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1 | <HTML> |
2 | <!-- | |
3 | Copyright (c) Jeremy Siek 2000 | |
4 | ||
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) | |
8 | --> | |
9 | <Head> | |
10 | <Title>Boost Graph Library: Push-Relabel Maximum Flow</Title> | |
11 | <BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b" | |
12 | ALINK="#ff0000"> | |
13 | <IMG SRC="../../../boost.png" | |
14 | ALT="C++ Boost" width="277" height="86"> | |
15 | ||
16 | <BR Clear> | |
17 | ||
18 | <H1><A NAME="sec:push_relabel_max_flow"> | |
19 | <TT>push_relabel_max_flow</TT> | |
20 | </H1> | |
21 | ||
22 | <P> | |
23 | <PRE> | |
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>) | |
31 | ||
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) | |
42 | </PRE> | |
43 | ||
44 | <P> | |
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>. | |
53 | ||
54 | <p> | |
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> | |
65 | to 0. | |
66 | ||
67 | <p> | |
68 | This algorithm was developed by <a | |
69 | href="./bibliography.html#goldberg85:_new_max_flow_algor">Goldberg</a>. | |
70 | ||
71 | ||
72 | <H3>Complexity</H3> | |
73 | ||
74 | The time complexity is <i>O(V<sup>3</sup>)</i>. | |
75 | ||
76 | ||
77 | <H3>Where Defined</H3> | |
78 | ||
79 | <P> | |
80 | <a href="../../../boost/graph/push_relabel_max_flow.hpp"><TT>boost/graph/push_relabel_max_flow.hpp</TT></a> | |
81 | ||
82 | <P> | |
83 | ||
84 | <h3>Parameters</h3> | |
85 | ||
86 | IN: <tt>VertexListGraph& g</tt> | |
87 | <blockquote> | |
88 | A directed graph. The | |
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 | |
92 | be in the graph. | |
93 | </blockquote> | |
94 | ||
95 | IN: <tt>vertex_descriptor src</tt> | |
96 | <blockquote> | |
97 | The source vertex for the flow network graph. | |
98 | </blockquote> | |
99 | ||
100 | IN: <tt>vertex_descriptor sink</tt> | |
101 | <blockquote> | |
102 | The sink vertex for the flow network graph. | |
103 | </blockquote> | |
104 | ||
105 | <h3>Named Parameters</h3> | |
106 | ||
107 | IN: <tt>capacity_map(EdgeCapacityMap cap)</tt> | |
108 | <blockquote> | |
109 | The edge capacity property map. The type must be a model of a | |
110 | constant <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> | |
114 | </blockquote> | |
115 | ||
116 | OUT: <tt>residual_capacity_map(ResidualCapacityEdgeMap res)</tt> | |
117 | <blockquote> | |
118 | The edge residual capacity property map. The type must be a model of | |
119 | a mutable <a | |
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> | |
123 | </blockquote> | |
124 | ||
125 | IN: <tt>reverse_edge_map(ReverseEdgeMap rev)</tt> | |
126 | <blockquote> | |
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 | |
129 | constant <a | |
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> | |
133 | </blockquote> | |
134 | ||
135 | IN: <tt>vertex_index_map(VertexIndexMap index_map)</tt> | |
136 | <blockquote> | |
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>adjacency_list</tt> with <tt>VertexList=listS</tt> does | |
145 | not have an internal <tt>vertex_index</tt> property. | |
146 | <br> | |
147 | </blockquote> | |
148 | ||
149 | ||
150 | <h3>Example</h3> | |
151 | ||
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>. | |
156 | ||
157 | <pre> | |
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> | |
164 | ||
165 | int | |
166 | main() | |
167 | { | |
168 | using namespace boost; | |
169 | ||
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> > > | |
176 | > Graph; | |
177 | ||
178 | Graph g; | |
179 | long flow; | |
180 | ||
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); | |
187 | ||
188 | Traits::vertex_descriptor s, t; | |
189 | read_dimacs_max_flow(g, capacity, rev, s, t); | |
190 | ||
191 | flow = push_relabel_max_flow(g, s, t); | |
192 | ||
193 | std::cout << "c The total flow:" << std::endl; | |
194 | std::cout << "s " << flow << std::endl << std::endl; | |
195 | ||
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; | |
204 | return 0; | |
205 | } | |
206 | </pre> | |
207 | The output is: | |
208 | <pre> | |
209 | c The total flow: | |
210 | s 4 | |
211 | ||
212 | c flow values: | |
213 | f 0 1 4 | |
214 | f 1 2 4 | |
215 | f 2 3 2 | |
216 | f 2 4 2 | |
217 | f 3 1 0 | |
218 | f 3 6 2 | |
219 | f 4 5 3 | |
220 | f 5 6 0 | |
221 | f 5 7 3 | |
222 | f 6 4 1 | |
223 | f 6 7 1 | |
224 | </pre> | |
225 | ||
226 | <h3>See Also</h3> | |
227 | ||
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>. | |
230 | ||
231 | <br> | |
232 | <HR> | |
233 | <TABLE> | |
234 | <TR valign=top> | |
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>) | |
237 | </TD></TR></TABLE> | |
238 | ||
239 | </BODY> | |
240 | </HTML> | |
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244 | --> | |
245 | <!-- LocalWords: typename VertexListGraph CapacityEdgeMap ReverseEdgeMap gt | |
246 | --> | |
247 | <!-- LocalWords: ResidualCapacityEdgeMap VertexIndexMap src rev ColorMap pred | |
248 | --> | |
249 | <!-- LocalWords: PredEdgeMap tt href html hpp ul li nbsp br LvaluePropertyMap | |
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