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| <Head> |
| <Title>Boost Graph Library: Breadth-First Search</Title> |
| <BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b" |
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| <BR Clear> |
| |
| <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> |
| <HR> |
| <TABLE> |
| <TR valign=top> |
| <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>) |
| </TD></TR></TABLE> |
| |
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