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@@ -18,12 +18,14 @@
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///\ingroup graph_concepts
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///\file
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///\brief The concept of Undirected Graphs.
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///\brief The concept of undirected graphs.
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#ifndef LEMON_CONCEPTS_GRAPH_H
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#define LEMON_CONCEPTS_GRAPH_H
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#include <lemon/concepts/graph_components.h>
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#include <lemon/concepts/maps.h>
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#include <lemon/concept_check.h>
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#include <lemon/core.h>
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namespace lemon {
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@@ -31,63 +33,74 @@
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/// \ingroup graph_concepts
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///
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/// \brief Class describing the concept of Undirected Graphs.
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/// \brief Class describing the concept of undirected graphs.
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///
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/// This class describes the common interface of all Undirected
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/// Graphs.
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/// This class describes the common interface of all undirected
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/// graphs.
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///
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/// As all concept describing classes it provides only interface
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/// without any sensible implementation. So any algorithm for
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/// undirected graph should compile with this class, but it will not
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/// Like all concept classes, it only provides an interface
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/// without any sensible implementation. So any general algorithm for
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/// undirected graphs should compile with this class, but it will not
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/// run properly, of course.
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/// An actual graph implementation like \ref ListGraph or
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/// \ref SmartGraph may have additional functionality.
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///
|
44 |
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/// The LEMON undirected graphs also fulfill the concept of
|
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/// directed graphs (\ref lemon::concepts::Digraph "Digraph
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/// Concept"). Each edges can be seen as two opposite
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/// directed arc and consequently the undirected graph can be
|
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/// seen as the direceted graph of these directed arcs. The
|
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/// Graph has the Edge inner class for the edges and
|
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/// the Arc type for the directed arcs. The Arc type is
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/// convertible to Edge or inherited from it so from a directed
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/// arc we can get the represented edge.
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/// The undirected graphs also fulfill the concept of \ref Digraph
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/// "directed graphs", since each edge can also be regarded as two
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/// oppositely directed arcs.
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/// Undirected graphs provide an Edge type for the undirected edges and
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/// an Arc type for the directed arcs. The Arc type is convertible to
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/// Edge or inherited from it, i.e. the corresponding edge can be
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/// obtained from an arc.
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/// EdgeIt and EdgeMap classes can be used for the edges, while ArcIt
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/// and ArcMap classes can be used for the arcs (just like in digraphs).
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/// Both InArcIt and OutArcIt iterates on the same edges but with
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/// opposite direction. IncEdgeIt also iterates on the same edges
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/// as OutArcIt and InArcIt, but it is not convertible to Arc,
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/// only to Edge.
|
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///
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/// In the sense of the LEMON each edge has a default
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/// direction (it should be in every computer implementation,
|
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/// because the order of edge's nodes defines an
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/// orientation). With the default orientation we can define that
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/// the directed arc is forward or backward directed. With the \c
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/// direction() and \c direct() function we can get the direction
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/// of the directed arc and we can direct an edge.
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/// In LEMON, each undirected edge has an inherent orientation.
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/// Thus it can defined if an arc is forward or backward oriented in
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/// an undirected graph with respect to this default oriantation of
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/// the represented edge.
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/// With the direction() and direct() functions the direction
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/// of an arc can be obtained and set, respectively.
|
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///
|
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/// The EdgeIt is an iterator for the edges. We can use
|
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/// the EdgeMap to map values for the edges. The InArcIt and
|
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/// OutArcIt iterates on the same edges but with opposite
|
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/// direction. The IncEdgeIt iterates also on the same edges
|
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/// as the OutArcIt and InArcIt but it is not convertible to Arc just
|
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/// to Edge.
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/// Only nodes and edges can be added to or removed from an undirected
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/// graph and the corresponding arcs are added or removed automatically.
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///
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/// \sa Digraph
|
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class Graph {
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private:
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/// Graphs are \e not copy constructible. Use DigraphCopy instead.
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Graph(const Graph&) {}
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/// \brief Assignment of a graph to another one is \e not allowed.
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/// Use DigraphCopy instead.
|
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void operator=(const Graph&) {}
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|
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public:
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/// \brief The undirected graph should be tagged by the
|
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/// UndirectedTag.
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/// Default constructor.
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Graph() {}
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|
|
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/// \brief Undirected graphs should be tagged with \c UndirectedTag.
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86 |
///
|
73 |
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/// The undirected graph should be tagged by the UndirectedTag. This
|
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/// tag helps the enable_if technics to make compile time
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|
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/// Undirected graphs should be tagged with \c UndirectedTag.
|
|
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///
|
|
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/// This tag helps the \c enable_if technics to make compile time
|
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90 |
/// specializations for undirected graphs.
|
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91 |
typedef True UndirectedTag;
|
77 |
92 |
|
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|
/// \brief The base type of node iterators,
|
79 |
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/// or in other words, the trivial node iterator.
|
80 |
|
///
|
81 |
|
/// This is the base type of each node iterator,
|
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/// thus each kind of node iterator converts to this.
|
83 |
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/// More precisely each kind of node iterator should be inherited
|
84 |
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/// from the trivial node iterator.
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/// The node type of the graph
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|
94 |
|
|
95 |
/// This class identifies a node of the graph. It also serves
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|
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/// as a base class of the node iterators,
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|
97 |
/// thus they convert to this type.
|
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98 |
class Node {
|
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99 |
public:
|
87 |
100 |
/// Default constructor
|
88 |
101 |
|
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/// @warning The default constructor sets the iterator
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90 |
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/// to an undefined value.
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/// Default constructor.
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|
103 |
/// \warning It sets the object to an undefined value.
|
91 |
104 |
Node() { }
|
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105 |
/// Copy constructor.
|
93 |
106 |
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@@ -95,40 +108,40 @@
|
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///
|
96 |
109 |
Node(const Node&) { }
|
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110 |
|
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/// Invalid constructor \& conversion.
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111 |
/// %Invalid constructor \& conversion.
|
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112 |
|
100 |
|
/// This constructor initializes the iterator to be invalid.
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|
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/// Initializes the object to be invalid.
|
101 |
114 |
/// \sa Invalid for more details.
|
102 |
115 |
Node(Invalid) { }
|
103 |
116 |
/// Equality operator
|
104 |
117 |
|
|
118 |
/// Equality operator.
|
|
119 |
///
|
105 |
120 |
/// Two iterators are equal if and only if they point to the
|
106 |
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/// same object or both are invalid.
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/// same object or both are \c INVALID.
|
107 |
122 |
bool operator==(Node) const { return true; }
|
108 |
123 |
|
109 |
124 |
/// Inequality operator
|
110 |
125 |
|
111 |
|
/// \sa operator==(Node n)
|
112 |
|
///
|
|
126 |
/// Inequality operator.
|
113 |
127 |
bool operator!=(Node) const { return true; }
|
114 |
128 |
|
115 |
129 |
/// Artificial ordering operator.
|
116 |
130 |
|
117 |
|
/// To allow the use of graph descriptors as key type in std::map or
|
118 |
|
/// similar associative container we require this.
|
|
131 |
/// Artificial ordering operator.
|
119 |
132 |
///
|
120 |
|
/// \note This operator only have to define some strict ordering of
|
|
133 |
/// \note This operator only has to define some strict ordering of
|
121 |
134 |
/// the items; this order has nothing to do with the iteration
|
122 |
135 |
/// ordering of the items.
|
123 |
136 |
bool operator<(Node) const { return false; }
|
124 |
137 |
|
125 |
138 |
};
|
126 |
139 |
|
127 |
|
/// This iterator goes through each node.
|
|
140 |
/// Iterator class for the nodes.
|
128 |
141 |
|
129 |
|
/// This iterator goes through each node.
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|
142 |
/// This iterator goes through each node of the graph.
|
130 |
143 |
/// Its usage is quite simple, for example you can count the number
|
131 |
|
/// of nodes in graph \c g of type \c Graph like this:
|
|
144 |
/// of nodes in a graph \c g of type \c %Graph like this:
|
132 |
145 |
///\code
|
133 |
146 |
/// int count=0;
|
134 |
147 |
/// for (Graph::NodeIt n(g); n!=INVALID; ++n) ++count;
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@@ -137,30 +150,28 @@
|
137 |
150 |
public:
|
138 |
151 |
/// Default constructor
|
139 |
152 |
|
140 |
|
/// @warning The default constructor sets the iterator
|
141 |
|
/// to an undefined value.
|
|
153 |
/// Default constructor.
|
|
154 |
/// \warning It sets the iterator to an undefined value.
|
142 |
155 |
NodeIt() { }
|
143 |
156 |
/// Copy constructor.
|
144 |
157 |
|
145 |
158 |
/// Copy constructor.
|
146 |
159 |
///
|
147 |
160 |
NodeIt(const NodeIt& n) : Node(n) { }
|
148 |
|
/// Invalid constructor \& conversion.
|
|
161 |
/// %Invalid constructor \& conversion.
|
149 |
162 |
|
150 |
|
/// Initialize the iterator to be invalid.
|
|
163 |
/// Initializes the iterator to be invalid.
|
151 |
164 |
/// \sa Invalid for more details.
|
152 |
165 |
NodeIt(Invalid) { }
|
153 |
166 |
/// Sets the iterator to the first node.
|
154 |
167 |
|
155 |
|
/// Sets the iterator to the first node of \c g.
|
|
168 |
/// Sets the iterator to the first node of the given digraph.
|
156 |
169 |
///
|
157 |
|
NodeIt(const Graph&) { }
|
158 |
|
/// Node -> NodeIt conversion.
|
|
170 |
explicit NodeIt(const Graph&) { }
|
|
171 |
/// Sets the iterator to the given node.
|
159 |
172 |
|
160 |
|
/// Sets the iterator to the node of \c the graph pointed by
|
161 |
|
/// the trivial iterator.
|
162 |
|
/// This feature necessitates that each time we
|
163 |
|
/// iterate the arc-set, the iteration order is the same.
|
|
173 |
/// Sets the iterator to the given node of the given digraph.
|
|
174 |
///
|
164 |
175 |
NodeIt(const Graph&, const Node&) { }
|
165 |
176 |
/// Next node.
|
166 |
177 |
|
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@@ -170,54 +181,55 @@
|
170 |
181 |
};
|
171 |
182 |
|
172 |
183 |
|
173 |
|
/// The base type of the edge iterators.
|
|
184 |
/// The edge type of the graph
|
174 |
185 |
|
175 |
|
/// The base type of the edge iterators.
|
176 |
|
///
|
|
186 |
/// This class identifies an edge of the graph. It also serves
|
|
187 |
/// as a base class of the edge iterators,
|
|
188 |
/// thus they will convert to this type.
|
177 |
189 |
class Edge {
|
178 |
190 |
public:
|
179 |
191 |
/// Default constructor
|
180 |
192 |
|
181 |
|
/// @warning The default constructor sets the iterator
|
182 |
|
/// to an undefined value.
|
|
193 |
/// Default constructor.
|
|
194 |
/// \warning It sets the object to an undefined value.
|
183 |
195 |
Edge() { }
|
184 |
196 |
/// Copy constructor.
|
185 |
197 |
|
186 |
198 |
/// Copy constructor.
|
187 |
199 |
///
|
188 |
200 |
Edge(const Edge&) { }
|
189 |
|
/// Initialize the iterator to be invalid.
|
|
201 |
/// %Invalid constructor \& conversion.
|
190 |
202 |
|
191 |
|
/// Initialize the iterator to be invalid.
|
192 |
|
///
|
|
203 |
/// Initializes the object to be invalid.
|
|
204 |
/// \sa Invalid for more details.
|
193 |
205 |
Edge(Invalid) { }
|
194 |
206 |
/// Equality operator
|
195 |
207 |
|
|
208 |
/// Equality operator.
|
|
209 |
///
|
196 |
210 |
/// Two iterators are equal if and only if they point to the
|
197 |
|
/// same object or both are invalid.
|
|
211 |
/// same object or both are \c INVALID.
|
198 |
212 |
bool operator==(Edge) const { return true; }
|
199 |
213 |
/// Inequality operator
|
200 |
214 |
|
201 |
|
/// \sa operator==(Edge n)
|
202 |
|
///
|
|
215 |
/// Inequality operator.
|
203 |
216 |
bool operator!=(Edge) const { return true; }
|
204 |
217 |
|
205 |
218 |
/// Artificial ordering operator.
|
206 |
219 |
|
207 |
|
/// To allow the use of graph descriptors as key type in std::map or
|
208 |
|
/// similar associative container we require this.
|
|
220 |
/// Artificial ordering operator.
|
209 |
221 |
///
|
210 |
|
/// \note This operator only have to define some strict ordering of
|
211 |
|
/// the items; this order has nothing to do with the iteration
|
212 |
|
/// ordering of the items.
|
|
222 |
/// \note This operator only has to define some strict ordering of
|
|
223 |
/// the edges; this order has nothing to do with the iteration
|
|
224 |
/// ordering of the edges.
|
213 |
225 |
bool operator<(Edge) const { return false; }
|
214 |
226 |
};
|
215 |
227 |
|
216 |
|
/// This iterator goes through each edge.
|
|
228 |
/// Iterator class for the edges.
|
217 |
229 |
|
218 |
|
/// This iterator goes through each edge of a graph.
|
|
230 |
/// This iterator goes through each edge of the graph.
|
219 |
231 |
/// Its usage is quite simple, for example you can count the number
|
220 |
|
/// of edges in a graph \c g of type \c Graph as follows:
|
|
232 |
/// of edges in a graph \c g of type \c %Graph as follows:
|
221 |
233 |
///\code
|
222 |
234 |
/// int count=0;
|
223 |
235 |
/// for(Graph::EdgeIt e(g); e!=INVALID; ++e) ++count;
|
... |
... |
@@ -226,219 +238,217 @@
|
226 |
238 |
public:
|
227 |
239 |
/// Default constructor
|
228 |
240 |
|
229 |
|
/// @warning The default constructor sets the iterator
|
230 |
|
/// to an undefined value.
|
|
241 |
/// Default constructor.
|
|
242 |
/// \warning It sets the iterator to an undefined value.
|
231 |
243 |
EdgeIt() { }
|
232 |
244 |
/// Copy constructor.
|
233 |
245 |
|
234 |
246 |
/// Copy constructor.
|
235 |
247 |
///
|
236 |
248 |
EdgeIt(const EdgeIt& e) : Edge(e) { }
|
237 |
|
/// Initialize the iterator to be invalid.
|
|
249 |
/// %Invalid constructor \& conversion.
|
238 |
250 |
|
239 |
|
/// Initialize the iterator to be invalid.
|
|
251 |
/// Initializes the iterator to be invalid.
|
|
252 |
/// \sa Invalid for more details.
|
|
253 |
EdgeIt(Invalid) { }
|
|
254 |
/// Sets the iterator to the first edge.
|
|
255 |
|
|
256 |
/// Sets the iterator to the first edge of the given graph.
|
240 |
257 |
///
|
241 |
|
EdgeIt(Invalid) { }
|
242 |
|
/// This constructor sets the iterator to the first edge.
|
|
258 |
explicit EdgeIt(const Graph&) { }
|
|
259 |
/// Sets the iterator to the given edge.
|
243 |
260 |
|
244 |
|
/// This constructor sets the iterator to the first edge.
|
245 |
|
EdgeIt(const Graph&) { }
|
246 |
|
/// Edge -> EdgeIt conversion
|
247 |
|
|
248 |
|
/// Sets the iterator to the value of the trivial iterator.
|
249 |
|
/// This feature necessitates that each time we
|
250 |
|
/// iterate the edge-set, the iteration order is the
|
251 |
|
/// same.
|
|
261 |
/// Sets the iterator to the given edge of the given graph.
|
|
262 |
///
|
252 |
263 |
EdgeIt(const Graph&, const Edge&) { }
|
253 |
264 |
/// Next edge
|
254 |
265 |
|
255 |
266 |
/// Assign the iterator to the next edge.
|
|
267 |
///
|
256 |
268 |
EdgeIt& operator++() { return *this; }
|
257 |
269 |
};
|
258 |
270 |
|
259 |
|
/// \brief This iterator goes trough the incident undirected
|
260 |
|
/// arcs of a node.
|
261 |
|
///
|
262 |
|
/// This iterator goes trough the incident edges
|
263 |
|
/// of a certain node of a graph. You should assume that the
|
264 |
|
/// loop arcs will be iterated twice.
|
265 |
|
///
|
|
271 |
/// Iterator class for the incident edges of a node.
|
|
272 |
|
|
273 |
/// This iterator goes trough the incident undirected edges
|
|
274 |
/// of a certain node of a graph.
|
266 |
275 |
/// Its usage is quite simple, for example you can compute the
|
267 |
|
/// degree (i.e. count the number of incident arcs of a node \c n
|
268 |
|
/// in graph \c g of type \c Graph as follows.
|
|
276 |
/// degree (i.e. the number of incident edges) of a node \c n
|
|
277 |
/// in a graph \c g of type \c %Graph as follows.
|
269 |
278 |
///
|
270 |
279 |
///\code
|
271 |
280 |
/// int count=0;
|
272 |
281 |
/// for(Graph::IncEdgeIt e(g, n); e!=INVALID; ++e) ++count;
|
273 |
282 |
///\endcode
|
|
283 |
///
|
|
284 |
/// \warning Loop edges will be iterated twice.
|
274 |
285 |
class IncEdgeIt : public Edge {
|
275 |
286 |
public:
|
276 |
287 |
/// Default constructor
|
277 |
288 |
|
278 |
|
/// @warning The default constructor sets the iterator
|
279 |
|
/// to an undefined value.
|
|
289 |
/// Default constructor.
|
|
290 |
/// \warning It sets the iterator to an undefined value.
|
280 |
291 |
IncEdgeIt() { }
|
281 |
292 |
/// Copy constructor.
|
282 |
293 |
|
283 |
294 |
/// Copy constructor.
|
284 |
295 |
///
|
285 |
296 |
IncEdgeIt(const IncEdgeIt& e) : Edge(e) { }
|
286 |
|
/// Initialize the iterator to be invalid.
|
|
297 |
/// %Invalid constructor \& conversion.
|
287 |
298 |
|
288 |
|
/// Initialize the iterator to be invalid.
|
|
299 |
/// Initializes the iterator to be invalid.
|
|
300 |
/// \sa Invalid for more details.
|
|
301 |
IncEdgeIt(Invalid) { }
|
|
302 |
/// Sets the iterator to the first incident edge.
|
|
303 |
|
|
304 |
/// Sets the iterator to the first incident edge of the given node.
|
289 |
305 |
///
|
290 |
|
IncEdgeIt(Invalid) { }
|
291 |
|
/// This constructor sets the iterator to first incident arc.
|
|
306 |
IncEdgeIt(const Graph&, const Node&) { }
|
|
307 |
/// Sets the iterator to the given edge.
|
292 |
308 |
|
293 |
|
/// This constructor set the iterator to the first incident arc of
|
294 |
|
/// the node.
|
295 |
|
IncEdgeIt(const Graph&, const Node&) { }
|
296 |
|
/// Edge -> IncEdgeIt conversion
|
|
309 |
/// Sets the iterator to the given edge of the given graph.
|
|
310 |
///
|
|
311 |
IncEdgeIt(const Graph&, const Edge&) { }
|
|
312 |
/// Next incident edge
|
297 |
313 |
|
298 |
|
/// Sets the iterator to the value of the trivial iterator \c e.
|
299 |
|
/// This feature necessitates that each time we
|
300 |
|
/// iterate the arc-set, the iteration order is the same.
|
301 |
|
IncEdgeIt(const Graph&, const Edge&) { }
|
302 |
|
/// Next incident arc
|
303 |
|
|
304 |
|
/// Assign the iterator to the next incident arc
|
|
314 |
/// Assign the iterator to the next incident edge
|
305 |
315 |
/// of the corresponding node.
|
306 |
316 |
IncEdgeIt& operator++() { return *this; }
|
307 |
317 |
};
|
308 |
318 |
|
309 |
|
/// The directed arc type.
|
|
319 |
/// The arc type of the graph
|
310 |
320 |
|
311 |
|
/// The directed arc type. It can be converted to the
|
312 |
|
/// edge or it should be inherited from the undirected
|
313 |
|
/// edge.
|
|
321 |
/// This class identifies a directed arc of the graph. It also serves
|
|
322 |
/// as a base class of the arc iterators,
|
|
323 |
/// thus they will convert to this type.
|
314 |
324 |
class Arc {
|
315 |
325 |
public:
|
316 |
326 |
/// Default constructor
|
317 |
327 |
|
318 |
|
/// @warning The default constructor sets the iterator
|
319 |
|
/// to an undefined value.
|
|
328 |
/// Default constructor.
|
|
329 |
/// \warning It sets the object to an undefined value.
|
320 |
330 |
Arc() { }
|
321 |
331 |
/// Copy constructor.
|
322 |
332 |
|
323 |
333 |
/// Copy constructor.
|
324 |
334 |
///
|
325 |
335 |
Arc(const Arc&) { }
|
326 |
|
/// Initialize the iterator to be invalid.
|
|
336 |
/// %Invalid constructor \& conversion.
|
327 |
337 |
|
328 |
|
/// Initialize the iterator to be invalid.
|
329 |
|
///
|
|
338 |
/// Initializes the object to be invalid.
|
|
339 |
/// \sa Invalid for more details.
|
330 |
340 |
Arc(Invalid) { }
|
331 |
341 |
/// Equality operator
|
332 |
342 |
|
|
343 |
/// Equality operator.
|
|
344 |
///
|
333 |
345 |
/// Two iterators are equal if and only if they point to the
|
334 |
|
/// same object or both are invalid.
|
|
346 |
/// same object or both are \c INVALID.
|
335 |
347 |
bool operator==(Arc) const { return true; }
|
336 |
348 |
/// Inequality operator
|
337 |
349 |
|
338 |
|
/// \sa operator==(Arc n)
|
339 |
|
///
|
|
350 |
/// Inequality operator.
|
340 |
351 |
bool operator!=(Arc) const { return true; }
|
341 |
352 |
|
342 |
353 |
/// Artificial ordering operator.
|
343 |
354 |
|
344 |
|
/// To allow the use of graph descriptors as key type in std::map or
|
345 |
|
/// similar associative container we require this.
|
|
355 |
/// Artificial ordering operator.
|
346 |
356 |
///
|
347 |
|
/// \note This operator only have to define some strict ordering of
|
348 |
|
/// the items; this order has nothing to do with the iteration
|
349 |
|
/// ordering of the items.
|
|
357 |
/// \note This operator only has to define some strict ordering of
|
|
358 |
/// the arcs; this order has nothing to do with the iteration
|
|
359 |
/// ordering of the arcs.
|
350 |
360 |
bool operator<(Arc) const { return false; }
|
351 |
361 |
|
352 |
|
/// Converison to Edge
|
|
362 |
/// Converison to \c Edge
|
|
363 |
|
|
364 |
/// Converison to \c Edge.
|
|
365 |
///
|
353 |
366 |
operator Edge() const { return Edge(); }
|
354 |
367 |
};
|
355 |
|
/// This iterator goes through each directed arc.
|
356 |
368 |
|
357 |
|
/// This iterator goes through each arc of a graph.
|
|
369 |
/// Iterator class for the arcs.
|
|
370 |
|
|
371 |
/// This iterator goes through each directed arc of the graph.
|
358 |
372 |
/// Its usage is quite simple, for example you can count the number
|
359 |
|
/// of arcs in a graph \c g of type \c Graph as follows:
|
|
373 |
/// of arcs in a graph \c g of type \c %Graph as follows:
|
360 |
374 |
///\code
|
361 |
375 |
/// int count=0;
|
362 |
|
/// for(Graph::ArcIt e(g); e!=INVALID; ++e) ++count;
|
|
376 |
/// for(Graph::ArcIt a(g); a!=INVALID; ++a) ++count;
|
363 |
377 |
///\endcode
|
364 |
378 |
class ArcIt : public Arc {
|
365 |
379 |
public:
|
366 |
380 |
/// Default constructor
|
367 |
381 |
|
368 |
|
/// @warning The default constructor sets the iterator
|
369 |
|
/// to an undefined value.
|
|
382 |
/// Default constructor.
|
|
383 |
/// \warning It sets the iterator to an undefined value.
|
370 |
384 |
ArcIt() { }
|
371 |
385 |
/// Copy constructor.
|
372 |
386 |
|
373 |
387 |
/// Copy constructor.
|
374 |
388 |
///
|
375 |
389 |
ArcIt(const ArcIt& e) : Arc(e) { }
|
376 |
|
/// Initialize the iterator to be invalid.
|
|
390 |
/// %Invalid constructor \& conversion.
|
377 |
391 |
|
378 |
|
/// Initialize the iterator to be invalid.
|
|
392 |
/// Initializes the iterator to be invalid.
|
|
393 |
/// \sa Invalid for more details.
|
|
394 |
ArcIt(Invalid) { }
|
|
395 |
/// Sets the iterator to the first arc.
|
|
396 |
|
|
397 |
/// Sets the iterator to the first arc of the given graph.
|
379 |
398 |
///
|
380 |
|
ArcIt(Invalid) { }
|
381 |
|
/// This constructor sets the iterator to the first arc.
|
|
399 |
explicit ArcIt(const Graph &g) { ignore_unused_variable_warning(g); }
|
|
400 |
/// Sets the iterator to the given arc.
|
382 |
401 |
|
383 |
|
/// This constructor sets the iterator to the first arc of \c g.
|
384 |
|
///@param g the graph
|
385 |
|
ArcIt(const Graph &g) { ignore_unused_variable_warning(g); }
|
386 |
|
/// Arc -> ArcIt conversion
|
387 |
|
|
388 |
|
/// Sets the iterator to the value of the trivial iterator \c e.
|
389 |
|
/// This feature necessitates that each time we
|
390 |
|
/// iterate the arc-set, the iteration order is the same.
|
|
402 |
/// Sets the iterator to the given arc of the given graph.
|
|
403 |
///
|
391 |
404 |
ArcIt(const Graph&, const Arc&) { }
|
392 |
405 |
///Next arc
|
393 |
406 |
|
394 |
407 |
/// Assign the iterator to the next arc.
|
|
408 |
///
|
395 |
409 |
ArcIt& operator++() { return *this; }
|
396 |
410 |
};
|
397 |
411 |
|
398 |
|
/// This iterator goes trough the outgoing directed arcs of a node.
|
|
412 |
/// Iterator class for the outgoing arcs of a node.
|
399 |
413 |
|
400 |
|
/// This iterator goes trough the \e outgoing arcs of a certain node
|
401 |
|
/// of a graph.
|
|
414 |
/// This iterator goes trough the \e outgoing directed arcs of a
|
|
415 |
/// certain node of a graph.
|
402 |
416 |
/// Its usage is quite simple, for example you can count the number
|
403 |
417 |
/// of outgoing arcs of a node \c n
|
404 |
|
/// in graph \c g of type \c Graph as follows.
|
|
418 |
/// in a graph \c g of type \c %Graph as follows.
|
405 |
419 |
///\code
|
406 |
420 |
/// int count=0;
|
407 |
|
/// for (Graph::OutArcIt e(g, n); e!=INVALID; ++e) ++count;
|
|
421 |
/// for (Digraph::OutArcIt a(g, n); a!=INVALID; ++a) ++count;
|
408 |
422 |
///\endcode
|
409 |
|
|
410 |
423 |
class OutArcIt : public Arc {
|
411 |
424 |
public:
|
412 |
425 |
/// Default constructor
|
413 |
426 |
|
414 |
|
/// @warning The default constructor sets the iterator
|
415 |
|
/// to an undefined value.
|
|
427 |
/// Default constructor.
|
|
428 |
/// \warning It sets the iterator to an undefined value.
|
416 |
429 |
OutArcIt() { }
|
417 |
430 |
/// Copy constructor.
|
418 |
431 |
|
419 |
432 |
/// Copy constructor.
|
420 |
433 |
///
|
421 |
434 |
OutArcIt(const OutArcIt& e) : Arc(e) { }
|
422 |
|
/// Initialize the iterator to be invalid.
|
|
435 |
/// %Invalid constructor \& conversion.
|
423 |
436 |
|
424 |
|
/// Initialize the iterator to be invalid.
|
|
437 |
/// Initializes the iterator to be invalid.
|
|
438 |
/// \sa Invalid for more details.
|
|
439 |
OutArcIt(Invalid) { }
|
|
440 |
/// Sets the iterator to the first outgoing arc.
|
|
441 |
|
|
442 |
/// Sets the iterator to the first outgoing arc of the given node.
|
425 |
443 |
///
|
426 |
|
OutArcIt(Invalid) { }
|
427 |
|
/// This constructor sets the iterator to the first outgoing arc.
|
428 |
|
|
429 |
|
/// This constructor sets the iterator to the first outgoing arc of
|
430 |
|
/// the node.
|
431 |
|
///@param n the node
|
432 |
|
///@param g the graph
|
433 |
444 |
OutArcIt(const Graph& n, const Node& g) {
|
434 |
445 |
ignore_unused_variable_warning(n);
|
435 |
446 |
ignore_unused_variable_warning(g);
|
436 |
447 |
}
|
437 |
|
/// Arc -> OutArcIt conversion
|
|
448 |
/// Sets the iterator to the given arc.
|
438 |
449 |
|
439 |
|
/// Sets the iterator to the value of the trivial iterator.
|
440 |
|
/// This feature necessitates that each time we
|
441 |
|
/// iterate the arc-set, the iteration order is the same.
|
|
450 |
/// Sets the iterator to the given arc of the given graph.
|
|
451 |
///
|
442 |
452 |
OutArcIt(const Graph&, const Arc&) { }
|
443 |
453 |
///Next outgoing arc
|
444 |
454 |
|
... |
... |
@@ -447,69 +457,66 @@
|
447 |
457 |
OutArcIt& operator++() { return *this; }
|
448 |
458 |
};
|
449 |
459 |
|
450 |
|
/// This iterator goes trough the incoming directed arcs of a node.
|
|
460 |
/// Iterator class for the incoming arcs of a node.
|
451 |
461 |
|
452 |
|
/// This iterator goes trough the \e incoming arcs of a certain node
|
453 |
|
/// of a graph.
|
|
462 |
/// This iterator goes trough the \e incoming directed arcs of a
|
|
463 |
/// certain node of a graph.
|
454 |
464 |
/// Its usage is quite simple, for example you can count the number
|
455 |
|
/// of outgoing arcs of a node \c n
|
456 |
|
/// in graph \c g of type \c Graph as follows.
|
|
465 |
/// of incoming arcs of a node \c n
|
|
466 |
/// in a graph \c g of type \c %Graph as follows.
|
457 |
467 |
///\code
|
458 |
468 |
/// int count=0;
|
459 |
|
/// for(Graph::InArcIt e(g, n); e!=INVALID; ++e) ++count;
|
|
469 |
/// for (Digraph::InArcIt a(g, n); a!=INVALID; ++a) ++count;
|
460 |
470 |
///\endcode
|
461 |
|
|
462 |
471 |
class InArcIt : public Arc {
|
463 |
472 |
public:
|
464 |
473 |
/// Default constructor
|
465 |
474 |
|
466 |
|
/// @warning The default constructor sets the iterator
|
467 |
|
/// to an undefined value.
|
|
475 |
/// Default constructor.
|
|
476 |
/// \warning It sets the iterator to an undefined value.
|
468 |
477 |
InArcIt() { }
|
469 |
478 |
/// Copy constructor.
|
470 |
479 |
|
471 |
480 |
/// Copy constructor.
|
472 |
481 |
///
|
473 |
482 |
InArcIt(const InArcIt& e) : Arc(e) { }
|
474 |
|
/// Initialize the iterator to be invalid.
|
|
483 |
/// %Invalid constructor \& conversion.
|
475 |
484 |
|
476 |
|
/// Initialize the iterator to be invalid.
|
|
485 |
/// Initializes the iterator to be invalid.
|
|
486 |
/// \sa Invalid for more details.
|
|
487 |
InArcIt(Invalid) { }
|
|
488 |
/// Sets the iterator to the first incoming arc.
|
|
489 |
|
|
490 |
/// Sets the iterator to the first incoming arc of the given node.
|
477 |
491 |
///
|
478 |
|
InArcIt(Invalid) { }
|
479 |
|
/// This constructor sets the iterator to first incoming arc.
|
480 |
|
|
481 |
|
/// This constructor set the iterator to the first incoming arc of
|
482 |
|
/// the node.
|
483 |
|
///@param n the node
|
484 |
|
///@param g the graph
|
485 |
492 |
InArcIt(const Graph& g, const Node& n) {
|
486 |
493 |
ignore_unused_variable_warning(n);
|
487 |
494 |
ignore_unused_variable_warning(g);
|
488 |
495 |
}
|
489 |
|
/// Arc -> InArcIt conversion
|
|
496 |
/// Sets the iterator to the given arc.
|
490 |
497 |
|
491 |
|
/// Sets the iterator to the value of the trivial iterator \c e.
|
492 |
|
/// This feature necessitates that each time we
|
493 |
|
/// iterate the arc-set, the iteration order is the same.
|
|
498 |
/// Sets the iterator to the given arc of the given graph.
|
|
499 |
///
|
494 |
500 |
InArcIt(const Graph&, const Arc&) { }
|
495 |
501 |
/// Next incoming arc
|
496 |
502 |
|
497 |
|
/// Assign the iterator to the next inarc of the corresponding node.
|
498 |
|
///
|
|
503 |
/// Assign the iterator to the next
|
|
504 |
/// incoming arc of the corresponding node.
|
499 |
505 |
InArcIt& operator++() { return *this; }
|
500 |
506 |
};
|
501 |
507 |
|
502 |
|
/// \brief Reference map of the nodes to type \c T.
|
|
508 |
/// \brief Standard graph map type for the nodes.
|
503 |
509 |
///
|
504 |
|
/// Reference map of the nodes to type \c T.
|
|
510 |
/// Standard graph map type for the nodes.
|
|
511 |
/// It conforms to the ReferenceMap concept.
|
505 |
512 |
template<class T>
|
506 |
513 |
class NodeMap : public ReferenceMap<Node, T, T&, const T&>
|
507 |
514 |
{
|
508 |
515 |
public:
|
509 |
516 |
|
510 |
|
///\e
|
511 |
|
NodeMap(const Graph&) { }
|
512 |
|
///\e
|
|
517 |
/// Constructor
|
|
518 |
explicit NodeMap(const Graph&) { }
|
|
519 |
/// Constructor with given initial value
|
513 |
520 |
NodeMap(const Graph&, T) { }
|
514 |
521 |
|
515 |
522 |
private:
|
... |
... |
@@ -524,18 +531,20 @@
|
524 |
531 |
}
|
525 |
532 |
};
|
526 |
533 |
|
527 |
|
/// \brief Reference map of the arcs to type \c T.
|
|
534 |
/// \brief Standard graph map type for the arcs.
|
528 |
535 |
///
|
529 |
|
/// Reference map of the arcs to type \c T.
|
|
536 |
/// Standard graph map type for the arcs.
|
|
537 |
/// It conforms to the ReferenceMap concept.
|
530 |
538 |
template<class T>
|
531 |
539 |
class ArcMap : public ReferenceMap<Arc, T, T&, const T&>
|
532 |
540 |
{
|
533 |
541 |
public:
|
534 |
542 |
|
535 |
|
///\e
|
536 |
|
ArcMap(const Graph&) { }
|
537 |
|
///\e
|
|
543 |
/// Constructor
|
|
544 |
explicit ArcMap(const Graph&) { }
|
|
545 |
/// Constructor with given initial value
|
538 |
546 |
ArcMap(const Graph&, T) { }
|
|
547 |
|
539 |
548 |
private:
|
540 |
549 |
///Copy constructor
|
541 |
550 |
ArcMap(const ArcMap& em) :
|
... |
... |
@@ -548,18 +557,20 @@
|
548 |
557 |
}
|
549 |
558 |
};
|
550 |
559 |
|
551 |
|
/// Reference map of the edges to type \c T.
|
552 |
|
|
553 |
|
/// Reference map of the edges to type \c T.
|
|
560 |
/// \brief Standard graph map type for the edges.
|
|
561 |
///
|
|
562 |
/// Standard graph map type for the edges.
|
|
563 |
/// It conforms to the ReferenceMap concept.
|
554 |
564 |
template<class T>
|
555 |
565 |
class EdgeMap : public ReferenceMap<Edge, T, T&, const T&>
|
556 |
566 |
{
|
557 |
567 |
public:
|
558 |
568 |
|
559 |
|
///\e
|
560 |
|
EdgeMap(const Graph&) { }
|
561 |
|
///\e
|
|
569 |
/// Constructor
|
|
570 |
explicit EdgeMap(const Graph&) { }
|
|
571 |
/// Constructor with given initial value
|
562 |
572 |
EdgeMap(const Graph&, T) { }
|
|
573 |
|
563 |
574 |
private:
|
564 |
575 |
///Copy constructor
|
565 |
576 |
EdgeMap(const EdgeMap& em) :
|
... |
... |
@@ -572,107 +583,124 @@
|
572 |
583 |
}
|
573 |
584 |
};
|
574 |
585 |
|
575 |
|
/// \brief Direct the given edge.
|
|
586 |
/// \brief The first node of the edge.
|
576 |
587 |
///
|
577 |
|
/// Direct the given edge. The returned arc source
|
578 |
|
/// will be the given node.
|
579 |
|
Arc direct(const Edge&, const Node&) const {
|
580 |
|
return INVALID;
|
581 |
|
}
|
582 |
|
|
583 |
|
/// \brief Direct the given edge.
|
|
588 |
/// Returns the first node of the given edge.
|
584 |
589 |
///
|
585 |
|
/// Direct the given edge. The returned arc
|
586 |
|
/// represents the given edge and the direction comes
|
587 |
|
/// from the bool parameter. The source of the edge and
|
588 |
|
/// the directed arc is the same when the given bool is true.
|
589 |
|
Arc direct(const Edge&, bool) const {
|
590 |
|
return INVALID;
|
591 |
|
}
|
592 |
|
|
593 |
|
/// \brief Returns true if the arc has default orientation.
|
594 |
|
///
|
595 |
|
/// Returns whether the given directed arc is same orientation as
|
596 |
|
/// the corresponding edge's default orientation.
|
597 |
|
bool direction(Arc) const { return true; }
|
598 |
|
|
599 |
|
/// \brief Returns the opposite directed arc.
|
600 |
|
///
|
601 |
|
/// Returns the opposite directed arc.
|
602 |
|
Arc oppositeArc(Arc) const { return INVALID; }
|
603 |
|
|
604 |
|
/// \brief Opposite node on an arc
|
605 |
|
///
|
606 |
|
/// \return The opposite of the given node on the given edge.
|
607 |
|
Node oppositeNode(Node, Edge) const { return INVALID; }
|
608 |
|
|
609 |
|
/// \brief First node of the edge.
|
610 |
|
///
|
611 |
|
/// \return The first node of the given edge.
|
612 |
|
///
|
613 |
|
/// Naturally edges don't have direction and thus
|
614 |
|
/// don't have source and target node. However we use \c u() and \c v()
|
615 |
|
/// methods to query the two nodes of the arc. The direction of the
|
616 |
|
/// arc which arises this way is called the inherent direction of the
|
617 |
|
/// edge, and is used to define the "default" direction
|
618 |
|
/// of the directed versions of the arcs.
|
|
590 |
/// Edges don't have source and target nodes, however methods
|
|
591 |
/// u() and v() are used to query the two end-nodes of an edge.
|
|
592 |
/// The orientation of an edge that arises this way is called
|
|
593 |
/// the inherent direction, it is used to define the default
|
|
594 |
/// direction for the corresponding arcs.
|
619 |
595 |
/// \sa v()
|
620 |
596 |
/// \sa direction()
|
621 |
597 |
Node u(Edge) const { return INVALID; }
|
622 |
598 |
|
623 |
|
/// \brief Second node of the edge.
|
|
599 |
/// \brief The second node of the edge.
|
624 |
600 |
///
|
625 |
|
/// \return The second node of the given edge.
|
|
601 |
/// Returns the second node of the given edge.
|
626 |
602 |
///
|
627 |
|
/// Naturally edges don't have direction and thus
|
628 |
|
/// don't have source and target node. However we use \c u() and \c v()
|
629 |
|
/// methods to query the two nodes of the arc. The direction of the
|
630 |
|
/// arc which arises this way is called the inherent direction of the
|
631 |
|
/// edge, and is used to define the "default" direction
|
632 |
|
/// of the directed versions of the arcs.
|
|
603 |
/// Edges don't have source and target nodes, however methods
|
|
604 |
/// u() and v() are used to query the two end-nodes of an edge.
|
|
605 |
/// The orientation of an edge that arises this way is called
|
|
606 |
/// the inherent direction, it is used to define the default
|
|
607 |
/// direction for the corresponding arcs.
|
633 |
608 |
/// \sa u()
|
634 |
609 |
/// \sa direction()
|
635 |
610 |
Node v(Edge) const { return INVALID; }
|
636 |
611 |
|
637 |
|
/// \brief Source node of the directed arc.
|
|
612 |
/// \brief The source node of the arc.
|
|
613 |
///
|
|
614 |
/// Returns the source node of the given arc.
|
638 |
615 |
Node source(Arc) const { return INVALID; }
|
639 |
616 |
|
640 |
|
/// \brief Target node of the directed arc.
|
|
617 |
/// \brief The target node of the arc.
|
|
618 |
///
|
|
619 |
/// Returns the target node of the given arc.
|
641 |
620 |
Node target(Arc) const { return INVALID; }
|
642 |
621 |
|
643 |
|
/// \brief Returns the id of the node.
|
|
622 |
/// \brief The ID of the node.
|
|
623 |
///
|
|
624 |
/// Returns the ID of the given node.
|
644 |
625 |
int id(Node) const { return -1; }
|
645 |
626 |
|
646 |
|
/// \brief Returns the id of the edge.
|
|
627 |
/// \brief The ID of the edge.
|
|
628 |
///
|
|
629 |
/// Returns the ID of the given edge.
|
647 |
630 |
int id(Edge) const { return -1; }
|
648 |
631 |
|
649 |
|
/// \brief Returns the id of the arc.
|
|
632 |
/// \brief The ID of the arc.
|
|
633 |
///
|
|
634 |
/// Returns the ID of the given arc.
|
650 |
635 |
int id(Arc) const { return -1; }
|
651 |
636 |
|
652 |
|
/// \brief Returns the node with the given id.
|
|
637 |
/// \brief The node with the given ID.
|
653 |
638 |
///
|
654 |
|
/// \pre The argument should be a valid node id in the graph.
|
|
639 |
/// Returns the node with the given ID.
|
|
640 |
/// \pre The argument should be a valid node ID in the graph.
|
655 |
641 |
Node nodeFromId(int) const { return INVALID; }
|
656 |
642 |
|
657 |
|
/// \brief Returns the edge with the given id.
|
|
643 |
/// \brief The edge with the given ID.
|
658 |
644 |
///
|
659 |
|
/// \pre The argument should be a valid edge id in the graph.
|
|
645 |
/// Returns the edge with the given ID.
|
|
646 |
/// \pre The argument should be a valid edge ID in the graph.
|
660 |
647 |
Edge edgeFromId(int) const { return INVALID; }
|
661 |
648 |
|
662 |
|
/// \brief Returns the arc with the given id.
|
|
649 |
/// \brief The arc with the given ID.
|
663 |
650 |
///
|
664 |
|
/// \pre The argument should be a valid arc id in the graph.
|
|
651 |
/// Returns the arc with the given ID.
|
|
652 |
/// \pre The argument should be a valid arc ID in the graph.
|
665 |
653 |
Arc arcFromId(int) const { return INVALID; }
|
666 |
654 |
|
667 |
|
/// \brief Returns an upper bound on the node IDs.
|
|
655 |
/// \brief An upper bound on the node IDs.
|
|
656 |
///
|
|
657 |
/// Returns an upper bound on the node IDs.
|
668 |
658 |
int maxNodeId() const { return -1; }
|
669 |
659 |
|
670 |
|
/// \brief Returns an upper bound on the edge IDs.
|
|
660 |
/// \brief An upper bound on the edge IDs.
|
|
661 |
///
|
|
662 |
/// Returns an upper bound on the edge IDs.
|
671 |
663 |
int maxEdgeId() const { return -1; }
|
672 |
664 |
|
673 |
|
/// \brief Returns an upper bound on the arc IDs.
|
|
665 |
/// \brief An upper bound on the arc IDs.
|
|
666 |
///
|
|
667 |
/// Returns an upper bound on the arc IDs.
|
674 |
668 |
int maxArcId() const { return -1; }
|
675 |
669 |
|
|
670 |
/// \brief The direction of the arc.
|
|
671 |
///
|
|
672 |
/// Returns \c true if the direction of the given arc is the same as
|
|
673 |
/// the inherent orientation of the represented edge.
|
|
674 |
bool direction(Arc) const { return true; }
|
|
675 |
|
|
676 |
/// \brief Direct the edge.
|
|
677 |
///
|
|
678 |
/// Direct the given edge. The returned arc
|
|
679 |
/// represents the given edge and its direction comes
|
|
680 |
/// from the bool parameter. If it is \c true, then the direction
|
|
681 |
/// of the arc is the same as the inherent orientation of the edge.
|
|
682 |
Arc direct(Edge, bool) const {
|
|
683 |
return INVALID;
|
|
684 |
}
|
|
685 |
|
|
686 |
/// \brief Direct the edge.
|
|
687 |
///
|
|
688 |
/// Direct the given edge. The returned arc represents the given
|
|
689 |
/// edge and its source node is the given node.
|
|
690 |
Arc direct(Edge, Node) const {
|
|
691 |
return INVALID;
|
|
692 |
}
|
|
693 |
|
|
694 |
/// \brief The oppositely directed arc.
|
|
695 |
///
|
|
696 |
/// Returns the oppositely directed arc representing the same edge.
|
|
697 |
Arc oppositeArc(Arc) const { return INVALID; }
|
|
698 |
|
|
699 |
/// \brief The opposite node on the edge.
|
|
700 |
///
|
|
701 |
/// Returns the opposite node on the given edge.
|
|
702 |
Node oppositeNode(Node, Edge) const { return INVALID; }
|
|
703 |
|
676 |
704 |
void first(Node&) const {}
|
677 |
705 |
void next(Node&) const {}
|
678 |
706 |
|
... |
... |
@@ -705,47 +733,39 @@
|
705 |
733 |
// Dummy parameter.
|
706 |
734 |
int maxId(Arc) const { return -1; }
|
707 |
735 |
|
708 |
|
/// \brief Base node of the iterator
|
|
736 |
/// \brief The base node of the iterator.
|
709 |
737 |
///
|
710 |
|
/// Returns the base node (the source in this case) of the iterator
|
711 |
|
Node baseNode(OutArcIt e) const {
|
712 |
|
return source(e);
|
713 |
|
}
|
714 |
|
/// \brief Running node of the iterator
|
|
738 |
/// Returns the base node of the given incident edge iterator.
|
|
739 |
Node baseNode(IncEdgeIt) const { return INVALID; }
|
|
740 |
|
|
741 |
/// \brief The running node of the iterator.
|
715 |
742 |
///
|
716 |
|
/// Returns the running node (the target in this case) of the
|
717 |
|
/// iterator
|
718 |
|
Node runningNode(OutArcIt e) const {
|
719 |
|
return target(e);
|
720 |
|
}
|
|
743 |
/// Returns the running node of the given incident edge iterator.
|
|
744 |
Node runningNode(IncEdgeIt) const { return INVALID; }
|
721 |
745 |
|
722 |
|
/// \brief Base node of the iterator
|
|
746 |
/// \brief The base node of the iterator.
|
723 |
747 |
///
|
724 |
|
/// Returns the base node (the target in this case) of the iterator
|
725 |
|
Node baseNode(InArcIt e) const {
|
726 |
|
return target(e);
|
727 |
|
}
|
728 |
|
/// \brief Running node of the iterator
|
|
748 |
/// Returns the base node of the given outgoing arc iterator
|
|
749 |
/// (i.e. the source node of the corresponding arc).
|
|
750 |
Node baseNode(OutArcIt) const { return INVALID; }
|
|
751 |
|
|
752 |
/// \brief The running node of the iterator.
|
729 |
753 |
///
|
730 |
|
/// Returns the running node (the source in this case) of the
|
731 |
|
/// iterator
|
732 |
|
Node runningNode(InArcIt e) const {
|
733 |
|
return source(e);
|
734 |
|
}
|
|
754 |
/// Returns the running node of the given outgoing arc iterator
|
|
755 |
/// (i.e. the target node of the corresponding arc).
|
|
756 |
Node runningNode(OutArcIt) const { return INVALID; }
|
735 |
757 |
|
736 |
|
/// \brief Base node of the iterator
|
|
758 |
/// \brief The base node of the iterator.
|
737 |
759 |
///
|
738 |
|
/// Returns the base node of the iterator
|
739 |
|
Node baseNode(IncEdgeIt) const {
|
740 |
|
return INVALID;
|
741 |
|
}
|
|
760 |
/// Returns the base node of the given incomming arc iterator
|
|
761 |
/// (i.e. the target node of the corresponding arc).
|
|
762 |
Node baseNode(InArcIt) const { return INVALID; }
|
742 |
763 |
|
743 |
|
/// \brief Running node of the iterator
|
|
764 |
/// \brief The running node of the iterator.
|
744 |
765 |
///
|
745 |
|
/// Returns the running node of the iterator
|
746 |
|
Node runningNode(IncEdgeIt) const {
|
747 |
|
return INVALID;
|
748 |
|
}
|
|
766 |
/// Returns the running node of the given incomming arc iterator
|
|
767 |
/// (i.e. the source node of the corresponding arc).
|
|
768 |
Node runningNode(InArcIt) const { return INVALID; }
|
749 |
769 |
|
750 |
770 |
template <typename _Graph>
|
751 |
771 |
struct Constraints {
|