isl_bernstein.c

   1 /*
   2  * Copyright 2006-2007 Universiteit Leiden
   3  * Copyright 2008-2009 Katholieke Universiteit Leuven
   4  * Copyright 2010      INRIA Saclay
   5  *
   6  * Use of this software is governed by the GNU LGPLv2.1 license
   7  *
   8  * Written by Sven Verdoolaege, Leiden Institute of Advanced Computer Science,
   9  * Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
  10  * and K.U.Leuven, Departement Computerwetenschappen, Celestijnenlaan 200A,
  11  * B-3001 Leuven, Belgium
  12  * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
  13  * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France
  14  */
  15
  16 #include <isl_ctx_private.h>
  17 #include <isl_map_private.h>
  18 #include <isl/set.h>
  19 #include <isl/seq.h>
  20 #include <isl_morph.h>
  21 #include <isl_factorization.h>
  22 #include <isl_vertices_private.h>
  23 #include <isl_polynomial_private.h>
  24 #include <isl_bernstein.h>
  25
  26 struct bernstein_data {
  27         enum isl_fold type;
  28         isl_qpolynomial *poly;
  29         int check_tight;
  30
  31         isl_cell *cell;
  32
  33         isl_qpolynomial_fold *fold;
  34         isl_qpolynomial_fold *fold_tight;
  35         isl_pw_qpolynomial_fold *pwf;
  36         isl_pw_qpolynomial_fold *pwf_tight;
  37 };
  38
  39 static int vertex_is_integral(__isl_keep isl_basic_set *vertex)
  40 {
  41         unsigned nvar;
  42         unsigned nparam;
  43         int i;
  44
  45         nvar = isl_basic_set_dim(vertex, isl_dim_set);
  46         nparam = isl_basic_set_dim(vertex, isl_dim_param);
  47         for (i = 0; i < nvar; ++i) {
  48                 int r = nvar - 1 - i;
  49                 if (!isl_int_is_one(vertex->eq[r][1 + nparam + i]) &&
  50                     !isl_int_is_negone(vertex->eq[r][1 + nparam + i]))
  51                         return 0;
  52         }
  53
  54         return 1;
  55 }
  56
  57 static __isl_give isl_qpolynomial *vertex_coordinate(
  58         __isl_keep isl_basic_set *vertex, int i, __isl_take isl_space *dim)
  59 {
  60         unsigned nvar;
  61         unsigned nparam;
  62         int r;
  63         isl_int denom;
  64         isl_qpolynomial *v;
  65
  66         nvar = isl_basic_set_dim(vertex, isl_dim_set);
  67         nparam = isl_basic_set_dim(vertex, isl_dim_param);
  68         r = nvar - 1 - i;
  69
  70         isl_int_init(denom);
  71         isl_int_set(denom, vertex->eq[r][1 + nparam + i]);
  72         isl_assert(vertex->ctx, !isl_int_is_zero(denom), goto error);
  73
  74         if (isl_int_is_pos(denom))
  75                 isl_seq_neg(vertex->eq[r], vertex->eq[r],
  76                                 1 + isl_basic_set_total_dim(vertex));
  77         else
  78                 isl_int_neg(denom, denom);
  79
  80         v = isl_qpolynomial_from_affine(dim, vertex->eq[r], denom);
  81         isl_int_clear(denom);
  82
  83         return v;
  84 error:
  85         isl_space_free(dim);
  86         isl_int_clear(denom);
  87         return NULL;
  88 }
  89
  90 /* Check whether the bound associated to the selection "k" is tight,
  91  * which is the case if we select exactly one vertex and if that vertex
  92  * is integral for all values of the parameters.
  93  */
  94 static int is_tight(int *k, int n, int d, isl_cell *cell)
  95 {
  96         int i;
  97
  98         for (i = 0; i < n; ++i) {
  99                 int v;
 100                 if (k[i] != d) {
 101                         if (k[i])
 102                                 return 0;
 103                         continue;
 104                 }
 105                 v = cell->ids[n - 1 - i];
 106                 return vertex_is_integral(cell->vertices->v[v].vertex);
 107         }
 108
 109         return 0;
 110 }
 111
 112 static void add_fold(__isl_take isl_qpolynomial *b, __isl_keep isl_set *dom,
 113         int *k, int n, int d, struct bernstein_data *data)
 114 {
 115         isl_qpolynomial_fold *fold;
 116
 117         fold = isl_qpolynomial_fold_alloc(data->type, b);
 118
 119         if (data->check_tight && is_tight(k, n, d, data->cell))
 120                 data->fold_tight = isl_qpolynomial_fold_fold_on_domain(dom,
 121                                                         data->fold_tight, fold);
 122         else
 123                 data->fold = isl_qpolynomial_fold_fold_on_domain(dom,
 124                                                         data->fold, fold);
 125 }
 126
 127 /* Extract the coefficients of the Bernstein base polynomials and store
 128  * them in data->fold and data->fold_tight.
 129  *
 130  * In particular, the coefficient of each monomial
 131  * of multi-degree (k[0], k[1], ..., k[n-1]) is divided by the corresponding
 132  * multinomial coefficient d!/k[0]! k[1]! ... k[n-1]!
 133  *
 134  * c[i] contains the coefficient of the selected powers of the first i+1 vars.
 135  * multinom[i] contains the partial multinomial coefficient.
 136  */
 137 static void extract_coefficients(isl_qpolynomial *poly,
 138         __isl_keep isl_set *dom, struct bernstein_data *data)
 139 {
 140         int i;
 141         int d;
 142         int n;
 143         isl_ctx *ctx;
 144         isl_qpolynomial **c = NULL;
 145         int *k = NULL;
 146         int *left = NULL;
 147         isl_vec *multinom = NULL;
 148
 149         if (!poly)
 150                 return;
 151
 152         ctx = isl_qpolynomial_get_ctx(poly);
 153         n = isl_qpolynomial_dim(poly, isl_dim_in);
 154         d = isl_qpolynomial_degree(poly);
 155         isl_assert(ctx, n >= 2, return);
 156
 157         c = isl_calloc_array(ctx, isl_qpolynomial *, n);
 158         k = isl_alloc_array(ctx, int, n);
 159         left = isl_alloc_array(ctx, int, n);
 160         multinom = isl_vec_alloc(ctx, n);
 161         if (!c || !k || !left || !multinom)
 162                 goto error;
 163
 164         isl_int_set_si(multinom->el[0], 1);
 165         for (k[0] = d; k[0] >= 0; --k[0]) {
 166                 int i = 1;
 167                 isl_qpolynomial_free(c[0]);
 168                 c[0] = isl_qpolynomial_coeff(poly, isl_dim_in, n - 1, k[0]);
 169                 left[0] = d - k[0];
 170                 k[1] = -1;
 171                 isl_int_set(multinom->el[1], multinom->el[0]);
 172                 while (i > 0) {
 173                         if (i == n - 1) {
 174                                 int j;
 175                                 isl_space *dim;
 176                                 isl_qpolynomial *b;
 177                                 isl_qpolynomial *f;
 178                                 for (j = 2; j <= left[i - 1]; ++j)
 179                                         isl_int_divexact_ui(multinom->el[i],
 180                                                 multinom->el[i], j);
 181                                 b = isl_qpolynomial_coeff(c[i - 1], isl_dim_in,
 182                                         n - 1 - i, left[i - 1]);
 183                                 b = isl_qpolynomial_project_domain_on_params(b);
 184                                 dim = isl_qpolynomial_get_domain_space(b);
 185                                 f = isl_qpolynomial_rat_cst_on_domain(dim, ctx->one,
 186                                         multinom->el[i]);
 187                                 b = isl_qpolynomial_mul(b, f);
 188                                 k[n - 1] = left[n - 2];
 189                                 add_fold(b, dom, k, n, d, data);
 190                                 --i;
 191                                 continue;
 192                         }
 193                         if (k[i] >= left[i - 1]) {
 194                                 --i;
 195                                 continue;
 196                         }
 197                         ++k[i];
 198                         if (k[i])
 199                                 isl_int_divexact_ui(multinom->el[i],
 200                                         multinom->el[i], k[i]);
 201                         isl_qpolynomial_free(c[i]);
 202                         c[i] = isl_qpolynomial_coeff(c[i - 1], isl_dim_in,
 203                                         n - 1 - i, k[i]);
 204                         left[i] = left[i - 1] - k[i];
 205                         k[i + 1] = -1;
 206                         isl_int_set(multinom->el[i + 1], multinom->el[i]);
 207                         ++i;
 208                 }
 209                 isl_int_mul_ui(multinom->el[0], multinom->el[0], k[0]);
 210         }
 211
 212         for (i = 0; i < n; ++i)
 213                 isl_qpolynomial_free(c[i]);
 214
 215         isl_vec_free(multinom);
 216         free(left);
 217         free(k);
 218         free(c);
 219         return;
 220 error:
 221         isl_vec_free(multinom);
 222         free(left);
 223         free(k);
 224         if (c)
 225                 for (i = 0; i < n; ++i)
 226                         isl_qpolynomial_free(c[i]);
 227         free(c);
 228         return;
 229 }
 230
 231 /* Perform bernstein expansion on the parametric vertices that are active
 232  * on "cell".
 233  *
 234  * data->poly has been homogenized in the calling function.
 235  *
 236  * We plug in the barycentric coordinates for the set variables
 237  *
 238  *              \vec x = \sum_i \alpha_i v_i(\vec p)
 239  *
 240  * and the constant "1 = \sum_i \alpha_i" for the homogeneous dimension.
 241  * Next, we extract the coefficients of the Bernstein base polynomials.
 242  */
 243 static int bernstein_coefficients_cell(__isl_take isl_cell *cell, void *user)
 244 {
 245         int i, j;
 246         struct bernstein_data *data = (struct bernstein_data *)user;
 247         isl_space *dim_param;
 248         isl_space *dim_dst;
 249         isl_qpolynomial *poly = data->poly;
 250         unsigned nvar;
 251         int n_vertices;
 252         isl_qpolynomial **subs;
 253         isl_pw_qpolynomial_fold *pwf;
 254         isl_set *dom;
 255         isl_ctx *ctx;
 256
 257         if (!poly)
 258                 goto error;
 259
 260         nvar = isl_qpolynomial_dim(poly, isl_dim_in) - 1;
 261         n_vertices = cell->n_vertices;
 262
 263         ctx = isl_qpolynomial_get_ctx(poly);
 264         if (n_vertices > nvar + 1 && ctx->opt->bernstein_triangulate)
 265                 return isl_cell_foreach_simplex(cell,
 266                                             &bernstein_coefficients_cell, user);
 267
 268         subs = isl_alloc_array(ctx, isl_qpolynomial *, 1 + nvar);
 269         if (!subs)
 270                 goto error;
 271
 272         dim_param = isl_basic_set_get_space(cell->dom);
 273         dim_dst = isl_qpolynomial_get_domain_space(poly);
 274         dim_dst = isl_space_add_dims(dim_dst, isl_dim_set, n_vertices);
 275
 276         for (i = 0; i < 1 + nvar; ++i)
 277                 subs[i] = isl_qpolynomial_zero_on_domain(isl_space_copy(dim_dst));
 278
 279         for (i = 0; i < n_vertices; ++i) {
 280                 isl_qpolynomial *c;
 281                 c = isl_qpolynomial_var_on_domain(isl_space_copy(dim_dst), isl_dim_set,
 282                                         1 + nvar + i);
 283                 for (j = 0; j < nvar; ++j) {
 284                         int k = cell->ids[i];
 285                         isl_qpolynomial *v;
 286                         v = vertex_coordinate(cell->vertices->v[k].vertex, j,
 287                                                 isl_space_copy(dim_param));
 288                         v = isl_qpolynomial_add_dims(v, isl_dim_in,
 289                                                         1 + nvar + n_vertices);
 290                         v = isl_qpolynomial_mul(v, isl_qpolynomial_copy(c));
 291                         subs[1 + j] = isl_qpolynomial_add(subs[1 + j], v);
 292                 }
 293                 subs[0] = isl_qpolynomial_add(subs[0], c);
 294         }
 295         isl_space_free(dim_dst);
 296
 297         poly = isl_qpolynomial_copy(poly);
 298
 299         poly = isl_qpolynomial_add_dims(poly, isl_dim_in, n_vertices);
 300         poly = isl_qpolynomial_substitute(poly, isl_dim_in, 0, 1 + nvar, subs);
 301         poly = isl_qpolynomial_drop_dims(poly, isl_dim_in, 0, 1 + nvar);
 302
 303         data->cell = cell;
 304         dom = isl_set_from_basic_set(isl_basic_set_copy(cell->dom));
 305         data->fold = isl_qpolynomial_fold_empty(data->type, isl_space_copy(dim_param));
 306         data->fold_tight = isl_qpolynomial_fold_empty(data->type, dim_param);
 307         extract_coefficients(poly, dom, data);
 308
 309         pwf = isl_pw_qpolynomial_fold_alloc(data->type, isl_set_copy(dom),
 310                                             data->fold);
 311         data->pwf = isl_pw_qpolynomial_fold_fold(data->pwf, pwf);
 312         pwf = isl_pw_qpolynomial_fold_alloc(data->type, dom, data->fold_tight);
 313         data->pwf_tight = isl_pw_qpolynomial_fold_fold(data->pwf_tight, pwf);
 314
 315         isl_qpolynomial_free(poly);
 316         isl_cell_free(cell);
 317         for (i = 0; i < 1 + nvar; ++i)
 318                 isl_qpolynomial_free(subs[i]);
 319         free(subs);
 320         return 0;
 321 error:
 322         isl_cell_free(cell);
 323         return -1;
 324 }
 325
 326 /* Base case of applying bernstein expansion.
 327  *
 328  * We compute the chamber decomposition of the parametric polytope "bset"
 329  * and then perform bernstein expansion on the parametric vertices
 330  * that are active on each chamber.
 331  */
 332 static __isl_give isl_pw_qpolynomial_fold *bernstein_coefficients_base(
 333         __isl_take isl_basic_set *bset,
 334         __isl_take isl_qpolynomial *poly, struct bernstein_data *data, int *tight)
 335 {
 336         unsigned nvar;
 337         isl_space *dim;
 338         isl_pw_qpolynomial_fold *pwf;
 339         isl_vertices *vertices;
 340         int covers;
 341
 342         nvar = isl_basic_set_dim(bset, isl_dim_set);
 343         if (nvar == 0) {
 344                 isl_set *dom;
 345                 isl_qpolynomial_fold *fold;
 346
 347                 fold = isl_qpolynomial_fold_alloc(data->type, poly);
 348                 dom = isl_set_from_basic_set(bset);
 349                 if (tight)
 350                         *tight = 1;
 351                 pwf = isl_pw_qpolynomial_fold_alloc(data->type, dom, fold);
 352                 return isl_pw_qpolynomial_fold_project_domain_on_params(pwf);
 353         }
 354
 355         if (isl_qpolynomial_is_zero(poly)) {
 356                 isl_set *dom;
 357                 isl_qpolynomial_fold *fold;
 358                 fold = isl_qpolynomial_fold_alloc(data->type, poly);
 359                 dom = isl_set_from_basic_set(bset);
 360                 pwf = isl_pw_qpolynomial_fold_alloc(data->type, dom, fold);
 361                 if (tight)
 362                         *tight = 1;
 363                 return isl_pw_qpolynomial_fold_project_domain_on_params(pwf);
 364         }
 365
 366         dim = isl_basic_set_get_space(bset);
 367         dim = isl_space_params(dim);
 368         dim = isl_space_from_domain(dim);
 369         dim = isl_space_add_dims(dim, isl_dim_set, 1);
 370         data->pwf = isl_pw_qpolynomial_fold_zero(isl_space_copy(dim), data->type);
 371         data->pwf_tight = isl_pw_qpolynomial_fold_zero(dim, data->type);
 372         data->poly = isl_qpolynomial_homogenize(isl_qpolynomial_copy(poly));
 373         vertices = isl_basic_set_compute_vertices(bset);
 374         isl_vertices_foreach_disjoint_cell(vertices,
 375                 &bernstein_coefficients_cell, data);
 376         isl_vertices_free(vertices);
 377         isl_qpolynomial_free(data->poly);
 378
 379         isl_basic_set_free(bset);
 380         isl_qpolynomial_free(poly);
 381
 382         covers = isl_pw_qpolynomial_fold_covers(data->pwf_tight, data->pwf);
 383         if (covers < 0)
 384                 goto error;
 385
 386         if (tight)
 387                 *tight = covers;
 388
 389         if (covers) {
 390                 isl_pw_qpolynomial_fold_free(data->pwf);
 391                 return data->pwf_tight;
 392         }
 393
 394         data->pwf = isl_pw_qpolynomial_fold_fold(data->pwf, data->pwf_tight);
 395
 396         return data->pwf;
 397 error:
 398         isl_pw_qpolynomial_fold_free(data->pwf_tight);
 399         isl_pw_qpolynomial_fold_free(data->pwf);
 400         return NULL;
 401 }
 402
 403 /* Apply bernstein expansion recursively by working in on len[i]
 404  * set variables at a time, with i ranging from n_group - 1 to 0.
 405  */
 406 static __isl_give isl_pw_qpolynomial_fold *bernstein_coefficients_recursive(
 407         __isl_take isl_pw_qpolynomial *pwqp,
 408         int n_group, int *len, struct bernstein_data *data, int *tight)
 409 {
 410         int i;
 411         unsigned nparam;
 412         unsigned nvar;
 413         isl_pw_qpolynomial_fold *pwf;
 414
 415         if (!pwqp)
 416                 return NULL;
 417
 418         nparam = isl_pw_qpolynomial_dim(pwqp, isl_dim_param);
 419         nvar = isl_pw_qpolynomial_dim(pwqp, isl_dim_in);
 420
 421         pwqp = isl_pw_qpolynomial_move_dims(pwqp, isl_dim_param, nparam,
 422                                         isl_dim_in, 0, nvar - len[n_group - 1]);
 423         pwf = isl_pw_qpolynomial_bound(pwqp, data->type, tight);
 424
 425         for (i = n_group - 2; i >= 0; --i) {
 426                 nparam = isl_pw_qpolynomial_fold_dim(pwf, isl_dim_param);
 427                 pwf = isl_pw_qpolynomial_fold_move_dims(pwf, isl_dim_in, 0,
 428                                 isl_dim_param, nparam - len[i], len[i]);
 429                 if (tight && !*tight)
 430                         tight = NULL;
 431                 pwf = isl_pw_qpolynomial_fold_bound(pwf, tight);
 432         }
 433
 434         return pwf;
 435 }
 436
 437 static __isl_give isl_pw_qpolynomial_fold *bernstein_coefficients_factors(
 438         __isl_take isl_basic_set *bset,
 439         __isl_take isl_qpolynomial *poly, struct bernstein_data *data, int *tight)
 440 {
 441         isl_factorizer *f;
 442         isl_set *set;
 443         isl_pw_qpolynomial *pwqp;
 444         isl_pw_qpolynomial_fold *pwf;
 445
 446         f = isl_basic_set_factorizer(bset);
 447         if (!f)
 448                 goto error;
 449         if (f->n_group == 0) {
 450                 isl_factorizer_free(f);
 451                 return  bernstein_coefficients_base(bset, poly, data, tight);
 452         }
 453
 454         set = isl_set_from_basic_set(bset);
 455         pwqp = isl_pw_qpolynomial_alloc(set, poly);
 456         pwqp = isl_pw_qpolynomial_morph_domain(pwqp, isl_morph_copy(f->morph));
 457
 458         pwf = bernstein_coefficients_recursive(pwqp, f->n_group, f->len, data,
 459                                                 tight);
 460
 461         isl_factorizer_free(f);
 462
 463         return pwf;
 464 error:
 465         isl_basic_set_free(bset);
 466         isl_qpolynomial_free(poly);
 467         return NULL;
 468 }
 469
 470 static __isl_give isl_pw_qpolynomial_fold *bernstein_coefficients_full_recursive(
 471         __isl_take isl_basic_set *bset,
 472         __isl_take isl_qpolynomial *poly, struct bernstein_data *data, int *tight)
 473 {
 474         int i;
 475         int *len;
 476         unsigned nvar;
 477         isl_pw_qpolynomial_fold *pwf;
 478         isl_set *set;
 479         isl_pw_qpolynomial *pwqp;
 480
 481         if (!bset || !poly)
 482                 goto error;
 483
 484         nvar = isl_basic_set_dim(bset, isl_dim_set);
 485
 486         len = isl_alloc_array(bset->ctx, int, nvar);
 487         if (!len)
 488                 goto error;
 489
 490         for (i = 0; i < nvar; ++i)
 491                 len[i] = 1;
 492
 493         set = isl_set_from_basic_set(bset);
 494         pwqp = isl_pw_qpolynomial_alloc(set, poly);
 495
 496         pwf = bernstein_coefficients_recursive(pwqp, nvar, len, data, tight);
 497
 498         free(len);
 499
 500         return pwf;
 501 error:
 502         isl_basic_set_free(bset);
 503         isl_qpolynomial_free(poly);
 504         return NULL;
 505 }
 506
 507 /* Compute a bound on the polynomial defined over the parametric polytope
 508  * using bernstein expansion and store the result
 509  * in bound->pwf and bound->pwf_tight.
 510  *
 511  * If bernstein_recurse is set to ISL_BERNSTEIN_FACTORS, we check if
 512  * the polytope can be factorized and apply bernstein expansion recursively
 513  * on the factors.
 514  * If bernstein_recurse is set to ISL_BERNSTEIN_INTERVALS, we apply
 515  * bernstein expansion recursively on each dimension.
 516  * Otherwise, we apply bernstein expansion on the entire polytope.
 517  */
 518 int isl_qpolynomial_bound_on_domain_bernstein(__isl_take isl_basic_set *bset,
 519         __isl_take isl_qpolynomial *poly, struct isl_bound *bound)
 520 {
 521         struct bernstein_data data;
 522         isl_pw_qpolynomial_fold *pwf;
 523         unsigned nvar;
 524         int tight = 0;
 525         int *tp = bound->check_tight ? &tight : NULL;
 526
 527         if (!bset || !poly)
 528                 goto error;
 529
 530         data.type = bound->type;
 531         data.check_tight = bound->check_tight;
 532
 533         nvar = isl_basic_set_dim(bset, isl_dim_set);
 534
 535         if (bset->ctx->opt->bernstein_recurse & ISL_BERNSTEIN_FACTORS)
 536                 pwf = bernstein_coefficients_factors(bset, poly, &data, tp);
 537         else if (nvar > 1 &&
 538             (bset->ctx->opt->bernstein_recurse & ISL_BERNSTEIN_INTERVALS))
 539                 pwf = bernstein_coefficients_full_recursive(bset, poly, &data, tp);
 540         else
 541                 pwf = bernstein_coefficients_base(bset, poly, &data, tp);
 542
 543         if (tight)
 544                 bound->pwf_tight = isl_pw_qpolynomial_fold_fold(bound->pwf_tight, pwf);
 545         else
 546                 bound->pwf = isl_pw_qpolynomial_fold_fold(bound->pwf, pwf);
 547
 548         return 0;
 549 error:
 550         isl_basic_set_free(bset);
 551         isl_qpolynomial_free(poly);
 552         return -1;
 553 }