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 <H1>
 Topologies for Graph Drawing
 </H1>

 <P>

 <h3>Synopsis</h3>

 <pre>
 template&lt;std::size_t Dims&gt; class <a href="#convex_topology">convex_topology</a>;
 template&lt;std::size_t Dims, typename RandomNumberGenerator = minstd_rand&gt; class <a href="#hypercube_topology">hypercube_topology</a>;
 template&lt;typename RandomNumberGenerator = minstd_rand&gt; class <a href="#square_topology">square_topology</a>;
 template&lt;typename RandomNumberGenerator = minstd_rand&gt; class <a href="#cube_topology">cube_topology</a>;
 template&lt;std::size_t Dims, typename RandomNumberGenerator = minstd_rand&gt; class <a href="#ball_topology">ball_topology</a>;
 template&lt;typename RandomNumberGenerator = minstd_rand&gt; class <a href="#circle_topology">circle_topology</a>;
 template&lt;typename RandomNumberGenerator = minstd_rand&gt; class <a href="#sphere_topology">sphere_topology</a>;
 template&lt;typename RandomNumberGenerator = minstd_rand&gt; class <a href="#heart_topology">heart_topology</a>;
 </pre>

 <h3>Summary</h3>

 <p> <a href="#topologies">Various topologies</a> are provided that
 produce different, interesting results for graph layout algorithms. The <a
 href="#square_topology">square topology</a> can be used for normal
 display of graphs or distributing vertices for parallel computation on
 a process array, for instance. Other topologies, such as the <a
 href="#sphere_topology">sphere topology</a> (or N-dimensional <a
 href="#ball_topology">ball topology</a>) make sense for different
 problems, whereas the <a href="#heart_topology">heart topology</a> is
 just plain fun. One can also <a href="#topology-concept">define a
 topology</a> to suit other particular needs. <br>

 <a name="topologies"><h3>Topologies</h3></a>
 A topology is a description of a space on which layout can be
 performed. Some common two, three, and multidimensional topologies
 are provided, or you may create your own so long as it meets the
 requirements of the <a href="#topology-concept">Topology concept</a>.

 <a name="topology-concept"><h4>Topology Concept</h4></a> Let
 <tt>Topology</tt> be a model of the Topology concept and let
 <tt>space</tt> be an object of type <tt>Topology</tt>. <tt>p1</tt> and
 <tt>p2</tt> are objects of associated type <tt>point_type</tt> (see
 below). The following expressions must be valid:

 <table border="1">
   <tr>
     <th>Expression</th>
     <th>Type</th>
     <th>Description</th>
   </tr>
   <tr>
     <td><tt>Topology::point_type</tt></td>
     <td>type</td>
     <td>The type of points in the space.</td>
   </tr>
   <tr>
     <td><tt>space.random_point()</tt></td>
     <td>point_type</td>
     <td>Returns a random point (usually uniformly distributed) within
     the space.</td>
   </tr>
   <tr>
     <td><tt>space.distance(p1, p2)</tt></td>
     <td>double</td>
     <td>Get a quantity representing the distance between <tt>p1</tt>
     and <tt>p2</tt> using a path going completely inside the space.
     This only needs to have the same &lt; relation as actual
     distances, and does not need to satisfy the other properties of a
     norm in a Banach space.</td>
   </tr>
   <tr>
     <td><tt>space.move_position_toward(p1, fraction, p2)</tt></td>
     <td>point_type</td>
     <td>Returns a point that is a fraction of the way from <tt>p1</tt>
     to <tt>p2</tt>, moving along a "line" in the space according to
     the distance measure. <tt>fraction</tt> is a <tt>double</tt>
     between 0 and 1, inclusive.</td>
   </tr>
 </table>

 <a name="convex_topology"><h3>Class template <tt>convex_topology</tt></h3></a>

 <p>Class template <tt>convex_topology</tt> implements the basic
 distance and point movement functions for any convex topology in
 <tt>Dims</tt> dimensions. It is not itself a topology, but is intended
 as a base class that any convex topology can derive from. The derived
 topology need only provide a suitable <tt>random_point</tt> function
 that returns a random point within the space.

 <pre>
 template&lt;std::size_t Dims&gt;
 class convex_topology
 {
   struct point
   {
     point() { }
     double& operator[](std::size_t i) {return values[i];}
     const double& operator[](std::size_t i) const {return values[i];}

   private:
     double values[Dims];
   };

  public:
   typedef point point_type;

   double distance(point a, point b) const;
   point move_position_toward(point a, double fraction, point b) const;
 };
 </pre>

 <a name="hypercube_topology"><h3>Class template <tt>hypercube_topology</tt></h3></a>

 <p>Class template <tt>hypercube_topology</tt> implements a
 <tt>Dims</tt>-dimensional hypercube. It is a convex topology whose
 points are drawn from a random number generator of type
 <tt>RandomNumberGenerator</tt>. The <tt>hypercube_topology</tt> can
 be constructed with a given random number generator; if omitted, a
 new, default-constructed random number generator will be used. The
 resulting layout will be contained within the hypercube, whose sides
 measure 2*<tt>scaling</tt> long (points will fall in the range
 [-<tt>scaling</tt>, <tt>scaling</tt>] in each dimension).

 <pre>
 template&lt;std::size_t Dims, typename RandomNumberGenerator = minstd_rand&gt;
 class hypercube_topology : public <a href="#convex_topology">convex_topology</a>&lt;Dims&gt;
 {
 public:
   explicit hypercube_topology(double scaling = 1.0);
   hypercube_topology(RandomNumberGenerator& gen, double scaling = 1.0);
   point_type random_point() const;
 };
 </pre>

 <a name="square_topology"><h3>Class template <tt>square_topology</tt></h3></a>

 <p>Class template <tt>square_topology</tt> is a two-dimensional
 hypercube topology.

 <pre>
 template&lt;typename RandomNumberGenerator = minstd_rand&gt;
 class square_topology : public <a href="#hypercube_topology">hypercube_topology</a>&lt;2, RandomNumberGenerator&gt;
 {
  public:
   explicit square_topology(double scaling = 1.0);
   square_topology(RandomNumberGenerator& gen, double scaling = 1.0);
 };
 </pre>

 <a name="cube_topology"><h3>Class template <tt>cube_topology</tt></h3></a>

 <p>Class template <tt>cube_topology</tt> is a two-dimensional
 hypercube topology.

 <pre>
 template&lt;typename RandomNumberGenerator = minstd_rand&gt;
 class cube_topology : public <a href="#hypercube_topology">hypercube_topology</a>&lt;3, RandomNumberGenerator&gt;
 {
  public:
   explicit cube_topology(double scaling = 1.0);
   cube_topology(RandomNumberGenerator& gen, double scaling = 1.0);
 };
 </pre>

 <a name="ball_topology"><h3>Class template <tt>ball_topology</tt></h3></a>

 <p>Class template <tt>ball_topology</tt> implements a
 <tt>Dims</tt>-dimensional ball. It is a convex topology whose points
 are drawn from a random number generator of type
 <tt>RandomNumberGenerator</tt> but reside inside the ball. The
 <tt>ball_topology</tt> can be constructed with a given random number
 generator; if omitted, a new, default-constructed random number
 generator will be used. The resulting layout will be contained within
 the ball with the given <tt>radius</tt>.

 <pre>
 template&lt;std::size_t Dims, typename RandomNumberGenerator = minstd_rand&gt;
 class ball_topology : public <a href="#convex_topology">convex_topology</a>&lt;Dims&gt;
 {
 public:
   explicit ball_topology(double radius = 1.0);
   ball_topology(RandomNumberGenerator& gen, double radius = 1.0);
   point_type random_point() const;
 };
 </pre>

 <a name="circle_topology"><h3>Class template <tt>circle_topology</tt></h3></a>

 <p>Class template <tt>circle_topology</tt> is a two-dimensional
 ball topology.

 <pre>
 template&lt;typename RandomNumberGenerator = minstd_rand&gt;
 class circle_topology : public <a href="#ball_topology">ball_topology</a>&lt;2, RandomNumberGenerator&gt;
 {
  public:
   explicit circle_topology(double radius = 1.0);
   circle_topology(RandomNumberGenerator& gen, double radius = 1.0);
 };
 </pre>

 <a name="sphere_topology"><h3>Class template <tt>sphere_topology</tt></h3></a>

 <p>Class template <tt>sphere_topology</tt> is a two-dimensional
 ball topology.

 <pre>
 template&lt;typename RandomNumberGenerator = minstd_rand&gt;
 class sphere_topology : public <a href="#ball_topology">ball_topology</a>&lt;3, RandomNumberGenerator&gt;
 {
  public:
   explicit sphere_topology(double radius = 1.0);
   sphere_topology(RandomNumberGenerator& gen, double radius = 1.0);
 };
 </pre>

 <a name="heart_topology"><h3>Class template <tt>heart_topology</tt></h3></a>

 <p>Class template <tt>heart_topology</tt> is topology in the shape of
 a heart. It serves as an example of a non-convex, nontrivial topology
 for layout.

 <pre>
 template&lt;typename RandomNumberGenerator = minstd_rand&gt;
 class heart_topology
 {
  public:
   typedef <em>unspecified</em> point_type;

   heart_topology();
   heart_topology(RandomNumberGenerator& gen);
   point_type random_point() const;
   double distance(point_type a, point_type b) const;
   point_type move_position_toward(point_type a, double fraction, point_type b) const;
 };
 </pre>

 <br>
 <HR>
 <TABLE>
 <TR valign=top>
 <TD nowrap>Copyright &copy; 2004, 2010 Trustees of Indiana University</TD><TD>
 Jeremiah Willcock, Indiana University (<script language="Javascript">address("osl.iu.edu", "jewillco")</script>)<br>
 <A HREF="http://www.boost.org/people/doug_gregor.html">Doug Gregor</A>, Indiana University (<script language="Javascript">address("cs.indiana.edu", "dgregor")</script>)<br>
   <A HREF="http://www.osl.iu.edu/~lums">Andrew Lumsdaine</A>,
 Indiana University (<script language="Javascript">address("osl.iu.edu", "lums")</script>)
 </TD></TR></TABLE>

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	<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
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	<IMG SRC="../../../boost.png"
	ALT="C++ Boost" width="277" height="86">

	<BR Clear>

	<H1>
	Topologies for Graph Drawing
	</H1>

	<P>

	<h3>Synopsis</h3>

	<pre>
	template<std::size_t Dims> class <a href="#convex_topology">convex_topology</a>;
	template<std::size_t Dims, typename RandomNumberGenerator = minstd_rand> class <a href="#hypercube_topology">hypercube_topology</a>;
	template<typename RandomNumberGenerator = minstd_rand> class <a href="#square_topology">square_topology</a>;
	template<typename RandomNumberGenerator = minstd_rand> class <a href="#cube_topology">cube_topology</a>;
	template<std::size_t Dims, typename RandomNumberGenerator = minstd_rand> class <a href="#ball_topology">ball_topology</a>;
	template<typename RandomNumberGenerator = minstd_rand> class <a href="#circle_topology">circle_topology</a>;
	template<typename RandomNumberGenerator = minstd_rand> class <a href="#sphere_topology">sphere_topology</a>;
	template<typename RandomNumberGenerator = minstd_rand> class <a href="#heart_topology">heart_topology</a>;
	</pre>

	<h3>Summary</h3>

	<p> <a href="#topologies">Various topologies</a> are provided that
	produce different, interesting results for graph layout algorithms. The <a
	href="#square_topology">square topology</a> can be used for normal
	display of graphs or distributing vertices for parallel computation on
	a process array, for instance. Other topologies, such as the <a
	href="#sphere_topology">sphere topology</a> (or N-dimensional <a
	href="#ball_topology">ball topology</a>) make sense for different
	problems, whereas the <a href="#heart_topology">heart topology</a> is
	just plain fun. One can also <a href="#topology-concept">define a
	topology</a> to suit other particular needs. <br>

	<a name="topologies"><h3>Topologies</h3></a>
	A topology is a description of a space on which layout can be
	performed. Some common two, three, and multidimensional topologies
	are provided, or you may create your own so long as it meets the
	requirements of the <a href="#topology-concept">Topology concept</a>.

	<a name="topology-concept"><h4>Topology Concept</h4></a> Let
	<tt>Topology</tt> be a model of the Topology concept and let
	<tt>space</tt> be an object of type <tt>Topology</tt>. <tt>p1</tt> and
	<tt>p2</tt> are objects of associated type <tt>point_type</tt> (see
	below). The following expressions must be valid:

	<table border="1">
	<tr>
	<th>Expression</th>
	<th>Type</th>
	<th>Description</th>
	</tr>
	<tr>
	<td><tt>Topology::point_type</tt></td>
	<td>type</td>
	<td>The type of points in the space.</td>
	</tr>
	<tr>
	<td><tt>space.random_point()</tt></td>
	<td>point_type</td>
	<td>Returns a random point (usually uniformly distributed) within
	the space.</td>
	</tr>
	<tr>
	<td><tt>space.distance(p1, p2)</tt></td>
	<td>double</td>
	<td>Get a quantity representing the distance between <tt>p1</tt>
	and <tt>p2</tt> using a path going completely inside the space.
	This only needs to have the same < relation as actual
	distances, and does not need to satisfy the other properties of a
	norm in a Banach space.</td>
	</tr>
	<tr>
	<td><tt>space.move_position_toward(p1, fraction, p2)</tt></td>
	<td>point_type</td>
	<td>Returns a point that is a fraction of the way from <tt>p1</tt>
	to <tt>p2</tt>, moving along a "line" in the space according to
	the distance measure. <tt>fraction</tt> is a <tt>double</tt>
	between 0 and 1, inclusive.</td>
	</tr>
	</table>

	<a name="convex_topology"><h3>Class template <tt>convex_topology</tt></h3></a>

	<p>Class template <tt>convex_topology</tt> implements the basic
	distance and point movement functions for any convex topology in
	<tt>Dims</tt> dimensions. It is not itself a topology, but is intended
	as a base class that any convex topology can derive from. The derived
	topology need only provide a suitable <tt>random_point</tt> function
	that returns a random point within the space.

	<pre>
	template<std::size_t Dims>
	class convex_topology
	{
	struct point
	{
	point() { }
	double& operator[](std::size_t i) {return values[i];}
	const double& operator[](std::size_t i) const {return values[i];}

	private:
	double values[Dims];
	};

	public:
	typedef point point_type;

	double distance(point a, point b) const;
	point move_position_toward(point a, double fraction, point b) const;
	};
	</pre>

	<a name="hypercube_topology"><h3>Class template <tt>hypercube_topology</tt></h3></a>

	<p>Class template <tt>hypercube_topology</tt> implements a
	<tt>Dims</tt>-dimensional hypercube. It is a convex topology whose
	points are drawn from a random number generator of type
	<tt>RandomNumberGenerator</tt>. The <tt>hypercube_topology</tt> can
	be constructed with a given random number generator; if omitted, a
	new, default-constructed random number generator will be used. The
	resulting layout will be contained within the hypercube, whose sides
	measure 2*<tt>scaling</tt> long (points will fall in the range
	[-<tt>scaling</tt>, <tt>scaling</tt>] in each dimension).

	<pre>
	template<std::size_t Dims, typename RandomNumberGenerator = minstd_rand>
	class hypercube_topology : public <a href="#convex_topology">convex_topology</a><Dims>
	{
	public:
	explicit hypercube_topology(double scaling = 1.0);
	hypercube_topology(RandomNumberGenerator& gen, double scaling = 1.0);
	point_type random_point() const;
	};
	</pre>

	<a name="square_topology"><h3>Class template <tt>square_topology</tt></h3></a>

	<p>Class template <tt>square_topology</tt> is a two-dimensional
	hypercube topology.

	<pre>
	template<typename RandomNumberGenerator = minstd_rand>
	class square_topology : public <a href="#hypercube_topology">hypercube_topology</a><2, RandomNumberGenerator>
	{
	public:
	explicit square_topology(double scaling = 1.0);
	square_topology(RandomNumberGenerator& gen, double scaling = 1.0);
	};
	</pre>

	<a name="cube_topology"><h3>Class template <tt>cube_topology</tt></h3></a>

	<p>Class template <tt>cube_topology</tt> is a two-dimensional
	hypercube topology.

	<pre>
	template<typename RandomNumberGenerator = minstd_rand>
	class cube_topology : public <a href="#hypercube_topology">hypercube_topology</a><3, RandomNumberGenerator>
	{
	public:
	explicit cube_topology(double scaling = 1.0);
	cube_topology(RandomNumberGenerator& gen, double scaling = 1.0);
	};
	</pre>

	<a name="ball_topology"><h3>Class template <tt>ball_topology</tt></h3></a>

	<p>Class template <tt>ball_topology</tt> implements a
	<tt>Dims</tt>-dimensional ball. It is a convex topology whose points
	are drawn from a random number generator of type
	<tt>RandomNumberGenerator</tt> but reside inside the ball. The
	<tt>ball_topology</tt> can be constructed with a given random number
	generator; if omitted, a new, default-constructed random number
	generator will be used. The resulting layout will be contained within
	the ball with the given <tt>radius</tt>.

	<pre>
	template<std::size_t Dims, typename RandomNumberGenerator = minstd_rand>
	class ball_topology : public <a href="#convex_topology">convex_topology</a><Dims>
	{
	public:
	explicit ball_topology(double radius = 1.0);
	ball_topology(RandomNumberGenerator& gen, double radius = 1.0);
	point_type random_point() const;
	};
	</pre>

	<a name="circle_topology"><h3>Class template <tt>circle_topology</tt></h3></a>

	<p>Class template <tt>circle_topology</tt> is a two-dimensional
	ball topology.

	<pre>
	template<typename RandomNumberGenerator = minstd_rand>
	class circle_topology : public <a href="#ball_topology">ball_topology</a><2, RandomNumberGenerator>
	{
	public:
	explicit circle_topology(double radius = 1.0);
	circle_topology(RandomNumberGenerator& gen, double radius = 1.0);
	};
	</pre>

	<a name="sphere_topology"><h3>Class template <tt>sphere_topology</tt></h3></a>

	<p>Class template <tt>sphere_topology</tt> is a two-dimensional
	ball topology.

	<pre>
	template<typename RandomNumberGenerator = minstd_rand>
	class sphere_topology : public <a href="#ball_topology">ball_topology</a><3, RandomNumberGenerator>
	{
	public:
	explicit sphere_topology(double radius = 1.0);
	sphere_topology(RandomNumberGenerator& gen, double radius = 1.0);
	};
	</pre>

	<a name="heart_topology"><h3>Class template <tt>heart_topology</tt></h3></a>

	<p>Class template <tt>heart_topology</tt> is topology in the shape of
	a heart. It serves as an example of a non-convex, nontrivial topology
	for layout.

	<pre>
	template<typename RandomNumberGenerator = minstd_rand>
	class heart_topology
	{
	public:
	typedef <em>unspecified</em> point_type;

	heart_topology();
	heart_topology(RandomNumberGenerator& gen);
	point_type random_point() const;
	double distance(point_type a, point_type b) const;
	point_type move_position_toward(point_type a, double fraction, point_type b) const;
	};
	</pre>

	<br>
	<HR>
	<TABLE>
	<TR valign=top>
	<TD nowrap>Copyright © 2004, 2010 Trustees of Indiana University</TD><TD>
	Jeremiah Willcock, Indiana University (<script language="Javascript">address("osl.iu.edu", "jewillco")</script>)<br>
	<A HREF="http://www.boost.org/people/doug_gregor.html">Doug Gregor</A>, Indiana University (<script language="Javascript">address("cs.indiana.edu", "dgregor")</script>)<br>
	<A HREF="http://www.osl.iu.edu/~lums">Andrew Lumsdaine</A>,
	Indiana University (<script language="Javascript">address("osl.iu.edu", "lums")</script>)
	</TD></TR></TABLE>

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	</HTML>