<HTML> <!-- Copyright (c) Jeremy Siek 2000 Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) --> <Head> <Title>Boost Graph Library: Push-Relabel Maximum Flow</Title> <BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b" ALINK="#ff0000"> <IMG SRC="../../../boost.png" ALT="C++ Boost" width="277" height="86"> <BR Clear> <H1><A NAME="sec:push_relabel_max_flow"> <TT>push_relabel_max_flow</TT> </H1> <P> <PRE> <i>// named parameter version</i> template <class Graph, class P, class T, class R> typename property_traits<CapacityEdgeMap>::value_type push_relabel_max_flow(Graph& g, typename graph_traits<Graph>::vertex_descriptor src, typename graph_traits<Graph>::vertex_descriptor sink, const bgl_named_params<P, T, R>& params = <i>all defaults</i>) <i>// non-named parameter version</i> template <class Graph, class CapacityEdgeMap, class ResidualCapacityEdgeMap, class ReverseEdgeMap, class VertexIndexMap> typename property_traits<CapacityEdgeMap>::value_type push_relabel_max_flow(Graph& g, typename graph_traits<Graph>::vertex_descriptor src, typename graph_traits<Graph>::vertex_descriptor sink, CapacityEdgeMap cap, ResidualCapacityEdgeMap res, ReverseEdgeMap rev, VertexIndexMap index_map) </PRE> <P> The <tt>push_relabel_max_flow()</tt> function calculates the maximum flow of a network. See Section <a href="./graph_theory_review.html#sec:network-flow-algorithms">Network Flow Algorithms</a> for a description of maximum flow. The calculated maximum flow will be the return value of the function. The function also calculates the flow values <i>f(u,v)</i> for all <i>(u,v)</i> in <i>E</i>, which are returned in the form of the residual capacity <i>r(u,v) = c(u,v) - f(u,v)</i>. <p> There are several special requirements on the input graph and property map parameters for this algorithm. First, the directed graph <i>G=(V,E)</i> that represents the network must be augmented to include the reverse edge for every edge in <i>E</i>. That is, the input graph should be <i>G<sub>in</sub> = (V,{E U E<sup>T</sup>})</i>. The <tt>ReverseEdgeMap</tt> argument <tt>rev</tt> must map each edge in the original graph to its reverse edge, that is <i>(u,v) -> (v,u)</i> for all <i>(u,v)</i> in <i>E</i>. The <tt>CapacityEdgeMap</tt> argument <tt>cap</tt> must map each edge in <i>E</i> to a positive number, and each edge in <i>E<sup>T</sup></i> to 0. <p> This algorithm was developed by <a href="./bibliography.html#goldberg85:_new_max_flow_algor">Goldberg</a>. <H3>Complexity</H3> The time complexity is <i>O(V<sup>3</sup>)</i>. <H3>Where Defined</H3> <P> <a href="../../../boost/graph/push_relabel_max_flow.hpp"><TT>boost/graph/preflow_push_max_flow.hpp</TT></a> <P> <h3>Parameters</h3> IN: <tt>VertexListGraph& g</tt> <blockquote> A directed graph. The graph's type must be a model of <a href="./VertexListGraph.html">Vertex List Graph</a>. For each edge <i>(u,v)</i> in the graph, the reverse edge <i>(v,u)</i> must also be in the graph. </blockquote> IN: <tt>vertex_descriptor src</tt> <blockquote> The source vertex for the flow network graph. </blockquote> IN: <tt>vertex_descriptor sink</tt> <blockquote> The sink vertex for the flow network graph. </blockquote> <h3>Named Parameters</h3> IN: <tt>capacity_map(EdgeCapacityMap cap)</tt> <blockquote> The edge capacity property map. The type must be a model of a constant <a href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>. The key type of the map must be the graph's edge descriptor type.<br> <b>Default:</b> <tt>get(edge_capacity, g)</tt> </blockquote> OUT: <tt>residual_capacity_map(ResidualCapacityEdgeMap res)</tt> <blockquote> The edge residual capacity property map. The type must be a model of a mutable <a href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>. The key type of the map must be the graph's edge descriptor type.<br> <b>Default:</b> <tt>get(edge_residual_capacity, g)</tt> </blockquote> IN: <tt>reverse_edge_map(ReverseEdgeMap rev)</tt> <blockquote> An edge property map that maps every edge <i>(u,v)</i> in the graph to the reverse edge <i>(v,u)</i>. The map must be a model of constant <a href="../../property_map/doc/LvaluePropertyMap.html">Lvalue Property Map</a>. The key type of the map must be the graph's edge descriptor type.<br> <b>Default:</b> <tt>get(edge_reverse, g)</tt> </blockquote> IN: <tt>vertex_index_map(VertexIndexMap index_map)</tt> <blockquote> Maps each vertex of the graph to a unique integer in the range <tt>[0, num_vertices(g))</tt>. The map must be a model of constant <a href="../../property_map/doc/LvaluePropertyMap.html">LvaluePropertyMap</a>. The key type of the map must be the graph's vertex descriptor type.<br> <b>Default:</b> <tt>get(vertex_index, g)</tt> Note: if you use this default, make sure your graph has an internal <tt>vertex_index</tt> property. For example, <tt>adjacenty_list</tt> with <tt>VertexList=listS</tt> does not have an internal <tt>vertex_index</tt> property. <br> </blockquote> <h3>Example</h3> This reads in an example maximum flow problem (a graph with edge capacities) from a file in the DIMACS format. The source for this example can be found in <a href="../example/max_flow.cpp"><tt>example/max_flow.cpp</tt></a>. <pre> #include <boost/config.hpp> #include <iostream> #include <string> #include <boost/graph/push_relabel_map_flow.hpp> #include <boost/graph/adjacency_list.hpp> #include <boost/graph/read_dimacs.hpp> int main() { using namespace boost; typedef adjacency_list_traits<vecS, vecS, directedS> Traits; typedef adjacency_list<vecS, vecS, directedS, property<vertex_name_t, std::string>, property<edge_capacity_t, long, property<edge_residual_capacity_t, long, property<edge_reverse_t, Traits::edge_descriptor> > > > Graph; Graph g; long flow; property_map<Graph, edge_capacity_t>::type capacity = get(edge_capacity, g); property_map<Graph, edge_reverse_t>::type rev = get(edge_reverse, g); property_map<Graph, edge_residual_capacity_t>::type residual_capacity = get(edge_residual_capacity, g); Traits::vertex_descriptor s, t; read_dimacs_max_flow(g, capacity, rev, s, t); flow = push_relabel_max_flow(g, s, t); std::cout << "c The total flow:" << std::endl; std::cout << "s " << flow << std::endl << std::endl; std::cout << "c flow values:" << std::endl; graph_traits<Graph>::vertex_iterator u_iter, u_end; graph_traits<Graph>::out_edge_iterator ei, e_end; for (tie(u_iter, u_end) = vertices(g); u_iter != u_end; ++u_iter) for (tie(ei, e_end) = out_edges(*u_iter, g); ei != e_end; ++ei) if (capacity[*ei] > 0) std::cout << "f " << *u_iter << " " << target(*ei, g) << " " << (capacity[*ei] - residual_capacity[*ei]) << std::endl; return 0; } </pre> The output is: <pre> c The total flow: s 4 c flow values: f 0 1 4 f 1 2 4 f 2 3 2 f 2 4 2 f 3 1 0 f 3 6 2 f 4 5 3 f 5 6 0 f 5 7 3 f 6 4 1 f 6 7 1 </pre> <h3>See Also</h3> <a href="./edmonds_karp_max_flow.html"><tt>edmonds_karp_max_flow()</tt></a><br> <a href="./boykov_kolmogorov_max_flow.html"><tt>boykov_kolmogorov_max_flow()</tt></a>. <br> <HR> <TABLE> <TR valign=top> <TD nowrap>Copyright © 2000-2001</TD><TD> <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>) </TD></TR></TABLE> </BODY> </HTML> <!-- LocalWords: HTML Siek BGCOLOR ffffff ee VLINK ALINK ff IMG SRC preflow --> <!-- LocalWords: gif ALT BR sec TT DIV CELLPADDING TR TD PRE lt --> <!-- LocalWords: typename VertexListGraph CapacityEdgeMap ReverseEdgeMap gt --> <!-- LocalWords: ResidualCapacityEdgeMap VertexIndexMap src rev ColorMap pred --> <!-- LocalWords: PredEdgeMap tt href html hpp ul li nbsp br LvaluePropertyMap --> <!-- LocalWords: num ColorValue DIMACS cpp pre config iostream dimacs int std --> <!-- LocalWords: namespace vecS directedS cout endl iter ei HR valign nowrap --> <!-- LocalWords: jeremy siek htm Univ mailto jsiek lsc edu -->