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