CVT
Centroidal Voronoi Tessellations

CVT is a C++ library which creates Centroidal Voronoi Tessellation (CVT) datasets.

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.

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 USER routine in the CVT library, and then specify the appropriate values init=3 and sample=3.

The USER routine returns a set of sample points from the region of interest. The default USER 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.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

CVT is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

BOX_BEHNKEN, 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;

CCVT_BOX, a C++ program which computes a CVT with some points forced to lie on the boundary.

CVT, a dataset directory which contains files describing a number of CVT's.

CVT_DATASET, a C++ program which creates a CVT dataset.

FAURE, a C++ library which computes Faure sequences.

GRID, a C++ library which computes points on a grid.

HALTON, a C++ library which computes Halton sequences.

HAMMERSLEY, a C++ library which computes Hammersley sequences.

HEX_GRID, a C++ library which computes sets of points in a 2D hexagonal grid.

IHS, a C++ library which computes improved Latin Hypercube datasets.

LATIN_CENTER, a C++ library which computes Latin square data choosing the center value.

LATIN_EDGE, a C++ library which computes Latin square data choosing the edge value.

LATIN_RANDOM, a C++ library which computes Latin square data choosing a random value in the square.

NIEDERREITER2, a C++ library which computes Niederreiter sequences with base 2.

NORMAL, a C++ library which computes elements of a sequence of pseudorandom normally distributed values.

SOBOL, a C++ library which computes Sobol sequences.

UNIFORM, a C++ library which computes uniform random values.

VAN_DER_CORPUT, a C++ library which computes van der Corput sequences.

Reference:

1. Franz Aurenhammer,
   Voronoi diagrams - a study of a fundamental geometric data structure,
   ACM Computing Surveys,
   Volume 23, Number 3, pages 345-405, September 1991.
2. Paul Bratley, Bennett Fox, Linus Schrage,
   A Guide to Simulation,
   Springer Verlag, pages 201-202, 1983.
3. John Burkardt, Max Gunzburger, Janet Peterson, Rebecca Brannon,
   User Manual and Supporting Information for Library of Codes for Centroidal Voronoi Placement and Associated Zeroth, First, and Second Moment Determination,
   Sandia National Laboratories Technical Report SAND2002-0099,
   February 2002.
4. Qiang Du, Vance Faber, Max Gunzburger,
   Centroidal Voronoi Tessellations: Applications and Algorithms,
   SIAM Review,
   Volume 41, 1999, pages 637-676.
5. Bennett Fox,
   Algorithm 647:
   Implementation and Relative Efficiency of Quasirandom Sequence Generators,
   ACM Transactions on Mathematical Software,
   Volume 12, Number 4, pages 362-376, 1986.
6. John Halton,
   On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals,
   Numerische Mathematik,
   Volume 2, pages 84-90, 1960.
7. Lili Ju, Qiang Du, Max Gunzburger,
   Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations,
   Parallel Computing,
   Volume 28, 2002, pages 1477-1500.

Source Code:

• cvt.cpp, the source code;
• cvt.hpp, the include file;

Examples and Tests:

• cvt_test.cpp, a sample calling program.
• cvt_test.txt, the output file.
• cvt_circle.txt, a set of 100 CVT points in a circle, generated using the built in USER routine.
• cvt_circle.png, a PNG image of the 100 CVT points in a circle.

List of Routines:

• CH_CAP capitalizes a single character.
• CH_EQI is true if two characters are equal, disregarding case.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• CVT computes a Centroidal Voronoi Tessellation.
• CVT_ENERGY computes the CVT energy of a dataset.
• CVT_ITERATE takes one step of the CVT iteration.
• CVT_SAMPLE returns sample points.
• CVT_WRITE writes a CVT dataset to a file.
• DATA_READ reads generator coordinate data from a file.
• DIGIT_TO_CH returns the base 10 digit character corresponding to a digit.
• FIND_CLOSEST finds the nearest R point to each S point.
• GET_SEED returns a random seed for the random number generator.
• HALHAM_LEAP_CHECK checks LEAP for a Halton or Hammersley sequence.
• HALHAM_N_CHECK checks N for a Halton or Hammersley sequence.
• HALHAM_DIM_NUM_CHECK checks DIM_NUM for a Halton or Hammersley sequence.
• HALHAM_SEED_CHECK checks SEED for a Halton or Hammersley sequence.
• HALHAM_STEP_CHECK checks STEP for a Halton or Hammersley sequence.
• HALTON_BASE_CHECK checks BASE for a Halton sequence.
• I4_LOG_10 returns the whole part of the logarithm base 10 of an integer.
• I4_MAX returns the maximum of two integers.
• I4_MIN returns the smaller of two integers.
• I4_TO_HALTON_SEQUENCE computes N elements of a leaped Halton subsequence.
• I4_TO_S converts an integer to a string.
• PRIME returns any of the first PRIME_MAX prime numbers.
• R8_EPSILON returns the R8 round off unit.
• R8_HUGE returns a "huge" R8.
• R8MAT_TRANSPOSE_PRINT prints an R8MAT, transposed.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
• R8MAT_UNIFORM_01 returns a unit pseudorandom R8MAT.
• RANDOM_INITIALIZE initializes the RANDOM random number generator.
• S_EQI reports whether two strings are equal, ignoring case.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING returns the current YMDHMS date as a string.
• TUPLE_NEXT_FAST computes the next element of a tuple space, "fast".
• USER samples points in a user-specified region with given density.

You can go up one level to the C++ source codes.