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CGAL 4.3 - CGAL and the Boost Graph Library
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CGAL 4.3 - CGAL and the Boost Graph Library
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Many geometric data structures can be interpreted as graphs, as they consist of vertices, edges and faces. This is the case for the halfedge data structure, for the polyhedron, for arrangements and for triangulations. With means of duality one can also interpret faces as vertices and edges between adjacent faces as edges of the dual graph.
As the scope of CGAL is geometry and not graph algorithms, we provide the necessary classes and functions that allow to use the algorithms of the Boost Graph Library (BGL) [1] for CGAL data structures.
The algorithms of the Bgl operate on models of the various graph concepts. The traits class boost::graph_traits allows the algorithms to determine the types of vertices and edges. Free functions that operator on graphs allow the algorithms to obtain, for example, the source vertex of an edge, or all edges incident to a vertex. The algorithms use property maps to associate information to vertices and edges. The algorithms allow visitors to register callbacks that will be called during the execution of the algorithms. Finally, the graph algorithms use the named parameter mechanism, which allows to pass the arguments in arbitrary order.
The Bgl introduces several graph concepts, which have different sets of characteristics and requirements, as for example whether one can enumerate all vertices or all edges, whether one only can get the outgoing edges of a vertex, or also the ingoing edges, or whether one can add and remove vertices and edges or not.
Graph concepts in the Bgl manual: http://www.boost.org/libs/graph/doc/graph_concepts.html
The algorithms determine types with the help of the traits class boost::graph_traits. Such types are the vertex_descriptor which is equivalent to a vertex handle in CGAL data structures, the vertex_iterator which is similar to the vertex iterators in CGAL data structures, and the out_edge_iterator which is similar to edge circulators, which allow to enumerate the edges incident to a vertex. The latter two are similar and not equivalent, because their value type is a vertex_descriptor, whereas in CGAL handles, iterators, and cicrulators all have the same value type, namely the vertex type. Given a graph type G the declaration of a vertex descriptor looks as follows:
The graph traits in the Bgl manual: http://www.boost.org/libs/graph/doc/graph_traits.html
The algorithms obtain incidence information with the help of global functions like
pair<vertex_iterator,vertex_iterator> vertices(const Graph& g); for getting an iterator range which allows to enumerate all vertices, orint num_vertices(const Graph&); for getting the number of vertices of a graph, orvertex_descriptor source(edge_descriptor, const Graph&, for getting the source vertex of an edge.Note, that the way we have written the types is a simplification, that is in reality the signature of the first of the above functions is
The free functions required for graph concepts: http://www.boost.org/libs/graph/doc/graph_concepts.html
Another feature used heavily in the Bgl is the property map which is offered by the Boost Property Map Library. Property maps are used to attach information to vertices and edges. It is again a traits class and some free functions for obtaining the property map from a graph, and for getting and putting properties.
The free functions are get and put. The first one is overloaded. One version allows to obtain a property map for a given property tag. For example m = get(g, boost::vertex_index) gives us a property map that associates an index in the range [0, num_vertices(g)) to each vertex descriptor of the graph. The second version of the get function allows to read it as follows for a vertex descriptor vd: int vdi = get(m, vd). Just as get allows to read data, put allows to write them. For example, the Dijksta's shortest path algorithm writes the predecessor of each vertex, as well as the distance to the source in such a property map.
The data themselves may be stored in the vertex or edge, or they may be stored in an external data structure, or they may be computed on the fly. This is an "implementation detail" of the particular property map.
Property maps in the Boost manuals: http://www.boost.org/libs/property_map/doc/property_map.html
Visitors are ojects that provide functions that get called at specified event points by the algorithm they visit. The notion of visitors is a design pattern, and also used in CGAL, e.g., the Arr_observer<Arrangement> in the arrangement package.
The functions as well as the event points are library specific. Event points in graph algorithms are, for example, when a vertex is traversed the first time, or when all outgoing edges of a vertex are traversed.
Visitors in the Bgl manual: http://www.boost.org/libs/graph/doc/visitor_concepts.html
The algorithms of the Bgl often have many parameters. Although the default value is most often appropriate, one has to write them explicitly, if one only wants to deviate from the default for the last one. The solution to this problem is to first write a tag and then the parameter, which for Dijkstra's shortest path algorithm, might look as follows:
The named parameters in the example use the tags predecessor_map and distance_map and they are concatenated with the dot operator.
Named parameters in the Bgl manual: http://www.boost.org/libs/graph/doc/bgl_named_params.html
CGAL provides the partial specializations and free functions such that several data structures become model of some of the Bgl graph concepts. Furthermore, we define the new graph concept HalfedgeGraph, a traits class halfedge_graph_traits, and free functions for accessing opposite edges as well as the clockwise and counterclockwise neighbor of an edge around a given vertex.
These extensions are used by the surface simplification algorithms which follow the design of the Bgl as sketched in the previous section.
As we interface two libraries we have to explain what resides in which namespace, and what naming conventions we apply to what.
Partial specializations of the boost::graph_traits<Graph> for the CGAL package Package are in the namespace boost and in the headerfile <CGAL/boost/graph/graph_traits_Package.h>.
The halfedge_graph_traits class is in the namespace CGAL, but it is not capitalized as the boost::graph_traits is not. The same holds for the types and enums for vertex and edge properties.
The class Polyhedron_3 is model of the graph concept. Furthermore this chapter introduces a new graph concept, the HalfedgeGraph.
The example code computes the minimum spanning tree on a polyhedral surface. More examples can be found in Chapter Chapter_Triangulated_Surface_Mesh_Simplification on surface mesh simplification.
File BGL_polyhedron_3/kruskal.cpp
The following example program shows a call to the Bgl Kruskal's minimum spanning tree algorithm accessing the id() field stored in a Polyhedron vertex.
The main function illustrates the access to the id() field.
File BGL_polyhedron_3/kruskal_with_stored_id.cpp
Triangulations have vertices and faces. Edges are pairs of a face and the index of the edge. Particular care has to be taken with the infinite vertex, and its incident edges. One can either use a boost::filtered_graph, which makes the infinite edges invisible, or one can have a property map that returns an infinite length for these edges.
A classical example for an algorithm that is a combination of computational geometry and graph theory is the Euclidean Minimum Spanning Tree for a point set in the plane. It can be computed by running the minimum spanning tree algorithm on a Delaunay triangulation of the point set.
In the following example we create a Delaunay triangulation and run Kruskal's minimum spanning tree algorithm on it. Because the vertex handles of the triangulation are not indices in an array, we have to provide a property map that maps vertex handles to int's in the range [0, t.number_of_vertices()).
File BGL_triangulation_2/emst.cpp
The algorithms of the Bgl extensively use of the indices of vertices. In the previous example we stored the index in a std::map and turned that map in a property map. This property map was then passed as argument to the shortest path function.
If the user does not pass explicitly a property map, the graph algorithms use the property map returned by the call boost::get(boost::vertex_index,ft). This property map assumes that the vertex has a member function id() that returns a reference to an int. Therefore CGAL offers a class Triangulation_vertex_base_with_id_2. It is in the users responsibility to set the indices properly.
The example further illustrates that the graph traits also works for the Delaunay triangulation.
File BGL_triangulation_2/dijkstra_with_internal_properties.cpp
For the arrangements CGAL offers the graph traits for the arrangement itself as well as for its dual graph.
Arrangement instances are adapted to boost graphs by specializing the boost:graph_traits template for Arrangement_2 instances. The graph-traits states the graph concepts that the arrangement class models (see below) and defines the types required by these concepts.
In this specialization the Arrangement_2 vertices correspond to the graph vertices, where two vertices are adjacent if there is at least one halfedge connecting them. More precisely, Arrangement_2::Vertex_handle is the graph-vertex type, while Arrangement_2::Halfedge_handle is the graph-edge type. As halfedges are directed, we consider the graph to be directed as well. Moreover, as several interior-disjoint \( x\)-monotone curves (say circular arcs) may share two common endpoints, inducing an arrangement with two vertices that are connected with several edges, we allow parallel edges in our boost graph.
Given an Arrangement_2 instance, we can efficiently traverse its vertices and halfedges. Thus, the arrangement graph is a model of the concepts VertexListGraph and EdgeListGraph introduced by the bgl. At the same time, we use an iterator adapter of the circulator over the halfedges incident to a vertex (Halfedge_around_vertex_circulator - see Section Traversal Methods for an Arrangement Vertex of the chaper on arrangements), so it is possible to go over the ingoing and outgoing edges of a vertex in linear time. Thus, our arrangement graph is a model of the concept BidirectionalGraph (this concept refines IncidenceGraph, which requires only the traversal of outgoing edges).
It is important to notice that the vertex descriptors we use are Vertex_handle objects and not vertex indices. However, in order to gain more efficiency in most Bgl algorithm, it is better to have them indexed \( 0, 1, \ldots, (n-1)\), where \( n\) is the number of vertices. We therefore introduce the Arr_vertex_index_map class-template, which maintains a mapping of vertex handles to indices, as required by the sc{bgl}. An instance of this class must be attached to a valid arrangement vertex when it is created. It uses the notification mechanism (see Section The Notification Mechanism) to automatically maintain the mapping of vertices to indices, even when new vertices are inserted into the arrangement or existing vertices are removed.
In most algorithm provided by the bgl, the output is given by property maps, such that each map entry corresponds to a vertex. For example, when we compute the shortest paths from a given source vertex \( s\) to all other vertices we can obtain a map of distances and a map of predecessors - namely for each \( v\) vertex we have its distance from \( s\) and a descriptor of the vertex that precedes \( v\) in the shortest path from \( s\). If the vertex descriptors are simply indices, one can use vectors to efficiently represent the property maps. As this is not the case with the arrangement graph, we offer the Arr_vertex_property_map<Arrangement,Type> template allows for an efficient mapping of Vertex_handle objects to properties of type Type. Note however that unlike the Arr_vertex_index_map class, the vertex property-map class is not kept synchronized with the number of vertices in the arrangement, so it should not be reused in calls to Bgl functions in case the arrangement is modified in between these calls.
ex_bgl_primal_adapter.cpp and ex_bgl_dual_adapter.cpp. The breadth-first visit times for the arrangement faces, starting from the unbounded face \( f_0\), are shown is brackets. In the following example we construct an arrangement of 7 line segments, as shown in Figure 72.1, then use Dijkstra's shortest-paths algorithm from the Bgl to compute the graph distance of all vertices from the leftmost vertex in the arrangement \( v_0\). Note the usage of the Arr_vertex_index_map and the Arr_vertex_property_map classes. The latter one, instantiated by the type double is used to map vertices to their distances from \( v_0\).
File BGL_arrangement_2/primal.cpp
It is possible to give a dual graph representation for an arrangement instance, such that each arrangement face corresponds to a graph vertex and two vertices are adjacent iff the corresponding faces share a common edge on their boundaries. This is done by specializing the boost:graph_traits template for Dual<Arrangement_2> instances, where Dual<Arrangement_2> is a template specialization that gives a dual interpretation to an arrangement instance.
In dual representation, Arrangement_2::Face_handle is the graph-vertex type, while Arrangement_2::Halfedge_handle is the graph-edge type. We treat the graph edges as directed, such that a halfedge e is directed from \( f_1\), which is its incident face, to \( f_2\), which is the incident face of its twin halfedge. As two arrangement faces may share more than a single edge on their boundary, we allow parallel edges in our boost graph. As is the case in the primal graph, the dual arrangement graph is also a model of the concepts VertexListGraph, EdgeListGraph and BidirectionalGraph (thus also of IncidenceGraph).
Since we use Face_handle objects as the vertex descriptors, we define the Arr_face_index_map<Arrangement> class-template, which maintains an efficient mapping of face handles to indices. We also provide the template Arr_face_property_map<Arrangement,Type> for associating arbitrary data with the arrangement faces.
In the following example we construct the same arrangement as in example ex_bgl_primal_adapter.cpp (see Figure 29.29), and perform breadth-first search on the graph faces, starting from the unbounded face. We extend the Dcel faces with an unsigned integer, marking the discover time of the face and use a breadth-first-search visitor to obtain these times and update the faces accordingly:
File BGL_arrangement_2/dual.cpp
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