194 lines
7.5 KiB
HTML
194 lines
7.5 KiB
HTML
<HTML>
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<!--
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Copyright (c) Jeremy Siek 2000
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt)
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-->
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<Head>
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<Title>Boost Graph Library: Isomorphism</Title>
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<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
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ALINK="#ff0000">
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<IMG SRC="../../../boost.png"
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ALT="C++ Boost" width="277" height="86">
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<BR Clear>
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<H1>
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<img src="figs/python.gif" alt="(Python)"/>
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<TT>isomorphism</TT>
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</H1>
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<PRE>
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<i>// named parameter version</i>
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template <class Graph1, class Graph2, class P, class T, class R>
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bool isomorphism(const Graph1& g1, const Graph2& g2,
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const bgl_named_params<P,T,R>& params = <i>all defaults</i>)
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<i>// non-named parameter version</i>
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template <typename Graph1, typename Graph2, typename IsoMap,
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typename VertexInvariant1, typename VertexInvariant2,
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typename V1Map, typename V2Map>
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bool isomorphism(const Graph1& g1, const Graph2& g2,
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IsoMap f, VertexInvariant1 invariant2, VertexInvariant2 invariant2,
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std::size_t max_invariant, VertexIndex1Map i1_map, VertexIndex2Map i2_map)
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</pre>
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<p>
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An <b><i>isomorphism</i></b> is a 1-to-1 mapping of the vertices in
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one graph to the vertices of another graph such that adjacency is
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preserved. Another words, given graphs <i>G<sub>1</sub> =
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(V<sub>1</sub>,E<sub>1</sub>)</i> and <i>G<sub>2</sub> =
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(V<sub>2</sub>,E<sub>2</sub>)</i> an isomorphism is a function
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<i>f</i> such that for all pairs of vertices <i>a,b</i> in
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<i>V<sub>1</sub></i>, edge <i>(a,b)</i> is in <i>E<sub>1</sub></i> if
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and only if edge <i>(f(a),f(b))</i> is in <i>E<sub>2</sub></i>.
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</p>
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<p>
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This function returns <tt>true</tt> if there exists an isomorphism
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between graph 1 and graph 2 and <tt>false</tt> otherwise. Also, if a
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isomorphism map named parameter is provided then an isomorphism is
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recorded in the map.
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</p>
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<p>
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The current implementation is based on descriptions of a backtracking
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algorithm in [<a
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href="./bibliography.html#fortin96:_graph_iso_prob">46</a>,<a
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href="./bibliography.html#reingold77:_combin_algo">48</a>]. The file
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<a href="./isomorphism-impl.pdf">isomorphism-impl.pdf</a> contains a (somewhat out-of-date)
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"literate" description of the implementation. The algorithm
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used is simple but slow. A more efficient (and much more complex)
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algorithm is described in [<a
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href="./bibliography.html#mckay81:_pract_graph_iso">47</a>]. When (and
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if) a version of this algorithm is ported to the BGL interface it
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should replace the current algorithm.
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</p>
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<H3>Where Defined</H3>
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<a href="../../../boost/graph/isomorphism.hpp"><TT>boost/graph/isomorphism.hpp</TT></a>
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<h3>Parameters</h3>
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IN: <tt>const Graph1& g1</tt>
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<blockquote>
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A directed or undirected graph. The graph type must model of <a
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href="./VertexListGraph.html">Vertex List Graph</a> and <a
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href="./EdgeListGraph.html">Edge List Graph</a>.
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</blockquote>
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IN: <tt>const Graph2& g2</tt>
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<blockquote>
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A directed or undirected graph. The graph type must model <a
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href="./BidirectionalGraph.html">Bidirectional Graph</a> and <a
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href="./VertexListGraph.html">Vertex List Graph</a>.
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</blockquote>
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<h3>Named Parameters</h3>
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OUT: <tt>isomorphism_map(IsoMap f)</tt>
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<blockquote>
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The mapping from vertices in graph 1 to vertices in graph 2. This must
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be a <a href="../../property_map/doc/ReadWritePropertyMap.html">Read/Write
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Property Map</a>.<br> <b>Default:</b> an <a
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href="../../property_map/doc/iterator_property_map.html"><tt>iterator_property_map</tt></a>
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constructed from a <tt>std::vector</tt> of graph 2's vertex
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descriptor type and the vertex index map for graph 1.<br>
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<b>Python</b>: Must be a <tt>vertex_vertex_map</tt> for the first graph.
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</blockquote>
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IN: <tt>vertex_invariant1(VertexInvariant1 i1)</tt>
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IN: <tt>vertex_invariant2(VertexInvariant2 i2)</tt>
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<blockquote>
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Mappings from vertices to integers which restrict which vertices may be
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considered isomorphic. If a candidate isomorphism maps <i>v1</i> to <i>v2</i>
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but <i>i1</i>(<i>v1</i>) != <i>i2</i>(<i>v2</i>), that candidate is rejected.
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This mapping can be used either to speed up the search (as is done by the
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default value, which requires that the degrees of <i>v1</i> and <i>v2</i> are
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equal) or to impose extra conditions on the result. The
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<tt>VertexInvariant1</tt> and <tt>VertexInvariant2</tt> types must model <a
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href="http://www.boost.org/sgi/stl/UnaryFunction.html">UnaryFunction</a>, with
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the argument type of <tt>vertex_invariant1</tt> being <tt>Graph1</tt>'s vertex
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descriptor type, the argument type of <tt>vertex_invariant2</tt> being
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<tt>Graph2</tt>'s vertex descriptor type, and both functions having integral
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result types. The values returned by these two functions must be in the range
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[0, <tt>vertex_max_invariant</tt>).
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<br>
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<b>Default:</b> <tt>degree_vertex_invariant</tt> for both arguments<br>
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<b>Python</b>: Unsupported parameter.
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</blockquote>
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IN: <tt>vertex_max_invariant(std::size_t max_invariant)</tt>
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<blockquote>
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An upper bound on the possible values returned from either
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vertex_invariant1 or vertex_invariant2.
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<br>
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<b>Default:</b> <tt>vertex_invariant2.max()</tt>. The default
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<tt>vertex_invariant2</tt> parameter, an instance of
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<tt>degree_vertex_invariant</tt>, defines this function to
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return <tt>num_vertices(g2) * (num_vertices(g2)+1)</tt>.<br>
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<b>Python</b>: Unsupported parameter.
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</blockquote>
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IN: <tt>vertex_index1_map(VertexIndex1Map i1_map)</tt>
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<blockquote>
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This maps each vertex to an integer in the range <tt>[0,
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num_vertices(g))</tt>. This is necessary for efficient updates of the
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heap data structure when an edge is relaxed. The type
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<tt>VertexIndex1Map</tt> must be a model of <a
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href="../../property_map/doc/ReadablePropertyMap.html">Readable Property
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Map</a>. The value type of the map must be an integer type. The vertex
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descriptor type of graph 1 needs to be usable as the key type of the
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map.<br>
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<b>Default:</b> <tt>get(vertex_index, g1)</tt>
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Note: if you use this default, make sure your graph has
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an internal <tt>vertex_index</tt> property. For example,
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<tt>adjacency_list</tt> with <tt>VertexList=listS</tt> does
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not have an internal <tt>vertex_index</tt> property.
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<br>
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<b>Python</b>: Unsupported parameter.
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</blockquote>
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IN: <tt>vertex_index2_map(VertexIndex2Map i2_map)</tt>
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<blockquote>
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This maps each vertex to an integer in the range <tt>[0,
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num_vertices(g))</tt>. This is necessary for efficient updates of the
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heap data structure when an edge is relaxed. The type
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<tt>VertexIndex2Map</tt> must be a model of <a
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href="../../property_map/doc/ReadablePropertyMap.html">Readable Property
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Map</a>. The value type of the map must be an integer type. The vertex
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descriptor type of graph 2 needs to be usable as the key type of the
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map.<br>
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<b>Default:</b> <tt>get(vertex_index, g2)</tt>
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Note: if you use this default, make sure your graph has
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an internal <tt>vertex_index</tt> property. For example,
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<tt>adjacency_list</tt> with <tt>VertexList=listS</tt> does
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not have an internal <tt>vertex_index</tt> property.
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<br>
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<b>Python</b>: Unsupported parameter.
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</blockquote>
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<h3>Complexity</h3>
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The worst-case time complexity is <i>O(|V|!)</i>.
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<h3>Example</h3>
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See <a href="../example/isomorphism.cpp"><tt>libs/graph/example/isomorphism.cpp</tt></a>.
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<br>
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<HR>
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<TABLE>
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<TR valign=top>
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<TD nowrap>Copyright © 2000-2001</TD><TD>
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<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>)
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</TD></TR></TABLE>
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</BODY>
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</HTML>
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