isl_pw_qpolynomial{,_fold}_gist: use simple hull instead of convex hull
[isl.git] / isl_sample.c
blob83a61a039a2823742be96be8a497da39ce78951a
1 /*
2 * Copyright 2008-2009 Katholieke Universiteit Leuven
4 * Use of this software is governed by the GNU LGPLv2.1 license
6 * Written by Sven Verdoolaege, K.U.Leuven, Departement
7 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
8 */
10 #include "isl_sample.h"
11 #include "isl_sample_piplib.h"
12 #include "isl_vec.h"
13 #include "isl_mat.h"
14 #include "isl_seq.h"
15 #include "isl_map_private.h"
16 #include "isl_equalities.h"
17 #include "isl_tab.h"
18 #include "isl_basis_reduction.h"
19 #include <isl_point_private.h>
21 static struct isl_vec *empty_sample(struct isl_basic_set *bset)
23 struct isl_vec *vec;
25 vec = isl_vec_alloc(bset->ctx, 0);
26 isl_basic_set_free(bset);
27 return vec;
30 /* Construct a zero sample of the same dimension as bset.
31 * As a special case, if bset is zero-dimensional, this
32 * function creates a zero-dimensional sample point.
34 static struct isl_vec *zero_sample(struct isl_basic_set *bset)
36 unsigned dim;
37 struct isl_vec *sample;
39 dim = isl_basic_set_total_dim(bset);
40 sample = isl_vec_alloc(bset->ctx, 1 + dim);
41 if (sample) {
42 isl_int_set_si(sample->el[0], 1);
43 isl_seq_clr(sample->el + 1, dim);
45 isl_basic_set_free(bset);
46 return sample;
49 static struct isl_vec *interval_sample(struct isl_basic_set *bset)
51 int i;
52 isl_int t;
53 struct isl_vec *sample;
55 bset = isl_basic_set_simplify(bset);
56 if (!bset)
57 return NULL;
58 if (isl_basic_set_fast_is_empty(bset))
59 return empty_sample(bset);
60 if (bset->n_eq == 0 && bset->n_ineq == 0)
61 return zero_sample(bset);
63 sample = isl_vec_alloc(bset->ctx, 2);
64 isl_int_set_si(sample->block.data[0], 1);
66 if (bset->n_eq > 0) {
67 isl_assert(bset->ctx, bset->n_eq == 1, goto error);
68 isl_assert(bset->ctx, bset->n_ineq == 0, goto error);
69 if (isl_int_is_one(bset->eq[0][1]))
70 isl_int_neg(sample->el[1], bset->eq[0][0]);
71 else {
72 isl_assert(bset->ctx, isl_int_is_negone(bset->eq[0][1]),
73 goto error);
74 isl_int_set(sample->el[1], bset->eq[0][0]);
76 isl_basic_set_free(bset);
77 return sample;
80 isl_int_init(t);
81 if (isl_int_is_one(bset->ineq[0][1]))
82 isl_int_neg(sample->block.data[1], bset->ineq[0][0]);
83 else
84 isl_int_set(sample->block.data[1], bset->ineq[0][0]);
85 for (i = 1; i < bset->n_ineq; ++i) {
86 isl_seq_inner_product(sample->block.data,
87 bset->ineq[i], 2, &t);
88 if (isl_int_is_neg(t))
89 break;
91 isl_int_clear(t);
92 if (i < bset->n_ineq) {
93 isl_vec_free(sample);
94 return empty_sample(bset);
97 isl_basic_set_free(bset);
98 return sample;
99 error:
100 isl_basic_set_free(bset);
101 isl_vec_free(sample);
102 return NULL;
105 static struct isl_mat *independent_bounds(struct isl_basic_set *bset)
107 int i, j, n;
108 struct isl_mat *dirs = NULL;
109 struct isl_mat *bounds = NULL;
110 unsigned dim;
112 if (!bset)
113 return NULL;
115 dim = isl_basic_set_n_dim(bset);
116 bounds = isl_mat_alloc(bset->ctx, 1+dim, 1+dim);
117 if (!bounds)
118 return NULL;
120 isl_int_set_si(bounds->row[0][0], 1);
121 isl_seq_clr(bounds->row[0]+1, dim);
122 bounds->n_row = 1;
124 if (bset->n_ineq == 0)
125 return bounds;
127 dirs = isl_mat_alloc(bset->ctx, dim, dim);
128 if (!dirs) {
129 isl_mat_free(bounds);
130 return NULL;
132 isl_seq_cpy(dirs->row[0], bset->ineq[0]+1, dirs->n_col);
133 isl_seq_cpy(bounds->row[1], bset->ineq[0], bounds->n_col);
134 for (j = 1, n = 1; n < dim && j < bset->n_ineq; ++j) {
135 int pos;
137 isl_seq_cpy(dirs->row[n], bset->ineq[j]+1, dirs->n_col);
139 pos = isl_seq_first_non_zero(dirs->row[n], dirs->n_col);
140 if (pos < 0)
141 continue;
142 for (i = 0; i < n; ++i) {
143 int pos_i;
144 pos_i = isl_seq_first_non_zero(dirs->row[i], dirs->n_col);
145 if (pos_i < pos)
146 continue;
147 if (pos_i > pos)
148 break;
149 isl_seq_elim(dirs->row[n], dirs->row[i], pos,
150 dirs->n_col, NULL);
151 pos = isl_seq_first_non_zero(dirs->row[n], dirs->n_col);
152 if (pos < 0)
153 break;
155 if (pos < 0)
156 continue;
157 if (i < n) {
158 int k;
159 isl_int *t = dirs->row[n];
160 for (k = n; k > i; --k)
161 dirs->row[k] = dirs->row[k-1];
162 dirs->row[i] = t;
164 ++n;
165 isl_seq_cpy(bounds->row[n], bset->ineq[j], bounds->n_col);
167 isl_mat_free(dirs);
168 bounds->n_row = 1+n;
169 return bounds;
172 static void swap_inequality(struct isl_basic_set *bset, int a, int b)
174 isl_int *t = bset->ineq[a];
175 bset->ineq[a] = bset->ineq[b];
176 bset->ineq[b] = t;
179 /* Skew into positive orthant and project out lineality space.
181 * We perform a unimodular transformation that turns a selected
182 * maximal set of linearly independent bounds into constraints
183 * on the first dimensions that impose that these first dimensions
184 * are non-negative. In particular, the constraint matrix is lower
185 * triangular with positive entries on the diagonal and negative
186 * entries below.
187 * If "bset" has a lineality space then these constraints (and therefore
188 * all constraints in bset) only involve the first dimensions.
189 * The remaining dimensions then do not appear in any constraints and
190 * we can select any value for them, say zero. We therefore project
191 * out this final dimensions and plug in the value zero later. This
192 * is accomplished by simply dropping the final columns of
193 * the unimodular transformation.
195 static struct isl_basic_set *isl_basic_set_skew_to_positive_orthant(
196 struct isl_basic_set *bset, struct isl_mat **T)
198 struct isl_mat *U = NULL;
199 struct isl_mat *bounds = NULL;
200 int i, j;
201 unsigned old_dim, new_dim;
203 *T = NULL;
204 if (!bset)
205 return NULL;
207 isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
208 isl_assert(bset->ctx, bset->n_div == 0, goto error);
209 isl_assert(bset->ctx, bset->n_eq == 0, goto error);
211 old_dim = isl_basic_set_n_dim(bset);
212 /* Try to move (multiples of) unit rows up. */
213 for (i = 0, j = 0; i < bset->n_ineq; ++i) {
214 int pos = isl_seq_first_non_zero(bset->ineq[i]+1, old_dim);
215 if (pos < 0)
216 continue;
217 if (isl_seq_first_non_zero(bset->ineq[i]+1+pos+1,
218 old_dim-pos-1) >= 0)
219 continue;
220 if (i != j)
221 swap_inequality(bset, i, j);
222 ++j;
224 bounds = independent_bounds(bset);
225 if (!bounds)
226 goto error;
227 new_dim = bounds->n_row - 1;
228 bounds = isl_mat_left_hermite(bounds, 1, &U, NULL);
229 if (!bounds)
230 goto error;
231 U = isl_mat_drop_cols(U, 1 + new_dim, old_dim - new_dim);
232 bset = isl_basic_set_preimage(bset, isl_mat_copy(U));
233 if (!bset)
234 goto error;
235 *T = U;
236 isl_mat_free(bounds);
237 return bset;
238 error:
239 isl_mat_free(bounds);
240 isl_mat_free(U);
241 isl_basic_set_free(bset);
242 return NULL;
245 /* Find a sample integer point, if any, in bset, which is known
246 * to have equalities. If bset contains no integer points, then
247 * return a zero-length vector.
248 * We simply remove the known equalities, compute a sample
249 * in the resulting bset, using the specified recurse function,
250 * and then transform the sample back to the original space.
252 static struct isl_vec *sample_eq(struct isl_basic_set *bset,
253 struct isl_vec *(*recurse)(struct isl_basic_set *))
255 struct isl_mat *T;
256 struct isl_vec *sample;
258 if (!bset)
259 return NULL;
261 bset = isl_basic_set_remove_equalities(bset, &T, NULL);
262 sample = recurse(bset);
263 if (!sample || sample->size == 0)
264 isl_mat_free(T);
265 else
266 sample = isl_mat_vec_product(T, sample);
267 return sample;
270 /* Return a matrix containing the equalities of the tableau
271 * in constraint form. The tableau is assumed to have
272 * an associated bset that has been kept up-to-date.
274 static struct isl_mat *tab_equalities(struct isl_tab *tab)
276 int i, j;
277 int n_eq;
278 struct isl_mat *eq;
279 struct isl_basic_set *bset;
281 if (!tab)
282 return NULL;
284 bset = isl_tab_peek_bset(tab);
285 isl_assert(tab->mat->ctx, bset, return NULL);
287 n_eq = tab->n_var - tab->n_col + tab->n_dead;
288 if (tab->empty || n_eq == 0)
289 return isl_mat_alloc(tab->mat->ctx, 0, tab->n_var);
290 if (n_eq == tab->n_var)
291 return isl_mat_identity(tab->mat->ctx, tab->n_var);
293 eq = isl_mat_alloc(tab->mat->ctx, n_eq, tab->n_var);
294 if (!eq)
295 return NULL;
296 for (i = 0, j = 0; i < tab->n_con; ++i) {
297 if (tab->con[i].is_row)
298 continue;
299 if (tab->con[i].index >= 0 && tab->con[i].index >= tab->n_dead)
300 continue;
301 if (i < bset->n_eq)
302 isl_seq_cpy(eq->row[j], bset->eq[i] + 1, tab->n_var);
303 else
304 isl_seq_cpy(eq->row[j],
305 bset->ineq[i - bset->n_eq] + 1, tab->n_var);
306 ++j;
308 isl_assert(bset->ctx, j == n_eq, goto error);
309 return eq;
310 error:
311 isl_mat_free(eq);
312 return NULL;
315 /* Compute and return an initial basis for the bounded tableau "tab".
317 * If the tableau is either full-dimensional or zero-dimensional,
318 * the we simply return an identity matrix.
319 * Otherwise, we construct a basis whose first directions correspond
320 * to equalities.
322 static struct isl_mat *initial_basis(struct isl_tab *tab)
324 int n_eq;
325 struct isl_mat *eq;
326 struct isl_mat *Q;
328 tab->n_unbounded = 0;
329 tab->n_zero = n_eq = tab->n_var - tab->n_col + tab->n_dead;
330 if (tab->empty || n_eq == 0 || n_eq == tab->n_var)
331 return isl_mat_identity(tab->mat->ctx, 1 + tab->n_var);
333 eq = tab_equalities(tab);
334 eq = isl_mat_left_hermite(eq, 0, NULL, &Q);
335 if (!eq)
336 return NULL;
337 isl_mat_free(eq);
339 Q = isl_mat_lin_to_aff(Q);
340 return Q;
343 /* Given a tableau representing a set, find and return
344 * an integer point in the set, if there is any.
346 * We perform a depth first search
347 * for an integer point, by scanning all possible values in the range
348 * attained by a basis vector, where an initial basis may have been set
349 * by the calling function. Otherwise an initial basis that exploits
350 * the equalities in the tableau is created.
351 * tab->n_zero is currently ignored and is clobbered by this function.
353 * The tableau is allowed to have unbounded direction, but then
354 * the calling function needs to set an initial basis, with the
355 * unbounded directions last and with tab->n_unbounded set
356 * to the number of unbounded directions.
357 * Furthermore, the calling functions needs to add shifted copies
358 * of all constraints involving unbounded directions to ensure
359 * that any feasible rational value in these directions can be rounded
360 * up to yield a feasible integer value.
361 * In particular, let B define the given basis x' = B x
362 * and let T be the inverse of B, i.e., X = T x'.
363 * Let a x + c >= 0 be a constraint of the set represented by the tableau,
364 * or a T x' + c >= 0 in terms of the given basis. Assume that
365 * the bounded directions have an integer value, then we can safely
366 * round up the values for the unbounded directions if we make sure
367 * that x' not only satisfies the original constraint, but also
368 * the constraint "a T x' + c + s >= 0" with s the sum of all
369 * negative values in the last n_unbounded entries of "a T".
370 * The calling function therefore needs to add the constraint
371 * a x + c + s >= 0. The current function then scans the first
372 * directions for an integer value and once those have been found,
373 * it can compute "T ceil(B x)" to yield an integer point in the set.
374 * Note that during the search, the first rows of B may be changed
375 * by a basis reduction, but the last n_unbounded rows of B remain
376 * unaltered and are also not mixed into the first rows.
378 * The search is implemented iteratively. "level" identifies the current
379 * basis vector. "init" is true if we want the first value at the current
380 * level and false if we want the next value.
382 * The initial basis is the identity matrix. If the range in some direction
383 * contains more than one integer value, we perform basis reduction based
384 * on the value of ctx->opt->gbr
385 * - ISL_GBR_NEVER: never perform basis reduction
386 * - ISL_GBR_ONCE: only perform basis reduction the first
387 * time such a range is encountered
388 * - ISL_GBR_ALWAYS: always perform basis reduction when
389 * such a range is encountered
391 * When ctx->opt->gbr is set to ISL_GBR_ALWAYS, then we allow the basis
392 * reduction computation to return early. That is, as soon as it
393 * finds a reasonable first direction.
395 struct isl_vec *isl_tab_sample(struct isl_tab *tab)
397 unsigned dim;
398 unsigned gbr;
399 struct isl_ctx *ctx;
400 struct isl_vec *sample;
401 struct isl_vec *min;
402 struct isl_vec *max;
403 enum isl_lp_result res;
404 int level;
405 int init;
406 int reduced;
407 struct isl_tab_undo **snap;
409 if (!tab)
410 return NULL;
411 if (tab->empty)
412 return isl_vec_alloc(tab->mat->ctx, 0);
414 if (!tab->basis)
415 tab->basis = initial_basis(tab);
416 if (!tab->basis)
417 return NULL;
418 isl_assert(tab->mat->ctx, tab->basis->n_row == tab->n_var + 1,
419 return NULL);
420 isl_assert(tab->mat->ctx, tab->basis->n_col == tab->n_var + 1,
421 return NULL);
423 ctx = tab->mat->ctx;
424 dim = tab->n_var;
425 gbr = ctx->opt->gbr;
427 if (tab->n_unbounded == tab->n_var) {
428 sample = isl_tab_get_sample_value(tab);
429 sample = isl_mat_vec_product(isl_mat_copy(tab->basis), sample);
430 sample = isl_vec_ceil(sample);
431 sample = isl_mat_vec_inverse_product(isl_mat_copy(tab->basis),
432 sample);
433 return sample;
436 if (isl_tab_extend_cons(tab, dim + 1) < 0)
437 return NULL;
439 min = isl_vec_alloc(ctx, dim);
440 max = isl_vec_alloc(ctx, dim);
441 snap = isl_alloc_array(ctx, struct isl_tab_undo *, dim);
443 if (!min || !max || !snap)
444 goto error;
446 level = 0;
447 init = 1;
448 reduced = 0;
450 while (level >= 0) {
451 int empty = 0;
452 if (init) {
453 res = isl_tab_min(tab, tab->basis->row[1 + level],
454 ctx->one, &min->el[level], NULL, 0);
455 if (res == isl_lp_empty)
456 empty = 1;
457 isl_assert(ctx, res != isl_lp_unbounded, goto error);
458 if (res == isl_lp_error)
459 goto error;
460 if (!empty && isl_tab_sample_is_integer(tab))
461 break;
462 isl_seq_neg(tab->basis->row[1 + level] + 1,
463 tab->basis->row[1 + level] + 1, dim);
464 res = isl_tab_min(tab, tab->basis->row[1 + level],
465 ctx->one, &max->el[level], NULL, 0);
466 isl_seq_neg(tab->basis->row[1 + level] + 1,
467 tab->basis->row[1 + level] + 1, dim);
468 isl_int_neg(max->el[level], max->el[level]);
469 if (res == isl_lp_empty)
470 empty = 1;
471 isl_assert(ctx, res != isl_lp_unbounded, goto error);
472 if (res == isl_lp_error)
473 goto error;
474 if (!empty && isl_tab_sample_is_integer(tab))
475 break;
476 if (!empty && !reduced &&
477 ctx->opt->gbr != ISL_GBR_NEVER &&
478 isl_int_lt(min->el[level], max->el[level])) {
479 unsigned gbr_only_first;
480 if (ctx->opt->gbr == ISL_GBR_ONCE)
481 ctx->opt->gbr = ISL_GBR_NEVER;
482 tab->n_zero = level;
483 gbr_only_first = ctx->opt->gbr_only_first;
484 ctx->opt->gbr_only_first =
485 ctx->opt->gbr == ISL_GBR_ALWAYS;
486 tab = isl_tab_compute_reduced_basis(tab);
487 ctx->opt->gbr_only_first = gbr_only_first;
488 if (!tab || !tab->basis)
489 goto error;
490 reduced = 1;
491 continue;
493 reduced = 0;
494 snap[level] = isl_tab_snap(tab);
495 } else
496 isl_int_add_ui(min->el[level], min->el[level], 1);
498 if (empty || isl_int_gt(min->el[level], max->el[level])) {
499 level--;
500 init = 0;
501 if (level >= 0)
502 if (isl_tab_rollback(tab, snap[level]) < 0)
503 goto error;
504 continue;
506 isl_int_neg(tab->basis->row[1 + level][0], min->el[level]);
507 tab = isl_tab_add_valid_eq(tab, tab->basis->row[1 + level]);
508 isl_int_set_si(tab->basis->row[1 + level][0], 0);
509 if (level + tab->n_unbounded < dim - 1) {
510 ++level;
511 init = 1;
512 continue;
514 break;
517 if (level >= 0) {
518 sample = isl_tab_get_sample_value(tab);
519 if (!sample)
520 goto error;
521 if (tab->n_unbounded && !isl_int_is_one(sample->el[0])) {
522 sample = isl_mat_vec_product(isl_mat_copy(tab->basis),
523 sample);
524 sample = isl_vec_ceil(sample);
525 sample = isl_mat_vec_inverse_product(
526 isl_mat_copy(tab->basis), sample);
528 } else
529 sample = isl_vec_alloc(ctx, 0);
531 ctx->opt->gbr = gbr;
532 isl_vec_free(min);
533 isl_vec_free(max);
534 free(snap);
535 return sample;
536 error:
537 ctx->opt->gbr = gbr;
538 isl_vec_free(min);
539 isl_vec_free(max);
540 free(snap);
541 return NULL;
544 /* Given a basic set that is known to be bounded, find and return
545 * an integer point in the basic set, if there is any.
547 * After handling some trivial cases, we construct a tableau
548 * and then use isl_tab_sample to find a sample, passing it
549 * the identity matrix as initial basis.
551 static struct isl_vec *sample_bounded(struct isl_basic_set *bset)
553 unsigned dim;
554 struct isl_ctx *ctx;
555 struct isl_vec *sample;
556 struct isl_tab *tab = NULL;
558 if (!bset)
559 return NULL;
561 if (isl_basic_set_fast_is_empty(bset))
562 return empty_sample(bset);
564 dim = isl_basic_set_total_dim(bset);
565 if (dim == 0)
566 return zero_sample(bset);
567 if (dim == 1)
568 return interval_sample(bset);
569 if (bset->n_eq > 0)
570 return sample_eq(bset, sample_bounded);
572 ctx = bset->ctx;
574 tab = isl_tab_from_basic_set(bset);
575 if (tab && tab->empty) {
576 isl_tab_free(tab);
577 ISL_F_SET(bset, ISL_BASIC_SET_EMPTY);
578 sample = isl_vec_alloc(bset->ctx, 0);
579 isl_basic_set_free(bset);
580 return sample;
583 if (isl_tab_track_bset(tab, isl_basic_set_copy(bset)) < 0)
584 goto error;
585 if (!ISL_F_ISSET(bset, ISL_BASIC_SET_NO_IMPLICIT))
586 if (isl_tab_detect_implicit_equalities(tab) < 0)
587 goto error;
589 sample = isl_tab_sample(tab);
590 if (!sample)
591 goto error;
593 if (sample->size > 0) {
594 isl_vec_free(bset->sample);
595 bset->sample = isl_vec_copy(sample);
598 isl_basic_set_free(bset);
599 isl_tab_free(tab);
600 return sample;
601 error:
602 isl_basic_set_free(bset);
603 isl_tab_free(tab);
604 return NULL;
607 /* Given a basic set "bset" and a value "sample" for the first coordinates
608 * of bset, plug in these values and drop the corresponding coordinates.
610 * We do this by computing the preimage of the transformation
612 * [ 1 0 ]
613 * x = [ s 0 ] x'
614 * [ 0 I ]
616 * where [1 s] is the sample value and I is the identity matrix of the
617 * appropriate dimension.
619 static struct isl_basic_set *plug_in(struct isl_basic_set *bset,
620 struct isl_vec *sample)
622 int i;
623 unsigned total;
624 struct isl_mat *T;
626 if (!bset || !sample)
627 goto error;
629 total = isl_basic_set_total_dim(bset);
630 T = isl_mat_alloc(bset->ctx, 1 + total, 1 + total - (sample->size - 1));
631 if (!T)
632 goto error;
634 for (i = 0; i < sample->size; ++i) {
635 isl_int_set(T->row[i][0], sample->el[i]);
636 isl_seq_clr(T->row[i] + 1, T->n_col - 1);
638 for (i = 0; i < T->n_col - 1; ++i) {
639 isl_seq_clr(T->row[sample->size + i], T->n_col);
640 isl_int_set_si(T->row[sample->size + i][1 + i], 1);
642 isl_vec_free(sample);
644 bset = isl_basic_set_preimage(bset, T);
645 return bset;
646 error:
647 isl_basic_set_free(bset);
648 isl_vec_free(sample);
649 return NULL;
652 /* Given a basic set "bset", return any (possibly non-integer) point
653 * in the basic set.
655 static struct isl_vec *rational_sample(struct isl_basic_set *bset)
657 struct isl_tab *tab;
658 struct isl_vec *sample;
660 if (!bset)
661 return NULL;
663 tab = isl_tab_from_basic_set(bset);
664 sample = isl_tab_get_sample_value(tab);
665 isl_tab_free(tab);
667 isl_basic_set_free(bset);
669 return sample;
672 /* Given a linear cone "cone" and a rational point "vec",
673 * construct a polyhedron with shifted copies of the constraints in "cone",
674 * i.e., a polyhedron with "cone" as its recession cone, such that each
675 * point x in this polyhedron is such that the unit box positioned at x
676 * lies entirely inside the affine cone 'vec + cone'.
677 * Any rational point in this polyhedron may therefore be rounded up
678 * to yield an integer point that lies inside said affine cone.
680 * Denote the constraints of cone by "<a_i, x> >= 0" and the rational
681 * point "vec" by v/d.
682 * Let b_i = <a_i, v>. Then the affine cone 'vec + cone' is given
683 * by <a_i, x> - b/d >= 0.
684 * The polyhedron <a_i, x> - ceil{b/d} >= 0 is a subset of this affine cone.
685 * We prefer this polyhedron over the actual affine cone because it doesn't
686 * require a scaling of the constraints.
687 * If each of the vertices of the unit cube positioned at x lies inside
688 * this polyhedron, then the whole unit cube at x lies inside the affine cone.
689 * We therefore impose that x' = x + \sum e_i, for any selection of unit
690 * vectors lies inside the polyhedron, i.e.,
692 * <a_i, x'> - ceil{b/d} = <a_i, x> + sum a_i - ceil{b/d} >= 0
694 * The most stringent of these constraints is the one that selects
695 * all negative a_i, so the polyhedron we are looking for has constraints
697 * <a_i, x> + sum_{a_i < 0} a_i - ceil{b/d} >= 0
699 * Note that if cone were known to have only non-negative rays
700 * (which can be accomplished by a unimodular transformation),
701 * then we would only have to check the points x' = x + e_i
702 * and we only have to add the smallest negative a_i (if any)
703 * instead of the sum of all negative a_i.
705 static struct isl_basic_set *shift_cone(struct isl_basic_set *cone,
706 struct isl_vec *vec)
708 int i, j, k;
709 unsigned total;
711 struct isl_basic_set *shift = NULL;
713 if (!cone || !vec)
714 goto error;
716 isl_assert(cone->ctx, cone->n_eq == 0, goto error);
718 total = isl_basic_set_total_dim(cone);
720 shift = isl_basic_set_alloc_dim(isl_basic_set_get_dim(cone),
721 0, 0, cone->n_ineq);
723 for (i = 0; i < cone->n_ineq; ++i) {
724 k = isl_basic_set_alloc_inequality(shift);
725 if (k < 0)
726 goto error;
727 isl_seq_cpy(shift->ineq[k] + 1, cone->ineq[i] + 1, total);
728 isl_seq_inner_product(shift->ineq[k] + 1, vec->el + 1, total,
729 &shift->ineq[k][0]);
730 isl_int_cdiv_q(shift->ineq[k][0],
731 shift->ineq[k][0], vec->el[0]);
732 isl_int_neg(shift->ineq[k][0], shift->ineq[k][0]);
733 for (j = 0; j < total; ++j) {
734 if (isl_int_is_nonneg(shift->ineq[k][1 + j]))
735 continue;
736 isl_int_add(shift->ineq[k][0],
737 shift->ineq[k][0], shift->ineq[k][1 + j]);
741 isl_basic_set_free(cone);
742 isl_vec_free(vec);
744 return isl_basic_set_finalize(shift);
745 error:
746 isl_basic_set_free(shift);
747 isl_basic_set_free(cone);
748 isl_vec_free(vec);
749 return NULL;
752 /* Given a rational point vec in a (transformed) basic set,
753 * such that cone is the recession cone of the original basic set,
754 * "round up" the rational point to an integer point.
756 * We first check if the rational point just happens to be integer.
757 * If not, we transform the cone in the same way as the basic set,
758 * pick a point x in this cone shifted to the rational point such that
759 * the whole unit cube at x is also inside this affine cone.
760 * Then we simply round up the coordinates of x and return the
761 * resulting integer point.
763 static struct isl_vec *round_up_in_cone(struct isl_vec *vec,
764 struct isl_basic_set *cone, struct isl_mat *U)
766 unsigned total;
768 if (!vec || !cone || !U)
769 goto error;
771 isl_assert(vec->ctx, vec->size != 0, goto error);
772 if (isl_int_is_one(vec->el[0])) {
773 isl_mat_free(U);
774 isl_basic_set_free(cone);
775 return vec;
778 total = isl_basic_set_total_dim(cone);
779 cone = isl_basic_set_preimage(cone, U);
780 cone = isl_basic_set_remove_dims(cone, 0, total - (vec->size - 1));
782 cone = shift_cone(cone, vec);
784 vec = rational_sample(cone);
785 vec = isl_vec_ceil(vec);
786 return vec;
787 error:
788 isl_mat_free(U);
789 isl_vec_free(vec);
790 isl_basic_set_free(cone);
791 return NULL;
794 /* Concatenate two integer vectors, i.e., two vectors with denominator
795 * (stored in element 0) equal to 1.
797 static struct isl_vec *vec_concat(struct isl_vec *vec1, struct isl_vec *vec2)
799 struct isl_vec *vec;
801 if (!vec1 || !vec2)
802 goto error;
803 isl_assert(vec1->ctx, vec1->size > 0, goto error);
804 isl_assert(vec2->ctx, vec2->size > 0, goto error);
805 isl_assert(vec1->ctx, isl_int_is_one(vec1->el[0]), goto error);
806 isl_assert(vec2->ctx, isl_int_is_one(vec2->el[0]), goto error);
808 vec = isl_vec_alloc(vec1->ctx, vec1->size + vec2->size - 1);
809 if (!vec)
810 goto error;
812 isl_seq_cpy(vec->el, vec1->el, vec1->size);
813 isl_seq_cpy(vec->el + vec1->size, vec2->el + 1, vec2->size - 1);
815 isl_vec_free(vec1);
816 isl_vec_free(vec2);
818 return vec;
819 error:
820 isl_vec_free(vec1);
821 isl_vec_free(vec2);
822 return NULL;
825 /* Drop all constraints in bset that involve any of the dimensions
826 * first to first+n-1.
828 static struct isl_basic_set *drop_constraints_involving
829 (struct isl_basic_set *bset, unsigned first, unsigned n)
831 int i;
833 if (!bset)
834 return NULL;
836 bset = isl_basic_set_cow(bset);
838 for (i = bset->n_ineq - 1; i >= 0; --i) {
839 if (isl_seq_first_non_zero(bset->ineq[i] + 1 + first, n) == -1)
840 continue;
841 isl_basic_set_drop_inequality(bset, i);
844 return bset;
847 /* Give a basic set "bset" with recession cone "cone", compute and
848 * return an integer point in bset, if any.
850 * If the recession cone is full-dimensional, then we know that
851 * bset contains an infinite number of integer points and it is
852 * fairly easy to pick one of them.
853 * If the recession cone is not full-dimensional, then we first
854 * transform bset such that the bounded directions appear as
855 * the first dimensions of the transformed basic set.
856 * We do this by using a unimodular transformation that transforms
857 * the equalities in the recession cone to equalities on the first
858 * dimensions.
860 * The transformed set is then projected onto its bounded dimensions.
861 * Note that to compute this projection, we can simply drop all constraints
862 * involving any of the unbounded dimensions since these constraints
863 * cannot be combined to produce a constraint on the bounded dimensions.
864 * To see this, assume that there is such a combination of constraints
865 * that produces a constraint on the bounded dimensions. This means
866 * that some combination of the unbounded dimensions has both an upper
867 * bound and a lower bound in terms of the bounded dimensions, but then
868 * this combination would be a bounded direction too and would have been
869 * transformed into a bounded dimensions.
871 * We then compute a sample value in the bounded dimensions.
872 * If no such value can be found, then the original set did not contain
873 * any integer points and we are done.
874 * Otherwise, we plug in the value we found in the bounded dimensions,
875 * project out these bounded dimensions and end up with a set with
876 * a full-dimensional recession cone.
877 * A sample point in this set is computed by "rounding up" any
878 * rational point in the set.
880 * The sample points in the bounded and unbounded dimensions are
881 * then combined into a single sample point and transformed back
882 * to the original space.
884 __isl_give isl_vec *isl_basic_set_sample_with_cone(
885 __isl_take isl_basic_set *bset, __isl_take isl_basic_set *cone)
887 unsigned total;
888 unsigned cone_dim;
889 struct isl_mat *M, *U;
890 struct isl_vec *sample;
891 struct isl_vec *cone_sample;
892 struct isl_ctx *ctx;
893 struct isl_basic_set *bounded;
895 if (!bset || !cone)
896 goto error;
898 ctx = bset->ctx;
899 total = isl_basic_set_total_dim(cone);
900 cone_dim = total - cone->n_eq;
902 M = isl_mat_sub_alloc(bset->ctx, cone->eq, 0, cone->n_eq, 1, total);
903 M = isl_mat_left_hermite(M, 0, &U, NULL);
904 if (!M)
905 goto error;
906 isl_mat_free(M);
908 U = isl_mat_lin_to_aff(U);
909 bset = isl_basic_set_preimage(bset, isl_mat_copy(U));
911 bounded = isl_basic_set_copy(bset);
912 bounded = drop_constraints_involving(bounded, total - cone_dim, cone_dim);
913 bounded = isl_basic_set_drop_dims(bounded, total - cone_dim, cone_dim);
914 sample = sample_bounded(bounded);
915 if (!sample || sample->size == 0) {
916 isl_basic_set_free(bset);
917 isl_basic_set_free(cone);
918 isl_mat_free(U);
919 return sample;
921 bset = plug_in(bset, isl_vec_copy(sample));
922 cone_sample = rational_sample(bset);
923 cone_sample = round_up_in_cone(cone_sample, cone, isl_mat_copy(U));
924 sample = vec_concat(sample, cone_sample);
925 sample = isl_mat_vec_product(U, sample);
926 return sample;
927 error:
928 isl_basic_set_free(cone);
929 isl_basic_set_free(bset);
930 return NULL;
933 static void vec_sum_of_neg(struct isl_vec *v, isl_int *s)
935 int i;
937 isl_int_set_si(*s, 0);
939 for (i = 0; i < v->size; ++i)
940 if (isl_int_is_neg(v->el[i]))
941 isl_int_add(*s, *s, v->el[i]);
944 /* Given a tableau "tab", a tableau "tab_cone" that corresponds
945 * to the recession cone and the inverse of a new basis U = inv(B),
946 * with the unbounded directions in B last,
947 * add constraints to "tab" that ensure any rational value
948 * in the unbounded directions can be rounded up to an integer value.
950 * The new basis is given by x' = B x, i.e., x = U x'.
951 * For any rational value of the last tab->n_unbounded coordinates
952 * in the update tableau, the value that is obtained by rounding
953 * up this value should be contained in the original tableau.
954 * For any constraint "a x + c >= 0", we therefore need to add
955 * a constraint "a x + c + s >= 0", with s the sum of all negative
956 * entries in the last elements of "a U".
958 * Since we are not interested in the first entries of any of the "a U",
959 * we first drop the columns of U that correpond to bounded directions.
961 static int tab_shift_cone(struct isl_tab *tab,
962 struct isl_tab *tab_cone, struct isl_mat *U)
964 int i;
965 isl_int v;
966 struct isl_basic_set *bset = NULL;
968 if (tab && tab->n_unbounded == 0) {
969 isl_mat_free(U);
970 return 0;
972 isl_int_init(v);
973 if (!tab || !tab_cone || !U)
974 goto error;
975 bset = isl_tab_peek_bset(tab_cone);
976 U = isl_mat_drop_cols(U, 0, tab->n_var - tab->n_unbounded);
977 for (i = 0; i < bset->n_ineq; ++i) {
978 int ok;
979 struct isl_vec *row = NULL;
980 if (isl_tab_is_equality(tab_cone, tab_cone->n_eq + i))
981 continue;
982 row = isl_vec_alloc(bset->ctx, tab_cone->n_var);
983 if (!row)
984 goto error;
985 isl_seq_cpy(row->el, bset->ineq[i] + 1, tab_cone->n_var);
986 row = isl_vec_mat_product(row, isl_mat_copy(U));
987 if (!row)
988 goto error;
989 vec_sum_of_neg(row, &v);
990 isl_vec_free(row);
991 if (isl_int_is_zero(v))
992 continue;
993 tab = isl_tab_extend(tab, 1);
994 isl_int_add(bset->ineq[i][0], bset->ineq[i][0], v);
995 ok = isl_tab_add_ineq(tab, bset->ineq[i]) >= 0;
996 isl_int_sub(bset->ineq[i][0], bset->ineq[i][0], v);
997 if (!ok)
998 goto error;
1001 isl_mat_free(U);
1002 isl_int_clear(v);
1003 return 0;
1004 error:
1005 isl_mat_free(U);
1006 isl_int_clear(v);
1007 return -1;
1010 /* Compute and return an initial basis for the possibly
1011 * unbounded tableau "tab". "tab_cone" is a tableau
1012 * for the corresponding recession cone.
1013 * Additionally, add constraints to "tab" that ensure
1014 * that any rational value for the unbounded directions
1015 * can be rounded up to an integer value.
1017 * If the tableau is bounded, i.e., if the recession cone
1018 * is zero-dimensional, then we just use inital_basis.
1019 * Otherwise, we construct a basis whose first directions
1020 * correspond to equalities, followed by bounded directions,
1021 * i.e., equalities in the recession cone.
1022 * The remaining directions are then unbounded.
1024 int isl_tab_set_initial_basis_with_cone(struct isl_tab *tab,
1025 struct isl_tab *tab_cone)
1027 struct isl_mat *eq;
1028 struct isl_mat *cone_eq;
1029 struct isl_mat *U, *Q;
1031 if (!tab || !tab_cone)
1032 return -1;
1034 if (tab_cone->n_col == tab_cone->n_dead) {
1035 tab->basis = initial_basis(tab);
1036 return tab->basis ? 0 : -1;
1039 eq = tab_equalities(tab);
1040 if (!eq)
1041 return -1;
1042 tab->n_zero = eq->n_row;
1043 cone_eq = tab_equalities(tab_cone);
1044 eq = isl_mat_concat(eq, cone_eq);
1045 if (!eq)
1046 return -1;
1047 tab->n_unbounded = tab->n_var - (eq->n_row - tab->n_zero);
1048 eq = isl_mat_left_hermite(eq, 0, &U, &Q);
1049 if (!eq)
1050 return -1;
1051 isl_mat_free(eq);
1052 tab->basis = isl_mat_lin_to_aff(Q);
1053 if (tab_shift_cone(tab, tab_cone, U) < 0)
1054 return -1;
1055 if (!tab->basis)
1056 return -1;
1057 return 0;
1060 /* Compute and return a sample point in bset using generalized basis
1061 * reduction. We first check if the input set has a non-trivial
1062 * recession cone. If so, we perform some extra preprocessing in
1063 * sample_with_cone. Otherwise, we directly perform generalized basis
1064 * reduction.
1066 static struct isl_vec *gbr_sample(struct isl_basic_set *bset)
1068 unsigned dim;
1069 struct isl_basic_set *cone;
1071 dim = isl_basic_set_total_dim(bset);
1073 cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));
1075 if (cone->n_eq < dim)
1076 return isl_basic_set_sample_with_cone(bset, cone);
1078 isl_basic_set_free(cone);
1079 return sample_bounded(bset);
1082 static struct isl_vec *pip_sample(struct isl_basic_set *bset)
1084 struct isl_mat *T;
1085 struct isl_ctx *ctx;
1086 struct isl_vec *sample;
1088 bset = isl_basic_set_skew_to_positive_orthant(bset, &T);
1089 if (!bset)
1090 return NULL;
1092 ctx = bset->ctx;
1093 sample = isl_pip_basic_set_sample(bset);
1095 if (sample && sample->size != 0)
1096 sample = isl_mat_vec_product(T, sample);
1097 else
1098 isl_mat_free(T);
1100 return sample;
1103 static struct isl_vec *basic_set_sample(struct isl_basic_set *bset, int bounded)
1105 struct isl_ctx *ctx;
1106 unsigned dim;
1107 if (!bset)
1108 return NULL;
1110 ctx = bset->ctx;
1111 if (isl_basic_set_fast_is_empty(bset))
1112 return empty_sample(bset);
1114 dim = isl_basic_set_n_dim(bset);
1115 isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
1116 isl_assert(ctx, bset->n_div == 0, goto error);
1118 if (bset->sample && bset->sample->size == 1 + dim) {
1119 int contains = isl_basic_set_contains(bset, bset->sample);
1120 if (contains < 0)
1121 goto error;
1122 if (contains) {
1123 struct isl_vec *sample = isl_vec_copy(bset->sample);
1124 isl_basic_set_free(bset);
1125 return sample;
1128 isl_vec_free(bset->sample);
1129 bset->sample = NULL;
1131 if (bset->n_eq > 0)
1132 return sample_eq(bset, bounded ? isl_basic_set_sample_bounded
1133 : isl_basic_set_sample_vec);
1134 if (dim == 0)
1135 return zero_sample(bset);
1136 if (dim == 1)
1137 return interval_sample(bset);
1139 switch (bset->ctx->opt->ilp_solver) {
1140 case ISL_ILP_PIP:
1141 return pip_sample(bset);
1142 case ISL_ILP_GBR:
1143 return bounded ? sample_bounded(bset) : gbr_sample(bset);
1145 isl_assert(bset->ctx, 0, );
1146 error:
1147 isl_basic_set_free(bset);
1148 return NULL;
1151 __isl_give isl_vec *isl_basic_set_sample_vec(__isl_take isl_basic_set *bset)
1153 return basic_set_sample(bset, 0);
1156 /* Compute an integer sample in "bset", where the caller guarantees
1157 * that "bset" is bounded.
1159 struct isl_vec *isl_basic_set_sample_bounded(struct isl_basic_set *bset)
1161 return basic_set_sample(bset, 1);
1164 __isl_give isl_basic_set *isl_basic_set_from_vec(__isl_take isl_vec *vec)
1166 int i;
1167 int k;
1168 struct isl_basic_set *bset = NULL;
1169 struct isl_ctx *ctx;
1170 unsigned dim;
1172 if (!vec)
1173 return NULL;
1174 ctx = vec->ctx;
1175 isl_assert(ctx, vec->size != 0, goto error);
1177 bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
1178 if (!bset)
1179 goto error;
1180 dim = isl_basic_set_n_dim(bset);
1181 for (i = dim - 1; i >= 0; --i) {
1182 k = isl_basic_set_alloc_equality(bset);
1183 if (k < 0)
1184 goto error;
1185 isl_seq_clr(bset->eq[k], 1 + dim);
1186 isl_int_neg(bset->eq[k][0], vec->el[1 + i]);
1187 isl_int_set(bset->eq[k][1 + i], vec->el[0]);
1189 bset->sample = vec;
1191 return bset;
1192 error:
1193 isl_basic_set_free(bset);
1194 isl_vec_free(vec);
1195 return NULL;
1198 __isl_give isl_basic_map *isl_basic_map_sample(__isl_take isl_basic_map *bmap)
1200 struct isl_basic_set *bset;
1201 struct isl_vec *sample_vec;
1203 bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
1204 sample_vec = isl_basic_set_sample_vec(bset);
1205 if (!sample_vec)
1206 goto error;
1207 if (sample_vec->size == 0) {
1208 struct isl_basic_map *sample;
1209 sample = isl_basic_map_empty_like(bmap);
1210 isl_vec_free(sample_vec);
1211 isl_basic_map_free(bmap);
1212 return sample;
1214 bset = isl_basic_set_from_vec(sample_vec);
1215 return isl_basic_map_overlying_set(bset, bmap);
1216 error:
1217 isl_basic_map_free(bmap);
1218 return NULL;
1221 __isl_give isl_basic_map *isl_map_sample(__isl_take isl_map *map)
1223 int i;
1224 isl_basic_map *sample = NULL;
1226 if (!map)
1227 goto error;
1229 for (i = 0; i < map->n; ++i) {
1230 sample = isl_basic_map_sample(isl_basic_map_copy(map->p[i]));
1231 if (!sample)
1232 goto error;
1233 if (!ISL_F_ISSET(sample, ISL_BASIC_MAP_EMPTY))
1234 break;
1235 isl_basic_map_free(sample);
1237 if (i == map->n)
1238 sample = isl_basic_map_empty_like_map(map);
1239 isl_map_free(map);
1240 return sample;
1241 error:
1242 isl_map_free(map);
1243 return NULL;
1246 __isl_give isl_basic_set *isl_set_sample(__isl_take isl_set *set)
1248 return (isl_basic_set *) isl_map_sample((isl_map *)set);
1251 __isl_give isl_point *isl_basic_set_sample_point(__isl_take isl_basic_set *bset)
1253 isl_vec *vec;
1254 isl_dim *dim;
1256 dim = isl_basic_set_get_dim(bset);
1257 bset = isl_basic_set_underlying_set(bset);
1258 vec = isl_basic_set_sample_vec(bset);
1260 return isl_point_alloc(dim, vec);
1263 __isl_give isl_point *isl_set_sample_point(__isl_take isl_set *set)
1265 int i;
1266 isl_point *pnt;
1268 if (!set)
1269 return NULL;
1271 for (i = 0; i < set->n; ++i) {
1272 pnt = isl_basic_set_sample_point(isl_basic_set_copy(set->p[i]));
1273 if (!pnt)
1274 goto error;
1275 if (!isl_point_is_void(pnt))
1276 break;
1277 isl_point_free(pnt);
1279 if (i == set->n)
1280 pnt = isl_point_void(isl_set_get_dim(set));
1282 isl_set_free(set);
1283 return pnt;
1284 error:
1285 isl_set_free(set);
1286 return NULL;