339 lines
11 KiB
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339 lines
11 KiB
HTML
<HTML>
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<!--
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Copyright (c) Jeremy Siek 2000, 2001
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt)
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-->
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<Head>
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<Title>Boost Graph Library: Breadth-First Search</Title>
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<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
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ALINK="#ff0000">
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<IMG SRC="../../../boost.png"
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ALT="C++ Boost" width="277" height="86">
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<BR Clear>
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<H1><A NAME="sec:bfs">
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<img src="figs/python.gif" alt="(Python)"/>
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<TT>breadth_first_search</TT>
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</H1>
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<P>
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<PRE>
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<i>// named parameter version</i>
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template <class Graph, class P, class T, class R>
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void breadth_first_search(Graph& G,
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typename graph_traits<Graph>::vertex_descriptor s,
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const bgl_named_params<P, T, R>& params);
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<i>// non-named parameter version</i>
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template <class Graph, class Buffer, class BFSVisitor,
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class ColorMap>
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void breadth_first_search(const Graph& g,
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typename graph_traits<Graph>::vertex_descriptor s,
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Buffer& Q, BFSVisitor vis, ColorMap color);
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</PRE>
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<p>
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The <tt>breadth_first_search()</tt> function performs a breadth-first
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traversal [<a href="./bibliography.html#moore59">49</a>] of a directed
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or undirected graph. A breadth-first traversal visits vertices that
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are closer to the source before visiting vertices that are further
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away. In this context ``distance'' is defined as the number of edges
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in the shortest path from the source vertex. The
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<tt>breadth_first_search()</tt> function can be used to compute the
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shortest path from the source to all reachable vertices and the
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resulting shortest-path distances. For more definitions related to BFS
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see section <a href="./graph_theory_review.html#sec:bfs-algorithm">
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Breadth-First Search</a>.
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</p>
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<p>
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BFS uses two data structures to to implement the traversal: a color
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marker for each vertex and a queue. White vertices are undiscovered
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while gray vertices are discovered but have undiscovered adjacent
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vertices. Black vertices are discovered and are adjacent to only other
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black or gray vertices. The algorithm proceeds by removing a vertex
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</i>u</i> from the queue and examining each out-edge <i>(u,v)</i>. If an
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adjacent vertex <i>v</i> is not already discovered, it is colored gray and
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placed in the queue. After all of the out-edges are examined, vertex
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<i>u</i> is colored black and the process is repeated. Pseudo-code for the
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BFS algorithm is a listed below.
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</p>
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<table>
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<tr>
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<td valign="top">
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<pre>
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BFS(<i>G</i>, <i>s</i>)
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<b>for</b> each vertex <i>u in V[G]</i>
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<i>color[u] :=</i> WHITE
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<i>d[u] := infinity</i>
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<i>p[u] := u</i>
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<b>end for</b>
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<i>color[s] :=</i> GRAY
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<i>d[s] := 0</i>
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ENQUEUE(<i>Q</i>, <i>s</i>)
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<b>while</b> (<i>Q != Ø</i>)
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<i>u :=</i> DEQUEUE(Q)
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<b>for</b> each vertex <i>v in Adj[u]</i>
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<b>if</b> (<i>color[v] =</i> WHITE)
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<i>color[v] :=</i> GRAY
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<i>d[v] := d[u] + 1</i>
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<i>p[v] := u</i>
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ENQUEUE(<i>Q</i>, <i>v</i>)
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<b>else</b>
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<b>if</b> (<i>color[v] =</i> GRAY)
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...
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<b>else</b>
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...
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<b>end for</b>
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<i>color[u] :=</i> BLACK
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<b>end while</b>
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return (<i>d</i>, <i>p</i>)
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</pre>
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</td>
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<td valign="top">
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<pre>
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initialize vertex <i>u</i>
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discover vertex <i>s</i>
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examine vertex <i>u</i>
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examine edge <i>(u,v)</i>
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<i>(u,v)</i> is a tree edge
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discover vertex <i>v</i>
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<i>(u,v)</i> is a non-tree edge
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<i>(u,v)</i> has a gray target
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<i>(u,v)</i> has a black target
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finish vertex <i>u</i>
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</pre>
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</tr>
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</table>
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The <tt>breadth_first_search()</tt> function can be extended with
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user-defined actions that will be called a certain event points. The
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actions must be provided in the form of a visitor object, that is, an
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object who's type meets the requirements for a <a
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href="./BFSVisitor.html">BFS Visitor</a>. In the above pseudo-code,
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the event points are the labels on the right. Also a description of
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each event point is given below. By default, the
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<tt>breadth_first_search()</tt> function does not carry out any
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actions, not even recording distances or predecessors. However these
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can be easily added using the <a
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href="./distance_recorder.html"><tt>distance_recorder</tt></a> and <a
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href="./predecessor_recorder.html"><tt>predecessor_recorder</tt></a>
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event visitors.
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<H3>Where Defined</H3>
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<P>
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<a href="../../../boost/graph/breadth_first_search.hpp"><TT>boost/graph/breadth_first_search.hpp</TT></a>
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<P>
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<h3>Parameters</h3>
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IN: <tt>Graph& g</tt>
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<blockquote>
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A directed or undirected graph. The graph type must
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be a model of <a href="./VertexListGraph.html">Vertex List Graph</a>
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and <a href="./IncidenceGraph.html">Incidence Graph</a>.<br>
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<b>Python</b>: The parameter is named <tt>graph</tt>.
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</blockquote>
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IN: <tt>vertex_descriptor s</tt>
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<blockquote>
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The source vertex where the search is started.<br>
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<b>Python</b>: The parameter is named <tt>root_vertex</tt>.
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</blockquote>
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<h3>Named Parameters</h3>
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IN: <tt>visitor(BFSVisitor vis)</tt>
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<blockquote>
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A visitor object that is invoked inside the algorithm at the
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event-points specified by the <a href="BFSVisitor.html">BFS
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Visitor</a> concept. The visitor object is passed by value <a
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href="#1">[1]</a>.<br> <b>Default:</b>
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<tt>bfs_visitor<null_visitor></tt> <br>
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<b>Python</b>: The parameter should be an object that derives from
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the <a href="BFSVisitor.html#python"><tt>BFSVisitor</tt></a> type of the graph.
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</blockquote>
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UTIL/OUT: <tt>color_map(ColorMap color)</tt>
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<blockquote>
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This is used by the algorithm to keep track of its progress through
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the graph. The user need not initialize the color map before calling
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<tt>breadth_first_search()</tt> since the algorithm initializes the
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color of every vertex to white at the start of the algorihtm. If you
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need to perform multiple breadth-first searches on a graph (for
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example, if there are some disconnected components) then use the <a
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href="./breadth_first_visit.html"><tt>breadth_first_visit()</tt></a>
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function and do your own color initialization.
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<p>The type <tt>ColorMap</tt> must be a model of <a
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href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
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Property Map</a> and its key type must be the graph's vertex
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descriptor type and the value type of the color map must model
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<a href="./ColorValue.html">ColorValue</a>.<br>
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<b>Default:</b> an <a
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href="../../property_map/doc/iterator_property_map.html">
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</tt>iterator_property_map</tt></a> created from a
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<tt>std::vector</tt> of <tt>default_color_type</tt> of size
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<tt>num_vertices(g)</tt> and using the <tt>i_map</tt> for the index
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map.<br>
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<b>Python</b>: The color map must be a <tt>vertex_color_map</tt> for
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the graph.
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</blockquote>
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IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt>
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<blockquote>
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This maps each vertex to an integer in the range <tt>[0,
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num_vertices(g))</tt>. This parameter is only necessary when the
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default color property map is used. The type <tt>VertexIndexMap</tt>
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must be a model of <a
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href="../../property_map/doc/ReadablePropertyMap.html">Readable Property
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Map</a>. The value type of the map must be an integer type. The
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vertex descriptor type of the graph needs to be usable as the key
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type of the map.<br>
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<b>Default:</b> <tt>get(vertex_index, g)</tt>.
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Note: if you use this default, make sure your graph has
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an internal <tt>vertex_index</tt> property. For example,
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<tt>adjacency_list</tt> with <tt>VertexList=listS</tt> does
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not have an internal <tt>vertex_index</tt> property.<br>
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<b>Python</b>: Unsupported parameter.
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</blockquote>
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UTIL: <tt>buffer(Buffer& Q)</tt>
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<blockquote>
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The queue used to determine the order in which vertices will be
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discovered. If a FIFO queue is used, then the traversal will
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be according to the usual BFS ordering. Other types of queues
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can be used, but the traversal order will be different.
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For example Dijkstra's algorithm can be implemented
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using a priority queue. The type <tt>Buffer</tt> must be a model of
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<a href="./Buffer.html">Buffer</a>.<br> The <tt>value_type</tt>
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of the buffer must be the <tt>vertex_descriptor</tt> type for the graph.<br>
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<b>Default:</b> <tt>boost::queue</tt><br>
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<b>Python</b>: The buffer must derive from the <a
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href="./Buffer.html">Buffer</a> type for the graph.
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</blockquote>
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<H3><A NAME="SECTION001330300000000000000">
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Complexity</A>
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</H3>
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<P>
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The time complexity is <i>O(E + V)</i>.
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<P>
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<h3>Visitor Event Points</h3>
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<ul>
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<li><b><tt>vis.initialize_vertex(v, g)</tt></b> is invoked on every vertex
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before the start of the search.
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<li><b><tt>vis.examine_vertex(u, g)</tt></b>r is invoked in each
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vertex as it is removed from the queue.
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<li><b><tt>vis.examine_edge(e, g)</tt></b> is invoked on every out-edge
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of each vertex immediately after the vertex is removed from the queue.
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<li><b><tt>vis.tree_edge(e, g)</tt></b> is invoked (in addition to
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<tt>examine_edge()</tt>) if the edge is a tree edge. The
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target vertex of edge <tt>e</tt> is discovered at this time.
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<li><b><tt>vis.discover_vertex(u, g)</tt></b> is invoked the first time the
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algorithm encounters vertex <i>u</i>. All vertices closer to the
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source vertex have been discovered, and vertices further from the
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source have not yet been discovered.
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<li><b><tt>vis.non_tree_edge(e, g)</tt></b> is invoked (in addition to
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<tt>examine_edge()</tt>) if the edge is not a tree edge.
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<li><b><tt>vis.gray_target(e, g)</tt></b> is invoked (in addition to
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<tt>non_tree_edge()</tt>) if the target vertex is colored gray at the
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time of examination. The color gray indicates that
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the vertex is currently in the queue.
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<li><b><tt>vis.black_target(e, g)</tt></b> is invoked (in addition to
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<tt>non_tree_edge()</tt>) if the target vertex is colored black at the
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time of examination. The color black indicates that the
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vertex is no longer in the queue.
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<li><b><tt>vis.finish_vertex(u, g)</tt></b> is invoked after all of the out
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edges of <i>u</i> have been examined and all of the adjacent vertices
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have been discovered.
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</ul>
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<H3><A NAME="SECTION001330400000000000000">
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Example</A>
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</H3>
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<P>
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The example in <a
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href="../example/bfs-example.cpp"><TT>example/bfs-example.cpp</TT></a>
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demonstrates using the BGL Breadth-first search algorithm on the graph
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from <A HREF="./graph_theory_review.html#fig:bfs-example">Figure
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6</A>. The file
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<a href="../example/bfs-example2.cpp"><TT>example/bfs-example2.cpp</TT></a>
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contains the same example, except that the <tt>adacency_list</tt>
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class used has <tt>VertexList</tt> and <tt>EdgeList</tt> set
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to <tt>listS</tt>.
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</P>
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<h3>See Also</h3>
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<a href="./bfs_visitor.html"><tt>bfs_visitor</tt></a> and
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<a href="./depth_first_search.html"><tt>depth_first_search()</tt></a>
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<h3>Notes</h3>
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<p><a name="1">[1]</a>
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Since the visitor parameter is passed by value, if your visitor
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contains state then any changes to the state during the algorithm
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will be made to a copy of the visitor object, not the visitor object
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passed in. Therefore you may want the visitor to hold this state by
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pointer or reference.
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<br>
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<HR>
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<TABLE>
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<TR valign=top>
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<TD nowrap>Copyright © 2000-2001</TD><TD>
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<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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</TD></TR></TABLE>
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</BODY>
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</HTML>
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