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<Title>Boost Graph Library: Breadth-First Search</Title>
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<H1><A NAME="sec:bfs">
<img src="figs/python.gif" alt="(Python)"/>
<TT>breadth_first_search</TT>
</H1>
<P>
<PRE>
<i>// named parameter version</i>
template <class Graph, class P, class T, class R>
void breadth_first_search(Graph& G,
typename graph_traits<Graph>::vertex_descriptor s,
const bgl_named_params<P, T, R>& params);
<i>// non-named parameter version</i>
template <class Graph, class Buffer, class BFSVisitor,
class ColorMap>
void breadth_first_search(const Graph& g,
typename graph_traits<Graph>::vertex_descriptor s,
Buffer& Q, BFSVisitor vis, ColorMap color);
</PRE>
<p>
The <tt>breadth_first_search()</tt> function performs a breadth-first
traversal [<a href="./bibliography.html#moore59">49</a>] of a directed
or undirected graph. A breadth-first traversal visits vertices that
are closer to the source before visiting vertices that are further
away. In this context ``distance'' is defined as the number of edges
in the shortest path from the source vertex. The
<tt>breadth_first_search()</tt> function can be used to compute the
shortest path from the source to all reachable vertices and the
resulting shortest-path distances. For more definitions related to BFS
see section <a href="./graph_theory_review.html#sec:bfs-algorithm">
Breadth-First Search</a>.
</p>
<p>
BFS uses two data structures to to implement the traversal: a color
marker for each vertex and a queue. White vertices are undiscovered
while gray vertices are discovered but have undiscovered adjacent
vertices. Black vertices are discovered and are adjacent to only other
black or gray vertices. The algorithm proceeds by removing a vertex
</i>u</i> from the queue and examining each out-edge <i>(u,v)</i>. If an
adjacent vertex <i>v</i> is not already discovered, it is colored gray and
placed in the queue. After all of the out-edges are examined, vertex
<i>u</i> is colored black and the process is repeated. Pseudo-code for the
BFS algorithm is a listed below.
</p>
<table>
<tr>
<td valign="top">
<pre>
BFS(<i>G</i>, <i>s</i>)
<b>for</b> each vertex <i>u in V[G]</i>
<i>color[u] :=</i> WHITE
<i>d[u] := infinity</i>
<i>p[u] := u</i>
<b>end for</b>
<i>color[s] :=</i> GRAY
<i>d[s] := 0</i>
ENQUEUE(<i>Q</i>, <i>s</i>)
<b>while</b> (<i>Q != Ø</i>)
<i>u :=</i> DEQUEUE(Q)
<b>for</b> each vertex <i>v in Adj[u]</i>
<b>if</b> (<i>color[v] =</i> WHITE)
<i>color[v] :=</i> GRAY
<i>d[v] := d[u] + 1</i>
<i>p[v] := u</i>
ENQUEUE(<i>Q</i>, <i>v</i>)
<b>else</b>
<b>if</b> (<i>color[v] =</i> GRAY)
...
<b>else</b>
...
<b>end for</b>
<i>color[u] :=</i> BLACK
<b>end while</b>
return (<i>d</i>, <i>p</i>)
</pre>
</td>
<td valign="top">
<pre>
initialize vertex <i>u</i>
discover vertex <i>s</i>
examine vertex <i>u</i>
examine edge <i>(u,v)</i>
<i>(u,v)</i> is a tree edge
discover vertex <i>v</i>
<i>(u,v)</i> is a non-tree edge
<i>(u,v)</i> has a gray target
<i>(u,v)</i> has a black target
finish vertex <i>u</i>
</pre>
</tr>
</table>
The <tt>breadth_first_search()</tt> function can be extended with
user-defined actions that will be called a certain event points. The
actions must be provided in the form of a visitor object, that is, an
object who's type meets the requirements for a <a
href="./BFSVisitor.html">BFS Visitor</a>. In the above pseudo-code,
the event points are the labels on the right. Also a description of
each event point is given below. By default, the
<tt>breadth_first_search()</tt> function does not carry out any
actions, not even recording distances or predecessors. However these
can be easily added using the <a
href="./distance_recorder.html"><tt>distance_recorder</tt></a> and <a
href="./predecessor_recorder.html"><tt>predecessor_recorder</tt></a>
event visitors.
<H3>Where Defined</H3>
<P>
<a href="../../../boost/graph/breadth_first_search.hpp"><TT>boost/graph/breadth_first_search.hpp</TT></a>
<P>
<h3>Parameters</h3>
IN: <tt>Graph& g</tt>
<blockquote>
A directed or undirected graph. The graph type must
be a model of <a href="./VertexListGraph.html">Vertex List Graph</a>
and <a href="./IncidenceGraph.html">Incidence Graph</a>.<br>
<b>Python</b>: The parameter is named <tt>graph</tt>.
</blockquote>
IN: <tt>vertex_descriptor s</tt>
<blockquote>
The source vertex where the search is started.<br>
<b>Python</b>: The parameter is named <tt>root_vertex</tt>.
</blockquote>
<h3>Named Parameters</h3>
IN: <tt>visitor(BFSVisitor vis)</tt>
<blockquote>
A visitor object that is invoked inside the algorithm at the
event-points specified by the <a href="BFSVisitor.html">BFS
Visitor</a> concept. The visitor object is passed by value <a
href="#1">[1]</a>.<br> <b>Default:</b>
<tt>bfs_visitor<null_visitor></tt> <br>
<b>Python</b>: The parameter should be an object that derives from
the <a href="BFSVisitor.html#python"><tt>BFSVisitor</tt></a> type of the graph.
</blockquote>
UTIL/OUT: <tt>color_map(ColorMap color)</tt>
<blockquote>
This is used by the algorithm to keep track of its progress through
the graph. The user need not initialize the color map before calling
<tt>breadth_first_search()</tt> since the algorithm initializes the
color of every vertex to white at the start of the algorihtm. If you
need to perform multiple breadth-first searches on a graph (for
example, if there are some disconnected components) then use the <a
href="./breadth_first_visit.html"><tt>breadth_first_visit()</tt></a>
function and do your own color initialization.
<p>The type <tt>ColorMap</tt> must be a model of <a
href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
Property Map</a> and its key type must be the graph's vertex
descriptor type and the value type of the color map must model
<a href="./ColorValue.html">ColorValue</a>.<br>
<b>Default:</b> an <a
href="../../property_map/doc/iterator_property_map.html">
</tt>iterator_property_map</tt></a> created from a
<tt>std::vector</tt> of <tt>default_color_type</tt> of size
<tt>num_vertices(g)</tt> and using the <tt>i_map</tt> for the index
map.<br>
<b>Python</b>: The color map must be a <tt>vertex_color_map</tt> for
the graph.
</blockquote>
IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt>
<blockquote>
This maps each vertex to an integer in the range <tt>[0,
num_vertices(g))</tt>. This parameter is only necessary when the
default color property map is used. The type <tt>VertexIndexMap</tt>
must be a model of <a
href="../../property_map/doc/ReadablePropertyMap.html">Readable Property
Map</a>. The value type of the map must be an integer type. The
vertex descriptor type of the graph needs to be usable as the key
type of the map.<br>
<b>Default:</b> <tt>get(vertex_index, g)</tt>.
Note: if you use this default, make sure your graph has
an internal <tt>vertex_index</tt> property. For example,
<tt>adjacenty_list</tt> with <tt>VertexList=listS</tt> does
not have an internal <tt>vertex_index</tt> property.<br>
<b>Python</b>: Unsupported parameter.
</blockquote>
UTIL: <tt>buffer(Buffer& Q)</tt>
<blockquote>
The queue used to determine the order in which vertices will be
discovered. If a FIFO queue is used, then the traversal will
be according to the usual BFS ordering. Other types of queues
can be used, but the traversal order will be different.
For example Dijkstra's algorithm can be implemented
using a priority queue. The type <tt>Buffer</tt> must be a model of
<a href="./Buffer.html">Buffer</a>.<br> The <tt>value_type</tt>
of the buffer must be the <tt>vertex_descriptor</tt> type for the graph.<br>
<b>Default:</b> <tt>boost::queue</tt><br>
<b>Python</b>: The buffer must derive from the <a
href="./Buffer.html">Buffer</a> type for the graph.
</blockquote>
<H3><A NAME="SECTION001330300000000000000">
Complexity</A>
</H3>
<P>
The time complexity is <i>O(E + V)</i>.
<P>
<h3>Visitor Event Points</h3>
<ul>
<li><b><tt>vis.initialize_vertex(v, g)</tt></b> is invoked on every vertex
before the start of the search.
<li><b><tt>vis.examine_vertex(u, g)</tt></b>r is invoked in each
vertex as it is removed from the queue.
<li><b><tt>vis.examine_edge(e, g)</tt></b> is invoked on every out-edge
of each vertex immediately after the vertex is removed from the queue.
<li><b><tt>vis.tree_edge(e, g)</tt></b> is invoked (in addition to
<tt>examine_edge()</tt>) if the edge is a tree edge. The
target vertex of edge <tt>e</tt> is discovered at this time.
<li><b><tt>vis.discover_vertex(u, g)</tt></b> is invoked the first time the
algorithm encounters vertex <i>u</i>. All vertices closer to the
source vertex have been discovered, and vertices further from the
source have not yet been discovered.
<li><b><tt>vis.non_tree_edge(e, g)</tt></b> is invoked (in addition to
<tt>examine_edge()</tt>) if the edge is not a tree edge.
<li><b><tt>vis.gray_target(e, g)</tt></b> is invoked (in addition to
<tt>non_tree_edge()</tt>) if the target vertex is colored gray at the
time of examination. The color gray indicates that
the vertex is currently in the queue.
<li><b><tt>vis.black_target(e, g)</tt></b> is invoked (in addition to
<tt>non_tree_edge()</tt>) if the target vertex is colored black at the
time of examination. The color black indicates that the
vertex is no longer in the queue.
<li><b><tt>vis.finish_vertex(u, g)</tt></b> is invoked after all of the out
edges of <i>u</i> have been examined and all of the adjacent vertices
have been discovered.
</ul>
<H3><A NAME="SECTION001330400000000000000">
Example</A>
</H3>
<P>
The example in <a
href="../example/bfs-example.cpp"><TT>example/bfs-example.cpp</TT></a>
demonstrates using the BGL Breadth-first search algorithm on the graph
from <A HREF="./graph_theory_review.html#fig:bfs-example">Figure
5</A>. The file
<a href="../example/bfs-example2.cpp"><TT>example/bfs-example2.cpp</TT></a>
contains the same example, except that the <tt>adacency_list</tt>
class used has <tt>VertexList</tt> and <tt>EdgeList</tt> set
to <tt>listS</tt>.
</P>
<h3>See Also</h3>
<a href="./bfs_visitor.html"><tt>bfs_visitor</tt></a> and
<a href="./depth_first_search.html"><tt>depth_first_search()</tt></a>
<h3>Notes</h3>
<p><a name="1">[1]</a>
Since the visitor parameter is passed by value, if your visitor
contains state then any changes to the state during the algorithm
will be made to a copy of the visitor object, not the visitor object
passed in. Therefore you may want the visitor to hold this state by
pointer or reference.
<br>
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<TD nowrap>Copyright © 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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