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| <title>Boost Graph Library: VF2 (Sub)Graph Isomorphism</title> |
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| <p><img src="../../../boost.png" alt="C++ Boost" width="277" height="86"></p> |
| <h1> |
| <tt>vf2_subgraph_iso</tt> |
| </h1> |
| <pre> |
| <em class="comment">// All defaults interface</em> |
| template <typename GraphSmall, |
| typename GraphLarge, |
| typename SubGraphIsoMapCallback> |
| bool vf2_subgraph_iso(const GraphSmall& graph_small, |
| const GraphLarge& graph_large, |
| SubGraphIsoMapCallback user_callback) |
| |
| |
| <em class="comment">// Named parameter version</em> |
| template <typename GraphSmall, |
| typename GraphLarge, |
| typename VertexOrderSmall, |
| typename SubGraphIsoMapCallback, |
| typename Param, |
| typename Tag, |
| typename Rest> |
| bool vf2_subgraph_iso(const GraphSmall& graph_small, |
| const GraphLarge& graph_large, |
| SubGraphIsoMapCallback user_callback, |
| const VertexOrderSmall& vertex_order_small, |
| const bgl_named_params<Param, Tag, Rest>& params) |
| |
| |
| <em class="comment">// Non-named parameter version</em> |
| template <typename GraphSmall, |
| typename GraphLarge, |
| typename <a href="../../property_map/doc/ReadablePropertyMap.html">IndexMapSmall</a>, |
| typename <a href="../../property_map/doc/ReadablePropertyMap.html">IndexMapLarge</a>, |
| typename VertexOrderSmall, |
| typename <a href="http://www.sgi.com/tech/stl/BinaryFunction.html">EdgeEquivalencePredicate</a>, |
| typename <a href="http://www.sgi.com/tech/stl/BinaryFunction.html">VertexEquivalencePredicate</a>, |
| typename SubGraphIsoMapCallback> |
| bool vf2_subgraph_iso(const GraphSmall& graph_small, |
| const GraphLarge& graph_large, |
| SubGraphIsoMapCallback user_callback, |
| IndexMapSmall index_map_small, |
| IndexMapLarge index_map_large, |
| const VertexOrderSmall& vertex_order_small, |
| EdgeEquivalencePredicate edge_comp, |
| VertexEquivalencePredicate vertex_comp) |
| </pre> |
| <p> |
| An isomorphism between two graphs <em>G<sub>1</sub>=(V<sub>1</sub>, E<sub>1</sub>)</em> |
| and <em>G<sub>2</sub>=(V<sub>2</sub>, E<sub>2</sub>)</em> is a |
| bijective mapping <em>M</em> of the vertices of one graph to vertices of the other |
| graph that preserves the edge structure of the graphs. <em>M</em> is said to be a |
| graph-subgraph isomorphism if and only if <em>M</em> is an isomorphism between |
| <em>G<sub>1</sub></em> and a subgraph of <em>G<sub>2</sub></em>. |
| An induced subgraph of a graph <em>G = (V, E)</em> is a normal subgraph |
| <em>G' = (V', E')</em> with the extra condition that all edges of <em>G</em> |
| which have both endpoints in <em>V'</em> are in <em>E'</em>. |
| </p> |
| |
| <p> |
| This function finds all induced subgraph isomorphisms between |
| graphs <tt>graph_small</tt> and <tt>graph_large</tt> and outputs them to |
| <tt>user_callback</tt>. It continues until <tt>user_callback</tt> |
| returns false or the search space has been fully explored. <tt>vf2_subgraph_iso</tt> |
| returns true if a graph-subgraph isomorphism exists and false otherwise. |
| <tt>EdgeEquivalencePredicate</tt> and |
| <tt>VertexEquivalencePredicate</tt> predicates are used to test whether |
| edges and vertices are equivalent. To use property maps for equivalence, |
| see |
| <tt><a href="./mcgregor_common_subgraphs.html#make_property_map_equivalent"> |
| make_property_map_equivalent</a></tt> |
| function. By default <tt><a href="./mcgregor_common_subgraphs.html#always_equivalent"> |
| always_equivalent</a></tt> is used, which returns |
| true for any pair of vertices or edges. |
| </p> |
| <p> |
| The current implementation is based on the <em>VF2</em> algorithm, |
| introduced by Cordella et al. An in-depth description of the algorithm is |
| given in [<a href="#cordella2001">1</a>] and [<a href="#cordella2004">2</a>] |
| and references therein. The original code by P. Foggia and collaborators can |
| be found at [<a href="#foggia_etal">3</a>]. In brief, the process of |
| finding a mapping between the two graphs <em>G<sub>1</sub></em> and |
| <em>G<sub>2</sub></em> determines the isomorphism mapping <em>M</em>, |
| which associates vertices <em>G<sub>1</sub></em> with vertices of |
| <em>G<sub>2</sub></em> and vice versa. It can be described by means of a |
| state space representation which is created by the algorithm |
| while exploring the search graph in depth-first fashion. |
| Each state <em>s</em> of the matching process |
| can be associated with a partial mapping <em>M(s)</em>. At each level, |
| the algorithm computes the set of the vertex pairs that are candidates to |
| be added to the current state <em>s</em>. If a pair of vertices |
| (<em>v, w</em>) is feasible, the mapping is extended and the associated |
| successor state <em>s'</em> is computed. |
| The whole procedure is then repeated for state <em>s'</em>. |
| </p> |
| |
| <h3>Where Defined</h3> |
| <p> |
| <a href="../../../boost/graph/vf2_sub_graph_iso.hpp"> |
| <tt>boost/graph/vf2_sub_graph_iso.hpp</tt></a><br> |
| All functions are defined in the boost namespace. |
| </p> |
| |
| <h3>Parameters</h3> |
| |
| <p>IN: <tt>const GraphSmall& graph_small</tt></p> |
| <blockquote> |
| <p> |
| The (first) smaller graph (fewer vertices) of the pair to be tested for |
| isomorphism. The type <tt>GraphSmall</tt> must be a |
| model of |
| <a href="./VertexListGraph.html">Vertex List Graph</a>, |
| <a href="./EdgeListGraph.html">Edge List Graph</a>, |
| <a href="./BidirectionalGraph.html">Bidirectional Graph</a> and |
| <a href="./AdjacencyMatrix.html">Adjacency Matrix</a>. |
| The edge descriptors <tt>graph_traits<GraphSmall>::edge_descriptor</tt> |
| must be <a href="http://www.sgi.com/tech/stl/LessThanComparable.html"> |
| LessThan Comparable</a>, cf. also the remark <a href="#notes">below</a>. |
| </p> |
| </blockquote> |
| |
| <p>IN: <tt>const GraphLarge& graph_large</tt></p> |
| <blockquote> |
| <p> |
| The (second) larger graph to be tested. |
| Type <tt>GraphLarge</tt> must be a model of |
| <a href="./VertexListGraph.html">Vertex List Graph</a>, |
| <a href="./EdgeListGraph.html">Edge List Graph</a>, |
| <a href="./BidirectionalGraph.html">Bidirectional Graph</a> and |
| <a href="./AdjacencyMatrix.html">Adjacency Matrix</a>. |
| The edge descriptors <tt>graph_traits<GraphLarge>::edge_descriptor</tt> |
| must be <a href="http://www.sgi.com/tech/stl/LessThanComparable.html"> |
| LessThan Comparable</a>, cf. also the remark <a href="#notes">below</a>. |
| </p> |
| </blockquote> |
| |
| <p>OUT: <tt>SubGraphIsoMapCallback user_callback</tt></p> |
| <blockquote> |
| <p> |
| A function object to be called when a graph-subgraph isomorphism has been discovered. The |
| <tt>operator()</tt> must have following form: |
| </p> |
| <pre> |
| template <typename CorrespondenceMap1To2, |
| typename CorrespondenceMap2To1> |
| bool operator()(CorrespondenceMap1To2 f, CorrespondenceMap2To1 g) const |
| </pre> |
| <p> |
| Both the <tt>CorrespondenceMap1To2</tt> |
| and <tt>CorresondenceMap2To1</tt> types are models |
| of <a href="../../property_map/doc/ReadablePropertyMap.html">Readable |
| Property Map</a> and map equivalent vertices across the two |
| graphs given to <tt>vf2_subgraph_iso</tt> (or <tt>vf2_graph_iso</tt> or |
| <tt>vf2_subgraph_mono</tt>). For instance, if <tt>v</tt> is |
| from <tt>graph_small</tt>, <tt>w</tt> is from <tt>graph_large</tt>, |
| and the vertices can be considered equivalent, |
| then <tt>get(f, v)</tt> will be <tt>w</tt> and <tt>get(g, w)</tt> |
| will be <tt>v</tt>. If any vertex, say <tt>v</tt> in <tt>graph_small</tt>, |
| does not match a vertex in <tt>graph_large</tt> , |
| then <tt>get(f, v)</tt> will be <tt>graph_traits<GraphLarge>::null_vertex()</tt>. |
| Likewise for any unmatched vertices from <tt>graph_large</tt>, |
| <tt>get(g, w)</tt> will be <tt>graph_traits<GraphSmall>::null_vertex()</tt>. |
| |
| Returning false from the callback will abort the search immediately. Otherwise, |
| the entire search space will be explored. A "default" print callback |
| is provided as a <a href="#vf2_callback">utility function</a>. |
| </p> |
| </blockquote> |
| |
| <p>IN: <tt>const VertexOrderSmall& vertex_order_small</tt></p> |
| <blockquote> |
| <p> |
| The ordered vertices of the smaller (first) graph <tt>graph_small</tt>. |
| During the matching process the vertices are examined in the order given by |
| <tt>vertex_order_small</tt>. Type <tt>VertexOrderSmall</tt> must be a model |
| of <a href="http://www.sgi.com/tech/stl/Container.html">ContainerConcept</a> |
| with value type |
| <tt>graph_traits<GraphSmall>::vertex_descriptor</tt>. |
| <br> |
| <b>Default</b> The vertices are ordered by multiplicity of |
| in/out degrees. |
| </p> |
| </blockquote> |
| |
| <h3>Named Parameters</h3> |
| |
| <p>IN: <tt>vertex_index1(IndexMapSmall index_map_small)</tt></p> |
| <blockquote> |
| <p> |
| This maps each vertex to an integer in the range <tt>[0, num_vertices(graph_small))</tt>. |
| Type <tt>IndexMapSmall</tt> must be a model of |
| <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. |
| <br> |
| <b>Default:</b> <tt>get(vertex_index, graph_small)</tt> |
| </p> |
| </blockquote> |
| |
| <p>IN: <tt>vertex_index2(IndexMapLarge index_map_large)</tt></p> |
| <blockquote> |
| <p> |
| This maps each vertex to an integer in the range <tt>[0, num_vertices(graph_large))</tt>. |
| Type <tt>IndexMapLarge</tt> must be a model of |
| <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map</a>. |
| <br> |
| <b>Default:</b> <tt>get(vertex_index, graph_large)</tt> |
| </p> |
| </blockquote> |
| |
| <p>IN: <tt>edges_equivalent(EdgeEquivalencePredicate edge_comp)</tt></p> |
| <blockquote> |
| <p> |
| This function object is used to determine if edges between the two graphs |
| <tt>graph_small</tt> and <tt>graph_large</tt> are equivalent. |
| Type <tt>EdgeEquivalencePredicate</tt> must be a model of |
| <a href="http://www.sgi.com/tech/stl/BinaryPredicate.html">Binary |
| Predicate</a> and have argument types of |
| <tt>graph_traits<GraphSmall>::edge_descriptor</tt> and |
| <tt>graph_traits<GraphLarge>::edge_descriptor</tt>. |
| The edge descriptors must be <a href="http://www.sgi.com/tech/stl/LessThanComparable.html"> |
| LessThan Comparable</a>. A return value of true indicates that the edges are equivalent. |
| <br> |
| <b>Default:</b> <tt><a href="./mcgregor_common_subgraphs.html#always_equivalent"> |
| always_equivalent</a></tt> |
| </p> |
| </blockquote> |
| |
| <p>IN: <tt>vertices_equivalent(VertexEquivalencePredicate vertex_comp)</tt></p> |
| <blockquote> |
| <p> |
| This function object is used to determine if vertices between the two graphs |
| <tt>graph_small</tt> and <tt>graph_large</tt> are equivalent. |
| Type <tt>VertexEquivalencePredicate</tt> must be a model of |
| <a href="http://www.sgi.com/tech/stl/BinaryPredicate.html">Binary Predicate</a> |
| and have argument types of |
| <tt>graph_traits<GraphSmall>::vertex_descriptor</tt> and |
| <tt>graph_traits<GraphLarge>::vertex_descriptor</tt>. A return value of true |
| indicates that the vertices are equivalent. |
| <br> |
| <b>Default:</b> <tt><a href="./mcgregor_common_subgraphs.html#always_equivalent"> |
| always_equivalent</a></tt> |
| </p> |
| </blockquote> |
| |
| <h3>Related Functions</h3> |
| <p> |
| Non-named parameter, named-parameter and all default parameter versions of |
| function |
| </p> |
| <p><tt>vf2_graph_iso(...)</tt></p> |
| <p><tt>vf2_subgraph_mono(...)</tt></p> |
| <p> |
| for isomorphism and (not necessarily induced) subgraph isomorphism testing, |
| taking the same parameters as the corresponding functions <tt>vf2_subgraph_iso</tt> |
| for induced subgraph isomorphism testing. |
| For <tt>vf2_graph_iso</tt> the algorithm finds all isomorphism mappings between |
| graphs <tt>graph1</tt> and <tt>graph2</tt> and outputs them to |
| <tt>user_callback</tt>. |
| For <tt>vf2_graph_mono</tt> the algorithm finds all mappings of <tt>graph_small</tt> |
| to subgraphs of <tt>graph_large</tt>. |
| Note that, as opposed to <tt>vf2_subgraph_iso</tt>, |
| these subgraphs need not to be induced subgraphs. |
| </p> |
| <p> |
| Both algorithms continues until <tt>user_callback</tt> |
| returns false or the search space has been fully explored. As before, |
| <tt>EdgeEquivalencePredicate</tt> and |
| <tt>VertexEquivalencePredicate</tt> predicates are used to test |
| whether edges and vertices are equivalent. By default |
| <tt>always_equivalent</tt> is used. |
| </p> |
| |
| <h3>Utility Functions and Structs</h3> |
| <p> |
| <tt id="vf2_callback"> |
| template<typename Graph1, |
| typename Graph2> |
| struct vf2_print_callback |
| </tt> |
| </p> |
| <blockquote> |
| <p> |
| Callback function object that prints out the correspondences between vertices |
| of <tt>Graph1</tt> and <tt>Graph2</tt>. The constructor takes |
| the two graphs <em>G<sub>1</sub></em> and <em>G<sub>2</sub></em>. |
| </p> |
| </blockquote> |
| |
| <pre> |
| template<typename Graph> |
| std::vector<typename graph_traits<Graph>::vertex_descriptor> |
| vertex_order_by_mult(const Graph& graph) |
| </pre> |
| <blockquote> |
| <p> |
| Returns a vector containing the vertices of a graph, sorted |
| by multiplicity of in/out degrees. |
| </p> |
| </blockquote> |
| |
| <pre> |
| <em class="comment">// Variant of verify_subgraph_iso with all default parameters</em> |
| template<typename Graph1, |
| typename Graph2, |
| typename CorresponenceMap1To2> |
| inline bool verify_vf2_subgraph_iso(const Graph1& graph1, const Graph2& graph2, |
| const CorresponenceMap1To2 f) |
| |
| |
| <em class="comment">// Verifies a graph (sub)graph isomorphism map</em> |
| template<typename Graph1, |
| typename Graph2, |
| typename CorresponenceMap1To2, |
| typename EdgeEquivalencePredicate, |
| typename VertexEquivalencePredicate> |
| inline bool verify_vf2_subgraph_iso(const Graph1& graph1, const Graph2& graph2, |
| const CorresponenceMap1To2 f, |
| EdgeEquivalencePredicate edge_comp, |
| VertexEquivalencePredicate vertex_comp) |
| </pre> |
| <blockquote> |
| <p> |
| This function can be used to verify a (sub)graph isomorphism |
| mapping <em>f</em>. The parameters are analogous to |
| function <tt>vf2_subgraph_iso</tt> (<tt>vf2_graph_iso</tt>). |
| </p> |
| </blockquote> |
| |
| |
| <h3>Complexity</h3> |
| <p> |
| Spatial and time complexity are given in [<a href="#cordella2004">2</a>]. The spatial |
| complexity of VF2 is of order <em>O(V)</em>, where V is the (maximum) number |
| of vertices of the two graphs. Time complexity is <em>O(V<sup>2</sup>)</em> in the best case and |
| <em>O(V!·V)</em> in the worst case. |
| </p> |
| |
| <h3>Examples</h3> |
| |
| <h4>Example 1</h4> |
| <p> |
| In the example below, a small graph <tt>graph1</tt> and a larger graph |
| <tt>graph2</tt> are defined. Here small and large refers to the number of |
| vertices of the graphs. <tt>vf2_subgraph_iso</tt> computes all the |
| subgraph isomorphism mappings between the two graphs and outputs them |
| via <tt>callback</tt>. |
| </p> |
| <pre> |
| typedef adjacency_list<setS, vecS, bidirectionalS> graph_type; |
| |
| <em class="comment">// Build graph1</em> |
| int num_vertices1 = 8; graph_type graph1(num_vertices1); |
| add_edge(0, 6, graph1); add_edge(0, 7, graph1); |
| add_edge(1, 5, graph1); add_edge(1, 7, graph1); |
| add_edge(2, 4, graph1); add_edge(2, 5, graph1); add_edge(2, 6, graph1); |
| add_edge(3, 4, graph1); |
| |
| <em class="comment">// Build graph2</em> |
| int num_vertices2 = 9; graph_type graph2(num_vertices2); |
| add_edge(0, 6, graph2); add_edge(0, 8, graph2); |
| add_edge(1, 5, graph2); add_edge(1, 7, graph2); |
| add_edge(2, 4, graph2); add_edge(2, 7, graph2); add_edge(2, 8, graph2); |
| add_edge(3, 4, graph2); add_edge(3, 5, graph2); add_edge(3, 6, graph2); |
| <em class="comment"> |
| // Create callback to print mappings</em> |
| vf2_print_callback<graph_type, graph_type> callback(graph1, graph2); |
| |
| <em class="comment"> |
| // Print out all subgraph isomorphism mappings between graph1 and graph2. |
| // Vertices and edges are assumed to be always equivalent.</em> |
| vf2_subgraph_iso(graph1, graph2, callback); |
| </pre> |
| <p> |
| The complete example can be found under |
| <a href="../example/vf2_sub_graph_iso_example.cpp"><tt>examples/vf2_sub_graph_iso_example.cpp</tt></a>. |
| <br> |
| </p> |
| |
| <h4>Example 2</h4> |
| <p> |
| In this example, the subgraph isomorphism mappings between multi-graphs are computed. The vertices |
| and edges of the multi-graphs are distinguished using property maps. |
| </p> |
| <pre> |
| <em class="comment">// Define edge and vertex properties</em> |
| typedef property<edge_name_t, char> edge_property; |
| typedef property<vertex_name_t, char, property<vertex_index_t, int> > vertex_property; |
| |
| <em class="comment">// Using a vecS graphs => the index maps are implicit.</em> |
| typedef adjacency_list<vecS, vecS, bidirectionalS, vertex_property, edge_property> graph_type; |
| |
| <em class="comment">// Create graph1</em> |
| graph_type graph1; |
| <em class="comment">// Add vertices... </em> |
| add_vertex(vertex_property('a'), graph1); |
| ... |
| <em class="comment">//... and edges </em> |
| add_edge(0, 1, edge_property('b'), graph1); |
| add_edge(0, 1, edge_property('b'), graph1); |
| ... |
| |
| <em class="comment">// Create graph2 </em> |
| graph_type graph2; |
| add_vertex(vertex_property('a'), graph2); |
| ... |
| add_edge(0, 1, edge_property('a'), graph2); |
| ... |
| </pre> |
| |
| <p> |
| To distinguish vertices and edges with property maps, binary predicates are created using the |
| <tt><a href="./mcgregor_common_subgraphs.html#make_property_map_equivalent"> |
| make_property_map_equivalent</a></tt> function: |
| </p> |
| |
| <pre> |
| <em class="comment">// Create the vertex binary predicate</em> |
| typedef property_map<graph_type, vertex_name_t>::type vertex_name_map_t; |
| typedef property_map_equivalent<vertex_name_map_t, vertex_name_map_t> vertex_comp_t; |
| vertex_comp_t vertex_comp = |
| make_property_map_equivalent(get(vertex_name, graph1), get(vertex_name, graph2)); |
| |
| <em class="comment">// Create the vertex binary predicate</em> |
| typedef property_map<graph_type, edge_name_t>::type edge_name_map_t; |
| typedef property_map_equivalent<edge_name_map_t, edge_name_map_t> edge_comp_t; |
| edge_comp_t edge_comp = |
| make_property_map_equivalent(get(edge_name, graph1), get(edge_name, graph2)); |
| </pre> |
| |
| <p> |
| Finally, a callback function object is created and the subgraph isomorphism mappings are |
| computed: |
| </p> |
| <pre> |
| <em class="comment">// Create callback</em> |
| vf2_print_callback<graph_type, graph_type> callback(graph1, graph2); |
| |
| <em class="comment"> |
| // Print out all subgraph isomorphism mappings between graph1 and graph2. |
| // Function vertex_order_by_mult is used to compute the order of |
| // vertices of graph1. This is the order in which the vertices are examined |
| // during the matching process.</em> |
| vf2_subgraph_iso(graph1, graph2, callback, vertex_order_by_mult(graph1), |
| edges_equivalent(edge_comp).vertices_equivalent(vertex_comp)); |
| </pre> |
| |
| <p> |
| For the complete example, see |
| <a href="../example/vf2_sub_graph_iso_multi_example.cpp"> |
| <tt>examples/vf2_sub_graph_iso_multi_example.cpp</tt></a>. |
| <br> |
| </p> |
| |
| <h3 id="notes">Notes</h3> |
| <p> |
| If the <tt>EdgeList</tt> allows for parallel edges, e.g. <tt>vecS</tt>, the |
| algorithm does some bookkeeping of already identified edges. Matched edges |
| are temporarily stored using <tt>std::set</tt> as container, requiring that |
| <tt>edge_descriptor</tt> are <a href="http://www.sgi.com/tech/stl/LessThanComparable.html"> |
| LessThan Comparable</a>. In contrast, if instead you enforce the absence of |
| parallel edges, e.g. by using <tt>setS</tt>, the lookup function falls back |
| to <tt>edge()</tt> without performing any bookkeeping. |
| </p> |
| |
| <h3>Bibliography</h3> |
| |
| <dl> |
| <dt><a name="cordella2001">1</a></dt> |
| <dd> |
| L. P. Cordella, P. Foggia, C. Sansone, and M. Vento. |
| <br><em>An improved algorithm for matching large graphs</em>. |
| <br>In: 3rd IAPR-TC15 Workshop on Graph-based Representations in Pattern Recognition, pp. 149-159, Cuen, 2001. |
| <p></p> |
| </dd> |
| <dt><a name="cordella2004">2</a></dt> |
| <dd> |
| L. P. Cordella, P. Foggia, C. Sansone, and M. Vento. |
| <br><em>A (Sub)Graph Isomorphism Algorithm for Matching Large Graphs</em>. |
| <br>IEEE Trans. Pattern Anal. Mach. Intell., vol. 26, no. 10, pp. 1367-1372, 2004. |
| <p></p> |
| </dd> |
| <dt><a name="foggia_etal">3</a></dt> |
| <dd> |
| <a href="http://www.cs.sunysb.edu/~algorith/implement/vflib/implement.shtml"> |
| <tt>http://www.cs.sunysb.edu/~algorith/implement/vflib/implement.shtml</tt></a> |
| <p></p> |
| </dd> |
| </dl> |
| <hr> |
| <p> |
| Copyright © 2012, Flavio De Lorenzi |
| (<a href="mailto:fdlorenzi@gmail.com">fdlorenzi@gmail.com</a>) <br /> |
| Copyright © 2013, Jakob Lykke Andersen, University of Southern Denmark |
| (<a href="mailto:jlandersen@imada.sdu.dk">jlandersen@imada.sdu.dk</a>) |
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