1 #include "isl_sample.h"
2 #include "isl_sample_piplib.h"
6 #include "isl_map_private.h"
7 #include "isl_equalities.h"
9 #include "isl_basis_reduction.h"
11 static struct isl_vec *empty_sample(struct isl_basic_set *bset)
15 vec = isl_vec_alloc(bset->ctx, 0);
16 isl_basic_set_free(bset);
20 /* Construct a zero sample of the same dimension as bset.
21 * As a special case, if bset is zero-dimensional, this
22 * function creates a zero-dimensional sample point.
24 static struct isl_vec *zero_sample(struct isl_basic_set *bset)
27 struct isl_vec *sample;
29 dim = isl_basic_set_total_dim(bset);
30 sample = isl_vec_alloc(bset->ctx, 1 + dim);
32 isl_int_set_si(sample->el[0], 1);
33 isl_seq_clr(sample->el + 1, dim);
35 isl_basic_set_free(bset);
39 static struct isl_vec *interval_sample(struct isl_basic_set *bset)
43 struct isl_vec *sample;
45 bset = isl_basic_set_simplify(bset);
48 if (isl_basic_set_fast_is_empty(bset))
49 return empty_sample(bset);
50 if (bset->n_eq == 0 && bset->n_ineq == 0)
51 return zero_sample(bset);
53 sample = isl_vec_alloc(bset->ctx, 2);
54 isl_int_set_si(sample->block.data[0], 1);
57 isl_assert(bset->ctx, bset->n_eq == 1, goto error);
58 isl_assert(bset->ctx, bset->n_ineq == 0, goto error);
59 if (isl_int_is_one(bset->eq[0][1]))
60 isl_int_neg(sample->el[1], bset->eq[0][0]);
62 isl_assert(bset->ctx, isl_int_is_negone(bset->eq[0][1]),
64 isl_int_set(sample->el[1], bset->eq[0][0]);
66 isl_basic_set_free(bset);
71 if (isl_int_is_one(bset->ineq[0][1]))
72 isl_int_neg(sample->block.data[1], bset->ineq[0][0]);
74 isl_int_set(sample->block.data[1], bset->ineq[0][0]);
75 for (i = 1; i < bset->n_ineq; ++i) {
76 isl_seq_inner_product(sample->block.data,
77 bset->ineq[i], 2, &t);
78 if (isl_int_is_neg(t))
82 if (i < bset->n_ineq) {
84 return empty_sample(bset);
87 isl_basic_set_free(bset);
90 isl_basic_set_free(bset);
95 static struct isl_mat *independent_bounds(struct isl_basic_set *bset)
98 struct isl_mat *dirs = NULL;
99 struct isl_mat *bounds = NULL;
105 dim = isl_basic_set_n_dim(bset);
106 bounds = isl_mat_alloc(bset->ctx, 1+dim, 1+dim);
110 isl_int_set_si(bounds->row[0][0], 1);
111 isl_seq_clr(bounds->row[0]+1, dim);
114 if (bset->n_ineq == 0)
117 dirs = isl_mat_alloc(bset->ctx, dim, dim);
119 isl_mat_free(bounds);
122 isl_seq_cpy(dirs->row[0], bset->ineq[0]+1, dirs->n_col);
123 isl_seq_cpy(bounds->row[1], bset->ineq[0], bounds->n_col);
124 for (j = 1, n = 1; n < dim && j < bset->n_ineq; ++j) {
127 isl_seq_cpy(dirs->row[n], bset->ineq[j]+1, dirs->n_col);
129 pos = isl_seq_first_non_zero(dirs->row[n], dirs->n_col);
132 for (i = 0; i < n; ++i) {
134 pos_i = isl_seq_first_non_zero(dirs->row[i], dirs->n_col);
139 isl_seq_elim(dirs->row[n], dirs->row[i], pos,
141 pos = isl_seq_first_non_zero(dirs->row[n], dirs->n_col);
149 isl_int *t = dirs->row[n];
150 for (k = n; k > i; --k)
151 dirs->row[k] = dirs->row[k-1];
155 isl_seq_cpy(bounds->row[n], bset->ineq[j], bounds->n_col);
162 static void swap_inequality(struct isl_basic_set *bset, int a, int b)
164 isl_int *t = bset->ineq[a];
165 bset->ineq[a] = bset->ineq[b];
169 /* Skew into positive orthant and project out lineality space.
171 * We perform a unimodular transformation that turns a selected
172 * maximal set of linearly independent bounds into constraints
173 * on the first dimensions that impose that these first dimensions
174 * are non-negative. In particular, the constraint matrix is lower
175 * triangular with positive entries on the diagonal and negative
177 * If "bset" has a lineality space then these constraints (and therefore
178 * all constraints in bset) only involve the first dimensions.
179 * The remaining dimensions then do not appear in any constraints and
180 * we can select any value for them, say zero. We therefore project
181 * out this final dimensions and plug in the value zero later. This
182 * is accomplished by simply dropping the final columns of
183 * the unimodular transformation.
185 static struct isl_basic_set *isl_basic_set_skew_to_positive_orthant(
186 struct isl_basic_set *bset, struct isl_mat **T)
188 struct isl_mat *U = NULL;
189 struct isl_mat *bounds = NULL;
191 unsigned old_dim, new_dim;
197 isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
198 isl_assert(bset->ctx, bset->n_div == 0, goto error);
199 isl_assert(bset->ctx, bset->n_eq == 0, goto error);
201 old_dim = isl_basic_set_n_dim(bset);
202 /* Try to move (multiples of) unit rows up. */
203 for (i = 0, j = 0; i < bset->n_ineq; ++i) {
204 int pos = isl_seq_first_non_zero(bset->ineq[i]+1, old_dim);
207 if (isl_seq_first_non_zero(bset->ineq[i]+1+pos+1,
211 swap_inequality(bset, i, j);
214 bounds = independent_bounds(bset);
217 new_dim = bounds->n_row - 1;
218 bounds = isl_mat_left_hermite(bounds, 1, &U, NULL);
221 U = isl_mat_drop_cols(U, 1 + new_dim, old_dim - new_dim);
222 bset = isl_basic_set_preimage(bset, isl_mat_copy(U));
226 isl_mat_free(bounds);
229 isl_mat_free(bounds);
231 isl_basic_set_free(bset);
235 /* Find a sample integer point, if any, in bset, which is known
236 * to have equalities. If bset contains no integer points, then
237 * return a zero-length vector.
238 * We simply remove the known equalities, compute a sample
239 * in the resulting bset, using the specified recurse function,
240 * and then transform the sample back to the original space.
242 static struct isl_vec *sample_eq(struct isl_basic_set *bset,
243 struct isl_vec *(*recurse)(struct isl_basic_set *))
246 struct isl_vec *sample;
251 bset = isl_basic_set_remove_equalities(bset, &T, NULL);
252 sample = recurse(bset);
253 if (!sample || sample->size == 0)
256 sample = isl_mat_vec_product(T, sample);
260 /* Return a matrix containing the equalities of the tableau
261 * in constraint form. The tableau is assumed to have
262 * an associated bset that has been kept up-to-date.
264 static struct isl_mat *tab_equalities(struct isl_tab *tab)
269 struct isl_basic_set *bset;
274 isl_assert(tab->mat->ctx, tab->bset, return NULL);
277 n_eq = tab->n_var - tab->n_col + tab->n_dead;
278 if (tab->empty || n_eq == 0)
279 return isl_mat_alloc(tab->mat->ctx, 0, tab->n_var);
280 if (n_eq == tab->n_var)
281 return isl_mat_identity(tab->mat->ctx, tab->n_var);
283 eq = isl_mat_alloc(tab->mat->ctx, n_eq, tab->n_var);
286 for (i = 0, j = 0; i < tab->n_con; ++i) {
287 if (tab->con[i].is_row)
289 if (tab->con[i].index >= 0 && tab->con[i].index >= tab->n_dead)
292 isl_seq_cpy(eq->row[j], bset->eq[i] + 1, tab->n_var);
294 isl_seq_cpy(eq->row[j],
295 bset->ineq[i - bset->n_eq] + 1, tab->n_var);
298 isl_assert(bset->ctx, j == n_eq, goto error);
305 /* Compute and return an initial basis for the bounded tableau "tab".
307 * If the tableau is either full-dimensional or zero-dimensional,
308 * the we simply return an identity matrix.
309 * Otherwise, we construct a basis whose first directions correspond
312 static struct isl_mat *initial_basis(struct isl_tab *tab)
318 n_eq = tab->n_var - tab->n_col + tab->n_dead;
319 if (tab->empty || n_eq == 0 || n_eq == tab->n_var)
320 return isl_mat_identity(tab->mat->ctx, 1 + tab->n_var);
322 eq = tab_equalities(tab);
323 eq = isl_mat_left_hermite(eq, 0, NULL, &Q);
328 Q = isl_mat_lin_to_aff(Q);
332 /* Given a tableau representing a set, find and return
333 * an integer point in the set, if there is any.
335 * We perform a depth first search
336 * for an integer point, by scanning all possible values in the range
337 * attained by a basis vector, where an initial basis may have been set
338 * by the calling function. Otherwise an initial basis that exploits
339 * the equalities in the tableau is created.
340 * tab->n_zero is currently ignored and is clobbered by this function.
342 * The tableau is allowed to have unbounded direction, but then
343 * the calling function needs to set an initial basis, with the
344 * unbounded directions last and with tab->n_unbounded set
345 * to the number of unbounded directions.
346 * Furthermore, the calling functions needs to add shifted copies
347 * of all constraints involving unbounded directions to ensure
348 * that any feasible rational value in these directions can be rounded
349 * up to yield a feasible integer value.
350 * In particular, let B define the given basis x' = B x
351 * and let T be the inverse of B, i.e., X = T x'.
352 * Let a x + c >= 0 be a constraint of the set represented by the tableau,
353 * or a T x' + c >= 0 in terms of the given basis. Assume that
354 * the bounded directions have an integer value, then we can safely
355 * round up the values for the unbounded directions if we make sure
356 * that x' not only satisfies the original constraint, but also
357 * the constraint "a T x' + c + s >= 0" with s the sum of all
358 * negative values in the last n_unbounded entries of "a T".
359 * The calling function therefore needs to add the constraint
360 * a x + c + s >= 0. The current function then scans the first
361 * directions for an integer value and once those have been found,
362 * it can compute "T ceil(B x)" to yield an integer point in the set.
363 * Note that during the search, the first rows of B may be changed
364 * by a basis reduction, but the last n_unbounded rows of B remain
365 * unaltered and are also not mixed into the first rows.
367 * The search is implemented iteratively. "level" identifies the current
368 * basis vector. "init" is true if we want the first value at the current
369 * level and false if we want the next value.
371 * The initial basis is the identity matrix. If the range in some direction
372 * contains more than one integer value, we perform basis reduction based
373 * on the value of ctx->gbr
374 * - ISL_GBR_NEVER: never perform basis reduction
375 * - ISL_GBR_ONCE: only perform basis reduction the first
376 * time such a range is encountered
377 * - ISL_GBR_ALWAYS: always perform basis reduction when
378 * such a range is encountered
380 * When ctx->gbr is set to ISL_GBR_ALWAYS, then we allow the basis
381 * reduction computation to return early. That is, as soon as it
382 * finds a reasonable first direction.
384 struct isl_vec *isl_tab_sample(struct isl_tab *tab)
389 struct isl_vec *sample;
392 enum isl_lp_result res;
396 struct isl_tab_undo **snap;
401 return isl_vec_alloc(tab->mat->ctx, 0);
404 tab->basis = initial_basis(tab);
407 isl_assert(tab->mat->ctx, tab->basis->n_row == tab->n_var + 1,
409 isl_assert(tab->mat->ctx, tab->basis->n_col == tab->n_var + 1,
416 if (tab->n_unbounded == tab->n_var) {
417 sample = isl_tab_get_sample_value(tab);
418 sample = isl_mat_vec_product(isl_mat_copy(tab->basis), sample);
419 sample = isl_vec_ceil(sample);
420 sample = isl_mat_vec_inverse_product(isl_mat_copy(tab->basis),
425 if (isl_tab_extend_cons(tab, dim + 1) < 0)
428 min = isl_vec_alloc(ctx, dim);
429 max = isl_vec_alloc(ctx, dim);
430 snap = isl_alloc_array(ctx, struct isl_tab_undo *, dim);
432 if (!min || !max || !snap)
442 res = isl_tab_min(tab, tab->basis->row[1 + level],
443 ctx->one, &min->el[level], NULL, 0);
444 if (res == isl_lp_empty)
446 isl_assert(ctx, res != isl_lp_unbounded, goto error);
447 if (res == isl_lp_error)
449 if (!empty && isl_tab_sample_is_integer(tab))
451 isl_seq_neg(tab->basis->row[1 + level] + 1,
452 tab->basis->row[1 + level] + 1, dim);
453 res = isl_tab_min(tab, tab->basis->row[1 + level],
454 ctx->one, &max->el[level], NULL, 0);
455 isl_seq_neg(tab->basis->row[1 + level] + 1,
456 tab->basis->row[1 + level] + 1, dim);
457 isl_int_neg(max->el[level], max->el[level]);
458 if (res == isl_lp_empty)
460 isl_assert(ctx, res != isl_lp_unbounded, goto error);
461 if (res == isl_lp_error)
463 if (!empty && isl_tab_sample_is_integer(tab))
465 if (!empty && !reduced && ctx->gbr != ISL_GBR_NEVER &&
466 isl_int_lt(min->el[level], max->el[level])) {
467 unsigned gbr_only_first;
468 if (ctx->gbr == ISL_GBR_ONCE)
469 ctx->gbr = ISL_GBR_NEVER;
471 gbr_only_first = ctx->gbr_only_first;
472 ctx->gbr_only_first =
473 ctx->gbr == ISL_GBR_ALWAYS;
474 tab = isl_tab_compute_reduced_basis(tab);
475 ctx->gbr_only_first = gbr_only_first;
476 if (!tab || !tab->basis)
482 snap[level] = isl_tab_snap(tab);
484 isl_int_add_ui(min->el[level], min->el[level], 1);
486 if (empty || isl_int_gt(min->el[level], max->el[level])) {
490 if (isl_tab_rollback(tab, snap[level]) < 0)
494 isl_int_neg(tab->basis->row[1 + level][0], min->el[level]);
495 tab = isl_tab_add_valid_eq(tab, tab->basis->row[1 + level]);
496 isl_int_set_si(tab->basis->row[1 + level][0], 0);
497 if (level + tab->n_unbounded < dim - 1) {
506 sample = isl_tab_get_sample_value(tab);
509 if (tab->n_unbounded && !isl_int_is_one(sample->el[0])) {
510 sample = isl_mat_vec_product(isl_mat_copy(tab->basis),
512 sample = isl_vec_ceil(sample);
513 sample = isl_mat_vec_inverse_product(
514 isl_mat_copy(tab->basis), sample);
517 sample = isl_vec_alloc(ctx, 0);
532 /* Given a basic set that is known to be bounded, find and return
533 * an integer point in the basic set, if there is any.
535 * After handling some trivial cases, we construct a tableau
536 * and then use isl_tab_sample to find a sample, passing it
537 * the identity matrix as initial basis.
539 static struct isl_vec *sample_bounded(struct isl_basic_set *bset)
543 struct isl_vec *sample;
544 struct isl_tab *tab = NULL;
549 if (isl_basic_set_fast_is_empty(bset))
550 return empty_sample(bset);
552 dim = isl_basic_set_total_dim(bset);
554 return zero_sample(bset);
556 return interval_sample(bset);
558 return sample_eq(bset, sample_bounded);
562 tab = isl_tab_from_basic_set(bset);
563 if (!ISL_F_ISSET(bset, ISL_BASIC_SET_NO_IMPLICIT))
564 tab = isl_tab_detect_implicit_equalities(tab);
568 tab->bset = isl_basic_set_copy(bset);
570 sample = isl_tab_sample(tab);
574 if (sample->size > 0) {
575 isl_vec_free(bset->sample);
576 bset->sample = isl_vec_copy(sample);
579 isl_basic_set_free(bset);
583 isl_basic_set_free(bset);
588 /* Given a basic set "bset" and a value "sample" for the first coordinates
589 * of bset, plug in these values and drop the corresponding coordinates.
591 * We do this by computing the preimage of the transformation
597 * where [1 s] is the sample value and I is the identity matrix of the
598 * appropriate dimension.
600 static struct isl_basic_set *plug_in(struct isl_basic_set *bset,
601 struct isl_vec *sample)
607 if (!bset || !sample)
610 total = isl_basic_set_total_dim(bset);
611 T = isl_mat_alloc(bset->ctx, 1 + total, 1 + total - (sample->size - 1));
615 for (i = 0; i < sample->size; ++i) {
616 isl_int_set(T->row[i][0], sample->el[i]);
617 isl_seq_clr(T->row[i] + 1, T->n_col - 1);
619 for (i = 0; i < T->n_col - 1; ++i) {
620 isl_seq_clr(T->row[sample->size + i], T->n_col);
621 isl_int_set_si(T->row[sample->size + i][1 + i], 1);
623 isl_vec_free(sample);
625 bset = isl_basic_set_preimage(bset, T);
628 isl_basic_set_free(bset);
629 isl_vec_free(sample);
633 /* Given a basic set "bset", return any (possibly non-integer) point
636 static struct isl_vec *rational_sample(struct isl_basic_set *bset)
639 struct isl_vec *sample;
644 tab = isl_tab_from_basic_set(bset);
645 sample = isl_tab_get_sample_value(tab);
648 isl_basic_set_free(bset);
653 /* Given a linear cone "cone" and a rational point "vec",
654 * construct a polyhedron with shifted copies of the constraints in "cone",
655 * i.e., a polyhedron with "cone" as its recession cone, such that each
656 * point x in this polyhedron is such that the unit box positioned at x
657 * lies entirely inside the affine cone 'vec + cone'.
658 * Any rational point in this polyhedron may therefore be rounded up
659 * to yield an integer point that lies inside said affine cone.
661 * Denote the constraints of cone by "<a_i, x> >= 0" and the rational
662 * point "vec" by v/d.
663 * Let b_i = <a_i, v>. Then the affine cone 'vec + cone' is given
664 * by <a_i, x> - b/d >= 0.
665 * The polyhedron <a_i, x> - ceil{b/d} >= 0 is a subset of this affine cone.
666 * We prefer this polyhedron over the actual affine cone because it doesn't
667 * require a scaling of the constraints.
668 * If each of the vertices of the unit cube positioned at x lies inside
669 * this polyhedron, then the whole unit cube at x lies inside the affine cone.
670 * We therefore impose that x' = x + \sum e_i, for any selection of unit
671 * vectors lies inside the polyhedron, i.e.,
673 * <a_i, x'> - ceil{b/d} = <a_i, x> + sum a_i - ceil{b/d} >= 0
675 * The most stringent of these constraints is the one that selects
676 * all negative a_i, so the polyhedron we are looking for has constraints
678 * <a_i, x> + sum_{a_i < 0} a_i - ceil{b/d} >= 0
680 * Note that if cone were known to have only non-negative rays
681 * (which can be accomplished by a unimodular transformation),
682 * then we would only have to check the points x' = x + e_i
683 * and we only have to add the smallest negative a_i (if any)
684 * instead of the sum of all negative a_i.
686 static struct isl_basic_set *shift_cone(struct isl_basic_set *cone,
692 struct isl_basic_set *shift = NULL;
697 isl_assert(cone->ctx, cone->n_eq == 0, goto error);
699 total = isl_basic_set_total_dim(cone);
701 shift = isl_basic_set_alloc_dim(isl_basic_set_get_dim(cone),
704 for (i = 0; i < cone->n_ineq; ++i) {
705 k = isl_basic_set_alloc_inequality(shift);
708 isl_seq_cpy(shift->ineq[k] + 1, cone->ineq[i] + 1, total);
709 isl_seq_inner_product(shift->ineq[k] + 1, vec->el + 1, total,
711 isl_int_cdiv_q(shift->ineq[k][0],
712 shift->ineq[k][0], vec->el[0]);
713 isl_int_neg(shift->ineq[k][0], shift->ineq[k][0]);
714 for (j = 0; j < total; ++j) {
715 if (isl_int_is_nonneg(shift->ineq[k][1 + j]))
717 isl_int_add(shift->ineq[k][0],
718 shift->ineq[k][0], shift->ineq[k][1 + j]);
722 isl_basic_set_free(cone);
725 return isl_basic_set_finalize(shift);
727 isl_basic_set_free(shift);
728 isl_basic_set_free(cone);
733 /* Given a rational point vec in a (transformed) basic set,
734 * such that cone is the recession cone of the original basic set,
735 * "round up" the rational point to an integer point.
737 * We first check if the rational point just happens to be integer.
738 * If not, we transform the cone in the same way as the basic set,
739 * pick a point x in this cone shifted to the rational point such that
740 * the whole unit cube at x is also inside this affine cone.
741 * Then we simply round up the coordinates of x and return the
742 * resulting integer point.
744 static struct isl_vec *round_up_in_cone(struct isl_vec *vec,
745 struct isl_basic_set *cone, struct isl_mat *U)
749 if (!vec || !cone || !U)
752 isl_assert(vec->ctx, vec->size != 0, goto error);
753 if (isl_int_is_one(vec->el[0])) {
755 isl_basic_set_free(cone);
759 total = isl_basic_set_total_dim(cone);
760 cone = isl_basic_set_preimage(cone, U);
761 cone = isl_basic_set_remove_dims(cone, 0, total - (vec->size - 1));
763 cone = shift_cone(cone, vec);
765 vec = rational_sample(cone);
766 vec = isl_vec_ceil(vec);
771 isl_basic_set_free(cone);
775 /* Concatenate two integer vectors, i.e., two vectors with denominator
776 * (stored in element 0) equal to 1.
778 static struct isl_vec *vec_concat(struct isl_vec *vec1, struct isl_vec *vec2)
784 isl_assert(vec1->ctx, vec1->size > 0, goto error);
785 isl_assert(vec2->ctx, vec2->size > 0, goto error);
786 isl_assert(vec1->ctx, isl_int_is_one(vec1->el[0]), goto error);
787 isl_assert(vec2->ctx, isl_int_is_one(vec2->el[0]), goto error);
789 vec = isl_vec_alloc(vec1->ctx, vec1->size + vec2->size - 1);
793 isl_seq_cpy(vec->el, vec1->el, vec1->size);
794 isl_seq_cpy(vec->el + vec1->size, vec2->el + 1, vec2->size - 1);
806 /* Drop all constraints in bset that involve any of the dimensions
807 * first to first+n-1.
809 static struct isl_basic_set *drop_constraints_involving
810 (struct isl_basic_set *bset, unsigned first, unsigned n)
817 bset = isl_basic_set_cow(bset);
819 for (i = bset->n_ineq - 1; i >= 0; --i) {
820 if (isl_seq_first_non_zero(bset->ineq[i] + 1 + first, n) == -1)
822 isl_basic_set_drop_inequality(bset, i);
828 /* Give a basic set "bset" with recession cone "cone", compute and
829 * return an integer point in bset, if any.
831 * If the recession cone is full-dimensional, then we know that
832 * bset contains an infinite number of integer points and it is
833 * fairly easy to pick one of them.
834 * If the recession cone is not full-dimensional, then we first
835 * transform bset such that the bounded directions appear as
836 * the first dimensions of the transformed basic set.
837 * We do this by using a unimodular transformation that transforms
838 * the equalities in the recession cone to equalities on the first
841 * The transformed set is then projected onto its bounded dimensions.
842 * Note that to compute this projection, we can simply drop all constraints
843 * involving any of the unbounded dimensions since these constraints
844 * cannot be combined to produce a constraint on the bounded dimensions.
845 * To see this, assume that there is such a combination of constraints
846 * that produces a constraint on the bounded dimensions. This means
847 * that some combination of the unbounded dimensions has both an upper
848 * bound and a lower bound in terms of the bounded dimensions, but then
849 * this combination would be a bounded direction too and would have been
850 * transformed into a bounded dimensions.
852 * We then compute a sample value in the bounded dimensions.
853 * If no such value can be found, then the original set did not contain
854 * any integer points and we are done.
855 * Otherwise, we plug in the value we found in the bounded dimensions,
856 * project out these bounded dimensions and end up with a set with
857 * a full-dimensional recession cone.
858 * A sample point in this set is computed by "rounding up" any
859 * rational point in the set.
861 * The sample points in the bounded and unbounded dimensions are
862 * then combined into a single sample point and transformed back
863 * to the original space.
865 __isl_give isl_vec *isl_basic_set_sample_with_cone(
866 __isl_take isl_basic_set *bset, __isl_take isl_basic_set *cone)
870 struct isl_mat *M, *U;
871 struct isl_vec *sample;
872 struct isl_vec *cone_sample;
874 struct isl_basic_set *bounded;
880 total = isl_basic_set_total_dim(cone);
881 cone_dim = total - cone->n_eq;
883 M = isl_mat_sub_alloc(bset->ctx, cone->eq, 0, cone->n_eq, 1, total);
884 M = isl_mat_left_hermite(M, 0, &U, NULL);
889 U = isl_mat_lin_to_aff(U);
890 bset = isl_basic_set_preimage(bset, isl_mat_copy(U));
892 bounded = isl_basic_set_copy(bset);
893 bounded = drop_constraints_involving(bounded, total - cone_dim, cone_dim);
894 bounded = isl_basic_set_drop_dims(bounded, total - cone_dim, cone_dim);
895 sample = sample_bounded(bounded);
896 if (!sample || sample->size == 0) {
897 isl_basic_set_free(bset);
898 isl_basic_set_free(cone);
902 bset = plug_in(bset, isl_vec_copy(sample));
903 cone_sample = rational_sample(bset);
904 cone_sample = round_up_in_cone(cone_sample, cone, isl_mat_copy(U));
905 sample = vec_concat(sample, cone_sample);
906 sample = isl_mat_vec_product(U, sample);
909 isl_basic_set_free(cone);
910 isl_basic_set_free(bset);
914 static void vec_sum_of_neg(struct isl_vec *v, isl_int *s)
918 isl_int_set_si(*s, 0);
920 for (i = 0; i < v->size; ++i)
921 if (isl_int_is_neg(v->el[i]))
922 isl_int_add(*s, *s, v->el[i]);
925 /* Given a tableau "tab", a tableau "tab_cone" that corresponds
926 * to the recession cone and the inverse of a new basis U = inv(B),
927 * with the unbounded directions in B last,
928 * add constraints to "tab" that ensure any rational value
929 * in the unbounded directions can be rounded up to an integer value.
931 * The new basis is given by x' = B x, i.e., x = U x'.
932 * For any rational value of the last tab->n_unbounded coordinates
933 * in the update tableau, the value that is obtained by rounding
934 * up this value should be contained in the original tableau.
935 * For any constraint "a x + c >= 0", we therefore need to add
936 * a constraint "a x + c + s >= 0", with s the sum of all negative
937 * entries in the last elements of "a U".
939 * Since we are not interested in the first entries of any of the "a U",
940 * we first drop the columns of U that correpond to bounded directions.
942 static int tab_shift_cone(struct isl_tab *tab,
943 struct isl_tab *tab_cone, struct isl_mat *U)
947 struct isl_basic_set *bset = NULL;
949 if (tab && tab->n_unbounded == 0) {
954 if (!tab || !tab_cone || !U)
956 bset = tab_cone->bset;
957 U = isl_mat_drop_cols(U, 0, tab->n_var - tab->n_unbounded);
958 for (i = 0; i < bset->n_ineq; ++i) {
959 struct isl_vec *row = NULL;
960 if (isl_tab_is_equality(tab_cone, tab_cone->n_eq + i))
962 row = isl_vec_alloc(bset->ctx, tab_cone->n_var);
965 isl_seq_cpy(row->el, bset->ineq[i] + 1, tab_cone->n_var);
966 row = isl_vec_mat_product(row, isl_mat_copy(U));
969 vec_sum_of_neg(row, &v);
971 if (isl_int_is_zero(v))
973 tab = isl_tab_extend(tab, 1);
974 isl_int_add(bset->ineq[i][0], bset->ineq[i][0], v);
975 tab = isl_tab_add_ineq(tab, bset->ineq[i]);
976 isl_int_sub(bset->ineq[i][0], bset->ineq[i][0], v);
990 /* Compute and return an initial basis for the possibly
991 * unbounded tableau "tab". "tab_cone" is a tableau
992 * for the corresponding recession cone.
993 * Additionally, add constraints to "tab" that ensure
994 * that any rational value for the unbounded directions
995 * can be rounded up to an integer value.
997 * If the tableau is bounded, i.e., if the recession cone
998 * is zero-dimensional, then we just use inital_basis.
999 * Otherwise, we construct a basis whose first directions
1000 * correspond to equalities, followed by bounded directions,
1001 * i.e., equalities in the recession cone.
1002 * The remaining directions are then unbounded.
1004 int isl_tab_set_initial_basis_with_cone(struct isl_tab *tab,
1005 struct isl_tab *tab_cone)
1008 struct isl_mat *cone_eq;
1009 struct isl_mat *U, *Q;
1011 if (!tab || !tab_cone)
1014 if (tab_cone->n_col == tab_cone->n_dead) {
1015 tab->basis = initial_basis(tab);
1016 return tab->basis ? 0 : -1;
1019 eq = tab_equalities(tab);
1022 tab->n_zero = eq->n_row;
1023 cone_eq = tab_equalities(tab_cone);
1024 eq = isl_mat_concat(eq, cone_eq);
1027 tab->n_unbounded = tab->n_var - (eq->n_row - tab->n_zero);
1028 eq = isl_mat_left_hermite(eq, 0, &U, &Q);
1032 tab->basis = isl_mat_lin_to_aff(Q);
1033 if (tab_shift_cone(tab, tab_cone, U) < 0)
1040 /* Compute and return a sample point in bset using generalized basis
1041 * reduction. We first check if the input set has a non-trivial
1042 * recession cone. If so, we perform some extra preprocessing in
1043 * sample_with_cone. Otherwise, we directly perform generalized basis
1046 static struct isl_vec *gbr_sample(struct isl_basic_set *bset)
1049 struct isl_basic_set *cone;
1051 dim = isl_basic_set_total_dim(bset);
1053 cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));
1055 if (cone->n_eq < dim)
1056 return isl_basic_set_sample_with_cone(bset, cone);
1058 isl_basic_set_free(cone);
1059 return sample_bounded(bset);
1062 static struct isl_vec *pip_sample(struct isl_basic_set *bset)
1065 struct isl_ctx *ctx;
1066 struct isl_vec *sample;
1068 bset = isl_basic_set_skew_to_positive_orthant(bset, &T);
1073 sample = isl_pip_basic_set_sample(bset);
1075 if (sample && sample->size != 0)
1076 sample = isl_mat_vec_product(T, sample);
1083 static struct isl_vec *basic_set_sample(struct isl_basic_set *bset, int bounded)
1085 struct isl_ctx *ctx;
1091 if (isl_basic_set_fast_is_empty(bset))
1092 return empty_sample(bset);
1094 dim = isl_basic_set_n_dim(bset);
1095 isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
1096 isl_assert(ctx, bset->n_div == 0, goto error);
1098 if (bset->sample && bset->sample->size == 1 + dim) {
1099 int contains = isl_basic_set_contains(bset, bset->sample);
1103 struct isl_vec *sample = isl_vec_copy(bset->sample);
1104 isl_basic_set_free(bset);
1108 isl_vec_free(bset->sample);
1109 bset->sample = NULL;
1112 return sample_eq(bset, bounded ? isl_basic_set_sample_bounded
1113 : isl_basic_set_sample_vec);
1115 return zero_sample(bset);
1117 return interval_sample(bset);
1119 switch (bset->ctx->ilp_solver) {
1121 return pip_sample(bset);
1123 return bounded ? sample_bounded(bset) : gbr_sample(bset);
1125 isl_assert(bset->ctx, 0, );
1127 isl_basic_set_free(bset);
1131 __isl_give isl_vec *isl_basic_set_sample_vec(__isl_take isl_basic_set *bset)
1133 return basic_set_sample(bset, 0);
1136 /* Compute an integer sample in "bset", where the caller guarantees
1137 * that "bset" is bounded.
1139 struct isl_vec *isl_basic_set_sample_bounded(struct isl_basic_set *bset)
1141 return basic_set_sample(bset, 1);
1144 __isl_give isl_basic_set *isl_basic_set_from_vec(__isl_take isl_vec *vec)
1148 struct isl_basic_set *bset = NULL;
1149 struct isl_ctx *ctx;
1155 isl_assert(ctx, vec->size != 0, goto error);
1157 bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
1160 dim = isl_basic_set_n_dim(bset);
1161 for (i = dim - 1; i >= 0; --i) {
1162 k = isl_basic_set_alloc_equality(bset);
1165 isl_seq_clr(bset->eq[k], 1 + dim);
1166 isl_int_neg(bset->eq[k][0], vec->el[1 + i]);
1167 isl_int_set(bset->eq[k][1 + i], vec->el[0]);
1173 isl_basic_set_free(bset);
1178 __isl_give isl_basic_map *isl_basic_map_sample(__isl_take isl_basic_map *bmap)
1180 struct isl_basic_set *bset;
1181 struct isl_vec *sample_vec;
1183 bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
1184 sample_vec = isl_basic_set_sample_vec(bset);
1187 if (sample_vec->size == 0) {
1188 struct isl_basic_map *sample;
1189 sample = isl_basic_map_empty_like(bmap);
1190 isl_vec_free(sample_vec);
1191 isl_basic_map_free(bmap);
1194 bset = isl_basic_set_from_vec(sample_vec);
1195 return isl_basic_map_overlying_set(bset, bmap);
1197 isl_basic_map_free(bmap);
1201 __isl_give isl_basic_map *isl_map_sample(__isl_take isl_map *map)
1204 isl_basic_map *sample = NULL;
1209 for (i = 0; i < map->n; ++i) {
1210 sample = isl_basic_map_sample(isl_basic_map_copy(map->p[i]));
1213 if (!ISL_F_ISSET(sample, ISL_BASIC_MAP_EMPTY))
1215 isl_basic_map_free(sample);
1218 sample = isl_basic_map_empty_like_map(map);
1226 __isl_give isl_basic_set *isl_set_sample(__isl_take isl_set *set)
1228 return (isl_basic_set *) isl_map_sample((isl_map *)set);