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 MIT 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_vec_private.h>
  17 #include <isl_options_private.h>
  18 #include "isl_basis_reduction.h"
  19
  20 static void save_alpha(GBR_LP *lp, int first, int n, GBR_type *alpha)
  21 {
  22         int i;
  23
  24         for (i = 0; i < n; ++i)
  25                 GBR_lp_get_alpha(lp, first + i, &alpha[i]);
  26 }
  27
  28 /* Compute a reduced basis for the set represented by the tableau "tab".
  29  * tab->basis, which must be initialized by the calling function to an affine
  30  * unimodular basis, is updated to reflect the reduced basis.
  31  * The first tab->n_zero rows of the basis (ignoring the constant row)
  32  * are assumed to correspond to equalities and are left untouched.
  33  * tab->n_zero is updated to reflect any additional equalities that
  34  * have been detected in the first rows of the new basis.
  35  * The final tab->n_unbounded rows of the basis are assumed to correspond
  36  * to unbounded directions and are also left untouched.
  37  * In particular this means that the remaining rows are assumed to
  38  * correspond to bounded directions.
  39  *
  40  * This function implements the algorithm described in
  41  * "An Implementation of the Generalized Basis Reduction Algorithm
  42  *  for Integer Programming" of Cook el al. to compute a reduced basis.
  43  * We use \epsilon = 1/4.
  44  *
  45  * If ctx->opt->gbr_only_first is set, the user is only interested
  46  * in the first direction.  In this case we stop the basis reduction when
  47  * the width in the first direction becomes smaller than 2.
  48  */
  49 struct isl_tab *isl_tab_compute_reduced_basis(struct isl_tab *tab)
  50 {
  51         unsigned dim;
  52         struct isl_ctx *ctx;
  53         struct isl_mat *B;
  54         int i;
  55         GBR_LP *lp = NULL;
  56         GBR_type F_old, alpha, F_new;
  57         int row;
  58         isl_int tmp;
  59         struct isl_vec *b_tmp;
  60         GBR_type *F = NULL;
  61         GBR_type *alpha_buffer[2] = { NULL, NULL };
  62         GBR_type *alpha_saved;
  63         GBR_type F_saved;
  64         int use_saved = 0;
  65         isl_int mu[2];
  66         GBR_type mu_F[2];
  67         GBR_type two;
  68         GBR_type one;
  69         int empty = 0;
  70         int fixed = 0;
  71         int fixed_saved = 0;
  72         int mu_fixed[2];
  73         int n_bounded;
  74         int gbr_only_first;
  75
  76         if (!tab)
  77                 return NULL;
  78
  79         if (tab->empty)
  80                 return tab;
  81
  82         ctx = tab->mat->ctx;
  83         gbr_only_first = ctx->opt->gbr_only_first;
  84         dim = tab->n_var;
  85         B = tab->basis;
  86         if (!B)
  87                 return tab;
  88
  89         n_bounded = dim - tab->n_unbounded;
  90         if (n_bounded <= tab->n_zero + 1)
  91                 return tab;
  92
  93         isl_int_init(tmp);
  94         isl_int_init(mu[0]);
  95         isl_int_init(mu[1]);
  96
  97         GBR_init(alpha);
  98         GBR_init(F_old);
  99         GBR_init(F_new);
 100         GBR_init(F_saved);
 101         GBR_init(mu_F[0]);
 102         GBR_init(mu_F[1]);
 103         GBR_init(two);
 104         GBR_init(one);
 105
 106         b_tmp = isl_vec_alloc(ctx, dim);
 107         if (!b_tmp)
 108                 goto error;
 109
 110         F = isl_alloc_array(ctx, GBR_type, n_bounded);
 111         alpha_buffer[0] = isl_alloc_array(ctx, GBR_type, n_bounded);
 112         alpha_buffer[1] = isl_alloc_array(ctx, GBR_type, n_bounded);
 113         alpha_saved = alpha_buffer[0];
 114
 115         if (!F || !alpha_buffer[0] || !alpha_buffer[1])
 116                 goto error;
 117
 118         for (i = 0; i < n_bounded; ++i) {
 119                 GBR_init(F[i]);
 120                 GBR_init(alpha_buffer[0][i]);
 121                 GBR_init(alpha_buffer[1][i]);
 122         }
 123
 124         GBR_set_ui(two, 2);
 125         GBR_set_ui(one, 1);
 126
 127         lp = GBR_lp_init(tab);
 128         if (!lp)
 129                 goto error;
 130
 131         i = tab->n_zero;
 132
 133         GBR_lp_set_obj(lp, B->row[1+i]+1, dim);
 134         ctx->stats->gbr_solved_lps++;
 135         if (GBR_lp_solve(lp) < 0)
 136                 goto error;
 137         GBR_lp_get_obj_val(lp, &F[i]);
 138
 139         if (GBR_lt(F[i], one)) {
 140                 if (!GBR_is_zero(F[i])) {
 141                         empty = GBR_lp_cut(lp, B->row[1+i]+1);
 142                         if (empty)
 143                                 goto done;
 144                         GBR_set_ui(F[i], 0);
 145                 }
 146                 tab->n_zero++;
 147         }
 148
 149         do {
 150                 if (i+1 == tab->n_zero) {
 151                         GBR_lp_set_obj(lp, B->row[1+i+1]+1, dim);
 152                         ctx->stats->gbr_solved_lps++;
 153                         if (GBR_lp_solve(lp) < 0)
 154                                 goto error;
 155                         GBR_lp_get_obj_val(lp, &F_new);
 156                         fixed = GBR_lp_is_fixed(lp);
 157                         GBR_set_ui(alpha, 0);
 158                 } else
 159                 if (use_saved) {
 160                         row = GBR_lp_next_row(lp);
 161                         GBR_set(F_new, F_saved);
 162                         fixed = fixed_saved;
 163                         GBR_set(alpha, alpha_saved[i]);
 164                 } else {
 165                         row = GBR_lp_add_row(lp, B->row[1+i]+1, dim);
 166                         GBR_lp_set_obj(lp, B->row[1+i+1]+1, dim);
 167                         ctx->stats->gbr_solved_lps++;
 168                         if (GBR_lp_solve(lp) < 0)
 169                                 goto error;
 170                         GBR_lp_get_obj_val(lp, &F_new);
 171                         fixed = GBR_lp_is_fixed(lp);
 172
 173                         GBR_lp_get_alpha(lp, row, &alpha);
 174
 175                         if (i > 0)
 176                                 save_alpha(lp, row-i, i, alpha_saved);
 177
 178                         if (GBR_lp_del_row(lp) < 0)
 179                                 goto error;
 180                 }
 181                 GBR_set(F[i+1], F_new);
 182
 183                 GBR_floor(mu[0], alpha);
 184                 GBR_ceil(mu[1], alpha);
 185
 186                 if (isl_int_eq(mu[0], mu[1]))
 187                         isl_int_set(tmp, mu[0]);
 188                 else {
 189                         int j;
 190
 191                         for (j = 0; j <= 1; ++j) {
 192                                 isl_int_set(tmp, mu[j]);
 193                                 isl_seq_combine(b_tmp->el,
 194                                                 ctx->one, B->row[1+i+1]+1,
 195                                                 tmp, B->row[1+i]+1, dim);
 196                                 GBR_lp_set_obj(lp, b_tmp->el, dim);
 197                                 ctx->stats->gbr_solved_lps++;
 198                                 if (GBR_lp_solve(lp) < 0)
 199                                         goto error;
 200                                 GBR_lp_get_obj_val(lp, &mu_F[j]);
 201                                 mu_fixed[j] = GBR_lp_is_fixed(lp);
 202                                 if (i > 0)
 203                                         save_alpha(lp, row-i, i, alpha_buffer[j]);
 204                         }
 205
 206                         if (GBR_lt(mu_F[0], mu_F[1]))
 207                                 j = 0;
 208                         else
 209                                 j = 1;
 210
 211                         isl_int_set(tmp, mu[j]);
 212                         GBR_set(F_new, mu_F[j]);
 213                         fixed = mu_fixed[j];
 214                         alpha_saved = alpha_buffer[j];
 215                 }
 216                 isl_seq_combine(B->row[1+i+1]+1, ctx->one, B->row[1+i+1]+1,
 217                                 tmp, B->row[1+i]+1, dim);
 218
 219                 if (i+1 == tab->n_zero && fixed) {
 220                         if (!GBR_is_zero(F[i+1])) {
 221                                 empty = GBR_lp_cut(lp, B->row[1+i+1]+1);
 222                                 if (empty)
 223                                         goto done;
 224                                 GBR_set_ui(F[i+1], 0);
 225                         }
 226                         tab->n_zero++;
 227                 }
 228
 229                 GBR_set(F_old, F[i]);
 230
 231                 use_saved = 0;
 232                 /* mu_F[0] = 4 * F_new; mu_F[1] = 3 * F_old */
 233                 GBR_set_ui(mu_F[0], 4);
 234                 GBR_mul(mu_F[0], mu_F[0], F_new);
 235                 GBR_set_ui(mu_F[1], 3);
 236                 GBR_mul(mu_F[1], mu_F[1], F_old);
 237                 if (GBR_lt(mu_F[0], mu_F[1])) {
 238                         B = isl_mat_swap_rows(B, 1 + i, 1 + i + 1);
 239                         if (i > tab->n_zero) {
 240                                 use_saved = 1;
 241                                 GBR_set(F_saved, F_new);
 242                                 fixed_saved = fixed;
 243                                 if (GBR_lp_del_row(lp) < 0)
 244                                         goto error;
 245                                 --i;
 246                         } else {
 247                                 GBR_set(F[tab->n_zero], F_new);
 248                                 if (gbr_only_first && GBR_lt(F[tab->n_zero], two))
 249                                         break;
 250
 251                                 if (fixed) {
 252                                         if (!GBR_is_zero(F[tab->n_zero])) {
 253                                                 empty = GBR_lp_cut(lp, B->row[1+tab->n_zero]+1);
 254                                                 if (empty)
 255                                                         goto done;
 256                                                 GBR_set_ui(F[tab->n_zero], 0);
 257                                         }
 258                                         tab->n_zero++;
 259                                 }
 260                         }
 261                 } else {
 262                         GBR_lp_add_row(lp, B->row[1+i]+1, dim);
 263                         ++i;
 264                 }
 265         } while (i < n_bounded - 1);
 266
 267         if (0) {
 268 done:
 269                 if (empty < 0) {
 270 error:
 271                         isl_mat_free(B);
 272                         B = NULL;
 273                 }
 274         }
 275
 276         GBR_lp_delete(lp);
 277
 278         if (alpha_buffer[1])
 279                 for (i = 0; i < n_bounded; ++i) {
 280                         GBR_clear(F[i]);
 281                         GBR_clear(alpha_buffer[0][i]);
 282                         GBR_clear(alpha_buffer[1][i]);
 283                 }
 284         free(F);
 285         free(alpha_buffer[0]);
 286         free(alpha_buffer[1]);
 287
 288         isl_vec_free(b_tmp);
 289
 290         GBR_clear(alpha);
 291         GBR_clear(F_old);
 292         GBR_clear(F_new);
 293         GBR_clear(F_saved);
 294         GBR_clear(mu_F[0]);
 295         GBR_clear(mu_F[1]);
 296         GBR_clear(two);
 297         GBR_clear(one);
 298
 299         isl_int_clear(tmp);
 300         isl_int_clear(mu[0]);
 301         isl_int_clear(mu[1]);
 302
 303         tab->basis = B;
 304
 305         return tab;
 306 }
 307
 308 /* Compute an affine form of a reduced basis of the given basic
 309  * non-parametric set, which is assumed to be bounded and not
 310  * include any integer divisions.
 311  * The first column and the first row correspond to the constant term.
 312  *
 313  * If the input contains any equalities, we first create an initial
 314  * basis with the equalities first.  Otherwise, we start off with
 315  * the identity matrix.
 316  */
 317 struct isl_mat *isl_basic_set_reduced_basis(struct isl_basic_set *bset)
 318 {
 319         struct isl_mat *basis;
 320         struct isl_tab *tab;
 321
 322         if (!bset)
 323                 return NULL;
 324
 325         if (isl_basic_set_dim(bset, isl_dim_div) != 0)
 326                 isl_die(bset->ctx, isl_error_invalid,
 327                         "no integer division allowed", return NULL);
 328         if (isl_basic_set_dim(bset, isl_dim_param) != 0)
 329                 isl_die(bset->ctx, isl_error_invalid,
 330                         "no parameters allowed", return NULL);
 331
 332         tab = isl_tab_from_basic_set(bset, 0);
 333         if (!tab)
 334                 return NULL;
 335
 336         if (bset->n_eq == 0)
 337                 tab->basis = isl_mat_identity(bset->ctx, 1 + tab->n_var);
 338         else {
 339                 isl_mat *eq;
 340                 unsigned nvar = isl_basic_set_total_dim(bset);
 341                 eq = isl_mat_sub_alloc6(bset->ctx, bset->eq, 0, bset->n_eq,
 342                                         1, nvar);
 343                 eq = isl_mat_left_hermite(eq, 0, NULL, &tab->basis);
 344                 tab->basis = isl_mat_lin_to_aff(tab->basis);
 345                 tab->n_zero = bset->n_eq;
 346                 isl_mat_free(eq);
 347         }
 348         tab = isl_tab_compute_reduced_basis(tab);
 349         if (!tab)
 350                 return NULL;
 351
 352         basis = isl_mat_copy(tab->basis);
 353
 354         isl_tab_free(tab);
 355
 356         return basis;
 357 }