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: 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> 24template <class <a href="./Graph.html">Graph</a>, class P, class T, class R> 25typename detail::edge_capacity_value<Graph, P, T, R>::value_type 26edmonds_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> 32template <class <a href="./Graph.html">Graph</a>, 33 class CapacityEdgeMap, class ResidualCapacityEdgeMap, 34 class ReverseEdgeMap, class ColorMap, class PredEdgeMap> 35typename property_traits<CapacityEdgeMap>::value_type 36edmonds_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> 44The <tt>edmonds_karp_max_flow()</tt> function calculates the maximum flow 45of a network. See Section <a 46href="./graph_theory_review.html#sec:network-flow-algorithms">Network 47Flow Algorithms</a> for a description of maximum flow. The calculated 48maximum flow will be the return value of the function. The function 49also 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> 54There are several special requirements on the input graph and property 55map parameters for this algorithm. First, the directed graph 56<i>G=(V,E)</i> that represents the network must be augmented to 57include the reverse edge for every edge in <i>E</i>. That is, the 58input graph should be <i>G<sub>in</sub> = (V,{E U 59E<sup>T</sup>})</i>. The <tt>ReverseEdgeMap</tt> argument <tt>rev</tt> 60must 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> 64to 0. 65 66<p> 67The algorithm is due to <a 68href="./bibliography.html#edmonds72:_improvements_netflow">Edmonds and 69Karp</a>, though we are using the variation called the ``labeling 70algorithm'' described in <a 71href="./bibliography.html#ahuja93:_network_flows">Network Flows</a>. 72 73<p> 74This algorithm provides a very simple and easy to implement solution to 75the maximum flow problem. However, there are several reasons why this 76algorithm is not as good as the <a 77href="./push_relabel_max_flow.html"><tt>push_relabel_max_flow()</tt></a> 78or the <a 79href="./boykov_kolmogorov_max_flow.html"><tt>boykov_kolmogorov_max_flow()</tt></a> 80algorithm. 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 102IN: <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 111IN: <tt>vertex_descriptor src</tt> 112<blockquote> 113 The source vertex for the flow network graph. 114</blockquote> 115 116IN: <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 124IN: <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 133OUT: <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 143IN: <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 153UTIL: <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 171UTIL: <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 186IN: <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 205The time complexity is <i>O(V E<sup>2</sup>)</i> in the general case 206or <i>O(V E U)</i> if capacity values are integers bounded by 207some constant <i>U</i>. 208 209<h3>Example</h3> 210 211The program in <a 212href="../example/edmonds-karp-eg.cpp"><tt>example/edmonds-karp-eg.cpp</tt></a> 213reads an example maximum flow problem (a graph with edge capacities) 214from 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> 231</HTML> 232<!-- LocalWords: HTML Siek Edmonds BGCOLOR ffffff ee VLINK ALINK ff IMG SRC 233 --> 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 --> 242<!-- LocalWords: num ColorValue DIMACS cpp pre config iostream dimacs int std 243 --> 244<!-- LocalWords: namespace vecS directedS cout endl iter ei HR valign nowrap 245 --> 246<!-- LocalWords: jeremy siek htm Univ mailto jsiek lsc edu 247p --> 248
