CVT_DATASET
Generate CVT Datasets

CVT_DATASET is a set of MATLAB routines which create a CVT dataset and write it to a file.

The program is interactive, and allows the user to choose the parameters that define the dataset.

Normally, data is computed in the unit hypercube, with uniform density. However, if you wish to work in a more interesting geometry, or to control the density function, it is necessary to modify the USER routine in the CVT library, and then direct CVT_DATASET to use that routine for initialization and sampling.

The data that the user may set includes:

• DIM_NUM, the spatial dimension,
• N, the number of points to generate,
• SEED, a seed to use for random number generation;
• INIT, initialize the points:
  • file, by reading data from file;
  • 'GRID', picking points from a grid;
  • 'HALTON', from a Halton sequence;
  • 'RANDOM', using FORTRAN RANDOM function;
  • 'UNIFORM', using a simple uniform RNG;
  • 'USER', by calling the "user" routine;
• IT_MAX, the maximum number of iterations;
• IT_FIXED, the number of iterations for which each set of sample points should be used (between 1 and IT_MAX);
• SAMPLE, how to conduct the sampling:
  • 'GRID', picking points from a grid;
  • 'HALTON', from a Halton sequence;
  • 'RANDOM', using FORTRAN RANDOM function;
  • 'UNIFORM', using a simple uniform RNG;
  • 'USER', by calling the "user" routine;
• SAMPLE_NUM, the number of sampling points;
• BATCH, the number of sampling points to create at one time (between 1 and SAMPLE_NUM);
• OUTPUT, a file in which to store the data.

A "CVT" is a Centroidal Voronoi Tessellation. Essentially, a CVT is a set of sample points in a (finite) region with the property that each point is the centroid of its Voronoi subregion. A "random" set of sample points will not have this property. However, it is possible to begin with a random set of sample points, and drive it towards a CVT set, by applying an iterative refinement process.

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 subregions. 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.

A reasonable set of input data might be:

        2             spatial dimension is 2
        10            compute 10 points
        123456789     seed for random numbers
        'uniform'     initialize by UNIFORM
        40            40 iterations
        0.0           zero tolerance; won't stop early
        'uniform'     sample using UNIFORM
        10000         use 10,000 sample points on each iteration
        1000          create 1,000 sample points at a time
        -1            stop; don't want to define another set.
      

Once these parameters are set, the program generates the data and writes it to a file. The user may then specify another set of input data, or terminate the program.

Related Data and Programs:

CCVT_BOX is a MATLAB program which constructs a modified CVT in which some points are forced to lie on the boundary.

CCVT_REFLECT is a MATLAB program which tries to construct a modified CVT in which some points are forced to lie on the boundary, using a reflection idea.

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

CVT is a dataset directory which contains a variety of examples of CVT datasets.

CVT_1D_NONUNIFORM is a MATLAB program which constructs a CVT in one dimension, under a nonuniform density function.

CVT_DATASET is also available in a C++ version. and a FORTRAN90 version.

CVT_DEMO is a MATLAB program which is an interactic graphic demonstration of a CVT calculation.

TABLE is the format used for the output of CVT_DATASET.

TABLE_DISCREPANCY is an executable C++ program which can read a TABLE file of points (presumed to lie in the unit hypercube) and compute bounds on the star discrepancy, a measure of dispersion.

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,
   ../../pdf/aurenhammer.pdf
2. 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.
   Online ordering
3. Qiang Du, Vance Faber, Max Gunzburger,
   Centroidal Voronoi Tessellations: Applications and Algorithms,
   SIAM Review,
   Volume 41, 1999, pages 637-676.
4. 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_dataset.m, the main program;

Examples and Tests:

Test 1 computes 85 CVT points in 2 dimensions, using uniform initialization, a seed of 123456789, 40 iterations, a zero tolerance, uniform sampling, 10,000 sample points in batches of 1000:

• input01.txt, the input defining the dataset;
• output01.txt, the printed response to the commands;
• cvt01.txt, the dataset created by the commands;
• cvt01.png, a PNG image of the dataset;

Test 2 repeats test 1, but with 80 iterations:

• input02.txt, the input defining the dataset;
• output02.txt, the printed response to the commands;
• cvt02.txt, the dataset created by the commands;
• cvt02.png, a PNG image of the dataset;

Test 3 repeats test 1, but with 1,000,000 sample points:

• input03.txt, the input defining the dataset;
• output03.txt, the printed response to the commands;
• cvt03.txt, the dataset created by the commands;
• cvt03.png, a PNG image of the dataset;

Test 4 repeats test 1, but with Halton sampling:

• input04.txt, the input defining the dataset;
• output04.txt, the printed response to the commands;
• cvt04.txt, the dataset created by the commands;
• cvt04.png, a PNG image of the dataset;

Test 5 repeats test 1, but with Grid sampling:

• input05.txt, the input defining the dataset;
• output05.txt, the printed response to the commands;
• cvt05.txt, the dataset created by the commands;
• cvt05.png, a PNG image of the dataset;

Test 6 repeats test 1, but with Random sampling:

• input06.txt, the input defining the dataset;
• output06.txt, the printed response to the commands;
• cvt06.txt, the dataset created by the commands;
• cvt06.png, a PNG image of the dataset;

Test 7 repeats test 1, but with a seed of 987654321:

• input07.txt, the input defining the dataset;
• output07.txt, the printed response to the commands;
• cvt07.txt, the dataset created by the commands;
• cvt07.png, a PNG image of the dataset;

Test 8 repeats test 1, but with a batch size of 5:

• input08.txt, the input defining the dataset;
• output08.txt, the printed response to the commands;
• cvt08.txt, the dataset created by the commands;
• cvt08.png, a PNG image of the dataset;

Test 9 computes 100 CVT points in 3 dimensions, using uniform initialization, a seed of 123456789, 40 iterations, a tolerance of 0.000001, uniform sampling, 10,000 sample points in batches of 1000:

• input09.txt, the input defining the dataset;
• output09.txt, the printed response to the commands;
• cvt09.txt, the dataset created by the commands;

Test 10 investigates the unstable CVT formed by a Cartesian grid of 100 points in 2D. Starting from this unstable solution, the iteration proceeds towards a more "hexagonal" pattern :

• input10.txt, the input defining the dataset;
• output10.txt, the printed response to the commands;
• cvt10.txt, the dataset created by the commands;
• cvt10.png, a PNG image of the dataset;

Test 11 shows how the user may specify the initial point locations in a file. 15 points are specified in 2D:

• input11.txt, the input defining the dataset;
• output11.txt, the printed response to the commands;
• initial11.txt, the initial dataset, created by the user;
• initial11.png, a PNG image of the initial dataset;
• cvt11.txt, the dataset created by the commands;
• cvt11.png, a PNG image of the dataset;

Test 12:

• input12.txt, the input defining the dataset;
• output12.txt, the printed response to the commands;
• cvt12.txt, the dataset created by the commands;

Test 13:

• input13.txt, the input defining the dataset;
• output13.txt, the printed response to the commands;
• cvt13.txt, the dataset created by the commands;

Test 14 shows how the user may refer to the USER routine for a different geometry. The default USER routine is set up to sample the unit circle in 2D. 100 points are requested:

• input14.txt, the input defining the dataset;
• output14.txt, the printed response to the commands;
• cvt14.txt, the dataset created by the commands;
• cvt14.png, a PNG image of the dataset;

You can go up one level to the MATLAB source codes.