212 lines
8.1 KiB
HTML
212 lines
8.1 KiB
HTML
<HTML>
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<!--
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Copyright (c) Jeremy Siek 2000
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt)
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-->
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<Head>
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<Title>Johnson All Pairs 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="../../../boost.png"
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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:johnson"></A>
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<TT>johnson_all_pairs_shortest_paths</TT>
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</H1>
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<PRE>
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<i>// named parameter version</i>
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template <class VertexAndEdgeListGraph, class DistanceMatrix,
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class VertexID, class Weight, class BinaryPredicate,
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class BinaryFunction, class Infinity, class DistanceZero>
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bool
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johnson_all_pairs_shortest_paths(VertexAndEdgeListGraph& g1,
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DistanceMatrix& D,
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VertexID id1, Weight w1, const BinaryPredicate& compare,
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const BinaryFunction& combine, const Infinity& inf,
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DistanceZero zero);
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template <typename Graph, typename DistanceMatrix, typename P, typename T, typename R>
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bool johnson_all_pairs_shortest_paths(Graph& g, DistanceMatrix& D,
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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 <typename Graph, typename DistanceMatrix,
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typename VertexIndex, typename WeightMap, typename DT>
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bool
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johnson_all_pairs_shortest_paths(VertexAndEdgeListGraph& g1,
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DistanceMatrix& D,
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VertexIndex i_map, WeightMap w_map, DT zero)
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</PRE>
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<P>
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This algorithm finds the shortest distance between every pair of
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vertices in the graph. The algorithm returns false if there is a
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negative weight cycle in the graph and true otherwise. The distance
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between each pair of vertices is stored in the distance matrix
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<TT>D</TT>. This is one of the more time intensive graph algorithms,
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having a time complexity of <i>O(V E log V)</i>.
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<P>This algorithm should be used to compute shortest paths between
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every pair of vertices for sparse graphs. For dense graphs, use <a
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href="floyd_warshall_shortest.html"><code>floyd_warshall_all_pairs_shortest_paths</code></a>.
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<H3>Where Defined</H3>
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<P>
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<a href="../../../boost/graph/johnson_all_pairs_shortest.hpp"><TT>boost/graph/johnson_all_pairs_shortest.hpp</TT></a>
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<h3>Parameters</h3>
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IN: <tt>Graph& g</tt>
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<blockquote>
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A directed or undirected graph. The graph type must be a model of
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<a href="VertexListGraph.html">Vertex List Graph</a>,
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<a href="EdgeListGraph.html">Edge List Graph</a>,
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and <a href="IncidenceGraph.html">Incidence Graph</a>.
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</blockquote>
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OUT: <tt>DistanceMatrix& D</tt>
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<blockquote>
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The length of the shortest path between each pair of vertices
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<i>u,v</i> in the graph is stored in <tt>D[u][v]</tt>. The tuple of
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types (<tt>DistanceMatrix, vertices_size_type, D</tt>) must be a model
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of <a href="BasicMatrix.html">BasicMatrix</a> where <tt>D</tt> is the
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value type of the <tt>DistanceMap</tt>. There must be implicit conversions
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between the value type of the distance matrix and the value type of the weight
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map.
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</blockquote>
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<h3>Named Parameters</h3>
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IN: <tt>weight_map(WeightMap w_map)</tt>
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<blockquote>
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The weight or "length" of each edge in the graph.
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The type <tt>WeightMap</tt> must be a model of
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<a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The edge descriptor type of
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the graph needs to be usable as the key type for the weight
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map. The value type of the weight map must support a subtraction operation.<br>
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<b>Default:</b> <tt>get(edge_weight, g)</tt>
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</blockquote>
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UTIL: <tt>weight_map2(WeightMap2 w2_map)</tt>
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<blockquote>
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This parameter is no longer needed, and will be ignored.
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</blockquote>
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IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt>
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<blockquote>
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This maps each vertex to an integer in the range <tt>[0,
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num_vertices(g))</tt>. This is necessary for efficient updates of the
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heap data structure in the internal call to Dijkstra's algorithm. The type
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<tt>VertexIndexMap</tt> must be a model of
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<a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The value type of the map must be an
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integer type. The vertex descriptor type of the graph needs to be
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usable as the key type of the map.<br>
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<b>Default:</b> <tt>get(vertex_index, g)</tt>
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Note: if you use this default, make sure your graph has
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an internal <tt>vertex_index</tt> property. For example,
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<tt>adjacency_list</tt> with <tt>VertexList=listS</tt> does
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not have an internal <tt>vertex_index</tt> property.
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<br>
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</blockquote>
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UTIL: <tt>distance_map(DistanceMap d_map)</tt>
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<blockquote>
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This parameter is no longer needed, and will be ignored.
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</blockquote>
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IN: <tt>distance_compare(CompareFunction cmp)</tt>
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<blockquote>
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This function is use to compare distances to determine
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which vertex is closer to the source vertex.
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The <tt>CompareFunction</tt> type must be a model of
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<a href="http://www.boost.org/sgi/stl/BinaryPredicate.html">Binary Predicate</a>
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and have argument types that
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match the value type of the <tt>WeightMap</tt> property map.<br>
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<b>Default:</b> <tt>std::less<DT></tt> with
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<tt>DT=typename property_traits<WeightMap>::value_type</tt>
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</blockquote>
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IN: <tt>distance_combine(CombineFunction cmb)</tt>
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<blockquote>
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This function is used to combine distances to compute the distance
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of a path. The <tt>CombineFunction</tt> type must be a model of <a
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href="http://www.boost.org/sgi/stl/BinaryFunction.html">Binary
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Function</a>. Both argument types and the return type of the binary function
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must match the value type of the <tt>WeightMap</tt> property map. This operation is required to act as the sum operation for the weight type; in particular, it must be the inverse of the binary <tt>-</tt> operator on that type.<br>
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<b>Default:</b> <tt>std::plus<DT></tt> with
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<tt>DT=typename property_traits<WeightMap>::value_type</tt>
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</blockquote>
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IN: <tt>distance_inf(DT inf)</tt>
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<blockquote>
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This value is used to initialize the distance for each
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vertex before the start of the algorithm.
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The type <tt>DT</tt> must be the value type of the <tt>WeightMap</tt>.<br>
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<b>Default:</b> <tt>std::numeric_limits::max()</tt>
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</blockquote>
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IN: <tt>distance_zero(DT zero)</tt>
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<blockquote>
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This value is used to initialize the distance for the source
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vertex before the start of the algorithm. The type <tt>DT</tt>
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must be the value type of the <tt>WeightMap</tt>.<br>
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<b>Default:</b> <tt>0</tt>
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</blockquote>
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UTIL/OUT: <tt>color_map(ColorMap c_map)</tt>
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<blockquote>
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This is used during the execution of the algorithm to mark the
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vertices. The vertices start out white and become gray when they are
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inserted in the queue. They then turn black when they are removed
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from the queue. At the end of the algorithm, vertices reachable from
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the source vertex will have been colored black. All other vertices
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will still be white. The type <tt>ColorMap</tt> must be a model of
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<a href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
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Property Map</a>. A vertex descriptor must be usable as the key type
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of the map, and the value type of the map must be a model of
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<a href="./ColorValue.html">Color Value</a>.<br>
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<b>Default:</b> an <a
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href="../../property_map/doc/iterator_property_map.html">
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<tt>iterator_property_map</tt></a> created from a <tt>std::vector</tt>
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of <tt>default_color_type</tt> of size <tt>num_vertices(g)</tt> and
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using the <tt>i_map</tt> for the index map.
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</blockquote>
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<h3>Complexity</h3>
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The time complexity is <i>O(V E log V)</i>.
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<H3>Example</H3>
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<P>
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The file <a
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href="../example/johnson-eg.cpp"><tt>example/johnson-eg.cpp</tt></a> applies
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Johnson's algorithm for all-pairs shortest paths to the example graph
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from page 568 of the CLR [<A
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HREF="bibliography.html#clr90">8</A>].
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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 © 2000-2001</TD><TD>
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<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>)
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</TD></TR></TABLE>
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</BODY>
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</HTML>
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