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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 &lt;class Graph, class P, class T, class R&gt;
void breadth_first_search(Graph& G, 
  typename graph_traits&lt;Graph&gt;::vertex_descriptor s, 
  const bgl_named_params&lt;P, T, R&gt;&amp; params);

<i>// non-named parameter version</i>
template &lt;class Graph, class Buffer, class BFSVisitor, 
	  class ColorMap&gt;
void breadth_first_search(const Graph&amp; g, 
   typename graph_traits&lt;Graph&gt;::vertex_descriptor s, 
   Buffer&amp; 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 != &Oslash;</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&amp; 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&lt;null_visitor&gt;</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&amp; 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>
<HR>
<TABLE>
<TR valign=top>
<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>)
</TD></TR></TABLE>

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