alpar@440: /* -*- mode: C++; indent-tabs-mode: nil; -*-
alpar@345:  *
alpar@440:  * This file is a part of LEMON, a generic C++ optimization library.
alpar@345:  *
alpar@440:  * Copyright (C) 2003-2009
alpar@345:  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
alpar@345:  * (Egervary Research Group on Combinatorial Optimization, EGRES).
alpar@345:  *
alpar@345:  * Permission to use, modify and distribute this software is granted
alpar@345:  * provided that this copyright notice appears in all copies. For
alpar@345:  * precise terms see the accompanying LICENSE file.
alpar@345:  *
alpar@345:  * This software is provided "AS IS" with no warranty of any kind,
alpar@345:  * express or implied, and with no claim as to its suitability for any
alpar@345:  * purpose.
alpar@345:  *
alpar@345:  */
alpar@345: 
alpar@345: #ifndef LEMON_SUURBALLE_H
alpar@345: #define LEMON_SUURBALLE_H
alpar@345: 
alpar@345: ///\ingroup shortest_path
alpar@345: ///\file
alpar@345: ///\brief An algorithm for finding arc-disjoint paths between two
alpar@345: /// nodes having minimum total length.
alpar@345: 
alpar@345: #include <vector>
kpeter@623: #include <limits>
alpar@345: #include <lemon/bin_heap.h>
alpar@345: #include <lemon/path.h>
deba@519: #include <lemon/list_graph.h>
deba@519: #include <lemon/maps.h>
alpar@345: 
alpar@345: namespace lemon {
alpar@345: 
alpar@345:   /// \addtogroup shortest_path
alpar@345:   /// @{
alpar@345: 
kpeter@346:   /// \brief Algorithm for finding arc-disjoint paths between two nodes
kpeter@346:   /// having minimum total length.
alpar@345:   ///
alpar@345:   /// \ref lemon::Suurballe "Suurballe" implements an algorithm for
alpar@345:   /// finding arc-disjoint paths having minimum total length (cost)
kpeter@346:   /// from a given source node to a given target node in a digraph.
alpar@345:   ///
kpeter@623:   /// Note that this problem is a special case of the \ref min_cost_flow
kpeter@623:   /// "minimum cost flow problem". This implementation is actually an
kpeter@623:   /// efficient specialized version of the \ref CapacityScaling
kpeter@853:   /// "successive shortest path" algorithm directly for this problem.
kpeter@623:   /// Therefore this class provides query functions for flow values and
kpeter@623:   /// node potentials (the dual solution) just like the minimum cost flow
kpeter@623:   /// algorithms.
alpar@345:   ///
kpeter@559:   /// \tparam GR The digraph type the algorithm runs on.
kpeter@623:   /// \tparam LEN The type of the length map.
kpeter@623:   /// The default value is <tt>GR::ArcMap<int></tt>.
alpar@345:   ///
kpeter@852:   /// \warning Length values should be \e non-negative.
alpar@345:   ///
kpeter@853:   /// \note For finding \e node-disjoint paths, this algorithm can be used
kpeter@623:   /// along with the \ref SplitNodes adaptor.
kpeter@346: #ifdef DOXYGEN
kpeter@559:   template <typename GR, typename LEN>
kpeter@346: #else
kpeter@623:   template < typename GR,
kpeter@559:              typename LEN = typename GR::template ArcMap<int> >
kpeter@346: #endif
alpar@345:   class Suurballe
alpar@345:   {
kpeter@559:     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
alpar@345: 
alpar@345:     typedef ConstMap<Arc, int> ConstArcMap;
kpeter@559:     typedef typename GR::template NodeMap<Arc> PredMap;
alpar@345: 
alpar@345:   public:
alpar@345: 
kpeter@559:     /// The type of the digraph the algorithm runs on.
kpeter@559:     typedef GR Digraph;
kpeter@559:     /// The type of the length map.
kpeter@559:     typedef LEN LengthMap;
kpeter@559:     /// The type of the lengths.
kpeter@559:     typedef typename LengthMap::Value Length;
kpeter@623: #ifdef DOXYGEN
kpeter@623:     /// The type of the flow map.
kpeter@623:     typedef GR::ArcMap<int> FlowMap;
kpeter@623:     /// The type of the potential map.
kpeter@623:     typedef GR::NodeMap<Length> PotentialMap;
kpeter@623: #else
alpar@345:     /// The type of the flow map.
alpar@345:     typedef typename Digraph::template ArcMap<int> FlowMap;
alpar@345:     /// The type of the potential map.
alpar@345:     typedef typename Digraph::template NodeMap<Length> PotentialMap;
kpeter@623: #endif
kpeter@623: 
alpar@345:     /// The type of the path structures.
kpeter@623:     typedef SimplePath<GR> Path;
alpar@345: 
alpar@345:   private:
alpar@440: 
kpeter@623:     // ResidualDijkstra is a special implementation of the
kpeter@623:     // Dijkstra algorithm for finding shortest paths in the
kpeter@623:     // residual network with respect to the reduced arc lengths
kpeter@623:     // and modifying the node potentials according to the
kpeter@623:     // distance of the nodes.
alpar@345:     class ResidualDijkstra
alpar@345:     {
alpar@345:       typedef typename Digraph::template NodeMap<int> HeapCrossRef;
alpar@345:       typedef BinHeap<Length, HeapCrossRef> Heap;
alpar@345: 
alpar@345:     private:
alpar@345: 
alpar@345:       const Digraph &_graph;
kpeter@853:       const LengthMap &_length;
alpar@345:       const FlowMap &_flow;
kpeter@853:       PotentialMap &_pi;
alpar@345:       PredMap &_pred;
alpar@345:       Node _s;
alpar@345:       Node _t;
kpeter@853:       
kpeter@853:       PotentialMap _dist;
kpeter@853:       std::vector<Node> _proc_nodes;
alpar@345: 
alpar@345:     public:
alpar@345: 
kpeter@853:       // Constructor
kpeter@853:       ResidualDijkstra(Suurballe &srb) :
kpeter@853:         _graph(srb._graph), _length(srb._length),
kpeter@853:         _flow(*srb._flow), _pi(*srb._potential), _pred(srb._pred), 
kpeter@853:         _s(srb._s), _t(srb._t), _dist(_graph) {}
kpeter@853:         
kpeter@853:       // Run the algorithm and return true if a path is found
kpeter@853:       // from the source node to the target node.
kpeter@853:       bool run(int cnt) {
kpeter@853:         return cnt == 0 ? startFirst() : start();
kpeter@853:       }
alpar@345: 
kpeter@853:     private:
kpeter@853:     
kpeter@853:       // Execute the algorithm for the first time (the flow and potential
kpeter@853:       // functions have to be identically zero).
kpeter@853:       bool startFirst() {
alpar@345:         HeapCrossRef heap_cross_ref(_graph, Heap::PRE_HEAP);
alpar@345:         Heap heap(heap_cross_ref);
alpar@345:         heap.push(_s, 0);
alpar@345:         _pred[_s] = INVALID;
alpar@345:         _proc_nodes.clear();
alpar@345: 
kpeter@346:         // Process nodes
alpar@345:         while (!heap.empty() && heap.top() != _t) {
alpar@345:           Node u = heap.top(), v;
kpeter@853:           Length d = heap.prio(), dn;
alpar@345:           _dist[u] = heap.prio();
kpeter@853:           _proc_nodes.push_back(u);
alpar@345:           heap.pop();
kpeter@853: 
kpeter@853:           // Traverse outgoing arcs
kpeter@853:           for (OutArcIt e(_graph, u); e != INVALID; ++e) {
kpeter@853:             v = _graph.target(e);
kpeter@853:             switch(heap.state(v)) {
kpeter@853:               case Heap::PRE_HEAP:
kpeter@853:                 heap.push(v, d + _length[e]);
kpeter@853:                 _pred[v] = e;
kpeter@853:                 break;
kpeter@853:               case Heap::IN_HEAP:
kpeter@853:                 dn = d + _length[e];
kpeter@853:                 if (dn < heap[v]) {
kpeter@853:                   heap.decrease(v, dn);
kpeter@853:                   _pred[v] = e;
kpeter@853:                 }
kpeter@853:                 break;
kpeter@853:               case Heap::POST_HEAP:
kpeter@853:                 break;
kpeter@853:             }
kpeter@853:           }
kpeter@853:         }
kpeter@853:         if (heap.empty()) return false;
kpeter@853: 
kpeter@853:         // Update potentials of processed nodes
kpeter@853:         Length t_dist = heap.prio();
kpeter@853:         for (int i = 0; i < int(_proc_nodes.size()); ++i)
kpeter@853:           _pi[_proc_nodes[i]] = _dist[_proc_nodes[i]] - t_dist;
kpeter@853:         return true;
kpeter@853:       }
kpeter@853: 
kpeter@853:       // Execute the algorithm.
kpeter@853:       bool start() {
kpeter@853:         HeapCrossRef heap_cross_ref(_graph, Heap::PRE_HEAP);
kpeter@853:         Heap heap(heap_cross_ref);
kpeter@853:         heap.push(_s, 0);
kpeter@853:         _pred[_s] = INVALID;
kpeter@853:         _proc_nodes.clear();
kpeter@853: 
kpeter@853:         // Process nodes
kpeter@853:         while (!heap.empty() && heap.top() != _t) {
kpeter@853:           Node u = heap.top(), v;
kpeter@853:           Length d = heap.prio() + _pi[u], dn;
kpeter@853:           _dist[u] = heap.prio();
alpar@345:           _proc_nodes.push_back(u);
kpeter@853:           heap.pop();
alpar@345: 
kpeter@346:           // Traverse outgoing arcs
alpar@345:           for (OutArcIt e(_graph, u); e != INVALID; ++e) {
alpar@345:             if (_flow[e] == 0) {
alpar@345:               v = _graph.target(e);
alpar@345:               switch(heap.state(v)) {
kpeter@853:                 case Heap::PRE_HEAP:
kpeter@853:                   heap.push(v, d + _length[e] - _pi[v]);
alpar@345:                   _pred[v] = e;
kpeter@853:                   break;
kpeter@853:                 case Heap::IN_HEAP:
kpeter@853:                   dn = d + _length[e] - _pi[v];
kpeter@853:                   if (dn < heap[v]) {
kpeter@853:                     heap.decrease(v, dn);
kpeter@853:                     _pred[v] = e;
kpeter@853:                   }
kpeter@853:                   break;
kpeter@853:                 case Heap::POST_HEAP:
kpeter@853:                   break;
alpar@345:               }
alpar@345:             }
alpar@345:           }
alpar@345: 
kpeter@346:           // Traverse incoming arcs
alpar@345:           for (InArcIt e(_graph, u); e != INVALID; ++e) {
alpar@345:             if (_flow[e] == 1) {
alpar@345:               v = _graph.source(e);
alpar@345:               switch(heap.state(v)) {
kpeter@853:                 case Heap::PRE_HEAP:
kpeter@853:                   heap.push(v, d - _length[e] - _pi[v]);
alpar@345:                   _pred[v] = e;
kpeter@853:                   break;
kpeter@853:                 case Heap::IN_HEAP:
kpeter@853:                   dn = d - _length[e] - _pi[v];
kpeter@853:                   if (dn < heap[v]) {
kpeter@853:                     heap.decrease(v, dn);
kpeter@853:                     _pred[v] = e;
kpeter@853:                   }
kpeter@853:                   break;
kpeter@853:                 case Heap::POST_HEAP:
kpeter@853:                   break;
alpar@345:               }
alpar@345:             }
alpar@345:           }
alpar@345:         }
alpar@345:         if (heap.empty()) return false;
alpar@345: 
kpeter@346:         // Update potentials of processed nodes
alpar@345:         Length t_dist = heap.prio();
alpar@345:         for (int i = 0; i < int(_proc_nodes.size()); ++i)
kpeter@853:           _pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - t_dist;
alpar@345:         return true;
alpar@345:       }
alpar@345: 
alpar@345:     }; //class ResidualDijkstra
alpar@345: 
alpar@345:   private:
alpar@345: 
kpeter@346:     // The digraph the algorithm runs on
alpar@345:     const Digraph &_graph;
alpar@345:     // The length map
alpar@345:     const LengthMap &_length;
alpar@440: 
alpar@345:     // Arc map of the current flow
alpar@345:     FlowMap *_flow;
alpar@345:     bool _local_flow;
alpar@345:     // Node map of the current potentials
alpar@345:     PotentialMap *_potential;
alpar@345:     bool _local_potential;
alpar@345: 
alpar@345:     // The source node
kpeter@853:     Node _s;
alpar@345:     // The target node
kpeter@853:     Node _t;
alpar@345: 
alpar@345:     // Container to store the found paths
kpeter@853:     std::vector<Path> _paths;
alpar@345:     int _path_num;
alpar@345: 
alpar@345:     // The pred arc map
alpar@345:     PredMap _pred;
alpar@345: 
alpar@345:   public:
alpar@345: 
alpar@345:     /// \brief Constructor.
alpar@345:     ///
alpar@345:     /// Constructor.
alpar@345:     ///
kpeter@623:     /// \param graph The digraph the algorithm runs on.
alpar@345:     /// \param length The length (cost) values of the arcs.
kpeter@623:     Suurballe( const Digraph &graph,
kpeter@623:                const LengthMap &length ) :
kpeter@623:       _graph(graph), _length(length), _flow(0), _local_flow(false),
kpeter@623:       _potential(0), _local_potential(false), _pred(graph)
kpeter@852:     {}
alpar@345: 
alpar@345:     /// Destructor.
alpar@345:     ~Suurballe() {
alpar@345:       if (_local_flow) delete _flow;
alpar@345:       if (_local_potential) delete _potential;
alpar@345:     }
alpar@345: 
kpeter@346:     /// \brief Set the flow map.
alpar@345:     ///
kpeter@346:     /// This function sets the flow map.
kpeter@623:     /// If it is not used before calling \ref run() or \ref init(),
kpeter@623:     /// an instance will be allocated automatically. The destructor
kpeter@623:     /// deallocates this automatically allocated map, of course.
alpar@345:     ///
kpeter@623:     /// The found flow contains only 0 and 1 values, since it is the
kpeter@623:     /// union of the found arc-disjoint paths.
alpar@345:     ///
kpeter@559:     /// \return <tt>(*this)</tt>
alpar@345:     Suurballe& flowMap(FlowMap &map) {
alpar@345:       if (_local_flow) {
alpar@345:         delete _flow;
alpar@345:         _local_flow = false;
alpar@345:       }
alpar@345:       _flow = &map;
alpar@345:       return *this;
alpar@345:     }
alpar@345: 
kpeter@346:     /// \brief Set the potential map.
alpar@345:     ///
kpeter@346:     /// This function sets the potential map.
kpeter@623:     /// If it is not used before calling \ref run() or \ref init(),
kpeter@623:     /// an instance will be allocated automatically. The destructor
kpeter@623:     /// deallocates this automatically allocated map, of course.
alpar@345:     ///
kpeter@623:     /// The node potentials provide the dual solution of the underlying
kpeter@623:     /// \ref min_cost_flow "minimum cost flow problem".
alpar@345:     ///
kpeter@559:     /// \return <tt>(*this)</tt>
alpar@345:     Suurballe& potentialMap(PotentialMap &map) {
alpar@345:       if (_local_potential) {
alpar@345:         delete _potential;
alpar@345:         _local_potential = false;
alpar@345:       }
alpar@345:       _potential = &map;
alpar@345:       return *this;
alpar@345:     }
alpar@345: 
kpeter@584:     /// \name Execution Control
alpar@345:     /// The simplest way to execute the algorithm is to call the run()
alpar@345:     /// function.
alpar@345:     /// \n
alpar@345:     /// If you only need the flow that is the union of the found
alpar@345:     /// arc-disjoint paths, you may call init() and findFlow().
alpar@345: 
alpar@345:     /// @{
alpar@345: 
kpeter@346:     /// \brief Run the algorithm.
alpar@345:     ///
kpeter@346:     /// This function runs the algorithm.
alpar@345:     ///
kpeter@623:     /// \param s The source node.
kpeter@623:     /// \param t The target node.
alpar@345:     /// \param k The number of paths to be found.
alpar@345:     ///
kpeter@346:     /// \return \c k if there are at least \c k arc-disjoint paths from
kpeter@346:     /// \c s to \c t in the digraph. Otherwise it returns the number of
alpar@345:     /// arc-disjoint paths found.
alpar@345:     ///
kpeter@623:     /// \note Apart from the return value, <tt>s.run(s, t, k)</tt> is
kpeter@623:     /// just a shortcut of the following code.
alpar@345:     /// \code
kpeter@623:     ///   s.init(s);
kpeter@623:     ///   s.findFlow(t, k);
alpar@345:     ///   s.findPaths();
alpar@345:     /// \endcode
kpeter@623:     int run(const Node& s, const Node& t, int k = 2) {
kpeter@623:       init(s);
kpeter@623:       findFlow(t, k);
alpar@345:       findPaths();
alpar@345:       return _path_num;
alpar@345:     }
alpar@345: 
kpeter@346:     /// \brief Initialize the algorithm.
alpar@345:     ///
kpeter@346:     /// This function initializes the algorithm.
kpeter@623:     ///
kpeter@623:     /// \param s The source node.
kpeter@623:     void init(const Node& s) {
kpeter@853:       _s = s;
kpeter@623: 
kpeter@346:       // Initialize maps
alpar@345:       if (!_flow) {
alpar@345:         _flow = new FlowMap(_graph);
alpar@345:         _local_flow = true;
alpar@345:       }
alpar@345:       if (!_potential) {
alpar@345:         _potential = new PotentialMap(_graph);
alpar@345:         _local_potential = true;
alpar@345:       }
alpar@345:       for (ArcIt e(_graph); e != INVALID; ++e) (*_flow)[e] = 0;
alpar@345:       for (NodeIt n(_graph); n != INVALID; ++n) (*_potential)[n] = 0;
alpar@345:     }
alpar@345: 
kpeter@623:     /// \brief Execute the algorithm to find an optimal flow.
alpar@345:     ///
kpeter@346:     /// This function executes the successive shortest path algorithm to
kpeter@623:     /// find a minimum cost flow, which is the union of \c k (or less)
alpar@345:     /// arc-disjoint paths.
alpar@345:     ///
kpeter@623:     /// \param t The target node.
kpeter@623:     /// \param k The number of paths to be found.
kpeter@623:     ///
kpeter@346:     /// \return \c k if there are at least \c k arc-disjoint paths from
kpeter@623:     /// the source node to the given node \c t in the digraph.
kpeter@623:     /// Otherwise it returns the number of arc-disjoint paths found.
alpar@345:     ///
alpar@345:     /// \pre \ref init() must be called before using this function.
kpeter@623:     int findFlow(const Node& t, int k = 2) {
kpeter@853:       _t = t;
kpeter@853:       ResidualDijkstra dijkstra(*this);
kpeter@623: 
kpeter@346:       // Find shortest paths
alpar@345:       _path_num = 0;
alpar@345:       while (_path_num < k) {
kpeter@346:         // Run Dijkstra
kpeter@853:         if (!dijkstra.run(_path_num)) break;
alpar@345:         ++_path_num;
alpar@345: 
kpeter@346:         // Set the flow along the found shortest path
kpeter@853:         Node u = _t;
alpar@345:         Arc e;
alpar@345:         while ((e = _pred[u]) != INVALID) {
alpar@345:           if (u == _graph.target(e)) {
alpar@345:             (*_flow)[e] = 1;
alpar@345:             u = _graph.source(e);
alpar@345:           } else {
alpar@345:             (*_flow)[e] = 0;
alpar@345:             u = _graph.target(e);
alpar@345:           }
alpar@345:         }
alpar@345:       }
alpar@345:       return _path_num;
alpar@345:     }
alpar@440: 
kpeter@346:     /// \brief Compute the paths from the flow.
alpar@345:     ///
kpeter@853:     /// This function computes arc-disjoint paths from the found minimum
kpeter@853:     /// cost flow, which is the union of them.
alpar@345:     ///
alpar@345:     /// \pre \ref init() and \ref findFlow() must be called before using
alpar@345:     /// this function.
alpar@345:     void findPaths() {
alpar@345:       FlowMap res_flow(_graph);
kpeter@346:       for(ArcIt a(_graph); a != INVALID; ++a) res_flow[a] = (*_flow)[a];
alpar@345: 
kpeter@853:       _paths.clear();
kpeter@853:       _paths.resize(_path_num);
alpar@345:       for (int i = 0; i < _path_num; ++i) {
kpeter@853:         Node n = _s;
kpeter@853:         while (n != _t) {
alpar@345:           OutArcIt e(_graph, n);
alpar@345:           for ( ; res_flow[e] == 0; ++e) ;
alpar@345:           n = _graph.target(e);
kpeter@853:           _paths[i].addBack(e);
alpar@345:           res_flow[e] = 0;
alpar@345:         }
alpar@345:       }
alpar@345:     }
alpar@345: 
alpar@345:     /// @}
alpar@345: 
alpar@345:     /// \name Query Functions
kpeter@346:     /// The results of the algorithm can be obtained using these
alpar@345:     /// functions.
alpar@345:     /// \n The algorithm should be executed before using them.
alpar@345: 
alpar@345:     /// @{
alpar@345: 
kpeter@623:     /// \brief Return the total length of the found paths.
kpeter@623:     ///
kpeter@623:     /// This function returns the total length of the found paths, i.e.
kpeter@623:     /// the total cost of the found flow.
kpeter@623:     /// The complexity of the function is O(e).
kpeter@623:     ///
kpeter@623:     /// \pre \ref run() or \ref findFlow() must be called before using
kpeter@623:     /// this function.
kpeter@623:     Length totalLength() const {
kpeter@623:       Length c = 0;
kpeter@623:       for (ArcIt e(_graph); e != INVALID; ++e)
kpeter@623:         c += (*_flow)[e] * _length[e];
kpeter@623:       return c;
kpeter@623:     }
kpeter@623: 
kpeter@623:     /// \brief Return the flow value on the given arc.
kpeter@623:     ///
kpeter@623:     /// This function returns the flow value on the given arc.
kpeter@623:     /// It is \c 1 if the arc is involved in one of the found arc-disjoint
kpeter@623:     /// paths, otherwise it is \c 0.
kpeter@623:     ///
kpeter@623:     /// \pre \ref run() or \ref findFlow() must be called before using
kpeter@623:     /// this function.
kpeter@623:     int flow(const Arc& arc) const {
kpeter@623:       return (*_flow)[arc];
kpeter@623:     }
kpeter@623: 
kpeter@623:     /// \brief Return a const reference to an arc map storing the
alpar@345:     /// found flow.
alpar@345:     ///
kpeter@623:     /// This function returns a const reference to an arc map storing
kpeter@346:     /// the flow that is the union of the found arc-disjoint paths.
alpar@345:     ///
kpeter@346:     /// \pre \ref run() or \ref findFlow() must be called before using
kpeter@346:     /// this function.
alpar@345:     const FlowMap& flowMap() const {
alpar@345:       return *_flow;
alpar@345:     }
alpar@345: 
kpeter@346:     /// \brief Return the potential of the given node.
alpar@345:     ///
kpeter@346:     /// This function returns the potential of the given node.
kpeter@623:     /// The node potentials provide the dual solution of the
kpeter@623:     /// underlying \ref min_cost_flow "minimum cost flow problem".
alpar@345:     ///
kpeter@346:     /// \pre \ref run() or \ref findFlow() must be called before using
kpeter@346:     /// this function.
alpar@345:     Length potential(const Node& node) const {
alpar@345:       return (*_potential)[node];
alpar@345:     }
alpar@345: 
kpeter@623:     /// \brief Return a const reference to a node map storing the
kpeter@623:     /// found potentials (the dual solution).
alpar@345:     ///
kpeter@623:     /// This function returns a const reference to a node map storing
kpeter@623:     /// the found potentials that provide the dual solution of the
kpeter@623:     /// underlying \ref min_cost_flow "minimum cost flow problem".
alpar@345:     ///
kpeter@346:     /// \pre \ref run() or \ref findFlow() must be called before using
kpeter@346:     /// this function.
kpeter@623:     const PotentialMap& potentialMap() const {
kpeter@623:       return *_potential;
alpar@345:     }
alpar@345: 
kpeter@346:     /// \brief Return the number of the found paths.
alpar@345:     ///
kpeter@346:     /// This function returns the number of the found paths.
alpar@345:     ///
kpeter@346:     /// \pre \ref run() or \ref findFlow() must be called before using
kpeter@346:     /// this function.
alpar@345:     int pathNum() const {
alpar@345:       return _path_num;
alpar@345:     }
alpar@345: 
kpeter@346:     /// \brief Return a const reference to the specified path.
alpar@345:     ///
kpeter@346:     /// This function returns a const reference to the specified path.
alpar@345:     ///
kpeter@623:     /// \param i The function returns the <tt>i</tt>-th path.
alpar@345:     /// \c i must be between \c 0 and <tt>%pathNum()-1</tt>.
alpar@345:     ///
kpeter@346:     /// \pre \ref run() or \ref findPaths() must be called before using
kpeter@346:     /// this function.
kpeter@851:     const Path& path(int i) const {
kpeter@853:       return _paths[i];
alpar@345:     }
alpar@345: 
alpar@345:     /// @}
alpar@345: 
alpar@345:   }; //class Suurballe
alpar@345: 
alpar@345:   ///@}
alpar@345: 
alpar@345: } //namespace lemon
alpar@345: 
alpar@345: #endif //LEMON_SUURBALLE_H