isl_ilp.c

   1 #include "isl_ilp.h"
   2 #include "isl_map_private.h"
   3 #include "isl_sample.h"
   4 #include "isl_seq.h"
   5 #include "isl_equalities.h"
   6
   7 /* Given a basic set "bset", construct a basic set U such that for
   8  * each element x in U, the whole unit box positioned at x is inside
   9  * the given basic set.
  10  * Note that U may not contain all points that satisfy this property.
  11  *
  12  * We simply add the sum of all negative coefficients to the constant
  13  * term.  This ensures that if x satisfies the resulting constraints,
  14  * then x plus any sum of unit vectors satisfies the original constraints.
  15  */
  16 static struct isl_basic_set *unit_box_base_points(struct isl_basic_set *bset)
  17 {
  18         int i, j, k;
  19         struct isl_basic_set *unit_box = NULL;
  20         unsigned total;
  21
  22         if (!bset)
  23                 goto error;
  24
  25         if (bset->n_eq != 0) {
  26                 unit_box = isl_basic_set_empty_like(bset);
  27                 isl_basic_set_free(bset);
  28                 return unit_box;
  29         }
  30
  31         total = isl_basic_set_total_dim(bset);
  32         unit_box = isl_basic_set_alloc_dim(isl_basic_set_get_dim(bset),
  33                                         0, 0, bset->n_ineq);
  34
  35         for (i = 0; i < bset->n_ineq; ++i) {
  36                 k = isl_basic_set_alloc_inequality(unit_box);
  37                 if (k < 0)
  38                         goto error;
  39                 isl_seq_cpy(unit_box->ineq[k], bset->ineq[i], 1 + total);
  40                 for (j = 0; j < total; ++j) {
  41                         if (isl_int_is_nonneg(unit_box->ineq[k][1 + j]))
  42                                 continue;
  43                         isl_int_add(unit_box->ineq[k][0],
  44                                 unit_box->ineq[k][0], unit_box->ineq[k][1 + j]);
  45                 }
  46         }
  47
  48         isl_basic_set_free(bset);
  49         return unit_box;
  50 error:
  51         isl_basic_set_free(bset);
  52         isl_basic_set_free(unit_box);
  53         return NULL;
  54 }
  55
  56 /* Find an integer point in "bset", preferably one that is
  57  * close to minimizing "f".
  58  *
  59  * We first check if we can easily put unit boxes inside bset.
  60  * If so, we take the best base point of any of the unit boxes we can find
  61  * and round it up to the nearest integer.
  62  * If not, we simply pick any integer point in "bset".
  63  */
  64 static struct isl_vec *initial_solution(struct isl_basic_set *bset, isl_int *f)
  65 {
  66         enum isl_lp_result res;
  67         struct isl_basic_set *unit_box;
  68         struct isl_vec *sol;
  69
  70         unit_box = unit_box_base_points(isl_basic_set_copy(bset));
  71
  72         res = isl_basic_set_solve_lp(unit_box, 0, f, bset->ctx->one,
  73                                         NULL, NULL, &sol);
  74         if (res == isl_lp_ok) {
  75                 isl_basic_set_free(unit_box);
  76                 return isl_vec_ceil(sol);
  77         }
  78
  79         isl_basic_set_free(unit_box);
  80
  81         return isl_basic_set_sample(isl_basic_set_copy(bset));
  82 }
  83
  84 /* Restrict "bset" to those points with values for f in the interval [l, u].
  85  */
  86 static struct isl_basic_set *add_bounds(struct isl_basic_set *bset,
  87         isl_int *f, isl_int l, isl_int u)
  88 {
  89         int k;
  90         unsigned total;
  91
  92         total = isl_basic_set_total_dim(bset);
  93         bset = isl_basic_set_extend_constraints(bset, 0, 2);
  94
  95         k = isl_basic_set_alloc_inequality(bset);
  96         if (k < 0)
  97                 goto error;
  98         isl_seq_cpy(bset->ineq[k], f, 1 + total);
  99         isl_int_sub(bset->ineq[k][0], bset->ineq[k][0], l);
 100
 101         k = isl_basic_set_alloc_inequality(bset);
 102         if (k < 0)
 103                 goto error;
 104         isl_seq_neg(bset->ineq[k], f, 1 + total);
 105         isl_int_add(bset->ineq[k][0], bset->ineq[k][0], u);
 106
 107         return bset;
 108 error:
 109         isl_basic_set_free(bset);
 110         return NULL;
 111 }
 112
 113 /* Find an integer point in "bset" that minimizes f (if any).
 114  * If sol_p is not NULL then the integer point is returned in *sol_p.
 115  * The optimal value of f is returned in *opt.
 116  *
 117  * The algorithm maintains a currently best solution and an interval [l, u]
 118  * of values of f for which integer solutions could potentially still be found.
 119  * The initial value of the best solution so far is any solution.
 120  * The initial value of l is minimal value of f over the rationals
 121  * (rounded up to the nearest integer).
 122  * The initial value of u is the value of f at the current solution minus 1.
 123  *
 124  * We perform a number of steps until l > u.
 125  * In each step, we look for an integer point with value in either
 126  * the whole interval [l, u] or half of the interval [l, l+floor(u-l-1/2)].
 127  * The choice depends on whether we have found an integer point in the
 128  * previous step.  If so, we look for the next point in half of the remaining
 129  * interval.
 130  * If we find a point, the current solution is updated and u is set
 131  * to its value minus 1.
 132  * If no point can be found, we update l to the upper bound of the interval
 133  * we checked (u or l+floor(u-l-1/2)) plus 1.
 134  */
 135 static enum isl_lp_result solve_ilp(struct isl_basic_set *bset,
 136                                       isl_int *f, isl_int *opt,
 137                                       struct isl_vec **sol_p)
 138 {
 139         enum isl_lp_result res;
 140         isl_int l, u, tmp;
 141         struct isl_vec *sol;
 142         int divide = 1;
 143
 144         res = isl_basic_set_solve_lp(bset, 0, f, bset->ctx->one,
 145                                         opt, NULL, &sol);
 146         if (res == isl_lp_ok && isl_int_is_one(sol->el[0])) {
 147                 if (sol_p)
 148                         *sol_p = sol;
 149                 else
 150                         isl_vec_free(sol);
 151                 return isl_lp_ok;
 152         }
 153         isl_vec_free(sol);
 154         if (res == isl_lp_error || res == isl_lp_empty)
 155                 return res;
 156
 157         sol = initial_solution(bset, f);
 158         if (!sol)
 159                 return isl_lp_error;
 160         if (sol->size == 0) {
 161                 isl_vec_free(sol);
 162                 return isl_lp_empty;
 163         }
 164         if (res == isl_lp_unbounded) {
 165                 isl_vec_free(sol);
 166                 return isl_lp_unbounded;
 167         }
 168
 169         isl_int_init(l);
 170         isl_int_init(u);
 171         isl_int_init(tmp);
 172
 173         isl_int_set(l, *opt);
 174
 175         isl_seq_inner_product(f, sol->el, sol->size, opt);
 176         isl_int_sub_ui(u, *opt, 1);
 177
 178         while (isl_int_le(l, u)) {
 179                 struct isl_basic_set *slice;
 180                 struct isl_vec *sample;
 181
 182                 if (!divide)
 183                         isl_int_set(tmp, u);
 184                 else {
 185                         isl_int_sub(tmp, u, l);
 186                         isl_int_fdiv_q_ui(tmp, tmp, 2);
 187                         isl_int_add(tmp, tmp, l);
 188                 }
 189                 slice = add_bounds(isl_basic_set_copy(bset), f, l, tmp);
 190                 sample = isl_basic_set_sample(slice);
 191                 if (!sample) {
 192                         isl_vec_free(sol);
 193                         sol = NULL;
 194                         res = isl_lp_error;
 195                         break;
 196                 }
 197                 if (sample->size > 0) {
 198                         isl_vec_free(sol);
 199                         sol = sample;
 200                         isl_seq_inner_product(f, sol->el, sol->size, opt);
 201                         isl_int_sub_ui(u, *opt, 1);
 202                         divide = 1;
 203                 } else {
 204                         isl_vec_free(sample);
 205                         if (!divide)
 206                                 break;
 207                         isl_int_add_ui(l, tmp, 1);
 208                         divide = 0;
 209                 }
 210         }
 211
 212         isl_int_clear(l);
 213         isl_int_clear(u);
 214         isl_int_clear(tmp);
 215
 216         if (sol_p)
 217                 *sol_p = sol;
 218         else
 219                 isl_vec_free(sol);
 220
 221         return res;
 222 }
 223
 224 enum isl_lp_result solve_ilp_with_eq(struct isl_basic_set *bset, int max,
 225                                       isl_int *f, isl_int *opt,
 226                                       struct isl_vec **sol_p)
 227 {
 228         unsigned dim;
 229         enum isl_lp_result res;
 230         struct isl_mat *T = NULL;
 231         struct isl_vec *v;
 232
 233         dim = isl_basic_set_total_dim(bset);
 234         v = isl_vec_alloc(bset->ctx, 1 + dim);
 235         if (!v)
 236                 goto error;
 237         isl_seq_cpy(v->el, f, 1 + dim);
 238         bset = isl_basic_set_remove_equalities(bset, &T, NULL);
 239         v = isl_vec_mat_product(v, isl_mat_copy(T));
 240         if (!v)
 241                 goto error;
 242         res = isl_basic_set_solve_ilp(bset, max, v->el, opt, sol_p);
 243         isl_vec_free(v);
 244         if (res == isl_lp_ok && *sol_p) {
 245                 *sol_p = isl_mat_vec_product(T, *sol_p);
 246                 if (!*sol_p)
 247                         res = isl_lp_error;
 248         } else
 249                 isl_mat_free(T);
 250         return res;
 251 error:
 252         isl_mat_free(T);
 253         isl_basic_set_free(bset);
 254         return isl_lp_error;
 255 }
 256
 257 /* Find an integer point in "bset" that minimizes (or maximizes if max is set)
 258  * f (if any).
 259  * If sol_p is not NULL then the integer point is returned in *sol_p.
 260  * The optimal value of f is returned in *opt.
 261  *
 262  * If there is any equality among the points in "bset", then we first
 263  * project it out.  Otherwise, we continue with solve_ilp above.
 264  */
 265 enum isl_lp_result isl_basic_set_solve_ilp(struct isl_basic_set *bset, int max,
 266                                       isl_int *f, isl_int *opt,
 267                                       struct isl_vec **sol_p)
 268 {
 269         unsigned dim;
 270         enum isl_lp_result res;
 271
 272         if (!bset)
 273                 return isl_lp_error;
 274         if (sol_p)
 275                 *sol_p = NULL;
 276
 277         isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
 278
 279         if (bset->n_eq)
 280                 return solve_ilp_with_eq(bset, max, f, opt, sol_p);
 281
 282         dim = isl_basic_set_total_dim(bset);
 283
 284         if (max)
 285                 isl_seq_neg(f, f, 1 + dim);
 286
 287         res = solve_ilp(bset, f, opt, sol_p);
 288
 289         if (max) {
 290                 isl_seq_neg(f, f, 1 + dim);
 291                 isl_int_neg(*opt, *opt);
 292         }
 293
 294         return res;
 295 error:
 296         isl_basic_set_free(bset);
 297         return isl_lp_error;
 298 }