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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 <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>
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<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>)
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