3 Copyright (c) Jeremy Siek 2000
5 Distributed under the Boost Software License, Version 1.0.
6 (See accompanying file LICENSE_1_0.txt or copy at
7 http://www.boost.org/LICENSE_1_0.txt)
10 <Title>Boost Graph Library: Push-Relabel Maximum Flow</Title>
11 <BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
13 <IMG SRC="../../../boost.png"
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18 <H1><A NAME="sec:push_relabel_max_flow">
19 <TT>push_relabel_max_flow</TT>
24 <i>// named parameter version</i>
25 template <class Graph, class P, class T, class R>
26 typename property_traits<CapacityEdgeMap>::value_type
27 push_relabel_max_flow(Graph& g,
28 typename graph_traits<Graph>::vertex_descriptor src,
29 typename graph_traits<Graph>::vertex_descriptor sink,
30 const bgl_named_params<P, T, R>& params = <i>all defaults</i>)
32 <i>// non-named parameter version</i>
33 template <class Graph,
34 class CapacityEdgeMap, class ResidualCapacityEdgeMap,
35 class ReverseEdgeMap, class VertexIndexMap>
36 typename property_traits<CapacityEdgeMap>::value_type
37 push_relabel_max_flow(Graph& g,
38 typename graph_traits<Graph>::vertex_descriptor src,
39 typename graph_traits<Graph>::vertex_descriptor sink,
40 CapacityEdgeMap cap, ResidualCapacityEdgeMap res,
41 ReverseEdgeMap rev, VertexIndexMap index_map)
45 The <tt>push_relabel_max_flow()</tt> function calculates the maximum flow
46 of a network. See Section <a
47 href="./graph_theory_review.html#sec:network-flow-algorithms">Network
48 Flow Algorithms</a> for a description of maximum flow. The calculated
49 maximum flow will be the return value of the function. The function
50 also calculates the flow values <i>f(u,v)</i> for all <i>(u,v)</i> in
51 <i>E</i>, which are returned in the form of the residual capacity
52 <i>r(u,v) = c(u,v) - f(u,v)</i>.
55 There are several special requirements on the input graph and property
56 map parameters for this algorithm. First, the directed graph
57 <i>G=(V,E)</i> that represents the network must be augmented to
58 include the reverse edge for every edge in <i>E</i>. That is, the
59 input graph should be <i>G<sub>in</sub> = (V,{E U
60 E<sup>T</sup>})</i>. The <tt>ReverseEdgeMap</tt> argument <tt>rev</tt>
61 must map each edge in the original graph to its reverse edge, that is
62 <i>(u,v) -> (v,u)</i> for all <i>(u,v)</i> in <i>E</i>. The
63 <tt>CapacityEdgeMap</tt> argument <tt>cap</tt> must map each edge in
64 <i>E</i> to a positive number, and each edge in <i>E<sup>T</sup></i>
68 This algorithm was developed by <a
69 href="./bibliography.html#goldberg85:_new_max_flow_algor">Goldberg</a>.
74 The time complexity is <i>O(V<sup>3</sup>)</i>.
77 <H3>Where Defined</H3>
80 <a href="../../../boost/graph/push_relabel_max_flow.hpp"><TT>boost/graph/preflow_push_max_flow.hpp</TT></a>
86 IN: <tt>VertexListGraph& g</tt>
89 graph's type must be a model of <a
90 href="./VertexListGraph.html">Vertex List Graph</a>. For each edge
91 <i>(u,v)</i> in the graph, the reverse edge <i>(v,u)</i> must also
95 IN: <tt>vertex_descriptor src</tt>
97 The source vertex for the flow network graph.
100 IN: <tt>vertex_descriptor sink</tt>
102 The sink vertex for the flow network graph.
105 <h3>Named Parameters</h3>
107 IN: <tt>capacity_map(EdgeCapacityMap cap)</tt>
109 The edge capacity property map. The type must be a model of a
111 href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>. The
112 key type of the map must be the graph's edge descriptor type.<br>
113 <b>Default:</b> <tt>get(edge_capacity, g)</tt>
116 OUT: <tt>residual_capacity_map(ResidualCapacityEdgeMap res)</tt>
118 The edge residual capacity property map. The type must be a model of
120 href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>. The
121 key type of the map must be the graph's edge descriptor type.<br>
122 <b>Default:</b> <tt>get(edge_residual_capacity, g)</tt>
125 IN: <tt>reverse_edge_map(ReverseEdgeMap rev)</tt>
127 An edge property map that maps every edge <i>(u,v)</i> in the graph
128 to the reverse edge <i>(v,u)</i>. The map must be a model of
130 href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>. The
131 key type of the map must be the graph's edge descriptor type.<br>
132 <b>Default:</b> <tt>get(edge_reverse, g)</tt>
135 IN: <tt>vertex_index_map(VertexIndexMap index_map)</tt>
137 Maps each vertex of the graph to a unique integer in the range
138 <tt>[0, num_vertices(g))</tt>. The map must be a model of constant <a
139 href="../../property_map/doc/LvaluePropertyMap.html">LvaluePropertyMap</a>. The
140 key type of the map must be the graph's vertex descriptor type.<br>
141 <b>Default:</b> <tt>get(vertex_index, g)</tt>
142 Note: if you use this default, make sure your graph has
143 an internal <tt>vertex_index</tt> property. For example,
144 <tt>adjacenty_list</tt> with <tt>VertexList=listS</tt> does
145 not have an internal <tt>vertex_index</tt> property.
152 This reads in an example maximum flow problem (a graph with edge
153 capacities) from a file in the DIMACS format. The source for this
154 example can be found in <a
155 href="../example/max_flow.cpp"><tt>example/max_flow.cpp</tt></a>.
158 #include <boost/config.hpp>
159 #include <iostream>
160 #include <string>
161 #include <boost/graph/push_relabel_max_flow.hpp>
162 #include <boost/graph/adjacency_list.hpp>
163 #include <boost/graph/read_dimacs.hpp>
168 using namespace boost;
170 typedef adjacency_list_traits<vecS, vecS, directedS> Traits;
171 typedef adjacency_list<vecS, vecS, directedS,
172 property<vertex_name_t, std::string>,
173 property<edge_capacity_t, long,
174 property<edge_residual_capacity_t, long,
175 property<edge_reverse_t, Traits::edge_descriptor> > >
181 property_map<Graph, edge_capacity_t>::type
182 capacity = get(edge_capacity, g);
183 property_map<Graph, edge_reverse_t>::type
184 rev = get(edge_reverse, g);
185 property_map<Graph, edge_residual_capacity_t>::type
186 residual_capacity = get(edge_residual_capacity, g);
188 Traits::vertex_descriptor s, t;
189 read_dimacs_max_flow(g, capacity, rev, s, t);
191 flow = push_relabel_max_flow(g, s, t);
193 std::cout << "c The total flow:" << std::endl;
194 std::cout << "s " << flow << std::endl << std::endl;
196 std::cout << "c flow values:" << std::endl;
197 graph_traits<Graph>::vertex_iterator u_iter, u_end;
198 graph_traits<Graph>::out_edge_iterator ei, e_end;
199 for (boost::tie(u_iter, u_end) = vertices(g); u_iter != u_end; ++u_iter)
200 for (boost::tie(ei, e_end) = out_edges(*u_iter, g); ei != e_end; ++ei)
201 if (capacity[*ei] > 0)
202 std::cout << "f " << *u_iter << " " << target(*ei, g) << " "
203 << (capacity[*ei] - residual_capacity[*ei]) << std::endl;
228 <a href="./edmonds_karp_max_flow.html"><tt>edmonds_karp_max_flow()</tt></a><br>
229 <a href="./boykov_kolmogorov_max_flow.html"><tt>boykov_kolmogorov_max_flow()</tt></a>.
235 <TD nowrap>Copyright © 2000-2001</TD><TD>
236 <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>)
241 <!-- LocalWords: HTML Siek BGCOLOR ffffff ee VLINK ALINK ff IMG SRC preflow
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