248 lines
9.1 KiB
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248 lines
9.1 KiB
HTML
<HTML>
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<!--
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Copyright (c) Jeremy Siek 2000
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt)
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-->
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<Head>
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<Title>Boost Graph Library: Edmonds-Karp Maximum Flow</Title>
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<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
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ALINK="#ff0000">
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<IMG SRC="../../../boost.png"
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ALT="C++ Boost" width="277" height="86">
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<BR Clear>
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<H1><A NAME="sec:edmonds_karp_max_flow">
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<TT>edmonds_karp_max_flow</TT>
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</H1>
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<PRE>
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<i>// named parameter version</i>
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template <class <a href="./Graph.html">Graph</a>, class P, class T, class R>
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typename detail::edge_capacity_value<Graph, P, T, R>::value_type
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edmonds_karp_max_flow(Graph& g,
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typename graph_traits<Graph>::vertex_descriptor src,
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typename graph_traits<Graph>::vertex_descriptor sink,
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const bgl_named_params<P, T, R>& params = <i>all defaults</i>)
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<i>// non-named parameter version</i>
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template <class <a href="./Graph.html">Graph</a>,
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class CapacityEdgeMap, class ResidualCapacityEdgeMap,
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class ReverseEdgeMap, class ColorMap, class PredEdgeMap>
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typename property_traits<CapacityEdgeMap>::value_type
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edmonds_karp_max_flow(Graph& g,
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typename graph_traits<Graph>::vertex_descriptor src,
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typename graph_traits<Graph>::vertex_descriptor sink,
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CapacityEdgeMap cap, ResidualCapacityEdgeMap res, ReverseEdgeMap rev,
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ColorMap color, PredEdgeMap pred)
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</PRE>
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<P>
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The <tt>edmonds_karp_max_flow()</tt> function calculates the maximum flow
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of a network. See Section <a
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href="./graph_theory_review.html#sec:network-flow-algorithms">Network
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Flow Algorithms</a> for a description of maximum flow. The calculated
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maximum flow will be the return value of the function. The function
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also calculates the flow values <i>f(u,v)</i> for all <i>(u,v)</i> in
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<i>E</i>, which are returned in the form of the residual capacity
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<i>r(u,v) = c(u,v) - f(u,v)</i>.
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<p>
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There are several special requirements on the input graph and property
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map parameters for this algorithm. First, the directed graph
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<i>G=(V,E)</i> that represents the network must be augmented to
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include the reverse edge for every edge in <i>E</i>. That is, the
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input graph should be <i>G<sub>in</sub> = (V,{E U
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E<sup>T</sup>})</i>. The <tt>ReverseEdgeMap</tt> argument <tt>rev</tt>
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must map each edge in the original graph to its reverse edge, that is
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<i>(u,v) -> (v,u)</i> for all <i>(u,v)</i> in <i>E</i>. The
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<tt>CapacityEdgeMap</tt> argument <tt>cap</tt> must map each edge in
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<i>E</i> to a positive number, and each edge in <i>E<sup>T</sup></i>
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to 0.
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<p>
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The algorithm is due to <a
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href="./bibliography.html#edmonds72:_improvements_netflow">Edmonds and
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Karp</a>, though we are using the variation called the ``labeling
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algorithm'' described in <a
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href="./bibliography.html#ahuja93:_network_flows">Network Flows</a>.
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<p>
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This algorithm provides a very simple and easy to implement solution to
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the maximum flow problem. However, there are several reasons why this
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algorithm is not as good as the <a
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href="./push_relabel_max_flow.html"><tt>push_relabel_max_flow()</tt></a>
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or the <a
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href="./boykov_kolmogorov_max_flow.html"><tt>boykov_kolmogorov_max_flow()</tt></a>
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algorithm.
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<ul>
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<li>In the non-integer capacity case, the time complexity is <i>O(V
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E<sup>2</sup>)</i> which is worse than the time complexity of the
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push-relabel algorithm <i>O(V<sup>2</sup>E<sup>1/2</sup>)</i>
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for all but the sparsest of graphs.</li>
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<li>In the integer capacity case, if the capacity bound <i>U</i> is
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very large then the algorithm will take a long time.</li>
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</ul>
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<H3>Where Defined</H3>
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<P>
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<a href="../../../boost/graph/edmonds_karp_max_flow.hpp"><TT>boost/graph/edmonds_karp_max_flow.hpp</TT></a>
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<P>
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<h3>Parameters</h3>
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IN: <tt>Graph& g</tt>
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<blockquote>
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A directed graph. The
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graph's type must be a model of <a
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href="./VertexListGraph.html">VertexListGraph</a> and <a href="./IncidenceGraph.html">IncidenceGraph</a> For each edge
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<i>(u,v)</i> in the graph, the reverse edge <i>(v,u)</i> must also
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be in the graph.
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</blockquote>
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IN: <tt>vertex_descriptor src</tt>
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<blockquote>
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The source vertex for the flow network graph.
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</blockquote>
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IN: <tt>vertex_descriptor sink</tt>
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<blockquote>
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The sink vertex for the flow network graph.
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</blockquote>
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<h3>Named Parameters</h3>
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IN: <tt>capacity_map(CapacityEdgeMap cap)</tt>
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<blockquote>
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The edge capacity property map. The type must be a model of a
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constant <a
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href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>. The
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key type of the map must be the graph's edge descriptor type.<br>
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<b>Default:</b> <tt>get(edge_capacity, g)</tt>
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</blockquote>
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OUT: <tt>residual_capacity_map(ResidualCapacityEdgeMap res)</tt>
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<blockquote>
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This maps edges to their residual capacity. The type must be a model
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of a mutable <a
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href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property
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Map</a>. The key type of the map must be the graph's edge descriptor
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type.<br>
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<b>Default:</b> <tt>get(edge_residual_capacity, g)</tt>
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</blockquote>
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IN: <tt>reverse_edge_map(ReverseEdgeMap rev)</tt>
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<blockquote>
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An edge property map that maps every edge <i>(u,v)</i> in the graph
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to the reverse edge <i>(v,u)</i>. The map must be a model of
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constant <a href="../../property_map/doc/LvaluePropertyMap.html">Lvalue
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Property Map</a>. The key type of the map must be the graph's edge
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descriptor type.<br>
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<b>Default:</b> <tt>get(edge_reverse, g)</tt>
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</blockquote>
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UTIL: <tt>color_map(ColorMap color)</tt>
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<blockquote>
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Used by the algorithm to keep track of progress during the
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breadth-first search stage. At the end of the algorithm, the white
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vertices define the minimum cut set. The map must be a model of
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mutable <a
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href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>.
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The key type of the map should be the graph's vertex descriptor type, and
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the value type must be a model of <a
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href="./ColorValue.html">ColorValue</a>.<br>
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<b>Default:</b> an <a
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href="../../property_map/doc/iterator_property_map.html">
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<tt>iterator_property_map</tt></a> created from a <tt>std::vector</tt>
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of <tt>default_color_type</tt> of size <tt>num_vertices(g)</tt> and
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using the <tt>i_map</tt> for the index map.
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</blockquote>
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UTIL: <tt>predecessor_map(PredEdgeMap pred)</tt>
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<blockquote>
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Use by the algorithm to store augmenting paths. The map must be a
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model of mutable <a
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href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>.
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The key type must be the graph's vertex descriptor type and the
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value type must be the graph's edge descriptor type.<br>
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<b>Default:</b> an <a
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href="../../property_map/doc/iterator_property_map.html">
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<tt>iterator_property_map</tt></a> created from a <tt>std::vector</tt>
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of edge descriptors of size <tt>num_vertices(g)</tt> and
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using the <tt>i_map</tt> for the index map.
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</blockquote>
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IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt>
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<blockquote>
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Maps each vertex of the graph to a unique integer in the range
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<tt>[0, num_vertices(g))</tt>. This property map is only needed
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if the default for the color or predecessor map is used.
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The vertex index map must be a model of <a
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href="../../property_map/doc/ReadablePropertyMap.html">Readable Property
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Map</a>. The key type of the map must be the graph's vertex
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descriptor type.<br>
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<b>Default:</b> <tt>get(vertex_index, g)</tt>
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Note: if you use this default, make sure your graph has
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an internal <tt>vertex_index</tt> property. For example,
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<tt>adjacency_list</tt> with <tt>VertexList=listS</tt> does
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not have an internal <tt>vertex_index</tt> property.
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</blockquote>
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<h3>Complexity</h3>
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The time complexity is <i>O(V E<sup>2</sup>)</i> in the general case
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or <i>O(V E U)</i> if capacity values are integers bounded by
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some constant <i>U</i>.
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<h3>Example</h3>
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The program in <a
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href="../example/edmonds-karp-eg.cpp"><tt>example/edmonds-karp-eg.cpp</tt></a>
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reads an example maximum flow problem (a graph with edge capacities)
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from a file in the DIMACS format and computes the maximum flow.
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<h3>See Also</h3>
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<a href="./push_relabel_max_flow.html"><tt>push_relabel_max_flow()</tt></a><br>
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<a href="./boykov_kolmogorov_max_flow.html"><tt>boykov_kolmogorov_max_flow()</tt></a>.
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<br>
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<HR>
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<TABLE>
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<TR valign=top>
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<TD nowrap>Copyright © 2000-2001</TD><TD>
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<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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</TD></TR></TABLE>
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</BODY>
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</HTML>
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<!-- LocalWords: HTML Siek Edmonds BGCOLOR ffffff ee VLINK ALINK ff IMG SRC
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<!-- LocalWords: gif ALT BR sec edmonds karp TT DIV CELLPADDING TR TD PRE lt
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<!-- LocalWords: typename VertexListGraph CapacityEdgeMap ReverseEdgeMap gt
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-->
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<!-- LocalWords: ResidualCapacityEdgeMap VertexIndexMap src rev ColorMap pred
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-->
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<!-- LocalWords: PredEdgeMap tt href html hpp ul li nbsp br LvaluePropertyMap
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-->
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<!-- LocalWords: num ColorValue DIMACS cpp pre config iostream dimacs int std
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-->
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<!-- LocalWords: namespace vecS directedS cout endl iter ei HR valign nowrap
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p -->
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