[u] = heap.prio();
   245           _proc_nodes.push_back(u);
   246           heap.pop();
   247 
   248           // Traverse outgoing residual arcs
   249           int last_out = _geq ? _first_out[u+1] : _first_out[u+1] - 1;
   250           for (int a = _first_out[u]; a != last_out; ++a) {
   251             if (_res_cap[a] < delta) continue;
   252             v = _target[a];
   253             switch (heap.state(v)) {
   254               case Heap::PRE_HEAP:
   255                 heap.push(v, d + _cost[a] - _pi[v]);
   256                 _pred[v] = a;
   257                 break;
   258               case Heap::IN_HEAP:
   259                 dn = d + _cost[a] - _pi[v];
   260                 if (dn < heap[v]) {
   261                   heap.decrease(v, dn);
   262                   _pred[v] = a;
   263                 }
   264                 break;
   265               case Heap::POST_HEAP:
   266                 break;
   267             }
   268           }
   269         }
   270         if (heap.empty()) return -1;
   271 
   272         // Update potentials of processed nodes
   273         int t = heap.top();
   274         Cost dt = heap.prio();
   275         for (int i = 0; i < int(_proc_nodes.size()); ++i) {
   276           _pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - dt;
   277         }
   278 
   279         return t;
   280       }
   281 
   282     }; //class ResidualDijkstra
   283 
   284   public:
   285 
   286     /// \name Named Template Parameters
   287     /// @{
   288 
   289     template <typename T>
   290     struct SetHeapTraits : public Traits {
   291       typedef T Heap;
   292     };
   293 
   294     /// \brief \ref named-templ-param "Named parameter" for setting
   295     /// \c Heap type.
   296     ///
   297     /// \ref named-templ-param "Named parameter" for setting \c Heap
   298     /// type, which is used for internal Dijkstra computations.
   299     /// It must conform to the \ref lemon::concepts::Heap "Heap" concept,
   300     /// its priority type must be \c Cost and its cross reference type
   301     /// must be \ref RangeMap "RangeMap<int>".
   302     template <typename T>
   303     struct SetHeap
   304       : public CapacityScaling<GR, V, C, SetHeapTraits<T> > {
   305       typedef  CapacityScaling<GR, V, C, SetHeapTraits<T> > Create;
   306     };
   307 
   308     /// @}
   309 
   310   protected:
   311 
   312     CapacityScaling() {}
   313 
   314   public:
   315 
   316     /// \brief Constructor.
   317     ///
   318     /// The constructor of the class.
   319     ///
   320     /// \param graph The digraph the algorithm runs on.
   321     CapacityScaling(const GR& graph) :
   322       _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
   323       INF(std::numeric_limits<Value>::has_infinity ?
   324           std::numeric_limits<Value>::infinity() :
   325           std::numeric_limits<Value>::max())
   326     {
   327       // Check the number types
   328       LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
   329         "The flow type of CapacityScaling must be signed");
   330       LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
   331         "The cost type of CapacityScaling must be signed");
   332 
   333       // Reset data structures
   334       reset();
   335     }
   336 
   337     /// \name Parameters
   338     /// The parameters of the algorithm can be specified using these
   339     /// functions.
   340 
   341     /// @{
   342 
   343     /// \brief Set the lower bounds on the arcs.
   344     ///
   345     /// This function sets the lower bounds on the arcs.
   346     /// If it is not used before calling \ref run(), the lower bounds
   347     /// will be set to zero on all arcs.
   348     ///
   349     /// \param map An arc map storing the lower bounds.
   350     /// Its \c Value type must be convertible to the \c Value type
   351     /// of the algorithm.
   352     ///
   353     /// \return <tt>(*this)</tt>
   354     template <typename LowerMap>
   355     CapacityScaling& lowerMap(const LowerMap& map) {
   356       _have_lower = true;
   357       for (ArcIt a(_graph); a != INVALID; ++a) {
   358         _lower[_arc_idf[a]] = map[a];
   359         _lower[_arc_idb[a]] = map[a];
   360       }
   361       return *this;
   362     }
   363 
   364     /// \brief Set the upper bounds (capacities) on the arcs.
   365     ///
   366     /// This function sets the upper bounds (capacities) on the arcs.
   367     /// If it is not used before calling \ref run(), the upper bounds
   368     /// will be set to \ref INF on all arcs (i.e. the flow value will be
   369     /// unbounded from above).
   370     ///
   371     /// \param map An arc map storing the upper bounds.
   372     /// Its \c Value type must be convertible to the \c Value type
   373     /// of the algorithm.
   374     ///
   375     /// \return <tt>(*this)</tt>
   376     template<typename UpperMap>
   377     CapacityScaling& upperMap(const UpperMap& map) {
   378       for (ArcIt a(_graph); a != INVALID; ++a) {
   379         _upper[_arc_idf[a]] = map[a];
   380       }
   381       return *this;
   382     }
   383 
   384     /// \brief Set the costs of the arcs.
   385     ///
   386     /// This function sets the costs of the arcs.
   387     /// If it is not used before calling \ref run(), the costs
   388     /// will be set to \c 1 on all arcs.
   389     ///
   390     /// \param map An arc map storing the costs.
   391     /// Its \c Value type must be convertible to the \c Cost type
   392     /// of the algorithm.
   393     ///
   394     /// \return <tt>(*this)</tt>
   395     template<typename CostMap>
   396     CapacityScaling& costMap(const CostMap& map) {
   397       for (ArcIt a(_graph); a != INVALID; ++a) {
   398         _cost[_arc_idf[a]] =  map[a];
   399         _cost[_arc_idb[a]] = -map[a];
   400       }
   401       return *this;
   402     }
   403 
   404     /// \brief Set the supply values of the nodes.
   405     ///
   406     /// This function sets the supply values of the nodes.
   407     /// If neither this function nor \ref stSupply() is used before
   408     /// calling \ref run(), the supply of each node will be set to zero.
   409     ///
   410     /// \param map A node map storing the supply values.
   411     /// Its \c Value type must be convertible to the \c Value type
   412     /// of the algorithm.
   413     ///
   414     /// \return <tt>(*this)</tt>
   415     template<typename SupplyMap>
   416     CapacityScaling& supplyMap(const SupplyMap& map) {
   417       for (NodeIt n(_graph); n != INVALID; ++n) {
   418         _supply[_node_id[n]] = map[n];
   419       }
   420       return *this;
   421     }
   422 
   423     /// \brief Set single source and target nodes and a supply value.
   424     ///
   425     /// This function sets a single source node and a single target node
   426     /// and the required flow value.
   427     /// If neither this function nor \ref supplyMap() is used before
   428     /// calling \ref run(), the supply of each node will be set to zero.
   429     ///
   430     /// Using this function has the same effect as using \ref supplyMap()
   431     /// with a map in which \c k is assigned to \c s, \c -k is
   432     /// assigned to \c t and all other nodes have zero supply value.
   433     ///
   434     /// \param s The source node.
   435     /// \param t The target node.
   436     /// \param k The required amount of flow from node \c s to node \c t
   437     /// (i.e. the supply of \c s and the demand of \c t).
   438     ///
   439     /// \return <tt>(*this)</tt>
   440     CapacityScaling& stSupply(const Node& s, const Node& t, Value k) {
   441       for (int i = 0; i != _node_num; ++i) {
   442         _supply[i] = 0;
   443       }
   444       _supply[_node_id[s]] =  k;
   445       _supply[_node_id[t]] = -k;
   446       return *this;
   447     }
   448 
   449     /// @}
   450 
   451     /// \name Execution control
   452     /// The algorithm can be executed using \ref run().
   453 
   454     /// @{
   455 
   456     /// \brief Run the algorithm.
   457     ///
   458     /// This function runs the algorithm.
   459     /// The paramters can be specified using functions \ref lowerMap(),
   460     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
   461     /// For example,
   462     /// \code
   463     ///   CapacityScaling<ListDigraph> cs(graph);
   464     ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
   465     ///     .supplyMap(sup).run();
   466     /// \endcode
   467     ///
   468     /// This function can be called more than once. All the given parameters
   469     /// are kept for the next call, unless \ref resetParams() or \ref reset()
   470     /// is used, thus only the modified parameters have to be set again.
   471     /// If the underlying digraph was also modified after the construction
   472     /// of the class (or the last \ref reset() call), then the \ref reset()
   473     /// function must be called.
   474     ///
   475     /// \param factor The capacity scaling factor. It must be larger than
   476     /// one to use scaling. If it is less or equal to one, then scaling
   477     /// will be disabled.
   478     ///
   479     /// \return \c INFEASIBLE if no feasible flow exists,
   480     /// \n \c OPTIMAL if the problem has optimal solution
   481     /// (i.e. it is feasible and bounded), and the algorithm has found
   482     /// optimal flow and node potentials (primal and dual solutions),
   483     /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
   484     /// and infinite upper bound. It means that the objective function
   485     /// is unbounded on that arc, however, note that it could actually be
   486     /// bounded over the feasible flows, but this algroithm cannot handle
   487     /// these cases.
   488     ///
   489     /// \see ProblemType
   490     /// \see resetParams(), reset()
   491     ProblemType run(int factor = 4) {
   492       _factor = factor;
   493       ProblemType pt = init();
   494       if (pt != OPTIMAL) return pt;
   495       return start();
   496     }
   497 
   498     /// \brief Reset all the parameters that have been given before.
   499     ///
   500     /// This function resets all the paramaters that have been given
   501     /// before using functions \ref lowerMap(), \ref upperMap(),
   502     /// \ref costMap(), \ref supplyMap(), \ref stSupply().
   503     ///
   504     /// It is useful for multiple \ref run() calls. Basically, all the given
   505     /// parameters are kept for the next \ref run() call, unless
   506     /// \ref resetParams() or \ref reset() is used.
   507     /// If the underlying digraph was also modified after the construction
   508     /// of the class or the last \ref reset() call, then the \ref reset()
   509     /// function must be used, otherwise \ref resetParams() is sufficient.
   510     ///
   511     /// For example,
   512     /// \code
   513     ///   CapacityScaling<ListDigraph> cs(graph);
   514     ///
   515     ///   // First run
   516     ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
   517     ///     .supplyMap(sup).run();
   518     ///
   519     ///   // Run again with modified cost map (resetParams() is not called,
   520     ///   // so only the cost map have to be set again)
   521     ///   cost[e] += 100;
   522     ///   cs.costMap(cost).run();
   523     ///
   524     ///   // Run again from scratch using resetParams()
   525     ///   // (the lower bounds will be set to zero on all arcs)
   526     ///   cs.resetParams();
   527     ///   cs.upperMap(capacity).costMap(cost)
   528     ///     .supplyMap(sup).run();
   529     /// \endcode
   530     ///
   531     /// \return <tt>(*this)</tt>
   532     ///
   533     /// \see reset(), run()
   534     CapacityScaling& resetParams() {
   535       for (int i = 0; i != _node_num; ++i) {
   536         _supply[i] = 0;
   537       }
   538       for (int j = 0; j != _res_arc_num; ++j) {
   539         _lower[j] = 0;
   540         _upper[j] = INF;
   541         _cost[j] = _forward[j] ? 1 : -1;
   542       }
   543       _have_lower = false;
   544       return *this;
   545     }
   546 
   547     /// \brief Reset the internal data structures and all the parameters
   548     /// that have been given before.
   549     ///
   550     /// This function resets the internal data structures and all the
   551     /// paramaters that have been given before using functions \ref lowerMap(),
   552     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
   553     ///
   554     /// It is useful for multiple \ref run() calls. Basically, all the given
   555     /// parameters are kept for the next \ref run() call, unless
   556     /// \ref resetParams() or \ref reset() is used.
   557     /// If the underlying digraph was also modified after the construction
   558     /// of the class or the last \ref reset() call, then the \ref reset()
   559     /// function must be used, otherwise \ref resetParams() is sufficient.
   560     ///
   561     /// See \ref resetParams() for examples.
   562     ///
   563     /// \return <tt>(*this)</tt>
   564     ///
   565     /// \see resetParams(), run()
   566     CapacityScaling& reset() {
   567       // Resize vectors
   568       _node_num = countNodes(_graph);
   569       _arc_num = countArcs(_graph);
   570       _res_arc_num = 2 * (_arc_num + _node_num);
   571       _root = _node_num;
   572       ++_node_num;
   573 
   574       _first_out.resize(_node_num + 1);
   575       _forward.resize(_res_arc_num);
   576       _source.resize(_res_arc_num);
   577       _target.resize(_res_arc_num);
   578       _reverse.resize(_res_arc_num);
   579 
   580       _lower.resize(_res_arc_num);
   581       _upper.resize(_res_arc_num);
   582       _cost.resize(_res_arc_num);
   583       _supply.resize(_node_num);
   584 
   585       _res_cap.resize(_res_arc_num);
   586       _pi.resize(_node_num);
   587       _excess.resize(_node_num);
   588       _pred.resize(_node_num);
   589 
   590       // Copy the graph
   591       int i = 0, j = 0, k = 2 * _arc_num + _node_num - 1;
   592       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   593         _node_id[n] = i;
   594       }
   595       i = 0;
   596       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   597         _first_out[i] = j;
   598         for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
   599           _arc_idf[a] = j;
   600           _forward[j] = true;
   601           _source[j] = i;
   602           _target[j] = _node_id[_graph.runningNode(a)];
   603         }
   604         for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
   605           _arc_idb[a] = j;
   606           _forward[j] = false;
   607           _source[j] = i;
   608           _target[j] = _node_id[_graph.runningNode(a)];
   609         }
   610         _forward[j] = false;
   611         _source[j] = i;
   612         _target[j] = _root;
   613         _reverse[j] = k;
   614         _forward[k] = true;
   615         _source[k] = _root;
   616         _target[k] = i;
   617         _reverse[k] = j;
   618         ++j; ++k;
   619       }
   620       _first_out[i] = j;
   621       _first_out[_node_num] = k;
   622       for (ArcIt a(_graph); a != INVALID; ++a) {
   623         int fi = _arc_idf[a];
   624         int bi = _arc_idb[a];
   625         _reverse[fi] = bi;
   626         _reverse[bi] = fi;
   627       }
   628 
   629       // Reset parameters
   630       resetParams();
   631       return *this;
   632     }
   633 
   634     /// @}
   635 
   636     /// \name Query Functions
   637     /// The results of the algorithm can be obtained using these
   638     /// functions.\n
   639     /// The \ref run() function must be called before using them.
   640 
   641     /// @{
   642 
   643     /// \brief Return the total cost of the found flow.
   644     ///
   645     /// This function returns the total cost of the found flow.
   646     /// Its complexity is O(e).
   647     ///
   648     /// \note The return type of the function can be specified as a
   649     /// template parameter. For example,
   650     /// \code
   651     ///   cs.totalCost<double>();
   652     /// \endcode
   653     /// It is useful if the total cost cannot be stored in the \c Cost
   654     /// type of the algorithm, which is the default return type of the
   655     /// function.
   656     ///
   657     /// \pre \ref run() must be called before using this function.
   658     template <typename Number>
   659     Number totalCost() const {
   660       Number c = 0;
   661       for (ArcIt a(_graph); a != INVALID; ++a) {
   662         int i = _arc_idb[a];
   663         c += static_cast<Number>(_res_cap[i]) *
   664              (-static_cast<Number>(_cost[i]));
   665       }
   666       return c;
   667     }
   668 
   669 #ifndef DOXYGEN
   670     Cost totalCost() const {
   671       return totalCost<Cost>();
   672     }
   673 #endif
   674 
   675     /// \brief Return the flow on the given arc.
   676     ///
   677     /// This function returns the flow on the given arc.
   678     ///
   679     /// \pre \ref run() must be called before using this function.
   680     Value flow(const Arc& a) const {
   681       return _res_cap[_arc_idb[a]];
   682     }
   683 
   684     /// \brief Copy the flow values (the primal solution) into the
   685     /// given map.
   686     ///
   687     /// This function copies the flow value on each arc into the given
   688     /// map. The \c Value type of the algorithm must be convertible to
   689     /// the \c Value type of the map.
   690     ///
   691     /// \pre \ref run() must be called before using this function.
   692     template <typename FlowMap>
   693     void flowMap(FlowMap &map) const {
   694       for (ArcIt a(_graph); a != INVALID; ++a) {
   695         map.set(a, _res_cap[_arc_idb[a]]);
   696       }
   697     }
   698 
   699     /// \brief Return the potential (dual value) of the given node.
   700     ///
   701     /// This function returns the potential (dual value) of the
   702     /// given node.
   703     ///
   704     /// \pre \ref run() must be called before using this function.
   705     Cost potential(const Node& n) const {
   706       return _pi[_node_id[n]];
   707     }
   708 
   709     /// \brief Copy the potential values (the dual solution) into the
   710     /// given map.
   711     ///
   712     /// This function copies the potential (dual value) of each node
   713     /// into the given map.
   714     /// The \c Cost type of the algorithm must be convertible to the
   715     /// \c Value type of the map.
   716     ///
   717     /// \pre \ref run() must be called before using this function.
   718     template <typename PotentialMap>
   719     void potentialMap(PotentialMap &map) const {
   720       for (NodeIt n(_graph); n != INVALID; ++n) {
   721         map.set(n, _pi[_node_id[n]]);
   722       }
   723     }
   724 
   725     /// @}
   726 
   727   private:
   728 
   729     // Initialize the algorithm
   730     ProblemType init() {
   731       if (_node_num <= 1) return INFEASIBLE;
   732 
   733       // Check the sum of supply values
   734       _sum_supply = 0;
   735       for (int i = 0; i != _root; ++i) {
   736         _sum_supply += _supply[i];
   737       }
   738       if (_sum_supply > 0) return INFEASIBLE;
   739 
   740       // Initialize vectors
   741       for (int i = 0; i != _root; ++i) {
   742         _pi[i] = 0;
   743         _excess[i] = _supply[i];
   744       }
   745 
   746       // Remove non-zero lower bounds
   747       const Value MAX = std::numeric_limits<Value>::max();
   748       int last_out;
   749       if (_have_lower) {
   750         for (int i = 0; i != _root; ++i) {
   751           last_out = _first_out[i+1];
   752           for (int j = _first_out[i]; j != last_out; ++j) {
   753             if (_forward[j]) {
   754               Value c = _lower[j];
   755               if (c >= 0) {
   756                 _res_cap[j] = _upper[j] < MAX ? _upper[j] - c : INF;
   757               } else {
   758                 _res_cap[j] = _upper[j] < MAX + c ? _upper[j] - c : INF;
   759               }
   760               _excess[i] -= c;
   761               _excess[_target[j]] += c;
   762             } else {
   763               _res_cap[j] = 0;
   764             }
   765           }
   766         }
   767       } else {
   768         for (int j = 0; j != _res_arc_num; ++j) {
   769           _res_cap[j] = _forward[j] ? _upper[j] : 0;
   770         }
   771       }
   772 
   773       // Handle negative costs
   774       for (int i = 0; i != _root; ++i) {
   775         last_out = _first_out[i+1] - 1;
   776         for (int j = _first_out[i]; j != last_out; ++j) {
   777           Value rc = _res_cap[j];
   778           if (_cost[j] < 0 && rc > 0) {
   779             if (rc >= MAX) return UNBOUNDED;
   780             _excess[i] -= rc;
   781             _excess[_target[j]] += rc;
   782             _res_cap[j] = 0;
   783             _res_cap[_reverse[j]] += rc;
   784           }
   785         }
   786       }
   787 
   788       // Handle GEQ supply type
   789       if (_sum_supply < 0) {
   790         _pi[_root] = 0;
   791         _excess[_root] = -_sum_supply;
   792         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
   793           int ra = _reverse[a];
   794           _res_cap[a] = -_sum_supply + 1;
   795           _res_cap[ra] = 0;
   796           _cost[a] = 0;
   797           _cost[ra] = 0;
   798         }
   799       } else {
   800         _pi[_root] = 0;
   801         _excess[_root] = 0;
   802         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
   803           int ra = _reverse[a];
   804           _res_cap[a] = 1;
   805           _res_cap[ra] = 0;
   806           _cost[a] = 0;
   807           _cost[ra] = 0;
   808         }
   809       }
   810 
   811       // Initialize delta value
   812       if (_factor > 1) {
   813         // With scaling
   814         Value max_sup = 0, max_dem = 0, max_cap = 0;
   815         for (int i = 0; i != _root; ++i) {
   816           Value ex = _excess[i];
   817           if ( ex > max_sup) max_sup =  ex;
   818           if (-ex > max_dem) max_dem = -ex;
   819           int last_out = _first_out[i+1] - 1;
   820           for (int j = _first_out[i]; j != last_out; ++j) {
   821             if (_res_cap[j] > max_cap) max_cap = _res_cap[j];
   822           }
   823         }
   824         max_sup = std::min(std::min(max_sup, max_dem), max_cap);
   825         for (_delta = 1; 2 * _delta <= max_sup; _delta *= 2) ;
   826       } else {
   827         // Without scaling
   828         _delta = 1;
   829       }
   830 
   831       return OPTIMAL;
   832     }
   833 
   834     ProblemType start() {
   835       // Execute the algorithm
   836       ProblemType pt;
   837       if (_delta > 1)
   838         pt = startWithScaling();
   839       else
   840         pt = startWithoutScaling();
   841 
   842       // Handle non-zero lower bounds
   843       if (_have_lower) {
   844         int limit = _first_out[_root];
   845         for (int j = 0; j != limit; ++j) {
   846           if (!_forward[j]) _res_cap[j] += _lower[j];
   847         }
   848       }
   849 
   850       // Shift potentials if necessary
   851       Cost pr = _pi[_root];
   852       if (_sum_supply < 0 || pr > 0) {
   853         for (int i = 0; i != _node_num; ++i) {
   854           _pi[i] -= pr;
   855         }
   856       }
   857 
   858       return pt;
   859     }
   860 
   861     // Execute the capacity scaling algorithm
   862     ProblemType startWithScaling() {
   863       // Perform capacity scaling phases
   864       int s, t;
   865       ResidualDijkstra _dijkstra(*this);
   866       while (true) {
   867         // Saturate all arcs not satisfying the optimality condition
   868         int last_out;
   869         for (int u = 0; u != _node_num; ++u) {
   870           last_out = _sum_supply < 0 ?
   871             _first_out[u+1] : _first_out[u+1] - 1;
   872           for (int a = _first_out[u]; a != last_out; ++a) {
   873             int v = _target[a];
   874             Cost c = _cost[a] + _pi[u] - _pi[v];
   875             Value rc = _res_cap[a];
   876             if (c < 0 && rc >= _delta) {
   877               _excess[u] -= rc;
   878               _excess[v] += rc;
   879               _res_cap[a] = 0;
   880               _res_cap[_reverse[a]] += rc;
   881             }
   882           }
   883         }
   884 
   885         // Find excess nodes and deficit nodes
   886         _excess_nodes.clear();
   887         _deficit_nodes.clear();
   888         for (int u = 0; u != _node_num; ++u) {
   889           Value ex = _excess[u];
   890           if (ex >=  _delta) _excess_nodes.push_back(u);
   891           if (ex <= -_delta) _deficit_nodes.push_back(u);
   892         }
   893         int next_node = 0, next_def_node = 0;
   894 
   895         // Find augmenting shortest paths
   896         while (next_node < int(_excess_nodes.size())) {
   897           // Check deficit nodes
   898           if (_delta > 1) {
   899             bool delta_deficit = false;
   900             for ( ; next_def_node < int(_deficit_nodes.size());
   901                     ++next_def_node ) {
   902               if (_excess[_deficit_nodes[next_def_node]] <= -_delta) {
   903                 delta_deficit = true;
   904                 break;
   905               }
   906             }
   907             if (!delta_deficit) break;
   908           }
   909 
   910           // Run Dijkstra in the residual network
   911           s = _excess_nodes[next_node];
   912           if ((t = _dijkstra.run(s, _delta)) == -1) {
   913             if (_delta > 1) {
   914               ++next_node;
   915               continue;
   916             }
   917             return INFEASIBLE;
   918           }
   919 
   920           // Augment along a shortest path from s to t
   921           Value d = std::min(_excess[s], -_excess[t]);
   922           int u = t;
   923           int a;
   924           if (d > _delta) {
   925             while ((a = _pred[u]) != -1) {
   926               if (_res_cap[a] < d) d = _res_cap[a];
   927               u = _source[a];
   928             }
   929           }
   930           u = t;
   931           while ((a = _pred[u]) != -1) {
   932             _res_cap[a] -= d;
   933             _res_cap[_reverse[a]] += d;
   934             u = _source[a];
   935           }
   936           _excess[s] -= d;
   937           _excess[t] += d;
   938 
   939           if (_excess[s] < _delta) ++next_node;
   940         }
   941 
   942         if (_delta == 1) break;
   943         _delta = _delta <= _factor ? 1 : _delta / _factor;
   944       }
   945 
   946       return OPTIMAL;
   947     }
   948 
   949     // Execute the successive shortest path algorithm
   950     ProblemType startWithoutScaling() {
   951       // Find excess nodes
   952       _excess_nodes.clear();
   953       for (int i = 0; i != _node_num; ++i) {
   954         if (_excess[i] > 0) _excess_nodes.push_back(i);
   955       }
   956       if (_excess_nodes.size() == 0) return OPTIMAL;
   957       int next_node = 0;
   958 
   959       // Find shortest paths
   960       int s, t;
   961       ResidualDijkstra _dijkstra(*this);
   962       while ( _excess[_excess_nodes[next_node]] > 0 ||
   963               ++next_node < int(_excess_nodes.size()) )
   964       {
   965         // Run Dijkstra in the residual network
   966         s = _excess_nodes[next_node];
   967         if ((t = _dijkstra.run(s)) == -1) return INFEASIBLE;
   968 
   969         // Augment along a shortest path from s to t
   970         Value d = std::min(_excess[s], -_excess[t]);
   971         int u = t;
   972         int a;
   973         if (d > 1) {
   974           while ((a = _pred[u]) != -1) {
   975             if (_res_cap[a] < d) d = _res_cap[a];
   976             u = _source[a];
   977           }
   978         }
   979         u = t;
   980         while ((a = _pred[u]) != -1) {
   981           _res_cap[a] -= d;
   982           _res_cap[_reverse[a]] += d;
   983           u = _source[a];
   984         }
   985         _excess[s] -= d;
   986         _excess[t] += d;
   987       }
   988 
   989       return OPTIMAL;
   990     }
   991 
   992   }; //class CapacityScaling
   993 
   994   ///@}
   995 
   996 } //namespace lemon
   997 
   998 #endif //LEMON_CAPACITY_SCALING_H