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<html>
<head>
<title>
CVT - Centroidal Voronoi Tessellations
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
CVT <br> Centroidal Voronoi Tessellations
</h1>
<hr>
<p>
<b>CVT</b>
is a C++ library which
creates Centroidal Voronoi Tessellation (CVT) datasets.
</p>
<p>
The generation of a CVT dataset is of necessity more complicated than
for a quasirandom sequence. An iteration is involved, so there
must be an initial assignment for the generators, and then a
number of iterations. Moreover, in each iteration, estimates must
be made of the volume and location of the Voronoi cells. This is
typically done by Monte Carlo sampling. The accuracy of the resulting
CVT depends in part on the number of sampling points and the number
of iterations taken.
</p>
<p>
The library is mostly used to generate a dataset of points
uniformly distributed in the unit hypersquare. However, a user
may be interested in computations with other geometries or
point densities. To do this, the user needs to replace the
<b>USER</b> routine in the <b>CVT</b> library, and then specify
the appropriate values <b>init=3</b> and <b>sample=3</b>.
</p>
<p>
The <b>USER</b> routine returns a set of sample points from the
region of interest. The default <b>USER</b> routine samples points
uniformly from the unit circle. But other geometries are
easy to set up. Changing the point density simply requires
weighting the sampling in the region.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>CVT</b> is available in
<a href = "../../cpp_src/cvt/cvt.html">a C++ version</a> and
<a href = "../../f_src/cvt/cvt.html">a FORTRAN90 version</a> and
<a href = "../../m_src/cvt/cvt.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/box_behnken/box_behnken.html">
BOX_BEHNKEN</a>,
a C++ library which
computes a Box-Behnken design,
that is, a set of arguments to sample the behavior
of a function of multiple parameters;
</p>
<p>
<a href = "../../cpp_src/ccvt_box/ccvt_box.html">
CCVT_BOX</a>,
a C++ program which
computes a CVT with some points
forced to lie on the boundary.
</p>
<p>
<a href = "../../datasets/cvt/cvt.html">
CVT</a>,
a dataset directory which
contains files describing a number of CVT's.
</p>
<p>
<a href = "../../cpp_src/cvt_dataset/cvt_dataset.html">
CVT_DATASET</a>,
a C++ program which
creates a CVT dataset.
</p>
<p>
<a href = "../../cpp_src/faure/faure.html">
FAURE</a>,
a C++ library which
computes Faure sequences.
</p>
<p>
<a href = "../../cpp_src/grid/grid.html">
GRID</a>,
a C++ library which
computes points on a grid.
</p>
<p>
<a href = "../../cpp_src/halton/halton.html">
HALTON</a>,
a C++ library which
computes Halton
sequences.
</p>
<p>
<a href = "../../cpp_src/hammersley/hammersley.html">
HAMMERSLEY</a>,
a C++ library which
computes Hammersley
sequences.
</p>
<p>
<a href = "../../cpp_src/hex_grid/hex_grid.html">
HEX_GRID</a>,
a C++ library which
computes sets of points in a 2D hexagonal grid.
</p>
<p>
<a href = "../../cpp_src/ihs/ihs.html">
IHS</a>,
a C++ library which
computes improved Latin Hypercube datasets.
</p>
<p>
<a href = "../../cpp_src/latin_center/latin_center.html">
LATIN_CENTER</a>,
a C++ library which
computes Latin square data choosing the center value.
</p>
<p>
<a href = "../../cpp_src/latin_edge/latin_edge.html">
LATIN_EDGE</a>,
a C++ library which
computes Latin square data choosing the edge value.
</p>
<p>
<a href = "../../cpp_src/latin_random/latin_random.html">
LATIN_RANDOM</a>,
a C++ library which
computes Latin square data choosing a random value in the square.
</p>
<p>
<a href = "../../cpp_src/niederreiter2/niederreiter2.html">
NIEDERREITER2</a>,
a C++ library which
computes Niederreiter sequences with base 2.
</p>
<p>
<a href = "../../cpp_src/normal/normal.html">
NORMAL</a>,
a C++ library which
computes elements of a
sequence of pseudorandom normally distributed values.
</p>
<p>
<a href = "../../cpp_src/sobol/sobol.html">
SOBOL</a>,
a C++ library which
computes Sobol sequences.
</p>
<p>
<a href = "../../cpp_src/uniform/uniform.html">
UNIFORM</a>,
a C++ library which
computes uniform random values.
</p>
<p>
<a href = "../../cpp_src/van_der_corput/van_der_corput.html">
VAN_DER_CORPUT</a>,
a C++ library which
computes van der Corput sequences.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Franz Aurenhammer,<br>
Voronoi diagrams -
a study of a fundamental geometric data structure,<br>
ACM Computing Surveys,<br>
Volume 23, Number 3, pages 345-405, September 1991.
</li>
<li>
Paul Bratley, Bennett Fox, Linus Schrage,<br>
A Guide to Simulation,<br>
Springer Verlag, pages 201-202, 1983.
</li>
<li>
John Burkardt, Max Gunzburger, Janet Peterson, Rebecca Brannon,<br>
User Manual and Supporting Information for Library of Codes
for Centroidal Voronoi Placement and Associated Zeroth,
First, and Second Moment Determination,<br>
Sandia National Laboratories Technical Report SAND2002-0099,<br>
February 2002.
</li>
<li>
Qiang Du, Vance Faber, Max Gunzburger,<br>
Centroidal Voronoi Tessellations: Applications and Algorithms,<br>
SIAM Review,<br>
Volume 41, 1999, pages 637-676.
</li>
<li>
Bennett Fox,<br>
Algorithm 647:<br>
Implementation and Relative Efficiency of Quasirandom
Sequence Generators,<br>
ACM Transactions on Mathematical Software,<br>
Volume 12, Number 4, pages 362-376, 1986.
</li>
<li>
John Halton,<br>
On the efficiency of certain quasi-random sequences of points
in evaluating multi-dimensional integrals,<br>
Numerische Mathematik,<br>
Volume 2, pages 84-90, 1960.
</li>
<li>
Lili Ju, Qiang Du, Max Gunzburger,<br>
Probabilistic methods for centroidal Voronoi tessellations
and their parallel implementations,<br>
Parallel Computing,<br>
Volume 28, 2002, pages 1477-1500.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "cvt.cpp">cvt.cpp</a>, the source code;
</li>
<li>
<a href = "cvt.hpp">cvt.hpp</a>, the include file;
</li>
<li>
<a href = "cvt.sh">cvt.sh</a>,
commands to compile the source code;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "cvt_prb.cpp">cvt_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "cvt_prb.sh">cvt_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "cvt_prb_output.txt">cvt_prb_output.txt</a>,
the output file.
</li>
<li>
<a href = "cvt_circle.txt">cvt_circle.txt</a>,
a set of 100 CVT points in a circle, generated using
the built in <b>USER</b> routine.
</li>
<li>
<a href = "cvt_circle.png">cvt_circle.png</a>,
a PNG image of
the 100 CVT points in a circle.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is true if two characters are equal, disregarding case.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>CVT</b> computes a Centroidal Voronoi Tessellation.
</li>
<li>
<b>CVT_ENERGY</b> computes the CVT energy of a dataset.
</li>
<li>
<b>CVT_ITERATE</b> takes one step of the CVT iteration.
</li>
<li>
<b>CVT_SAMPLE</b> returns sample points.
</li>
<li>
<b>CVT_WRITE</b> writes a CVT dataset to a file.
</li>
<li>
<b>DATA_READ</b> reads generator coordinate data from a file.
</li>
<li>
<b>DIGIT_TO_CH</b> returns the base 10 digit character corresponding to a digit.
</li>
<li>
<b>FIND_CLOSEST</b> finds the nearest R point to each S point.
</li>
<li>
<b>GET_SEED</b> returns a random seed for the random number generator.
</li>
<li>
<b>HALHAM_LEAP_CHECK</b> checks LEAP for a Halton or Hammersley sequence.
</li>
<li>
<b>HALHAM_N_CHECK</b> checks N for a Halton or Hammersley sequence.
</li>
<li>
<b>HALHAM_DIM_NUM_CHECK</b> checks DIM_NUM for a Halton or Hammersley sequence.
</li>
<li>
<b>HALHAM_SEED_CHECK</b> checks SEED for a Halton or Hammersley sequence.
</li>
<li>
<b>HALHAM_STEP_CHECK</b> checks STEP for a Halton or Hammersley sequence.
</li>
<li>
<b>HALTON_BASE_CHECK</b> checks BASE for a Halton sequence.
</li>
<li>
<b>I4_LOG_10</b> returns the whole part of the logarithm base 10 of an integer.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two integers.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two integers.
</li>
<li>
<b>I4_TO_HALTON_SEQUENCE</b> computes N elements of a leaped Halton subsequence.
</li>
<li>
<b>I4_TO_S</b> converts an integer to a string.
</li>
<li>
<b>PRIME</b> returns any of the first PRIME_MAX prime numbers.
</li>
<li>
<b>R8_EPSILON</b> returns the R8 round off unit.
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT</b> prints an R8MAT, transposed.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>R8MAT_UNIFORM_01</b> returns a unit pseudorandom R8MAT.
</li>
<li>
<b>RANDOM_INITIALIZE</b> initializes the RANDOM random number generator.
</li>
<li>
<b>S_EQI</b> reports whether two strings are equal, ignoring case.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</li>
<li>
<b>S_TO_R8</b> reads an R8 from a string.
</li>
<li>
<b>S_TO_R8VEC</b> reads an R8VEC from a string.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TIMESTRING</b> returns the current YMDHMS date as a string.
</li>
<li>
<b>TUPLE_NEXT_FAST</b> computes the next element of a tuple space, "fast".
</li>
<li>
<b>USER</b> samples points in a user-specified region with given density.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 10 November 2006.
</i>
<!-- John Burkardt -->
</body>
</html>