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 <Head>
 <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>
 <TABLE>
 <TR valign=top>
 <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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	<Head>
	<Title>Boost Graph Library: Edmonds-Karp Maximum Flow</Title>
	<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
	ALINK="#ff0000">
	<IMG SRC="../../../boost.png"
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	<BR Clear>

	<H1><A NAME="sec:edmonds_karp_max_flow">
	<TT>edmonds_karp_max_flow</TT>
	</H1>

	<PRE>
	<i>// named parameter version</i>
	template <class <a href="./Graph.html">Graph</a>, class P, class T, class R>
	typename detail::edge_capacity_value<Graph, P, T, R>::value_type
	edmonds_karp_max_flow(Graph& g,
	typename graph_traits<Graph>::vertex_descriptor src,
	typename graph_traits<Graph>::vertex_descriptor sink,
	const bgl_named_params<P, T, R>& params = <i>all defaults</i>)

	<i>// non-named parameter version</i>
	template <class <a href="./Graph.html">Graph</a>,
	class CapacityEdgeMap, class ResidualCapacityEdgeMap,
	class ReverseEdgeMap, class ColorMap, class PredEdgeMap>
	typename property_traits<CapacityEdgeMap>::value_type
	edmonds_karp_max_flow(Graph& g,
	typename graph_traits<Graph>::vertex_descriptor src,
	typename graph_traits<Graph>::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& 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>
	<TABLE>
	<TR valign=top>
	<TD nowrap>Copyright © 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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