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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 | ||
34 | template <typename Graph, typename DistanceMatrix, typename P, typename T, typename R> | |
35 | bool 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> | |
39 | template <typename Graph, typename DistanceMatrix, | |
40 | typename VertexIndex, typename WeightMap, typename DT> | |
41 | bool | |
42 | johnson_all_pairs_shortest_paths(VertexAndEdgeListGraph& g1, | |
43 | DistanceMatrix& D, | |
44 | VertexIndex i_map, WeightMap w_map, DT zero) | |
45 | </PRE> | |
46 | ||
47 | <P> | |
48 | This algorithm finds the shortest distance between every pair of | |
49 | vertices in the graph. The algorithm returns false if there is a | |
50 | negative weight cycle in the graph and true otherwise. The distance | |
51 | between each pair of vertices is stored in the distance matrix | |
52 | <TT>D</TT>. This is one of the more time intensive graph algorithms, | |
53 | having a time complexity of <i>O(V E log V)</i>. | |
54 | ||
55 | <P>This algorithm should be used to compute shortest paths between | |
56 | every pair of vertices for sparse graphs. For dense graphs, use <a | |
57 | href="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 | ||
67 | IN: <tt>Graph& g</tt> | |
68 | <blockquote> | |
69 | A 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>, | |
72 | and <a href="IncidenceGraph.html">Incidence Graph</a>. | |
73 | </blockquote> | |
74 | ||
75 | OUT: <tt>DistanceMatrix& D</tt> | |
76 | <blockquote> | |
77 | The 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 | |
79 | types (<tt>DistanceMatrix, vertices_size_type, D</tt>) must be a model | |
80 | of <a href="BasicMatrix.html">BasicMatrix</a> where <tt>D</tt> is the | |
81 | value type of the <tt>DistanceMap</tt>. There must be implicit conversions | |
82 | between the value type of the distance matrix and the value type of the weight | |
83 | map. | |
84 | </blockquote> | |
85 | ||
86 | ||
87 | <h3>Named Parameters</h3> | |
88 | ||
89 | IN: <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 | ||
99 | UTIL: <tt>weight_map2(WeightMap2 w2_map)</tt> | |
100 | <blockquote> | |
101 | This parameter is no longer needed, and will be ignored. | |
102 | </blockquote> | |
103 | ||
104 | IN: <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 | ||
122 | UTIL: <tt>distance_map(DistanceMap d_map)</tt> | |
123 | <blockquote> | |
124 | This parameter is no longer needed, and will be ignored. | |
125 | </blockquote> | |
126 | ||
127 | IN: <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.sgi.com/tech/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 | ||
139 | IN: <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.sgi.com/tech/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 | ||
150 | IN: <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 | ||
158 | IN: <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 | ||
166 | UTIL/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 | ||
188 | The time complexity is <i>O(V E log V)</i>. | |
189 | ||
190 | ||
191 | ||
192 | <H3>Example</H3> | |
193 | ||
194 | <P> | |
195 | The file <a | |
196 | href="../example/johnson-eg.cpp"><tt>example/johnson-eg.cpp</tt></a> applies | |
197 | Johnson's algorithm for all-pairs shortest paths to the example graph | |
198 | from page 568 of the CLR [<A | |
199 | HREF="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> |