/* -*- mode: C++; indent-tabs-mode: nil; -*-

  *

  * This file is a part of LEMON, a generic C++ optimization library.

  *

  * Copyright (C) 2003-2010

  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

  * (Egervary Research Group on Combinatorial Optimization, EGRES).

  *

  * Permission to use, modify and distribute this software is granted

  * provided that this copyright notice appears in all copies. For

  * precise terms see the accompanying LICENSE file.

  *

  * This software is provided "AS IS" with no warranty of any kind,

  * express or implied, and with no claim as to its suitability for any

  * purpose.

  *

  */

 #ifndef LEMON_SUURBALLE_H

 #define LEMON_SUURBALLE_H

 ///\ingroup shortest_path

 ///\file

 ///\brief An algorithm for finding arc-disjoint paths between two

 /// nodes having minimum total length.

 #include <vector>

 #include <limits>

 #include <lemon/bin_heap.h>

 #include <lemon/path.h>

 #include <lemon/list_graph.h>

 #include <lemon/dijkstra.h>

 #include <lemon/maps.h>

 namespace lemon {

   /// \brief Default traits class of Suurballe algorithm.

   ///

   /// Default traits class of Suurballe algorithm.

   /// \tparam GR The digraph type the algorithm runs on.

   /// \tparam LEN The type of the length map.

   /// The default value is <tt>GR::ArcMap<int></tt>.

 #ifdef DOXYGEN

   template <typename GR, typename LEN>

 #else

   template < typename GR,

              typename LEN = typename GR::template ArcMap<int> >

 #endif

   struct SuurballeDefaultTraits

   {

     /// The type of the digraph.

     typedef GR Digraph;

     /// The type of the length map.

     typedef LEN LengthMap;

     /// The type of the lengths.

     typedef typename LEN::Value Length;

     /// The type of the flow map.

     typedef typename GR::template ArcMap<int> FlowMap;

     /// The type of the potential map.

     typedef typename GR::template NodeMap<Length> PotentialMap;

     /// \brief The path type

     ///

     /// The type used for storing the found arc-disjoint paths.

     /// It must conform to the \ref lemon::concepts::Path "Path" concept

     /// and it must have an \c addBack() function.

     typedef lemon::Path<Digraph> Path;

     /// The cross reference type used for the heap.

     typedef typename GR::template NodeMap<int> HeapCrossRef;

     /// \brief The heap type used for internal Dijkstra computations.

     ///

     /// The type of the heap used for internal Dijkstra computations.

     /// It must conform to the \ref lemon::concepts::Heap "Heap" concept

     /// and its priority type must be \c Length.

     typedef BinHeap<Length, HeapCrossRef> Heap;

   };

   /// \addtogroup shortest_path

   /// @{

   /// \brief Algorithm for finding arc-disjoint paths between two nodes

   /// having minimum total length.

   ///

   /// \ref lemon::Suurballe "Suurballe" implements an algorithm for

   /// finding arc-disjoint paths having minimum total length (cost)

   /// from a given source node to a given target node in a digraph.

   ///

   /// Note that this problem is a special case of the \ref min_cost_flow

   /// "minimum cost flow problem". This implementation is actually an

   /// efficient specialized version of the \ref CapacityScaling

   /// "successive shortest path" algorithm directly for this problem.

   /// Therefore this class provides query functions for flow values and

   /// node potentials (the dual solution) just like the minimum cost flow

   /// algorithms.

   ///

   /// \tparam GR The digraph type the algorithm runs on.

   /// \tparam LEN The type of the length map.

   /// The default value is <tt>GR::ArcMap<int></tt>.

   ///

   /// \warning Length values should be \e non-negative.

   ///

   /// \note For finding \e node-disjoint paths, this algorithm can be used

   /// along with the \ref SplitNodes adaptor.

 #ifdef DOXYGEN

   template <typename GR, typename LEN, typename TR>

 #else

   template < typename GR,

              typename LEN = typename GR::template ArcMap<int>,

              typename TR = SuurballeDefaultTraits<GR, LEN> >

 #endif

   class Suurballe

   {

     TEMPLATE_DIGRAPH_TYPEDEFS(GR);

     typedef ConstMap<Arc, int> ConstArcMap;

     typedef typename GR::template NodeMap<Arc> PredMap;

   public:

     /// The type of the digraph.

     typedef typename TR::Digraph Digraph;

     /// The type of the length map.

     typedef typename TR::LengthMap LengthMap;

     /// The type of the lengths.

     typedef typename TR::Length Length;

     /// The type of the flow map.

     typedef typename TR::FlowMap FlowMap;

     /// The type of the potential map.

     typedef typename TR::PotentialMap PotentialMap;

     /// The type of the path structures.

     typedef typename TR::Path Path;

     /// The cross reference type used for the heap.

     typedef typename TR::HeapCrossRef HeapCrossRef;

     /// The heap type used for internal Dijkstra computations.

     typedef typename TR::Heap Heap;

     /// The \ref SuurballeDefaultTraits "traits class" of the algorithm.

     typedef TR Traits;

   private:

     // ResidualDijkstra is a special implementation of the

     // Dijkstra algorithm for finding shortest paths in the

     // residual network with respect to the reduced arc lengths

     // and modifying the node potentials according to the

     // distance of the nodes.

     class ResidualDijkstra

     {

     private:

       const Digraph &_graph;

       const LengthMap &_length;

       const FlowMap &_flow;

       PotentialMap &_pi;

       PredMap &_pred;

       Node _s;

       Node _t;

       PotentialMap _dist;

       std::vector<Node> _proc_nodes;

     public:

       // Constructor

       ResidualDijkstra(Suurballe &srb) :

         _graph(srb._graph), _length(srb._length),

         _flow(*srb._flow), _pi(*srb._potential), _pred(srb._pred),

         _s(srb._s), _t(srb._t), _dist(_graph) {}

       // Run the algorithm and return true if a path is found

       // from the source node to the target node.

       bool run(int cnt) {

         return cnt == 0 ? startFirst() : start();

       }

     private:

       // Execute the algorithm for the first time (the flow and potential

       // functions have to be identically zero).

       bool startFirst() {

         HeapCrossRef heap_cross_ref(_graph, Heap::PRE_HEAP);

         Heap heap(heap_cross_ref);

         heap.push(_s, 0);

         _pred[_s] = INVALID;

         _proc_nodes.clear();

         // Process nodes

         while (!heap.empty() && heap.top() != _t) {

           Node u = heap.top(), v;

           Length d = heap.prio(), dn;

           _dist[u] = heap.prio();

           _proc_nodes.push_back(u);

           heap.pop();

           // Traverse outgoing arcs

           for (OutArcIt e(_graph, u); e != INVALID; ++e) {

             v = _graph.target(e);

             switch(heap.state(v)) {

               case Heap::PRE_HEAP:

                 heap.push(v, d + _length[e]);

                 _pred[v] = e;

                 break;

               case Heap::IN_HEAP:

                 dn = d + _length[e];

                 if (dn < heap[v]) {

                   heap.decrease(v, dn);

                   _pred[v] = e;

                 }

                 break;

               case Heap::POST_HEAP:

                 break;

             }

           }

         }

         if (heap.empty()) return false;

         // Update potentials of processed nodes

         Length t_dist = heap.prio();

         for (int i = 0; i < int(_proc_nodes.size()); ++i)

           _pi[_proc_nodes[i]] = _dist[_proc_nodes[i]] - t_dist;

         return true;

       }

       // Execute the algorithm.

       bool start() {

         HeapCrossRef heap_cross_ref(_graph, Heap::PRE_HEAP);

         Heap heap(heap_cross_ref);

         heap.push(_s, 0);

         _pred[_s] = INVALID;

         _proc_nodes.clear();

         // Process nodes

         while (!heap.empty() && heap.top() != _t) {

           Node u = heap.top(), v;

           Length d = heap.prio() + _pi[u], dn;

           _dist[u] = heap.prio();

           _proc_nodes.push_back(u);

           heap.pop();

           // Traverse outgoing arcs

           for (OutArcIt e(_graph, u); e != INVALID; ++e) {

             if (_flow[e] == 0) {

               v = _graph.target(e);

               switch(heap.state(v)) {

                 case Heap::PRE_HEAP:

                   heap.push(v, d + _length[e] - _pi[v]);

                   _pred[v] = e;

                   break;

                 case Heap::IN_HEAP:

                   dn = d + _length[e] - _pi[v];

                   if (dn < heap[v]) {

                     heap.decrease(v, dn);

                     _pred[v] = e;

                   }

                   break;

                 case Heap::POST_HEAP:

                   break;

               }

             }

           }

           // Traverse incoming arcs

           for (InArcIt e(_graph, u); e != INVALID; ++e) {

             if (_flow[e] == 1) {

               v = _graph.source(e);

               switch(heap.state(v)) {

                 case Heap::PRE_HEAP:

                   heap.push(v, d - _length[e] - _pi[v]);

                   _pred[v] = e;

                   break;

                 case Heap::IN_HEAP:

                   dn = d - _length[e] - _pi[v];

                   if (dn < heap[v]) {

                     heap.decrease(v, dn);

                     _pred[v] = e;

                   }

                   break;

                 case Heap::POST_HEAP:

                   break;

               }

             }

           }

         }

         if (heap.empty()) return false;

         // Update potentials of processed nodes

         Length t_dist = heap.prio();

         for (int i = 0; i < int(_proc_nodes.size()); ++i)

           _pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - t_dist;

         return true;

       }

     }; //class ResidualDijkstra

   public:

     /// \name Named Template Parameters

     /// @{

     template <typename T>

     struct SetFlowMapTraits : public Traits {

       typedef T FlowMap;

     };

     /// \brief \ref named-templ-param "Named parameter" for setting

     /// \c FlowMap type.

     ///

     /// \ref named-templ-param "Named parameter" for setting

     /// \c FlowMap type.

     template <typename T>

     struct SetFlowMap

       : public Suurballe<GR, LEN, SetFlowMapTraits<T> > {

       typedef Suurballe<GR, LEN, SetFlowMapTraits<T> > Create;

     };

     template <typename T>

     struct SetPotentialMapTraits : public Traits {

       typedef T PotentialMap;

     };

     /// \brief \ref named-templ-param "Named parameter" for setting

     /// \c PotentialMap type.

     ///

     /// \ref named-templ-param "Named parameter" for setting

     /// \c PotentialMap type.

     template <typename T>

     struct SetPotentialMap

       : public Suurballe<GR, LEN, SetPotentialMapTraits<T> > {

       typedef Suurballe<GR, LEN, SetPotentialMapTraits<T> > Create;

     };

     template <typename T>

     struct SetPathTraits : public Traits {

       typedef T Path;

     };

     /// \brief \ref named-templ-param "Named parameter" for setting

     /// \c %Path type.

     ///

     /// \ref named-templ-param "Named parameter" for setting \c %Path type.

     /// It must conform to the \ref lemon::concepts::Path "Path" concept

     /// and it must have an \c addBack() function.

     template <typename T>

     struct SetPath

       : public Suurballe<GR, LEN, SetPathTraits<T> > {

       typedef Suurballe<GR, LEN, SetPathTraits<T> > Create;

     };

     template <typename H, typename CR>

     struct SetHeapTraits : public Traits {

       typedef H Heap;

       typedef CR HeapCrossRef;

     };

     /// \brief \ref named-templ-param "Named parameter" for setting

     /// \c Heap and \c HeapCrossRef types.

     ///

     /// \ref named-templ-param "Named parameter" for setting \c Heap

     /// and \c HeapCrossRef types with automatic allocation.

     /// They will be used for internal Dijkstra computations.

     /// The heap type must conform to the \ref lemon::concepts::Heap "Heap"

     /// concept and its priority type must be \c Length.

     template <typename H,

               typename CR = typename Digraph::template NodeMap<int> >

     struct SetHeap

       : public Suurballe<GR, LEN, SetHeapTraits<H, CR> > {

       typedef Suurballe<GR, LEN, SetHeapTraits<H, CR> > Create;

     };

     /// @}

   private:

     // The digraph the algorithm runs on

     const Digraph &_graph;

     // The length map

     const LengthMap &_length;

     // Arc map of the current flow

     FlowMap *_flow;

     bool _local_flow;

     // Node map of the current potentials

     PotentialMap *_potential;

     bool _local_potential;

     // The source node

     Node _s;

     // The target node

     Node _t;

     // Container to store the found paths

     std::vector<Path> _paths;

     int _path_num;

     // The pred arc map

     PredMap _pred;

     // Data for full init

     PotentialMap *_init_dist;

     PredMap *_init_pred;

     bool _full_init;

   protected:

     Suurballe() {}

   public:

     /// \brief Constructor.

     ///

     /// Constructor.

     ///

     /// \param graph The digraph the algorithm runs on.

     /// \param length The length (cost) values of the arcs.

     Suurballe( const Digraph &graph,

                const LengthMap &length ) :

       _graph(graph), _length(length), _flow(0), _local_flow(false),

       _potential(0), _local_potential(false), _pred(graph),

       _init_dist(0), _init_pred(0)

     {}

     /// Destructor.

     ~Suurballe() {

       if (_local_flow) delete _flow;

       if (_local_potential) delete _potential;

       delete _init_dist;

       delete _init_pred;

     }

     /// \brief Set the flow map.

     ///

     /// This function sets the flow map.

     /// If it is not used before calling \ref run() or \ref init(),

     /// an instance will be allocated automatically. The destructor

     /// deallocates this automatically allocated map, of course.

     ///

     /// The found flow contains only 0 and 1 values, since it is the

     /// union of the found arc-disjoint paths.

     ///

     /// \return <tt>(*this)</tt>

     Suurballe& flowMap(FlowMap &map) {

       if (_local_flow) {

         delete _flow;

         _local_flow = false;

       }

       _flow = ↦

       return *this;

     }

     /// \brief Set the potential map.

     ///

     /// This function sets the potential map.

     /// If it is not used before calling \ref run() or \ref init(),

     /// an instance will be allocated automatically. The destructor

     /// deallocates this automatically allocated map, of course.

     ///

     /// The node potentials provide the dual solution of the underlying

     /// \ref min_cost_flow "minimum cost flow problem".

     ///

     /// \return <tt>(*this)</tt>

     Suurballe& potentialMap(PotentialMap &map) {

       if (_local_potential) {

         delete _potential;

         _local_potential = false;

       }

       _potential = ↦

       return *this;

     }

     /// \name Execution Control

     /// The simplest way to execute the algorithm is to call the run()

     /// function.\n

     /// If you need to execute the algorithm many times using the same

     /// source node, then you may call fullInit() once and start()

     /// for each target node.\n

     /// If you only need the flow that is the union of the found

     /// arc-disjoint paths, then you may call findFlow() instead of

     /// start().

     /// @{

     /// \brief Run the algorithm.

     ///

     /// This function runs the algorithm.

     ///

     /// \param s The source node.

     /// \param t The target node.

     /// \param k The number of paths to be found.

     ///

     /// \return \c k if there are at least \c k arc-disjoint paths from

     /// \c s to \c t in the digraph. Otherwise it returns the number of

     /// arc-disjoint paths found.

     ///

     /// \note Apart from the return value, <tt>s.run(s, t, k)</tt> is

     /// just a shortcut of the following code.

     /// \code

     ///   s.init(s);

     ///   s.start(t, k);

     /// \endcode

     int run(const Node& s, const Node& t, int k = 2) {

       init(s);

       start(t, k);

       return _path_num;

     }

     /// \brief Initialize the algorithm.

     ///

     /// This function initializes the algorithm with the given source node.

     ///

     /// \param s The source node.

     void init(const Node& s) {

       _s = s;

       // Initialize maps

       if (!_flow) {

         _flow = new FlowMap(_graph);

         _local_flow = true;

       }

       if (!_potential) {

         _potential = new PotentialMap(_graph);

         _local_potential = true;

       }

       _full_init = false;

     }

     /// \brief Initialize the algorithm and perform Dijkstra.

     ///

     /// This function initializes the algorithm and performs a full

     /// Dijkstra search from the given source node. It makes consecutive

     /// executions of \ref start() "start(t, k)" faster, since they

     /// have to perform %Dijkstra only k-1 times.

     ///

     /// This initialization is usually worth using instead of \ref init()

     /// if the algorithm is executed many times using the same source node.

     ///

     /// \param s The source node.

     void fullInit(const Node& s) {

       // Initialize maps

       init(s);

       if (!_init_dist) {

         _init_dist = new PotentialMap(_graph);

       }

       if (!_init_pred) {

         _init_pred = new PredMap(_graph);

       }

       // Run a full Dijkstra

       typename Dijkstra<Digraph, LengthMap>

         ::template SetStandardHeap<Heap>

         ::template SetDistMap<PotentialMap>

         ::template SetPredMap<PredMap>

         ::Create dijk(_graph, _length);

       dijk.distMap(*_init_dist).predMap(*_init_pred);

       dijk.run(s);

       _full_init = true;

     }

     /// \brief Execute the algorithm.

     ///

     /// This function executes the algorithm.

     ///

     /// \param t The target node.

     /// \param k The number of paths to be found.

     ///

     /// \return \c k if there are at least \c k arc-disjoint paths from

     /// \c s to \c t in the digraph. Otherwise it returns the number of

     /// arc-disjoint paths found.

     ///

     /// \note Apart from the return value, <tt>s.start(t, k)</tt> is

     /// just a shortcut of the following code.

     /// \code

     ///   s.findFlow(t, k);

     ///   s.findPaths();

     /// \endcode

     int start(const Node& t, int k = 2) {

       findFlow(t, k);

       findPaths();

       return _path_num;

     }

     /// \brief Execute the algorithm to find an optimal flow.

     ///

     /// This function executes the successive shortest path algorithm to

     /// find a minimum cost flow, which is the union of \c k (or less)

     /// arc-disjoint paths.

     ///

     /// \param t The target node.

     /// \param k The number of paths to be found.

     ///

     /// \return \c k if there are at least \c k arc-disjoint paths from

     /// the source node to the given node \c t in the digraph.

     /// Otherwise it returns the number of arc-disjoint paths found.

     ///

     /// \pre \ref init() must be called before using this function.

     int findFlow(const Node& t, int k = 2) {

       _t = t;

       ResidualDijkstra dijkstra(*this);

       // Initialization

       for (ArcIt e(_graph); e != INVALID; ++e) {

         (*_flow)[e] = 0;

       }

       if (_full_init) {

         for (NodeIt n(_graph); n != INVALID; ++n) {

           (*_potential)[n] = (*_init_dist)[n];

         }

         Node u = _t;

         Arc e;

         while ((e = (*_init_pred)[u]) != INVALID) {

           (*_flow)[e] = 1;

           u = _graph.source(e);

         }

         _path_num = 1;

       } else {

         for (NodeIt n(_graph); n != INVALID; ++n) {

           (*_potential)[n] = 0;

         }

         _path_num = 0;

       }

       // Find shortest paths

       while (_path_num < k) {

         // Run Dijkstra

         if (!dijkstra.run(_path_num)) break;

         ++_path_num;

         // Set the flow along the found shortest path

         Node u = _t;

         Arc e;

         while ((e = _pred[u]) != INVALID) {

           if (u == _graph.target(e)) {

             (*_flow)[e] = 1;

             u = _graph.source(e);

           } else {

             (*_flow)[e] = 0;

             u = _graph.target(e);

           }

         }

       }

       return _path_num;

     }

     /// \brief Compute the paths from the flow.

     ///

     /// This function computes arc-disjoint paths from the found minimum

     /// cost flow, which is the union of them.

     ///

     /// \pre \ref init() and \ref findFlow() must be called before using

     /// this function.

     void findPaths() {

       FlowMap res_flow(_graph);

       for(ArcIt a(_graph); a != INVALID; ++a) res_flow[a] = (*_flow)[a];

       _paths.clear();

       _paths.resize(_path_num);

       for (int i = 0; i < _path_num; ++i) {

         Node n = _s;

         while (n != _t) {

           OutArcIt e(_graph, n);

           for ( ; res_flow[e] == 0; ++e) ;

           n = _graph.target(e);

           _paths[i].addBack(e);

           res_flow[e] = 0;

         }

       }

     }

     /// @}

     /// \name Query Functions

     /// The results of the algorithm can be obtained using these

     /// functions.

     /// \n The algorithm should be executed before using them.

     /// @{

     /// \brief Return the total length of the found paths.

     ///

     /// This function returns the total length of the found paths, i.e.

     /// the total cost of the found flow.

     /// The complexity of the function is O(e).

     ///

     /// \pre \ref run() or \ref findFlow() must be called before using

     /// this function.

     Length totalLength() const {

       Length c = 0;

       for (ArcIt e(_graph); e != INVALID; ++e)

         c += (*_flow)[e] * _length[e];

       return c;

     }

     /// \brief Return the flow value on the given arc.

     ///

     /// This function returns the flow value on the given arc.

     /// It is \c 1 if the arc is involved in one of the found arc-disjoint

     /// paths, otherwise it is \c 0.

     ///

     /// \pre \ref run() or \ref findFlow() must be called before using

     /// this function.

     int flow(const Arc& arc) const {

       return (*_flow)[arc];

     }

     /// \brief Return a const reference to an arc map storing the

     /// found flow.

     ///

     /// This function returns a const reference to an arc map storing

     /// the flow that is the union of the found arc-disjoint paths.

     ///

     /// \pre \ref run() or \ref findFlow() must be called before using

     /// this function.

     const FlowMap& flowMap() const {

       return *_flow;

     }

     /// \brief Return the potential of the given node.

     ///

     /// This function returns the potential of the given node.

     /// The node potentials provide the dual solution of the

     /// underlying \ref min_cost_flow "minimum cost flow problem".

     ///

     /// \pre \ref run() or \ref findFlow() must be called before using

     /// this function.

     Length potential(const Node& node) const {

       return (*_potential)[node];

     }

     /// \brief Return a const reference to a node map storing the

     /// found potentials (the dual solution).

     ///

     /// This function returns a const reference to a node map storing

     /// the found potentials that provide the dual solution of the

     /// underlying \ref min_cost_flow "minimum cost flow problem".

     ///

     /// \pre \ref run() or \ref findFlow() must be called before using

     /// this function.

     const PotentialMap& potentialMap() const {

       return *_potential;

     }

     /// \brief Return the number of the found paths.

     ///

     /// This function returns the number of the found paths.

     ///

     /// \pre \ref run() or \ref findFlow() must be called before using

     /// this function.

     int pathNum() const {

       return _path_num;

     }

     /// \brief Return a const reference to the specified path.

     ///

     /// This function returns a const reference to the specified path.

     ///

     /// \param i The function returns the <tt>i</tt>-th path.

     /// \c i must be between \c 0 and <tt>%pathNum()-1</tt>.

     ///

     /// \pre \ref run() or \ref findPaths() must be called before using

     /// this function.

     const Path& path(int i) const {

       return _paths[i];

     }

     /// @}

   }; //class Suurballe

   ///@}

 } //namespace lemon

 #endif //LEMON_SUURBALLE_H