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11 KiB
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-- Copyright (c) Jeremy Siek 2000
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<Head>
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<Title>Bellman Ford Shortest Paths</Title>
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<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
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ALINK="#ff0000">
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<IMG SRC="../../../c++boost.gif"
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ALT="C++ Boost" width="277" height="86">
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<BR Clear>
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<H1><A NAME="sec:bellman-ford"></A>
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<TT>bellman_ford_shortest_paths</TT>
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</H1>
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<P>
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<PRE>
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<i>// named paramter version</i>
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template <class <a href="./EdgeListGraph.html">EdgeListGraph</a>, class Size, class P, class T, class R>
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bool bellman_ford_shortest_paths(EdgeListGraph& g, Size N,
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const bgl_named_params<P, T, R>& params = <i>all defaults</i>)
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<i>// non-named parameter version</i>
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template <class <a href="./EdgeListGraph.html">EdgeListGraph</a>, class Size, class WeightMap,
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class PredecessorMap, class DistanceMap,
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class <a href="http://www.sgi.com/tech/stl/BinaryFunction.html">BinaryFunction</a>, class <a href="http://www.sgi.com/tech/stl/BinaryPredicate.html">BinaryPredicate</a>,
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class <a href="./BellmanFordVisitor.html">BellmanFordVisitor</a>>
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bool bellman_ford_shortest_paths(EdgeListGraph& g, Size N,
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WeightMap weight, PredecessorMap pred, DistanceMap distance,
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BinaryFunction combine, BinaryPredicate compare, BellmanFordVisitor v)
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</PRE>
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<P>
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The Bellman-Ford algorithm [<A
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HREF="bibliography.html#bellman58">4</A>,<A
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HREF="bibliography.html#ford62:_flows">11</A>,<A
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HREF="bibliography.html#lawler76:_comb_opt">20</A>,<A
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HREF="bibliography.html#clr90">8</A>] solves the single-source
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shortest paths problem for a graph with both positive and negative
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edge weights. For the definition of the shortest paths problem see
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Section <A
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HREF="./graph_theory_review.html#sec:shortest-paths-algorithms">Shortest-Paths
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Algorithms</A>.
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If you only need to solve the shortest paths problem for positive edge
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weights, Dijkstra's algorithm provides a more efficient
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alternative. If all the edge weights are all equal to one then breadth-first
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search provides an even more efficient alternative.
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</p>
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<p>
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Before calling the <tt>bellman_ford_shortest_paths()</tt> function,
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the user must assign the source vertex a distance of zero and all
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other vertices a distance of infinity. The Bellman-Ford algorithm
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proceeds by looping through all of the edges in the graph, applying
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the relaxation operation to each edge. In the following pseudo-code,
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<i>v</i> is a vertex adjacent to <i>u</i>, <i>w</i> maps edges to
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their weight, and <i>d</i> is a distance map that records the length
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of the shortest path to each vertex seen so far. <i>p</i> is a
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predecessor map which records the parent of each vertex, which will
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ultimately be the parent in the shortest paths tree
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</p>
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<table>
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<tr>
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<td valign="top">
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<pre>
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RELAX(<i>u</i>, <i>v</i>, <i>w</i>, <i>d</i>, <i>p</i>)
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<b>if</b> (<i>w(u,v) + d[u] < d[v]</i>)
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<i>d[v] := w(u,v) + d[u]</i>
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<i>p[v] := u</i>
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<b>else</b>
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...
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</pre>
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</td>
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<td valign="top">
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<pre>
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relax edge <i>(u,v)</i>
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edge <i>(u,v)</i> is not relaxed
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</pre>
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</td>
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</tr>
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</table>
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<p>
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The algorithm repeats this loop <i>|V|</i> times after which it is
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guaranteed that the distances to each vertex have been reduced to the
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minimum possible unless there is a negative cycle in the graph. If
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there is a negative cycle, then there will be edges in the graph that
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were not properly minimized. That is, there will be edges <i>(u,v)</i> such
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that <i>w(u,v) + d[u] < d[v]</i>. The algorithm loops over the edges in
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the graph one final time to check if all the edges were minimized,
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returning <tt>true</tt> if they were and returning <tt>false</tt>
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otherwise.
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</p>
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<table>
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<tr>
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<td valign="top">
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<pre>
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BELLMAN-FORD(<i>G</i>)
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<b>for</b> each vertex <i>u in V</i>
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<i>d[u] := infinity</i>
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<i>p[u] := u</i>
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<b>end for</b>
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<b>for</b> <i>i := 1</i> <b>to</b> <i>|V|-1</i>
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<b>for</b> each edge <i>(u,v) in E</i>
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RELAX(<i>u</i>, <i>v</i>, <i>w</i>, <i>d</i>, <i>p</i>)
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<b>end for</b>
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<b>end for</b>
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<b>for</b> each edge <i>(u,v) in E</i>
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<b>if</b> (<i>w(u,v) + d[u] < d[v]</i>)
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<b>return</b> (false, , )
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<b>else</b>
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...
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<b>end for</b>
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<b>return</b> (true, <i>p</i>, <i>d</i>)
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</pre>
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</td>
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<td valign="top">
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<pre>
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initialize vertex <i>u</i>
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examine edge <i>(u,v)</i>
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edge <i>(u,v)</i> was not minimized
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edge <i>(u,v)</i> was minimized
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</pre>
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</td>
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</tr>
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</table>
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There are two main options for obtaining output from the
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<tt>bellman_ford_shortest_paths()</tt> function. If the user provides
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a distance property map through the <tt>distance_map()</tt> parameter
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then the shortest distance from the source vertex to every other
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vertex in the graph will be recorded in the distance map (provided the
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function returns <tt>true</tt>). The second option is recording the
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shortest paths tree in the <tt>predecessor_map()</tt>. For each vertex
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<i>u in V</i>, <i>p[u]</i> will be the predecessor of <i>u</i> in the
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shortest paths tree (unless <i>p[u] = u</i>, in which case <i>u</i> is
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either the source vertex or a vertex unreachable from the source). In
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addition to these two options, the user can provide there own
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custom-made visitor that can takes actions during any of the
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algorithm's event points.
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<P>
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<h3>Parameters</h3>
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IN: <tt>EdgeListGraph& g</tt>
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<blockquote>
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A directed or undirected graph whose type must be a model of
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<a href="./EdgeListGraph.html">Edge List Graph</a>.
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</blockquote>
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IN: <tt>Size N</tt>
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<blockquote>
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The number of vertices in the graph. The type <tt>Size</tt> must
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be an integer type.
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</blockquote>
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<h3>Named Parameters</h3>
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IN: <tt>weight_map(WeightMap w)</tt>
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<blockquote>
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The weight (also know as ``length'' or ``cost'') of each edge in the
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graph. The <tt>WeightMap</tt> type must be a model of <a
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href="../../property_map/ReadablePropertyMap.html">Readable Property
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Map</a>. The key type for this property map must be the edge
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descriptor of the graph. The value type for the weight map must be
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<i>Addable</i> with the distance map's value type. <br>
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<b>Default:</b> <tt>get(edge_weight, g)</tt>
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</blockquote>
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OUT: <tt>predecessor_map(PredecessorMap p_map)</tt>
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<blockquote>
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The predecessor map records the edges in the minimum spanning
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tree. Upon completion of the algorithm, the edges <i>(p[u],u)</i>
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for all <i>u in V</i> are in the minimum spanning tree. If <i>p[u] =
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u</i> then <i>u</i> is either the source vertex or a vertex that is
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not reachable from the source. The <tt>PredecessorMap</tt> type
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must be a <a
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href="../../property_map/ReadWritePropertyMap.html">Read/Write
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Property Map</a> which key and vertex types the same as the vertex
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descriptor type of the graph.<br>
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<b>Default:</b> <tt>dummy_property_map</tt>
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</blockquote>
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IN/OUT: <tt>distance_map(DistanceMap d)</tt>
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<blockquote>
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The shortest path weight from the source vertex to each vertex in
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the graph <tt>g</tt> is recorded in this property map. The type
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<tt>DistanceMap</tt> must be a model of <a
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href="../../property_map/ReadWritePropertyMap.html">Read/Write
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Property Map</a>. The key type of the property map must be the
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vertex descriptor type of the graph, and the value type of the
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distance map must be <a
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href="http://www.sgi.com/tech/stl/LessThanComparable.html"> Less
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Than Comparable</a>.<br> <b>Default:</b> <tt>get(vertex_distance,
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g)</tt>
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</blockquote>
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IN: <tt>visitor(BellmanFordVisitor v)</tt>
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<blockquote>
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The visitor object, whose type must be a model of
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<a href="./BellmanFordVisitor.html">Bellman-Ford Visitor</a>.
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The visitor object is passed by value <a
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href="#1">[1]</a>.
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<br>
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<b>Default:</b> <tt>bellman_visitor<null_visitor></tt>
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</blockquote>
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IN: <tt>distance_combine(BinaryFunction combine)</tt>
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<blockquote>
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This function object replaces the role of addition in the relaxation
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step. The first argument type must match the distance map's value
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type and the second argument type must match the weight map's value
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type. The result type must be the same as the distance map's value
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type.<br>
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<b>Default:</b><tt>std::plus<D></tt>
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with <tt>D=typename property_traits<DistanceMap>::value_type</tt>.
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</blockquote>
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IN: <tt>distance_compare(BinaryPredicate compare)</tt>
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<blockquote>
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This function object replaces the role of the less-than operator
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that compares distances in the relaxation step. The argument types
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must match the distance map's value type.<br>
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<b>Default:</b> <tt>std::less<D></tt>
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with <tt>D=typename property_traits<DistanceMap>::value_type</tt>.
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</blockquote>
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<P>
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<H3>Complexity</H3>
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<P>
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The time complexity is <i>O(V E)</i>.
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<h3>Visitor Event Points</h3>
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<ul>
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<li><b><tt>vis.examine_edge(e, g)</tt></b> is invoked on every edge in
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the graph <i>|V|</i> times.
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<li><b><tt>vis.edge_relaxed(e, g)</tt></b> is invoked when the distance
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label for the target vertex is decreased. The edge <i>(u,v)</i> that
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participated in the last relaxation for vertex <i>v</i> is an edge in the
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shortest paths tree.
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<li><b><tt>vis.edge_not_relaxed(e, g)</tt></b> is invoked if the distance label
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for the target vertex is not decreased.
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<li><b><tt>vis.edge_minimized(e, g)</tt></b> is invoked during the
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second stage of the algorithm, during the test of whether each edge
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was minimized. If the edge is minimized then this function
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is invoked.
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<li><b><tt>vis.edge_not_minimized(e, g)</tt></b> is also invoked during the
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second stage of the algorithm, during the test of whether each edge
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was minimized. If the edge was not minimized, this function is
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invoked. This happens when there is a negative cycle in the graph.
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</ul>
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<H3>Example</H3>
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<P>
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An example of using the Bellman-Ford algorithm is in <a
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href="../example/bellman-example.cpp"><TT>examples/bellman-example.cpp</TT></a>.
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<h3>Notes</h3>
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<p><a name="1">[1]</a>
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Since the visitor parameter is passed by value, if your visitor
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contains state then any changes to the state during the algorithm
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will be made to a copy of the visitor object, not the visitor object
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passed in. Therefore you may want the visitor to hold this state by
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pointer or reference.
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<br>
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<HR>
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<TABLE>
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<TR valign=top>
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<TD nowrap>Copyright © 2000</TD><TD>
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<A HREF="../../../people/jeremy_siek.htm">Jeremy Siek</A>, Indiana University (<A HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)
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</TD></TR></TABLE>
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</BODY>
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</HTML>
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