257 lines
8.4 KiB
HTML
257 lines
8.4 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: Push-Relabel 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:push_relabel_max_flow">
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<TT>push_relabel_max_flow</TT>
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</H1>
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<P>
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<PRE>
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<i>// named parameter version</i>
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template <class Graph, class P, class T, class R>
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typename property_traits<CapacityEdgeMap>::value_type
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push_relabel_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 Graph,
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class CapacityEdgeMap, class ResidualCapacityEdgeMap,
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class ReverseEdgeMap, class VertexIndexMap>
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typename property_traits<CapacityEdgeMap>::value_type
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push_relabel_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,
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ReverseEdgeMap rev, VertexIndexMap index_map)
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</PRE>
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<P>
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The <tt>push_relabel_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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This algorithm was developed by <a
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href="./bibliography.html#goldberg85:_new_max_flow_algor">Goldberg</a>.
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<H3>Complexity</H3>
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The time complexity is <i>O(V<sup>3</sup>)</i>.
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<H3>Where Defined</H3>
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<P>
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<a href="../../../boost/graph/push_relabel_max_flow.hpp"><TT>boost/graph/push_relabel_max_flow.hpp</TT></a>
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<P>
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<h3>Parameters</h3>
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IN: <tt>VertexListGraph& 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">Vertex List Graph</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(EdgeCapacityMap 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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The edge residual capacity property map. The type must be a model of
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a mutable <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_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
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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_reverse, g)</tt>
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</blockquote>
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IN: <tt>vertex_index_map(VertexIndexMap index_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>. The map must be a model of constant <a
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href="../../property_map/doc/LvaluePropertyMap.html">LvaluePropertyMap</a>. The
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key type of the map must be the graph's vertex 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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<br>
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</blockquote>
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<h3>Example</h3>
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This reads in an example maximum flow problem (a graph with edge
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capacities) from a file in the DIMACS format. The source for this
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example can be found in <a
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href="../example/max_flow.cpp"><tt>example/max_flow.cpp</tt></a>.
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<pre>
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#include <boost/config.hpp>
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#include <iostream>
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#include <string>
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#include <boost/graph/push_relabel_max_flow.hpp>
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#include <boost/graph/adjacency_list.hpp>
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#include <boost/graph/read_dimacs.hpp>
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int
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main()
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{
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using namespace boost;
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typedef adjacency_list_traits<vecS, vecS, directedS> Traits;
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typedef adjacency_list<vecS, vecS, directedS,
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property<vertex_name_t, std::string>,
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property<edge_capacity_t, long,
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property<edge_residual_capacity_t, long,
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property<edge_reverse_t, Traits::edge_descriptor> > >
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> Graph;
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Graph g;
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long flow;
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property_map<Graph, edge_capacity_t>::type
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capacity = get(edge_capacity, g);
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property_map<Graph, edge_reverse_t>::type
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rev = get(edge_reverse, g);
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property_map<Graph, edge_residual_capacity_t>::type
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residual_capacity = get(edge_residual_capacity, g);
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Traits::vertex_descriptor s, t;
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read_dimacs_max_flow(g, capacity, rev, s, t);
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flow = push_relabel_max_flow(g, s, t);
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std::cout << "c The total flow:" << std::endl;
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std::cout << "s " << flow << std::endl << std::endl;
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std::cout << "c flow values:" << std::endl;
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graph_traits<Graph>::vertex_iterator u_iter, u_end;
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graph_traits<Graph>::out_edge_iterator ei, e_end;
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for (boost::tie(u_iter, u_end) = vertices(g); u_iter != u_end; ++u_iter)
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for (boost::tie(ei, e_end) = out_edges(*u_iter, g); ei != e_end; ++ei)
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if (capacity[*ei] > 0)
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std::cout << "f " << *u_iter << " " << target(*ei, g) << " "
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<< (capacity[*ei] - residual_capacity[*ei]) << std::endl;
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return 0;
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}
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</pre>
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The output is:
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<pre>
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c The total flow:
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s 4
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c flow values:
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f 0 1 4
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f 1 2 4
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f 2 3 2
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f 2 4 2
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f 3 1 0
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f 3 6 2
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f 4 5 3
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f 5 6 0
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f 5 7 3
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f 6 4 1
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f 6 7 1
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</pre>
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<h3>See Also</h3>
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<a href="./edmonds_karp_max_flow.html"><tt>edmonds_karp_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/people/jeremy_siek.htm">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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<!-- LocalWords: typename VertexListGraph CapacityEdgeMap ReverseEdgeMap gt
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<!-- LocalWords: ResidualCapacityEdgeMap VertexIndexMap src rev ColorMap pred
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