/* -*- 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.
 *
 */

namespace lemon {
/**
[PAGE]sec_undir_graphs[PAGE] Undirected Graphs

In \ref sec_basics, we have introduced a general digraph structure,
\ref ListDigraph. LEMON also contains undirected graph classes,
for example, \ref ListGraph is the undirected versions of \ref ListDigraph.

[SEC]sec_undir_graph_use[SEC] Working with Undirected Graphs

The \ref concepts::Graph "undirected graphs" also fulfill the concept of
\ref concepts::Digraph "directed graphs", in such a way that each
undirected \e edge of a graph can also be regarded as two oppositely
directed \e arcs. As a result, all directed graph algorithms automatically
run on undirected graphs, as well.

Undirected graphs provide an \c Edge type for the \e undirected \e edges
and an \c Arc type for the \e directed \e arcs. The \c Arc type is
convertible to \c Edge (or inherited from it), thus the corresponding
edge can always be obtained from an arc.
Of course, only nodes and edges can be added to or removed from an undirected
graph and the corresponding arcs are added or removed automatically
(there are twice as many arcs as edges)

For example,
\code
  ListGraph g;

  ListGraph::Node a = g.addNode();
  ListGraph::Node b = g.addNode();
  ListGraph::Node c = g.addNode();

  ListGraph::Edge e = g.addEdge(a,b);
  g.addEdge(b,c);
  g.addEdge(c,a);
\endcode

Each edge has an inherent orientation, thus it can be defined whether
an arc is forward or backward oriented in an undirected graph with respect
to this default orientation of the represented edge.
The direction of an arc can be obtained and set using the functions
\ref concepts::Graph::direction() "direction()" and
\ref concepts::Graph::direct() "direct()", respectively.

For example,
\code
  ListGraph::Arc a1 = g.direct(e, true);    // a1 is the forward arc
  ListGraph::Arc a2 = g.direct(e, false);   // a2 is the backward arc

  if (a2 == g.oppositeArc(a1))
    std::cout << "a2 is the opposite of a1" << std::endl;
\endcode

The end nodes of an edge can be obtained using the functions
\ref concepts::Graph::source() "u()" and
\ref concepts::Graph::target() "v()", while the
\ref concepts::Graph::source() "source()" and
\ref concepts::Graph::target() "target()" can be used for arcs.

\code
  std::cout << "Edge " << g.id(e) << " connects node "
    << g.id(g.u(e)) << " and node " << g.id(g.v(e)) << std::endl;

  std::cout << "Arc " << g.id(a2) << " goes from node "
    << g.id(g.source(a2)) << " to node " << g.id(g.target(a2)) << std::endl;
\endcode

Similarly to the digraphs, the undirected graphs also provide iterators
\ref concepts::Graph::NodeIt "NodeIt", \ref concepts::Graph::ArcIt "ArcIt",
\ref concepts::Graph::OutArcIt "OutArcIt" and \ref concepts::Graph::InArcIt
"InArcIt", which can be used the same way.
However, they also have iterator classes for edges.
\ref concepts::Graph::EdgeIt "EdgeIt" traverses all edges in the graph and
\ref concepts::Graph::IncEdgeIt "IncEdgeIt" lists the incident edges of a
certain node.

For example, the degree of each node can be printed out like this:

\code
  for (ListGraph::NodeIt n(g); n != INVALID; ++n) {
    int cnt = 0;
    for (ListGraph::IncEdgeIt e(g, n); e != INVALID; ++e) {
      cnt++;
    }
    std::cout << "deg(" << g.id(n) << ") = " << cnt << std::endl;
  }
\endcode

In an undirected graph, both \ref concepts::Graph::OutArcIt "OutArcIt"
and \ref concepts::Graph::InArcIt "InArcIt" iterates on the same \e edges
but with opposite direction. They are convertible to both \c Arc and
\c Edge types. \ref concepts::Graph::IncEdgeIt "IncEdgeIt" also iterates
on these edges, but it is not convertible to \c Arc, only to \c Edge.

Apart from the node and arc maps, an undirected graph also defines
a member class for constructing edge maps. These maps can be
used in conjunction with both edges and arcs.

For example,
\code
  ListGraph::EdgeMap cost(g);
  cost[e] = 10;
  std::cout << cost[e] << std::endl;
  std::cout << cost[a1] << ", " << cost[a2] << std::endl;

  ListGraph::ArcMap arc_cost(g);
  arc_cost[a1] = cost[a1];
  arc_cost[a2] = 2 * cost[a2];
  // std::cout << arc_cost[e] << std::endl;   // this is not valid
  std::cout << arc_cost[a1] << ", " << arc_cost[a2] << std::endl;
\endcode


[SEC]sec_undir_graph_algs[SEC] Undirected Graph Algorithms

\todo This subsection is under construction.

If you would like to design an electric network minimizing the total length
of wires, then you might be looking for a minimum spanning tree in an
undirected graph.
This can be found using the \ref kruskal() "Kruskal" algorithm.

Let us suppose that the network is stored in a \ref ListGraph object \c g
with a cost map \c cost. We create a \c bool valued edge map \c tree_map or
a vector \c tree_vector for storing the tree that is found by the algorithm.
After that, we could call the \ref kruskal() function. It gives back the weight
of the minimum spanning tree and \c tree_map or \c tree_vector
will contain the found spanning tree.

If you want to store the arcs in a \c bool valued edge map, then you can use
the function as follows.

\code
  // Kruskal algorithm with edge map
  ListGraph::EdgeMap<bool> tree_map(g);
  std::cout << "The weight of the minimum spanning tree is "
            << kruskal(g, cost_map, tree_map) << std::endl;

  // Print the results
  std::cout << "Edges of the tree: " << std::endl;
  for (ListGraph::EdgeIt e(g); e != INVALID; ++e) {
    if (!tree_map[e]) continue;
    std::cout << "(" << g.id(g.u(e)) << ", " << g.id(g.v(e)) << ")\n";
  }
\endcode

If you would like to store the edges in a standard container, you can
do it like this.

\code
  // Kruskal algorithm with edge vector
  std::vector<ListGraph::Edge> tree_vector;
  std::cout << "The weight of the minimum spanning tree is "
            << kruskal(g, cost_map, std::back_inserter(tree_vector))
            << std::endl;

  // Print the results
  std::cout << "Edges of the tree: " << std::endl;
  for (unsigned i = 0; i != tree_vector.size(); ++i) {
    Edge e = tree_vector[i];
    std::cout << "(" << g.id(g.u(e)) << ", " << g.id(g.v(e)) << ")\n";
  }
\endcode

\todo \ref matching "matching algorithms".

[TRAILER]
*/
}