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- <!-- Copyright 2007 Aaron Windsor
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- Distributed under the Boost Software License, Version 1.0.
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- <Head>
- <Title>Boost Graph Library: Planar Face Traversal</Title>
- <BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
- ALINK="#ff0000">
- <IMG SRC="../../../boost.png"
- ALT="C++ Boost" width="277" height="86">
- <BR Clear>
- <H1>Planar Face Traversal</H1>
- <pre>
- template<typename Graph, typename PlanarEmbedding, typename PlanarFaceVisitor, typename EdgeIndexMap>
- void planar_face_traversal(const Graph& g, PlanarEmbedding embedding, PlanarFaceVisitor& visitor, EdgeIndexMap em);
- </pre>
- <p>
- A graph is <i>planar</i> if it can be drawn in two-dimensional space with no
- two of its edges crossing. Any embedding of a planar graph separates the plane
- into distinct regions that are bounded by sequences of edges in the graph.
- These regions are called <i>faces</i>.
- <br>
- <br>
- <table align="center" class="image">
- <caption align="bottom">
- <h5>A plane drawing of a graph (left), and the 8 faces defined by the planar
- embedding (right.) Each connected blue region in the image on the right is a
- face. The large blue region surrounding the graph is the <i>outer face</i>.
- </h5>
- </caption>
- <tr>
- <td>
- <img src="./figs/face_illustration.png">
- </td>
- </tr>
- <tr></tr>
- </table>
- <br>
- A traversal of the faces of a planar graph involves iterating through all faces
- of the graph, and on each face, iterating through all vertices and edges of the
- face. The iteration through all vertices and edges of each face follows a
- path around the border of the face.
- <p>
- In a biconnected graph, like the one shown above, each face is bounded by a
- cycle and each edge belongs to exactly two faces. For this reason, when
- <tt>planar_face_traversal</tt> is called on a biconnected graph, each edge will
- be visited exactly twice: once on each of two distinct faces, and no vertex
- will be visited more than once on a particular face. The output of
- <tt>planar_face_traversal</tt> on non-biconnected graphs is less intuitive -
- for example, if the graph
- consists solely of a path of vertices (and therefore a single face),
- <tt>planar_face_traversal</tt> will iterate <i>around</i> the path, visiting
- each edge twice and visiting some vertices more than once.
- <tt>planar_face_traversal</tt> does not visit isolated vertices.
- <p>
- Like other graph traversal algorithms in the Boost Graph Library, the planar
- face traversal is a generic traversal that can be customized by the
- redefinition of certain visitor event points. By defining an appropriate
- visitor, this traversal can be
- used to enumerate the faces of a planar graph, triangulate a planar graph, or
- even construct a dual of a planar graph.
- <br>
- <center>
- <img src="./figs/face_traversal_example.png">
- </center>
- <br>
- For example, on the above graph, an instance <tt>my_visitor</tt> of the
- following visitor:
- <pre>
- struct output_visitor: public planar_face_traversal_visitor
- {
- void begin_face() { std::cout << "New face: "; }
- template <typename Vertex> void next_vertex(Vertex v) { std::cout << v << " "; }
- void finish_face() { std::cout << std::endl; }
- };
- </pre>
- can be passed to the <tt>planar_face_traversal</tt> function:
- <pre>
- output_visitor my_visitor;
- planar_face_traversal(g, embed, my_visitor); //embed is a planar embedding of g
- </pre>
- and might produce the output
- <pre>
- New face: 1 2 5 4
- New face: 2 3 4 5
- New face: 3 0 1 4
- New face: 1 0 3 2
- </pre>
- <h3>Visitor Event Points</h3>
-
- <ul>
- <li><tt>visitor.begin_traversal()</tt>: called once before any faces are
- visited.
- <li><tt>visitor.begin_face()</tt>: called once, for each face, before any
- vertex or edge on that face has been visited.
- <li><tt>visitor.end_face()</tt>: called once, for each face, after all vertices
- and all edges on that face have been visited.
- <li><tt>visitor.next_vertex(Vertex v)</tt>: called once on each vertex in the
- current face (the start and end of which are designated by calls to
- <tt>begin_face()</tt> and <tt>end_face()</tt>, respectively) in order
- according to the order established by the planar embedding.
- <li><tt>visitor.next_edge(Edge e)</tt>: called once on each edge in the current
- face (the start and end of which are designated by calls to
- <tt>begin_face()</tt> and <tt>end_face()</tt>, respectively) in order
- according to the order established by the planar embedding.
- <li><tt>visitor.end_traversal()</tt>: called once after all faces have been
- visited.
- </ul>
- Although <tt>next_vertex</tt> is guaranteed to be called in sequence for each
- vertex as the traversal moves around a face and <tt>next_edge</tt> is
- guaranteed to be called in sequence for each edge as the traversal moves
- around a face, there's no guarantee about the order in which
- <tt>next_vertex</tt> and <tt>next_edge</tt> are called with respect to each
- other in between calls to <tt>begin_face</tt> and <tt>end_face</tt>. These
- calls may be interleaved, all vertex visits may precede all edge visits, or
- vise-versa.
- <p>
- <tt>planar_face_traversal</tt> iterates over a copy of the edges of the input
- graph, so it is safe to add edges to the graph during visitor event points.
- <h3>Complexity</h3>
- If all of the visitor event points run in constant time, the traversal takes
- time <i>O(n + m)</i> for a planar graph with <i>n</i> vertices and <i>m</i>
- edges. Note that
- in a simple planar graph with <i>f</i> faces, <i>m</i> edges, and <i>n</i>
- vertices, both <i>f</i> and <i>m</i> are <i>O(n)</i>.
- <H3>Where Defined</H3>
- <P>
- <a href="../../../boost/graph/planar_face_traversal.hpp">
- <TT>boost/graph/planar_face_traversal.hpp</TT>
- </a>
- <h3>Parameters</h3>
- IN: <tt>Graph& g</tt>
- <blockquote>
- An undirected graph. The graph type must
- be a model of <a href="VertexAndEdgeListGraph.html">VertexAndEdgeListGraph</a>
- </blockquote>
- IN: <tt>PlanarEmbedding</tt>
- <blockquote>
- A model of <a href="PlanarEmbedding.html">PlanarEmbedding</a>.
- </blockquote>
- IN: <tt>PlanarFaceVisitor</tt>
- <blockquote>
- A model of <a href="PlanarFaceVisitor.html">PlanarFaceVisitor</a>.
- </blockquote>
- IN: <tt>EdgeIndexMap vm</tt>
- <blockquote>
- A <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map
- </a> that maps edges from <tt>g</tt> to distinct integers in the range
- <tt>[0, num_edges(g) )</tt><br>
- <b>Default</b>: <tt>get(edge_index,g)</tt><br>
- </blockquote>
- <H3>Example</H3>
- <P>
- <a href="../example/planar_face_traversal.cpp">
- <TT>examples/planar_face_traversal.cpp</TT></a>
- <h3>See Also</h3>
- <p>
- <ul>
- <li><a href="./planar_graphs.html">Planar Graphs in the Boost Graph Library</a>
- <li><a href="./PlanarFaceVisitor.html">PlanarFaceVisitor</a> concept.
- </ul>
- <br>
- <HR>
- Copyright © 2007 Aaron Windsor (<a href="mailto:aaron.windsor@gmail.com">
- aaron.windsor@gmail.com</a>)
- </BODY>
- </HTML>
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