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
10 #include "isl_sample.h"
11 #include "isl_sample_piplib.h"
15 #include "isl_map_private.h"
16 #include "isl_equalities.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)
25 vec = isl_vec_alloc(bset->ctx, 0);
26 isl_basic_set_free(bset);
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)
37 struct isl_vec *sample;
39 dim = isl_basic_set_total_dim(bset);
40 sample = isl_vec_alloc(bset->ctx, 1 + dim);
42 isl_int_set_si(sample->el[0], 1);
43 isl_seq_clr(sample->el + 1, dim);
45 isl_basic_set_free(bset);
49 static struct isl_vec *interval_sample(struct isl_basic_set *bset)
53 struct isl_vec *sample;
55 bset = isl_basic_set_simplify(bset);
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);
68 isl_int_set_si(sample->block.data[0], 1);
71 isl_assert(bset->ctx, bset->n_eq == 1, goto error);
72 isl_assert(bset->ctx, bset->n_ineq == 0, goto error);
73 if (isl_int_is_one(bset->eq[0][1]))
74 isl_int_neg(sample->el[1], bset->eq[0][0]);
76 isl_assert(bset->ctx, isl_int_is_negone(bset->eq[0][1]),
78 isl_int_set(sample->el[1], bset->eq[0][0]);
80 isl_basic_set_free(bset);
85 if (isl_int_is_one(bset->ineq[0][1]))
86 isl_int_neg(sample->block.data[1], bset->ineq[0][0]);
88 isl_int_set(sample->block.data[1], bset->ineq[0][0]);
89 for (i = 1; i < bset->n_ineq; ++i) {
90 isl_seq_inner_product(sample->block.data,
91 bset->ineq[i], 2, &t);
92 if (isl_int_is_neg(t))
96 if (i < bset->n_ineq) {
98 return empty_sample(bset);
101 isl_basic_set_free(bset);
104 isl_basic_set_free(bset);
105 isl_vec_free(sample);
109 static struct isl_mat *independent_bounds(struct isl_basic_set *bset)
112 struct isl_mat *dirs = NULL;
113 struct isl_mat *bounds = NULL;
119 dim = isl_basic_set_n_dim(bset);
120 bounds = isl_mat_alloc(bset->ctx, 1+dim, 1+dim);
124 isl_int_set_si(bounds->row[0][0], 1);
125 isl_seq_clr(bounds->row[0]+1, dim);
128 if (bset->n_ineq == 0)
131 dirs = isl_mat_alloc(bset->ctx, dim, dim);
133 isl_mat_free(bounds);
136 isl_seq_cpy(dirs->row[0], bset->ineq[0]+1, dirs->n_col);
137 isl_seq_cpy(bounds->row[1], bset->ineq[0], bounds->n_col);
138 for (j = 1, n = 1; n < dim && j < bset->n_ineq; ++j) {
141 isl_seq_cpy(dirs->row[n], bset->ineq[j]+1, dirs->n_col);
143 pos = isl_seq_first_non_zero(dirs->row[n], dirs->n_col);
146 for (i = 0; i < n; ++i) {
148 pos_i = isl_seq_first_non_zero(dirs->row[i], dirs->n_col);
153 isl_seq_elim(dirs->row[n], dirs->row[i], pos,
155 pos = isl_seq_first_non_zero(dirs->row[n], dirs->n_col);
163 isl_int *t = dirs->row[n];
164 for (k = n; k > i; --k)
165 dirs->row[k] = dirs->row[k-1];
169 isl_seq_cpy(bounds->row[n], bset->ineq[j], bounds->n_col);
176 static void swap_inequality(struct isl_basic_set *bset, int a, int b)
178 isl_int *t = bset->ineq[a];
179 bset->ineq[a] = bset->ineq[b];
183 /* Skew into positive orthant and project out lineality space.
185 * We perform a unimodular transformation that turns a selected
186 * maximal set of linearly independent bounds into constraints
187 * on the first dimensions that impose that these first dimensions
188 * are non-negative. In particular, the constraint matrix is lower
189 * triangular with positive entries on the diagonal and negative
191 * If "bset" has a lineality space then these constraints (and therefore
192 * all constraints in bset) only involve the first dimensions.
193 * The remaining dimensions then do not appear in any constraints and
194 * we can select any value for them, say zero. We therefore project
195 * out this final dimensions and plug in the value zero later. This
196 * is accomplished by simply dropping the final columns of
197 * the unimodular transformation.
199 static struct isl_basic_set *isl_basic_set_skew_to_positive_orthant(
200 struct isl_basic_set *bset, struct isl_mat **T)
202 struct isl_mat *U = NULL;
203 struct isl_mat *bounds = NULL;
205 unsigned old_dim, new_dim;
211 isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
212 isl_assert(bset->ctx, bset->n_div == 0, goto error);
213 isl_assert(bset->ctx, bset->n_eq == 0, goto error);
215 old_dim = isl_basic_set_n_dim(bset);
216 /* Try to move (multiples of) unit rows up. */
217 for (i = 0, j = 0; i < bset->n_ineq; ++i) {
218 int pos = isl_seq_first_non_zero(bset->ineq[i]+1, old_dim);
221 if (isl_seq_first_non_zero(bset->ineq[i]+1+pos+1,
225 swap_inequality(bset, i, j);
228 bounds = independent_bounds(bset);
231 new_dim = bounds->n_row - 1;
232 bounds = isl_mat_left_hermite(bounds, 1, &U, NULL);
235 U = isl_mat_drop_cols(U, 1 + new_dim, old_dim - new_dim);
236 bset = isl_basic_set_preimage(bset, isl_mat_copy(U));
240 isl_mat_free(bounds);
243 isl_mat_free(bounds);
245 isl_basic_set_free(bset);
249 /* Find a sample integer point, if any, in bset, which is known
250 * to have equalities. If bset contains no integer points, then
251 * return a zero-length vector.
252 * We simply remove the known equalities, compute a sample
253 * in the resulting bset, using the specified recurse function,
254 * and then transform the sample back to the original space.
256 static struct isl_vec *sample_eq(struct isl_basic_set *bset,
257 struct isl_vec *(*recurse)(struct isl_basic_set *))
260 struct isl_vec *sample;
265 bset = isl_basic_set_remove_equalities(bset, &T, NULL);
266 sample = recurse(bset);
267 if (!sample || sample->size == 0)
270 sample = isl_mat_vec_product(T, sample);
274 /* Return a matrix containing the equalities of the tableau
275 * in constraint form. The tableau is assumed to have
276 * an associated bset that has been kept up-to-date.
278 static struct isl_mat *tab_equalities(struct isl_tab *tab)
283 struct isl_basic_set *bset;
288 bset = isl_tab_peek_bset(tab);
289 isl_assert(tab->mat->ctx, bset, return NULL);
291 n_eq = tab->n_var - tab->n_col + tab->n_dead;
292 if (tab->empty || n_eq == 0)
293 return isl_mat_alloc(tab->mat->ctx, 0, tab->n_var);
294 if (n_eq == tab->n_var)
295 return isl_mat_identity(tab->mat->ctx, tab->n_var);
297 eq = isl_mat_alloc(tab->mat->ctx, n_eq, tab->n_var);
300 for (i = 0, j = 0; i < tab->n_con; ++i) {
301 if (tab->con[i].is_row)
303 if (tab->con[i].index >= 0 && tab->con[i].index >= tab->n_dead)
306 isl_seq_cpy(eq->row[j], bset->eq[i] + 1, tab->n_var);
308 isl_seq_cpy(eq->row[j],
309 bset->ineq[i - bset->n_eq] + 1, tab->n_var);
312 isl_assert(bset->ctx, j == n_eq, goto error);
319 /* Compute and return an initial basis for the bounded tableau "tab".
321 * If the tableau is either full-dimensional or zero-dimensional,
322 * the we simply return an identity matrix.
323 * Otherwise, we construct a basis whose first directions correspond
326 static struct isl_mat *initial_basis(struct isl_tab *tab)
332 tab->n_unbounded = 0;
333 tab->n_zero = n_eq = tab->n_var - tab->n_col + tab->n_dead;
334 if (tab->empty || n_eq == 0 || n_eq == tab->n_var)
335 return isl_mat_identity(tab->mat->ctx, 1 + tab->n_var);
337 eq = tab_equalities(tab);
338 eq = isl_mat_left_hermite(eq, 0, NULL, &Q);
343 Q = isl_mat_lin_to_aff(Q);
347 /* Given a tableau representing a set, find and return
348 * an integer point in the set, if there is any.
350 * We perform a depth first search
351 * for an integer point, by scanning all possible values in the range
352 * attained by a basis vector, where an initial basis may have been set
353 * by the calling function. Otherwise an initial basis that exploits
354 * the equalities in the tableau is created.
355 * tab->n_zero is currently ignored and is clobbered by this function.
357 * The tableau is allowed to have unbounded direction, but then
358 * the calling function needs to set an initial basis, with the
359 * unbounded directions last and with tab->n_unbounded set
360 * to the number of unbounded directions.
361 * Furthermore, the calling functions needs to add shifted copies
362 * of all constraints involving unbounded directions to ensure
363 * that any feasible rational value in these directions can be rounded
364 * up to yield a feasible integer value.
365 * In particular, let B define the given basis x' = B x
366 * and let T be the inverse of B, i.e., X = T x'.
367 * Let a x + c >= 0 be a constraint of the set represented by the tableau,
368 * or a T x' + c >= 0 in terms of the given basis. Assume that
369 * the bounded directions have an integer value, then we can safely
370 * round up the values for the unbounded directions if we make sure
371 * that x' not only satisfies the original constraint, but also
372 * the constraint "a T x' + c + s >= 0" with s the sum of all
373 * negative values in the last n_unbounded entries of "a T".
374 * The calling function therefore needs to add the constraint
375 * a x + c + s >= 0. The current function then scans the first
376 * directions for an integer value and once those have been found,
377 * it can compute "T ceil(B x)" to yield an integer point in the set.
378 * Note that during the search, the first rows of B may be changed
379 * by a basis reduction, but the last n_unbounded rows of B remain
380 * unaltered and are also not mixed into the first rows.
382 * The search is implemented iteratively. "level" identifies the current
383 * basis vector. "init" is true if we want the first value at the current
384 * level and false if we want the next value.
386 * The initial basis is the identity matrix. If the range in some direction
387 * contains more than one integer value, we perform basis reduction based
388 * on the value of ctx->opt->gbr
389 * - ISL_GBR_NEVER: never perform basis reduction
390 * - ISL_GBR_ONCE: only perform basis reduction the first
391 * time such a range is encountered
392 * - ISL_GBR_ALWAYS: always perform basis reduction when
393 * such a range is encountered
395 * When ctx->opt->gbr is set to ISL_GBR_ALWAYS, then we allow the basis
396 * reduction computation to return early. That is, as soon as it
397 * finds a reasonable first direction.
399 struct isl_vec *isl_tab_sample(struct isl_tab *tab)
404 struct isl_vec *sample;
407 enum isl_lp_result res;
411 struct isl_tab_undo **snap;
416 return isl_vec_alloc(tab->mat->ctx, 0);
419 tab->basis = initial_basis(tab);
422 isl_assert(tab->mat->ctx, tab->basis->n_row == tab->n_var + 1,
424 isl_assert(tab->mat->ctx, tab->basis->n_col == tab->n_var + 1,
431 if (tab->n_unbounded == tab->n_var) {
432 sample = isl_tab_get_sample_value(tab);
433 sample = isl_mat_vec_product(isl_mat_copy(tab->basis), sample);
434 sample = isl_vec_ceil(sample);
435 sample = isl_mat_vec_inverse_product(isl_mat_copy(tab->basis),
440 if (isl_tab_extend_cons(tab, dim + 1) < 0)
443 min = isl_vec_alloc(ctx, dim);
444 max = isl_vec_alloc(ctx, dim);
445 snap = isl_alloc_array(ctx, struct isl_tab_undo *, dim);
447 if (!min || !max || !snap)
457 res = isl_tab_min(tab, tab->basis->row[1 + level],
458 ctx->one, &min->el[level], NULL, 0);
459 if (res == isl_lp_empty)
461 isl_assert(ctx, res != isl_lp_unbounded, goto error);
462 if (res == isl_lp_error)
464 if (!empty && isl_tab_sample_is_integer(tab))
466 isl_seq_neg(tab->basis->row[1 + level] + 1,
467 tab->basis->row[1 + level] + 1, dim);
468 res = isl_tab_min(tab, tab->basis->row[1 + level],
469 ctx->one, &max->el[level], NULL, 0);
470 isl_seq_neg(tab->basis->row[1 + level] + 1,
471 tab->basis->row[1 + level] + 1, dim);
472 isl_int_neg(max->el[level], max->el[level]);
473 if (res == isl_lp_empty)
475 isl_assert(ctx, res != isl_lp_unbounded, goto error);
476 if (res == isl_lp_error)
478 if (!empty && isl_tab_sample_is_integer(tab))
480 if (!empty && !reduced &&
481 ctx->opt->gbr != ISL_GBR_NEVER &&
482 isl_int_lt(min->el[level], max->el[level])) {
483 unsigned gbr_only_first;
484 if (ctx->opt->gbr == ISL_GBR_ONCE)
485 ctx->opt->gbr = ISL_GBR_NEVER;
487 gbr_only_first = ctx->opt->gbr_only_first;
488 ctx->opt->gbr_only_first =
489 ctx->opt->gbr == ISL_GBR_ALWAYS;
490 tab = isl_tab_compute_reduced_basis(tab);
491 ctx->opt->gbr_only_first = gbr_only_first;
492 if (!tab || !tab->basis)
498 snap[level] = isl_tab_snap(tab);
500 isl_int_add_ui(min->el[level], min->el[level], 1);
502 if (empty || isl_int_gt(min->el[level], max->el[level])) {
506 if (isl_tab_rollback(tab, snap[level]) < 0)
510 isl_int_neg(tab->basis->row[1 + level][0], min->el[level]);
511 if (isl_tab_add_valid_eq(tab, tab->basis->row[1 + level]) < 0)
513 isl_int_set_si(tab->basis->row[1 + level][0], 0);
514 if (level + tab->n_unbounded < dim - 1) {
523 sample = isl_tab_get_sample_value(tab);
526 if (tab->n_unbounded && !isl_int_is_one(sample->el[0])) {
527 sample = isl_mat_vec_product(isl_mat_copy(tab->basis),
529 sample = isl_vec_ceil(sample);
530 sample = isl_mat_vec_inverse_product(
531 isl_mat_copy(tab->basis), sample);
534 sample = isl_vec_alloc(ctx, 0);
549 /* Given a basic set that is known to be bounded, find and return
550 * an integer point in the basic set, if there is any.
552 * After handling some trivial cases, we construct a tableau
553 * and then use isl_tab_sample to find a sample, passing it
554 * the identity matrix as initial basis.
556 static struct isl_vec *sample_bounded(struct isl_basic_set *bset)
560 struct isl_vec *sample;
561 struct isl_tab *tab = NULL;
566 if (isl_basic_set_fast_is_empty(bset))
567 return empty_sample(bset);
569 dim = isl_basic_set_total_dim(bset);
571 return zero_sample(bset);
573 return interval_sample(bset);
575 return sample_eq(bset, sample_bounded);
579 tab = isl_tab_from_basic_set(bset);
580 if (tab && tab->empty) {
582 ISL_F_SET(bset, ISL_BASIC_SET_EMPTY);
583 sample = isl_vec_alloc(bset->ctx, 0);
584 isl_basic_set_free(bset);
588 if (isl_tab_track_bset(tab, isl_basic_set_copy(bset)) < 0)
590 if (!ISL_F_ISSET(bset, ISL_BASIC_SET_NO_IMPLICIT))
591 if (isl_tab_detect_implicit_equalities(tab) < 0)
594 sample = isl_tab_sample(tab);
598 if (sample->size > 0) {
599 isl_vec_free(bset->sample);
600 bset->sample = isl_vec_copy(sample);
603 isl_basic_set_free(bset);
607 isl_basic_set_free(bset);
612 /* Given a basic set "bset" and a value "sample" for the first coordinates
613 * of bset, plug in these values and drop the corresponding coordinates.
615 * We do this by computing the preimage of the transformation
621 * where [1 s] is the sample value and I is the identity matrix of the
622 * appropriate dimension.
624 static struct isl_basic_set *plug_in(struct isl_basic_set *bset,
625 struct isl_vec *sample)
631 if (!bset || !sample)
634 total = isl_basic_set_total_dim(bset);
635 T = isl_mat_alloc(bset->ctx, 1 + total, 1 + total - (sample->size - 1));
639 for (i = 0; i < sample->size; ++i) {
640 isl_int_set(T->row[i][0], sample->el[i]);
641 isl_seq_clr(T->row[i] + 1, T->n_col - 1);
643 for (i = 0; i < T->n_col - 1; ++i) {
644 isl_seq_clr(T->row[sample->size + i], T->n_col);
645 isl_int_set_si(T->row[sample->size + i][1 + i], 1);
647 isl_vec_free(sample);
649 bset = isl_basic_set_preimage(bset, T);
652 isl_basic_set_free(bset);
653 isl_vec_free(sample);
657 /* Given a basic set "bset", return any (possibly non-integer) point
660 static struct isl_vec *rational_sample(struct isl_basic_set *bset)
663 struct isl_vec *sample;
668 tab = isl_tab_from_basic_set(bset);
669 sample = isl_tab_get_sample_value(tab);
672 isl_basic_set_free(bset);
677 /* Given a linear cone "cone" and a rational point "vec",
678 * construct a polyhedron with shifted copies of the constraints in "cone",
679 * i.e., a polyhedron with "cone" as its recession cone, such that each
680 * point x in this polyhedron is such that the unit box positioned at x
681 * lies entirely inside the affine cone 'vec + cone'.
682 * Any rational point in this polyhedron may therefore be rounded up
683 * to yield an integer point that lies inside said affine cone.
685 * Denote the constraints of cone by "<a_i, x> >= 0" and the rational
686 * point "vec" by v/d.
687 * Let b_i = <a_i, v>. Then the affine cone 'vec + cone' is given
688 * by <a_i, x> - b/d >= 0.
689 * The polyhedron <a_i, x> - ceil{b/d} >= 0 is a subset of this affine cone.
690 * We prefer this polyhedron over the actual affine cone because it doesn't
691 * require a scaling of the constraints.
692 * If each of the vertices of the unit cube positioned at x lies inside
693 * this polyhedron, then the whole unit cube at x lies inside the affine cone.
694 * We therefore impose that x' = x + \sum e_i, for any selection of unit
695 * vectors lies inside the polyhedron, i.e.,
697 * <a_i, x'> - ceil{b/d} = <a_i, x> + sum a_i - ceil{b/d} >= 0
699 * The most stringent of these constraints is the one that selects
700 * all negative a_i, so the polyhedron we are looking for has constraints
702 * <a_i, x> + sum_{a_i < 0} a_i - ceil{b/d} >= 0
704 * Note that if cone were known to have only non-negative rays
705 * (which can be accomplished by a unimodular transformation),
706 * then we would only have to check the points x' = x + e_i
707 * and we only have to add the smallest negative a_i (if any)
708 * instead of the sum of all negative a_i.
710 static struct isl_basic_set *shift_cone(struct isl_basic_set *cone,
716 struct isl_basic_set *shift = NULL;
721 isl_assert(cone->ctx, cone->n_eq == 0, goto error);
723 total = isl_basic_set_total_dim(cone);
725 shift = isl_basic_set_alloc_dim(isl_basic_set_get_dim(cone),
728 for (i = 0; i < cone->n_ineq; ++i) {
729 k = isl_basic_set_alloc_inequality(shift);
732 isl_seq_cpy(shift->ineq[k] + 1, cone->ineq[i] + 1, total);
733 isl_seq_inner_product(shift->ineq[k] + 1, vec->el + 1, total,
735 isl_int_cdiv_q(shift->ineq[k][0],
736 shift->ineq[k][0], vec->el[0]);
737 isl_int_neg(shift->ineq[k][0], shift->ineq[k][0]);
738 for (j = 0; j < total; ++j) {
739 if (isl_int_is_nonneg(shift->ineq[k][1 + j]))
741 isl_int_add(shift->ineq[k][0],
742 shift->ineq[k][0], shift->ineq[k][1 + j]);
746 isl_basic_set_free(cone);
749 return isl_basic_set_finalize(shift);
751 isl_basic_set_free(shift);
752 isl_basic_set_free(cone);
757 /* Given a rational point vec in a (transformed) basic set,
758 * such that cone is the recession cone of the original basic set,
759 * "round up" the rational point to an integer point.
761 * We first check if the rational point just happens to be integer.
762 * If not, we transform the cone in the same way as the basic set,
763 * pick a point x in this cone shifted to the rational point such that
764 * the whole unit cube at x is also inside this affine cone.
765 * Then we simply round up the coordinates of x and return the
766 * resulting integer point.
768 static struct isl_vec *round_up_in_cone(struct isl_vec *vec,
769 struct isl_basic_set *cone, struct isl_mat *U)
773 if (!vec || !cone || !U)
776 isl_assert(vec->ctx, vec->size != 0, goto error);
777 if (isl_int_is_one(vec->el[0])) {
779 isl_basic_set_free(cone);
783 total = isl_basic_set_total_dim(cone);
784 cone = isl_basic_set_preimage(cone, U);
785 cone = isl_basic_set_remove_dims(cone, 0, total - (vec->size - 1));
787 cone = shift_cone(cone, vec);
789 vec = rational_sample(cone);
790 vec = isl_vec_ceil(vec);
795 isl_basic_set_free(cone);
799 /* Concatenate two integer vectors, i.e., two vectors with denominator
800 * (stored in element 0) equal to 1.
802 static struct isl_vec *vec_concat(struct isl_vec *vec1, struct isl_vec *vec2)
808 isl_assert(vec1->ctx, vec1->size > 0, goto error);
809 isl_assert(vec2->ctx, vec2->size > 0, goto error);
810 isl_assert(vec1->ctx, isl_int_is_one(vec1->el[0]), goto error);
811 isl_assert(vec2->ctx, isl_int_is_one(vec2->el[0]), goto error);
813 vec = isl_vec_alloc(vec1->ctx, vec1->size + vec2->size - 1);
817 isl_seq_cpy(vec->el, vec1->el, vec1->size);
818 isl_seq_cpy(vec->el + vec1->size, vec2->el + 1, vec2->size - 1);
830 /* Drop all constraints in bset that involve any of the dimensions
831 * first to first+n-1.
833 static struct isl_basic_set *drop_constraints_involving
834 (struct isl_basic_set *bset, unsigned first, unsigned n)
838 bset = isl_basic_set_cow(bset);
843 for (i = bset->n_ineq - 1; i >= 0; --i) {
844 if (isl_seq_first_non_zero(bset->ineq[i] + 1 + first, n) == -1)
846 isl_basic_set_drop_inequality(bset, i);
852 /* Give a basic set "bset" with recession cone "cone", compute and
853 * return an integer point in bset, if any.
855 * If the recession cone is full-dimensional, then we know that
856 * bset contains an infinite number of integer points and it is
857 * fairly easy to pick one of them.
858 * If the recession cone is not full-dimensional, then we first
859 * transform bset such that the bounded directions appear as
860 * the first dimensions of the transformed basic set.
861 * We do this by using a unimodular transformation that transforms
862 * the equalities in the recession cone to equalities on the first
865 * The transformed set is then projected onto its bounded dimensions.
866 * Note that to compute this projection, we can simply drop all constraints
867 * involving any of the unbounded dimensions since these constraints
868 * cannot be combined to produce a constraint on the bounded dimensions.
869 * To see this, assume that there is such a combination of constraints
870 * that produces a constraint on the bounded dimensions. This means
871 * that some combination of the unbounded dimensions has both an upper
872 * bound and a lower bound in terms of the bounded dimensions, but then
873 * this combination would be a bounded direction too and would have been
874 * transformed into a bounded dimensions.
876 * We then compute a sample value in the bounded dimensions.
877 * If no such value can be found, then the original set did not contain
878 * any integer points and we are done.
879 * Otherwise, we plug in the value we found in the bounded dimensions,
880 * project out these bounded dimensions and end up with a set with
881 * a full-dimensional recession cone.
882 * A sample point in this set is computed by "rounding up" any
883 * rational point in the set.
885 * The sample points in the bounded and unbounded dimensions are
886 * then combined into a single sample point and transformed back
887 * to the original space.
889 __isl_give isl_vec *isl_basic_set_sample_with_cone(
890 __isl_take isl_basic_set *bset, __isl_take isl_basic_set *cone)
894 struct isl_mat *M, *U;
895 struct isl_vec *sample;
896 struct isl_vec *cone_sample;
898 struct isl_basic_set *bounded;
904 total = isl_basic_set_total_dim(cone);
905 cone_dim = total - cone->n_eq;
907 M = isl_mat_sub_alloc(bset->ctx, cone->eq, 0, cone->n_eq, 1, total);
908 M = isl_mat_left_hermite(M, 0, &U, NULL);
913 U = isl_mat_lin_to_aff(U);
914 bset = isl_basic_set_preimage(bset, isl_mat_copy(U));
916 bounded = isl_basic_set_copy(bset);
917 bounded = drop_constraints_involving(bounded, total - cone_dim, cone_dim);
918 bounded = isl_basic_set_drop_dims(bounded, total - cone_dim, cone_dim);
919 sample = sample_bounded(bounded);
920 if (!sample || sample->size == 0) {
921 isl_basic_set_free(bset);
922 isl_basic_set_free(cone);
926 bset = plug_in(bset, isl_vec_copy(sample));
927 cone_sample = rational_sample(bset);
928 cone_sample = round_up_in_cone(cone_sample, cone, isl_mat_copy(U));
929 sample = vec_concat(sample, cone_sample);
930 sample = isl_mat_vec_product(U, sample);
933 isl_basic_set_free(cone);
934 isl_basic_set_free(bset);
938 static void vec_sum_of_neg(struct isl_vec *v, isl_int *s)
942 isl_int_set_si(*s, 0);
944 for (i = 0; i < v->size; ++i)
945 if (isl_int_is_neg(v->el[i]))
946 isl_int_add(*s, *s, v->el[i]);
949 /* Given a tableau "tab", a tableau "tab_cone" that corresponds
950 * to the recession cone and the inverse of a new basis U = inv(B),
951 * with the unbounded directions in B last,
952 * add constraints to "tab" that ensure any rational value
953 * in the unbounded directions can be rounded up to an integer value.
955 * The new basis is given by x' = B x, i.e., x = U x'.
956 * For any rational value of the last tab->n_unbounded coordinates
957 * in the update tableau, the value that is obtained by rounding
958 * up this value should be contained in the original tableau.
959 * For any constraint "a x + c >= 0", we therefore need to add
960 * a constraint "a x + c + s >= 0", with s the sum of all negative
961 * entries in the last elements of "a U".
963 * Since we are not interested in the first entries of any of the "a U",
964 * we first drop the columns of U that correpond to bounded directions.
966 static int tab_shift_cone(struct isl_tab *tab,
967 struct isl_tab *tab_cone, struct isl_mat *U)
971 struct isl_basic_set *bset = NULL;
973 if (tab && tab->n_unbounded == 0) {
978 if (!tab || !tab_cone || !U)
980 bset = isl_tab_peek_bset(tab_cone);
981 U = isl_mat_drop_cols(U, 0, tab->n_var - tab->n_unbounded);
982 for (i = 0; i < bset->n_ineq; ++i) {
984 struct isl_vec *row = NULL;
985 if (isl_tab_is_equality(tab_cone, tab_cone->n_eq + i))
987 row = isl_vec_alloc(bset->ctx, tab_cone->n_var);
990 isl_seq_cpy(row->el, bset->ineq[i] + 1, tab_cone->n_var);
991 row = isl_vec_mat_product(row, isl_mat_copy(U));
994 vec_sum_of_neg(row, &v);
996 if (isl_int_is_zero(v))
998 tab = isl_tab_extend(tab, 1);
999 isl_int_add(bset->ineq[i][0], bset->ineq[i][0], v);
1000 ok = isl_tab_add_ineq(tab, bset->ineq[i]) >= 0;
1001 isl_int_sub(bset->ineq[i][0], bset->ineq[i][0], v);
1015 /* Compute and return an initial basis for the possibly
1016 * unbounded tableau "tab". "tab_cone" is a tableau
1017 * for the corresponding recession cone.
1018 * Additionally, add constraints to "tab" that ensure
1019 * that any rational value for the unbounded directions
1020 * can be rounded up to an integer value.
1022 * If the tableau is bounded, i.e., if the recession cone
1023 * is zero-dimensional, then we just use inital_basis.
1024 * Otherwise, we construct a basis whose first directions
1025 * correspond to equalities, followed by bounded directions,
1026 * i.e., equalities in the recession cone.
1027 * The remaining directions are then unbounded.
1029 int isl_tab_set_initial_basis_with_cone(struct isl_tab *tab,
1030 struct isl_tab *tab_cone)
1033 struct isl_mat *cone_eq;
1034 struct isl_mat *U, *Q;
1036 if (!tab || !tab_cone)
1039 if (tab_cone->n_col == tab_cone->n_dead) {
1040 tab->basis = initial_basis(tab);
1041 return tab->basis ? 0 : -1;
1044 eq = tab_equalities(tab);
1047 tab->n_zero = eq->n_row;
1048 cone_eq = tab_equalities(tab_cone);
1049 eq = isl_mat_concat(eq, cone_eq);
1052 tab->n_unbounded = tab->n_var - (eq->n_row - tab->n_zero);
1053 eq = isl_mat_left_hermite(eq, 0, &U, &Q);
1057 tab->basis = isl_mat_lin_to_aff(Q);
1058 if (tab_shift_cone(tab, tab_cone, U) < 0)
1065 /* Compute and return a sample point in bset using generalized basis
1066 * reduction. We first check if the input set has a non-trivial
1067 * recession cone. If so, we perform some extra preprocessing in
1068 * sample_with_cone. Otherwise, we directly perform generalized basis
1071 static struct isl_vec *gbr_sample(struct isl_basic_set *bset)
1074 struct isl_basic_set *cone;
1076 dim = isl_basic_set_total_dim(bset);
1078 cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));
1082 if (cone->n_eq < dim)
1083 return isl_basic_set_sample_with_cone(bset, cone);
1085 isl_basic_set_free(cone);
1086 return sample_bounded(bset);
1088 isl_basic_set_free(bset);
1092 static struct isl_vec *pip_sample(struct isl_basic_set *bset)
1095 struct isl_ctx *ctx;
1096 struct isl_vec *sample;
1098 bset = isl_basic_set_skew_to_positive_orthant(bset, &T);
1103 sample = isl_pip_basic_set_sample(bset);
1105 if (sample && sample->size != 0)
1106 sample = isl_mat_vec_product(T, sample);
1113 static struct isl_vec *basic_set_sample(struct isl_basic_set *bset, int bounded)
1115 struct isl_ctx *ctx;
1121 if (isl_basic_set_fast_is_empty(bset))
1122 return empty_sample(bset);
1124 dim = isl_basic_set_n_dim(bset);
1125 isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
1126 isl_assert(ctx, bset->n_div == 0, goto error);
1128 if (bset->sample && bset->sample->size == 1 + dim) {
1129 int contains = isl_basic_set_contains(bset, bset->sample);
1133 struct isl_vec *sample = isl_vec_copy(bset->sample);
1134 isl_basic_set_free(bset);
1138 isl_vec_free(bset->sample);
1139 bset->sample = NULL;
1142 return sample_eq(bset, bounded ? isl_basic_set_sample_bounded
1143 : isl_basic_set_sample_vec);
1145 return zero_sample(bset);
1147 return interval_sample(bset);
1149 switch (bset->ctx->opt->ilp_solver) {
1151 return pip_sample(bset);
1153 return bounded ? sample_bounded(bset) : gbr_sample(bset);
1155 isl_assert(bset->ctx, 0, );
1157 isl_basic_set_free(bset);
1161 __isl_give isl_vec *isl_basic_set_sample_vec(__isl_take isl_basic_set *bset)
1163 return basic_set_sample(bset, 0);
1166 /* Compute an integer sample in "bset", where the caller guarantees
1167 * that "bset" is bounded.
1169 struct isl_vec *isl_basic_set_sample_bounded(struct isl_basic_set *bset)
1171 return basic_set_sample(bset, 1);
1174 __isl_give isl_basic_set *isl_basic_set_from_vec(__isl_take isl_vec *vec)
1178 struct isl_basic_set *bset = NULL;
1179 struct isl_ctx *ctx;
1185 isl_assert(ctx, vec->size != 0, goto error);
1187 bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
1190 dim = isl_basic_set_n_dim(bset);
1191 for (i = dim - 1; i >= 0; --i) {
1192 k = isl_basic_set_alloc_equality(bset);
1195 isl_seq_clr(bset->eq[k], 1 + dim);
1196 isl_int_neg(bset->eq[k][0], vec->el[1 + i]);
1197 isl_int_set(bset->eq[k][1 + i], vec->el[0]);
1203 isl_basic_set_free(bset);
1208 __isl_give isl_basic_map *isl_basic_map_sample(__isl_take isl_basic_map *bmap)
1210 struct isl_basic_set *bset;
1211 struct isl_vec *sample_vec;
1213 bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
1214 sample_vec = isl_basic_set_sample_vec(bset);
1217 if (sample_vec->size == 0) {
1218 struct isl_basic_map *sample;
1219 sample = isl_basic_map_empty_like(bmap);
1220 isl_vec_free(sample_vec);
1221 isl_basic_map_free(bmap);
1224 bset = isl_basic_set_from_vec(sample_vec);
1225 return isl_basic_map_overlying_set(bset, bmap);
1227 isl_basic_map_free(bmap);
1231 __isl_give isl_basic_map *isl_map_sample(__isl_take isl_map *map)
1234 isl_basic_map *sample = NULL;
1239 for (i = 0; i < map->n; ++i) {
1240 sample = isl_basic_map_sample(isl_basic_map_copy(map->p[i]));
1243 if (!ISL_F_ISSET(sample, ISL_BASIC_MAP_EMPTY))
1245 isl_basic_map_free(sample);
1248 sample = isl_basic_map_empty_like_map(map);
1256 __isl_give isl_basic_set *isl_set_sample(__isl_take isl_set *set)
1258 return (isl_basic_set *) isl_map_sample((isl_map *)set);
1261 __isl_give isl_point *isl_basic_set_sample_point(__isl_take isl_basic_set *bset)
1266 dim = isl_basic_set_get_dim(bset);
1267 bset = isl_basic_set_underlying_set(bset);
1268 vec = isl_basic_set_sample_vec(bset);
1270 return isl_point_alloc(dim, vec);
1273 __isl_give isl_point *isl_set_sample_point(__isl_take isl_set *set)
1281 for (i = 0; i < set->n; ++i) {
1282 pnt = isl_basic_set_sample_point(isl_basic_set_copy(set->p[i]));
1285 if (!isl_point_is_void(pnt))
1287 isl_point_free(pnt);
1290 pnt = isl_point_void(isl_set_get_dim(set));