ceph/src/boost/libs/graph/doc/edmonds_karp_max_flow.html

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  10 <Title>Boost Graph Library: Edmonds-Karp Maximum Flow</Title>
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  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 &lt;class <a href="./Graph.html">Graph</a>, class P, class T, class R&gt;
  25 typename detail::edge_capacity_value&lt;Graph, P, T, R&gt;::value_type
  26 edmonds_karp_max_flow(Graph& g,
  27    typename graph_traits&lt;Graph&gt;::vertex_descriptor src,
  28    typename graph_traits&lt;Graph&gt;::vertex_descriptor sink,
  29    const bgl_named_params&lt;P, T, R&gt;&amp; params = <i>all defaults</i>)
  30
  31 <i>// non-named parameter version</i>
  32 template &lt;class <a href="./Graph.html">Graph</a>,
  33           class CapacityEdgeMap, class ResidualCapacityEdgeMap,
  34           class ReverseEdgeMap, class ColorMap, class PredEdgeMap&gt;
  35 typename property_traits&lt;CapacityEdgeMap&gt;::value_type
  36 edmonds_karp_max_flow(Graph&amp; g,
  37    typename graph_traits&lt;Graph&gt;::vertex_descriptor src,
  38    typename graph_traits&lt;Graph&gt;::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&amp; 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 &copy; 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
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