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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN"
"http://www.w3.org/TR/html4/strict.dtd">
<!--
Copyright (C) Flavio De Lorenzi 2012
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)
-->
<html>
<head>
<meta http-equiv="content-type" content="text/html; charset=iso-8859-15">
<title>Boost Graph Library: VF2 (Sub)Graph Isomorphism</title>
<style type="text/css">
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<body>
<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,
bool(*user_step_callback)() = &vf2_trivial_step_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,
bool(*user_step_callback)() = &vf2_trivial_step_callback)
<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.boost.org/sgi/stl/BinaryFunction.html">EdgeEquivalencePredicate</a>,
typename <a href="http://www.boost.org/sgi/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,
bool(*user_step_callback)() = &vf2_trivial_step_callback)
</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> or <tt>user_step_callback</tt>
returns false or the search space has been fully explored. <tt>vf2_subgraph_iso</tt>
returns true if a graph-subgraph isomorphism was found 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.boost.org/sgi/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.boost.org/sgi/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>OUT: <tt>bool(*user_step_callback)() = &vf2_trivial_step_callback</tt></p>
<blockquote>
<p>
A function to be called at each step of the search. If it returns false,
the search is terminated. The default always returns true. Unlike
<tt>user_callback</tt> (which is only called when a match is found),
this callback is called with high frequency; it may be used, for
example, to stop the search when a time limit is exceeded or for other
external reasons.
</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.boost.org/sgi/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.boost.org/sgi/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.boost.org/sgi/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.boost.org/sgi/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.boost.org/sgi/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>)
</p>
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