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Copyright (c) Jeremy Siek 2000
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<Head>
<Title>Using the Boost Graph Library</Title>
<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
ALINK="#ff0000">
<IMG SRC="../../../boost.png"
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<BR Clear>
<H1><A NAME="SECTION00830000000000000000"></A>
<A NAME="sec:using-adjacency-list"></A>
<BR>
Using <TT>adjacency_list</TT>
</H1>
This section describes the details of how use the
<tt>adjacency_list</tt> class. The presentation is divided into the
following topics:
<OL>
<li><A href="#sec:choosing-graph-type">Choosing the <TT>Edgelist</TT> and <TT>VertexList</TT></A>
<li><a href="#sec:directed-and-undirected">Directed and Undirected
Adjacency Lists</a>
<li><A href="#sec:adjacency-list-properties">Internal Properties</A>
<li><A href="#sec:custom-storage">Customizing the Adjacency List Storage</A>
</ol>
<P>
<H2><A NAME="SECTION00831000000000000000"></A>
<A NAME="sec:choosing-graph-type"></A>
<BR>
Choosing the <TT>Edgelist</TT> and <TT>VertexList</TT>
</H2>
<P>
This section focuses on how to decide which version of the <a
href="./adjacency_list.html"><TT>adjacency_list</TT></a> class to use
in different situations. The <TT>adjacency_list</TT> 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 <TT>OutEdgeList</TT>
and <TT>VertexList</TT>, which control the underlying data structures
that will be used to represent the graph. The choice of
<TT>OutEdgeList</TT> and <TT>VertexList</TT> affects the time complexity
of many of the graph operations and the space complexity of the graph
object.
<P>
BGL uses containers from the STL such as
<a href="http://www.boost.org/sgi/stl/Vector.html"><TT>std::vector</TT></a>,
<a href="http://www.boost.org/sgi/stl/List.html"><TT>std::list</TT></a>,
and <a href="http://www.boost.org/sgi/stl/set.html"><TT>std::set</TT></a>
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
<TT>OutEdgeList</TT> and <TT>VertexList</TT>.
<P>
<UL>
<LI><TT>vecS</TT> selects <TT>std::vector</TT>.</LI>
<LI><TT>listS</TT> selects <TT>std::list</TT>.</LI>
<LI><TT>slistS</TT> selects <TT>std::slist</TT>.</LI>
<LI><TT>setS</TT> selects <TT>std::set</TT>.</LI>
<LI><TT>multisetS</TT> selects <TT>std::multiset</TT>.</LI>
<LI><TT>hash_setS</TT> selects <TT>boost::unordered_set</TT>.</LI>
</UL>
<P>
<H3>Choosing the <TT>VertexList</TT> type</A></H3>
<P>
The <TT>VertexList</TT> 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 <a
href="http://www.boost.org/sgi/stl/Sequence.html">Sequence</a> or
<a
href="http://www.boost.org/sgi/stl/RandomAccessContainer.html">RandomAccessContainer</a>. In
general, <TT>listS</TT> is a good choice if you need to add and remove
vertices quickly. The price for this is extra space overhead compared
to choosing <TT>vecS</TT>.
<P>
<H4>Space Complexity</H4>
<P>
The <TT>std::list</TT> has a higher per-vertex space overhead than the
<TT>std::vector</TT>, storing three extra pointers per vertex.
<P>
<H4>Time Complexity</H4>
<P>
The choice of <TT>VertexList</TT> affects the time complexity of the
following operations.
<ul>
<li>
<pre>
add_vertex()
</PRE>
This operation is amortized constant time for both <TT>vecS</TT> and
<TT>listS</TT> (implemented with <TT>push_back()</TT>). However, when
the <TT>VertexList</TT> type is <TT>vecS</TT> the time for this
operation is occasionally large because the vector will be
reallocated and the whole graph copied.
<P></p>
<li>
<PRE>
remove_vertex()
</PRE>
This operation is constant time for <TT>listS</TT> and <i>O(V + E)</i> for
<TT>vecS</TT>. The large time complexity for <TT>vecS</TT> 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.
<P></P>
<li>
<PRE>
vertex()
</PRE>
This operation is constant time for <TT>vecS</TT> and for
<TT>listS</TT>.
</ul>
<P>
<H3><A NAME="SECTION00831200000000000000">
Choosing the <TT>OutEdgeList</TT> type</A>
</H3>
<P>
The <TT>OutEdgeList</TT> 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 <a
href="http://www.boost.org/sgi/stl/Sequence.html">Sequence</a> or for
<a
href="http://www.boost.org/sgi/stl/AssociativeContainer.html">AssociativeContainer</a>.
<P>
One of the first things to consider when choosing the
<TT>OutEdgeList</TT> is whether you want <TT>adjacency_list</TT> 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 <TT>setS</TT> or <TT>hash_setS</TT> 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 <a
href="http://www.boost.org/sgi/stl/Sequence.html">Sequence</a>
types: <TT>vecS</TT>, <TT>listS</TT>, or <TT>slistS</TT>.
You will also want to take into account the differences in time and space
complexity for the various graph operations. Below we use <i>V</i> for
the total number of vertices in the graph and <i>E</i> for the total
number of edges. Operations not discussed here are constant time.
<P>
<H4>Space Complexity</H4>
<P>
The selection of the <TT>OutEdgeList</TT> affects the amount of space
overhead per edge in the graph object. In the order of least space to
most space, the selectors are <TT>vecS</TT>, <TT>slistS</TT>,
<TT>listS</TT>, and <TT>setS</TT>.
<P>
<H4>Time Complexity</H4>
<P>
In the following description of the time complexity for various
operations, we use <i>E/V</i> inside of the ``big-O'' notation to
express the length of an out-edge list. Strictly speaking this is not
accurate because <i>E/V</i> merely gives the average number of edges
per vertex in a random graph. The worst-case number of out-edges for a
vertex is <i>V</i> (unless it is a multi-graph). For sparse graphs
<i>E/V</i> is typically much smaller than <i>V</i> and can be
considered a constant.
<P>
<P> <P>
<UL>
<LI>
<PRE>
add_edge()
</PRE>
When the <TT>OutEdgeList</TT> is a <a
href="http://www.boost.org/sgi/stl/UniqueAssociativeContainer.html">UniqueAssociativeContainer</a>
like <TT>std::set</TT> the absence of parallel edges is enforced when
an edge is added. The extra lookup involved has time complexity
<i>O(log(E/V))</i>. The <TT>OutEdgeList</TT> types that model <a
href="http://www.boost.org/sgi/stl/Sequence.html">Sequence</a> do
not perform this check and therefore <TT>add_edge()</TT> 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
<TT>OutEdgeList</TT>. The <TT>add_edge()</TT> for the sequence-based
<TT>OutEdgeList</TT> is implemented with <TT>push_front()</TT> or
<TT>push_back()</TT>. However, for <TT>std::list</TT> and
<TT>std::slist</TT> this operation will typically be faster than with
<TT>std::vector</TT> which occasionally reallocates and copies all
elements.
<p></p>
<li>
<PRE>
remove_edge()
</PRE>
For sequence-based <TT>OutEdgeList</TT> types this operation is
implemented with <TT>std::remove_if()</TT> which means the average
time is <i>E/V</i>. For set-based <TT>OutEdgeList</TT> types this is
implemented with the <TT>erase()</TT> member function, which has
average time <i>log(E/V)</i>.
<p></p>
<li>
<PRE>
edge()
</PRE>
The time complexity for this operation is <i>O(E/V)</i> when the
<TT>OutEdgeList</TT> type is a <a
href="http://www.boost.org/sgi/stl/Sequence.html">Sequence</a> and it
is <i>O(log(E/V))</i> when the <TT>OutEdgeList</TT> type is an <a
href="http://www.boost.org/sgi/stl/AssociativeContainer.html">AssociativeContainer</a>.
<p></p>
<li>
<PRE>
clear_vertex()
</PRE>
For directed graphs with sequence-based <TT>OutEdgeList</TT> types the time
complexity is <i>O(V + E)</i>, while for associative container based
<TT>OutEdgeList</TT> types the operation is faster, with time complexity
<i>O(V log(E/V))</i>. For undirected graphs this operation is
<i>O(E<sup>2</sup>/V<sup>2</sup>)</i> or <i>O(E/V log(E/V))</i>.
<p></p>
<li>
<PRE>
remove_vertex()
</PRE>
The time complexity for this operation is <i>O(V + E)</i> regardless of the
<TT>OutEdgeList</TT> type.
<p></p>
<li>
<PRE>
out_edge_iterator::operator++()
</PRE>
This operation is constant time for all the <TT>OneD</TT> 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
<TT>vecS</TT>, <TT>slistS</TT>, <TT>listS</TT>, <TT>setS</TT>,
<TT>hash_setS</TT>.
<p></p>
<li>
<PRE>
in_edge_iterator::operator++()
</PRE>
This operation is constant time and exhibits a similar speed
ordering as the <TT>out_edge_iterator</TT> with respect to
the <TT>OutEdgeList</TT> selection.
<p></p>
<li>
<PRE>
vertex_iterator::operator++()
</PRE>
This operation is constant time and fast (same speed as incrementing a
pointer). The selection of <TT>OneD</TT> does not affect the speed of
this operation.
<p></p>
<li>
<PRE>
edge_iterator::operator++()
</PRE>
This operation is constant time and exhibits a similar speed ordering
as the <TT>out_edge_iterator</TT> with respect to the <TT>OutEdgeList</TT>
selection. Traversing through the whole edge set is <i>O(V + E)</i>.
<p></p>
<li>
<PRE>
adjacency_iterator::operator++()
</PRE>
This operation is constant time and exhibits a similar speed
ordering as the <TT>out_edge_iterator</TT> with respect to
the <TT>OutEdgeList</TT> selection.
<p></p>
</ul>
<P>
<P>
<H2><a name="sec:directed-and-undirected">Directed and Undirected Adjacency Lists</H2>
<P>
The <TT>adjacency_list</TT> class can be used to represent both
directed and undirected graphs, depending on the argument passed to
the <TT>Directed</TT> template parameter. Selecting <TT>directedS</TT>
or <TT>bidirectionalS</TT> choose a directed graph, whereas
<TT>undirectedS</TT> selects the representation for an undirected
graph. See Section <A
HREF="graph_concepts.html#sec:undirected-graphs">Undirected Graphs</A>
for a description of the difference between directed and undirected
graphs in BGL. The <TT>bidirectionalS</TT> selector specifies that the
graph will provide the <TT>in_edges()</TT> function as well as the
<TT>out_edges()</TT> function. This imposes twice as much space
overhead per edge, which is why <TT>in_edges()</TT> is optional.
<P>
<H2><A NAME="sec:adjacency-list-properties"></A>
Internal Properties
</H2>
<P>
Properties can be attached to the vertices or edges of an
<TT>adjacency_list</TT> graph via the property interface. The template
parameters <TT>VertexProperty</TT> and <TT>EdgeProperty</TT> of the
<TT>adjacency_list</TT> class are meant to be filled by these interior
properties.
<p><b>NOTE</b>: The Boost Graph Library supports two interchangeable methods for
specifying interior properties: <a href="bundles.html">bundled properties</a>
and property lists. The former is easier to use and requires less effort,
whereas the latter is compatible with older, broken compilers and is
backward-compatible with Boost versions prior to 1.32.0. If you absolutely
require these compatibility features, read on to learn about property lists.
Otherwise, we strongly suggest that you read about the <a href="bundles.html">bundled
properties</a> mechanism.
<p>One may specify internal properties via property lists, which are build from instances of the
property class declared as follows.
<P>
<PRE>
template &lt;class PropertyTag, class T, class NextProperty = no_property&gt;
struct property;
</PRE>
<P>
The <a href="./PropertyTag.html"><TT>PropertyTag</TT></a> template
parameter is a tag class that simply identifies or gives a unique name
to the property. There are several predefined tags, and it is easy to
add more.
<P>
<PRE>
struct vertex_index_t { };
struct vertex_index1_t { };
struct vertex_index2_t { };
struct edge_index_t { };
struct graph_name_t { };
struct vertex_name_t { };
struct edge_name_t { };
struct edge_weight_t { };
struct edge_weight2_t { };
struct edge_capacity_t { };
struct edge_residual_capacity_t { };
struct edge_reverse_t { };
struct vertex_distance_t { };
struct vertex_root_t { };
struct vertex_all_t { };
struct edge_all_t { };
struct graph_all_t { };
struct vertex_color_t { };
struct vertex_rank_t { };
struct vertex_predecessor_t { };
struct vertex_isomorphism_t { };
struct vertex_invariant_t { };
struct vertex_invariant1_t { };
struct vertex_invariant2_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 { };
</PRE>
<P>
The <b><TT>T</TT></b> template parameter of <TT>property</TT>
specifies the type of the property values. The type <tt>T</tt> must be
<a
href="http://www.boost.org/sgi/stl/DefaultConstructible.html">Default
Constructible</a>, <a
href="../../utility/Assignable.html">Assignable</a>, and <a
href="../../utility/CopyConstructible.html">Copy Constructible</a>.
Like the containers of the C++ Standard Library, the property objects
of type <tt>T</tt> are held by-value inside of the graph.
<p>
The <b><TT>NextProperty</TT></b> parameter allows <TT>property</TT>
types to be nested, so that an arbitrary number of properties can be
attached to the same graph.
<P>
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 <TT>float</TT> and a name property with
values of type <TT>std::string</TT> to the vertices of the graph. We
have attached a weight property with values of type <TT>float</TT> to
the edges of the graph.
<P>
<PRE>
typedef property&lt;vertex_distance_t, float,
property&lt;vertex_name_t, std::string&gt; &gt; VertexProperty;
typedef property&lt;edge_weight_t, float&gt; EdgeProperty;
typedef adjacency_list&lt;mapS, vecS, undirectedS,
VertexProperty, EdgeProperty&gt; Graph;
Graph g(num_vertices); // construct a graph object
</PRE>
<P>
The property values are then read from and written to using property
maps. See Section <A HREF="using_property_maps.html#sec:interior-properties">Interior
Properties</A> for a description of how to obtain property maps
from a graph, and read Section <A
HREF="./using_property_maps.html">Property Maps</A> for how
to use property maps.
<P>
<H3><A NAME="sec:custom-edge-properties"></A>
Custom Edge Properties
</H3>
<P>
Creating your own property types and properties is easy; just define
a tag class for your new property. The property tag class will need to
define <tt>num</tt> with a unique integer ID, and <tt>kind</tt> which
should be either <tt>edge_property_tag</tt>,
<tt>vertex_property_tag</tt>, or <tt>graph_property_tag</tt>.
<P>
<PRE>
struct flow_t {
typedef edge_property_tag kind;
};
struct capacity_t {
typedef edge_property_tag kind;
};
</PRE>
<p>
You can also use enum's instead of struct's to create tag types. Create an enum
type for each property inside the boost namespace. The first part of the name of
the enum type must be <tt>edge</tt>, <tt>vertex</tt>, or <tt>graph</tt> followed
by an underscore, the new property name, and a <tt>_t</tt> at the end. Inside
the enum, define a value with the same name minus the <tt>_t</tt>. Then invoke
the <tt>BOOST_INSTALL_PROPERTY</tt> macro.
<pre>
namespace boost {
enum edge_flow_t { edge_flow };
enum edge_capacity_t { edge_capacity };
BOOST_INSTALL_PROPERTY(edge, flow);
BOOST_INSTALL_PROPERTY(edge, capacity);
}
</pre>
<P>
Now you can use your new property tag in the definition of properties just as
you would one of the builtin tags.
<P>
<PRE>
typedef property&lt;capacity_t, int&gt; Cap;
typedef property&lt;flow_t, int, Cap&gt; EdgeProperty;
typedef adjacency_list&lt;vecS, vecS, no_property, EdgeProperty&gt; Graph;
</PRE>
<P>
Just as before, the property maps for these properties can be
obtained from the graph via the
<TT>get(Property, g)</TT> function.
<P>
<PRE>
property_map&lt;Graph, capacity_t&gt;::type capacity
= get(capacity_t(), G);
property_map&lt;Graph, flow_t&gt;::type flow
= get(flow_t(), G);
</PRE>
<P>
The file <TT>edge_property.cpp</TT> shows the complete source
code for this example.
<P>
<H3><A NAME="SECTION00833200000000000000"></A>
<A NAME="sec:custom-vertex-properties"></A>
<BR>
Custom Vertex Properties
</H3>
<P>
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.
<P>
<PRE>
struct first_name_t {
typedef vertex_property_tag kind;
};
</PRE>
<P>
Now we can use the new tag in the <TT>property</TT> 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''.
<P>
<PRE>
typedef property&lt;first_name_t, std::string&gt; FirstNameProperty;
typedef adjacency_list&lt;vecS, vecS, directedS,
FirstNameProperty&gt; MyGraphType;
typedef pair&lt;int,int&gt; 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 &lt; 11; ++i)
add_edge(edge_array[i].first, edge_array[i].second, G);
property_map&lt;MyGraphType, first_name_t&gt;::type
name = get(first_name_t(), 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);
</PRE>
<P>
The <TT>who_owes_who()</TT> 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
<TT>property_map</TT> traits class. The <TT>const_type</TT>
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
<TT>property_traits</TT> class. In this example, we know that the
property's value type will be <TT>std::string</TT>, but written in this
generic fashion the <TT>who_owes_who()</TT> function could work with
other property value types.
<P>
<PRE>
template &lt;class EdgeIter, class Graph&gt;
void who_owes_who(EdgeIter first, EdgeIter last, const Graph&amp; G)
{
// Access the propety acessor type for this graph
typedef typename property_map&lt;Graph,
first_name_t&gt;::const_type NameMap;
NameMap name = get(first_name, G);
typedef typename boost::property_traits&lt;NameMap&gt;
::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 &lt;&lt; src_name &lt;&lt; " owes "
&lt;&lt; targ_name &lt;&lt; " some money" &lt;&lt; endl;
++first;
}
</PRE>
The output is:
<PRE>
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
</PRE>
The complete source code to this example is in the file
<TT>interior_property_map.cpp</TT>.
<P>
<H2><A NAME="sec:custom-storage"></A>
Customizing the Adjacency List Storage
</H2>
<P>
The <TT>adjacency_list</TT> 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. BGL provides selector
classes that allow the user to choose between several of the containers
from the STL. It is also possible to use your own container types.
When customizing the <TT>VertexList</TT> you need to define a container
generator as described below. When customizing the <TT>OutEdgeList</TT> you
will need to define a container generator and the parallel edge
traits. The file <TT>container_gen.cpp</TT> has an example of
how to use a custom storage type.
<P>
<H3><A NAME="SECTION00834100000000000000">
Container Generator</A>
</H3>
<P>
The <TT>adjacency_list</TT> class uses a traits class called
<TT>container_gen</TT> to map the <TT>OutEdgeList</TT> and <TT>VertexList</TT>
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 <TT>listS</TT> selector.
<P>
<PRE>
namespace boost {
template &lt;class Selector, class ValueType&gt;
struct container_gen { };
template &lt;class ValueType&gt;
struct container_gen&lt;listS, ValueType&gt; {
typedef std::list&lt;ValueType&gt; type;
};
}
</PRE>
<P>
To use some other container of your choice, define a
selector class and then specialize the <TT>container_gen</TT>
for your selector.
<P>
<PRE>
struct custom_containerS { }; // your selector
namespace boost {
// the specialization for your selector
template &lt;class ValueType&gt;
struct container_gen&lt;custom_containerS, ValueType&gt; {
typedef custom_container&lt;ValueType&gt; type;
};
}
</PRE>
<P>
There may also be situations when you want to use a container that has
more template parameters than just <TT>ValueType</TT>. 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
<TT>container_gen</TT>. 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 <TT>std::list</TT>.
<P>
<PRE>
template &lt;class Allocator&gt;
struct list_with_allocatorS { };
namespace boost {
template &lt;class Alloc, class ValueType&gt;
struct container_gen&lt;list_with_allocatorS&lt;Alloc&gt;, ValueType&gt;
{
typedef typename Alloc::template rebind&lt;ValueType&gt;::other Allocator;
typedef std::list&lt;ValueType, Allocator&gt; type;
};
}
// now you can define a graph using std::list
// and a specific allocator
typedef adjacency_list&lt; list_with_allocatorS&lt; std::allocator&lt;int&gt; &gt;, vecS, directedS&gt; MyGraph;
</PRE>
<P>
<H3><A NAME="SECTION00834300000000000000">
Push and Erase for the Custom Container</A>
</H3>
<P>
You must also tell the <TT>adjacency_list</TT> how elements can be
efficiently added and removed from the custom container. This is
accomplished by overloading the <TT>push()</TT> and <TT>erase()</TT>
functions for the custom container type. The <TT>push()</TT> function
should return an iterator pointing to the newly inserted element and a
<TT>bool</TT> flag saying whether the edge was inserted.
<PRE>
template &lt;class T&gt;
std::pair&lt;typename custom_container&lt;T&gt;::iterator, bool&gt;
push(custom_container&lt;T&gt;&amp; c, const T&amp; v)
{
// this implementation may need to change for your container
c.push_back(v);
return std::make_pair(boost::prior(c.end()), true);
}
template &lt;class T&gt;
void erase(custom_container&lt;T&gt;&amp; c, const T&amp; x)
{
// this implementation may need to change for your container
c.erase(std::remove(c.begin(), c.end(), x), c.end());
}
</PRE>
<P> There are default <TT>push()</TT> and <TT>erase()</TT> functions
implemented for the STL container types.
<H3><A NAME="SECTION00834200000000000000">
Parallel Edge Traits</A>
</H3>
<P>
When customizing the <TT>OutEdgeList</TT>, you must also specialize
the <TT>parallel_edge_traits</TT> class to specify whether the
container type allows parallel edges (and is a <a
href="http://www.boost.org/sgi/stl/Sequence.html">Sequence</a>) or if
the container does not allow parallel edges (and is an <a
href="http://www.boost.org/sgi/stl/AssociativeContainer.html">AssociativeContainer</a>).
<P>
<PRE>
template &lt;&gt;
struct parallel_edge_traits&lt;custom_containerS&gt; {
typedef allow_parallel_edge_tag type;
};
</PRE>
<br>
<HR>
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<TD nowrap>Copyright &copy; 2000-2001</TD><TD>
<A HREF="http://www.boost.org/people/jeremy_siek.htm">Jeremy Siek</A>, Indiana University (<A HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)
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