/* -*- C++ -*-

  * lemon/topology.h - Part of LEMON, a generic C++ optimization library

  *

  * Copyright (C) 2006 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_TOPOLOGY_H

 #define LEMON_TOPOLOGY_H

 #include <lemon/dfs.h>

 #include <lemon/bfs.h>

 #include <lemon/graph_utils.h>

 #include <lemon/graph_adaptor.h>

 #include <lemon/maps.h>

 #include <lemon/concept/graph.h>

 #include <lemon/concept/ugraph.h>

 #include <lemon/concept_check.h>

 #include <lemon/bin_heap.h>

 #include <lemon/linear_heap.h>

 #include <stack>

 #include <functional>

 /// \ingroup topology

 /// \file

 /// \brief Topology related algorithms

 ///

 /// Topology related algorithms

 namespace lemon {

   /// \ingroup topology

   ///

   /// \brief Check that the given undirected graph is connected.

   ///

   /// Check that the given undirected graph connected.

   /// \param graph The undirected graph.

   /// \return %True when there is path between any two nodes in the graph.

   /// \note By definition, the empty graph is connected.

   template <typename UGraph>

   bool connected(const UGraph& graph) {

     checkConcept<concept::UGraph, UGraph>();

     typedef typename UGraph::NodeIt NodeIt;

     if (NodeIt(graph) == INVALID) return true;

     Dfs<UGraph> dfs(graph);

     dfs.run(NodeIt(graph));

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

 	return false;

       }

     }

     return true;

   }

   /// \ingroup topology

   ///

   /// \brief Count the number of connected components of an undirected graph

   ///

   /// Count the number of connected components of an undirected graph

   ///

   /// \param graph The graph. It should be undirected.

   /// \return The number of components

   /// \note By definition, the empty graph consists

   /// of zero connected components.

   template <typename UGraph>

   int countConnectedComponents(const UGraph &graph) {

     checkConcept<concept::UGraph, UGraph>();

     typedef typename UGraph::Node Node;

     typedef typename UGraph::Edge Edge;

     typedef NullMap<Node, Edge> PredMap;

     typedef NullMap<Node, int> DistMap;

     int compNum = 0;

     typename Bfs<UGraph>::

       template DefPredMap<PredMap>::

       template DefDistMap<DistMap>::

       Create bfs(graph);

     PredMap predMap;

     bfs.predMap(predMap);

     DistMap distMap;

     bfs.distMap(distMap);

     bfs.init();

     for(typename UGraph::NodeIt n(graph); n != INVALID; ++n) {

       if (!bfs.reached(n)) {

 	bfs.addSource(n);

 	bfs.start();

 	++compNum;

       }

     }

     return compNum;

   }

   /// \ingroup topology

   ///

   /// \brief Find the connected components of an undirected graph

   ///

   /// Find the connected components of an undirected graph.

   ///

   /// \image html connected_components.png

   /// \image latex connected_components.eps "Connected components" width=\textwidth

   ///

   /// \param graph The graph. It should be undirected.

   /// \retval compMap A writable node map. The values will be set from 0 to

   /// the number of the connected components minus one. Each values of the map

   /// will be set exactly once, the values of a certain component will be

   /// set continuously.

   /// \return The number of components

   ///

   template <class UGraph, class NodeMap>

   int connectedComponents(const UGraph &graph, NodeMap &compMap) {

     checkConcept<concept::UGraph, UGraph>();

     typedef typename UGraph::Node Node;

     typedef typename UGraph::Edge Edge;

     checkConcept<concept::WriteMap<Node, int>, NodeMap>();

     typedef NullMap<Node, Edge> PredMap;

     typedef NullMap<Node, int> DistMap;

     int compNum = 0;

     typename Bfs<UGraph>::

       template DefPredMap<PredMap>::

       template DefDistMap<DistMap>::

       Create bfs(graph);

     PredMap predMap;

     bfs.predMap(predMap);

     DistMap distMap;

     bfs.distMap(distMap);

     bfs.init();

     for(typename UGraph::NodeIt n(graph); n != INVALID; ++n) {

       if(!bfs.reached(n)) {

 	bfs.addSource(n);

 	while (!bfs.emptyQueue()) {

 	  compMap.set(bfs.nextNode(), compNum);

 	  bfs.processNextNode();

 	}

 	++compNum;

       }

     }

     return compNum;

   }

   namespace _topology_bits {

     template <typename Graph, typename Iterator >

     struct LeaveOrderVisitor : public DfsVisitor<Graph> {

     public:

       typedef typename Graph::Node Node;

       LeaveOrderVisitor(Iterator it) : _it(it) {}

       void leave(const Node& node) {

 	*(_it++) = node;

       }

     private:

       Iterator _it;

     };

     template <typename Graph, typename Map>

     struct FillMapVisitor : public DfsVisitor<Graph> {

     public:

       typedef typename Graph::Node Node;

       typedef typename Map::Value Value;

       FillMapVisitor(Map& map, Value& value) 

 	: _map(map), _value(value) {}

       void reach(const Node& node) {

 	_map.set(node, _value);

       }

     private:

       Map& _map;

       Value& _value;

     };

     template <typename Graph, typename EdgeMap>

     struct StronglyConnectedCutEdgesVisitor : public DfsVisitor<Graph> {

     public:

       typedef typename Graph::Node Node;

       typedef typename Graph::Edge Edge;

       StronglyConnectedCutEdgesVisitor(const Graph& graph, EdgeMap& cutMap, 

 				       int& cutNum) 

 	: _graph(graph), _cutMap(cutMap), _cutNum(cutNum), 

 	  _compMap(graph), _num(0) {

       }

       void stop(const Node&) {

 	++_num;

       }

       void reach(const Node& node) {

 	_compMap.set(node, _num);

       }

       void examine(const Edge& edge) {

  	if (_compMap[_graph.source(edge)] != _compMap[_graph.target(edge)]) {

  	  _cutMap.set(edge, true);

  	  ++_cutNum;

  	}

       }

     private:

       const Graph& _graph;

       EdgeMap& _cutMap;

       int& _cutNum;

       typename Graph::template NodeMap<int> _compMap;

       int _num;

     };

   }

   /// \ingroup topology

   ///

   /// \brief Check that the given directed graph is strongly connected.

   ///

   /// Check that the given directed graph is strongly connected. The

   /// graph is strongly connected when any two nodes of the graph are

   /// connected with directed paths in both direction.

   /// \return %False when the graph is not strongly connected.

   /// \see connected

   ///

   /// \note By definition, the empty graph is strongly connected.

   template <typename Graph>

   bool stronglyConnected(const Graph& graph) {

     checkConcept<concept::StaticGraph, Graph>();

     if (NodeIt(graph) == INVALID) return true;

     typedef typename Graph::Node Node;

     typedef typename Graph::NodeIt NodeIt;

     using namespace _topology_bits;

     typedef DfsVisitor<Graph> Visitor;

     Visitor visitor;

     DfsVisit<Graph, Visitor> dfs(graph, visitor);

     dfs.init();

     dfs.addSource(NodeIt(graph));

     dfs.start();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

 	return false;

       }

     }

     typedef RevGraphAdaptor<const Graph> RGraph;

     RGraph rgraph(graph);

     typedef DfsVisitor<Graph> RVisitor;

     RVisitor rvisitor;

     DfsVisit<RGraph, RVisitor> rdfs(rgraph, rvisitor);

     rdfs.init();    

     rdfs.addSource(NodeIt(graph));

     rdfs.start();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!rdfs.reached(it)) {

 	return false;

       }

     }

     return true;

   }

   /// \ingroup topology

   ///

   /// \brief Count the strongly connected components of a directed graph

   ///

   /// Count the strongly connected components of a directed graph.

   /// The strongly connected components are the classes of an equivalence

   /// relation on the nodes of the graph. Two nodes are connected with

   /// directed paths in both direction.

   ///

   /// \param graph The graph.

   /// \return The number of components

   /// \note By definition, the empty graph has zero

   /// strongly connected components.

   template <typename Graph>

   int countStronglyConnectedComponents(const Graph& graph) {

     checkConcept<concept::StaticGraph, Graph>();

     using namespace _topology_bits;

     typedef typename Graph::Node Node;

     typedef typename Graph::Edge Edge;

     typedef typename Graph::NodeIt NodeIt;

     typedef typename Graph::EdgeIt EdgeIt;

     typedef std::vector<Node> Container;

     typedef typename Container::iterator Iterator;

     Container nodes(countNodes(graph));

     typedef LeaveOrderVisitor<Graph, Iterator> Visitor;

     Visitor visitor(nodes.begin());

     DfsVisit<Graph, Visitor> dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

 	dfs.addSource(it);

 	dfs.start();

       }

     }

     typedef typename Container::reverse_iterator RIterator;

     typedef RevGraphAdaptor<const Graph> RGraph;

     RGraph rgraph(graph);

     typedef DfsVisitor<Graph> RVisitor;

     RVisitor rvisitor;

     DfsVisit<RGraph, RVisitor> rdfs(rgraph, rvisitor);

     int compNum = 0;

     rdfs.init();

     for (RIterator it = nodes.rbegin(); it != nodes.rend(); ++it) {

       if (!rdfs.reached(*it)) {

 	rdfs.addSource(*it);

 	rdfs.start();

 	++compNum;

       }

     }

     return compNum;

   }

   /// \ingroup topology

   ///

   /// \brief Find the strongly connected components of a directed graph

   ///

   /// Find the strongly connected components of a directed graph.

   /// The strongly connected components are the classes of an equivalence

   /// relation on the nodes of the graph. Two nodes are in relationship

   /// when there are directed paths between them in both direction.

   ///

   /// \image html strongly_connected_components.png

   /// \image latex strongly_connected_components.eps "Strongly connected components" width=\textwidth

   ///

   /// \param graph The graph.

   /// \retval compMap A writable node map. The values will be set from 0 to

   /// the number of the strongly connected components minus one. Each values 

   /// of the map will be set exactly once, the values of a certain component 

   /// will be set continuously.

   /// \return The number of components

   ///

   template <typename Graph, typename NodeMap>

   int stronglyConnectedComponents(const Graph& graph, NodeMap& compMap) {

     checkConcept<concept::StaticGraph, Graph>();

     typedef typename Graph::Node Node;

     typedef typename Graph::NodeIt NodeIt;

     checkConcept<concept::WriteMap<Node, int>, NodeMap>();

     using namespace _topology_bits;

     typedef std::vector<Node> Container;

     typedef typename Container::iterator Iterator;

     Container nodes(countNodes(graph));

     typedef LeaveOrderVisitor<Graph, Iterator> Visitor;

     Visitor visitor(nodes.begin());

     DfsVisit<Graph, Visitor> dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

 	dfs.addSource(it);

 	dfs.start();

       }

     }

     typedef typename Container::reverse_iterator RIterator;

     typedef RevGraphAdaptor<const Graph> RGraph;

     RGraph rgraph(graph);

     int compNum = 0;

     typedef FillMapVisitor<RGraph, NodeMap> RVisitor;

     RVisitor rvisitor(compMap, compNum);

     DfsVisit<RGraph, RVisitor> rdfs(rgraph, rvisitor);

     rdfs.init();

     for (RIterator it = nodes.rbegin(); it != nodes.rend(); ++it) {

       if (!rdfs.reached(*it)) {

 	rdfs.addSource(*it);

 	rdfs.start();

 	++compNum;

       }

     }

     return compNum;

   }

   /// \ingroup topology

   ///

   /// \brief Find the cut edges of the strongly connected components.

   ///

   /// Find the cut edges of the strongly connected components.

   /// The strongly connected components are the classes of an equivalence

   /// relation on the nodes of the graph. Two nodes are in relationship

   /// when there are directed paths between them in both direction.

   /// The strongly connected components are separated by the cut edges.

   ///

   /// \param graph The graph.

   /// \retval cutMap A writable node map. The values will be set true when the

   /// edge is a cut edge.

   ///

   /// \return The number of cut edges

   template <typename Graph, typename EdgeMap>

   int stronglyConnectedCutEdges(const Graph& graph, EdgeMap& cutMap) {

     checkConcept<concept::StaticGraph, Graph>();

     typedef typename Graph::Node Node;

     typedef typename Graph::Edge Edge;

     typedef typename Graph::NodeIt NodeIt;

     checkConcept<concept::WriteMap<Edge, bool>, EdgeMap>();

     using namespace _topology_bits;

     typedef std::vector<Node> Container;

     typedef typename Container::iterator Iterator;

     Container nodes(countNodes(graph));

     typedef LeaveOrderVisitor<Graph, Iterator> Visitor;

     Visitor visitor(nodes.begin());

     DfsVisit<Graph, Visitor> dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

 	dfs.addSource(it);

 	dfs.start();

       }

     }

     typedef typename Container::reverse_iterator RIterator;

     typedef RevGraphAdaptor<const Graph> RGraph;

     RGraph rgraph(graph);

     int cutNum = 0;

     typedef StronglyConnectedCutEdgesVisitor<RGraph, EdgeMap> RVisitor;

     RVisitor rvisitor(rgraph, cutMap, cutNum);

     DfsVisit<RGraph, RVisitor> rdfs(rgraph, rvisitor);

     rdfs.init();

     for (RIterator it = nodes.rbegin(); it != nodes.rend(); ++it) {

       if (!rdfs.reached(*it)) {

 	rdfs.addSource(*it);

 	rdfs.start();

       }

     }

     return cutNum;

   }

   namespace _topology_bits {

     template <typename Graph>

     class CountBiNodeConnectedComponentsVisitor : public DfsVisitor<Graph> {

     public:

       typedef typename Graph::Node Node;

       typedef typename Graph::Edge Edge;

       typedef typename Graph::UEdge UEdge;

       CountBiNodeConnectedComponentsVisitor(const Graph& graph, int &compNum) 

 	: _graph(graph), _compNum(compNum), 

 	  _numMap(graph), _retMap(graph), _predMap(graph), _num(0) {}

       void start(const Node& node) {

 	_predMap.set(node, INVALID);

       }

       void reach(const Node& node) {

 	_numMap.set(node, _num);

 	_retMap.set(node, _num);

 	++_num;

       }

       void discover(const Edge& edge) {

 	_predMap.set(_graph.target(edge), _graph.source(edge));

       }

       void examine(const Edge& edge) {

 	if (_graph.source(edge) == _graph.target(edge) && 

 	    _graph.direction(edge)) {

 	  ++_compNum;

 	  return;

 	}

 	if (_predMap[_graph.source(edge)] == _graph.target(edge)) {

 	  return;

 	}

 	if (_retMap[_graph.source(edge)] > _numMap[_graph.target(edge)]) {

 	  _retMap.set(_graph.source(edge), _numMap[_graph.target(edge)]);

 	}

       }

       void backtrack(const Edge& edge) {

 	if (_retMap[_graph.source(edge)] > _retMap[_graph.target(edge)]) {

 	  _retMap.set(_graph.source(edge), _retMap[_graph.target(edge)]);

 	}  

 	if (_numMap[_graph.source(edge)] <= _retMap[_graph.target(edge)]) {

 	  ++_compNum;

 	}

       }

     private:

       const Graph& _graph;

       int& _compNum; 

       typename Graph::template NodeMap<int> _numMap;

       typename Graph::template NodeMap<int> _retMap;

       typename Graph::template NodeMap<Node> _predMap;

       int _num;

     };

     template <typename Graph, typename EdgeMap>

     class BiNodeConnectedComponentsVisitor : public DfsVisitor<Graph> {

     public:

       typedef typename Graph::Node Node;

       typedef typename Graph::Edge Edge;

       typedef typename Graph::UEdge UEdge;

       BiNodeConnectedComponentsVisitor(const Graph& graph, 

 				       EdgeMap& compMap, int &compNum) 

 	: _graph(graph), _compMap(compMap), _compNum(compNum), 

 	  _numMap(graph), _retMap(graph), _predMap(graph), _num(0) {}

       void start(const Node& node) {

 	_predMap.set(node, INVALID);

       }

       void reach(const Node& node) {

 	_numMap.set(node, _num);

 	_retMap.set(node, _num);

 	++_num;

       }

       void discover(const Edge& edge) {

 	Node target = _graph.target(edge);

 	_predMap.set(target, edge);

 	_edgeStack.push(edge);

       }

       void examine(const Edge& edge) {

 	Node source = _graph.source(edge);

 	Node target = _graph.target(edge);

 	if (source == target && _graph.direction(edge)) {

 	  _compMap.set(edge, _compNum);

 	  ++_compNum;

 	  return;

 	}

 	if (_numMap[target] < _numMap[source]) {

 	  if (_predMap[source] != _graph.oppositeEdge(edge)) {

 	    _edgeStack.push(edge);

 	  }

 	}

 	if (_predMap[source] != INVALID && 

 	    target == _graph.source(_predMap[source])) {

 	  return;

 	}

 	if (_retMap[source] > _numMap[target]) {

 	  _retMap.set(source, _numMap[target]);

 	}

       }

       void backtrack(const Edge& edge) {

 	Node source = _graph.source(edge);

 	Node target = _graph.target(edge);

 	if (_retMap[source] > _retMap[target]) {

 	  _retMap.set(source, _retMap[target]);

 	}  

 	if (_numMap[source] <= _retMap[target]) {

 	  while (_edgeStack.top() != edge) {

 	    _compMap.set(_edgeStack.top(), _compNum);

 	    _edgeStack.pop();

 	  }

 	  _compMap.set(edge, _compNum);

 	  _edgeStack.pop();

 	  ++_compNum;

 	}

       }

     private:

       const Graph& _graph;

       EdgeMap& _compMap;

       int& _compNum; 

       typename Graph::template NodeMap<int> _numMap;

       typename Graph::template NodeMap<int> _retMap;

       typename Graph::template NodeMap<Edge> _predMap;

       std::stack<UEdge> _edgeStack;

       int _num;

     };

     template <typename Graph, typename NodeMap>

     class BiNodeConnectedCutNodesVisitor : public DfsVisitor<Graph> {

     public:

       typedef typename Graph::Node Node;

       typedef typename Graph::Edge Edge;

       typedef typename Graph::UEdge UEdge;

       BiNodeConnectedCutNodesVisitor(const Graph& graph, NodeMap& cutMap,

 				     int& cutNum) 

 	: _graph(graph), _cutMap(cutMap), _cutNum(cutNum),

 	  _numMap(graph), _retMap(graph), _predMap(graph), _num(0) {}

       void start(const Node& node) {

 	_predMap.set(node, INVALID);

 	rootCut = false;

       }

       void reach(const Node& node) {

 	_numMap.set(node, _num);

 	_retMap.set(node, _num);

 	++_num;

       }

       void discover(const Edge& edge) {

 	_predMap.set(_graph.target(edge), _graph.source(edge));

       }

       void examine(const Edge& edge) {

 	if (_graph.source(edge) == _graph.target(edge) && 

 	    _graph.direction(edge)) {

 	  if (!_cutMap[_graph.source(edge)]) {

 	    _cutMap.set(_graph.source(edge), true);

 	    ++_cutNum;

 	  }

 	  return;

 	}

 	if (_predMap[_graph.source(edge)] == _graph.target(edge)) return;

 	if (_retMap[_graph.source(edge)] > _numMap[_graph.target(edge)]) {

 	  _retMap.set(_graph.source(edge), _numMap[_graph.target(edge)]);

 	}

       }

       void backtrack(const Edge& edge) {

 	if (_retMap[_graph.source(edge)] > _retMap[_graph.target(edge)]) {

 	  _retMap.set(_graph.source(edge), _retMap[_graph.target(edge)]);

 	}  

 	if (_numMap[_graph.source(edge)] <= _retMap[_graph.target(edge)]) {

 	  if (_predMap[_graph.source(edge)] != INVALID) {

 	    if (!_cutMap[_graph.source(edge)]) {

 	      _cutMap.set(_graph.source(edge), true);

 	      ++_cutNum;

 	    }

 	  } else if (rootCut) {

 	    if (!_cutMap[_graph.source(edge)]) {

 	      _cutMap.set(_graph.source(edge), true);

 	      ++_cutNum;

 	    }

 	  } else {

 	    rootCut = true;

 	  }

 	}

       }

     private:

       const Graph& _graph;

       NodeMap& _cutMap;

       int& _cutNum; 

       typename Graph::template NodeMap<int> _numMap;

       typename Graph::template NodeMap<int> _retMap;

       typename Graph::template NodeMap<Node> _predMap;

       std::stack<UEdge> _edgeStack;

       int _num;

       bool rootCut;

     };

   }

   template <typename UGraph>

   int countBiNodeConnectedComponents(const UGraph& graph);

   /// \ingroup topology

   ///

   /// \brief Checks the graph is bi-node-connected.

   ///

   /// This function checks that the undirected graph is bi-node-connected  

   /// graph. The graph is bi-node-connected if any two undirected edge is 

   /// on same circle.

   ///

   /// \param graph The graph.

   /// \return %True when the graph bi-node-connected.

   /// \todo Make it faster.

   template <typename UGraph>

   bool biNodeConnected(const UGraph& graph) {

     return countBiNodeConnectedComponents(graph) == 1;

   }

   /// \ingroup topology

   ///

   /// \brief Count the biconnected components.

   ///

   /// This function finds the bi-node-connected components in an undirected 

   /// graph. The biconnected components are the classes of an equivalence 

   /// relation on the undirected edges. Two undirected edge is in relationship

   /// when they are on same circle.

   ///

   /// \param graph The graph.

   /// \return The number of components.

   template <typename UGraph>

   int countBiNodeConnectedComponents(const UGraph& graph) {

     checkConcept<concept::UGraph, UGraph>();

     typedef typename UGraph::NodeIt NodeIt;

     using namespace _topology_bits;

     typedef CountBiNodeConnectedComponentsVisitor<UGraph> Visitor;

     int compNum = 0;

     Visitor visitor(graph, compNum);

     DfsVisit<UGraph, Visitor> dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

 	dfs.addSource(it);

 	dfs.start();

       }

     }

     return compNum;

   }

   /// \ingroup topology

   ///

   /// \brief Find the bi-node-connected components.

   ///

   /// This function finds the bi-node-connected components in an undirected 

   /// graph. The bi-node-connected components are the classes of an equivalence

   /// relation on the undirected edges. Two undirected edge are in relationship

   /// when they are on same circle.

   ///

   /// \image html node_biconnected_components.png

   /// \image latex node_biconnected_components.eps "bi-node-connected components" width=\textwidth

   ///

   /// \param graph The graph.

   /// \retval compMap A writable uedge map. The values will be set from 0

   /// to the number of the biconnected components minus one. Each values 

   /// of the map will be set exactly once, the values of a certain component 

   /// will be set continuously.

   /// \return The number of components.

   ///

   template <typename UGraph, typename UEdgeMap>

   int biNodeConnectedComponents(const UGraph& graph, 

 				UEdgeMap& compMap) {

     checkConcept<concept::UGraph, UGraph>();

     typedef typename UGraph::NodeIt NodeIt;

     typedef typename UGraph::UEdge UEdge;

     checkConcept<concept::WriteMap<UEdge, int>, UEdgeMap>();

     using namespace _topology_bits;

     typedef BiNodeConnectedComponentsVisitor<UGraph, UEdgeMap> Visitor;

     int compNum = 0;

     Visitor visitor(graph, compMap, compNum);

     DfsVisit<UGraph, Visitor> dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

 	dfs.addSource(it);

 	dfs.start();

       }

     }

     return compNum;

   }

   /// \ingroup topology

   ///

   /// \brief Find the bi-node-connected cut nodes.

   ///

   /// This function finds the bi-node-connected cut nodes in an undirected 

   /// graph. The bi-node-connected components are the classes of an equivalence

   /// relation on the undirected edges. Two undirected edges are in 

   /// relationship when they are on same circle. The biconnected components 

   /// are separted by nodes which are the cut nodes of the components.

   ///

   /// \param graph The graph.

   /// \retval cutMap A writable edge map. The values will be set true when

   /// the node separate two or more components.

   /// \return The number of the cut nodes.

   template <typename UGraph, typename NodeMap>

   int biNodeConnectedCutNodes(const UGraph& graph, NodeMap& cutMap) {

     checkConcept<concept::UGraph, UGraph>();

     typedef typename UGraph::Node Node;

     typedef typename UGraph::NodeIt NodeIt;

     checkConcept<concept::WriteMap<Node, bool>, NodeMap>();

     using namespace _topology_bits;

     typedef BiNodeConnectedCutNodesVisitor<UGraph, NodeMap> Visitor;

     int cutNum = 0;

     Visitor visitor(graph, cutMap, cutNum);

     DfsVisit<UGraph, Visitor> dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

 	dfs.addSource(it);

 	dfs.start();

       }

     }

     return cutNum;

   }

   namespace _topology_bits {

     template <typename Graph>

     class CountBiEdgeConnectedComponentsVisitor : public DfsVisitor<Graph> {

     public:

       typedef typename Graph::Node Node;

       typedef typename Graph::Edge Edge;

       typedef typename Graph::UEdge UEdge;

       CountBiEdgeConnectedComponentsVisitor(const Graph& graph, int &compNum) 

 	: _graph(graph), _compNum(compNum), 

 	  _numMap(graph), _retMap(graph), _predMap(graph), _num(0) {}

       void start(const Node& node) {

 	_predMap.set(node, INVALID);

       }

       void reach(const Node& node) {

 	_numMap.set(node, _num);

 	_retMap.set(node, _num);

 	++_num;

       }

       void leave(const Node& node) {

 	if (_numMap[node] <= _retMap[node]) {

 	  ++_compNum;

 	}	

       }

       void discover(const Edge& edge) {

 	_predMap.set(_graph.target(edge), edge);

       }

       void examine(const Edge& edge) {

 	if (_predMap[_graph.source(edge)] == _graph.oppositeEdge(edge)) {

 	  return;

 	}

 	if (_retMap[_graph.source(edge)] > _retMap[_graph.target(edge)]) {

 	  _retMap.set(_graph.source(edge), _retMap[_graph.target(edge)]);

 	}

       }

       void backtrack(const Edge& edge) {

 	if (_retMap[_graph.source(edge)] > _retMap[_graph.target(edge)]) {

 	  _retMap.set(_graph.source(edge), _retMap[_graph.target(edge)]);

 	}  

       }

     private:

       const Graph& _graph;

       int& _compNum; 

       typename Graph::template NodeMap<int> _numMap;

       typename Graph::template NodeMap<int> _retMap;

       typename Graph::template NodeMap<Edge> _predMap;

       int _num;

     };

     template <typename Graph, typename NodeMap>

     class BiEdgeConnectedComponentsVisitor : public DfsVisitor<Graph> {

     public:

       typedef typename Graph::Node Node;

       typedef typename Graph::Edge Edge;

       typedef typename Graph::UEdge UEdge;

       BiEdgeConnectedComponentsVisitor(const Graph& graph, 

 				       NodeMap& compMap, int &compNum) 

 	: _graph(graph), _compMap(compMap), _compNum(compNum), 

 	  _numMap(graph), _retMap(graph), _predMap(graph), _num(0) {}

       void start(const Node& node) {

 	_predMap.set(node, INVALID);

       }

       void reach(const Node& node) {

 	_numMap.set(node, _num);

 	_retMap.set(node, _num);

 	_nodeStack.push(node);

 	++_num;

       }

       void leave(const Node& node) {

 	if (_numMap[node] <= _retMap[node]) {

 	  while (_nodeStack.top() != node) {

 	    _compMap.set(_nodeStack.top(), _compNum);

 	    _nodeStack.pop();

 	  }

 	  _compMap.set(node, _compNum);

 	  _nodeStack.pop();

 	  ++_compNum;

 	}	

       }

       void discover(const Edge& edge) {

 	_predMap.set(_graph.target(edge), edge);

       }

       void examine(const Edge& edge) {

 	if (_predMap[_graph.source(edge)] == _graph.oppositeEdge(edge)) {

 	  return;

 	}

 	if (_retMap[_graph.source(edge)] > _retMap[_graph.target(edge)]) {

 	  _retMap.set(_graph.source(edge), _retMap[_graph.target(edge)]);

 	}

       }

       void backtrack(const Edge& edge) {

 	if (_retMap[_graph.source(edge)] > _retMap[_graph.target(edge)]) {

 	  _retMap.set(_graph.source(edge), _retMap[_graph.target(edge)]);

 	}  

       }

     private:

       const Graph& _graph;

       NodeMap& _compMap;

       int& _compNum; 

       typename Graph::template NodeMap<int> _numMap;

       typename Graph::template NodeMap<int> _retMap;

       typename Graph::template NodeMap<Edge> _predMap;

       std::stack<Node> _nodeStack;

       int _num;

     };

     template <typename Graph, typename EdgeMap>

     class BiEdgeConnectedCutEdgesVisitor : public DfsVisitor<Graph> {

     public:

       typedef typename Graph::Node Node;

       typedef typename Graph::Edge Edge;

       typedef typename Graph::UEdge UEdge;

       BiEdgeConnectedCutEdgesVisitor(const Graph& graph, 

 				     EdgeMap& cutMap, int &cutNum) 

 	: _graph(graph), _cutMap(cutMap), _cutNum(cutNum), 

 	  _numMap(graph), _retMap(graph), _predMap(graph), _num(0) {}

       void start(const Node& node) {

 	_predMap[node] = INVALID;

       }

       void reach(const Node& node) {

 	_numMap.set(node, _num);

 	_retMap.set(node, _num);

 	++_num;

       }

       void leave(const Node& node) {

 	if (_numMap[node] <= _retMap[node]) {

 	  if (_predMap[node] != INVALID) {

 	    _cutMap.set(_predMap[node], true);

 	    ++_cutNum;

 	  }

 	}	

       }

       void discover(const Edge& edge) {

 	_predMap.set(_graph.target(edge), edge);

       }

       void examine(const Edge& edge) {

 	if (_predMap[_graph.source(edge)] == _graph.oppositeEdge(edge)) {

 	  return;

 	}

 	if (_retMap[_graph.source(edge)] > _retMap[_graph.target(edge)]) {

 	  _retMap.set(_graph.source(edge), _retMap[_graph.target(edge)]);

 	}

       }

       void backtrack(const Edge& edge) {

 	if (_retMap[_graph.source(edge)] > _retMap[_graph.target(edge)]) {

 	  _retMap.set(_graph.source(edge), _retMap[_graph.target(edge)]);

 	}  

       }

     private:

       const Graph& _graph;

       EdgeMap& _cutMap;

       int& _cutNum; 

       typename Graph::template NodeMap<int> _numMap;

       typename Graph::template NodeMap<int> _retMap;

       typename Graph::template NodeMap<Edge> _predMap;

       int _num;

     };

   }

   template <typename UGraph>

   int countbiEdgeConnectedComponents(const UGraph& graph);

   /// \ingroup topology

   ///

   /// \brief Checks that the graph is bi-edge-connected.

   ///

   /// This function checks that the graph is bi-edge-connected. The undirected

   /// graph is bi-edge-connected when any two nodes are connected with two

   /// edge-disjoint paths.

   ///

   /// \param graph The undirected graph.

   /// \return The number of components.

   /// \todo Make it faster.

   template <typename UGraph>

   bool biEdgeConnected(const UGraph& graph) { 

     return countBiEdgeConnectedComponents(graph) == 1;

   }

   /// \ingroup topology

   ///

   /// \brief Count the bi-edge-connected components.

   ///

   /// This function count the bi-edge-connected components in an undirected 

   /// graph. The bi-edge-connected components are the classes of an equivalence

   /// relation on the nodes. Two nodes are in relationship when they are  

   /// connected with at least two edge-disjoint paths.

   ///

   /// \param graph The undirected graph.

   /// \return The number of components.

   template <typename UGraph>

   int countBiEdgeConnectedComponents(const UGraph& graph) { 

     checkConcept<concept::UGraph, UGraph>();

     typedef typename UGraph::NodeIt NodeIt;

     using namespace _topology_bits;

     typedef CountBiEdgeConnectedComponentsVisitor<UGraph> Visitor;

     int compNum = 0;

     Visitor visitor(graph, compNum);

     DfsVisit<UGraph, Visitor> dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

 	dfs.addSource(it);

 	dfs.start();

       }

     }

     return compNum;

   }

   /// \ingroup topology

   ///

   /// \brief Find the bi-edge-connected components.

   ///

   /// This function finds the bi-edge-connected components in an undirected 

   /// graph. The bi-edge-connected components are the classes of an equivalence

   /// relation on the nodes. Two nodes are in relationship when they are  

   /// connected at least two edge-disjoint paths.

   ///

   /// \image html edge_biconnected_components.png

   /// \image latex edge_biconnected_components.eps "bi-edge-connected components" width=\textwidth

   ///

   /// \param graph The graph.

   /// \retval compMap A writable node map. The values will be set from 0 to

   /// the number of the biconnected components minus one. Each values 

   /// of the map will be set exactly once, the values of a certain component 

   /// will be set continuously.

   /// \return The number of components.

   ///

   template <typename UGraph, typename NodeMap>

   int biEdgeConnectedComponents(const UGraph& graph, NodeMap& compMap) { 

     checkConcept<concept::UGraph, UGraph>();

     typedef typename UGraph::NodeIt NodeIt;

     typedef typename UGraph::Node Node;

     checkConcept<concept::WriteMap<Node, int>, NodeMap>();

     using namespace _topology_bits;

     typedef BiEdgeConnectedComponentsVisitor<UGraph, NodeMap> Visitor;

     int compNum = 0;

     Visitor visitor(graph, compMap, compNum);

     DfsVisit<UGraph, Visitor> dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

 	dfs.addSource(it);

 	dfs.start();

       }

     }

     return compNum;

   }

   /// \ingroup topology

   ///

   /// \brief Find the bi-edge-connected cut edges.

   ///

   /// This function finds the bi-edge-connected components in an undirected 

   /// graph. The bi-edge-connected components are the classes of an equivalence

   /// relation on the nodes. Two nodes are in relationship when they are 

   /// connected with at least two edge-disjoint paths. The bi-edge-connected 

   /// components are separted by edges which are the cut edges of the 

   /// components.

   ///

   /// \param graph The graph.

   /// \retval cutMap A writable node map. The values will be set true when the

   /// edge is a cut edge.

   /// \return The number of cut edges.

   template <typename UGraph, typename UEdgeMap>

   int biEdgeConnectedCutEdges(const UGraph& graph, UEdgeMap& cutMap) { 

     checkConcept<concept::UGraph, UGraph>();

     typedef typename UGraph::NodeIt NodeIt;

     typedef typename UGraph::UEdge UEdge;

     checkConcept<concept::WriteMap<UEdge, bool>, UEdgeMap>();

     using namespace _topology_bits;

     typedef BiEdgeConnectedCutEdgesVisitor<UGraph, UEdgeMap> Visitor;

     int cutNum = 0;

     Visitor visitor(graph, cutMap, cutNum);

     DfsVisit<UGraph, Visitor> dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

 	dfs.addSource(it);

 	dfs.start();

       }

     }

     return cutNum;

   }

   namespace _topology_bits {

     template <typename Graph, typename IntNodeMap>

     class TopologicalSortVisitor : public DfsVisitor<Graph> {

     public:

       typedef typename Graph::Node Node;

       typedef typename Graph::Edge edge;

       TopologicalSortVisitor(IntNodeMap& order, int num) 

 	: _order(order), _num(num) {}

       void leave(const Node& node) {

 	_order.set(node, --_num);

       }

     private:

       IntNodeMap& _order;

       int _num;

     };

   }

   /// \ingroup topology

   ///

   /// \brief Sort the nodes of a DAG into topolgical order.

   ///

   /// Sort the nodes of a DAG into topolgical order.

   ///

   /// \param graph The graph. It should be directed and acyclic.

   /// \retval order A writable node map. The values will be set from 0 to

   /// the number of the nodes in the graph minus one. Each values of the map

   /// will be set exactly once, the values  will be set descending order.

   ///

   /// \see checkedTopologicalSort

   /// \see dag

   template <typename Graph, typename NodeMap>

   void topologicalSort(const Graph& graph, NodeMap& order) {

     using namespace _topology_bits;

     checkConcept<concept::StaticGraph, Graph>();

     checkConcept<concept::WriteMap<typename Graph::Node, int>, NodeMap>();

     typedef typename Graph::Node Node;

     typedef typename Graph::NodeIt NodeIt;

     typedef typename Graph::Edge Edge;

     TopologicalSortVisitor<Graph, NodeMap> 

       visitor(order, countNodes(graph)); 

     DfsVisit<Graph, TopologicalSortVisitor<Graph, NodeMap> >

       dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

 	dfs.addSource(it);

 	dfs.start();

       }

     }    

   }

   /// \ingroup topology

   ///

   /// \brief Sort the nodes of a DAG into topolgical order.

   ///

   /// Sort the nodes of a DAG into topolgical order. It also checks

   /// that the given graph is DAG.

   ///

   /// \param graph The graph. It should be directed and acyclic.

   /// \retval order A readable - writable node map. The values will be set 

   /// from 0 to the number of the nodes in the graph minus one. Each values 

   /// of the map will be set exactly once, the values will be set descending 

   /// order.

   /// \return %False when the graph is not DAG.

   ///

   /// \see topologicalSort

   /// \see dag

   template <typename Graph, typename NodeMap>

   bool checkedTopologicalSort(const Graph& graph, NodeMap& order) {

     using namespace _topology_bits;

     checkConcept<concept::StaticGraph, Graph>();

     checkConcept<concept::ReadWriteMap<typename Graph::Node, int>, NodeMap>();

     typedef typename Graph::Node Node;

     typedef typename Graph::NodeIt NodeIt;

     typedef typename Graph::Edge Edge;

     order = constMap<Node, int, -1>();

     TopologicalSortVisitor<Graph, NodeMap> 

       visitor(order, countNodes(graph)); 

     DfsVisit<Graph, TopologicalSortVisitor<Graph, NodeMap> >

       dfs(graph, visitor);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

 	dfs.addSource(it);

 	while (!dfs.emptyQueue()) {

  	  Edge edge = dfs.nextEdge();

  	  Node target = graph.target(edge);

  	  if (dfs.reached(target) && order[target] == -1) {

  	    return false;

  	  }

  	  dfs.processNextEdge();

  	}

       }

     }

     return true;

   }

   /// \ingroup topology

   ///

   /// \brief Check that the given directed graph is a DAG.

   ///

   /// Check that the given directed graph is a DAG. The DAG is

   /// an Directed Acyclic Graph.

   /// \return %False when the graph is not DAG.

   /// \see acyclic

   template <typename Graph>

   bool dag(const Graph& graph) {

     checkConcept<concept::StaticGraph, Graph>();

     typedef typename Graph::Node Node;

     typedef typename Graph::NodeIt NodeIt;

     typedef typename Graph::Edge Edge;

     typedef typename Graph::template NodeMap<bool> ProcessedMap;

     typename Dfs<Graph>::template DefProcessedMap<ProcessedMap>::

       Create dfs(graph);

     ProcessedMap processed(graph);

     dfs.processedMap(processed);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

 	dfs.addSource(it);

 	while (!dfs.emptyQueue()) {

 	  Edge edge = dfs.nextEdge();

 	  Node target = graph.target(edge);

 	  if (dfs.reached(target) && !processed[target]) {

 	    return false;

 	  }

 	  dfs.processNextEdge();

 	}

       }

     }    

     return true;

   }

   /// \ingroup topology

   ///

   /// \brief Check that the given undirected graph is acyclic.

   ///

   /// Check that the given undirected graph acyclic.

   /// \param graph The undirected graph.

   /// \return %True when there is no circle in the graph.

   /// \see dag

   template <typename UGraph>

   bool acyclic(const UGraph& graph) {

     checkConcept<concept::UGraph, UGraph>();

     typedef typename UGraph::Node Node;

     typedef typename UGraph::NodeIt NodeIt;

     typedef typename UGraph::Edge Edge;

     Dfs<UGraph> dfs(graph);

     dfs.init();

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

 	dfs.addSource(it);

 	while (!dfs.emptyQueue()) {

 	  Edge edge = dfs.nextEdge();

 	  Node source = graph.source(edge);

 	  Node target = graph.target(edge);

 	  if (dfs.reached(target) && 

 	      dfs.predEdge(source) != graph.oppositeEdge(edge)) {

 	    return false;

 	  }

 	  dfs.processNextEdge();

 	}

       }

     }

     return true;

   }

   /// \ingroup topology

   ///

   /// \brief Check that the given undirected graph is tree.

   ///

   /// Check that the given undirected graph is tree.

   /// \param graph The undirected graph.

   /// \return %True when the graph is acyclic and connected.

   template <typename UGraph>

   bool tree(const UGraph& graph) {

     checkConcept<concept::UGraph, UGraph>();

     typedef typename UGraph::Node Node;

     typedef typename UGraph::NodeIt NodeIt;

     typedef typename UGraph::Edge Edge;

     Dfs<UGraph> dfs(graph);

     dfs.init();

     dfs.addSource(NodeIt(graph));

     while (!dfs.emptyQueue()) {

       Edge edge = dfs.nextEdge();

       Node source = graph.source(edge);

       Node target = graph.target(edge);

       if (dfs.reached(target) && 

 	  dfs.predEdge(source) != graph.oppositeEdge(edge)) {

 	return false;

       }

       dfs.processNextEdge();

     }

     for (NodeIt it(graph); it != INVALID; ++it) {

       if (!dfs.reached(it)) {

 	return false;

       }

     }    

     return true;

   }

   /// \ingroup topology

   ///

   /// \brief Check if the given undirected graph is bipartite or not

   ///

   /// The function checks if the given undirected \c graph graph is bipartite 

   /// or not. The \ref Bfs algorithm is used to calculate the result.

   /// \param graph The undirected graph.

   /// \return %True if \c graph is bipartite, %false otherwise.

   /// \sa bipartitePartitions

   ///

   /// \author Balazs Attila Mihaly  

   template<typename UGraph>

   inline bool bipartite(const UGraph &graph){

     checkConcept<concept::UGraph, UGraph>();

     typedef typename UGraph::NodeIt NodeIt;

     typedef typename UGraph::EdgeIt EdgeIt;

     Bfs<UGraph> bfs(graph);

     bfs.init();

     for(NodeIt i(graph);i!=INVALID;++i){

       if(!bfs.reached(i)){

 	bfs.run(i);

       }

     }

     for(EdgeIt i(graph);i!=INVALID;++i){

       if(bfs.dist(graph.source(i))==bfs.dist(graph.target(i)))return false;

     }

     return true;

   };

   /// \ingroup topology

   ///

   /// \brief Check if the given undirected graph is bipartite or not

   ///

   /// The function checks if the given undirected graph is bipartite 

   /// or not. The  \ref  Bfs  algorithm  is   used  to  calculate the result. 

   /// During the execution, the \c partMap will be set as the two 

   /// partitions of the graph.

   /// \param graph The undirected graph.

   /// \retval partMap A writable bool map of nodes. It will be set as the

   /// two partitions of the graph. 

   /// \return %True if \c graph is bipartite, %false otherwise.

   ///

   /// \author Balazs Attila Mihaly  

   ///

   /// \image html bipartite_partitions.png

   /// \image latex bipartite_partitions.eps "Bipartite partititions" width=\textwidth

   template<typename UGraph, typename NodeMap>

   inline bool bipartitePartitions(const UGraph &graph, NodeMap &partMap){

     checkConcept<concept::UGraph, UGraph>();

     typedef typename UGraph::Node Node;

     typedef typename UGraph::NodeIt NodeIt;

     typedef typename UGraph::EdgeIt EdgeIt;

     Bfs<UGraph> bfs(graph);

     bfs.init();

     for(NodeIt i(graph);i!=INVALID;++i){

       if(!bfs.reached(i)){

 	bfs.addSource(i);

 	for(Node j=bfs.processNextNode();!bfs.emptyQueue();

 	    j=bfs.processNextNode()){

 	  partMap.set(j,bfs.dist(j)%2==0);

 	}

       }

     }

     for(EdgeIt i(graph);i!=INVALID;++i){

       if(bfs.dist(graph.source(i)) == bfs.dist(graph.target(i)))return false;

     }

     return true;

   };

 } //namespace lemon

 #endif //LEMON_TOPOLOGY_H