1<HTML> 2<!-- 3 Copyright (c) Jeremy Siek 2000 4 5 Distributed under the Boost Software License, Version 1.0. 6 (See accompanying file LICENSE_1_0.txt or copy at 7 http://www.boost.org/LICENSE_1_0.txt) 8 --> 9<Head> 10<Title>Johnson All Pairs Shortest Paths</Title> 11<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b" 12 ALINK="#ff0000"> 13<IMG SRC="../../../boost.png" 14 ALT="C++ Boost" width="277" height="86"> 15 16<BR Clear> 17 18<H1><A NAME="sec:johnson"></A> 19<TT>johnson_all_pairs_shortest_paths</TT> 20</H1> 21 22<PRE> 23<i>// named parameter version</i> 24 template <class VertexAndEdgeListGraph, class DistanceMatrix, 25 class VertexID, class Weight, class BinaryPredicate, 26 class BinaryFunction, class Infinity, class DistanceZero> 27 bool 28 johnson_all_pairs_shortest_paths(VertexAndEdgeListGraph& g1, 29 DistanceMatrix& D, 30 VertexID id1, Weight w1, const BinaryPredicate& compare, 31 const BinaryFunction& combine, const Infinity& inf, 32 DistanceZero zero); 33 34template <typename Graph, typename DistanceMatrix, typename P, typename T, typename R> 35bool johnson_all_pairs_shortest_paths(Graph& g, DistanceMatrix& D, 36 const bgl_named_params<P, T, R>& params = <i>all defaults</i>) 37 38<i>// non-named parameter version</i> 39template <typename Graph, typename DistanceMatrix, 40 typename VertexIndex, typename WeightMap, typename DT> 41bool 42johnson_all_pairs_shortest_paths(VertexAndEdgeListGraph& g1, 43 DistanceMatrix& D, 44 VertexIndex i_map, WeightMap w_map, DT zero) 45</PRE> 46 47<P> 48This algorithm finds the shortest distance between every pair of 49vertices in the graph. The algorithm returns false if there is a 50negative weight cycle in the graph and true otherwise. The distance 51between each pair of vertices is stored in the distance matrix 52<TT>D</TT>. This is one of the more time intensive graph algorithms, 53having a time complexity of <i>O(V E log V)</i>. 54 55<P>This algorithm should be used to compute shortest paths between 56every pair of vertices for sparse graphs. For dense graphs, use <a 57href="floyd_warshall_shortest.html"><code>floyd_warshall_all_pairs_shortest_paths</code></a>. 58 59<H3>Where Defined</H3> 60 61<P> 62<a href="../../../boost/graph/johnson_all_pairs_shortest.hpp"><TT>boost/graph/johnson_all_pairs_shortest.hpp</TT></a> 63 64 65<h3>Parameters</h3> 66 67IN: <tt>Graph& g</tt> 68<blockquote> 69A directed or undirected graph. The graph type must be a model of 70<a href="VertexListGraph.html">Vertex List Graph</a>, 71<a href="EdgeListGraph.html">Edge List Graph</a>, 72and <a href="IncidenceGraph.html">Incidence Graph</a>. 73</blockquote> 74 75OUT: <tt>DistanceMatrix& D</tt> 76<blockquote> 77The length of the shortest path between each pair of vertices 78<i>u,v</i> in the graph is stored in <tt>D[u][v]</tt>. The tuple of 79types (<tt>DistanceMatrix, vertices_size_type, D</tt>) must be a model 80of <a href="BasicMatrix.html">BasicMatrix</a> where <tt>D</tt> is the 81value type of the <tt>DistanceMap</tt>. There must be implicit conversions 82between the value type of the distance matrix and the value type of the weight 83map. 84</blockquote> 85 86 87<h3>Named Parameters</h3> 88 89IN: <tt>weight_map(WeightMap w_map)</tt> 90<blockquote> 91 The weight or "length" of each edge in the graph. 92 The type <tt>WeightMap</tt> must be a model of 93 <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The edge descriptor type of 94 the graph needs to be usable as the key type for the weight 95 map. The value type of the weight map must support a subtraction operation.<br> 96 <b>Default:</b> <tt>get(edge_weight, g)</tt> 97</blockquote> 98 99UTIL: <tt>weight_map2(WeightMap2 w2_map)</tt> 100<blockquote> 101 This parameter is no longer needed, and will be ignored. 102</blockquote> 103 104IN: <tt>vertex_index_map(VertexIndexMap i_map)</tt> 105<blockquote> 106 This maps each vertex to an integer in the range <tt>[0, 107 num_vertices(g))</tt>. This is necessary for efficient updates of the 108 heap data structure in the internal call to Dijkstra's algorithm. The type 109 <tt>VertexIndexMap</tt> must be a model of 110 <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. The value type of the map must be an 111 integer type. The vertex descriptor type of the graph needs to be 112 usable as the key type of the map.<br> 113 <b>Default:</b> <tt>get(vertex_index, g)</tt> 114 Note: if you use this default, make sure your graph has 115 an internal <tt>vertex_index</tt> property. For example, 116 <tt>adjacency_list</tt> with <tt>VertexList=listS</tt> does 117 not have an internal <tt>vertex_index</tt> property. 118 <br> 119</blockquote> 120 121 122UTIL: <tt>distance_map(DistanceMap d_map)</tt> 123<blockquote> 124 This parameter is no longer needed, and will be ignored. 125</blockquote> 126 127IN: <tt>distance_compare(CompareFunction cmp)</tt> 128<blockquote> 129 This function is use to compare distances to determine 130 which vertex is closer to the source vertex. 131 The <tt>CompareFunction</tt> type must be a model of 132 <a href="http://www.boost.org/sgi/stl/BinaryPredicate.html">Binary Predicate</a> 133 and have argument types that 134 match the value type of the <tt>WeightMap</tt> property map.<br> 135 <b>Default:</b> <tt>std::less<DT></tt> with 136 <tt>DT=typename property_traits<WeightMap>::value_type</tt> 137</blockquote> 138 139IN: <tt>distance_combine(CombineFunction cmb)</tt> 140<blockquote> 141 This function is used to combine distances to compute the distance 142 of a path. The <tt>CombineFunction</tt> type must be a model of <a 143 href="http://www.boost.org/sgi/stl/BinaryFunction.html">Binary 144 Function</a>. Both argument types and the return type of the binary function 145 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> 146 <b>Default:</b> <tt>std::plus<DT></tt> with 147 <tt>DT=typename property_traits<WeightMap>::value_type</tt> 148</blockquote> 149 150IN: <tt>distance_inf(DT inf)</tt> 151<blockquote> 152 This value is used to initialize the distance for each 153 vertex before the start of the algorithm. 154 The type <tt>DT</tt> must be the value type of the <tt>WeightMap</tt>.<br> 155 <b>Default:</b> <tt>std::numeric_limits::max()</tt> 156</blockquote> 157 158IN: <tt>distance_zero(DT zero)</tt> 159<blockquote> 160 This value is used to initialize the distance for the source 161 vertex before the start of the algorithm. The type <tt>DT</tt> 162 must be the value type of the <tt>WeightMap</tt>.<br> 163 <b>Default:</b> <tt>0</tt> 164</blockquote> 165 166UTIL/OUT: <tt>color_map(ColorMap c_map)</tt> 167<blockquote> 168 This is used during the execution of the algorithm to mark the 169 vertices. The vertices start out white and become gray when they are 170 inserted in the queue. They then turn black when they are removed 171 from the queue. At the end of the algorithm, vertices reachable from 172 the source vertex will have been colored black. All other vertices 173 will still be white. The type <tt>ColorMap</tt> must be a model of 174 <a href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write 175 Property Map</a>. A vertex descriptor must be usable as the key type 176 of the map, and the value type of the map must be a model of 177 <a href="./ColorValue.html">Color Value</a>.<br> 178 <b>Default:</b> an <a 179 href="../../property_map/doc/iterator_property_map.html"> 180 <tt>iterator_property_map</tt></a> created from a <tt>std::vector</tt> 181 of <tt>default_color_type</tt> of size <tt>num_vertices(g)</tt> and 182 using the <tt>i_map</tt> for the index map. 183</blockquote> 184 185 186<h3>Complexity</h3> 187 188The time complexity is <i>O(V E log V)</i>. 189 190 191 192<H3>Example</H3> 193 194<P> 195The file <a 196href="../example/johnson-eg.cpp"><tt>example/johnson-eg.cpp</tt></a> applies 197Johnson's algorithm for all-pairs shortest paths to the example graph 198from page 568 of the CLR [<A 199HREF="bibliography.html#clr90">8</A>]. 200 201 202<br> 203<HR> 204<TABLE> 205<TR valign=top> 206<TD nowrap>Copyright © 2000-2001</TD><TD> 207<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>) 208</TD></TR></TABLE> 209 210</BODY> 211</HTML> 212
