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 <Head>
 <Title>Johnson All Pairs Shortest Paths</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:johnson">
 <TT>johnson_all_pairs_shortest_paths</TT>
 </H1>

 <PRE>
 <i>// named parameter version</i>
   template &lt;class VertexAndEdgeListGraph, class DistanceMatrix,
             class VertexID, class Weight, class BinaryPredicate,
             class BinaryFunction, class Infinity, class DistanceZero&gt;
   bool
   johnson_all_pairs_shortest_paths(VertexAndEdgeListGraph& g1,
                DistanceMatrix&amp; D,
                VertexID id1, Weight w1, const BinaryPredicate&amp; compare,
                const BinaryFunction&amp; combine, const Infinity&amp; inf,
                DistanceZero zero);

 template &lt;typename Graph, typename DistanceMatrix, typename P, typename T, typename R&gt;
 bool johnson_all_pairs_shortest_paths(Graph&amp; g, DistanceMatrix&amp; D,
   const bgl_named_params&lt;P, T, R&gt;&amp; params = <i>all defaults</i>)

 <i>// non-named parameter version</i>
 template &lt;typename Graph, typename DistanceMatrix,
           typename VertexIndex, typename WeightMap, typename DT&gt;
 bool
 johnson_all_pairs_shortest_paths(VertexAndEdgeListGraph&amp; g1,
   DistanceMatrix&amp; D,
   VertexIndex i_map, WeightMap w_map, DT zero)
 </PRE>

 <P>
 This algorithm finds the shortest distance between every pair of
 vertices in the graph. The algorithm returns false if there is a
 negative weight cycle in the graph and true otherwise. The distance
 between each pair of vertices is stored in the distance matrix
 <TT>D</TT>. This is one of the more time intensive graph algorithms,
 having a time complexity of <i>O(V E log V)</i>.

 <P>This algorithm should be used to compute shortest paths between
 every pair of vertices for sparse graphs. For dense graphs, use <a
 href="floyd_warshall_shortest.html"><code>floyd_warshall_all_pairs_shortest_paths</code></a>.

 <H3>Where Defined</H3>

 <P>
 <a href="../../../boost/graph/johnson_all_pairs_shortest.hpp"><TT>boost/graph/johnson_all_pairs_shortest.hpp</TT></a>


 <h3>Parameters</h3>

 IN: <tt>Graph&amp; g</tt>
 <blockquote>
 A directed or undirected graph. The graph type must be a model of
 <a href="VertexListGraph.html">Vertex List Graph</a>,
 <a href="EdgeListGraph.html">Edge List Graph</a>,
 and <a href="IncidenceGraph.html">Incidence Graph</a>.
 </blockquote>

 OUT: <tt>DistanceMatrix&amp; D</tt>
 <blockquote>
 The length of the shortest path between each pair of vertices
 <i>u,v</i> in the graph is stored in <tt>D[u][v]</tt>. The set of
 types {<tt>DistanceMatrix, vertices_size_type, D</tt>} must be a model
 of <a href="BasicMatrix.html">BasicMatrix</a> where <tt>D</tt> is the
 value type of the <tt>DistanceMap</tt>.
 </blockquote>


 <h3>Named Parameters</h3>

 IN: <tt>weight_map(WeightMap w_map)</tt>
 <blockquote>
   The weight or ``length'' of each edge in the graph.
   The type <tt>WeightMap</tt> must be a model of
   <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The edge descriptor type of
   the graph needs to be usable as the key type for the weight
   map. The value type for the map must be
   <i>Addable</i> with the value type of the distance map.<br>
   <b>Default:</b>  <tt>get(edge_weight, g)</tt>
 </blockquote>

 UTIL: <tt>weight_map2(WeightMap2 w2_map)</tt>
 <blockquote>
   This parameter is no longer needed, and will be ignored.
 </blockquote>

 IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt>
 <blockquote>
   This maps each vertex to an integer in the range <tt>[0,
     num_vertices(g))</tt>. This is necessary for efficient updates of the
   heap data structure in the internal call to Dijkstra's algorithm.  The type
   <tt>VertexIndexMap</tt> must be a model of
   <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The value type of the map must be an
   integer type. The vertex descriptor type of the graph needs to be
   usable as the key type of the map.<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>


 UTIL: <tt>distance_map(DistanceMap d_map)</tt>
 <blockquote>
   This parameter is no longer needed, and will be ignored.
 </blockquote>

 IN: <tt>distance_compare(CompareFunction cmp)</tt>
 <blockquote>
   This function is use to compare distances to determine
   which vertex is closer to the source vertex.
   The <tt>CompareFunction</tt> type must be a model of
   \stlconcept{BinaryPredicate} and have argument types that
   match the value type of the <tt>WeightMap</tt> property map.<br>
   <b>Default:</b> <tt>std::less&lt;DT&gt;</tt> with
   <tt>DT=typename&nbsp;property_traits&lt;WeightMap&gt;::value_type</tt>
 </blockquote>

 IN: <tt>distance_combine(CombineFunction cmb)</tt>
 <blockquote>
   This function is used to combine distances to compute the distance
   of a path. The <tt>CombineFunction</tt> type must be a model of <a
   href="http://www.sgi.com/tech/stl/BinaryFunction.html">Binary
   Function</a>. The first argument type of the binary function must
   match the value type of the <tt>DistanceMap</tt> property map and
   the second argument type must match the value type of the
   <tt>WeightMap</tt> property map.  The result type must be the same
   type as the distance value type.<br>
   <b>Default:</b> <tt>std::plus&lt;DT&gt;</tt> with
    <tt>DT=typename&nbsp;property_traits&lt;WeightMap&gt;::value_type</tt>
 </blockquote>

 IN: <tt>distance_inf(DT inf)</tt>
 <blockquote>
   This value is used to initialize the distance for each
   vertex before the start of the algorithm.
   The type <tt>DT</tt> must be the value type of the <tt>WeigthMap</tt>.<br>
   <b>Default:</b> <tt>std::numeric_limits::max()</tt>
 </blockquote>

 IN: <tt>distance_zero(DT zero)</tt>
 <blockquote>
   This value is used to initialize the distance for the source
   vertex before the start of the algorithm.  The type <tt>DT</tt>
   must be the value type of the <tt>WeigthMap</tt>.<br>
   <b>Default:</b> <tt>0</tt>
 </blockquote>

 UTIL/OUT: <tt>color_map(ColorMap c_map)</tt>
 <blockquote>
   This is used during the execution of the algorithm to mark the
   vertices. The vertices start out white and become gray when they are
   inserted in the queue. They then turn black when they are removed
   from the queue. At the end of the algorithm, vertices reachable from
   the source vertex will have been colored black. All other vertices
   will still be white. The type <tt>ColorMap</tt> must be a model of
   <a href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
   Property Map</a>. A vertex descriptor must be usable as the key type
   of the map, and the value type of the map must be a model of
   <a href="./ColorValue.html">Color Value</a>.<br>
   <b>Default:</b> an <a
   href="../../property_map/doc/iterator_property_map.html">
   <tt>iterator_property_map</tt></a> created from a <tt>std::vector</tt>
   of <tt>default_color_type</tt> of size <tt>num_vertices(g)</tt> and
   using the <tt>i_map</tt> for the index map.
 </blockquote>


 <h3>Complexity</h3>

 The time complexity is <i>O(V E log V)</i>.


 <H3>Example</H3>

 <P>
 The file <a
 href="../example/johnson-eg.cpp"><tt>example/johnson-eg.cpp</tt></a> applies
 Johnson's algorithm for all-pairs shortest paths to the example graph
 from page 568 of the CLR&nbsp;[<A
 HREF="bibliography.html#clr90">8</A>].


 <br>
 <HR>
 <TABLE>
 <TR valign=top>
 <TD nowrap>Copyright &copy; 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>
	<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>Johnson All Pairs Shortest Paths</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:johnson">
	<TT>johnson_all_pairs_shortest_paths</TT>
	</H1>

	<PRE>
	<i>// named parameter version</i>
	template <class VertexAndEdgeListGraph, class DistanceMatrix,
	class VertexID, class Weight, class BinaryPredicate,
	class BinaryFunction, class Infinity, class DistanceZero>
	bool
	johnson_all_pairs_shortest_paths(VertexAndEdgeListGraph& g1,
	DistanceMatrix& D,
	VertexID id1, Weight w1, const BinaryPredicate& compare,
	const BinaryFunction& combine, const Infinity& inf,
	DistanceZero zero);

	template <typename Graph, typename DistanceMatrix, typename P, typename T, typename R>
	bool johnson_all_pairs_shortest_paths(Graph& g, DistanceMatrix& D,
	const bgl_named_params<P, T, R>& params = <i>all defaults</i>)

	<i>// non-named parameter version</i>
	template <typename Graph, typename DistanceMatrix,
	typename VertexIndex, typename WeightMap, typename DT>
	bool
	johnson_all_pairs_shortest_paths(VertexAndEdgeListGraph& g1,
	DistanceMatrix& D,
	VertexIndex i_map, WeightMap w_map, DT zero)
	</PRE>

	<P>
	This algorithm finds the shortest distance between every pair of
	vertices in the graph. The algorithm returns false if there is a
	negative weight cycle in the graph and true otherwise. The distance
	between each pair of vertices is stored in the distance matrix
	<TT>D</TT>. This is one of the more time intensive graph algorithms,
	having a time complexity of <i>O(V E log V)</i>.

	<P>This algorithm should be used to compute shortest paths between
	every pair of vertices for sparse graphs. For dense graphs, use <a
	href="floyd_warshall_shortest.html"><code>floyd_warshall_all_pairs_shortest_paths</code></a>.

	<H3>Where Defined</H3>

	<P>
	<a href="../../../boost/graph/johnson_all_pairs_shortest.hpp"><TT>boost/graph/johnson_all_pairs_shortest.hpp</TT></a>


	<h3>Parameters</h3>

	IN: <tt>Graph& g</tt>
	<blockquote>
	A directed or undirected graph. The graph type must be a model of
	<a href="VertexListGraph.html">Vertex List Graph</a>,
	<a href="EdgeListGraph.html">Edge List Graph</a>,
	and <a href="IncidenceGraph.html">Incidence Graph</a>.
	</blockquote>

	OUT: <tt>DistanceMatrix& D</tt>
	<blockquote>
	The length of the shortest path between each pair of vertices
	<i>u,v</i> in the graph is stored in <tt>D[u][v]</tt>. The set of
	types {<tt>DistanceMatrix, vertices_size_type, D</tt>} must be a model
	of <a href="BasicMatrix.html">BasicMatrix</a> where <tt>D</tt> is the
	value type of the <tt>DistanceMap</tt>.
	</blockquote>


	<h3>Named Parameters</h3>

	IN: <tt>weight_map(WeightMap w_map)</tt>
	<blockquote>
	The weight or ``length'' of each edge in the graph.
	The type <tt>WeightMap</tt> must be a model of
	<a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The edge descriptor type of
	the graph needs to be usable as the key type for the weight
	map. The value type for the map must be
	<i>Addable</i> with the value type of the distance map.<br>
	<b>Default:</b> <tt>get(edge_weight, g)</tt>
	</blockquote>

	UTIL: <tt>weight_map2(WeightMap2 w2_map)</tt>
	<blockquote>
	This parameter is no longer needed, and will be ignored.
	</blockquote>

	IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt>
	<blockquote>
	This maps each vertex to an integer in the range <tt>[0,
	num_vertices(g))</tt>. This is necessary for efficient updates of the
	heap data structure in the internal call to Dijkstra's algorithm. The type
	<tt>VertexIndexMap</tt> must be a model of
	<a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The value type of the map must be an
	integer type. The vertex descriptor type of the graph needs to be
	usable as the key type of the map.<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>


	UTIL: <tt>distance_map(DistanceMap d_map)</tt>
	<blockquote>
	This parameter is no longer needed, and will be ignored.
	</blockquote>

	IN: <tt>distance_compare(CompareFunction cmp)</tt>
	<blockquote>
	This function is use to compare distances to determine
	which vertex is closer to the source vertex.
	The <tt>CompareFunction</tt> type must be a model of
	\stlconcept{BinaryPredicate} and have argument types that
	match the value type of the <tt>WeightMap</tt> property map.<br>
	<b>Default:</b> <tt>std::less<DT></tt> with
	<tt>DT=typename property_traits<WeightMap>::value_type</tt>
	</blockquote>

	IN: <tt>distance_combine(CombineFunction cmb)</tt>
	<blockquote>
	This function is used to combine distances to compute the distance
	of a path. The <tt>CombineFunction</tt> type must be a model of <a
	href="http://www.sgi.com/tech/stl/BinaryFunction.html">Binary
	Function</a>. The first argument type of the binary function must
	match the value type of the <tt>DistanceMap</tt> property map and
	the second argument type must match the value type of the
	<tt>WeightMap</tt> property map. The result type must be the same
	type as the distance value type.<br>
	<b>Default:</b> <tt>std::plus<DT></tt> with
	<tt>DT=typename property_traits<WeightMap>::value_type</tt>
	</blockquote>

	IN: <tt>distance_inf(DT inf)</tt>
	<blockquote>
	This value is used to initialize the distance for each
	vertex before the start of the algorithm.
	The type <tt>DT</tt> must be the value type of the <tt>WeigthMap</tt>.<br>
	<b>Default:</b> <tt>std::numeric_limits::max()</tt>
	</blockquote>

	IN: <tt>distance_zero(DT zero)</tt>
	<blockquote>
	This value is used to initialize the distance for the source
	vertex before the start of the algorithm. The type <tt>DT</tt>
	must be the value type of the <tt>WeigthMap</tt>.<br>
	<b>Default:</b> <tt>0</tt>
	</blockquote>

	UTIL/OUT: <tt>color_map(ColorMap c_map)</tt>
	<blockquote>
	This is used during the execution of the algorithm to mark the
	vertices. The vertices start out white and become gray when they are
	inserted in the queue. They then turn black when they are removed
	from the queue. At the end of the algorithm, vertices reachable from
	the source vertex will have been colored black. All other vertices
	will still be white. The type <tt>ColorMap</tt> must be a model of
	<a href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
	Property Map</a>. A vertex descriptor must be usable as the key type
	of the map, and the value type of the map must be a model of
	<a href="./ColorValue.html">Color Value</a>.<br>
	<b>Default:</b> an <a
	href="../../property_map/doc/iterator_property_map.html">
	<tt>iterator_property_map</tt></a> created from a <tt>std::vector</tt>
	of <tt>default_color_type</tt> of size <tt>num_vertices(g)</tt> and
	using the <tt>i_map</tt> for the index map.
	</blockquote>


	<h3>Complexity</h3>

	The time complexity is <i>O(V E log V)</i>.



	<H3>Example</H3>

	<P>
	The file <a
	href="../example/johnson-eg.cpp"><tt>example/johnson-eg.cpp</tt></a> applies
	Johnson's algorithm for all-pairs shortest paths to the example graph
	from page 568 of the CLR [<A
	HREF="bibliography.html#clr90">8</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>