lemon/capacity_scaling.h
author Alpar Juttner <alpar@cs.elte.hu>
Mon, 01 Mar 2010 07:51:45 +0100
changeset 850 e77b621e6e7e
parent 831 cc9e0c15d747
parent 839 f3bc4e9b5f3a
child 863 a93f1a27d831
permissions -rw-r--r--
Configurable glpk prefix in ./scripts/bootstrap.sh and ...
unneeded solver backends are explicitely switched off with --without-*
     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_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   /// Most of the parameters of the problem (except for the digraph)
    74   /// can be given using separate functions, and the algorithm can be
    75   /// executed using the \ref run() function. If some parameters are not
    76   /// specified, then default values will be used.
    77   ///
    78   /// \tparam GR The digraph type the algorithm runs on.
    79   /// \tparam V The number type used for flow amounts, capacity bounds
    80   /// and supply values in the algorithm. By default, it is \c int.
    81   /// \tparam C The number type used for costs and potentials in the
    82   /// algorithm. By default, it is the same as \c V.
    83   /// \tparam TR The traits class that defines various types used by the
    84   /// algorithm. By default, it is \ref CapacityScalingDefaultTraits
    85   /// "CapacityScalingDefaultTraits<GR, V, C>".
    86   /// In most cases, this parameter should not be set directly,
    87   /// consider to use the named template parameters instead.
    88   ///
    89   /// \warning Both number types must be signed and all input data must
    90   /// be integer.
    91   /// \warning This algorithm does not support negative costs for such
    92   /// arcs that have infinite upper bound.
    93 #ifdef DOXYGEN
    94   template <typename GR, typename V, typename C, typename TR>
    95 #else
    96   template < typename GR, typename V = int, typename C = V,
    97              typename TR = CapacityScalingDefaultTraits<GR, V, C> >
    98 #endif
    99   class CapacityScaling
   100   {
   101   public:
   102 
   103     /// The type of the digraph
   104     typedef typename TR::Digraph Digraph;
   105     /// The type of the flow amounts, capacity bounds and supply values
   106     typedef typename TR::Value Value;
   107     /// The type of the arc costs
   108     typedef typename TR::Cost Cost;
   109 
   110     /// The type of the heap used for internal Dijkstra computations
   111     typedef typename TR::Heap Heap;
   112 
   113     /// The \ref CapacityScalingDefaultTraits "traits class" of the algorithm
   114     typedef TR Traits;
   115 
   116   public:
   117 
   118     /// \brief Problem type constants for the \c run() function.
   119     ///
   120     /// Enum type containing the problem type constants that can be
   121     /// returned by the \ref run() function of the algorithm.
   122     enum ProblemType {
   123       /// The problem has no feasible solution (flow).
   124       INFEASIBLE,
   125       /// The problem has optimal solution (i.e. it is feasible and
   126       /// bounded), and the algorithm has found optimal flow and node
   127       /// potentials (primal and dual solutions).
   128       OPTIMAL,
   129       /// The digraph contains an arc of negative cost and infinite
   130       /// upper bound. It means that the objective function is unbounded
   131       /// on that arc, however, note that it could actually be bounded
   132       /// over the feasible flows, but this algroithm cannot handle
   133       /// these cases.
   134       UNBOUNDED
   135     };
   136   
   137   private:
   138 
   139     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
   140 
   141     typedef std::vector<int> IntVector;
   142     typedef std::vector<Value> ValueVector;
   143     typedef std::vector<Cost> CostVector;
   144     typedef std::vector<char> BoolVector;
   145     // Note: vector<char> is used instead of vector<bool> for efficiency reasons
   146 
   147   private:
   148 
   149     // Data related to the underlying digraph
   150     const GR &_graph;
   151     int _node_num;
   152     int _arc_num;
   153     int _res_arc_num;
   154     int _root;
   155 
   156     // Parameters of the problem
   157     bool _have_lower;
   158     Value _sum_supply;
   159 
   160     // Data structures for storing the digraph
   161     IntNodeMap _node_id;
   162     IntArcMap _arc_idf;
   163     IntArcMap _arc_idb;
   164     IntVector _first_out;
   165     BoolVector _forward;
   166     IntVector _source;
   167     IntVector _target;
   168     IntVector _reverse;
   169 
   170     // Node and arc data
   171     ValueVector _lower;
   172     ValueVector _upper;
   173     CostVector _cost;
   174     ValueVector _supply;
   175 
   176     ValueVector _res_cap;
   177     CostVector _pi;
   178     ValueVector _excess;
   179     IntVector _excess_nodes;
   180     IntVector _deficit_nodes;
   181 
   182     Value _delta;
   183     int _factor;
   184     IntVector _pred;
   185 
   186   public:
   187   
   188     /// \brief Constant for infinite upper bounds (capacities).
   189     ///
   190     /// Constant for infinite upper bounds (capacities).
   191     /// It is \c std::numeric_limits<Value>::infinity() if available,
   192     /// \c std::numeric_limits<Value>::max() otherwise.
   193     const Value INF;
   194 
   195   private:
   196 
   197     // Special implementation of the Dijkstra algorithm for finding
   198     // shortest paths in the residual network of the digraph with
   199     // respect to the reduced arc costs and modifying the node
   200     // potentials according to the found distance labels.
   201     class ResidualDijkstra
   202     {
   203     private:
   204 
   205       int _node_num;
   206       bool _geq;
   207       const IntVector &_first_out;
   208       const IntVector &_target;
   209       const CostVector &_cost;
   210       const ValueVector &_res_cap;
   211       const ValueVector &_excess;
   212       CostVector &_pi;
   213       IntVector &_pred;
   214       
   215       IntVector _proc_nodes;
   216       CostVector _dist;
   217       
   218     public:
   219 
   220       ResidualDijkstra(CapacityScaling& cs) :
   221         _node_num(cs._node_num), _geq(cs._sum_supply < 0),
   222         _first_out(cs._first_out), _target(cs._target), _cost(cs._cost),
   223         _res_cap(cs._res_cap), _excess(cs._excess), _pi(cs._pi),
   224         _pred(cs._pred), _dist(cs._node_num)
   225       {}
   226 
   227       int run(int s, Value delta = 1) {
   228         RangeMap<int> heap_cross_ref(_node_num, Heap::PRE_HEAP);
   229         Heap heap(heap_cross_ref);
   230         heap.push(s, 0);
   231         _pred[s] = -1;
   232         _proc_nodes.clear();
   233 
   234         // Process nodes
   235         while (!heap.empty() && _excess[heap.top()] > -delta) {
   236           int u = heap.top(), v;
   237           Cost d = heap.prio() + _pi[u], dn;
   238           _dist[u] = heap.prio();
   239           _proc_nodes.push_back(u);
   240           heap.pop();
   241 
   242           // Traverse outgoing residual arcs
   243           int last_out = _geq ? _first_out[u+1] : _first_out[u+1] - 1;
   244           for (int a = _first_out[u]; a != last_out; ++a) {
   245             if (_res_cap[a] < delta) continue;
   246             v = _target[a];
   247             switch (heap.state(v)) {
   248               case Heap::PRE_HEAP:
   249                 heap.push(v, d + _cost[a] - _pi[v]);
   250                 _pred[v] = a;
   251                 break;
   252               case Heap::IN_HEAP:
   253                 dn = d + _cost[a] - _pi[v];
   254                 if (dn < heap[v]) {
   255                   heap.decrease(v, dn);
   256                   _pred[v] = a;
   257                 }
   258                 break;
   259               case Heap::POST_HEAP:
   260                 break;
   261             }
   262           }
   263         }
   264         if (heap.empty()) return -1;
   265 
   266         // Update potentials of processed nodes
   267         int t = heap.top();
   268         Cost dt = heap.prio();
   269         for (int i = 0; i < int(_proc_nodes.size()); ++i) {
   270           _pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - dt;
   271         }
   272 
   273         return t;
   274       }
   275 
   276     }; //class ResidualDijkstra
   277 
   278   public:
   279 
   280     /// \name Named Template Parameters
   281     /// @{
   282 
   283     template <typename T>
   284     struct SetHeapTraits : public Traits {
   285       typedef T Heap;
   286     };
   287 
   288     /// \brief \ref named-templ-param "Named parameter" for setting
   289     /// \c Heap type.
   290     ///
   291     /// \ref named-templ-param "Named parameter" for setting \c Heap
   292     /// type, which is used for internal Dijkstra computations.
   293     /// It must conform to the \ref lemon::concepts::Heap "Heap" concept,
   294     /// its priority type must be \c Cost and its cross reference type
   295     /// must be \ref RangeMap "RangeMap<int>".
   296     template <typename T>
   297     struct SetHeap
   298       : public CapacityScaling<GR, V, C, SetHeapTraits<T> > {
   299       typedef  CapacityScaling<GR, V, C, SetHeapTraits<T> > Create;
   300     };
   301 
   302     /// @}
   303 
   304   public:
   305 
   306     /// \brief Constructor.
   307     ///
   308     /// The constructor of the class.
   309     ///
   310     /// \param graph The digraph the algorithm runs on.
   311     CapacityScaling(const GR& graph) :
   312       _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
   313       INF(std::numeric_limits<Value>::has_infinity ?
   314           std::numeric_limits<Value>::infinity() :
   315           std::numeric_limits<Value>::max())
   316     {
   317       // Check the number types
   318       LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
   319         "The flow type of CapacityScaling must be signed");
   320       LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
   321         "The cost type of CapacityScaling must be signed");
   322 
   323       // Reset data structures
   324       reset();
   325     }
   326 
   327     /// \name Parameters
   328     /// The parameters of the algorithm can be specified using these
   329     /// functions.
   330 
   331     /// @{
   332 
   333     /// \brief Set the lower bounds on the arcs.
   334     ///
   335     /// This function sets the lower bounds on the arcs.
   336     /// If it is not used before calling \ref run(), the lower bounds
   337     /// will be set to zero on all arcs.
   338     ///
   339     /// \param map An arc map storing the lower bounds.
   340     /// Its \c Value type must be convertible to the \c Value type
   341     /// of the algorithm.
   342     ///
   343     /// \return <tt>(*this)</tt>
   344     template <typename LowerMap>
   345     CapacityScaling& lowerMap(const LowerMap& map) {
   346       _have_lower = true;
   347       for (ArcIt a(_graph); a != INVALID; ++a) {
   348         _lower[_arc_idf[a]] = map[a];
   349         _lower[_arc_idb[a]] = map[a];
   350       }
   351       return *this;
   352     }
   353 
   354     /// \brief Set the upper bounds (capacities) on the arcs.
   355     ///
   356     /// This function sets the upper bounds (capacities) on the arcs.
   357     /// If it is not used before calling \ref run(), the upper bounds
   358     /// will be set to \ref INF on all arcs (i.e. the flow value will be
   359     /// unbounded from above).
   360     ///
   361     /// \param map An arc map storing the upper bounds.
   362     /// Its \c Value type must be convertible to the \c Value type
   363     /// of the algorithm.
   364     ///
   365     /// \return <tt>(*this)</tt>
   366     template<typename UpperMap>
   367     CapacityScaling& upperMap(const UpperMap& map) {
   368       for (ArcIt a(_graph); a != INVALID; ++a) {
   369         _upper[_arc_idf[a]] = map[a];
   370       }
   371       return *this;
   372     }
   373 
   374     /// \brief Set the costs of the arcs.
   375     ///
   376     /// This function sets the costs of the arcs.
   377     /// If it is not used before calling \ref run(), the costs
   378     /// will be set to \c 1 on all arcs.
   379     ///
   380     /// \param map An arc map storing the costs.
   381     /// Its \c Value type must be convertible to the \c Cost type
   382     /// of the algorithm.
   383     ///
   384     /// \return <tt>(*this)</tt>
   385     template<typename CostMap>
   386     CapacityScaling& costMap(const CostMap& map) {
   387       for (ArcIt a(_graph); a != INVALID; ++a) {
   388         _cost[_arc_idf[a]] =  map[a];
   389         _cost[_arc_idb[a]] = -map[a];
   390       }
   391       return *this;
   392     }
   393 
   394     /// \brief Set the supply values of the nodes.
   395     ///
   396     /// This function sets the supply values of the nodes.
   397     /// If neither this function nor \ref stSupply() is used before
   398     /// calling \ref run(), the supply of each node will be set to zero.
   399     ///
   400     /// \param map A node map storing the supply values.
   401     /// Its \c Value type must be convertible to the \c Value type
   402     /// of the algorithm.
   403     ///
   404     /// \return <tt>(*this)</tt>
   405     template<typename SupplyMap>
   406     CapacityScaling& supplyMap(const SupplyMap& map) {
   407       for (NodeIt n(_graph); n != INVALID; ++n) {
   408         _supply[_node_id[n]] = map[n];
   409       }
   410       return *this;
   411     }
   412 
   413     /// \brief Set single source and target nodes and a supply value.
   414     ///
   415     /// This function sets a single source node and a single target node
   416     /// and the required flow value.
   417     /// If neither this function nor \ref supplyMap() is used before
   418     /// calling \ref run(), the supply of each node will be set to zero.
   419     ///
   420     /// Using this function has the same effect as using \ref supplyMap()
   421     /// with such a map in which \c k is assigned to \c s, \c -k is
   422     /// assigned to \c t and all other nodes have zero supply value.
   423     ///
   424     /// \param s The source node.
   425     /// \param t The target node.
   426     /// \param k The required amount of flow from node \c s to node \c t
   427     /// (i.e. the supply of \c s and the demand of \c t).
   428     ///
   429     /// \return <tt>(*this)</tt>
   430     CapacityScaling& stSupply(const Node& s, const Node& t, Value k) {
   431       for (int i = 0; i != _node_num; ++i) {
   432         _supply[i] = 0;
   433       }
   434       _supply[_node_id[s]] =  k;
   435       _supply[_node_id[t]] = -k;
   436       return *this;
   437     }
   438     
   439     /// @}
   440 
   441     /// \name Execution control
   442     /// The algorithm can be executed using \ref run().
   443 
   444     /// @{
   445 
   446     /// \brief Run the algorithm.
   447     ///
   448     /// This function runs the algorithm.
   449     /// The paramters can be specified using functions \ref lowerMap(),
   450     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
   451     /// For example,
   452     /// \code
   453     ///   CapacityScaling<ListDigraph> cs(graph);
   454     ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
   455     ///     .supplyMap(sup).run();
   456     /// \endcode
   457     ///
   458     /// This function can be called more than once. All the given parameters
   459     /// are kept for the next call, unless \ref resetParams() or \ref reset()
   460     /// is used, thus only the modified parameters have to be set again.
   461     /// If the underlying digraph was also modified after the construction
   462     /// of the class (or the last \ref reset() call), then the \ref reset()
   463     /// function must be called.
   464     ///
   465     /// \param factor The capacity scaling factor. It must be larger than
   466     /// one to use scaling. If it is less or equal to one, then scaling
   467     /// will be disabled.
   468     ///
   469     /// \return \c INFEASIBLE if no feasible flow exists,
   470     /// \n \c OPTIMAL if the problem has optimal solution
   471     /// (i.e. it is feasible and bounded), and the algorithm has found
   472     /// optimal flow and node potentials (primal and dual solutions),
   473     /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
   474     /// and infinite upper bound. It means that the objective function
   475     /// is unbounded on that arc, however, note that it could actually be
   476     /// bounded over the feasible flows, but this algroithm cannot handle
   477     /// these cases.
   478     ///
   479     /// \see ProblemType
   480     /// \see resetParams(), reset()
   481     ProblemType run(int factor = 4) {
   482       _factor = factor;
   483       ProblemType pt = init();
   484       if (pt != OPTIMAL) return pt;
   485       return start();
   486     }
   487 
   488     /// \brief Reset all the parameters that have been given before.
   489     ///
   490     /// This function resets all the paramaters that have been given
   491     /// before using functions \ref lowerMap(), \ref upperMap(),
   492     /// \ref costMap(), \ref supplyMap(), \ref stSupply().
   493     ///
   494     /// It is useful for multiple \ref run() calls. Basically, all the given
   495     /// parameters are kept for the next \ref run() call, unless
   496     /// \ref resetParams() or \ref reset() is used.
   497     /// If the underlying digraph was also modified after the construction
   498     /// of the class or the last \ref reset() call, then the \ref reset()
   499     /// function must be used, otherwise \ref resetParams() is sufficient.
   500     ///
   501     /// For example,
   502     /// \code
   503     ///   CapacityScaling<ListDigraph> cs(graph);
   504     ///
   505     ///   // First run
   506     ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
   507     ///     .supplyMap(sup).run();
   508     ///
   509     ///   // Run again with modified cost map (resetParams() is not called,
   510     ///   // so only the cost map have to be set again)
   511     ///   cost[e] += 100;
   512     ///   cs.costMap(cost).run();
   513     ///
   514     ///   // Run again from scratch using resetParams()
   515     ///   // (the lower bounds will be set to zero on all arcs)
   516     ///   cs.resetParams();
   517     ///   cs.upperMap(capacity).costMap(cost)
   518     ///     .supplyMap(sup).run();
   519     /// \endcode
   520     ///
   521     /// \return <tt>(*this)</tt>
   522     ///
   523     /// \see reset(), run()
   524     CapacityScaling& resetParams() {
   525       for (int i = 0; i != _node_num; ++i) {
   526         _supply[i] = 0;
   527       }
   528       for (int j = 0; j != _res_arc_num; ++j) {
   529         _lower[j] = 0;
   530         _upper[j] = INF;
   531         _cost[j] = _forward[j] ? 1 : -1;
   532       }
   533       _have_lower = false;
   534       return *this;
   535     }
   536 
   537     /// \brief Reset the internal data structures and all the parameters
   538     /// that have been given before.
   539     ///
   540     /// This function resets the internal data structures and all the
   541     /// paramaters that have been given before using functions \ref lowerMap(),
   542     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
   543     ///
   544     /// It is useful for multiple \ref run() calls. Basically, all the given
   545     /// parameters are kept for the next \ref run() call, unless
   546     /// \ref resetParams() or \ref reset() is used.
   547     /// If the underlying digraph was also modified after the construction
   548     /// of the class or the last \ref reset() call, then the \ref reset()
   549     /// function must be used, otherwise \ref resetParams() is sufficient.
   550     ///
   551     /// See \ref resetParams() for examples.
   552     ///
   553     /// \return <tt>(*this)</tt>
   554     ///
   555     /// \see resetParams(), run()
   556     CapacityScaling& reset() {
   557       // Resize vectors
   558       _node_num = countNodes(_graph);
   559       _arc_num = countArcs(_graph);
   560       _res_arc_num = 2 * (_arc_num + _node_num);
   561       _root = _node_num;
   562       ++_node_num;
   563 
   564       _first_out.resize(_node_num + 1);
   565       _forward.resize(_res_arc_num);
   566       _source.resize(_res_arc_num);
   567       _target.resize(_res_arc_num);
   568       _reverse.resize(_res_arc_num);
   569 
   570       _lower.resize(_res_arc_num);
   571       _upper.resize(_res_arc_num);
   572       _cost.resize(_res_arc_num);
   573       _supply.resize(_node_num);
   574       
   575       _res_cap.resize(_res_arc_num);
   576       _pi.resize(_node_num);
   577       _excess.resize(_node_num);
   578       _pred.resize(_node_num);
   579 
   580       // Copy the graph
   581       int i = 0, j = 0, k = 2 * _arc_num + _node_num - 1;
   582       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   583         _node_id[n] = i;
   584       }
   585       i = 0;
   586       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
   587         _first_out[i] = j;
   588         for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
   589           _arc_idf[a] = j;
   590           _forward[j] = true;
   591           _source[j] = i;
   592           _target[j] = _node_id[_graph.runningNode(a)];
   593         }
   594         for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
   595           _arc_idb[a] = j;
   596           _forward[j] = false;
   597           _source[j] = i;
   598           _target[j] = _node_id[_graph.runningNode(a)];
   599         }
   600         _forward[j] = false;
   601         _source[j] = i;
   602         _target[j] = _root;
   603         _reverse[j] = k;
   604         _forward[k] = true;
   605         _source[k] = _root;
   606         _target[k] = i;
   607         _reverse[k] = j;
   608         ++j; ++k;
   609       }
   610       _first_out[i] = j;
   611       _first_out[_node_num] = k;
   612       for (ArcIt a(_graph); a != INVALID; ++a) {
   613         int fi = _arc_idf[a];
   614         int bi = _arc_idb[a];
   615         _reverse[fi] = bi;
   616         _reverse[bi] = fi;
   617       }
   618       
   619       // Reset parameters
   620       resetParams();
   621       return *this;
   622     }
   623 
   624     /// @}
   625 
   626     /// \name Query Functions
   627     /// The results of the algorithm can be obtained using these
   628     /// functions.\n
   629     /// The \ref run() function must be called before using them.
   630 
   631     /// @{
   632 
   633     /// \brief Return the total cost of the found flow.
   634     ///
   635     /// This function returns the total cost of the found flow.
   636     /// Its complexity is O(e).
   637     ///
   638     /// \note The return type of the function can be specified as a
   639     /// template parameter. For example,
   640     /// \code
   641     ///   cs.totalCost<double>();
   642     /// \endcode
   643     /// It is useful if the total cost cannot be stored in the \c Cost
   644     /// type of the algorithm, which is the default return type of the
   645     /// function.
   646     ///
   647     /// \pre \ref run() must be called before using this function.
   648     template <typename Number>
   649     Number totalCost() const {
   650       Number c = 0;
   651       for (ArcIt a(_graph); a != INVALID; ++a) {
   652         int i = _arc_idb[a];
   653         c += static_cast<Number>(_res_cap[i]) *
   654              (-static_cast<Number>(_cost[i]));
   655       }
   656       return c;
   657     }
   658 
   659 #ifndef DOXYGEN
   660     Cost totalCost() const {
   661       return totalCost<Cost>();
   662     }
   663 #endif
   664 
   665     /// \brief Return the flow on the given arc.
   666     ///
   667     /// This function returns the flow on the given arc.
   668     ///
   669     /// \pre \ref run() must be called before using this function.
   670     Value flow(const Arc& a) const {
   671       return _res_cap[_arc_idb[a]];
   672     }
   673 
   674     /// \brief Return the flow map (the primal solution).
   675     ///
   676     /// This function copies the flow value on each arc into the given
   677     /// map. The \c Value type of the algorithm must be convertible to
   678     /// the \c Value type of the map.
   679     ///
   680     /// \pre \ref run() must be called before using this function.
   681     template <typename FlowMap>
   682     void flowMap(FlowMap &map) const {
   683       for (ArcIt a(_graph); a != INVALID; ++a) {
   684         map.set(a, _res_cap[_arc_idb[a]]);
   685       }
   686     }
   687 
   688     /// \brief Return the potential (dual value) of the given node.
   689     ///
   690     /// This function returns the potential (dual value) of the
   691     /// given node.
   692     ///
   693     /// \pre \ref run() must be called before using this function.
   694     Cost potential(const Node& n) const {
   695       return _pi[_node_id[n]];
   696     }
   697 
   698     /// \brief Return the potential map (the dual solution).
   699     ///
   700     /// This function copies the potential (dual value) of each node
   701     /// into the given map.
   702     /// The \c Cost type of the algorithm must be convertible to the
   703     /// \c Value type of the map.
   704     ///
   705     /// \pre \ref run() must be called before using this function.
   706     template <typename PotentialMap>
   707     void potentialMap(PotentialMap &map) const {
   708       for (NodeIt n(_graph); n != INVALID; ++n) {
   709         map.set(n, _pi[_node_id[n]]);
   710       }
   711     }
   712 
   713     /// @}
   714 
   715   private:
   716 
   717     // Initialize the algorithm
   718     ProblemType init() {
   719       if (_node_num <= 1) return INFEASIBLE;
   720 
   721       // Check the sum of supply values
   722       _sum_supply = 0;
   723       for (int i = 0; i != _root; ++i) {
   724         _sum_supply += _supply[i];
   725       }
   726       if (_sum_supply > 0) return INFEASIBLE;
   727       
   728       // Initialize vectors
   729       for (int i = 0; i != _root; ++i) {
   730         _pi[i] = 0;
   731         _excess[i] = _supply[i];
   732       }
   733 
   734       // Remove non-zero lower bounds
   735       const Value MAX = std::numeric_limits<Value>::max();
   736       int last_out;
   737       if (_have_lower) {
   738         for (int i = 0; i != _root; ++i) {
   739           last_out = _first_out[i+1];
   740           for (int j = _first_out[i]; j != last_out; ++j) {
   741             if (_forward[j]) {
   742               Value c = _lower[j];
   743               if (c >= 0) {
   744                 _res_cap[j] = _upper[j] < MAX ? _upper[j] - c : INF;
   745               } else {
   746                 _res_cap[j] = _upper[j] < MAX + c ? _upper[j] - c : INF;
   747               }
   748               _excess[i] -= c;
   749               _excess[_target[j]] += c;
   750             } else {
   751               _res_cap[j] = 0;
   752             }
   753           }
   754         }
   755       } else {
   756         for (int j = 0; j != _res_arc_num; ++j) {
   757           _res_cap[j] = _forward[j] ? _upper[j] : 0;
   758         }
   759       }
   760 
   761       // Handle negative costs
   762       for (int i = 0; i != _root; ++i) {
   763         last_out = _first_out[i+1] - 1;
   764         for (int j = _first_out[i]; j != last_out; ++j) {
   765           Value rc = _res_cap[j];
   766           if (_cost[j] < 0 && rc > 0) {
   767             if (rc >= MAX) return UNBOUNDED;
   768             _excess[i] -= rc;
   769             _excess[_target[j]] += rc;
   770             _res_cap[j] = 0;
   771             _res_cap[_reverse[j]] += rc;
   772           }
   773         }
   774       }
   775       
   776       // Handle GEQ supply type
   777       if (_sum_supply < 0) {
   778         _pi[_root] = 0;
   779         _excess[_root] = -_sum_supply;
   780         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
   781           int ra = _reverse[a];
   782           _res_cap[a] = -_sum_supply + 1;
   783           _res_cap[ra] = 0;
   784           _cost[a] = 0;
   785           _cost[ra] = 0;
   786         }
   787       } else {
   788         _pi[_root] = 0;
   789         _excess[_root] = 0;
   790         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
   791           int ra = _reverse[a];
   792           _res_cap[a] = 1;
   793           _res_cap[ra] = 0;
   794           _cost[a] = 0;
   795           _cost[ra] = 0;
   796         }
   797       }
   798 
   799       // Initialize delta value
   800       if (_factor > 1) {
   801         // With scaling
   802         Value max_sup = 0, max_dem = 0, max_cap = 0;
   803         for (int i = 0; i != _root; ++i) {
   804           Value ex = _excess[i];
   805           if ( ex > max_sup) max_sup =  ex;
   806           if (-ex > max_dem) max_dem = -ex;
   807           int last_out = _first_out[i+1] - 1;
   808           for (int j = _first_out[i]; j != last_out; ++j) {
   809             if (_res_cap[j] > max_cap) max_cap = _res_cap[j];
   810           }
   811         }
   812         max_sup = std::min(std::min(max_sup, max_dem), max_cap);
   813         for (_delta = 1; 2 * _delta <= max_sup; _delta *= 2) ;
   814       } else {
   815         // Without scaling
   816         _delta = 1;
   817       }
   818 
   819       return OPTIMAL;
   820     }
   821 
   822     ProblemType start() {
   823       // Execute the algorithm
   824       ProblemType pt;
   825       if (_delta > 1)
   826         pt = startWithScaling();
   827       else
   828         pt = startWithoutScaling();
   829 
   830       // Handle non-zero lower bounds
   831       if (_have_lower) {
   832         int limit = _first_out[_root];
   833         for (int j = 0; j != limit; ++j) {
   834           if (!_forward[j]) _res_cap[j] += _lower[j];
   835         }
   836       }
   837 
   838       // Shift potentials if necessary
   839       Cost pr = _pi[_root];
   840       if (_sum_supply < 0 || pr > 0) {
   841         for (int i = 0; i != _node_num; ++i) {
   842           _pi[i] -= pr;
   843         }        
   844       }
   845       
   846       return pt;
   847     }
   848 
   849     // Execute the capacity scaling algorithm
   850     ProblemType startWithScaling() {
   851       // Perform capacity scaling phases
   852       int s, t;
   853       ResidualDijkstra _dijkstra(*this);
   854       while (true) {
   855         // Saturate all arcs not satisfying the optimality condition
   856         int last_out;
   857         for (int u = 0; u != _node_num; ++u) {
   858           last_out = _sum_supply < 0 ?
   859             _first_out[u+1] : _first_out[u+1] - 1;
   860           for (int a = _first_out[u]; a != last_out; ++a) {
   861             int v = _target[a];
   862             Cost c = _cost[a] + _pi[u] - _pi[v];
   863             Value rc = _res_cap[a];
   864             if (c < 0 && rc >= _delta) {
   865               _excess[u] -= rc;
   866               _excess[v] += rc;
   867               _res_cap[a] = 0;
   868               _res_cap[_reverse[a]] += rc;
   869             }
   870           }
   871         }
   872 
   873         // Find excess nodes and deficit nodes
   874         _excess_nodes.clear();
   875         _deficit_nodes.clear();
   876         for (int u = 0; u != _node_num; ++u) {
   877           Value ex = _excess[u];
   878           if (ex >=  _delta) _excess_nodes.push_back(u);
   879           if (ex <= -_delta) _deficit_nodes.push_back(u);
   880         }
   881         int next_node = 0, next_def_node = 0;
   882 
   883         // Find augmenting shortest paths
   884         while (next_node < int(_excess_nodes.size())) {
   885           // Check deficit nodes
   886           if (_delta > 1) {
   887             bool delta_deficit = false;
   888             for ( ; next_def_node < int(_deficit_nodes.size());
   889                     ++next_def_node ) {
   890               if (_excess[_deficit_nodes[next_def_node]] <= -_delta) {
   891                 delta_deficit = true;
   892                 break;
   893               }
   894             }
   895             if (!delta_deficit) break;
   896           }
   897 
   898           // Run Dijkstra in the residual network
   899           s = _excess_nodes[next_node];
   900           if ((t = _dijkstra.run(s, _delta)) == -1) {
   901             if (_delta > 1) {
   902               ++next_node;
   903               continue;
   904             }
   905             return INFEASIBLE;
   906           }
   907 
   908           // Augment along a shortest path from s to t
   909           Value d = std::min(_excess[s], -_excess[t]);
   910           int u = t;
   911           int a;
   912           if (d > _delta) {
   913             while ((a = _pred[u]) != -1) {
   914               if (_res_cap[a] < d) d = _res_cap[a];
   915               u = _source[a];
   916             }
   917           }
   918           u = t;
   919           while ((a = _pred[u]) != -1) {
   920             _res_cap[a] -= d;
   921             _res_cap[_reverse[a]] += d;
   922             u = _source[a];
   923           }
   924           _excess[s] -= d;
   925           _excess[t] += d;
   926 
   927           if (_excess[s] < _delta) ++next_node;
   928         }
   929 
   930         if (_delta == 1) break;
   931         _delta = _delta <= _factor ? 1 : _delta / _factor;
   932       }
   933 
   934       return OPTIMAL;
   935     }
   936 
   937     // Execute the successive shortest path algorithm
   938     ProblemType startWithoutScaling() {
   939       // Find excess nodes
   940       _excess_nodes.clear();
   941       for (int i = 0; i != _node_num; ++i) {
   942         if (_excess[i] > 0) _excess_nodes.push_back(i);
   943       }
   944       if (_excess_nodes.size() == 0) return OPTIMAL;
   945       int next_node = 0;
   946 
   947       // Find shortest paths
   948       int s, t;
   949       ResidualDijkstra _dijkstra(*this);
   950       while ( _excess[_excess_nodes[next_node]] > 0 ||
   951               ++next_node < int(_excess_nodes.size()) )
   952       {
   953         // Run Dijkstra in the residual network
   954         s = _excess_nodes[next_node];
   955         if ((t = _dijkstra.run(s)) == -1) return INFEASIBLE;
   956 
   957         // Augment along a shortest path from s to t
   958         Value d = std::min(_excess[s], -_excess[t]);
   959         int u = t;
   960         int a;
   961         if (d > 1) {
   962           while ((a = _pred[u]) != -1) {
   963             if (_res_cap[a] < d) d = _res_cap[a];
   964             u = _source[a];
   965           }
   966         }
   967         u = t;
   968         while ((a = _pred[u]) != -1) {
   969           _res_cap[a] -= d;
   970           _res_cap[_reverse[a]] += d;
   971           u = _source[a];
   972         }
   973         _excess[s] -= d;
   974         _excess[t] += d;
   975       }
   976 
   977       return OPTIMAL;
   978     }
   979 
   980   }; //class CapacityScaling
   981 
   982   ///@}
   983 
   984 } //namespace lemon
   985 
   986 #endif //LEMON_CAPACITY_SCALING_H