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<Title>Boost Graph Library: Edmonds-Karp Maximum Flow</Title>
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<H1><A NAME="sec:edmonds_karp_max_flow">
<TT>edmonds_karp_max_flow</TT>
</H1>

<PRE>
<i>// named parameter version</i>
template &lt;class <a href="./Graph.html">Graph</a>, class P, class T, class R&gt;
typename detail::edge_capacity_value&lt;Graph, P, T, R&gt;::value_type
edmonds_karp_max_flow(Graph& g, 
   typename graph_traits&lt;Graph&gt;::vertex_descriptor src,
   typename graph_traits&lt;Graph&gt;::vertex_descriptor sink,
   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="./Graph.html">Graph</a>, 
	  class CapacityEdgeMap, class ResidualCapacityEdgeMap,
	  class ReverseEdgeMap, class ColorMap, class PredEdgeMap&gt;
typename property_traits&lt;CapacityEdgeMap&gt;::value_type
edmonds_karp_max_flow(Graph&amp; g, 
   typename graph_traits&lt;Graph&gt;::vertex_descriptor src,
   typename graph_traits&lt;Graph&gt;::vertex_descriptor sink,
   CapacityEdgeMap cap, ResidualCapacityEdgeMap res, ReverseEdgeMap rev, 
   ColorMap color, PredEdgeMap pred)
</PRE>

<P>
The <tt>edmonds_karp_max_flow()</tt> function calculates the maximum flow
of a network. See Section <a
href="./graph_theory_review.html#sec:network-flow-algorithms">Network
Flow Algorithms</a> for a description of maximum flow.  The calculated
maximum flow will be the return value of the function. The function
also calculates the flow values <i>f(u,v)</i> for all <i>(u,v)</i> in
<i>E</i>, which are returned in the form of the residual capacity
<i>r(u,v) = c(u,v) - f(u,v)</i>. 

<p>
There are several special requirements on the input graph and property
map parameters for this algorithm. First, the directed graph
<i>G=(V,E)</i> that represents the network must be augmented to
include the reverse edge for every edge in <i>E</i>.  That is, the
input graph should be <i>G<sub>in</sub> = (V,{E U
E<sup>T</sup>})</i>. The <tt>ReverseEdgeMap</tt> argument <tt>rev</tt>
must map each edge in the original graph to its reverse edge, that is
<i>(u,v) -> (v,u)</i> for all <i>(u,v)</i> in <i>E</i>. The
<tt>CapacityEdgeMap</tt> argument <tt>cap</tt> must map each edge in
<i>E</i> to a positive number, and each edge in <i>E<sup>T</sup></i>
to 0.

<p>
The algorithm is due to <a
href="./bibliography.html#edmonds72:_improvements_netflow">Edmonds and
Karp</a>, though we are using the variation called the ``labeling
algorithm'' described in <a
href="./bibliography.html#ahuja93:_network_flows">Network Flows</a>.

<p>
This algorithm provides a very simple and easy to implement solution to
the maximum flow problem. However, there are several reasons why this
algorithm is not as good as the <a
href="./push_relabel_max_flow.html"><tt>push_relabel_max_flow()</tt></a>
or the <a
href="./boykov_kolmogorov_max_flow.html"><tt>boykov_kolmogorov_max_flow()</tt></a>
algorithm.

<ul>
  <li>In the non-integer capacity case, the time complexity is <i>O(V
   E<sup>2</sup>)</i> which is worse than the time complexity of the
   push-relabel algorithm <i>O(V<sup>2</sup>E<sup>1/2</sup>)</i>
   for all but the sparsest of graphs.</li>

  <li>In the integer capacity case, if the capacity bound <i>U</i> is
    very large then the algorithm will take a long time.</li>
</ul>


<H3>Where Defined</H3>

<P>
<a href="../../../boost/graph/edmonds_karp_max_flow.hpp"><TT>boost/graph/edmonds_karp_max_flow.hpp</TT></a>

<P>

<h3>Parameters</h3>

IN: <tt>Graph&amp; g</tt>
<blockquote>
  A directed graph. The
  graph's type must be a model of <a
  href="./VertexListGraph.html">VertexListGraph</a> and <a href="./IncidenceGraph.html">IncidenceGraph</a> For each edge
  <i>(u,v)</i> in the graph, the reverse edge <i>(v,u)</i> must also
  be in the graph.
</blockquote>

IN: <tt>vertex_descriptor src</tt>
<blockquote>
  The source vertex for the flow network graph.
</blockquote>
  
IN: <tt>vertex_descriptor sink</tt>
<blockquote>
  The sink vertex for the flow network graph.
</blockquote>
  
<h3>Named Parameters</h3>


IN: <tt>capacity_map(CapacityEdgeMap cap)</tt>
<blockquote>
  The edge capacity property map. The type must be a model of a
  constant <a
  href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>. The
  key type of the map must be the graph's edge descriptor type.<br>
  <b>Default:</b> <tt>get(edge_capacity, g)</tt>
</blockquote>
  
OUT: <tt>residual_capacity_map(ResidualCapacityEdgeMap res)</tt>
<blockquote>
  This maps edges to their residual capacity. The type must be a model
  of a mutable <a
  href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property
  Map</a>. The key type of the map must be the graph's edge descriptor
  type.<br>
  <b>Default:</b> <tt>get(edge_residual_capacity, g)</tt>
</blockquote>

IN: <tt>reverse_edge_map(ReverseEdgeMap rev)</tt>
<blockquote>
  An edge property map that maps every edge <i>(u,v)</i> in the graph
  to the reverse edge <i>(v,u)</i>. The map must be a model of
  constant <a href="../../property_map/doc/LvaluePropertyMap.html">Lvalue
  Property Map</a>. The key type of the map must be the graph's edge
  descriptor type.<br>
  <b>Default:</b> <tt>get(edge_reverse, g)</tt>
</blockquote>

UTIL: <tt>color_map(ColorMap color)</tt>
<blockquote>
  Used by the algorithm to keep track of progress during the
  breadth-first search stage. At the end of the algorithm, the white
  vertices define the minimum cut set. The map must be a model of
  mutable <a
  href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>.
  The key type of the map should be the graph's vertex descriptor type, and
  the value type must be a model of <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.
</blockquote>

UTIL: <tt>predecessor_map(PredEdgeMap pred)</tt>
<blockquote>
  Use by the algorithm to store augmenting paths. The map must be a
  model of mutable <a
  href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>.
  The key type must be the graph's vertex descriptor type and the
  value type must be the graph's edge descriptor type.<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 edge descriptors of size <tt>num_vertices(g)</tt> and
  using the <tt>i_map</tt> for the index map.
</blockquote>

IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt>
<blockquote>
  Maps each vertex of the graph to a unique integer in the range
  <tt>[0, num_vertices(g))</tt>. This property map is only needed
  if the default for the color or predecessor map is used.
  The vertex index map must be a model of <a
  href="../../property_map/doc/ReadablePropertyMap.html">Readable Property
  Map</a>. The key type of the map must be the graph's vertex
  descriptor type.<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.
</blockquote>


<h3>Complexity</h3>

The time complexity is <i>O(V E<sup>2</sup>)</i> in the general case
or <i>O(V E U)</i> if capacity values are integers bounded by
some constant <i>U</i>.

<h3>Example</h3>

The program in <a
href="../example/edmonds-karp-eg.cpp"><tt>example/edmonds-karp-eg.cpp</tt></a>
reads an example maximum flow problem (a graph with edge capacities)
from a file in the DIMACS format and computes the maximum flow.


<h3>See Also</h3>

<a href="./push_relabel_max_flow.html"><tt>push_relabel_max_flow()</tt></a><br>
<a href="./boykov_kolmogorov_max_flow.html"><tt>boykov_kolmogorov_max_flow()</tt></a>.

<br>
<HR>
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<TD nowrap>Copyright &copy; 2000-2001</TD><TD>
<A HREF="http://www.boost.org/users/people/jeremy_siek.html">Jeremy Siek</A>, Indiana University (<A HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)
</TD></TR></TABLE>

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