Boost Graph Library: Adjacency Matrix

adjacency_matrix<Directed, VertexProperty, 
                 EdgeProperty, GraphProperty,
                 Allocator>

The adjacency_matrix class implements the BGL graph interface using the traditional adjacency matrix storage format. For a graph with V vertices, a V x V matrix is used, where each element a_ij is a boolean flag that says whether there is an edge from vertex i to vertex j. Figure 1 shows the adjacency matrix representation of a graph.

**Figure 1:** Adjacency Matrix Representation of a Directed Graph.

The advantage of this matrix format over the adjacency list is that edge insertion and removal is constant time. There are several disadvantages. The first is that the amount of memory used is O(V²) instead of O(V + E) (where E is the number of edges). The second is that operations that traverse all the out-edges of each vertex (such as breadth-first search) run in O(V²) time instead of O(V + E) time for the adjacency list. In short, it is better to use the adjacency_matrix for dense graphs (where E is close to V²) and it is better to use adjacency_list for sparse graphs (where E is much smaller than V²). The adjacency_matrix class extends the traditional data-structure by allowing objects to be attached to vertices and edges using the same property template parameters supported by adjacency_list. These may be bundled properties or standard (backward-compatible) interior properties. The types of all property values must be Copy Constructible, Assignable and Default Constructible. In the case of an undirected graph, the adjacency_matrix. class does not use a full V x V matrix but instead uses a lower triangle (the diagonal and below) since the matrix for an undirected graph is symmetric. This reduces the storage to (V²)/2. Figure 2 shows an adjacency matrix representation of an undirected graph.

**Figure 1:** Adjacency Matrix Representation of an Undirected Graph.

Example

Creating the graph of Figure 1.

  enum { A, B, C, D, E, F, N };
  const char* name = "ABCDEF";
  
  typedef boost::adjacency_matrix<boost::directedS> Graph;
  Graph g(N);
  add_edge(B, C, g);
  add_edge(B, F, g);
  add_edge(C, A, g);
  add_edge(C, C, g);
  add_edge(D, E, g);
  add_edge(E, D, g);
  add_edge(F, A, g);

  std::cout << "vertex set: ";
  boost::print_vertices(g, name);
  std::cout << std::endl;

  std::cout << "edge set: ";
  boost::print_edges(g, name);
  std::cout << std::endl;

  std::cout << "out-edges: " << std::endl;
  boost::print_graph(g, name);
  std::cout << std::endl;

The output is:

  vertex set: A B C D E F 

  edge set: (B,C) (B,F) (C,A) (C,C) (D,E) (E,D) (F,A) 

  out-edges: 
  A --> 
  B --> C F 
  C --> A C 
  D --> E 
  E --> D 
  F --> A

Creating the graph of Figure 2.

  enum { A, B, C, D, E, F, N };
  const char* name = "ABCDEF";

  typedef boost::adjacency_matrix<boost::undirectedS> UGraph;
  UGraph ug(N);
  add_edge(B, C, ug);
  add_edge(B, F, ug);
  add_edge(C, A, ug);
  add_edge(D, E, ug);
  add_edge(F, A, ug);

  std::cout << "vertex set: ";
  boost::print_vertices(ug, name);
  std::cout << std::endl;

  std::cout << "edge set: ";
  boost::print_edges(ug, name);
  std::cout << std::endl;

  std::cout << "incident edges: " << std::endl;
  boost::print_graph(ug, name);
  std::cout << std::endl;

The output is:

  vertex set: A B C D E F 

  edge set: (C,A) (C,B) (E,D) (F,A) (F,B) 

  incident edges: 
  A <--> C F 
  B <--> C F 
  C <--> A B 
  D <--> E 
  E <--> D 
  F <--> A B

Where Defined

boost/graph/adjacency_matrix.hpp

Template Parameters

Parameter Description Default

Directed A selector to choose whether the graph is directed or undirected. The options are directedS and undirectedS. directedS

VertexProperty for specifying internal property storage. no_property

EdgeProperty for specifying internal property storage. no_property

GraphProperty for specifying property storage for the graph object. no_property

Parameter	Description	Default
`Directed`	A selector to choose whether the graph is directed or undirected. The options are `directedS` and `undirectedS`.	`directedS`
`VertexProperty`	for specifying internal property storage.	`no_property`
`EdgeProperty`	for specifying internal property storage.	`no_property`
`GraphProperty`	for specifying property storage for the graph object.	`no_property`

Model Of

VertexAndEdgeListGraph, AdjacencyMatrix, MutablePropertyGraph, CopyConstructible, and Assignable.

Associates Types

graph_traits<adjacency_matrix>::vertex_descriptor

The type for the vertex descriptors associated with the adjacency_matrix.
(Required by Graph.)

graph_traits<adjacency_matrix>::edge_descriptor

The type for the edge descriptors associated with the adjacency_matrix.
(Required by Graph.)

graph_traits<adjacency_matrix>::vertex_iterator

The type for the iterators returned by vertices(). The vertex iterator models RandomAccessIterator.
(Required by VertexListGraph.)

graph_traits<adjacency_matrix>::edge_iterator

The type for the iterators returned by edges(). This iterator models MultiPassInputIterator.
(Required by EdgeListGraph.)

graph_traits<adjacency_matrix>::out_edge_iterator

The type for the iterators returned by out_edges(). This iterator models MultiPassInputIterator.
(Required by IncidenceGraph.)

graph_traits<adjacency_matrix>::adjacency_iterator

The type for the iterators returned by adjacent_vertices(). This iterator models the same concept as the out-edge iterator.
(Required by AdjacencyGraph.)

graph_traits<adjacency_matrix>::directed_category

Provides information about whether the graph is directed (directed_tag) or undirected (undirected_tag).
(Required by Graph.)

graph_traits<adjacency_matrix>::edge_parallel_category

An adjacency matrix does not allow the insertion of parallel edges, so this type is always disallow_parallel_edge_tag.
(Required by Graph.)

graph_traits<adjacency_matrix>::vertices_size_type

The type used for dealing with the number of vertices in the graph.
(Required by VertexListGraph.)

graph_traits<adjacency_matrix>::edges_size_type

The type used for dealing with the number of edges in the graph.
(Required by EdgeListGraph.)

graph_traits<adjacency_matrix>::degree_size_type

The type used for dealing with the number of out-edges of a vertex.
(Required by IncidenceGraph.)

property_map<adjacency_matrix, PropertyTag>::type
property_map<adjacency_matrix, PropertyTag>::const_type

The map type for vertex or edge properties in the graph. The specific property is specified by the PropertyTag template argument, and must match one of the properties specified in the VertexProperty or EdgeProperty for the graph.
(Required by PropertyGraph.)

Member Functions

adjacency_matrix(vertices_size_type n,
                 const GraphProperty& p = GraphProperty())

Creates a graph object with n vertices and zero edges.
(Required by MutableGraph.)

template <typename EdgeIterator>
adjacency_matrix(EdgeIterator first,
                 EdgeIterator last,
                 vertices_size_type n,
                 const GraphProperty& p = GraphProperty())

Creates a graph object with n vertices with the edges specified in the edge list given by the range [first, last). The value type of the EdgeIterator must be a std::pair, where the type in the pair is an integer type. The integers will correspond to vertices, and they must all fall in the range of [0, n).
(Required by IteratorConstructibleGraph.)

template <typename EdgeIterator, typename EdgePropertyIterator>
adjacency_matrix(EdgeIterator first, EdgeIterator last,
                 EdgePropertyIterator ep_iter,
                 vertices_size_type n,
                 const GraphProperty& p = GraphProperty())

Creates a graph object with n vertices, with the edges specified in the edge list given by the range [first, last). The value type of the EdgeIterator must be a std::pair, where the type in the pair is an integer type. The integers will correspond to vertices, and they must all fall in the range of [0, n). The value_type of the ep_iter should be EdgeProperty.

Non-Member Functions

std::pair<vertex_iterator, vertex_iterator>
vertices(const adjacency_matrix& g)

Returns an iterator-range providing access to the vertex set of graph g. (Required by VertexListGraph.) std::pair<edge_iterator, edge_iterator> edges(const adjacency_matrix& g) Returns an iterator-range providing access to the edge set of graph g. (Required by EdgeListGraph.) std::pair<adjacency_iterator, adjacency_iterator> adjacent_vertices(vertex_descriptor v, const adjacency_matrix& g) Returns an iterator-range providing access to the vertices adjacent to vertex v in graph g. (Required by AdjacencyGraph.) std::pair<out_edge_iterator, out_edge_iterator> out_edges(vertex_descriptor v, const adjacency_matrix& g) Returns an iterator-range providing access to the out-edges of vertex v in graph g. If the graph is undirected, this iterator-range provides access to all edge incident on vertex v. (Required by IncidenceGraph.) vertex_descriptor source(edge_descriptor e, const adjacency_matrix& g) Returns the source vertex of edge e. (Required by IncidenceGraph.) vertex_descriptor target(edge_descriptor e, const adjacency_matrix& g) Returns the target vertex of edge e. (Required by IncidenceGraph.) degree_size_type out_degree(vertex_descriptor u, const adjacency_matrix& g) Returns the number of edges leaving vertex u. (Required by IncidenceGraph.) vertices_size_type num_vertices(const adjacency_matrix& g) Returns the number of vertices in the graph g. (Required by VertexListGraph.) edges_size_type num_edges(const adjacency_matrix& g) Returns the number of edges in the graph g. (Required by EdgeListGraph.) vertex_descriptor vertex(vertices_size_type n, const adjacency_matrix& g) Returns the nth vertex in the graph's vertex list. std::pair<edge_descriptor, bool> edge(vertex_descriptor u, vertex_descriptor v, const adjacency_matrix& g) Returns the edge connecting vertex u to vertex v in graph g. (Required by AdjacencyMatrix.) std::pair<edge_descriptor, bool> add_edge(vertex_descriptor u, vertex_descriptor v, adjacency_matrix& g) Adds edge (u,v) to the graph and returns the edge descriptor for the new edge. If the edge is already in the graph then a duplicate will not be added and the bool flag will be false. This operation does not invalidate any of the graph's iterators or descriptors. (Required by MutableGraph.) std::pair<edge_descriptor, bool> add_edge(vertex_descriptor u, vertex_descriptor v, const EdgeProperty& p, adjacency_matrix& g) Adds edge (u,v) to the graph and attaches p as the value of the edge's internal property storage. Also see the previous add_edge() member function for more details. void remove_edge(vertex_descriptor u, vertex_descriptor v, adjacency_matrix& g) Removes the edge (u,v) from the graph. (Required by MutableGraph.) void remove_edge(edge_descriptor e, adjacency_matrix& g) Removes the edge e from the graph. This is equivalent to calling remove_edge(source(e, g), target(e, g), g). (Required by MutableGraph.) void clear_vertex(vertex_descriptor u, adjacency_matrix& g) Removes all edges to and from vertex u. The vertex still appears in the vertex set of the graph. (Required by MutableGraph.) template <typename Property> property_map<adjacency_matrix, Property>::type get(Property, adjacency_matrix& g) template <typename Property> property_map<adjacency_matrix, Property>::const_type get(Property, const adjacency_matrix& g) Returns the property map object for the vertex property specified by Property. The Property must match one of the properties specified in the graph's VertexProperty template argument. (Required by PropertyGraph.) template <typename Property, typename X> typename property_traits< typename property_map<adjacency_matrix, Property>::const_type >::value_type get(Property, const adjacency_matrix& g, X x) This returns the property value for x, which is either a vertex or edge descriptor. (Required by PropertyGraph.) template <typename Property, typename X, typename Value> void put(Property, const adjacency_matrix& g, X x, const Value& value) This sets the property value for x to value. x is either a vertex or edge descriptor. Value must be convertible to typename property_traits<property_map<adjacency_matrix, Property>::type>::value_type. (Required by PropertyGraph.) template <typename GraphProperty, typename GraphProperty> typename property_value<GraphProperty, GraphProperty>::type& get_property(adjacency_matrix& g, GraphProperty) Return the property specified by GraphProperty that is attached to the graph object g. The property_value traits class is defined in boost/pending/property.hpp. template <typename GraphProperty, typename GraphProperty> const typename property_value<GraphProperty, GraphProperty>::type& get_property(const adjacency_matrix& g, GraphProperty) Return the property specified by GraphProperty that is attached to the graph object g. The property_value traits class is defined in boost/pending/property.hpp.