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[isl.git] / isl_sample.c
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1 /*
2 * Copyright 2008-2009 Katholieke Universiteit Leuven
4 * Use of this software is governed by the MIT license
6 * Written by Sven Verdoolaege, K.U.Leuven, Departement
7 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
8 */
10 #include <isl_ctx_private.h>
11 #include <isl_map_private.h>
12 #include "isl_sample.h"
13 #include <isl/vec.h>
14 #include <isl/mat.h>
15 #include <isl_seq.h>
16 #include "isl_equalities.h"
17 #include "isl_tab.h"
18 #include "isl_basis_reduction.h"
19 #include <isl_factorization.h>
20 #include <isl_point_private.h>
21 #include <isl_options_private.h>
22 #include <isl_vec_private.h>
24 static struct isl_vec *empty_sample(struct isl_basic_set *bset)
26 struct isl_vec *vec;
28 vec = isl_vec_alloc(bset->ctx, 0);
29 isl_basic_set_free(bset);
30 return vec;
33 /* Construct a zero sample of the same dimension as bset.
34 * As a special case, if bset is zero-dimensional, this
35 * function creates a zero-dimensional sample point.
37 static struct isl_vec *zero_sample(struct isl_basic_set *bset)
39 unsigned dim;
40 struct isl_vec *sample;
42 dim = isl_basic_set_total_dim(bset);
43 sample = isl_vec_alloc(bset->ctx, 1 + dim);
44 if (sample) {
45 isl_int_set_si(sample->el[0], 1);
46 isl_seq_clr(sample->el + 1, dim);
48 isl_basic_set_free(bset);
49 return sample;
52 static struct isl_vec *interval_sample(struct isl_basic_set *bset)
54 int i;
55 isl_int t;
56 struct isl_vec *sample;
58 bset = isl_basic_set_simplify(bset);
59 if (!bset)
60 return NULL;
61 if (isl_basic_set_plain_is_empty(bset))
62 return empty_sample(bset);
63 if (bset->n_eq == 0 && bset->n_ineq == 0)
64 return zero_sample(bset);
66 sample = isl_vec_alloc(bset->ctx, 2);
67 if (!sample)
68 goto error;
69 if (!bset)
70 return NULL;
71 isl_int_set_si(sample->block.data[0], 1);
73 if (bset->n_eq > 0) {
74 isl_assert(bset->ctx, bset->n_eq == 1, goto error);
75 isl_assert(bset->ctx, bset->n_ineq == 0, goto error);
76 if (isl_int_is_one(bset->eq[0][1]))
77 isl_int_neg(sample->el[1], bset->eq[0][0]);
78 else {
79 isl_assert(bset->ctx, isl_int_is_negone(bset->eq[0][1]),
80 goto error);
81 isl_int_set(sample->el[1], bset->eq[0][0]);
83 isl_basic_set_free(bset);
84 return sample;
87 isl_int_init(t);
88 if (isl_int_is_one(bset->ineq[0][1]))
89 isl_int_neg(sample->block.data[1], bset->ineq[0][0]);
90 else
91 isl_int_set(sample->block.data[1], bset->ineq[0][0]);
92 for (i = 1; i < bset->n_ineq; ++i) {
93 isl_seq_inner_product(sample->block.data,
94 bset->ineq[i], 2, &t);
95 if (isl_int_is_neg(t))
96 break;
98 isl_int_clear(t);
99 if (i < bset->n_ineq) {
100 isl_vec_free(sample);
101 return empty_sample(bset);
104 isl_basic_set_free(bset);
105 return sample;
106 error:
107 isl_basic_set_free(bset);
108 isl_vec_free(sample);
109 return NULL;
112 /* Find a sample integer point, if any, in bset, which is known
113 * to have equalities. If bset contains no integer points, then
114 * return a zero-length vector.
115 * We simply remove the known equalities, compute a sample
116 * in the resulting bset, using the specified recurse function,
117 * and then transform the sample back to the original space.
119 static struct isl_vec *sample_eq(struct isl_basic_set *bset,
120 struct isl_vec *(*recurse)(struct isl_basic_set *))
122 struct isl_mat *T;
123 struct isl_vec *sample;
125 if (!bset)
126 return NULL;
128 bset = isl_basic_set_remove_equalities(bset, &T, NULL);
129 sample = recurse(bset);
130 if (!sample || sample->size == 0)
131 isl_mat_free(T);
132 else
133 sample = isl_mat_vec_product(T, sample);
134 return sample;
137 /* Return a matrix containing the equalities of the tableau
138 * in constraint form. The tableau is assumed to have
139 * an associated bset that has been kept up-to-date.
141 static struct isl_mat *tab_equalities(struct isl_tab *tab)
143 int i, j;
144 int n_eq;
145 struct isl_mat *eq;
146 struct isl_basic_set *bset;
148 if (!tab)
149 return NULL;
151 bset = isl_tab_peek_bset(tab);
152 isl_assert(tab->mat->ctx, bset, return NULL);
154 n_eq = tab->n_var - tab->n_col + tab->n_dead;
155 if (tab->empty || n_eq == 0)
156 return isl_mat_alloc(tab->mat->ctx, 0, tab->n_var);
157 if (n_eq == tab->n_var)
158 return isl_mat_identity(tab->mat->ctx, tab->n_var);
160 eq = isl_mat_alloc(tab->mat->ctx, n_eq, tab->n_var);
161 if (!eq)
162 return NULL;
163 for (i = 0, j = 0; i < tab->n_con; ++i) {
164 if (tab->con[i].is_row)
165 continue;
166 if (tab->con[i].index >= 0 && tab->con[i].index >= tab->n_dead)
167 continue;
168 if (i < bset->n_eq)
169 isl_seq_cpy(eq->row[j], bset->eq[i] + 1, tab->n_var);
170 else
171 isl_seq_cpy(eq->row[j],
172 bset->ineq[i - bset->n_eq] + 1, tab->n_var);
173 ++j;
175 isl_assert(bset->ctx, j == n_eq, goto error);
176 return eq;
177 error:
178 isl_mat_free(eq);
179 return NULL;
182 /* Compute and return an initial basis for the bounded tableau "tab".
184 * If the tableau is either full-dimensional or zero-dimensional,
185 * the we simply return an identity matrix.
186 * Otherwise, we construct a basis whose first directions correspond
187 * to equalities.
189 static struct isl_mat *initial_basis(struct isl_tab *tab)
191 int n_eq;
192 struct isl_mat *eq;
193 struct isl_mat *Q;
195 tab->n_unbounded = 0;
196 tab->n_zero = n_eq = tab->n_var - tab->n_col + tab->n_dead;
197 if (tab->empty || n_eq == 0 || n_eq == tab->n_var)
198 return isl_mat_identity(tab->mat->ctx, 1 + tab->n_var);
200 eq = tab_equalities(tab);
201 eq = isl_mat_left_hermite(eq, 0, NULL, &Q);
202 if (!eq)
203 return NULL;
204 isl_mat_free(eq);
206 Q = isl_mat_lin_to_aff(Q);
207 return Q;
210 /* Compute the minimum of the current ("level") basis row over "tab"
211 * and store the result in position "level" of "min".
213 static enum isl_lp_result compute_min(isl_ctx *ctx, struct isl_tab *tab,
214 __isl_keep isl_vec *min, int level)
216 return isl_tab_min(tab, tab->basis->row[1 + level],
217 ctx->one, &min->el[level], NULL, 0);
220 /* Compute the maximum of the current ("level") basis row over "tab"
221 * and store the result in position "level" of "max".
223 static enum isl_lp_result compute_max(isl_ctx *ctx, struct isl_tab *tab,
224 __isl_keep isl_vec *max, int level)
226 enum isl_lp_result res;
227 unsigned dim = tab->n_var;
229 isl_seq_neg(tab->basis->row[1 + level] + 1,
230 tab->basis->row[1 + level] + 1, dim);
231 res = isl_tab_min(tab, tab->basis->row[1 + level],
232 ctx->one, &max->el[level], NULL, 0);
233 isl_seq_neg(tab->basis->row[1 + level] + 1,
234 tab->basis->row[1 + level] + 1, dim);
235 isl_int_neg(max->el[level], max->el[level]);
237 return res;
240 /* Perform a greedy search for an integer point in the set represented
241 * by "tab", given that the minimal rational value (rounded up to the
242 * nearest integer) at "level" is smaller than the maximal rational
243 * value (rounded down to the nearest integer).
245 * Return 1 if we have found an integer point (if tab->n_unbounded > 0
246 * then we may have only found integer values for the bounded dimensions
247 * and it is the responsibility of the caller to extend this solution
248 * to the unbounded dimensions).
249 * Return 0 if greedy search did not result in a solution.
250 * Return -1 if some error occurred.
252 * We assign a value half-way between the minimum and the maximum
253 * to the current dimension and check if the minimal value of the
254 * next dimension is still smaller than (or equal) to the maximal value.
255 * We continue this process until either
256 * - the minimal value (rounded up) is greater than the maximal value
257 * (rounded down). In this case, greedy search has failed.
258 * - we have exhausted all bounded dimensions, meaning that we have
259 * found a solution.
260 * - the sample value of the tableau is integral.
261 * - some error has occurred.
263 static int greedy_search(isl_ctx *ctx, struct isl_tab *tab,
264 __isl_keep isl_vec *min, __isl_keep isl_vec *max, int level)
266 struct isl_tab_undo *snap;
267 enum isl_lp_result res;
269 snap = isl_tab_snap(tab);
271 do {
272 isl_int_add(tab->basis->row[1 + level][0],
273 min->el[level], max->el[level]);
274 isl_int_fdiv_q_ui(tab->basis->row[1 + level][0],
275 tab->basis->row[1 + level][0], 2);
276 isl_int_neg(tab->basis->row[1 + level][0],
277 tab->basis->row[1 + level][0]);
278 if (isl_tab_add_valid_eq(tab, tab->basis->row[1 + level]) < 0)
279 return -1;
280 isl_int_set_si(tab->basis->row[1 + level][0], 0);
282 if (++level >= tab->n_var - tab->n_unbounded)
283 return 1;
284 if (isl_tab_sample_is_integer(tab))
285 return 1;
287 res = compute_min(ctx, tab, min, level);
288 if (res == isl_lp_error)
289 return -1;
290 if (res != isl_lp_ok)
291 isl_die(ctx, isl_error_internal,
292 "expecting bounded rational solution",
293 return -1);
294 res = compute_max(ctx, tab, max, level);
295 if (res == isl_lp_error)
296 return -1;
297 if (res != isl_lp_ok)
298 isl_die(ctx, isl_error_internal,
299 "expecting bounded rational solution",
300 return -1);
301 } while (isl_int_le(min->el[level], max->el[level]));
303 if (isl_tab_rollback(tab, snap) < 0)
304 return -1;
306 return 0;
309 /* Given a tableau representing a set, find and return
310 * an integer point in the set, if there is any.
312 * We perform a depth first search
313 * for an integer point, by scanning all possible values in the range
314 * attained by a basis vector, where an initial basis may have been set
315 * by the calling function. Otherwise an initial basis that exploits
316 * the equalities in the tableau is created.
317 * tab->n_zero is currently ignored and is clobbered by this function.
319 * The tableau is allowed to have unbounded direction, but then
320 * the calling function needs to set an initial basis, with the
321 * unbounded directions last and with tab->n_unbounded set
322 * to the number of unbounded directions.
323 * Furthermore, the calling functions needs to add shifted copies
324 * of all constraints involving unbounded directions to ensure
325 * that any feasible rational value in these directions can be rounded
326 * up to yield a feasible integer value.
327 * In particular, let B define the given basis x' = B x
328 * and let T be the inverse of B, i.e., X = T x'.
329 * Let a x + c >= 0 be a constraint of the set represented by the tableau,
330 * or a T x' + c >= 0 in terms of the given basis. Assume that
331 * the bounded directions have an integer value, then we can safely
332 * round up the values for the unbounded directions if we make sure
333 * that x' not only satisfies the original constraint, but also
334 * the constraint "a T x' + c + s >= 0" with s the sum of all
335 * negative values in the last n_unbounded entries of "a T".
336 * The calling function therefore needs to add the constraint
337 * a x + c + s >= 0. The current function then scans the first
338 * directions for an integer value and once those have been found,
339 * it can compute "T ceil(B x)" to yield an integer point in the set.
340 * Note that during the search, the first rows of B may be changed
341 * by a basis reduction, but the last n_unbounded rows of B remain
342 * unaltered and are also not mixed into the first rows.
344 * The search is implemented iteratively. "level" identifies the current
345 * basis vector. "init" is true if we want the first value at the current
346 * level and false if we want the next value.
348 * At the start of each level, we first check if we can find a solution
349 * using greedy search. If not, we continue with the exhaustive search.
351 * The initial basis is the identity matrix. If the range in some direction
352 * contains more than one integer value, we perform basis reduction based
353 * on the value of ctx->opt->gbr
354 * - ISL_GBR_NEVER: never perform basis reduction
355 * - ISL_GBR_ONCE: only perform basis reduction the first
356 * time such a range is encountered
357 * - ISL_GBR_ALWAYS: always perform basis reduction when
358 * such a range is encountered
360 * When ctx->opt->gbr is set to ISL_GBR_ALWAYS, then we allow the basis
361 * reduction computation to return early. That is, as soon as it
362 * finds a reasonable first direction.
364 struct isl_vec *isl_tab_sample(struct isl_tab *tab)
366 unsigned dim;
367 unsigned gbr;
368 struct isl_ctx *ctx;
369 struct isl_vec *sample;
370 struct isl_vec *min;
371 struct isl_vec *max;
372 enum isl_lp_result res;
373 int level;
374 int init;
375 int reduced;
376 struct isl_tab_undo **snap;
378 if (!tab)
379 return NULL;
380 if (tab->empty)
381 return isl_vec_alloc(tab->mat->ctx, 0);
383 if (!tab->basis)
384 tab->basis = initial_basis(tab);
385 if (!tab->basis)
386 return NULL;
387 isl_assert(tab->mat->ctx, tab->basis->n_row == tab->n_var + 1,
388 return NULL);
389 isl_assert(tab->mat->ctx, tab->basis->n_col == tab->n_var + 1,
390 return NULL);
392 ctx = tab->mat->ctx;
393 dim = tab->n_var;
394 gbr = ctx->opt->gbr;
396 if (tab->n_unbounded == tab->n_var) {
397 sample = isl_tab_get_sample_value(tab);
398 sample = isl_mat_vec_product(isl_mat_copy(tab->basis), sample);
399 sample = isl_vec_ceil(sample);
400 sample = isl_mat_vec_inverse_product(isl_mat_copy(tab->basis),
401 sample);
402 return sample;
405 if (isl_tab_extend_cons(tab, dim + 1) < 0)
406 return NULL;
408 min = isl_vec_alloc(ctx, dim);
409 max = isl_vec_alloc(ctx, dim);
410 snap = isl_alloc_array(ctx, struct isl_tab_undo *, dim);
412 if (!min || !max || !snap)
413 goto error;
415 level = 0;
416 init = 1;
417 reduced = 0;
419 while (level >= 0) {
420 if (init) {
421 int choice;
423 res = compute_min(ctx, tab, min, level);
424 if (res == isl_lp_error)
425 goto error;
426 if (res != isl_lp_ok)
427 isl_die(ctx, isl_error_internal,
428 "expecting bounded rational solution",
429 goto error);
430 if (isl_tab_sample_is_integer(tab))
431 break;
432 res = compute_max(ctx, tab, max, level);
433 if (res == isl_lp_error)
434 goto error;
435 if (res != isl_lp_ok)
436 isl_die(ctx, isl_error_internal,
437 "expecting bounded rational solution",
438 goto error);
439 if (isl_tab_sample_is_integer(tab))
440 break;
441 choice = isl_int_lt(min->el[level], max->el[level]);
442 if (choice) {
443 int g;
444 g = greedy_search(ctx, tab, min, max, level);
445 if (g < 0)
446 goto error;
447 if (g)
448 break;
450 if (!reduced && choice &&
451 ctx->opt->gbr != ISL_GBR_NEVER) {
452 unsigned gbr_only_first;
453 if (ctx->opt->gbr == ISL_GBR_ONCE)
454 ctx->opt->gbr = ISL_GBR_NEVER;
455 tab->n_zero = level;
456 gbr_only_first = ctx->opt->gbr_only_first;
457 ctx->opt->gbr_only_first =
458 ctx->opt->gbr == ISL_GBR_ALWAYS;
459 tab = isl_tab_compute_reduced_basis(tab);
460 ctx->opt->gbr_only_first = gbr_only_first;
461 if (!tab || !tab->basis)
462 goto error;
463 reduced = 1;
464 continue;
466 reduced = 0;
467 snap[level] = isl_tab_snap(tab);
468 } else
469 isl_int_add_ui(min->el[level], min->el[level], 1);
471 if (isl_int_gt(min->el[level], max->el[level])) {
472 level--;
473 init = 0;
474 if (level >= 0)
475 if (isl_tab_rollback(tab, snap[level]) < 0)
476 goto error;
477 continue;
479 isl_int_neg(tab->basis->row[1 + level][0], min->el[level]);
480 if (isl_tab_add_valid_eq(tab, tab->basis->row[1 + level]) < 0)
481 goto error;
482 isl_int_set_si(tab->basis->row[1 + level][0], 0);
483 if (level + tab->n_unbounded < dim - 1) {
484 ++level;
485 init = 1;
486 continue;
488 break;
491 if (level >= 0) {
492 sample = isl_tab_get_sample_value(tab);
493 if (!sample)
494 goto error;
495 if (tab->n_unbounded && !isl_int_is_one(sample->el[0])) {
496 sample = isl_mat_vec_product(isl_mat_copy(tab->basis),
497 sample);
498 sample = isl_vec_ceil(sample);
499 sample = isl_mat_vec_inverse_product(
500 isl_mat_copy(tab->basis), sample);
502 } else
503 sample = isl_vec_alloc(ctx, 0);
505 ctx->opt->gbr = gbr;
506 isl_vec_free(min);
507 isl_vec_free(max);
508 free(snap);
509 return sample;
510 error:
511 ctx->opt->gbr = gbr;
512 isl_vec_free(min);
513 isl_vec_free(max);
514 free(snap);
515 return NULL;
518 static struct isl_vec *sample_bounded(struct isl_basic_set *bset);
520 /* Compute a sample point of the given basic set, based on the given,
521 * non-trivial factorization.
523 static __isl_give isl_vec *factored_sample(__isl_take isl_basic_set *bset,
524 __isl_take isl_factorizer *f)
526 int i, n;
527 isl_vec *sample = NULL;
528 isl_ctx *ctx;
529 unsigned nparam;
530 unsigned nvar;
532 ctx = isl_basic_set_get_ctx(bset);
533 if (!ctx)
534 goto error;
536 nparam = isl_basic_set_dim(bset, isl_dim_param);
537 nvar = isl_basic_set_dim(bset, isl_dim_set);
539 sample = isl_vec_alloc(ctx, 1 + isl_basic_set_total_dim(bset));
540 if (!sample)
541 goto error;
542 isl_int_set_si(sample->el[0], 1);
544 bset = isl_morph_basic_set(isl_morph_copy(f->morph), bset);
546 for (i = 0, n = 0; i < f->n_group; ++i) {
547 isl_basic_set *bset_i;
548 isl_vec *sample_i;
550 bset_i = isl_basic_set_copy(bset);
551 bset_i = isl_basic_set_drop_constraints_involving(bset_i,
552 nparam + n + f->len[i], nvar - n - f->len[i]);
553 bset_i = isl_basic_set_drop_constraints_involving(bset_i,
554 nparam, n);
555 bset_i = isl_basic_set_drop(bset_i, isl_dim_set,
556 n + f->len[i], nvar - n - f->len[i]);
557 bset_i = isl_basic_set_drop(bset_i, isl_dim_set, 0, n);
559 sample_i = sample_bounded(bset_i);
560 if (!sample_i)
561 goto error;
562 if (sample_i->size == 0) {
563 isl_basic_set_free(bset);
564 isl_factorizer_free(f);
565 isl_vec_free(sample);
566 return sample_i;
568 isl_seq_cpy(sample->el + 1 + nparam + n,
569 sample_i->el + 1, f->len[i]);
570 isl_vec_free(sample_i);
572 n += f->len[i];
575 f->morph = isl_morph_inverse(f->morph);
576 sample = isl_morph_vec(isl_morph_copy(f->morph), sample);
578 isl_basic_set_free(bset);
579 isl_factorizer_free(f);
580 return sample;
581 error:
582 isl_basic_set_free(bset);
583 isl_factorizer_free(f);
584 isl_vec_free(sample);
585 return NULL;
588 /* Given a basic set that is known to be bounded, find and return
589 * an integer point in the basic set, if there is any.
591 * After handling some trivial cases, we construct a tableau
592 * and then use isl_tab_sample to find a sample, passing it
593 * the identity matrix as initial basis.
595 static struct isl_vec *sample_bounded(struct isl_basic_set *bset)
597 unsigned dim;
598 struct isl_ctx *ctx;
599 struct isl_vec *sample;
600 struct isl_tab *tab = NULL;
601 isl_factorizer *f;
603 if (!bset)
604 return NULL;
606 if (isl_basic_set_plain_is_empty(bset))
607 return empty_sample(bset);
609 dim = isl_basic_set_total_dim(bset);
610 if (dim == 0)
611 return zero_sample(bset);
612 if (dim == 1)
613 return interval_sample(bset);
614 if (bset->n_eq > 0)
615 return sample_eq(bset, sample_bounded);
617 f = isl_basic_set_factorizer(bset);
618 if (!f)
619 goto error;
620 if (f->n_group != 0)
621 return factored_sample(bset, f);
622 isl_factorizer_free(f);
624 ctx = bset->ctx;
626 tab = isl_tab_from_basic_set(bset, 1);
627 if (tab && tab->empty) {
628 isl_tab_free(tab);
629 ISL_F_SET(bset, ISL_BASIC_SET_EMPTY);
630 sample = isl_vec_alloc(bset->ctx, 0);
631 isl_basic_set_free(bset);
632 return sample;
635 if (!ISL_F_ISSET(bset, ISL_BASIC_SET_NO_IMPLICIT))
636 if (isl_tab_detect_implicit_equalities(tab) < 0)
637 goto error;
639 sample = isl_tab_sample(tab);
640 if (!sample)
641 goto error;
643 if (sample->size > 0) {
644 isl_vec_free(bset->sample);
645 bset->sample = isl_vec_copy(sample);
648 isl_basic_set_free(bset);
649 isl_tab_free(tab);
650 return sample;
651 error:
652 isl_basic_set_free(bset);
653 isl_tab_free(tab);
654 return NULL;
657 /* Given a basic set "bset" and a value "sample" for the first coordinates
658 * of bset, plug in these values and drop the corresponding coordinates.
660 * We do this by computing the preimage of the transformation
662 * [ 1 0 ]
663 * x = [ s 0 ] x'
664 * [ 0 I ]
666 * where [1 s] is the sample value and I is the identity matrix of the
667 * appropriate dimension.
669 static struct isl_basic_set *plug_in(struct isl_basic_set *bset,
670 struct isl_vec *sample)
672 int i;
673 unsigned total;
674 struct isl_mat *T;
676 if (!bset || !sample)
677 goto error;
679 total = isl_basic_set_total_dim(bset);
680 T = isl_mat_alloc(bset->ctx, 1 + total, 1 + total - (sample->size - 1));
681 if (!T)
682 goto error;
684 for (i = 0; i < sample->size; ++i) {
685 isl_int_set(T->row[i][0], sample->el[i]);
686 isl_seq_clr(T->row[i] + 1, T->n_col - 1);
688 for (i = 0; i < T->n_col - 1; ++i) {
689 isl_seq_clr(T->row[sample->size + i], T->n_col);
690 isl_int_set_si(T->row[sample->size + i][1 + i], 1);
692 isl_vec_free(sample);
694 bset = isl_basic_set_preimage(bset, T);
695 return bset;
696 error:
697 isl_basic_set_free(bset);
698 isl_vec_free(sample);
699 return NULL;
702 /* Given a basic set "bset", return any (possibly non-integer) point
703 * in the basic set.
705 static struct isl_vec *rational_sample(struct isl_basic_set *bset)
707 struct isl_tab *tab;
708 struct isl_vec *sample;
710 if (!bset)
711 return NULL;
713 tab = isl_tab_from_basic_set(bset, 0);
714 sample = isl_tab_get_sample_value(tab);
715 isl_tab_free(tab);
717 isl_basic_set_free(bset);
719 return sample;
722 /* Given a linear cone "cone" and a rational point "vec",
723 * construct a polyhedron with shifted copies of the constraints in "cone",
724 * i.e., a polyhedron with "cone" as its recession cone, such that each
725 * point x in this polyhedron is such that the unit box positioned at x
726 * lies entirely inside the affine cone 'vec + cone'.
727 * Any rational point in this polyhedron may therefore be rounded up
728 * to yield an integer point that lies inside said affine cone.
730 * Denote the constraints of cone by "<a_i, x> >= 0" and the rational
731 * point "vec" by v/d.
732 * Let b_i = <a_i, v>. Then the affine cone 'vec + cone' is given
733 * by <a_i, x> - b/d >= 0.
734 * The polyhedron <a_i, x> - ceil{b/d} >= 0 is a subset of this affine cone.
735 * We prefer this polyhedron over the actual affine cone because it doesn't
736 * require a scaling of the constraints.
737 * If each of the vertices of the unit cube positioned at x lies inside
738 * this polyhedron, then the whole unit cube at x lies inside the affine cone.
739 * We therefore impose that x' = x + \sum e_i, for any selection of unit
740 * vectors lies inside the polyhedron, i.e.,
742 * <a_i, x'> - ceil{b/d} = <a_i, x> + sum a_i - ceil{b/d} >= 0
744 * The most stringent of these constraints is the one that selects
745 * all negative a_i, so the polyhedron we are looking for has constraints
747 * <a_i, x> + sum_{a_i < 0} a_i - ceil{b/d} >= 0
749 * Note that if cone were known to have only non-negative rays
750 * (which can be accomplished by a unimodular transformation),
751 * then we would only have to check the points x' = x + e_i
752 * and we only have to add the smallest negative a_i (if any)
753 * instead of the sum of all negative a_i.
755 static struct isl_basic_set *shift_cone(struct isl_basic_set *cone,
756 struct isl_vec *vec)
758 int i, j, k;
759 unsigned total;
761 struct isl_basic_set *shift = NULL;
763 if (!cone || !vec)
764 goto error;
766 isl_assert(cone->ctx, cone->n_eq == 0, goto error);
768 total = isl_basic_set_total_dim(cone);
770 shift = isl_basic_set_alloc_space(isl_basic_set_get_space(cone),
771 0, 0, cone->n_ineq);
773 for (i = 0; i < cone->n_ineq; ++i) {
774 k = isl_basic_set_alloc_inequality(shift);
775 if (k < 0)
776 goto error;
777 isl_seq_cpy(shift->ineq[k] + 1, cone->ineq[i] + 1, total);
778 isl_seq_inner_product(shift->ineq[k] + 1, vec->el + 1, total,
779 &shift->ineq[k][0]);
780 isl_int_cdiv_q(shift->ineq[k][0],
781 shift->ineq[k][0], vec->el[0]);
782 isl_int_neg(shift->ineq[k][0], shift->ineq[k][0]);
783 for (j = 0; j < total; ++j) {
784 if (isl_int_is_nonneg(shift->ineq[k][1 + j]))
785 continue;
786 isl_int_add(shift->ineq[k][0],
787 shift->ineq[k][0], shift->ineq[k][1 + j]);
791 isl_basic_set_free(cone);
792 isl_vec_free(vec);
794 return isl_basic_set_finalize(shift);
795 error:
796 isl_basic_set_free(shift);
797 isl_basic_set_free(cone);
798 isl_vec_free(vec);
799 return NULL;
802 /* Given a rational point vec in a (transformed) basic set,
803 * such that cone is the recession cone of the original basic set,
804 * "round up" the rational point to an integer point.
806 * We first check if the rational point just happens to be integer.
807 * If not, we transform the cone in the same way as the basic set,
808 * pick a point x in this cone shifted to the rational point such that
809 * the whole unit cube at x is also inside this affine cone.
810 * Then we simply round up the coordinates of x and return the
811 * resulting integer point.
813 static struct isl_vec *round_up_in_cone(struct isl_vec *vec,
814 struct isl_basic_set *cone, struct isl_mat *U)
816 unsigned total;
818 if (!vec || !cone || !U)
819 goto error;
821 isl_assert(vec->ctx, vec->size != 0, goto error);
822 if (isl_int_is_one(vec->el[0])) {
823 isl_mat_free(U);
824 isl_basic_set_free(cone);
825 return vec;
828 total = isl_basic_set_total_dim(cone);
829 cone = isl_basic_set_preimage(cone, U);
830 cone = isl_basic_set_remove_dims(cone, isl_dim_set,
831 0, total - (vec->size - 1));
833 cone = shift_cone(cone, vec);
835 vec = rational_sample(cone);
836 vec = isl_vec_ceil(vec);
837 return vec;
838 error:
839 isl_mat_free(U);
840 isl_vec_free(vec);
841 isl_basic_set_free(cone);
842 return NULL;
845 /* Concatenate two integer vectors, i.e., two vectors with denominator
846 * (stored in element 0) equal to 1.
848 static struct isl_vec *vec_concat(struct isl_vec *vec1, struct isl_vec *vec2)
850 struct isl_vec *vec;
852 if (!vec1 || !vec2)
853 goto error;
854 isl_assert(vec1->ctx, vec1->size > 0, goto error);
855 isl_assert(vec2->ctx, vec2->size > 0, goto error);
856 isl_assert(vec1->ctx, isl_int_is_one(vec1->el[0]), goto error);
857 isl_assert(vec2->ctx, isl_int_is_one(vec2->el[0]), goto error);
859 vec = isl_vec_alloc(vec1->ctx, vec1->size + vec2->size - 1);
860 if (!vec)
861 goto error;
863 isl_seq_cpy(vec->el, vec1->el, vec1->size);
864 isl_seq_cpy(vec->el + vec1->size, vec2->el + 1, vec2->size - 1);
866 isl_vec_free(vec1);
867 isl_vec_free(vec2);
869 return vec;
870 error:
871 isl_vec_free(vec1);
872 isl_vec_free(vec2);
873 return NULL;
876 /* Give a basic set "bset" with recession cone "cone", compute and
877 * return an integer point in bset, if any.
879 * If the recession cone is full-dimensional, then we know that
880 * bset contains an infinite number of integer points and it is
881 * fairly easy to pick one of them.
882 * If the recession cone is not full-dimensional, then we first
883 * transform bset such that the bounded directions appear as
884 * the first dimensions of the transformed basic set.
885 * We do this by using a unimodular transformation that transforms
886 * the equalities in the recession cone to equalities on the first
887 * dimensions.
889 * The transformed set is then projected onto its bounded dimensions.
890 * Note that to compute this projection, we can simply drop all constraints
891 * involving any of the unbounded dimensions since these constraints
892 * cannot be combined to produce a constraint on the bounded dimensions.
893 * To see this, assume that there is such a combination of constraints
894 * that produces a constraint on the bounded dimensions. This means
895 * that some combination of the unbounded dimensions has both an upper
896 * bound and a lower bound in terms of the bounded dimensions, but then
897 * this combination would be a bounded direction too and would have been
898 * transformed into a bounded dimensions.
900 * We then compute a sample value in the bounded dimensions.
901 * If no such value can be found, then the original set did not contain
902 * any integer points and we are done.
903 * Otherwise, we plug in the value we found in the bounded dimensions,
904 * project out these bounded dimensions and end up with a set with
905 * a full-dimensional recession cone.
906 * A sample point in this set is computed by "rounding up" any
907 * rational point in the set.
909 * The sample points in the bounded and unbounded dimensions are
910 * then combined into a single sample point and transformed back
911 * to the original space.
913 __isl_give isl_vec *isl_basic_set_sample_with_cone(
914 __isl_take isl_basic_set *bset, __isl_take isl_basic_set *cone)
916 unsigned total;
917 unsigned cone_dim;
918 struct isl_mat *M, *U;
919 struct isl_vec *sample;
920 struct isl_vec *cone_sample;
921 struct isl_ctx *ctx;
922 struct isl_basic_set *bounded;
924 if (!bset || !cone)
925 goto error;
927 ctx = bset->ctx;
928 total = isl_basic_set_total_dim(cone);
929 cone_dim = total - cone->n_eq;
931 M = isl_mat_sub_alloc6(bset->ctx, cone->eq, 0, cone->n_eq, 1, total);
932 M = isl_mat_left_hermite(M, 0, &U, NULL);
933 if (!M)
934 goto error;
935 isl_mat_free(M);
937 U = isl_mat_lin_to_aff(U);
938 bset = isl_basic_set_preimage(bset, isl_mat_copy(U));
940 bounded = isl_basic_set_copy(bset);
941 bounded = isl_basic_set_drop_constraints_involving(bounded,
942 total - cone_dim, cone_dim);
943 bounded = isl_basic_set_drop_dims(bounded, total - cone_dim, cone_dim);
944 sample = sample_bounded(bounded);
945 if (!sample || sample->size == 0) {
946 isl_basic_set_free(bset);
947 isl_basic_set_free(cone);
948 isl_mat_free(U);
949 return sample;
951 bset = plug_in(bset, isl_vec_copy(sample));
952 cone_sample = rational_sample(bset);
953 cone_sample = round_up_in_cone(cone_sample, cone, isl_mat_copy(U));
954 sample = vec_concat(sample, cone_sample);
955 sample = isl_mat_vec_product(U, sample);
956 return sample;
957 error:
958 isl_basic_set_free(cone);
959 isl_basic_set_free(bset);
960 return NULL;
963 static void vec_sum_of_neg(struct isl_vec *v, isl_int *s)
965 int i;
967 isl_int_set_si(*s, 0);
969 for (i = 0; i < v->size; ++i)
970 if (isl_int_is_neg(v->el[i]))
971 isl_int_add(*s, *s, v->el[i]);
974 /* Given a tableau "tab", a tableau "tab_cone" that corresponds
975 * to the recession cone and the inverse of a new basis U = inv(B),
976 * with the unbounded directions in B last,
977 * add constraints to "tab" that ensure any rational value
978 * in the unbounded directions can be rounded up to an integer value.
980 * The new basis is given by x' = B x, i.e., x = U x'.
981 * For any rational value of the last tab->n_unbounded coordinates
982 * in the update tableau, the value that is obtained by rounding
983 * up this value should be contained in the original tableau.
984 * For any constraint "a x + c >= 0", we therefore need to add
985 * a constraint "a x + c + s >= 0", with s the sum of all negative
986 * entries in the last elements of "a U".
988 * Since we are not interested in the first entries of any of the "a U",
989 * we first drop the columns of U that correpond to bounded directions.
991 static int tab_shift_cone(struct isl_tab *tab,
992 struct isl_tab *tab_cone, struct isl_mat *U)
994 int i;
995 isl_int v;
996 struct isl_basic_set *bset = NULL;
998 if (tab && tab->n_unbounded == 0) {
999 isl_mat_free(U);
1000 return 0;
1002 isl_int_init(v);
1003 if (!tab || !tab_cone || !U)
1004 goto error;
1005 bset = isl_tab_peek_bset(tab_cone);
1006 U = isl_mat_drop_cols(U, 0, tab->n_var - tab->n_unbounded);
1007 for (i = 0; i < bset->n_ineq; ++i) {
1008 int ok;
1009 struct isl_vec *row = NULL;
1010 if (isl_tab_is_equality(tab_cone, tab_cone->n_eq + i))
1011 continue;
1012 row = isl_vec_alloc(bset->ctx, tab_cone->n_var);
1013 if (!row)
1014 goto error;
1015 isl_seq_cpy(row->el, bset->ineq[i] + 1, tab_cone->n_var);
1016 row = isl_vec_mat_product(row, isl_mat_copy(U));
1017 if (!row)
1018 goto error;
1019 vec_sum_of_neg(row, &v);
1020 isl_vec_free(row);
1021 if (isl_int_is_zero(v))
1022 continue;
1023 tab = isl_tab_extend(tab, 1);
1024 isl_int_add(bset->ineq[i][0], bset->ineq[i][0], v);
1025 ok = isl_tab_add_ineq(tab, bset->ineq[i]) >= 0;
1026 isl_int_sub(bset->ineq[i][0], bset->ineq[i][0], v);
1027 if (!ok)
1028 goto error;
1031 isl_mat_free(U);
1032 isl_int_clear(v);
1033 return 0;
1034 error:
1035 isl_mat_free(U);
1036 isl_int_clear(v);
1037 return -1;
1040 /* Compute and return an initial basis for the possibly
1041 * unbounded tableau "tab". "tab_cone" is a tableau
1042 * for the corresponding recession cone.
1043 * Additionally, add constraints to "tab" that ensure
1044 * that any rational value for the unbounded directions
1045 * can be rounded up to an integer value.
1047 * If the tableau is bounded, i.e., if the recession cone
1048 * is zero-dimensional, then we just use inital_basis.
1049 * Otherwise, we construct a basis whose first directions
1050 * correspond to equalities, followed by bounded directions,
1051 * i.e., equalities in the recession cone.
1052 * The remaining directions are then unbounded.
1054 int isl_tab_set_initial_basis_with_cone(struct isl_tab *tab,
1055 struct isl_tab *tab_cone)
1057 struct isl_mat *eq;
1058 struct isl_mat *cone_eq;
1059 struct isl_mat *U, *Q;
1061 if (!tab || !tab_cone)
1062 return -1;
1064 if (tab_cone->n_col == tab_cone->n_dead) {
1065 tab->basis = initial_basis(tab);
1066 return tab->basis ? 0 : -1;
1069 eq = tab_equalities(tab);
1070 if (!eq)
1071 return -1;
1072 tab->n_zero = eq->n_row;
1073 cone_eq = tab_equalities(tab_cone);
1074 eq = isl_mat_concat(eq, cone_eq);
1075 if (!eq)
1076 return -1;
1077 tab->n_unbounded = tab->n_var - (eq->n_row - tab->n_zero);
1078 eq = isl_mat_left_hermite(eq, 0, &U, &Q);
1079 if (!eq)
1080 return -1;
1081 isl_mat_free(eq);
1082 tab->basis = isl_mat_lin_to_aff(Q);
1083 if (tab_shift_cone(tab, tab_cone, U) < 0)
1084 return -1;
1085 if (!tab->basis)
1086 return -1;
1087 return 0;
1090 /* Compute and return a sample point in bset using generalized basis
1091 * reduction. We first check if the input set has a non-trivial
1092 * recession cone. If so, we perform some extra preprocessing in
1093 * sample_with_cone. Otherwise, we directly perform generalized basis
1094 * reduction.
1096 static struct isl_vec *gbr_sample(struct isl_basic_set *bset)
1098 unsigned dim;
1099 struct isl_basic_set *cone;
1101 dim = isl_basic_set_total_dim(bset);
1103 cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));
1104 if (!cone)
1105 goto error;
1107 if (cone->n_eq < dim)
1108 return isl_basic_set_sample_with_cone(bset, cone);
1110 isl_basic_set_free(cone);
1111 return sample_bounded(bset);
1112 error:
1113 isl_basic_set_free(bset);
1114 return NULL;
1117 static struct isl_vec *basic_set_sample(struct isl_basic_set *bset, int bounded)
1119 struct isl_ctx *ctx;
1120 unsigned dim;
1121 if (!bset)
1122 return NULL;
1124 ctx = bset->ctx;
1125 if (isl_basic_set_plain_is_empty(bset))
1126 return empty_sample(bset);
1128 dim = isl_basic_set_n_dim(bset);
1129 isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
1130 isl_assert(ctx, bset->n_div == 0, goto error);
1132 if (bset->sample && bset->sample->size == 1 + dim) {
1133 int contains = isl_basic_set_contains(bset, bset->sample);
1134 if (contains < 0)
1135 goto error;
1136 if (contains) {
1137 struct isl_vec *sample = isl_vec_copy(bset->sample);
1138 isl_basic_set_free(bset);
1139 return sample;
1142 isl_vec_free(bset->sample);
1143 bset->sample = NULL;
1145 if (bset->n_eq > 0)
1146 return sample_eq(bset, bounded ? isl_basic_set_sample_bounded
1147 : isl_basic_set_sample_vec);
1148 if (dim == 0)
1149 return zero_sample(bset);
1150 if (dim == 1)
1151 return interval_sample(bset);
1153 return bounded ? sample_bounded(bset) : gbr_sample(bset);
1154 error:
1155 isl_basic_set_free(bset);
1156 return NULL;
1159 __isl_give isl_vec *isl_basic_set_sample_vec(__isl_take isl_basic_set *bset)
1161 return basic_set_sample(bset, 0);
1164 /* Compute an integer sample in "bset", where the caller guarantees
1165 * that "bset" is bounded.
1167 struct isl_vec *isl_basic_set_sample_bounded(struct isl_basic_set *bset)
1169 return basic_set_sample(bset, 1);
1172 __isl_give isl_basic_set *isl_basic_set_from_vec(__isl_take isl_vec *vec)
1174 int i;
1175 int k;
1176 struct isl_basic_set *bset = NULL;
1177 struct isl_ctx *ctx;
1178 unsigned dim;
1180 if (!vec)
1181 return NULL;
1182 ctx = vec->ctx;
1183 isl_assert(ctx, vec->size != 0, goto error);
1185 bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
1186 if (!bset)
1187 goto error;
1188 dim = isl_basic_set_n_dim(bset);
1189 for (i = dim - 1; i >= 0; --i) {
1190 k = isl_basic_set_alloc_equality(bset);
1191 if (k < 0)
1192 goto error;
1193 isl_seq_clr(bset->eq[k], 1 + dim);
1194 isl_int_neg(bset->eq[k][0], vec->el[1 + i]);
1195 isl_int_set(bset->eq[k][1 + i], vec->el[0]);
1197 bset->sample = vec;
1199 return bset;
1200 error:
1201 isl_basic_set_free(bset);
1202 isl_vec_free(vec);
1203 return NULL;
1206 __isl_give isl_basic_map *isl_basic_map_sample(__isl_take isl_basic_map *bmap)
1208 struct isl_basic_set *bset;
1209 struct isl_vec *sample_vec;
1211 bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
1212 sample_vec = isl_basic_set_sample_vec(bset);
1213 if (!sample_vec)
1214 goto error;
1215 if (sample_vec->size == 0) {
1216 struct isl_basic_map *sample;
1217 sample = isl_basic_map_empty_like(bmap);
1218 isl_vec_free(sample_vec);
1219 isl_basic_map_free(bmap);
1220 return sample;
1222 bset = isl_basic_set_from_vec(sample_vec);
1223 return isl_basic_map_overlying_set(bset, bmap);
1224 error:
1225 isl_basic_map_free(bmap);
1226 return NULL;
1229 __isl_give isl_basic_set *isl_basic_set_sample(__isl_take isl_basic_set *bset)
1231 return isl_basic_map_sample(bset);
1234 __isl_give isl_basic_map *isl_map_sample(__isl_take isl_map *map)
1236 int i;
1237 isl_basic_map *sample = NULL;
1239 if (!map)
1240 goto error;
1242 for (i = 0; i < map->n; ++i) {
1243 sample = isl_basic_map_sample(isl_basic_map_copy(map->p[i]));
1244 if (!sample)
1245 goto error;
1246 if (!ISL_F_ISSET(sample, ISL_BASIC_MAP_EMPTY))
1247 break;
1248 isl_basic_map_free(sample);
1250 if (i == map->n)
1251 sample = isl_basic_map_empty_like_map(map);
1252 isl_map_free(map);
1253 return sample;
1254 error:
1255 isl_map_free(map);
1256 return NULL;
1259 __isl_give isl_basic_set *isl_set_sample(__isl_take isl_set *set)
1261 return (isl_basic_set *) isl_map_sample((isl_map *)set);
1264 __isl_give isl_point *isl_basic_set_sample_point(__isl_take isl_basic_set *bset)
1266 isl_vec *vec;
1267 isl_space *dim;
1269 dim = isl_basic_set_get_space(bset);
1270 bset = isl_basic_set_underlying_set(bset);
1271 vec = isl_basic_set_sample_vec(bset);
1273 return isl_point_alloc(dim, vec);
1276 __isl_give isl_point *isl_set_sample_point(__isl_take isl_set *set)
1278 int i;
1279 isl_point *pnt;
1281 if (!set)
1282 return NULL;
1284 for (i = 0; i < set->n; ++i) {
1285 pnt = isl_basic_set_sample_point(isl_basic_set_copy(set->p[i]));
1286 if (!pnt)
1287 goto error;
1288 if (!isl_point_is_void(pnt))
1289 break;
1290 isl_point_free(pnt);
1292 if (i == set->n)
1293 pnt = isl_point_void(isl_set_get_space(set));
1295 isl_set_free(set);
1296 return pnt;
1297 error:
1298 isl_set_free(set);
1299 return NULL;