basis_reduction_templ.c

   1 /*
   2  * Copyright 2006-2007 Universiteit Leiden
   3  * Copyright 2008-2009 Katholieke Universiteit Leuven
   4  *
   5  * Use of this software is governed by the GNU LGPLv2.1 license
   6  *
   7  * Written by Sven Verdoolaege, Leiden Institute of Advanced Computer Science,
   8  * Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
   9  * and K.U.Leuven, Departement Computerwetenschappen, Celestijnenlaan 200A,
  10  * B-3001 Leuven, Belgium
  11  */
  12
  13 #include <stdlib.h>
  14 #include <isl_ctx_private.h>
  15 #include <isl_map_private.h>
  16 #include "isl_basis_reduction.h"
  17
  18 static void save_alpha(GBR_LP *lp, int first, int n, GBR_type *alpha)
  19 {
  20         int i;
  21
  22         for (i = 0; i < n; ++i)
  23                 GBR_lp_get_alpha(lp, first + i, &alpha[i]);
  24 }
  25
  26 /* Compute a reduced basis for the set represented by the tableau "tab".
  27  * tab->basis, which must be initialized by the calling function to an affine
  28  * unimodular basis, is updated to reflect the reduced basis.
  29  * The first tab->n_zero rows of the basis (ignoring the constant row)
  30  * are assumed to correspond to equalities and are left untouched.
  31  * tab->n_zero is updated to reflect any additional equalities that
  32  * have been detected in the first rows of the new basis.
  33  * The final tab->n_unbounded rows of the basis are assumed to correspond
  34  * to unbounded directions and are also left untouched.
  35  * In particular this means that the remaining rows are assumed to
  36  * correspond to bounded directions.
  37  *
  38  * This function implements the algorithm described in
  39  * "An Implementation of the Generalized Basis Reduction Algorithm
  40  *  for Integer Programming" of Cook el al. to compute a reduced basis.
  41  * We use \epsilon = 1/4.
  42  *
  43  * If ctx->opt->gbr_only_first is set, the user is only interested
  44  * in the first direction.  In this case we stop the basis reduction when
  45  * the width in the first direction becomes smaller than 2.
  46  */
  47 struct isl_tab *isl_tab_compute_reduced_basis(struct isl_tab *tab)
  48 {
  49         unsigned dim;
  50         struct isl_ctx *ctx;
  51         struct isl_mat *B;
  52         int unbounded;
  53         int i;
  54         GBR_LP *lp = NULL;
  55         GBR_type F_old, alpha, F_new;
  56         int row;
  57         isl_int tmp;
  58         struct isl_vec *b_tmp;
  59         GBR_type *F = NULL;
  60         GBR_type *alpha_buffer[2] = { NULL, NULL };
  61         GBR_type *alpha_saved;
  62         GBR_type F_saved;
  63         int use_saved = 0;
  64         isl_int mu[2];
  65         GBR_type mu_F[2];
  66         GBR_type two;
  67         GBR_type one;
  68         int empty = 0;
  69         int fixed = 0;
  70         int fixed_saved = 0;
  71         int mu_fixed[2];
  72         int n_bounded;
  73         int gbr_only_first;
  74
  75         if (!tab)
  76                 return NULL;
  77
  78         if (tab->empty)
  79                 return tab;
  80
  81         ctx = tab->mat->ctx;
  82         gbr_only_first = ctx->opt->gbr_only_first;
  83         dim = tab->n_var;
  84         B = tab->basis;
  85         if (!B)
  86                 return tab;
  87
  88         n_bounded = dim - tab->n_unbounded;
  89         if (n_bounded <= tab->n_zero + 1)
  90                 return tab;
  91
  92         isl_int_init(tmp);
  93         isl_int_init(mu[0]);
  94         isl_int_init(mu[1]);
  95
  96         GBR_init(alpha);
  97         GBR_init(F_old);
  98         GBR_init(F_new);
  99         GBR_init(F_saved);
 100         GBR_init(mu_F[0]);
 101         GBR_init(mu_F[1]);
 102         GBR_init(two);
 103         GBR_init(one);
 104
 105         b_tmp = isl_vec_alloc(ctx, dim);
 106         if (!b_tmp)
 107                 goto error;
 108
 109         F = isl_alloc_array(ctx, GBR_type, n_bounded);
 110         alpha_buffer[0] = isl_alloc_array(ctx, GBR_type, n_bounded);
 111         alpha_buffer[1] = isl_alloc_array(ctx, GBR_type, n_bounded);
 112         alpha_saved = alpha_buffer[0];
 113
 114         if (!F || !alpha_buffer[0] || !alpha_buffer[1])
 115                 goto error;
 116
 117         for (i = 0; i < n_bounded; ++i) {
 118                 GBR_init(F[i]);
 119                 GBR_init(alpha_buffer[0][i]);
 120                 GBR_init(alpha_buffer[1][i]);
 121         }
 122
 123         GBR_set_ui(two, 2);
 124         GBR_set_ui(one, 1);
 125
 126         lp = GBR_lp_init(tab);
 127         if (!lp)
 128                 goto error;
 129
 130         i = tab->n_zero;
 131
 132         GBR_lp_set_obj(lp, B->row[1+i]+1, dim);
 133         ctx->stats->gbr_solved_lps++;
 134         unbounded = GBR_lp_solve(lp);
 135         isl_assert(ctx, !unbounded, goto error);
 136         GBR_lp_get_obj_val(lp, &F[i]);
 137
 138         if (GBR_lt(F[i], one)) {
 139                 if (!GBR_is_zero(F[i])) {
 140                         empty = GBR_lp_cut(lp, B->row[1+i]+1);
 141                         if (empty)
 142                                 goto done;
 143                         GBR_set_ui(F[i], 0);
 144                 }
 145                 tab->n_zero++;
 146         }
 147
 148         do {
 149                 if (i+1 == tab->n_zero) {
 150                         GBR_lp_set_obj(lp, B->row[1+i+1]+1, dim);
 151                         ctx->stats->gbr_solved_lps++;
 152                         unbounded = GBR_lp_solve(lp);
 153                         isl_assert(ctx, !unbounded, goto error);
 154                         GBR_lp_get_obj_val(lp, &F_new);
 155                         fixed = GBR_lp_is_fixed(lp);
 156                         GBR_set_ui(alpha, 0);
 157                 } else
 158                 if (use_saved) {
 159                         row = GBR_lp_next_row(lp);
 160                         GBR_set(F_new, F_saved);
 161                         fixed = fixed_saved;
 162                         GBR_set(alpha, alpha_saved[i]);
 163                 } else {
 164                         row = GBR_lp_add_row(lp, B->row[1+i]+1, dim);
 165                         GBR_lp_set_obj(lp, B->row[1+i+1]+1, dim);
 166                         ctx->stats->gbr_solved_lps++;
 167                         unbounded = GBR_lp_solve(lp);
 168                         isl_assert(ctx, !unbounded, goto error);
 169                         GBR_lp_get_obj_val(lp, &F_new);
 170                         fixed = GBR_lp_is_fixed(lp);
 171
 172                         GBR_lp_get_alpha(lp, row, &alpha);
 173
 174                         if (i > 0)
 175                                 save_alpha(lp, row-i, i, alpha_saved);
 176
 177                         if (GBR_lp_del_row(lp) < 0)
 178                                 goto error;
 179                 }
 180                 GBR_set(F[i+1], F_new);
 181
 182                 GBR_floor(mu[0], alpha);
 183                 GBR_ceil(mu[1], alpha);
 184
 185                 if (isl_int_eq(mu[0], mu[1]))
 186                         isl_int_set(tmp, mu[0]);
 187                 else {
 188                         int j;
 189
 190                         for (j = 0; j <= 1; ++j) {
 191                                 isl_int_set(tmp, mu[j]);
 192                                 isl_seq_combine(b_tmp->el,
 193                                                 ctx->one, B->row[1+i+1]+1,
 194                                                 tmp, B->row[1+i]+1, dim);
 195                                 GBR_lp_set_obj(lp, b_tmp->el, dim);
 196                                 ctx->stats->gbr_solved_lps++;
 197                                 unbounded = GBR_lp_solve(lp);
 198                                 isl_assert(ctx, !unbounded, goto error);
 199                                 GBR_lp_get_obj_val(lp, &mu_F[j]);
 200                                 mu_fixed[j] = GBR_lp_is_fixed(lp);
 201                                 if (i > 0)
 202                                         save_alpha(lp, row-i, i, alpha_buffer[j]);
 203                         }
 204
 205                         if (GBR_lt(mu_F[0], mu_F[1]))
 206                                 j = 0;
 207                         else
 208                                 j = 1;
 209
 210                         isl_int_set(tmp, mu[j]);
 211                         GBR_set(F_new, mu_F[j]);
 212                         fixed = mu_fixed[j];
 213                         alpha_saved = alpha_buffer[j];
 214                 }
 215                 isl_seq_combine(B->row[1+i+1]+1, ctx->one, B->row[1+i+1]+1,
 216                                 tmp, B->row[1+i]+1, dim);
 217
 218                 if (i+1 == tab->n_zero && fixed) {
 219                         if (!GBR_is_zero(F[i+1])) {
 220                                 empty = GBR_lp_cut(lp, B->row[1+i+1]+1);
 221                                 if (empty)
 222                                         goto done;
 223                                 GBR_set_ui(F[i+1], 0);
 224                         }
 225                         tab->n_zero++;
 226                 }
 227
 228                 GBR_set(F_old, F[i]);
 229
 230                 use_saved = 0;
 231                 /* mu_F[0] = 4 * F_new; mu_F[1] = 3 * F_old */
 232                 GBR_set_ui(mu_F[0], 4);
 233                 GBR_mul(mu_F[0], mu_F[0], F_new);
 234                 GBR_set_ui(mu_F[1], 3);
 235                 GBR_mul(mu_F[1], mu_F[1], F_old);
 236                 if (GBR_lt(mu_F[0], mu_F[1])) {
 237                         B = isl_mat_swap_rows(B, 1 + i, 1 + i + 1);
 238                         if (i > tab->n_zero) {
 239                                 use_saved = 1;
 240                                 GBR_set(F_saved, F_new);
 241                                 fixed_saved = fixed;
 242                                 if (GBR_lp_del_row(lp) < 0)
 243                                         goto error;
 244                                 --i;
 245                         } else {
 246                                 GBR_set(F[tab->n_zero], F_new);
 247                                 if (gbr_only_first && GBR_lt(F[tab->n_zero], two))
 248                                         break;
 249
 250                                 if (fixed) {
 251                                         if (!GBR_is_zero(F[tab->n_zero])) {
 252                                                 empty = GBR_lp_cut(lp, B->row[1+tab->n_zero]+1);
 253                                                 if (empty)
 254                                                         goto done;
 255                                                 GBR_set_ui(F[tab->n_zero], 0);
 256                                         }
 257                                         tab->n_zero++;
 258                                 }
 259                         }
 260                 } else {
 261                         GBR_lp_add_row(lp, B->row[1+i]+1, dim);
 262                         ++i;
 263                 }
 264         } while (i < n_bounded - 1);
 265
 266         if (0) {
 267 done:
 268                 if (empty < 0) {
 269 error:
 270                         isl_mat_free(B);
 271                         B = NULL;
 272                 }
 273         }
 274
 275         GBR_lp_delete(lp);
 276
 277         if (alpha_buffer[1])
 278                 for (i = 0; i < n_bounded; ++i) {
 279                         GBR_clear(F[i]);
 280                         GBR_clear(alpha_buffer[0][i]);
 281                         GBR_clear(alpha_buffer[1][i]);
 282                 }
 283         free(F);
 284         free(alpha_buffer[0]);
 285         free(alpha_buffer[1]);
 286
 287         isl_vec_free(b_tmp);
 288
 289         GBR_clear(alpha);
 290         GBR_clear(F_old);
 291         GBR_clear(F_new);
 292         GBR_clear(F_saved);
 293         GBR_clear(mu_F[0]);
 294         GBR_clear(mu_F[1]);
 295         GBR_clear(two);
 296         GBR_clear(one);
 297
 298         isl_int_clear(tmp);
 299         isl_int_clear(mu[0]);
 300         isl_int_clear(mu[1]);
 301
 302         tab->basis = B;
 303
 304         return tab;
 305 }
 306
 307 /* Compute an affine form of a reduced basis of the given basic
 308  * non-parametric set, which is assumed to be bounded and not
 309  * include any integer divisions.
 310  * The first column and the first row correspond to the constant term.
 311  *
 312  * If the input contains any equalities, we first create an initial
 313  * basis with the equalities first.  Otherwise, we start off with
 314  * the identity matrix.
 315  */
 316 struct isl_mat *isl_basic_set_reduced_basis(struct isl_basic_set *bset)
 317 {
 318         struct isl_mat *basis;
 319         struct isl_tab *tab;
 320
 321         if (!bset)
 322                 return NULL;
 323
 324         if (isl_basic_set_dim(bset, isl_dim_div) != 0)
 325                 isl_die(bset->ctx, isl_error_invalid,
 326                         "no integer division allowed", return NULL);
 327         if (isl_basic_set_dim(bset, isl_dim_param) != 0)
 328                 isl_die(bset->ctx, isl_error_invalid,
 329                         "no parameters allowed", return NULL);
 330
 331         tab = isl_tab_from_basic_set(bset);
 332         if (!tab)
 333                 return NULL;
 334
 335         if (bset->n_eq == 0)
 336                 tab->basis = isl_mat_identity(bset->ctx, 1 + tab->n_var);
 337         else {
 338                 isl_mat *eq;
 339                 unsigned nvar = isl_basic_set_total_dim(bset);
 340                 eq = isl_mat_sub_alloc6(bset->ctx, bset->eq, 0, bset->n_eq,
 341                                         1, nvar);
 342                 eq = isl_mat_left_hermite(eq, 0, NULL, &tab->basis);
 343                 tab->basis = isl_mat_lin_to_aff(tab->basis);
 344                 tab->n_zero = bset->n_eq;
 345                 isl_mat_free(eq);
 346         }
 347         tab = isl_tab_compute_reduced_basis(tab);
 348         if (!tab)
 349                 return NULL;
 350
 351         basis = isl_mat_copy(tab->basis);
 352
 353         isl_tab_free(tab);
 354
 355         return basis;
 356 }