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+/* randist/discrete.c
+ *
+ * Copyright (C) 1996, 1997, 1998, 1999, 2000 James Theiler, Brian Gough
+ *
+ * This program is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU General Public License as published by
+ * the Free Software Foundation; either version 2 of the License, or (at
+ * your option) any later version.
+ *
+ * This program is distributed in the hope that it will be useful, but
+ * WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+ * General Public License for more details.
+ *
+ * You should have received a copy of the GNU General Public License
+ * along with this program; if not, write to the Free Software
+ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
+ */
+
+/*
+ Random Discrete Events
+
+ Given K discrete events with different probabilities P[k]
+ produce a value k consistent with its probability.
+
+ This program is free software; you can redistribute it and/or
+ modify it under the terms of the GNU General Public License as
+ published by the Free Software Foundation; either version 2 of the
+ License, or (at your option) any later version.
+
+ This program is distributed in the hope that it will be useful, but
+ WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
+ General Public License for more details. You should have received
+ a copy of the GNU General Public License along with this program;
+ if not, write to the Free Foundation, Inc., 59 Temple Place, Suite
+ 330, Boston, MA 02111-1307 USA
+*/
+
+/*
+ * Based on: Alastair J Walker, An efficient method for generating
+ * discrete random variables with general distributions, ACM Trans
+ * Math Soft 3, 253-256 (1977). See also: D. E. Knuth, The Art of
+ * Computer Programming, Volume 2 (Seminumerical algorithms), 3rd
+ * edition, Addison-Wesley (1997), p120.
+
+ * Walker's algorithm does some preprocessing, and provides two
+ * arrays: floating point F[k] and integer A[k]. A value k is chosen
+ * from 0..K-1 with equal likelihood, and then a uniform random number
+ * u is compared to F[k]. If it is less than F[k], then k is
+ * returned. Otherwise, A[k] is returned.
+
+ * Walker's original paper describes an O(K^2) algorithm for setting
+ * up the F and A arrays. I found this disturbing since I wanted to
+ * use very large values of K. I'm sure I'm not the first to realize
+ * this, but in fact the preprocessing can be done in O(K) steps.
+
+ * A figure of merit for the preprocessing is the average value for
+ * the F[k]'s (that is, SUM_k F[k]/K); this corresponds to the
+ * probability that k is returned, instead of A[k], thereby saving a
+ * redirection. Walker's O(K^2) preprocessing will generally improve
+ * that figure of merit, compared to my cheaper O(K) method; from some
+ * experiments with a perl script, I get values of around 0.6 for my
+ * method and just under 0.75 for Walker's. Knuth has pointed out
+ * that finding _the_ optimum lookup tables, which maximize the
+ * average F[k], is a combinatorially difficult problem. But any
+ * valid preprocessing will still provide O(1) time for the call to
+ * gsl_ran_discrete(). I find that if I artificially set F[k]=1 --
+ * ie, better than optimum! -- I get a speedup of maybe 20%, so that's
+ * the maximum I could expect from the most expensive preprocessing.
+ * Folding in the difference of 0.6 vs 0.75, I'd estimate that the
+ * speedup would be less than 10%.
+
+ * I've not implemented it here, but one compromise is to sort the
+ * probabilities once, and then work from the two ends inward. This
+ * requires O(K log K), still lots cheaper than O(K^2), and from my
+ * experiments with the perl script, the figure of merit is within
+ * about 0.01 for K up to 1000, and no sign of diverging (in fact,
+ * they seemed to be converging, but it's hard to say with just a
+ * handful of runs).
+
+ * The O(K) algorithm goes through all the p_k's and decides if they
+ * are "smalls" or "bigs" according to whether they are less than or
+ * greater than the mean value 1/K. The indices to the smalls and the
+ * bigs are put in separate stacks, and then we work through the
+ * stacks together. For each small, we pair it up with the next big
+ * in the stack (Walker always wanted to pair up the smallest small
+ * with the biggest big). The small "borrows" from the big just
+ * enough to bring the small up to mean. This reduces the size of the
+ * big, so the (smaller) big is compared again to the mean, and if it
+ * is smaller, it gets "popped" from the big stack and "pushed" to the
+ * small stack. Otherwise, it stays put. Since every time we pop a
+ * small, we are able to deal with it right then and there, and we
+ * never have to pop more than K smalls, then the algorithm is O(K).
+
+ * This implementation sets up two separate stacks, and allocates K
+ * elements between them. Since neither stack ever grows, we do an
+ * extra O(K) pass through the data to determine how many smalls and
+ * bigs there are to begin with and allocate appropriately. In all
+ * there are 2*K*sizeof(double) transient bytes of memory that are
+ * used than returned, and K*(sizeof(int)+sizeof(double)) bytes used
+ * in the lookup table.
+
+ * Walker spoke of using two random numbers (an integer 0..K-1, and a
+ * floating point u in [0,1]), but Knuth points out that one can just
+ * use the integer and fractional parts of K*u where u is in [0,1].
+ * In fact, Knuth further notes that taking F'[k]=(k+F[k])/K, one can
+ * directly compare u to F'[k] without having to explicitly set
+ * u=K*u-int(K*u).
+
+ * Usage:
+
+ * Starting with an array of probabilities P, initialize and do
+ * preprocessing with a call to:
+
+ * gsl_rng *r;
+ * gsl_ran_discrete_t *f;
+ * f = gsl_ran_discrete_preproc(K,P);
+
+ * Then, whenever a random index 0..K-1 is desired, use
+
+ * k = gsl_ran_discrete(r,f);
+
+ * Note that several different randevent struct's can be
+ * simultaneously active.
+
+ * Aside: A very clever alternative approach is described in
+ * Abramowitz and Stegun, p 950, citing: Marsaglia, Random variables
+ * and computers, Proc Third Prague Conference in Probability Theory,
+ * 1962. A more accesible reference is: G. Marsaglia, Generating
+ * discrete random numbers in a computer, Comm ACM 6, 37-38 (1963).
+ * If anybody is interested, I (jt) have also coded up this version as
+ * part of another software package. However, I've done some
+ * comparisons, and the Walker method is both faster and more stingy
+ * with memory. So, in the end I decided not to include it with the
+ * GSL package.
+
+ * Written 26 Jan 1999, James Theiler, jt@lanl.gov
+ * Adapted to GSL, 30 Jan 1999, jt
+
+ */
+
+#include <config.h>
+#include <stdio.h> /* used for NULL, also fprintf(stderr,...) */
+#include <stdlib.h> /* used for malloc's */
+#include <math.h>
+#include <gsl/gsl_errno.h>
+#include <gsl/gsl_rng.h>
+#include <gsl/gsl_randist.h>
+#define DEBUG 0
+#define KNUTH_CONVENTION 1 /* Saves a few steps of arithmetic
+ * in the call to gsl_ran_discrete()
+ */
+
+/*** Begin Stack (this code is used just in this file) ***/
+
+/* Stack code converted to use unsigned indices (i.e. s->i == 0 now
+ means an empty stack, instead of -1), for consistency and to give a
+ bigger allowable range. BJG */
+
+typedef struct {
+ size_t N; /* max number of elts on stack */
+ size_t *v; /* array of values on the stack */
+ size_t i; /* index of top of stack */
+} gsl_stack_t;
+
+static gsl_stack_t *
+new_stack(size_t N) {
+ gsl_stack_t *s;
+ s = (gsl_stack_t *)malloc(sizeof(gsl_stack_t));
+ s->N = N;
+ s->i = 0; /* indicates stack is empty */
+ s->v = (size_t *)malloc(sizeof(size_t)*N);
+ return s;
+}
+
+static void
+push_stack(gsl_stack_t *s, size_t v)
+{
+ if ((s->i) >= (s->N)) {
+ fprintf(stderr,"Cannot push stack!\n");
+ abort(); /* FIXME: fatal!! */
+ }
+ (s->v)[s->i] = v;
+ s->i += 1;
+}
+
+static size_t pop_stack(gsl_stack_t *s)
+{
+ if ((s->i) == 0) {
+ fprintf(stderr,"Cannot pop stack!\n");
+ abort(); /* FIXME: fatal!! */
+ }
+ s->i -= 1;
+ return ((s->v)[s->i]);
+}
+
+static inline size_t size_stack(const gsl_stack_t *s)
+{
+ return s->i;
+}
+
+static void free_stack(gsl_stack_t *s)
+{
+ free((char *)(s->v));
+ free((char *)s);
+}
+
+/*** End Stack ***/
+
+
+/*** Begin Walker's Algorithm ***/
+
+gsl_ran_discrete_t *
+gsl_ran_discrete_preproc(size_t Kevents, const double *ProbArray)
+{
+ size_t k,b,s;
+ gsl_ran_discrete_t *g;
+ size_t nBigs, nSmalls;
+ gsl_stack_t *Bigs;
+ gsl_stack_t *Smalls;
+ double *E;
+ double pTotal = 0.0, mean, d;
+
+ if (Kevents < 1) {
+ /* Could probably treat Kevents=1 as a special case */
+
+ GSL_ERROR_VAL ("number of events must be a positive integer",
+ GSL_EINVAL, 0);
+ }
+
+ /* Make sure elements of ProbArray[] are positive.
+ * Won't enforce that sum is unity; instead will just normalize
+ */
+
+ for (k=0; k<Kevents; ++k) {
+ if (ProbArray[k] < 0) {
+ GSL_ERROR_VAL ("probabilities must be non-negative",
+ GSL_EINVAL, 0) ;
+ }
+ pTotal += ProbArray[k];
+ }
+
+ /* Begin setting up the main "object" (just a struct, no steroids) */
+ g = (gsl_ran_discrete_t *)malloc(sizeof(gsl_ran_discrete_t));
+ g->K = Kevents;
+ g->F = (double *)malloc(sizeof(double)*Kevents);
+ g->A = (size_t *)malloc(sizeof(size_t)*Kevents);
+
+ E = (double *)malloc(sizeof(double)*Kevents);
+
+ if (E==NULL) {
+ GSL_ERROR_VAL ("Cannot allocate memory for randevent", GSL_ENOMEM, 0);
+ }
+
+ for (k=0; k<Kevents; ++k) {
+ E[k] = ProbArray[k]/pTotal;
+ }
+
+ /* Now create the Bigs and the Smalls */
+ mean = 1.0/Kevents;
+ nSmalls=nBigs=0;
+ for (k=0; k<Kevents; ++k) {
+ if (E[k] < mean) ++nSmalls;
+ else ++nBigs;
+ }
+ Bigs = new_stack(nBigs);
+ Smalls = new_stack(nSmalls);
+ for (k=0; k<Kevents; ++k) {
+ if (E[k] < mean) {
+ push_stack(Smalls,k);
+ }
+ else {
+ push_stack(Bigs,k);
+ }
+ }
+ /* Now work through the smalls */
+ while (size_stack(Smalls) > 0) {
+ s = pop_stack(Smalls);
+ if (size_stack(Bigs) == 0) {
+ (g->A)[s]=s;
+ (g->F)[s]=1.0;
+ continue;
+ }
+ b = pop_stack(Bigs);
+ (g->A)[s]=b;
+ (g->F)[s]=Kevents*E[s];
+#if DEBUG
+ fprintf(stderr,"s=%2d, A=%2d, F=%.4f\n",s,(g->A)[s],(g->F)[s]);
+#endif
+ d = mean - E[s];
+ E[s] += d; /* now E[s] == mean */
+ E[b] -= d;
+ if (E[b] < mean) {
+ push_stack(Smalls,b); /* no longer big, join ranks of the small */
+ }
+ else if (E[b] > mean) {
+ push_stack(Bigs,b); /* still big, put it back where you found it */
+ }
+ else {
+ /* E[b]==mean implies it is finished too */
+ (g->A)[b]=b;
+ (g->F)[b]=1.0;
+ }
+ }
+ while (size_stack(Bigs) > 0) {
+ b = pop_stack(Bigs);
+ (g->A)[b]=b;
+ (g->F)[b]=1.0;
+ }
+ /* Stacks have been emptied, and A and F have been filled */
+
+ if ( size_stack(Smalls) != 0) {
+ GSL_ERROR_VAL ("Smalls stack has not been emptied",
+ GSL_ESANITY, 0 );
+ }
+
+#if 0
+ /* if 1, then artificially set all F[k]'s to unity. This will
+ * give wrong answers, but you'll get them faster. But, not
+ * that much faster (I get maybe 20%); that's an upper bound
+ * on what the optimal preprocessing would give.
+ */
+ for (k=0; k<Kevents; ++k) {
+ (g->F)[k] = 1.0;
+ }
+#endif
+
+#if KNUTH_CONVENTION
+ /* For convenience, set F'[k]=(k+F[k])/K */
+ /* This saves some arithmetic in gsl_ran_discrete(); I find that
+ * it doesn't actually make much difference.
+ */
+ for (k=0; k<Kevents; ++k) {
+ (g->F)[k] += k;
+ (g->F)[k] /= Kevents;
+ }
+#endif
+
+ free_stack(Bigs);
+ free_stack(Smalls);
+ free((char *)E);
+
+ return g;
+}
+
+size_t
+gsl_ran_discrete(const gsl_rng *r, const gsl_ran_discrete_t *g)
+{
+ size_t c=0;
+ double u,f;
+ u = gsl_rng_uniform(r);
+#if KNUTH_CONVENTION
+ c = (u*(g->K));
+#else
+ u *= g->K;
+ c = u;
+ u -= c;
+#endif
+ f = (g->F)[c];
+ /* fprintf(stderr,"c,f,u: %d %.4f %f\n",c,f,u); */
+ if (f == 1.0) return c;
+
+ if (u < f) {
+ return c;
+ }
+ else {
+ return (g->A)[c];
+ }
+}
+
+void gsl_ran_discrete_free(gsl_ran_discrete_t *g)
+{
+ free((char *)(g->A));
+ free((char *)(g->F));
+ free((char *)g);
+}
+
+double
+gsl_ran_discrete_pdf(size_t k, const gsl_ran_discrete_t *g)
+{
+ size_t i,K;
+ double f,p=0;
+ K= g->K;
+ if (k>K) return 0;
+ for (i=0; i<K; ++i) {
+ f = (g->F)[i];
+#if KNUTH_CONVENTION
+ f = K*f-i;
+#endif
+ if (i==k) {
+ p += f;
+ } else if (k == (g->A)[i]) {
+ p += 1.0 - f;
+ }
+ }
+ return p/K;
+}
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