CVT
Centroidal Voronoi Tessellations

CVT is a FORTRAN90 library which computes Centroidal Voronoi Tessellation (CVT) datasets.

The generation of a CVT dataset is of necessity more complicated than that of a quasirandom dataset. A quasirandom dataset is generally simply the first N values of a deterministic sequence. But to compute a CVT requires an iteration, 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:

CVT, a dataset directory which contains a variety of CVT datasets.

CVT_BASIS, a FORTRAN90 program which seeks to produce a small set of basis vectors from a large set of data vectors.

CVT_DATASET, a FORTRAN90 program which computes a CVT dataset.

CVT_DEMO, a MATLAB library which demonstrates the computation of a CVT.

CVT_FIXED, a FORTRAN90 library which computes CVT's in which some values are to be fixed.

CVTP, a FORTRAN90 library which creates a CVTP, that is, a Centroidal Voronoi Tessellation on a periodic domain.

FAURE, a FORTRAN90 library which computes elements of a Faure quasirandom sequence.

HALTON, a FORTRAN90 library which computes elements of a Halton Quasi Monte Carlo (QMC) sequence, using a simple interface.

HAMMERSLEY, a FORTRAN90 library which computes elements of a Hammersley quasirandom sequence.

HEX_GRID, a FORTRAN90 library which computes elements of a hexagonal grid dataset.

HEX_GRID_ANGLE, a FORTRAN90 library which computes elements of an angled hexagonal grid dataset.

IHS, a FORTRAN90 library which computes elements of an improved distributed Latin hypercube dataset.

LATIN_CENTER, a FORTRAN90 library which computes elements of a Latin Hypercube dataset, choosing center points.

LATIN_EDGE, a FORTRAN90 library which computes elements of a Latin Hypercube dataset, choosing edge points.

LATIN_RANDOM, a FORTRAN90 library which computes elements of a Latin Hypercube dataset, choosing points at random.

LATTICE_RULE, a FORTRAN90 library which approximates multidimensional integrals using lattice rules.

LCVT, a FORTRAN90 library which computes a latinized Centroidal Voronoi Tessellation.

NIEDERREITER2, a FORTRAN90 library which computes elements of a Niederreiter quasirandom sequence with base 2.

NORMAL, a FORTRAN90 library which computes elements of a sequence of pseudorandom normally distributed values.

SANDIA_CVT, a FORTRAN90 library which computes CVT's.

SOBOL, a FORTRAN90 library which computes elements of a Sobol quasirandom sequence.

UNIFORM, a FORTRAN90 library which computes elements of a uniform pseudorandom sequence.

VAN_DER_CORPUT, a FORTRAN90 library which computes elements of a van der Corput quasirandom sequence.

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.f90, the source code;

Examples and Tests:

• cvt_test.f90, 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 a case insensitive comparison of two characters for equality.
• 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.
• FIND_CLOSEST finds the nearest R point to each S point.
• GET_SEED returns a seed for the random number generator.
• GET_UNIT returns a free FORTRAN unit number.
• 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_TO_HALTON_SEQUENCE computes N elements of a leaped Halton subsequence.
• I4VEC_TRANSPOSE_PRINT prints an integer vector "transposed".
• PRIME returns any of the first PRIME_MAX prime numbers.
• R8MAT_TRANSPOSE_PRINT prints a real matrix, transposed.
• R8MAT_TRANSPOSE_PRINT_SOME prints some of a real matrix, transposed.
• R8MAT_UNIFORM_01 fills an array with pseudorandom numbers.
• RANDOM_INITIALIZE initializes the FORTRAN90 random number seed.
• S_CAP replaces any lowercase letters by uppercase ones in a string.
• S_EQI is a case insensitive comparison of two strings for equality.
• S_TO_I4 reads an integer value from a string.
• 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 writes the current YMDHMS date into 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 FORTRAN90 source codes.