\todo Clarify this section.

 In typical algorithms and applications related to graphs and networks,

 we usually encounter situations in which a specific alteration of a graph

 Alteration of standard containers need a very limited number of

 has to be considered.

 operations, these together satisfy the everyday requirements.

 If some nodes or arcs have to be hidden (maybe temporarily) or the reverse

 In the case of graph structures, different operations are needed which do

 oriented graph has to be used, then this is the case.

 not alter the physical graph, but gives another view. If some nodes or

 However, actually modifing physical storage of the graph or

 arcs have to be hidden or the reverse oriented graph have to be used, then

 making a copy of the graph structure along with the required maps

 this is the case. It also may happen that in a flow implementation

 could be rather expensive (in time or in memory usage) compared to the

 the residual graph can be accessed by another algorithm, or a node-set

 operations that should be performed on the altered graph.

 is to be shrunk for another algorithm.

 In such cases, the LEMON \e graph \e adaptor \e classes could be used.

 LEMON also provides a variety of graphs for these requirements called

 \ref graph_adaptors "graph adaptors". Adaptors cannot be used alone but only

 in conjunction with other graph representations.

 [SEC]sec_reverse_digraph[SEC] Reverse Oriented Digraph

 The main parts of LEMON are the different graph structures, generic

 Let us suppose that we have an instance \c g of a directed graph type, say

 graph algorithms, graph concepts, which couple them, and graph

 \ref ListDigraph and an algorithm

 adaptors. While the previous notions are more or less clear, the

 \code

 latter one needs further explanation. Graph adaptors are graph classes

   template <typename Digraph>

 which serve for considering graph structures in different ways.

   int algorithm(const Digraph&);

 \endcode

 A short example makes this much clearer.  Suppose that we have an

 is needed to run on the reverse oriented digraph.

 instance \c g of a directed graph type, say ListDigraph and an algorithm

 In this situation, a certain adaptor class

 \code

 \code

 template <typename Digraph>

   template <typename Digraph>

 int algorithm(const Digraph&);

   class ReverseDigraph;

 \endcode

 \endcode

 is needed to run on the reverse oriented graph.  It may be expensive

 can be used.

 (in time or in memory usage) to copy \c g with the reversed

 arcs.  In this case, an adaptor class is used, which (according

 The graph adaptors are special classes that serve for considering other graph

 to LEMON \ref concepts::Digraph "digraph concepts") works as a digraph.

 structures in different ways. They can be used exactly the same as "real"

 The adaptor uses the original digraph structure and digraph operations when

 graphs, i.e. they conform to the \ref graph_concepts "graph concepts", thus all

 methods of the reversed oriented graph are called.  This means that the adaptor

 generic algorithms can be performed on them. However, the adaptor classes

 have minor memory usage, and do not perform sophisticated algorithmic

 cannot be used alone but only in conjunction with actual graph representations.

 actions.  The purpose of it is to give a tool for the cases when a

 They do not alter the physical graph storage, they just give another view of it.

 graph have to be used in a specific alteration.  If this alteration is

 When the methods of the adaptors are called, they use the underlying

 obtained by a usual construction like filtering the node or the arc set or

 graph structures and their operations, thus these classes have only negligible

 considering a new orientation, then an adaptor is worthwhile to use.

 memory usage and do not perform sophisticated algorithmic actions.

 To come back to the reverse oriented graph, in this situation

 \code

 This technique yields convenient tools that help writing compact and elegant

 template<typename Digraph> class ReverseDigraph;

 code, and makes it possible to easily implement complex algorithms based on

 \endcode

 well tested standard components.

 template class can be used. The code looks as follows

 \code

 For solving the problem introduced above, we could use the follwing code.

 ListDigraph g;

 ReverseDigraph<ListDigraph> rg(g);

 \code

 int result = algorithm(rg);

   ListDigraph g;

 \endcode

   ReverseDigraph<ListDigraph> rg(g);

 During running the algorithm, the original digraph \c g is untouched.

   int result = algorithm(rg);

 This techniques give rise to an elegant code, and based on stable

 \endcode

 graph adaptors, complex algorithms can be implemented easily.

 Note that the original digraph \c g remains untouched during the whole

 In flow, circulation and matching problems, the residual

 procedure.

 graph is of particular importance. Combining an adaptor implementing

 this with shortest path algorithms or minimum mean cycle algorithms,

 LEMON also provides simple "creator functions" for the adaptor

 a range of weighted and cardinality optimization algorithms can be

 classes to make their usage even simpler.

 obtained. For other examples, the interested user is referred to the

 For example, \ref reverseDigraph() returns an instance of \ref ReverseDigraph,

 detailed documentation of particular adaptors.

 thus the above code can be written like this.

 The behavior of graph adaptors can be very different. Some of them keep

 \code

 capabilities of the original graph while in other cases this would be

   ListDigraph g;

 meaningless. This means that the concepts that they meet depend

   int result = algorithm(reverseDigraph(g));

 on the graph adaptor, and the wrapped graph.

 \endcode

 For example, if an arc of a reversed digraph is deleted, this is carried

 out by deleting the corresponding arc of the original digraph, thus the

 Another essential feature of the adaptors is that their \c Node and \c Arc

 adaptor modifies the original digraph.

 types convert to the original item types.

 However in case of a residual digraph, this operation has no sense.

 Therefore, the maps of the original graph can be used in connection with

 the adaptor.

 Let us stand one more example here to simplify your work.

 ReverseDigraph has constructor

 In the following code, Dijksta's algorithm is run on the reverse oriented

 \code

 graph but using the original node and arc maps.

 ReverseDigraph(Digraph& digraph);

 \endcode

 \code

 This means that in a situation, when a <tt>const %ListDigraph&</tt>

   ListDigraph g;

 reference to a graph is given, then it have to be instantiated with

   ListDigraph::ArcMap length(g);

 <tt>Digraph=const %ListDigraph</tt>.

   ListDigraph::NodeMap dist(g);

   ListDigraph::Node s = g.addNode();

   // add more nodes and arcs

   dijkstra(reverseDigraph(g), length).distMap(dist).run(s);

 \endcode

 In the above examples, we used \ref ReverseDigraph in such a way that the

 underlying digraph was not changed. However, the adaptor class can even be

 used for modifying the original graph structure.

 It allows adding and deleting arcs or nodes, and these operations are carried

 out by calling suitable functions of the underlying digraph (if it supports

 them).

 For this, \ref ReverseDigraph "ReverseDigraph<GR>" has a constructor of the

 following form.

 \code

   ReverseDigraph(GR& gr);

 \endcode

 This means that in a situation, when the modification of the original graph

 has to be avoided (e.g. it is given as a const reference), then the adaptor

 class has to be instantiated with \c GR set to be \c const type

 (e.g. <tt>GR = const %ListDigraph</tt>), as in the following example.

 <hr>

 \note Modification capabilities are not supported for all adaptors.

 E.g. for \ref ResidualDigraph (see \ref sec_other_adaptors "later"),

 The LEMON graph adaptor classes serve for considering graphs in

 this makes no sense.

 different ways. The adaptors can be used exactly the same as "real"

 graphs (i.e., they conform to the graph concepts), thus all generic

 As a more complex example, let us see how \ref ReverseDigraph can be used

 algorithms can be performed on them. However, the adaptor classes use

 together with a graph search algorithm to decide whether a directed graph is

 the underlying graph structures and operations when their methods are

 strongly connected or not.

 called, thus they have only negligible memory usage and do not perform

 We exploit the fact the a digraph is strongly connected if and only if

 sophisticated algorithmic actions. This technique yields convenient and

 for an arbitrarily selected node \c u, each other node is reachable from

 elegant tools for the cases when a graph has to be used in a specific

 \c u (along a directed path) and \c u is reachable from each node.

 alteration, but copying it would be too expensive (in time or in memory

 The latter condition is the same that each node is reachable from \c u

 usage) compared to the algorithm that should be executed on it. The

 in the reversed digraph.

 following example shows how the \ref ReverseDigraph adaptor can be used

 to run Dijksta's algorithm on the reverse oriented graph. Note that the

 \code

 maps of the original graph can be used in connection with the adaptor,

   template <typename Digraph>

 since the node and arc types of the adaptors convert to the original

   bool stronglyConnected(const Digraph& g) {

 item types.

     typedef typename Digraph::NodeIt NodeIt;

     NodeIt u(g);

 \code

     if (u == INVALID) return true;

 dijkstra(reverseDigraph(g), length).distMap(dist).run(s);

 \endcode

     // Run BFS on the original digraph

     Bfs<Digraph> bfs(g);

 Using \ref ReverseDigraph could be as efficient as working with the

     bfs.run(u);

 original graph, but not all adaptors can be so fast, of course. For

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

 example, the subgraph adaptors have to access filter maps for the nodes

       if (!bfs.reached(n)) return false;

 and/or the arcs, thus their iterators are significantly slower than the

     }

 original iterators. LEMON also provides some more complex adaptors, for

 instance, \ref SplitNodes, which can be used for splitting each node in

     // Run BFS on the reverse oriented digraph

 a directed graph and \ref ResidualDigraph for modeling the residual

     typedef ReverseDigraph<const Digraph> RDigraph;

 network for flow and matching problems.

     RDigraph rg(g);

     Bfs<RDigraph> rbfs(rg);

 Therefore, in cases when rather complex algorithms have to be used

     rbfs.run(u);

 on a subgraph (e.g. when the nodes and arcs have to be traversed several

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

 times), it could worth copying the altered graph into an efficient structure

       if (!rbfs.reached(n)) return false;

 and run the algorithm on it.

     }

     return true;

   }

 \endcode

 Note that we have to use the adaptor with '<tt>const Digraph</tt>' type, since

 \c g is a \c const reference to the original graph structure.

 The \ref stronglyConnected() function provided in LEMON has a quite

 similar implementation.

 [SEC]sec_subgraphs[SEC] Subgraph Adaptorts

 Another typical requirement is the use of certain subgraphs of a graph,

 or in other words, hiding nodes and/or arcs from a graph.

 LEMON provides several convenient adaptors for these purposes.

 \ref FilterArcs can be used when some arcs have to be hidden from a digraph.

 A \e filter \e map has to be given to the constructor, which assign \c bool

 values to the arcs specifying whether they have to be shown or not in the

 subgraph structure.

 Suppose we have a \ref ListDigraph structure \c g.

 Then we can construct a subgraph in which some arcs (\c a1, \c a2 etc.)

 are hidden as follows.

 \code

   ListDigraph::ArcMap filter(g, true);

   filter[a1] = false;

   filter[a2] = false;

   // ...

   FilterArcs<ListDigraph> subgraph(g, filter);

 \endcode

 The following more complex code runs Dijkstra's algorithm on a digraph

 that is obtained from another digraph by hiding all arcs having negative

 lengths.

 \code

   ListDigraph::ArcMap<int> length(g);

   ListDigraph::NodeMap<int> dist(g);

   dijkstra(filterArcs( g, lessMap(length, constMap<ListDigraph::Arc>(0)) ),

            length).distMap(dist).run(s);

 \endcode

 Note the extensive use of map adaptors and creator functions, which makes

 the code really compact and elegant.

 \note Implicit maps and graphs (e.g. created using functions) can only be

 used with the function-type interfaces of the algorithms, since they store

 only references for the used structures.

 \ref FilterEdges can be used for hiding edges from an undirected graph (like

 \ref FilterArcs is used for digraphs). \ref FilterNodes serves for filtering

 nodes along with the incident arcs or edges in a directed or undirected graph.

 If both arcs/edges and nodes have to be hidden, then you could use

 \ref SubDigraph or \ref SubGraph adaptors.

 \code

   ListGraph ug;

   ListGraph::NodeMap<bool> node_filter(ug);

   ListGraph::EdgeMap<bool> edge_filter(ug);

   SubGraph<ListGraph> sg(ug, node_filter, edge_filter);

 \endcode

 As you see, we needed two filter maps in this case: one for the nodes and

 another for the edges. If a node is hidden, then all of its incident edges

 are also considered to be hidden independently of their own filter values.

 The subgraph adaptors also make it possible to modify the filter values

 even after the construction of the adaptor class, thus the corresponding

 graph items can be hidden or shown on the fly.

 The adaptors store references to the filter maps, thus the map values can be

 set directly and even by using the \c enable(), \c disable() and \c status()

 functions.

 \code

   ListDigraph g;

   ListDigraph::Node x = g.addNode();

   ListDigraph::Node y = g.addNode();

   ListDigraph::Node z = g.addNode();

   ListDigraph::NodeMap<bool> filter(g, true);

   FilterNodes<ListDigraph> subgraph(g, filter);

   std::cout << countNodes(subgraph) << ", ";

   filter[x] = false;

   std::cout << countNodes(subgraph) << ", ";

   subgraph.enable(x);

   subgraph.disable(y);

   subgraph.status(z, !subgraph.status(z));

   std::cout << countNodes(subgraph) << std::endl;

 \endcode

 The above example prints out this line.

 \code

   3, 2, 1

 \endcode

 Similarly to \ref ReverseDigraph, the subgraph adaptors also allow the

 modification of the underlying graph structures unless the graph template

 parameter is set to be \c const type.

 Moreover the item types of the original graphs and the subgraphs are

 convertible to each other.

 The iterators of the subgraph adaptors use the iterators of the original

 graph structures in such a way that each item with \c false filter value

 is skipped. If both the node and arc sets are filtered, then the arc iterators

 check for each arc the status of its end nodes in addition to its own assigned

 filter value. If the arc or one of its end nodes is hidden, then the arc

 is left out and the next arc is considered.

 (It is the same for edges in undirected graphs.)

 Therefore, the iterators of these adaptors are significantly slower than the

 original iterators.

 Using adaptors, these efficiency aspects should be kept in mind.

 For example, if rather complex algorithms have to be performed on a

 subgraph (e.g. the nodes and arcs need to be traversed several times),

 then it could worth copying the altered graph into an efficient

 structure (e.g. \ref StaticDigraph) and run the algorithm on it.

     SmartDigraph temp_graph;

     StaticDigraph tmp_graph;

     ListDigraph::NodeMap<SmartDigraph::Node> node_ref(g);

     ListDigraph::NodeMap<StaticDigraph::Node> node_ref(g);

     digraphCopy(filterNodes(g, filter_map), temp_graph)

     digraphCopy(filterNodes(g, filter_map), tmp_graph)

     // use temp_graph

     // use tmp_graph

 <hr>

 \note Using \ref ReverseDigraph could be as efficient as working with the

 original graph, but most of the adaptors cannot be so fast, of course. 

 Another interesting adaptor in LEMON is \ref SplitNodes.

 It can be used for splitting each node into an in-node and an out-node

 in a directed graph. Formally, the adaptor replaces each node

 [SEC]sec_other_adaptors[SEC] Other Graph Adaptors

 u in the graph with two nodes, namely node u<sub>in</sub> and node

 u<sub>out</sub>. Each arc (u,c) in the original graph will correspond to an

 Two other practical adaptors are \ref Undirector and \ref Orienter.

 arc (u<sub>out</sub>,v<sub>in</sub>). The adaptor also adds an

 \ref Undirector makes an undirected graph from a digraph disregarding the

 additional bind arc (u<sub>in</sub>,u<sub>out</sub>) for each node u

 orientations of the arcs. More precisely, an arc of the original digraph

 of the original digraph.

 is considered as an edge (and two arcs, as well) in the adaptor.

 \ref Orienter can be used for the reverse alteration, it assigns a certain

 The aim of this class is to assign costs to the nodes when using

 orientation to each edge of an undirected graph to form a directed graph.

 algorithms which would otherwise consider arc costs only.

 A \c bool edge map of the underlying graph must be given to the constructor

 For example, let us suppose that we have a directed graph with costs

 of the class, which define the direction of the arcs in the created adaptor

 given for both the nodes and the arcs.

 (with respect to the inherent orientation of the original edges).

 Then Dijkstra's algorithm can be used in connection with \ref SplitNodes

 as follows.

 \code

   ListGraph graph;

   ListGraph::EdgeMap<bool> dir_map(graph, true);

   Orienter<ListGraph> directed_graph(graph, dir_map);

 \endcode

 LEMON also provides some more complex adaptors, for

 instance, \ref SplitNodes, which can be used for splitting each node of a

 directed graph into an in-node and an out-node.

 Formally, the adaptor replaces each node u in the graph with two nodes,

 namely u<sub>in</sub> and u<sub>out</sub>. Each arc (u,v) of the original

 graph will correspond to an arc (u<sub>out</sub>,v<sub>in</sub>).

 The adaptor also adds an additional bind arc (u<sub>in</sub>,u<sub>out</sub>)

 for each node u of the original digraph.

 The aim of this class is to assign costs or capacities to the nodes when using

 algorithms which would otherwise consider arc costs or capacities only.

 For example, let us suppose that we have a digraph \c g with costs assigned to

 both the nodes and the arcs. Then Dijkstra's algorithm can be used in

 connection with \ref SplitNodes as follows.

 Note that this problem can be solved more efficiently with

 \note This problem can also be solved using map adaptors to create

 map adaptors.

 an implicit arc map that assigns for each arc the sum of its cost

 and the cost of its target node. This map can be used with the original

 These techniques help writing compact and elegant code, and makes it possible

 graph more efficiently than using the above solution.

 to easily implement complex algorithms based on well tested standard components.

 For instance, in flow and matching problems the residual graph is of

 Another nice application is the problem of finding disjoint paths in

 particular importance.

 a digraph.

 Combining \ref ResidualDigraph adaptor with various algorithms, a

 The maximum number of \e edge \e disjoint paths from a source node to

 range of weighted and cardinality optimization methods can be obtained

 a sink node in a digraph can be easily computed using a maximum flow

 directly.

 algorithm with all arc capacities set to 1.

 On the other hand, \e node \e disjoint paths cannot be found directly

 using a standard algorithm.

 However, \ref SplitNodes adaptor makes it really simple.

 If a maximum flow computation is performed on this adaptor, then the

 bottleneck of the flow (i.e. the minimum cut) will be formed by bind arcs,

 thus the found flow will correspond to the union of some node disjoint

 paths in terms of the original digraph.

 In flow, circulation and matching problems, the residual network is of

 particular importance, which is implemented in \ref ResidualDigraph.

 Combining this adaptor with various algorithms, a range of weighted and

 cardinality optimization methods can be implemented easily.

 To construct a residual network, a digraph structure, a flow map and a

 capacity map have to be given to the constructor of the adaptor as shown

 in the following code.

 \code

   ListDigraph g;

   ListDigraph::ArcMap<int> flow(g);

   ListDigraph::ArcMap<int> capacity(g);

   ResidualDigraph<ListDigraph> res_graph(g, capacity, flow); 

 \endcode

 \note In fact, this class is implemented using two other adaptors:

 \ref Undirector and \ref FilterArcs.