132 lines
4.2 KiB
HTML
132 lines
4.2 KiB
HTML
<html><head><!-- Copyright 2007 Aaron Windsor
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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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--><title>Boost Graph Library: Planar Canonical Ordering</title>
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</head>
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<body alink="#ff0000"
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bgcolor="#ffffff"
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link="#0000ee"
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text="#000000"
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vlink="#551a8b">
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<img src="../../../boost.png" alt="C++ Boost" height="86" width="277">
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<br clear="">
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<h1>Planar Canonical Ordering</h1>
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<pre>template <typename Graph, typename PlanarEmbedding, typename OutputIterator, typename VertexIndexMap>
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void planar_canonical_ordering(const Graph& g, PlanarEmbedding embedding, OutputIterator ordering, VertexIndexMap vm);
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</pre>
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<p>
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A <i>planar canonical ordering</i> is an ordering <i>v<sub>1</sub>,
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v<sub>2</sub>, ..., v<sub>n</sub></i> of the vertices of a
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<a href="make_maximal_planar.html">maximal</a>
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<a href="planar_graphs.html">planar</a> graph having the property that, for
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each <i>k</i>, <i>3 <= k < n</i>, the graph induced by
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<i>v<sub>1</sub>, v<sub>2</sub>, ..., v<sub>k</sub></i>
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</p><ul>
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<li>is biconnected and contains the edge <i>{v<sub>1</sub>, v<sub>2</sub>}</i>
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on its outer face.
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</li><li>has any vertices in the range <i>v<sub>1</sub>, v<sub>2</sub>, ...,
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v<sub>k</sub></i> that are adjacent to <i>v<sub>(k+1)</sub></i> on its outer
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face, and these vertices form a path along the outer face.
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</li></ul>
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Let <i>G<sub>k</sub></i> be the graph induced by the first <i>k</i> vertices in
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the canonical ordering, along with all edges between any of the first <i>k</i>
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vertices. After <i>G<sub>k</sub></i> has been drawn, the <i>(k+1)</i>st vertex
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can be drawn easily without edge crossings, since it's adjacent only to a
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consecutive sequence of vertices on the outer face of <i>G<sub>k</sub></i>.
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<p>
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</p><blockquote>
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<center>
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<img src="./figs/canonical_ordering.png">
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</center>
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</blockquote>
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A planar canonical ordering exists for every maximal planar graph with at
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least 2 vertices. <tt>planar_canonical_ordering</tt> expects the input graph
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to have at least 2 vertices.
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<p>
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The planar canonical ordering is used as an input in some planar graph drawing
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algorithms, particularly those that create a straight line embedding.
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de Fraysseix, Pach, and Pollack
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[<a href="./bibliography.html#defraysseixpachpollack90">72</a>]
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first proved the
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existence of such an ordering and showed how to compute one in time
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<i>O(n)</i> on a maximal planar graph with <i>n</i> vertices.
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<h3>Complexity</h3>
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If the vertex index map provides constant-time access to indices, this
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function takes time <i>O(n + m)</i> for a planar graph with <i>n</i> vertices
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and <i>m</i> edges. Note that
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in a simple planar graph with <i>f</i> faces, <i>m</i> edges, and <i>n</i>
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vertices, both <i>f</i> and <i>m</i> are <i>O(n)</i>.
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<h3>Where Defined</h3>
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<p>
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<a href="../../../boost/graph/planar_canonical_ordering.hpp">
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<tt>boost/graph/planar_canonical_ordering.hpp</tt></a>
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</p><h3>Parameters</h3>
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IN: <tt>Graph& g</tt>
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<blockquote>
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An undirected graph. The graph type must be a model of
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<a href="VertexAndEdgeListGraph.html">VertexAndEdgeListGraph</a>.
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The graph must:
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<ul>
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<li>Be maximal planar.</li>
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<li>Have at least two vertices.</li>
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</ul>
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</blockquote>
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IN: <tt>PlanarEmbedding</tt>
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<blockquote>
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A model of <a href="PlanarEmbedding.html">PlanarEmbedding</a>.
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</blockquote>
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IN: <tt>OutputIterator</tt>
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<blockquote>
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An OutputIterator with <tt>value_type</tt> equal to
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<tt>graph_traits<Graph>::vertex_descriptor</tt>. The canonical ordering
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will be written to this iterator.
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</blockquote>
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IN: <tt>VertexIndexMap vm</tt>
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<blockquote>
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A <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property Map
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</a> that maps vertices from <tt>g</tt> to distinct integers in the range
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<tt>[0, num_vertices(g) )</tt><br>
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<b>Default</b>: <tt>get(vertex_index,g)</tt><br>
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</blockquote>
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<h3>Example</h3>
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<p>
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<a href="../example/canonical_ordering.cpp">
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<tt>examples/canonical_ordering.cpp</tt></a>
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</p><h3>See Also</h3>
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<p>
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<a href="planar_graphs.html">Planar Graphs in the Boost Graph Library</a>
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<br>
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</p><hr>
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Copyright <20> 2007 Aaron Windsor (<a href="mailto:aaron.windsor@gmail.com">
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aaron.windsor@gmail.com</a>)
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</body></html>
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