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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: Edmonds-Karp 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:edmonds_karp_max_flow"> | |
19 | <TT>edmonds_karp_max_flow</TT> | |
20 | </H1> | |
21 | ||
22 | <PRE> | |
23 | <i>// named parameter version</i> | |
24 | template <class <a href="./Graph.html">Graph</a>, class P, class T, class R> | |
25 | typename detail::edge_capacity_value<Graph, P, T, R>::value_type | |
26 | edmonds_karp_max_flow(Graph& g, | |
27 | typename graph_traits<Graph>::vertex_descriptor src, | |
28 | typename graph_traits<Graph>::vertex_descriptor sink, | |
29 | const bgl_named_params<P, T, R>& params = <i>all defaults</i>) | |
30 | ||
31 | <i>// non-named parameter version</i> | |
32 | template <class <a href="./Graph.html">Graph</a>, | |
33 | class CapacityEdgeMap, class ResidualCapacityEdgeMap, | |
34 | class ReverseEdgeMap, class ColorMap, class PredEdgeMap> | |
35 | typename property_traits<CapacityEdgeMap>::value_type | |
36 | edmonds_karp_max_flow(Graph& g, | |
37 | typename graph_traits<Graph>::vertex_descriptor src, | |
38 | typename graph_traits<Graph>::vertex_descriptor sink, | |
39 | CapacityEdgeMap cap, ResidualCapacityEdgeMap res, ReverseEdgeMap rev, | |
40 | ColorMap color, PredEdgeMap pred) | |
41 | </PRE> | |
42 | ||
43 | <P> | |
44 | The <tt>edmonds_karp_max_flow()</tt> function calculates the maximum flow | |
45 | of a network. See Section <a | |
46 | href="./graph_theory_review.html#sec:network-flow-algorithms">Network | |
47 | Flow Algorithms</a> for a description of maximum flow. The calculated | |
48 | maximum flow will be the return value of the function. The function | |
49 | also calculates the flow values <i>f(u,v)</i> for all <i>(u,v)</i> in | |
50 | <i>E</i>, which are returned in the form of the residual capacity | |
51 | <i>r(u,v) = c(u,v) - f(u,v)</i>. | |
52 | ||
53 | <p> | |
54 | There are several special requirements on the input graph and property | |
55 | map parameters for this algorithm. First, the directed graph | |
56 | <i>G=(V,E)</i> that represents the network must be augmented to | |
57 | include the reverse edge for every edge in <i>E</i>. That is, the | |
58 | input graph should be <i>G<sub>in</sub> = (V,{E U | |
59 | E<sup>T</sup>})</i>. The <tt>ReverseEdgeMap</tt> argument <tt>rev</tt> | |
60 | must map each edge in the original graph to its reverse edge, that is | |
61 | <i>(u,v) -> (v,u)</i> for all <i>(u,v)</i> in <i>E</i>. The | |
62 | <tt>CapacityEdgeMap</tt> argument <tt>cap</tt> must map each edge in | |
63 | <i>E</i> to a positive number, and each edge in <i>E<sup>T</sup></i> | |
64 | to 0. | |
65 | ||
66 | <p> | |
67 | The algorithm is due to <a | |
68 | href="./bibliography.html#edmonds72:_improvements_netflow">Edmonds and | |
69 | Karp</a>, though we are using the variation called the ``labeling | |
70 | algorithm'' described in <a | |
71 | href="./bibliography.html#ahuja93:_network_flows">Network Flows</a>. | |
72 | ||
73 | <p> | |
74 | This algorithm provides a very simple and easy to implement solution to | |
75 | the maximum flow problem. However, there are several reasons why this | |
76 | algorithm is not as good as the <a | |
77 | href="./push_relabel_max_flow.html"><tt>push_relabel_max_flow()</tt></a> | |
78 | or the <a | |
79 | href="./boykov_kolmogorov_max_flow.html"><tt>boykov_kolmogorov_max_flow()</tt></a> | |
80 | algorithm. | |
81 | ||
82 | <ul> | |
83 | <li>In the non-integer capacity case, the time complexity is <i>O(V | |
84 | E<sup>2</sup>)</i> which is worse than the time complexity of the | |
85 | push-relabel algorithm <i>O(V<sup>2</sup>E<sup>1/2</sup>)</i> | |
86 | for all but the sparsest of graphs.</li> | |
87 | ||
88 | <li>In the integer capacity case, if the capacity bound <i>U</i> is | |
89 | very large then the algorithm will take a long time.</li> | |
90 | </ul> | |
91 | ||
92 | ||
93 | <H3>Where Defined</H3> | |
94 | ||
95 | <P> | |
96 | <a href="../../../boost/graph/edmonds_karp_max_flow.hpp"><TT>boost/graph/edmonds_karp_max_flow.hpp</TT></a> | |
97 | ||
98 | <P> | |
99 | ||
100 | <h3>Parameters</h3> | |
101 | ||
102 | IN: <tt>Graph& g</tt> | |
103 | <blockquote> | |
104 | A directed graph. The | |
105 | graph's type must be a model of <a | |
106 | href="./VertexListGraph.html">VertexListGraph</a> and <a href="./IncidenceGraph.html">IncidenceGraph</a> For each edge | |
107 | <i>(u,v)</i> in the graph, the reverse edge <i>(v,u)</i> must also | |
108 | be in the graph. | |
109 | </blockquote> | |
110 | ||
111 | IN: <tt>vertex_descriptor src</tt> | |
112 | <blockquote> | |
113 | The source vertex for the flow network graph. | |
114 | </blockquote> | |
115 | ||
116 | IN: <tt>vertex_descriptor sink</tt> | |
117 | <blockquote> | |
118 | The sink vertex for the flow network graph. | |
119 | </blockquote> | |
120 | ||
121 | <h3>Named Parameters</h3> | |
122 | ||
123 | ||
124 | IN: <tt>capacity_map(CapacityEdgeMap cap)</tt> | |
125 | <blockquote> | |
126 | The edge capacity property map. The type must be a model of a | |
127 | constant <a | |
128 | href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>. The | |
129 | key type of the map must be the graph's edge descriptor type.<br> | |
130 | <b>Default:</b> <tt>get(edge_capacity, g)</tt> | |
131 | </blockquote> | |
132 | ||
133 | OUT: <tt>residual_capacity_map(ResidualCapacityEdgeMap res)</tt> | |
134 | <blockquote> | |
135 | This maps edges to their residual capacity. The type must be a model | |
136 | of a mutable <a | |
137 | href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property | |
138 | Map</a>. The key type of the map must be the graph's edge descriptor | |
139 | type.<br> | |
140 | <b>Default:</b> <tt>get(edge_residual_capacity, g)</tt> | |
141 | </blockquote> | |
142 | ||
143 | IN: <tt>reverse_edge_map(ReverseEdgeMap rev)</tt> | |
144 | <blockquote> | |
145 | An edge property map that maps every edge <i>(u,v)</i> in the graph | |
146 | to the reverse edge <i>(v,u)</i>. The map must be a model of | |
147 | constant <a href="../../property_map/doc/LvaluePropertyMap.html">Lvalue | |
148 | Property Map</a>. The key type of the map must be the graph's edge | |
149 | descriptor type.<br> | |
150 | <b>Default:</b> <tt>get(edge_reverse, g)</tt> | |
151 | </blockquote> | |
152 | ||
153 | UTIL: <tt>color_map(ColorMap color)</tt> | |
154 | <blockquote> | |
155 | Used by the algorithm to keep track of progress during the | |
156 | breadth-first search stage. At the end of the algorithm, the white | |
157 | vertices define the minimum cut set. The map must be a model of | |
158 | mutable <a | |
159 | href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>. | |
160 | The key type of the map should be the graph's vertex descriptor type, and | |
161 | the value type must be a model of <a | |
162 | href="./ColorValue.html">ColorValue</a>.<br> | |
163 | ||
164 | <b>Default:</b> an <a | |
165 | href="../../property_map/doc/iterator_property_map.html"> | |
166 | <tt>iterator_property_map</tt></a> created from a <tt>std::vector</tt> | |
167 | of <tt>default_color_type</tt> of size <tt>num_vertices(g)</tt> and | |
168 | using the <tt>i_map</tt> for the index map. | |
169 | </blockquote> | |
170 | ||
171 | UTIL: <tt>predecessor_map(PredEdgeMap pred)</tt> | |
172 | <blockquote> | |
173 | Use by the algorithm to store augmenting paths. The map must be a | |
174 | model of mutable <a | |
175 | href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>. | |
176 | The key type must be the graph's vertex descriptor type and the | |
177 | value type must be the graph's edge descriptor type.<br> | |
178 | ||
179 | <b>Default:</b> an <a | |
180 | href="../../property_map/doc/iterator_property_map.html"> | |
181 | <tt>iterator_property_map</tt></a> created from a <tt>std::vector</tt> | |
182 | of edge descriptors of size <tt>num_vertices(g)</tt> and | |
183 | using the <tt>i_map</tt> for the index map. | |
184 | </blockquote> | |
185 | ||
186 | IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt> | |
187 | <blockquote> | |
188 | Maps each vertex of the graph to a unique integer in the range | |
189 | <tt>[0, num_vertices(g))</tt>. This property map is only needed | |
190 | if the default for the color or predecessor map is used. | |
191 | The vertex index map must be a model of <a | |
192 | href="../../property_map/doc/ReadablePropertyMap.html">Readable Property | |
193 | Map</a>. The key type of the map must be the graph's vertex | |
194 | descriptor type.<br> | |
195 | <b>Default:</b> <tt>get(vertex_index, g)</tt> | |
196 | Note: if you use this default, make sure your graph has | |
197 | an internal <tt>vertex_index</tt> property. For example, | |
198 | <tt>adjacency_list</tt> with <tt>VertexList=listS</tt> does | |
199 | not have an internal <tt>vertex_index</tt> property. | |
200 | </blockquote> | |
201 | ||
202 | ||
203 | <h3>Complexity</h3> | |
204 | ||
205 | The time complexity is <i>O(V E<sup>2</sup>)</i> in the general case | |
206 | or <i>O(V E U)</i> if capacity values are integers bounded by | |
207 | some constant <i>U</i>. | |
208 | ||
209 | <h3>Example</h3> | |
210 | ||
211 | The program in <a | |
212 | href="../example/edmonds-karp-eg.cpp"><tt>example/edmonds-karp-eg.cpp</tt></a> | |
213 | reads an example maximum flow problem (a graph with edge capacities) | |
214 | from a file in the DIMACS format and computes the maximum flow. | |
215 | ||
216 | ||
217 | <h3>See Also</h3> | |
218 | ||
219 | <a href="./push_relabel_max_flow.html"><tt>push_relabel_max_flow()</tt></a><br> | |
220 | <a href="./boykov_kolmogorov_max_flow.html"><tt>boykov_kolmogorov_max_flow()</tt></a>. | |
221 | ||
222 | <br> | |
223 | <HR> | |
224 | <TABLE> | |
225 | <TR valign=top> | |
226 | <TD nowrap>Copyright © 2000-2001</TD><TD> | |
227 | <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>) | |
228 | </TD></TR></TABLE> | |
229 | ||
230 | </BODY> | |
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234 | <!-- LocalWords: gif ALT BR sec edmonds karp TT DIV CELLPADDING TR TD PRE lt | |
235 | --> | |
236 | <!-- LocalWords: typename VertexListGraph CapacityEdgeMap ReverseEdgeMap gt | |
237 | --> | |
238 | <!-- LocalWords: ResidualCapacityEdgeMap VertexIndexMap src rev ColorMap pred | |
239 | --> | |
240 | <!-- LocalWords: PredEdgeMap tt href html hpp ul li nbsp br LvaluePropertyMap | |
241 | --> | |
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