/* -*- 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]

*/

}