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<Title>Bellman Ford Shortest Paths</Title>
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<H1><A NAME="sec:bellman-ford"></A><img src="figs/python.gif" alt="(Python)"/>
<TT>bellman_ford_shortest_paths</TT>
</H1>

<P>
<PRE>
<i>// named paramter version</i>
template &lt;class <a href="./EdgeListGraph.html">EdgeListGraph</a>, class Size, class P, class T, class R&gt;
bool bellman_ford_shortest_paths(const EdgeListGraph&amp; g, Size N, 
  const bgl_named_params&lt;P, T, R&gt;&amp; params = <i>all defaults</i>);

template &lt;class <a href="./VertexAndEdgeListGraph.html">VertexAndEdgeListGraph</a>, class P, class T, class R&gt;
bool bellman_ford_shortest_paths(const VertexAndEdgeListGraph&amp; g,
  const bgl_named_params&lt;P, T, R&gt;&amp; params = <i>all defaults</i>);

<i>// non-named parameter version</i>
template &lt;class <a href="./EdgeListGraph.html">EdgeListGraph</a>, class Size, class WeightMap,
	  class PredecessorMap, class DistanceMap,
	  class <a href="http://www.sgi.com/tech/stl/BinaryFunction.html">BinaryFunction</a>, class <a href="http://www.sgi.com/tech/stl/BinaryPredicate.html">BinaryPredicate</a>,
	  class <a href="./BellmanFordVisitor.html">BellmanFordVisitor</a>&gt;
bool bellman_ford_shortest_paths(EdgeListGraph&amp; g, Size N, 
  WeightMap weight, PredecessorMap pred, DistanceMap distance, 
  BinaryFunction combine, BinaryPredicate compare, BellmanFordVisitor v)
</PRE>

<P>
The Bellman-Ford algorithm&nbsp;[<A
HREF="bibliography.html#bellman58">4</A>,<A
HREF="bibliography.html#ford62:_flows">11</A>,<A
HREF="bibliography.html#lawler76:_comb_opt">20</A>,<A
HREF="bibliography.html#clr90">8</A>] solves the single-source
shortest paths problem for a graph with both positive and negative
edge weights. For the definition of the shortest paths problem see
Section <A
HREF="./graph_theory_review.html#sec:shortest-paths-algorithms">Shortest-Paths
Algorithms</A>. 
If you only need to solve the shortest paths problem for positive edge
weights, Dijkstra's algorithm provides a more efficient
alternative. If all the edge weights are all equal to one then breadth-first
search provides an even more efficient alternative.
</p>

<p>
Before calling the <tt>bellman_ford_shortest_paths()</tt> function,
the user must assign the source vertex a distance of zero and all
other vertices a distance of infinity <i>unless</i> you are providing
a starting vertex. The Bellman-Ford algorithm
proceeds by looping through all of the edges in the graph, applying
the relaxation operation to each edge. In the following pseudo-code,
<i>v</i> is a vertex adjacent to <i>u</i>, <i>w</i> maps edges to
their weight, and <i>d</i> is a distance map that records the length
of the shortest path to each vertex seen so far. <i>p</i> is a
predecessor map which records the parent of each vertex, which will
ultimately be the parent in the shortest paths tree
</p>

<table>
<tr>
<td valign="top">
<pre>
RELAX(<i>u</i>, <i>v</i>, <i>w</i>, <i>d</i>, <i>p</i>)
  <b>if</b> (<i>w(u,v) + d[u] < d[v]</i>) 
    <i>d[v] := w(u,v) + d[u]</i>
    <i>p[v] := u</i>
  <b>else</b>
    ...
</pre>
</td>
<td valign="top">
<pre>


relax edge <i>(u,v)</i>


edge <i>(u,v)</i> is not relaxed 
</pre>
</td>
</tr>
</table>

<p>
The algorithm repeats this loop <i>|V|</i> times after which it is
guaranteed that the distances to each vertex have been reduced to the
minimum possible unless there is a negative cycle in the graph. If
there is a negative cycle, then there will be edges in the graph that
were not properly minimized. That is, there will be edges <i>(u,v)</i> such
that <i>w(u,v) + d[u] < d[v]</i>.  The algorithm loops over the edges in
the graph one final time to check if all the edges were minimized,
returning <tt>true</tt> if they were and returning <tt>false</tt>
otherwise.
</p>

<table>
<tr>
<td valign="top">
<pre>
BELLMAN-FORD(<i>G</i>)
  <i>// Optional initialization</i>
  <b>for</b> each vertex <i>u in V</i> 
    <i>d[u] := infinity</i>
    <i>p[u] := u</i> 
  <b>end for</b>
  <b>for</b> <i>i := 1</i> <b>to</b> <i>|V|-1</i> 
    <b>for</b> each edge <i>(u,v) in E</i> 
      RELAX(<i>u</i>, <i>v</i>, <i>w</i>, <i>d</i>, <i>p</i>)
    <b>end for</b>
  <b>end for</b>
  <b>for</b> each edge <i>(u,v) in E</i> 
    <b>if</b> (<i>w(u,v) + d[u] < d[v]</i>)
      <b>return</b> (false, , )
    <b>else</b> 
      ...
  <b>end for</b>
  <b>return</b> (true, <i>p</i>, <i>d</i>)
</pre>
</td>
<td valign="top">
<pre>


examine edge <i>(u,v)</i>


edge <i>(u,v)</i> was not minimized 

edge <i>(u,v)</i> was minimized 
</pre>
</td>
</tr>
</table>

There are two main options for obtaining output from the
<tt>bellman_ford_shortest_paths()</tt> function. If the user provides
a distance property map through the <tt>distance_map()</tt> parameter
then the shortest distance from the source vertex to every other
vertex in the graph will be recorded in the distance map (provided the
function returns <tt>true</tt>). The second option is recording the
shortest paths tree in the <tt>predecessor_map()</tt>. For each vertex
<i>u in V</i>, <i>p[u]</i> will be the predecessor of <i>u</i> in the
shortest paths tree (unless <i>p[u] = u</i>, in which case <i>u</i> is
either the source vertex or a vertex unreachable from the source).  In
addition to these two options, the user can provide her own
custom-made visitor that can take actions at any of the
algorithm's event points.

<P>

<h3>Parameters</h3>


IN: <tt>EdgeListGraph&amp; g</tt> 
<blockquote>
  A directed or undirected graph whose type must be a model of
  <a href="./EdgeListGraph.html">Edge List Graph</a>. If a root vertex is
  provided, then the graph must also model
  <a href="./VertexListGraph.html">Vertex List Graph</a>.<br>

  <b>Python</b>: The parameter is named <tt>graph</tt>.
</blockquote>

IN: <tt>Size N</tt>
<blockquote>
  The number of vertices in the graph. The type <tt>Size</tt> must
  be an integer type.<br>
  <b>Default:</b> <tt>num_vertices(g)</tt>.<br>

  <b>Python</b>: Unsupported parameter.
</blockquote>


<h3>Named Parameters</h3>

IN: <tt>weight_map(WeightMap w)</tt>
<blockquote>
  The weight (also know as ``length'' or ``cost'') of each edge in the
  graph.  The <tt>WeightMap</tt> type must be a model of <a
  href="../../property_map/doc/ReadablePropertyMap.html">Readable Property
  Map</a>.  The key type for this property map must be the edge
  descriptor of the graph.  The value type for the weight map must be
  <i>Addable</i> with the distance map's value type. <br>
  <b>Default:</b> <tt>get(edge_weight, g)</tt><br>

  <b>Python</b>: Must be an <tt>edge_double_map</tt> for the graph.<br>
  <b>Python default</b>: <tt>graph.get_edge_double_map("weight")</tt>
</blockquote>

OUT: <tt>predecessor_map(PredecessorMap p_map)</tt> 
<blockquote>
  The predecessor map records the edges in the minimum spanning
  tree. Upon completion of the algorithm, the edges <i>(p[u],u)</i>
  for all <i>u in V</i> are in the minimum spanning tree. If <i>p[u] =
  u</i> then <i>u</i> is either the source vertex or a vertex that is
  not reachable from the source.  The <tt>PredecessorMap</tt> type
  must be a <a
  href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
  Property Map</a> which key and vertex types the same as the vertex
  descriptor type of the graph.<br>
  <b>Default:</b> <tt>dummy_property_map</tt><br>

  <b>Python</b>: Must be a <tt>vertex_vertex_map</tt> for the graph.<br>
</blockquote>

IN/OUT: <tt>distance_map(DistanceMap d)</tt> 
<blockquote>
  The shortest path weight from the source vertex to each vertex in
  the graph <tt>g</tt> is recorded in this property map.  The type
  <tt>DistanceMap</tt> must be a model of <a
  href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
  Property Map</a>. The key type of the property map must be the
  vertex descriptor type of the graph, and the value type of the
  distance map must be <a
  href="http://www.sgi.com/tech/stl/LessThanComparable.html"> Less
  Than Comparable</a>.<br> <b>Default:</b> <tt>get(vertex_distance,
  g)</tt><br>
  <b>Python</b>: Must be a <tt>vertex_double_map</tt> for the graph.<br>
</blockquote>

IN: <tt>root_vertex(Vertex s)</tt>
<blockquote>
  The starting (or "root") vertex from which shortest paths will be
  computed. When provided, the distance map need not be initialized
  (the algorithm will perform the initialization itself). However, the
  graph must model <a href="./VertexListGraph.html">Vertex List
  Graph</a> when this parameter is provided.<br>
  <b>Default:</b> None; if omitted, the user must initialize the
  distance map.
</blockquote>

IN: <tt>visitor(BellmanFordVisitor v)</tt>
<blockquote>
  The visitor object, whose type must be a model of
  <a href="./BellmanFordVisitor.html">Bellman-Ford Visitor</a>.
  The visitor object is passed by value <a
  href="#1">[1]</a>.
<br>
  <b>Default:</b> <tt>bellman_visitor&lt;null_visitor&gt;</tt><br>

  <b>Python</b>: The parameter should be an object that derives from
  the <a
  href="BellmanFordVisitor.html#python"><tt>BellmanFordVisitor</tt></a> type
  of the graph.

</blockquote>

IN: <tt>distance_combine(BinaryFunction combine)</tt>
<blockquote>
  This function object replaces the role of addition in the relaxation
  step. The first argument type must match the distance map's value
  type and the second argument type must match the weight map's value
  type.  The result type must be the same as the distance map's value
  type.<br>
  <b>Default:</b><tt>std::plus&lt;D&gt;</tt>
  with <tt>D=typename&nbsp;property_traits&lt;DistanceMap&gt;::value_type</tt>.<br>

  <b>Python</b>: Unsupported parameter.
</blockquote>

IN: <tt>distance_compare(BinaryPredicate compare)</tt>
<blockquote>
  This function object replaces the role of the less-than operator
  that compares distances in the relaxation step. The argument types
  must match the distance map's value type.<br>
  <b>Default:</b> <tt>std::less&lt;D&gt;</tt>
  with <tt>D=typename&nbsp;property_traits&lt;DistanceMap&gt;::value_type</tt>.<br>

  <b>Python</b>: Unsupported parameter.
</blockquote>

<P>

<H3>Complexity</H3>

<P>
The time complexity is <i>O(V E)</i>.


<h3>Visitor Event Points</h3>

<ul>
<li><b><tt>vis.examine_edge(e, g)</tt></b>  is invoked on every edge in
  the graph <i>|V|</i> times.
<li><b><tt>vis.edge_relaxed(e, g)</tt></b>  is invoked when the distance
  label for the target vertex is decreased. The edge <i>(u,v)</i> that 
  participated in the last relaxation for vertex <i>v</i> is an edge in the
  shortest paths tree. 
<li><b><tt>vis.edge_not_relaxed(e, g)</tt></b>  is invoked if the distance label
  for the target vertex is not decreased. 
<li><b><tt>vis.edge_minimized(e, g)</tt></b>  is invoked during the
  second stage of the algorithm, during the test of whether each edge
  was minimized. If the edge is minimized then this function
  is invoked.
<li><b><tt>vis.edge_not_minimized(e, g)</tt></b>  is also invoked during the
  second stage of the algorithm, during the test of whether each edge
  was minimized.  If the edge was not minimized, this function is
  invoked.  This happens when there is a negative cycle in the graph.
</ul>

<H3>Example</H3>

<P>
An example of using the Bellman-Ford algorithm is in <a
href="../example/bellman-example.cpp"><TT>examples/bellman-example.cpp</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</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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