1<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN" 2 "http://www.w3.org/TR/html4/strict.dtd"> 3<!-- 4 Copyright (C) Flavio De Lorenzi 2012 5 6 Distributed under the Boost Software License, Version 1.0. 7 (See accompanying file LICENSE_1_0.txt or copy at 8 http://www.boost.org/LICENSE_1_0.txt) 9 --> 10<html> 11 <head> 12 <meta http-equiv="content-type" content="text/html; charset=iso-8859-15"> 13 <title>Boost Graph Library: VF2 (Sub)Graph Isomorphism</title> 14 <style type="text/css"> 15 <!-- 16 body { 17 color: black; 18 background-color: white; 19 } 20 21 .comment { 22 color: #333333; 23 } 24 25 a { 26 color: blue; 27 } 28 29 a:visited { 30 color: #551A8B; 31 } 32 --> 33 </style> 34 </head> 35 <body> 36 <p><img src="../../../boost.png" alt="C++ Boost" width="277" height="86"></p> 37 <h1> 38 <tt>vf2_subgraph_iso</tt> 39 </h1> 40 <pre> 41<em class="comment">// All defaults interface</em> 42template <typename GraphSmall, 43 typename GraphLarge, 44 typename SubGraphIsoMapCallback> 45bool vf2_subgraph_iso(const GraphSmall& graph_small, 46 const GraphLarge& graph_large, 47 SubGraphIsoMapCallback user_callback) 48 49 50<em class="comment">// Named parameter version</em> 51template <typename GraphSmall, 52 typename GraphLarge, 53 typename VertexOrderSmall, 54 typename SubGraphIsoMapCallback, 55 typename Param, 56 typename Tag, 57 typename Rest> 58bool vf2_subgraph_iso(const GraphSmall& graph_small, 59 const GraphLarge& graph_large, 60 SubGraphIsoMapCallback user_callback, 61 const VertexOrderSmall& vertex_order_small, 62 const bgl_named_params<Param, Tag, Rest>& params) 63 64 65<em class="comment">// Non-named parameter version</em> 66template <typename GraphSmall, 67 typename GraphLarge, 68 typename <a href="../../property_map/doc/ReadablePropertyMap.html">IndexMapSmall</a>, 69 typename <a href="../../property_map/doc/ReadablePropertyMap.html">IndexMapLarge</a>, 70 typename VertexOrderSmall, 71 typename <a href="http://www.boost.org/sgi/stl/BinaryFunction.html">EdgeEquivalencePredicate</a>, 72 typename <a href="http://www.boost.org/sgi/stl/BinaryFunction.html">VertexEquivalencePredicate</a>, 73 typename SubGraphIsoMapCallback> 74bool vf2_subgraph_iso(const GraphSmall& graph_small, 75 const GraphLarge& graph_large, 76 SubGraphIsoMapCallback user_callback, 77 IndexMapSmall index_map_small, 78 IndexMapLarge index_map_large, 79 const VertexOrderSmall& vertex_order_small, 80 EdgeEquivalencePredicate edge_comp, 81 VertexEquivalencePredicate vertex_comp) 82 </pre> 83 <p> 84 An isomorphism between two graphs <em>G<sub>1</sub>=(V<sub>1</sub>, E<sub>1</sub>)</em> 85 and <em>G<sub>2</sub>=(V<sub>2</sub>, E<sub>2</sub>)</em> is a 86 bijective mapping <em>M</em> of the vertices of one graph to vertices of the other 87 graph that preserves the edge structure of the graphs. <em>M</em> is said to be a 88 graph-subgraph isomorphism if and only if <em>M</em> is an isomorphism between 89 <em>G<sub>1</sub></em> and a subgraph of <em>G<sub>2</sub></em>. 90 An induced subgraph of a graph <em>G = (V, E)</em> is a normal subgraph 91 <em>G' = (V', E')</em> with the extra condition that all edges of <em>G</em> 92 which have both endpoints in <em>V'</em> are in <em>E'</em>. 93 </p> 94 95 <p> 96 This function finds all induced subgraph isomorphisms between 97 graphs <tt>graph_small</tt> and <tt>graph_large</tt> and outputs them to 98 <tt>user_callback</tt>. It continues until <tt>user_callback</tt> 99 returns false or the search space has been fully explored. <tt>vf2_subgraph_iso</tt> 100 returns true if a graph-subgraph isomorphism exists and false otherwise. 101 <tt>EdgeEquivalencePredicate</tt> and 102 <tt>VertexEquivalencePredicate</tt> predicates are used to test whether 103 edges and vertices are equivalent. To use property maps for equivalence, 104 see 105 <tt><a href="./mcgregor_common_subgraphs.html#make_property_map_equivalent"> 106 make_property_map_equivalent</a></tt> 107 function. By default <tt><a href="./mcgregor_common_subgraphs.html#always_equivalent"> 108 always_equivalent</a></tt> is used, which returns 109 true for any pair of vertices or edges. 110 </p> 111 <p> 112 The current implementation is based on the <em>VF2</em> algorithm, 113 introduced by Cordella et al. An in-depth description of the algorithm is 114 given in [<a href="#cordella2001">1</a>] and [<a href="#cordella2004">2</a>] 115 and references therein. The original code by P. Foggia and collaborators can 116 be found at [<a href="#foggia_etal">3</a>]. In brief, the process of 117 finding a mapping between the two graphs <em>G<sub>1</sub></em> and 118 <em>G<sub>2</sub></em> determines the isomorphism mapping <em>M</em>, 119 which associates vertices <em>G<sub>1</sub></em> with vertices of 120 <em>G<sub>2</sub></em> and vice versa. It can be described by means of a 121 state space representation which is created by the algorithm 122 while exploring the search graph in depth-first fashion. 123 Each state <em>s</em> of the matching process 124 can be associated with a partial mapping <em>M(s)</em>. At each level, 125 the algorithm computes the set of the vertex pairs that are candidates to 126 be added to the current state <em>s</em>. If a pair of vertices 127 (<em>v, w</em>) is feasible, the mapping is extended and the associated 128 successor state <em>s'</em> is computed. 129 The whole procedure is then repeated for state <em>s'</em>. 130 </p> 131 132 <h3>Where Defined</h3> 133 <p> 134 <a href="../../../boost/graph/vf2_sub_graph_iso.hpp"> 135 <tt>boost/graph/vf2_sub_graph_iso.hpp</tt></a><br> 136 All functions are defined in the boost namespace. 137 </p> 138 139 <h3>Parameters</h3> 140 141 <p>IN: <tt>const GraphSmall& graph_small</tt></p> 142 <blockquote> 143 <p> 144 The (first) smaller graph (fewer vertices) of the pair to be tested for 145 isomorphism. The type <tt>GraphSmall</tt> must be a 146 model of 147 <a href="./VertexListGraph.html">Vertex List Graph</a>, 148 <a href="./EdgeListGraph.html">Edge List Graph</a>, 149 <a href="./BidirectionalGraph.html">Bidirectional Graph</a> and 150 <a href="./AdjacencyMatrix.html">Adjacency Matrix</a>. 151 The edge descriptors <tt>graph_traits<GraphSmall>::edge_descriptor</tt> 152 must be <a href="http://www.boost.org/sgi/stl/LessThanComparable.html"> 153 LessThan Comparable</a>, cf. also the remark <a href="#notes">below</a>. 154 </p> 155 </blockquote> 156 157 <p>IN: <tt>const GraphLarge& graph_large</tt></p> 158 <blockquote> 159 <p> 160 The (second) larger graph to be tested. 161 Type <tt>GraphLarge</tt> must be a model of 162 <a href="./VertexListGraph.html">Vertex List Graph</a>, 163 <a href="./EdgeListGraph.html">Edge List Graph</a>, 164 <a href="./BidirectionalGraph.html">Bidirectional Graph</a> and 165 <a href="./AdjacencyMatrix.html">Adjacency Matrix</a>. 166 The edge descriptors <tt>graph_traits<GraphLarge>::edge_descriptor</tt> 167 must be <a href="http://www.boost.org/sgi/stl/LessThanComparable.html"> 168 LessThan Comparable</a>, cf. also the remark <a href="#notes">below</a>. 169 </p> 170 </blockquote> 171 172 <p>OUT: <tt>SubGraphIsoMapCallback user_callback</tt></p> 173 <blockquote> 174 <p> 175 A function object to be called when a graph-subgraph isomorphism has been discovered. The 176 <tt>operator()</tt> must have following form: 177 </p> 178 <pre> 179template <typename CorrespondenceMap1To2, 180 typename CorrespondenceMap2To1> 181bool operator()(CorrespondenceMap1To2 f, CorrespondenceMap2To1 g) const 182 </pre> 183 <p> 184 Both the <tt>CorrespondenceMap1To2</tt> 185 and <tt>CorresondenceMap2To1</tt> types are models 186 of <a href="../../property_map/doc/ReadablePropertyMap.html">Readable 187 Property Map</a> and map equivalent vertices across the two 188 graphs given to <tt>vf2_subgraph_iso</tt> (or <tt>vf2_graph_iso</tt> or 189 <tt>vf2_subgraph_mono</tt>). For instance, if <tt>v</tt> is 190 from <tt>graph_small</tt>, <tt>w</tt> is from <tt>graph_large</tt>, 191 and the vertices can be considered equivalent, 192 then <tt>get(f, v)</tt> will be <tt>w</tt> and <tt>get(g, w)</tt> 193 will be <tt>v</tt>. If any vertex, say <tt>v</tt> in <tt>graph_small</tt>, 194 does not match a vertex in <tt>graph_large</tt> , 195 then <tt>get(f, v)</tt> will be <tt>graph_traits<GraphLarge>::null_vertex()</tt>. 196 Likewise for any unmatched vertices from <tt>graph_large</tt>, 197 <tt>get(g, w)</tt> will be <tt>graph_traits<GraphSmall>::null_vertex()</tt>. 198 199 Returning false from the callback will abort the search immediately. Otherwise, 200 the entire search space will be explored. A "default" print callback 201 is provided as a <a href="#vf2_callback">utility function</a>. 202 </p> 203 </blockquote> 204 205 <p>IN: <tt>const VertexOrderSmall& vertex_order_small</tt></p> 206 <blockquote> 207 <p> 208 The ordered vertices of the smaller (first) graph <tt>graph_small</tt>. 209 During the matching process the vertices are examined in the order given by 210 <tt>vertex_order_small</tt>. Type <tt>VertexOrderSmall</tt> must be a model 211 of <a href="http://www.boost.org/sgi/stl/Container.html">ContainerConcept</a> 212 with value type 213 <tt>graph_traits<GraphSmall>::vertex_descriptor</tt>. 214 <br> 215 <b>Default</b> The vertices are ordered by multiplicity of 216 in/out degrees. 217 </p> 218 </blockquote> 219 220 <h3>Named Parameters</h3> 221 222 <p>IN: <tt>vertex_index1(IndexMapSmall index_map_small)</tt></p> 223 <blockquote> 224 <p> 225 This maps each vertex to an integer in the range <tt>[0, num_vertices(graph_small))</tt>. 226 Type <tt>IndexMapSmall</tt> must be a model of 227 <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. 228 <br> 229 <b>Default:</b> <tt>get(vertex_index, graph_small)</tt> 230 </p> 231 </blockquote> 232 233 <p>IN: <tt>vertex_index2(IndexMapLarge index_map_large)</tt></p> 234 <blockquote> 235 <p> 236 This maps each vertex to an integer in the range <tt>[0, num_vertices(graph_large))</tt>. 237 Type <tt>IndexMapLarge</tt> must be a model of 238 <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. 239 <br> 240 <b>Default:</b> <tt>get(vertex_index, graph_large)</tt> 241 </p> 242 </blockquote> 243 244 <p>IN: <tt>edges_equivalent(EdgeEquivalencePredicate edge_comp)</tt></p> 245 <blockquote> 246 <p> 247 This function object is used to determine if edges between the two graphs 248 <tt>graph_small</tt> and <tt>graph_large</tt> are equivalent. 249 Type <tt>EdgeEquivalencePredicate</tt> must be a model of 250 <a href="http://www.boost.org/sgi/stl/BinaryPredicate.html">Binary 251 Predicate</a> and have argument types of 252 <tt>graph_traits<GraphSmall>::edge_descriptor</tt> and 253 <tt>graph_traits<GraphLarge>::edge_descriptor</tt>. 254 The edge descriptors must be <a href="http://www.boost.org/sgi/stl/LessThanComparable.html"> 255 LessThan Comparable</a>. A return value of true indicates that the edges are equivalent. 256 <br> 257 <b>Default:</b> <tt><a href="./mcgregor_common_subgraphs.html#always_equivalent"> 258 always_equivalent</a></tt> 259 </p> 260 </blockquote> 261 262 <p>IN: <tt>vertices_equivalent(VertexEquivalencePredicate vertex_comp)</tt></p> 263 <blockquote> 264 <p> 265 This function object is used to determine if vertices between the two graphs 266 <tt>graph_small</tt> and <tt>graph_large</tt> are equivalent. 267 Type <tt>VertexEquivalencePredicate</tt> must be a model of 268 <a href="http://www.boost.org/sgi/stl/BinaryPredicate.html">Binary Predicate</a> 269 and have argument types of 270 <tt>graph_traits<GraphSmall>::vertex_descriptor</tt> and 271 <tt>graph_traits<GraphLarge>::vertex_descriptor</tt>. A return value of true 272 indicates that the vertices are equivalent. 273 <br> 274 <b>Default:</b> <tt><a href="./mcgregor_common_subgraphs.html#always_equivalent"> 275 always_equivalent</a></tt> 276 </p> 277 </blockquote> 278 279 <h3>Related Functions</h3> 280 <p> 281 Non-named parameter, named-parameter and all default parameter versions of 282 function 283 </p> 284 <p><tt>vf2_graph_iso(...)</tt></p> 285 <p><tt>vf2_subgraph_mono(...)</tt></p> 286 <p> 287 for isomorphism and (not necessarily induced) subgraph isomorphism testing, 288 taking the same parameters as the corresponding functions <tt>vf2_subgraph_iso</tt> 289 for induced subgraph isomorphism testing. 290 For <tt>vf2_graph_iso</tt> the algorithm finds all isomorphism mappings between 291 graphs <tt>graph1</tt> and <tt>graph2</tt> and outputs them to 292 <tt>user_callback</tt>. 293 For <tt>vf2_graph_mono</tt> the algorithm finds all mappings of <tt>graph_small</tt> 294 to subgraphs of <tt>graph_large</tt>. 295 Note that, as opposed to <tt>vf2_subgraph_iso</tt>, 296 these subgraphs need not to be induced subgraphs. 297 </p> 298 <p> 299 Both algorithms continues until <tt>user_callback</tt> 300 returns false or the search space has been fully explored. As before, 301 <tt>EdgeEquivalencePredicate</tt> and 302 <tt>VertexEquivalencePredicate</tt> predicates are used to test 303 whether edges and vertices are equivalent. By default 304 <tt>always_equivalent</tt> is used. 305 </p> 306 307 <h3>Utility Functions and Structs</h3> 308 <p> 309 <tt id="vf2_callback"> 310template<typename Graph1, 311 typename Graph2> 312struct vf2_print_callback 313 </tt> 314 </p> 315 <blockquote> 316 <p> 317 Callback function object that prints out the correspondences between vertices 318 of <tt>Graph1</tt> and <tt>Graph2</tt>. The constructor takes 319 the two graphs <em>G<sub>1</sub></em> and <em>G<sub>2</sub></em>. 320 </p> 321 </blockquote> 322 323 <pre> 324template<typename Graph> 325std::vector<typename graph_traits<Graph>::vertex_descriptor> 326 vertex_order_by_mult(const Graph& graph) 327 </pre> 328 <blockquote> 329 <p> 330 Returns a vector containing the vertices of a graph, sorted 331 by multiplicity of in/out degrees. 332 </p> 333 </blockquote> 334 335 <pre> 336<em class="comment">// Variant of verify_subgraph_iso with all default parameters</em> 337template<typename Graph1, 338 typename Graph2, 339 typename CorresponenceMap1To2> 340inline bool verify_vf2_subgraph_iso(const Graph1& graph1, const Graph2& graph2, 341 const CorresponenceMap1To2 f) 342 343 344<em class="comment">// Verifies a graph (sub)graph isomorphism map</em> 345template<typename Graph1, 346 typename Graph2, 347 typename CorresponenceMap1To2, 348 typename EdgeEquivalencePredicate, 349 typename VertexEquivalencePredicate> 350inline bool verify_vf2_subgraph_iso(const Graph1& graph1, const Graph2& graph2, 351 const CorresponenceMap1To2 f, 352 EdgeEquivalencePredicate edge_comp, 353 VertexEquivalencePredicate vertex_comp) 354 </pre> 355 <blockquote> 356 <p> 357 This function can be used to verify a (sub)graph isomorphism 358 mapping <em>f</em>. The parameters are analogous to 359 function <tt>vf2_subgraph_iso</tt> (<tt>vf2_graph_iso</tt>). 360 </p> 361 </blockquote> 362 363 364 <h3>Complexity</h3> 365 <p> 366 Spatial and time complexity are given in [<a href="#cordella2004">2</a>]. The spatial 367 complexity of VF2 is of order <em>O(V)</em>, where V is the (maximum) number 368 of vertices of the two graphs. Time complexity is <em>O(V<sup>2</sup>)</em> in the best case and 369 <em>O(V!·V)</em> in the worst case. 370 </p> 371 372 <h3>Examples</h3> 373 374 <h4>Example 1</h4> 375 <p> 376 In the example below, a small graph <tt>graph1</tt> and a larger graph 377 <tt>graph2</tt> are defined. Here small and large refers to the number of 378 vertices of the graphs. <tt>vf2_subgraph_iso</tt> computes all the 379 subgraph isomorphism mappings between the two graphs and outputs them 380 via <tt>callback</tt>. 381 </p> 382 <pre> 383typedef adjacency_list<setS, vecS, bidirectionalS> graph_type; 384 385<em class="comment">// Build graph1</em> 386int num_vertices1 = 8; graph_type graph1(num_vertices1); 387add_edge(0, 6, graph1); add_edge(0, 7, graph1); 388add_edge(1, 5, graph1); add_edge(1, 7, graph1); 389add_edge(2, 4, graph1); add_edge(2, 5, graph1); add_edge(2, 6, graph1); 390add_edge(3, 4, graph1); 391 392<em class="comment">// Build graph2</em> 393int num_vertices2 = 9; graph_type graph2(num_vertices2); 394add_edge(0, 6, graph2); add_edge(0, 8, graph2); 395add_edge(1, 5, graph2); add_edge(1, 7, graph2); 396add_edge(2, 4, graph2); add_edge(2, 7, graph2); add_edge(2, 8, graph2); 397add_edge(3, 4, graph2); add_edge(3, 5, graph2); add_edge(3, 6, graph2); 398<em class="comment"> 399// Create callback to print mappings</em> 400vf2_print_callback<graph_type, graph_type> callback(graph1, graph2); 401 402<em class="comment"> 403// Print out all subgraph isomorphism mappings between graph1 and graph2. 404// Vertices and edges are assumed to be always equivalent.</em> 405vf2_subgraph_iso(graph1, graph2, callback); 406 </pre> 407 <p> 408 The complete example can be found under 409 <a href="../example/vf2_sub_graph_iso_example.cpp"><tt>examples/vf2_sub_graph_iso_example.cpp</tt></a>. 410 <br> 411 </p> 412 413 <h4>Example 2</h4> 414 <p> 415 In this example, the subgraph isomorphism mappings between multi-graphs are computed. The vertices 416 and edges of the multi-graphs are distinguished using property maps. 417 </p> 418 <pre> 419<em class="comment">// Define edge and vertex properties</em> 420typedef property<edge_name_t, char> edge_property; 421typedef property<vertex_name_t, char, property<vertex_index_t, int> > vertex_property; 422 423<em class="comment">// Using a vecS graphs => the index maps are implicit.</em> 424typedef adjacency_list<vecS, vecS, bidirectionalS, vertex_property, edge_property> graph_type; 425 426<em class="comment">// Create graph1</em> 427graph_type graph1; 428<em class="comment">// Add vertices... </em> 429add_vertex(vertex_property('a'), graph1); 430... 431<em class="comment">//... and edges </em> 432add_edge(0, 1, edge_property('b'), graph1); 433add_edge(0, 1, edge_property('b'), graph1); 434... 435 436<em class="comment">// Create graph2 </em> 437graph_type graph2; 438add_vertex(vertex_property('a'), graph2); 439... 440add_edge(0, 1, edge_property('a'), graph2); 441... 442 </pre> 443 444 <p> 445 To distinguish vertices and edges with property maps, binary predicates are created using the 446 <tt><a href="./mcgregor_common_subgraphs.html#make_property_map_equivalent"> 447 make_property_map_equivalent</a></tt> function: 448 </p> 449 450 <pre> 451<em class="comment">// Create the vertex binary predicate</em> 452typedef property_map<graph_type, vertex_name_t>::type vertex_name_map_t; 453typedef property_map_equivalent<vertex_name_map_t, vertex_name_map_t> vertex_comp_t; 454vertex_comp_t vertex_comp = 455 make_property_map_equivalent(get(vertex_name, graph1), get(vertex_name, graph2)); 456 457<em class="comment">// Create the vertex binary predicate</em> 458typedef property_map<graph_type, edge_name_t>::type edge_name_map_t; 459typedef property_map_equivalent<edge_name_map_t, edge_name_map_t> edge_comp_t; 460edge_comp_t edge_comp = 461 make_property_map_equivalent(get(edge_name, graph1), get(edge_name, graph2)); 462 </pre> 463 464 <p> 465 Finally, a callback function object is created and the subgraph isomorphism mappings are 466 computed: 467 </p> 468 <pre> 469<em class="comment">// Create callback</em> 470vf2_print_callback<graph_type, graph_type> callback(graph1, graph2); 471 472<em class="comment"> 473// Print out all subgraph isomorphism mappings between graph1 and graph2. 474// Function vertex_order_by_mult is used to compute the order of 475// vertices of graph1. This is the order in which the vertices are examined 476// during the matching process.</em> 477vf2_subgraph_iso(graph1, graph2, callback, vertex_order_by_mult(graph1), 478 edges_equivalent(edge_comp).vertices_equivalent(vertex_comp)); 479 </pre> 480 481 <p> 482 For the complete example, see 483 <a href="../example/vf2_sub_graph_iso_multi_example.cpp"> 484 <tt>examples/vf2_sub_graph_iso_multi_example.cpp</tt></a>. 485 <br> 486 </p> 487 488 <h3 id="notes">Notes</h3> 489 <p> 490 If the <tt>EdgeList</tt> allows for parallel edges, e.g. <tt>vecS</tt>, the 491 algorithm does some bookkeeping of already identified edges. Matched edges 492 are temporarily stored using <tt>std::set</tt> as container, requiring that 493 <tt>edge_descriptor</tt> are <a href="http://www.boost.org/sgi/stl/LessThanComparable.html"> 494 LessThan Comparable</a>. In contrast, if instead you enforce the absence of 495 parallel edges, e.g. by using <tt>setS</tt>, the lookup function falls back 496 to <tt>edge()</tt> without performing any bookkeeping. 497 </p> 498 499 <h3>Bibliography</h3> 500 501 <dl> 502 <dt><a name="cordella2001">1</a></dt> 503 <dd> 504 L. P. Cordella, P. Foggia, C. Sansone, and M. Vento. 505 <br><em>An improved algorithm for matching large graphs</em>. 506 <br>In: 3rd IAPR-TC15 Workshop on Graph-based Representations in Pattern Recognition, pp. 149-159, Cuen, 2001. 507 <p></p> 508 </dd> 509 <dt><a name="cordella2004">2</a></dt> 510 <dd> 511 L. P. Cordella, P. Foggia, C. Sansone, and M. Vento. 512 <br><em>A (Sub)Graph Isomorphism Algorithm for Matching Large Graphs</em>. 513 <br>IEEE Trans. Pattern Anal. Mach. Intell., vol. 26, no. 10, pp. 1367-1372, 2004. 514 <p></p> 515 </dd> 516 <dt><a name="foggia_etal">3</a></dt> 517 <dd> 518 <a href="http://www.cs.sunysb.edu/~algorith/implement/vflib/implement.shtml"> 519 <tt>http://www.cs.sunysb.edu/~algorith/implement/vflib/implement.shtml</tt></a> 520 <p></p> 521 </dd> 522 </dl> 523 <hr> 524 <p> 525 Copyright © 2012, Flavio De Lorenzi 526 (<a href="mailto:fdlorenzi@gmail.com">fdlorenzi@gmail.com</a>) <br /> 527 Copyright © 2013, Jakob Lykke Andersen, University of Southern Denmark 528 (<a href="mailto:jlandersen@imada.sdu.dk">jlandersen@imada.sdu.dk</a>) 529 </p> 530 </body> 531</html> 532
