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rand.cpp
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1 //----------------------------------------------------------------------
2 // File: rand.cpp
3 // Programmer: Sunil Arya and David Mount
4 // Description: Routines for random point generation
5 // Last modified: 08/04/06 (Version 1.1.1)
6 //----------------------------------------------------------------------
7 // Copyright (c) 1997-2005 University of Maryland and Sunil Arya and
8 // David Mount. All Rights Reserved.
9 //
10 // This software and related documentation is part of the Approximate
11 // Nearest Neighbor Library (ANN). This software is provided under
12 // the provisions of the Lesser GNU Public License (LGPL). See the
13 // file ../ReadMe.txt for further information.
14 //
15 // The University of Maryland (U.M.) and the authors make no
16 // representations about the suitability or fitness of this software for
17 // any purpose. It is provided "as is" without express or implied
18 // warranty.
19 //----------------------------------------------------------------------
20 // History:
21 // Revision 0.1 03/04/98
22 // Initial release
23 // Revision 0.2 03/26/98
24 // Changed random/srandom declarations for SGI's.
25 // Revision 1.0 04/01/05
26 // annClusGauss centers distributed over [-1,1] rather than [0,1]
27 // Added annClusOrthFlats distribution
28 // Changed procedure names to avoid namespace conflicts
29 // Added annClusFlats distribution
30 // Added rand/srand option and fixed annRan0() initialization.
31 // Revision 1.1.1 08/04/06
32 // Added planted distribution
33 //----------------------------------------------------------------------
34 
35 #include "rand.h" // random generator declarations
36 
37 using namespace std; // make std:: accessible
38 
39 //----------------------------------------------------------------------
40 // Globals
41 //----------------------------------------------------------------------
42 int annIdum = 0; // used for random number generation
43 
44 //------------------------------------------------------------------------
45 // annRan0 - (safer) uniform random number generator
46 //
47 // The code given here is taken from "Numerical Recipes in C" by
48 // William Press, Brian Flannery, Saul Teukolsky, and William
49 // Vetterling. The task of the code is to do an additional randomizing
50 // shuffle on the system-supplied random number generator to make it
51 // safer to use.
52 //
53 // Returns a uniform deviate between 0.0 and 1.0 using the
54 // system-supplied routine random() or rand(). Set the global
55 // annIdum to any negative value to initialise or reinitialise
56 // the sequence.
57 //------------------------------------------------------------------------
58 
59 double annRan0()
60 {
61  const int TAB_SIZE = 97; // table size: any large number
62  int j;
63 
64  static double y, v[TAB_SIZE];
65  static int iff = 0;
66  const double RAN_DIVISOR = double(ANN_RAND_MAX + 1UL);
67  if (RAN_DIVISOR < 0) {
68  cout << "RAN_DIVISOR " << RAN_DIVISOR << endl;
69  exit(0);
70  }
71 
72  //--------------------------------------------------------------------
73  // As a precaution against misuse, we will always initialize on the
74  // first call, even if "annIdum" is not set negative. Determine
75  // "maxran", the next integer after the largest representable value
76  // of type int. We assume this is a factor of 2 smaller than the
77  // corresponding value of type unsigned int.
78  //--------------------------------------------------------------------
79 
80  if (annIdum < 0 || iff == 0) { // initialize
81  iff = 1;
82  ANN_SRAND(annIdum); // (re)seed the generator
83  annIdum = 1;
84 
85  for (j = 0; j < TAB_SIZE; j++) // exercise the system routine
86  ANN_RAND(); // (values intentionally ignored)
87 
88  for (j = 0; j < TAB_SIZE; j++) // then save TAB_SIZE-1 values
89  v[j] = ANN_RAND();
90  y = ANN_RAND(); // generate starting value
91  }
92 
93  //--------------------------------------------------------------------
94  // This is where we start if not initializing. Use the previously
95  // saved random number y to get an index j between 1 and TAB_SIZE-1.
96  // Then use the corresponding v[j] for both the next j and as the
97  // output number.
98  //--------------------------------------------------------------------
99 
100  j = int(TAB_SIZE * (y / RAN_DIVISOR));
101  y = v[j];
102  v[j] = ANN_RAND(); // refill the table entry
103  return y / RAN_DIVISOR;
104 }
105 
106 //------------------------------------------------------------------------
107 // annRanInt - generate a random integer from {0,1,...,n-1}
108 //
109 // If n == 0, then -1 is returned.
110 //------------------------------------------------------------------------
111 
112 static int annRanInt(
113  int n)
114 {
115  int r = (int) (annRan0()*n);
116  if (r == n) r--; // (in case annRan0() == 1 or n == 0)
117  return r;
118 }
119 
120 //------------------------------------------------------------------------
121 // annRanUnif - generate a random uniform in [lo,hi]
122 //------------------------------------------------------------------------
123 
124 static double annRanUnif(
125  double lo,
126  double hi)
127 {
128  return annRan0()*(hi-lo) + lo;
129 }
130 
131 //------------------------------------------------------------------------
132 // annRanGauss - Gaussian random number generator
133 // Returns a normally distributed deviate with zero mean and unit
134 // variance, using annRan0() as the source of uniform deviates.
135 //------------------------------------------------------------------------
136 
137 static double annRanGauss()
138 {
139  static int iset=0;
140  static double gset;
141 
142  if (iset == 0) { // we don't have a deviate handy
143  double v1, v2;
144  double r = 2.0;
145  while (r >= 1.0) {
146  //------------------------------------------------------------
147  // Pick two uniform numbers in the square extending from -1 to
148  // +1 in each direction, see if they are in the circle of radius
149  // 1. If not, try again
150  //------------------------------------------------------------
151  v1 = annRanUnif(-1, 1);
152  v2 = annRanUnif(-1, 1);
153  r = v1 * v1 + v2 * v2;
154  }
155  double fac = sqrt(-2.0 * log(r) / r);
156  //-----------------------------------------------------------------
157  // Now make the Box-Muller transformation to get two normal
158  // deviates. Return one and save the other for next time.
159  //-----------------------------------------------------------------
160  gset = v1 * fac;
161  iset = 1; // set flag
162  return v2 * fac;
163  }
164  else { // we have an extra deviate handy
165  iset = 0; // so unset the flag
166  return gset; // and return it
167  }
168 }
169 
170 //------------------------------------------------------------------------
171 // annRanLaplace - Laplacian random number generator
172 // Returns a Laplacian distributed deviate with zero mean and
173 // unit variance, using annRan0() as the source of uniform deviates.
174 //
175 // prob(x) = b/2 * exp(-b * |x|).
176 //
177 // b is chosen to be sqrt(2.0) so that the variance of the Laplacian
178 // distribution [2/(b^2)] becomes 1.
179 //------------------------------------------------------------------------
180 
181 static double annRanLaplace()
182 {
183  const double b = 1.4142136;
184 
185  double laprand = -log(annRan0()) / b;
186  double sign = annRan0();
187  if (sign < 0.5) laprand = -laprand;
188  return(laprand);
189 }
190 
191 //----------------------------------------------------------------------
192 // annUniformPts - Generate uniformly distributed points
193 // A uniform distribution over [-1,1].
194 //----------------------------------------------------------------------
195 
196 void annUniformPts( // uniform distribution
197  ANNpointArray pa, // point array (modified)
198  int n, // number of points
199  int dim) // dimension
200 {
201  for (int i = 0; i < n; i++) {
202  for (int d = 0; d < dim; d++) {
203  pa[i][d] = (ANNcoord) (annRanUnif(-1,1));
204  }
205  }
206 }
207 
208 //----------------------------------------------------------------------
209 // annGaussPts - Generate Gaussian distributed points
210 // A Gaussian distribution with zero mean and the given standard
211 // deviation.
212 //----------------------------------------------------------------------
213 
214 void annGaussPts( // Gaussian distribution
215  ANNpointArray pa, // point array (modified)
216  int n, // number of points
217  int dim, // dimension
218  double std_dev) // standard deviation
219 {
220  for (int i = 0; i < n; i++) {
221  for (int d = 0; d < dim; d++) {
222  pa[i][d] = (ANNcoord) (annRanGauss() * std_dev);
223  }
224  }
225 }
226 
227 //----------------------------------------------------------------------
228 // annLaplacePts - Generate Laplacian distributed points
229 // Generates a Laplacian distribution (zero mean and unit variance).
230 //----------------------------------------------------------------------
231 
232 void annLaplacePts( // Laplacian distribution
233  ANNpointArray pa, // point array (modified)
234  int n, // number of points
235  int dim) // dimension
236 {
237  for (int i = 0; i < n; i++) {
238  for (int d = 0; d < dim; d++) {
239  pa[i][d] = (ANNcoord) annRanLaplace();
240  }
241  }
242 }
243 
244 //----------------------------------------------------------------------
245 // annCoGaussPts - Generate correlated Gaussian distributed points
246 // Generates a Gauss-Markov distribution of zero mean and unit
247 // variance.
248 //----------------------------------------------------------------------
249 
250 void annCoGaussPts( // correlated-Gaussian distribution
251  ANNpointArray pa, // point array (modified)
252  int n, // number of points
253  int dim, // dimension
254  double correlation) // correlation
255 {
256  double std_dev_w = sqrt(1.0 - correlation * correlation);
257  for (int i = 0; i < n; i++) {
258  double previous = annRanGauss();
259  pa[i][0] = (ANNcoord) previous;
260  for (int d = 1; d < dim; d++) {
261  previous = correlation*previous + std_dev_w*annRanGauss();
262  pa[i][d] = (ANNcoord) previous;
263  }
264  }
265 }
266 
267 //----------------------------------------------------------------------
268 // annCoLaplacePts - Generate correlated Laplacian distributed points
269 // Generates a Laplacian-Markov distribution of zero mean and unit
270 // variance.
271 //----------------------------------------------------------------------
272 
273 void annCoLaplacePts( // correlated-Laplacian distribution
274  ANNpointArray pa, // point array (modified)
275  int n, // number of points
276  int dim, // dimension
277  double correlation) // correlation
278 {
279  double wn;
280  double corr_sq = correlation * correlation;
281 
282  for (int i = 0; i < n; i++) {
283  double previous = annRanLaplace();
284  pa[i][0] = (ANNcoord) previous;
285  for (int d = 1; d < dim; d++) {
286  double temp = annRan0();
287  if (temp < corr_sq)
288  wn = 0.0;
289  else
290  wn = annRanLaplace();
291  previous = correlation * previous + wn;
292  pa[i][d] = (ANNcoord) previous;
293  }
294  }
295 }
296 
297 //----------------------------------------------------------------------
298 // annClusGaussPts - Generate clusters of Gaussian distributed points
299 // Cluster centers are uniformly distributed over [-1,1], and the
300 // standard deviation within each cluster is fixed.
301 //
302 // Note: Once cluster centers have been set, they are not changed,
303 // unless new_clust = true. This is so that subsequent calls generate
304 // points from the same distribution. It follows, of course, that any
305 // attempt to change the dimension or number of clusters without
306 // generating new clusters is asking for trouble.
307 //
308 // Note: Cluster centers are not generated by a call to uniformPts().
309 // Although this could be done, it has been omitted for
310 // compatibility with annClusGaussPts() in the colored version,
311 // rand_c.cc.
312 //----------------------------------------------------------------------
313 
314 void annClusGaussPts( // clustered-Gaussian distribution
315  ANNpointArray pa, // point array (modified)
316  int n, // number of points
317  int dim, // dimension
318  int n_clus, // number of colors
319  ANNbool new_clust, // generate new clusters.
320  double std_dev) // standard deviation within clusters
321 {
322  static ANNpointArray clusters = NULL;// cluster storage
323 
324  if (clusters == NULL || new_clust) {// need new cluster centers
325  if (clusters != NULL) // clusters already exist
326  annDeallocPts(clusters); // get rid of them
327  clusters = annAllocPts(n_clus, dim);
328  // generate cluster center coords
329  for (int i = 0; i < n_clus; i++) {
330  for (int d = 0; d < dim; d++) {
331  clusters[i][d] = (ANNcoord) annRanUnif(-1,1);
332  }
333  }
334  }
335 
336  for (int i = 0; i < n; i++) {
337  int c = annRanInt(n_clus); // generate cluster index
338  for (int d = 0; d < dim; d++) {
339  pa[i][d] = (ANNcoord) (std_dev*annRanGauss() + clusters[c][d]);
340  }
341  }
342 }
343 
344 //----------------------------------------------------------------------
345 // annClusOrthFlats - points clustered along orthogonal flats
346 //
347 // This distribution consists of a collection points clustered
348 // among a collection of axis-aligned low dimensional flats in
349 // the hypercube [-1,1]^d. A set of n_clus orthogonal flats are
350 // generated, each whose dimension is a random number between 1
351 // and max_dim. The points are evenly distributed among the clusters.
352 // For each cluster, we generate points uniformly distributed along
353 // the flat within the hypercube.
354 //
355 // This is done as follows. Each cluster is defined by a d-element
356 // control vector whose components are either:
357 //
358 // CO_FLAG indicating that this component is to be generated
359 // uniformly in [-1,1],
360 // x a value other than CO_FLAG in the range [-1,1],
361 // which indicates that this coordinate is to be
362 // generated as x plus a Gaussian random deviation
363 // with the given standard deviation.
364 //
365 // The number of zero components is the dimension of the flat, which
366 // is a random integer in the range from 1 to max_dim. The points
367 // are disributed between clusters in nearly equal sized groups.
368 //
369 // Note: Once cluster centers have been set, they are not changed,
370 // unless new_clust = true. This is so that subsequent calls generate
371 // points from the same distribution. It follows, of course, that any
372 // attempt to change the dimension or number of clusters without
373 // generating new clusters is asking for trouble.
374 //
375 // To make this a bad scenario at query time, query points should be
376 // selected from a different distribution, e.g. uniform or Gaussian.
377 //
378 // We use a little programming trick to generate groups of roughly
379 // equal size. If n is the total number of points, and n_clus is
380 // the number of clusters, then the c-th cluster (0 <= c < n_clus)
381 // is given floor((n+c)/n_clus) points. It can be shown that this
382 // will exactly consume all n points.
383 //
384 // This procedure makes use of the utility procedure, genOrthFlat
385 // which generates points in one orthogonal flat, according to
386 // the given control vector.
387 //
388 //----------------------------------------------------------------------
389 const double CO_FLAG = 999; // special flag value
390 
391 static void genOrthFlat( // generate points on an orthog flat
392  ANNpointArray pa, // point array
393  int n, // number of points
394  int dim, // dimension
395  double *control, // control vector
396  double std_dev) // standard deviation
397 {
398  for (int i = 0; i < n; i++) { // generate each point
399  for (int d = 0; d < dim; d++) { // generate each coord
400  if (control[d] == CO_FLAG) // dimension on flat
401  pa[i][d] = (ANNcoord) annRanUnif(-1,1);
402  else // dimension off flat
403  pa[i][d] = (ANNcoord) (std_dev*annRanGauss() + control[d]);
404  }
405  }
406 }
407 
408 void annClusOrthFlats( // clustered along orthogonal flats
409  ANNpointArray pa, // point array (modified)
410  int n, // number of points
411  int dim, // dimension
412  int n_clus, // number of colors
413  ANNbool new_clust, // generate new clusters.
414  double std_dev, // standard deviation within clusters
415  int max_dim) // maximum dimension of the flats
416 {
417  static ANNpointArray control = NULL; // control vectors
418 
419  if (control == NULL || new_clust) { // need new cluster centers
420  if (control != NULL) { // clusters already exist
421  annDeallocPts(control); // get rid of them
422  }
423  control = annAllocPts(n_clus, dim);
424 
425  for (int c = 0; c < n_clus; c++) { // generate clusters
426  int n_dim = 1 + annRanInt(max_dim); // number of dimensions in flat
427  for (int d = 0; d < dim; d++) { // generate side locations
428  // prob. of picking next dim
429  double Prob = ((double) n_dim)/((double) (dim-d));
430  if (annRan0() < Prob) { // add this one to flat
431  control[c][d] = CO_FLAG; // flag this entry
432  n_dim--; // one fewer dim to fill
433  }
434  else { // don't take this one
435  control[c][d] = annRanUnif(-1,1);// random value in [-1,1]
436  }
437  }
438  }
439  }
440  int offset = 0; // offset in pa array
441  for (int c = 0; c < n_clus; c++) { // generate clusters
442  int pick = (n+c)/n_clus; // number of points to pick
443  // generate the points
444  genOrthFlat(pa+offset, pick, dim, control[c], std_dev);
445  offset += pick; // increment offset
446  }
447 }
448 
449 //----------------------------------------------------------------------
450 // annClusEllipsoids - points clustered around axis-aligned ellipsoids
451 //
452 // This distribution consists of a collection points clustered
453 // among a collection of low dimensional ellipsoids whose axes
454 // are alligned with the coordinate axes in the hypercube [-1,1]^d.
455 // The objective is to model distributions in which the points are
456 // distributed in lower dimensional subspaces, and within this
457 // lower dimensional space the points are distributed with a
458 // Gaussian distribution (with no correlation between the
459 // dimensions).
460 //
461 // The distribution is given the number of clusters or "colors"
462 // (n_clus), maximum number of dimensions (max_dim) of the lower
463 // dimensional subspace, a "small" standard deviation
464 // (std_dev_small), and a "large" standard deviation range
465 // (std_dev_lo, std_dev_hi).
466 //
467 // The algorithm generates n_clus cluster centers uniformly from
468 // the hypercube [-1,1]^d. For each cluster, it selects the
469 // dimension of the subspace as a random number r between 1 and
470 // max_dim. These are the dimensions of the ellipsoid. Then it
471 // generates a d-element std dev vector whose entries are the
472 // standard deviation for the coordinates of each cluster in the
473 // distribution. Among the d-element control vector, r randomly
474 // chosen values are chosen uniformly from the range [std_dev_lo,
475 // std_dev_hi]. The remaining values are set to std_dev_small.
476 //
477 // Note that annClusGaussPts is a special case of this in which
478 // max_dim = 0, and std_dev = std_dev_small.
479 //
480 // If the flag new_clust is set, then new cluster centers are
481 // generated.
482 //
483 // This procedure makes use of the utility procedure genGauss
484 // which generates points distributed according to a Gaussian
485 // distribution.
486 //
487 //----------------------------------------------------------------------
488 
489 static void genGauss( // generate points on a general Gaussian
490  ANNpointArray pa, // point array
491  int n, // number of points
492  int dim, // dimension
493  double *center, // center vector
494  double *std_dev) // standard deviation vector
495 {
496  for (int i = 0; i < n; i++) {
497  for (int d = 0; d < dim; d++) {
498  pa[i][d] = (ANNcoord) (std_dev[d]*annRanGauss() + center[d]);
499  }
500  }
501 }
502 
503 void annClusEllipsoids( // clustered around ellipsoids
504  ANNpointArray pa, // point array (modified)
505  int n, // number of points
506  int dim, // dimension
507  int n_clus, // number of colors
508  ANNbool new_clust, // generate new clusters.
509  double std_dev_small, // small standard deviation
510  double std_dev_lo, // low standard deviation for ellipses
511  double std_dev_hi, // high standard deviation for ellipses
512  int max_dim) // maximum dimension of the flats
513 {
514  static ANNpointArray centers = NULL; // cluster centers
515  static ANNpointArray std_dev = NULL; // standard deviations
516 
517  if (centers == NULL || new_clust) { // need new cluster centers
518  if (centers != NULL) // clusters already exist
519  annDeallocPts(centers); // get rid of them
520  if (std_dev != NULL) // std deviations already exist
521  annDeallocPts(std_dev); // get rid of them
522 
523  centers = annAllocPts(n_clus, dim); // alloc new clusters and devs
524  std_dev = annAllocPts(n_clus, dim);
525 
526  for (int i = 0; i < n_clus; i++) { // gen cluster center coords
527  for (int d = 0; d < dim; d++) {
528  centers[i][d] = (ANNcoord) annRanUnif(-1,1);
529  }
530  }
531  for (int c = 0; c < n_clus; c++) { // generate cluster std dev
532  int n_dim = 1 + annRanInt(max_dim); // number of dimensions in flat
533  for (int d = 0; d < dim; d++) { // generate std dev's
534  // prob. of picking next dim
535  double Prob = ((double) n_dim)/((double) (dim-d));
536  if (annRan0() < Prob) { // add this one to ellipse
537  // generate random std dev
538  std_dev[c][d] = annRanUnif(std_dev_lo, std_dev_hi);
539  n_dim--; // one fewer dim to fill
540  }
541  else { // don't take this one
542  std_dev[c][d] = std_dev_small;// use small std dev
543  }
544  }
545  }
546  }
547 
548  int offset = 0; // next slot to fill
549  for (int c = 0; c < n_clus; c++) { // generate clusters
550  int pick = (n+c)/n_clus; // number of points to pick
551  // generate the points
552  genGauss(pa+offset, pick, dim, centers[c], std_dev[c]);
553  offset += pick; // increment offset in array
554  }
555 }
556 
557 //----------------------------------------------------------------------
558 // annPlanted - Generates points from a "planted" distribution
559 // In high dimensional spaces, interpoint distances tend to be
560 // highly clustered around the mean value. Approximate nearest
561 // neighbor searching makes little sense in this context, unless it
562 // is the case that each query point is significantly closer to its
563 // nearest neighbor than to other points. Thus, the query points
564 // should be planted close to the data points. Given a source data
565 // set, this procedure generates a set of query points having this
566 // property.
567 //
568 // We are given a source data array and a standard deviation. We
569 // generate points as follows. We select a random point from the
570 // source data set, and we generate a Gaussian point centered about
571 // this random point and perturbed by a normal distributed random
572 // variable with mean zero and the given standard deviation along
573 // each coordinate.
574 //
575 // Note that this essentially the same a clustered Gaussian
576 // distribution, but where the cluster centers are given by the
577 // source data set.
578 //----------------------------------------------------------------------
579 
580 void annPlanted( // planted nearest neighbors
581  ANNpointArray pa, // point array (modified)
582  int n, // number of points
583  int dim, // dimension
584  ANNpointArray src, // source point array
585  int n_src, // source size
586  double std_dev) // standard deviation about source
587 {
588  for (int i = 0; i < n; i++) {
589  int c = annRanInt(n_src); // generate source index
590  for (int d = 0; d < dim; d++) {
591  pa[i][d] = (ANNcoord) (std_dev*annRanGauss() + src[c][d]);
592  }
593  }
594 }
ANNpoint * ANNpointArray
Definition: ANN.h:376
void annClusGaussPts(ANNpointArray pa, int n, int dim, int n_clus, ANNbool new_clust, double std_dev)
Definition: rand.cpp:314
double ANNcoord
Definition: ANN.h:158
static double annRanGauss()
Definition: rand.cpp:137
static double annRanLaplace()
Definition: rand.cpp:181
double std_dev_lo
Definition: ann_test.cpp:481
static void genOrthFlat(ANNpointArray pa, int n, int dim, double *control, double std_dev)
Definition: rand.cpp:391
DLL_API void annDeallocPts(ANNpointArray &pa)
Definition: ANN.cpp:133
void annPlanted(ANNpointArray pa, int n, int dim, ANNpointArray src, int n_src, double std_dev)
Definition: rand.cpp:580
void annClusEllipsoids(ANNpointArray pa, int n, int dim, int n_clus, ANNbool new_clust, double std_dev_small, double std_dev_lo, double std_dev_hi, int max_dim)
Definition: rand.cpp:503
void annLaplacePts(ANNpointArray pa, int n, int dim)
Definition: rand.cpp:232
void annGaussPts(ANNpointArray pa, int n, int dim, double std_dev)
Definition: rand.cpp:214
int annIdum
Definition: rand.cpp:42
void annCoGaussPts(ANNpointArray pa, int n, int dim, double correlation)
Definition: rand.cpp:250
int dim
Definition: ann_test.cpp:472
const double CO_FLAG
Definition: rand.cpp:389
int max_dim
Definition: ann_test.cpp:477
static int annRanInt(int n)
Definition: rand.cpp:112
ANNbool new_clust
Definition: ann_test.cpp:476
double annRan0()
Definition: rand.cpp:59
DLL_API ANNpointArray annAllocPts(int n, int dim)
Definition: ANN.cpp:117
ANNbool
Definition: ANN.h:132
double std_dev
Definition: ann_test.cpp:480
void annUniformPts(ANNpointArray pa, int n, int dim)
Definition: rand.cpp:196
static double annRanUnif(double lo, double hi)
Definition: rand.cpp:124
double std_dev_hi
Definition: ann_test.cpp:482
void annCoLaplacePts(ANNpointArray pa, int n, int dim, double correlation)
Definition: rand.cpp:273
void annClusOrthFlats(ANNpointArray pa, int n, int dim, int n_clus, ANNbool new_clust, double std_dev, int max_dim)
Definition: rand.cpp:408
static void genGauss(ANNpointArray pa, int n, int dim, double *center, double *std_dev)
Definition: rand.cpp:489

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