#
Using `adjacency_list`

This section describes the details of how use the `adjacency_list` class. The presentation is divided into the following topics:

1. Choosing the `Edgelist` and `VertexList`
2. Directed and Undirected Adjacency Lists
3. Internal Properties
4. Customizing the Adjacency List Storage

##
Choosing the `Edgelist` and `VertexList`

This section focuses on how to decide which version of the `adjacency_list` class to use in different situations. The `adjacency_list` is like a swiss-army knife in that it can be configured in many ways. The parameters that we will focus on in this section are `EdgeList` and `VertexList`, which control the underlying data structures that will be used to represent the graph. The choice of `EdgeList` and `VertexList` affects the time complexity of many of the graph operations and the space complexity of the graph object.

BGL uses containers from the STL such as `std::vector`, `std::list`, and `std::set` to represent the set of vertices and the adjacency structure (out-edges and in-edges) of the graph. There are several selector types that are used to specify the choice of container for `EdgeList` and `VertexList`.

- `vecS` selects `std::vector`.
- `listS` selects `std::list`.
- `slistS` selects `std::slist`.
- `setS` selects `std::set`.
- `multisetS` selects `std::multiset`.
- `hash_setS` selects `std::hash_set`.

### Choosing the `VertexList` type

The `VertexList` parameter determines what kind of container will be used to represent the vertex set, or two-dimensional structure of the graph. The container must model Sequence or RandomAccessContainer. In general, `listS` is a good choice if you need to add and remove vertices quickly. The price for this is extra space overhead compared to choosing `vecS`.

#### Space Complexity

The `std::list` has a higher per-vertex space overhead than the `std::vector`, storing three extra pointers per vertex.

#### Time Complexity

The choice of `VertexList` affects the time complexity of the following operations.

-     add_vertex()

  This operation is amortized constant time for both `vecS` and `listS` (implemented with `push_back()`). However, when the `VertexList` type is `vecS` the time for this operation is occasionally large because the vector will be reallocated and the whole graph copied.

-     remove_vertex()

  This operation is constant time for `listS` and O(V + E) for `vecS`. The large time complexity for `vecS` is because the vertex descriptors (which in this case are indices that correspond to the vertices' place in the vertex list) must be adjusted in the out-edges for the whole graph.

-     vertex()

  This operation is constant time for `vecS` and for `listS`.

### Choosing the `EdgeList` type

The `EdgeList` parameter determines what kind of container will be used to store the out-edges (and possibly in-edges) for each vertex in the graph. The containers used for edge lists must either satisfy the requirements for Sequence or for AssociativeContainer.

One of the first things to consider when choosing the `EdgeList` is whether you want `adjacency_list` to enforce the absence of parallel edges in the graph (that is, enforce that the graph not become a multi-graph). If you want this enforced then use the `setS` or `hash_setS` selectors. If you want to represent a multi-graph, or know that you will not be inserting parallel edges into the graph, then choose one of the Sequence types: `vecS`, `listS`, or `slistS`. You will also want to take into account the differences in time and space complexity for the various graph operations. Below we use V for the total number of vertices in the graph and E for the total number of edges. Operations not discussed here are constant time.

#### Space Complexity

The selection of the `EdgeList` affects the amount of space overhead per edge in the graph object. In the order of least space to most space, the selectors are `vecS`, `slistS`, `listS`, and `setS`.

#### Time Complexity

In the following description of the time complexity for various operations, we use E/V inside of the ``big-O'' notation to express the length of an out-edge list. Strictly speaking this is not accurate because E/V merely gives the average number of edges per vertex in a random graph. The worst-case number of out-edges for a vertex is V (unless it is a multi-graph). For sparse graphs E/V is typically much smaller than V and can be considered a constant.

-     add_edge()

  When the `EdgeList` is a UniqueAssociativeContainer like `std::set` the absence of parallel edges is enforced when an edge is added. The extra lookup involved has time complexity O(log(E/V)). The `EdgeList` types that model Sequence do not perform this check and therefore `add_edge()` is amortized constant time. This means that it if you don't care whether the graph has parallel edges, or know that the input to the graph does not contain them, then it is better to use the sequence-based `EdgeList`. The `add_edge()` for the sequence-based `EdgeList` is implemented with `push_front()` or `push_back()`. However, for `std::list` and `std::slist` this operation will typically be faster than with `std::vector` which occasionally reallocates and copies all elements.

-     remove_edge()

  For sequence-based `EdgeList` types this operation is implemented with `std::remove_if()` which means the average time is E/V. For set-based `EdgeList` types this is implemented with the `erase()` member function, which has average time log(E/V).

-     edge()

  The time complexity for this operation is O(E/V) when the `EdgeList` type is a Sequence and it is O(log(E/V)) when the `EdgeList` type is an AssociativeContainer.

-     clear_vertex()

  For directed graphs with sequence-based `EdgeList` types the time complexity is O(V + E), while for associative container based `EdgeList` types the operation is faster, with time complexity O(V log(E/V)). For undirected graphs this operation is O(E2/V2) or O(E/V log(E/V)).

-     remove_vertex()

  The time complexity for this operation is O(V + E) regardless of the `EdgeList` type.

-     out_edge_iterator::operator++()

  This operation is constant time for all the `OneD` types. However, there is a significant constant factor time difference between the various types, which is important since this operation is the work-horse of most graph algorithms. The speed of this operation in order of fastest to slowest is `vecS`, `slistS`, `listS`, `setS`, `hash_setS`.

-     in_edge_iterator::operator++()

  This operation is constant time and exhibits a similar speed ordering as the `out_edge_iterator` with respect to the `EdgeList` selection.

-     vertex_iterator::operator++()

  This operation is constant time and fast (same speed as incrementing a pointer). The selection of `OneD` does not affect the speed of this operation.

-     edge_iterator::operator++()

  This operation is constant time and exhibits a similar speed ordering as the `out_edge_iterator` with respect to the `EdgeList` selection. Traversing through the whole edge set is O(V + E).

-     adjacency_iterator::operator++()

  This operation is constant time and exhibits a similar speed ordering as the `out_edge_iterator` with respect to the `EdgeList` selection.

## Directed and Undirected Adjacency Lists

The `adjacency_list` class can be used to represent both directed and undirected graphs, depending on the argument passed to the `Directed` template parameter. Selecting `directedS` or `bidirectionalS` choose a directed graph, whereas `undirectedS` selects the representation for an undirected graph. See Section Undirected Graphs for a description of the difference between directed and undirected graphs in BGL. The `bidirectealS` selector specifies that the graph will provide the `in_edges()` function as well as the `out_edges()` function. This imposes twice as much space overhead per edge, which is why `in_edges()` is optional.

## Internal Properties

Properties can be attached to the vertices or edges of an `adjacency_list` graph via the property interface. The template parameters `VertexProperty` and `EdgeProperty` of the `adjacency_list` class are meant to be filled by the property class, which is declared as follows.

    template <class PropertyTag, class T, class NextProperty = no_property>
    struct property;

The `PropertyTag` is a tag class (empty class) that simply identifies or gives a unique name to the property. There are several predefined tags, and it is easy to add more.

      struct vertex_index_t { }
      struct edge_index_t { }
      struct graph_name_t { }
      struct vertex_name_t { }
      struct edge_name_t { }
      struct edge_weight_t { }
      struct vertex_distance_t { }
      struct vertex_color_t { }
      struct vertex_degree_t { }
      struct vertex_out_degree_t { }
      struct vertex_in_degree_t { }
      struct vertex_discover_time_t { }
      struct vertex_finish_time_t { }

The `T` parameter of `property` specifies the type of the property values. The `NextProperty` parameter allows `property` types to be nested, so that an arbitrary number of properties can be attached to the same graph.

The following code shows how a vertex and edge property type can be assembled and used to create a graph type. We have attached a distance property with values of type `float` and a name property with values of type `std::string` to the vertices of the graph. We have attached a weight property with values of type `float` to the edges of the graph.

      typedef property<vertex_distance_t, float,
                property<vertex_name_t, std::string> > VertexProperty;
      typedef property<edge_weight_t, float> EdgeProperty;

      typedef adjacency_list<mapS, vecS, undirectedS,
                             VertexProperty, EdgeProperty> Graph;

      Graph g(num_vertices); // construct a graph object

The property values are then read from and written to using property maps. See Section Interior Properties for a description of how to obtain property maps from a graph, and read Section ;Property Maps for how to use property maps.

### Custom Edge Properties

Creating a your own property types and properties is easy; just define a tag class for your new property. Here we define a tag class for capacity and flow properties which we will be attaching to the edges on the graph.

      struct capacity_tag { };
      struct flow_tag { };

Now you can use your new property tag in the definition of properties just as you would one of the builtin tags.

      typedef property<capacity_tag, int> Cap;
      typedef property<flow_tag, int, Cap> EdgeProperty;
      typedef adjacency_list<vecS, vecS, no_property, EdgeProperty> Graph;

Just as before, the property maps for these properties can be obtained from the graph via the `get(Property, g)` function.

      property_map<Graph, capacity_tag>::type capacity
        = get(capacity_tag(), G);
      property_map<Graph, flow_tag>::type flow
        = get(flow_tag(), G);

The file `edge_property.cpp` shows the complete source code for this example.

###
Custom Vertex Properties

Creating your own properties to attach to vertices is just as easy as for edges. Here we want to attach people's first names to the vertices in the graph.

      struct first_vertex_name_t { };

Now we can use the new tag in the `property` class and use that in the assembly of a graph type. The following code shows creating the graph type, and then creating the graph object. We fill in the edges and also assign names to the vertices. The edges will represent ``who owes who''.

      typedef property<first_vertex_name_t, std::string> FirstNameProperty;
      typedef adjacency_list<vecS, vecS, directedS,
                             FirstNameProperty> MyGraphType;

      typedef pair<int,int> Pair;
      Pair edge_array[11] = { Pair(0,1), Pair(0,2), Pair(0,3),
                              Pair(0,4), Pair(2,0), Pair(3,0),
                              Pair(2,4), Pair(3,1), Pair(3,4),
                              Pair(4,0), Pair(4,1) };

      MyGraphType G(5);
      for (int i = 0; i < 11; ++i)
        add_edge(edge_array[i].first, edge_array[i].second, G);

      property_map<MyGraphType, first_vertex_name_t>::type
        name = get(first_vertex_name, G);

      boost::put(name, 0, "Jeremy");
      boost::put(name, 1, "Rich");
      boost::put(name, 2, "Andrew");
      boost::put(name, 3, "Jeff");
      name[4] = "Kinis"; // you can use operator[] too

      who_owes_who(edges(G).first, edges(G).second, G);

The `who_owes_who()` function written for this example was implemented in a generic style. The input is templated so we do not know the actual graph type. To find out the type of the property map for our first name property, we need to use the `property_map` traits class. The `const_type` is used since the graph parameter is const. Once we have the property map type, we can deduce the value type of the property using the `property_traits` class. In this example, we know that the property's value type will be `std::string`, but written in this generic fashion the `who_owes_who()` function could work with other property value types.

      template <class EdgeIter, class Graph>
      void who_owes_who(EdgeIter first, EdgeIter last, const Graph& G)
      {
        // Access the propety acessor type for this graph
        typedef typename property_map<Graph,
          first_vertex_name_t>::const_type NameMap;
        NameMap name = get(first_vertex_name, G);

        typedef typename boost::property_traits<NameMap>
          ::value_type NameType;

        NameType src_name, targ_name;

        while (first != last) {
          src_name = boost::get(name, source(*first, G));
          targ_name = boost::get(name, target(*first, G));
          cout << src_name << " owes "
               << targ_name << " some money" << endl;
          ++first;
        }

The output is:

    Jeremy owes Rich some money
    Jeremy owes Andrew some money
    Jeremy owes Jeff some money
    Jeremy owes Kinis some money
    Andrew owes Jeremy some money
    Andrew owes Kinis some money
    Jeff owes Jeremy some money
    Jeff owes Rich some money
    Jeff owes Kinis some money
    Kinis owes Jeremy some money
    Kinis owes Rich some money

The complete source code to this example is in the file `interior_property_map.cpp`.

## Customizing the Adjacency List Storage

The `adjacency_list` is constructed out of two kinds of containers. One type of container to hold all the vertices in the graph, and another type of container for the out-edge list (and potentially in-edge list) for each vertex. BGLprovides selector classes that allow the user to choose between several of the container from the STL. It is also possible to use your own container types. When customizing the `VertexList` you need to define a container generator as described below. When customizing the `EdgeList` you will need to define a container generator and the parallel edge traits. The file `container_gen.cpp` has an example of how to use a custom storage types.

### Container Generator

The `adjacency_list` class uses a traits class called `container_gen` to map the `EdgeList` and `VertexList` selectors to the actual container types used for the graph storage. The default version of the traits class is listed below, along with an example of how the class is specialized for the `listS` selector.

    namespace boost {
      template <class Selector, class ValueType>
      struct container_gen { };

      template <class ValueType>
      struct container_gen<listS, ValueType> {
        typedef std::list<ValueType> type;
      };
    }

To use some other container of your choice, define a selector class and then specialize the `container_gen` for your selector.

      struct custom_containerS { }; // your selector

      namespace boost {
        // the specialization for your selector
        template <class ValueType>
        struct container_gen<custom_containerS, ValueType> {
          typedef custom_container<ValueType> type;
        };
      }

There may also be situations when you want to use a container that has more template parameters than just `ValueType`. For instance, you may want to supply the allocator type. One way to do this is to hard-code in the extra parameters within the specialization of `container_gen`. However, if you want more flexibility then you can add a template parameter to the selector class. In the code below we show how to create a selector that lets you specify the allocator to be used with the `std::list`.

      template <class Allocator>
      struct list_with_allocatorS { };

      namespace boost {
        template <class Alloc, class ValueType>
        struct container_gen<list_with_allocatorS<Alloc>, ValueType>
        {
          typedef typename Alloc::template rebind<ValueType>::other Allocator;
          typedef std::list<ValueType, Allocator> type;
        };
      }

      // now you can define a graph using std::list
      //   and a specific allocator
      typedef adjacency_list< list_with_allocatorS< std::allocator<int> >, vecS, directedS> MyGraph;

### Parallel Edge Traits

In addition to specializing the `container_gen` class, one must also specialize the `parallel_edge_traits` class to specify whether the container type allows parallel edges (and is a Sequence) or if the container does not allow parallel edges (and is an AssociativeContainer).

      template <class StorageSelector>
      struct parallel_edge_traits { };

      template <>
      struct parallel_edge_traits<vecS> {
        typedef allow_parallel_edge_tag type;
      };
      template <>
      struct parallel_edge_traits<setS> {
        typedef disallow_parallel_edge_tag type;
      };
      ...

### Push and Erase for the Edge List Container

One must also tell the `adjacency_list` how edges can be efficiently added and removed from the edge-list container. This is accomplished by overloading the `push()` and `erase()` functions for the custom container type. The `push()` function must return an iterator pointing to the newly inserted element and a `bool` flag saying whether the edge was inserted. The following default `push()` and `erase()` functions are already supplied for all STL container types. The family of `push_dispatch()` and `erase_dispatch()` function overloads handle the various ways inserting and erasing can be done with standard containers.

      template <class Container, class T>
      std::pair<typename Container::iterator, bool>
      push(Container& c, const T& v)
      {
        return push_dispatch(c, v, container_category(c));
      }

      template <class Container, class T>
      void erase(Container& c, const T& x)
      {
        erase_dispatch(c, x, container_category(c));
      }

Copyright © 2000 | Jeremy Siek, Univ.of Notre Dame (jsiek@lsc.nd.edu)