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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&amp; operator[](std::size_t i) {return values[i];}
    const double&amp; 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&amp; 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&amp; 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 three-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&amp; 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&amp; 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&amp; 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 three-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&amp; 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&amp; 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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	9	<Head>
	10	<Title>Boost Graph Library: Topologies for Graph Drawing</Title>
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	27	ALT="C++ Boost" width="277" height="86">
	28
	29	<BR Clear>
	30
	31	<H1>
	32	Topologies for Graph Drawing
	33	</H1>
	34
	35	<P>
	36
	37	<h3>Synopsis</h3>
	38
	39	<pre>
	40	template<std::size_t Dims> class <a href="#convex_topology">convex_topology</a>;
	41	template<std::size_t Dims, typename RandomNumberGenerator = minstd_rand> class <a href="#hypercube_topology">hypercube_topology</a>;
	42	template<typename RandomNumberGenerator = minstd_rand> class <a href="#square_topology">square_topology</a>;
	43	template<typename RandomNumberGenerator = minstd_rand> class <a href="#cube_topology">cube_topology</a>;
	44	template<std::size_t Dims, typename RandomNumberGenerator = minstd_rand> class <a href="#ball_topology">ball_topology</a>;
	45	template<typename RandomNumberGenerator = minstd_rand> class <a href="#circle_topology">circle_topology</a>;
	46	template<typename RandomNumberGenerator = minstd_rand> class <a href="#sphere_topology">sphere_topology</a>;
	47	template<typename RandomNumberGenerator = minstd_rand> class <a href="#heart_topology">heart_topology</a>;
	48	</pre>
	49
	50	<h3>Summary</h3>
	51
	52	<p> <a href="#topologies">Various topologies</a> are provided that
	53	produce different, interesting results for graph layout algorithms. The <a
	54	href="#square_topology">square topology</a> can be used for normal
	55	display of graphs or distributing vertices for parallel computation on
	56	a process array, for instance. Other topologies, such as the <a
	57	href="#sphere_topology">sphere topology</a> (or N-dimensional <a
	58	href="#ball_topology">ball topology</a>) make sense for different
	59	problems, whereas the <a href="#heart_topology">heart topology</a> is
	60	just plain fun. One can also <a href="#topology-concept">define a
	61	topology</a> to suit other particular needs. <br>
	62
	63	<a name="topologies"><h3>Topologies</h3></a>
	64	A topology is a description of a space on which layout can be
65	performed. Some common two, three, and multidimensional topologies
66	are provided, or you may create your own so long as it meets the
67	requirements of the <a href="#topology-concept">Topology concept</a>.
68
69	<a name="topology-concept"><h4>Topology Concept</h4></a> Let
70	<tt>Topology</tt> be a model of the Topology concept and let
71	<tt>space</tt> be an object of type <tt>Topology</tt>. <tt>p1</tt> and
72	<tt>p2</tt> are objects of associated type <tt>point_type</tt> (see
73	below). The following expressions must be valid:
74
75	<table border="1">
76	<tr>
77	<th>Expression</th>
78	<th>Type</th>
79	<th>Description</th>
80	</tr>
81	<tr>
82	<td><tt>Topology::point_type</tt></td>
83	<td>type</td>
84	<td>The type of points in the space.</td>
85	</tr>
86	<tr>
87	<td><tt>space.random_point()</tt></td>
88	<td>point_type</td>
89	<td>Returns a random point (usually uniformly distributed) within
90	the space.</td>
91	</tr>
92	<tr>
93	<td><tt>space.distance(p1, p2)</tt></td>
94	<td>double</td>
95	<td>Get a quantity representing the distance between <tt>p1</tt>
96	and <tt>p2</tt> using a path going completely inside the space.
97	This only needs to have the same < relation as actual
98	distances, and does not need to satisfy the other properties of a
99	norm in a Banach space.</td>
100	</tr>
101	<tr>
102	<td><tt>space.move_position_toward(p1, fraction, p2)</tt></td>
103	<td>point_type</td>
104	<td>Returns a point that is a fraction of the way from <tt>p1</tt>
105	to <tt>p2</tt>, moving along a "line" in the space according to
106	the distance measure. <tt>fraction</tt> is a <tt>double</tt>
107	between 0 and 1, inclusive.</td>
108	</tr>
109	</table>
110
111	<a name="convex_topology"><h3>Class template <tt>convex_topology</tt></h3></a>
112
113	<p>Class template <tt>convex_topology</tt> implements the basic
114	distance and point movement functions for any convex topology in
115	<tt>Dims</tt> dimensions. It is not itself a topology, but is intended
116	as a base class that any convex topology can derive from. The derived
117	topology need only provide a suitable <tt>random_point</tt> function
118	that returns a random point within the space.
119
120	<pre>
121	template<std::size_t Dims>
122	class convex_topology
123	{
124	struct point
125	{
126	point() { }
127	double& operator[](std::size_t i) {return values[i];}
128	const double& operator[](std::size_t i) const {return values[i];}
129
130	private:
131	double values[Dims];
132	};
133
134	public:
135	typedef point point_type;
136
137	double distance(point a, point b) const;
138	point move_position_toward(point a, double fraction, point b) const;
139	};
140	</pre>
141
142	<a name="hypercube_topology"><h3>Class template <tt>hypercube_topology</tt></h3></a>
143
144	<p>Class template <tt>hypercube_topology</tt> implements a
145	<tt>Dims</tt>-dimensional hypercube. It is a convex topology whose
146	points are drawn from a random number generator of type
147	<tt>RandomNumberGenerator</tt>. The <tt>hypercube_topology</tt> can
148	be constructed with a given random number generator; if omitted, a
149	new, default-constructed random number generator will be used. The
150	resulting layout will be contained within the hypercube, whose sides
151	measure 2*<tt>scaling</tt> long (points will fall in the range
152	[-<tt>scaling</tt>, <tt>scaling</tt>] in each dimension).
153
154	<pre>
155	template<std::size_t Dims, typename RandomNumberGenerator = minstd_rand>
156	class hypercube_topology : public <a href="#convex_topology">convex_topology</a><Dims>
157	{
158	public:
159	explicit hypercube_topology(double scaling = 1.0);
160	hypercube_topology(RandomNumberGenerator& gen, double scaling = 1.0);
161	point_type random_point() const;
162	};
163	</pre>
164
165	<a name="square_topology"><h3>Class template <tt>square_topology</tt></h3></a>
166
167	<p>Class template <tt>square_topology</tt> is a two-dimensional
168	hypercube topology.
169
170	<pre>
171	template<typename RandomNumberGenerator = minstd_rand>
172	class square_topology : public <a href="#hypercube_topology">hypercube_topology</a><2, RandomNumberGenerator>
173	{
174	public:
175	explicit square_topology(double scaling = 1.0);
176	square_topology(RandomNumberGenerator& gen, double scaling = 1.0);
177	};
178	</pre>
179
180	<a name="cube_topology"><h3>Class template <tt>cube_topology</tt></h3></a>
181
182	<p>Class template <tt>cube_topology</tt> is a three-dimensional
183	hypercube topology.
184
185	<pre>
186	template<typename RandomNumberGenerator = minstd_rand>
187	class cube_topology : public <a href="#hypercube_topology">hypercube_topology</a><3, RandomNumberGenerator>
188	{
189	public:
190	explicit cube_topology(double scaling = 1.0);
191	cube_topology(RandomNumberGenerator& gen, double scaling = 1.0);
192	};
193	</pre>
194
195	<a name="ball_topology"><h3>Class template <tt>ball_topology</tt></h3></a>
196
197	<p>Class template <tt>ball_topology</tt> implements a
198	<tt>Dims</tt>-dimensional ball. It is a convex topology whose points
199	are drawn from a random number generator of type
200	<tt>RandomNumberGenerator</tt> but reside inside the ball. The
201	<tt>ball_topology</tt> can be constructed with a given random number
202	generator; if omitted, a new, default-constructed random number
203	generator will be used. The resulting layout will be contained within
204	the ball with the given <tt>radius</tt>.
205
206	<pre>
207	template<std::size_t Dims, typename RandomNumberGenerator = minstd_rand>
208	class ball_topology : public <a href="#convex_topology">convex_topology</a><Dims>
209	{
210	public:
211	explicit ball_topology(double radius = 1.0);
212	ball_topology(RandomNumberGenerator& gen, double radius = 1.0);
213	point_type random_point() const;
214	};
215	</pre>
216
217	<a name="circle_topology"><h3>Class template <tt>circle_topology</tt></h3></a>
218
219	<p>Class template <tt>circle_topology</tt> is a two-dimensional
220	ball topology.
221
222	<pre>
223	template<typename RandomNumberGenerator = minstd_rand>
224	class circle_topology : public <a href="#ball_topology">ball_topology</a><2, RandomNumberGenerator>
225	{
226	public:
227	explicit circle_topology(double radius = 1.0);
228	circle_topology(RandomNumberGenerator& gen, double radius = 1.0);
229	};
230	</pre>
231
232	<a name="sphere_topology"><h3>Class template <tt>sphere_topology</tt></h3></a>
233
234	<p>Class template <tt>sphere_topology</tt> is a three-dimensional
235	ball topology.
236
237	<pre>
238	template<typename RandomNumberGenerator = minstd_rand>
239	class sphere_topology : public <a href="#ball_topology">ball_topology</a><3, RandomNumberGenerator>
240	{
241	public:
242	explicit sphere_topology(double radius = 1.0);
243	sphere_topology(RandomNumberGenerator& gen, double radius = 1.0);
244	};
245	</pre>
246
247	<a name="heart_topology"><h3>Class template <tt>heart_topology</tt></h3></a>
248
249	<p>Class template <tt>heart_topology</tt> is topology in the shape of
250	a heart. It serves as an example of a non-convex, nontrivial topology
251	for layout.
252
253	<pre>
254	template<typename RandomNumberGenerator = minstd_rand>
255	class heart_topology
256	{
257	public:
258	typedef <em>unspecified</em> point_type;
259
260	heart_topology();
261	heart_topology(RandomNumberGenerator& gen);
262	point_type random_point() const;
263	double distance(point_type a, point_type b) const;
264	point_type move_position_toward(point_type a, double fraction, point_type b) const;
265	};
266	</pre>
267
268	<br>
269	<HR>
270	<TABLE>
271	<TR valign=top>
272	<TD nowrap>Copyright © 2004, 2010 Trustees of Indiana University</TD><TD>
273	Jeremiah Willcock, Indiana University (<script language="Javascript">address("osl.iu.edu", "jewillco")</script>)<br>
274	<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>
275	<A HREF="http://www.osl.iu.edu/~lums">Andrew Lumsdaine</A>,
276	Indiana University (<script language="Javascript">address("osl.iu.edu", "lums")</script>)
277	</TD></TR></TABLE>
278
279	</BODY>
280	</HTML>