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_ctx *ctx,
96 struct isl_basic_set *bset)
99 struct isl_mat *dirs = NULL;
105 dim = isl_basic_set_n_dim(bset);
106 if (bset->n_ineq == 0)
107 return isl_mat_alloc(ctx, 0, dim);
109 dirs = isl_mat_alloc(ctx, dim, dim);
112 isl_seq_cpy(dirs->row[0], bset->ineq[0]+1, dirs->n_col);
113 for (j = 1, n = 1; n < dim && j < bset->n_ineq; ++j) {
116 isl_seq_cpy(dirs->row[n], bset->ineq[j]+1, dirs->n_col);
118 pos = isl_seq_first_non_zero(dirs->row[n], dirs->n_col);
121 for (i = 0; i < n; ++i) {
123 pos_i = isl_seq_first_non_zero(dirs->row[i], dirs->n_col);
128 isl_seq_elim(dirs->row[n], dirs->row[i], pos,
130 pos = isl_seq_first_non_zero(dirs->row[n], dirs->n_col);
138 isl_int *t = dirs->row[n];
139 for (k = n; k > i; --k)
140 dirs->row[k] = dirs->row[k-1];
149 /* Find a sample integer point, if any, in bset, which is known
150 * to have equalities. If bset contains no integer points, then
151 * return a zero-length vector.
152 * We simply remove the known equalities, compute a sample
153 * in the resulting bset, using the specified recurse function,
154 * and then transform the sample back to the original space.
156 static struct isl_vec *sample_eq(struct isl_basic_set *bset,
157 struct isl_vec *(*recurse)(struct isl_basic_set *))
160 struct isl_vec *sample;
167 bset = isl_basic_set_remove_equalities(bset, &T, NULL);
168 sample = recurse(bset);
169 if (!sample || sample->size == 0)
170 isl_mat_free(ctx, T);
172 sample = isl_mat_vec_product(ctx, T, sample);
176 /* Given a basic set "bset" and an affine function "f"/"denom",
177 * check if bset is bounded and non-empty and if so, return the minimal
178 * and maximal value attained by the affine function in "min" and "max".
179 * The minimal value is rounded up to the nearest integer, while the
180 * maximal value is rounded down.
181 * The return value indicates whether the set was empty or unbounded.
183 static enum isl_lp_result basic_set_range(struct isl_basic_set *bset,
184 isl_int *f, isl_int denom, isl_int *min, isl_int *max)
188 enum isl_lp_result res;
192 if (isl_basic_set_fast_is_empty(bset))
195 tab = isl_tab_from_basic_set(bset);
196 res = isl_tab_min(bset->ctx, tab, f, denom, min, NULL, 0);
197 if (res != isl_lp_ok) {
198 isl_tab_free(bset->ctx, tab);
201 dim = isl_basic_set_total_dim(bset);
202 isl_seq_neg(f, f, 1 + dim);
203 res = isl_tab_min(bset->ctx, tab, f, denom, max, NULL, 0);
204 isl_seq_neg(f, f, 1 + dim);
205 isl_int_neg(*max, *max);
207 isl_tab_free(bset->ctx, tab);
211 /* Perform a basis reduction on "bset" and return the inverse of
212 * the new basis, i.e., an affine mapping from the new coordinates to the old,
215 static struct isl_basic_set *basic_set_reduced(struct isl_basic_set *bset,
219 unsigned gbr_only_first;
227 gbr_only_first = ctx->gbr_only_first;
228 ctx->gbr_only_first = 1;
229 *T = isl_basic_set_reduced_basis(bset);
230 ctx->gbr_only_first = gbr_only_first;
232 *T = isl_mat_lin_to_aff(bset->ctx, *T);
233 *T = isl_mat_right_inverse(bset->ctx, *T);
235 bset = isl_basic_set_preimage(bset, isl_mat_copy(bset->ctx, *T));
241 isl_mat_free(ctx, *T);
246 static struct isl_vec *sample_bounded(struct isl_basic_set *bset);
248 /* Given a basic set "bset" whose first coordinate ranges between
249 * "min" and "max", step through all values from min to max, until
250 * the slice of bset with the first coordinate fixed to one of these
251 * values contains an integer point. If such a point is found, return it.
252 * If none of the slices contains any integer point, then bset itself
253 * doesn't contain any integer point and an empty sample is returned.
255 static struct isl_vec *sample_scan(struct isl_basic_set *bset,
256 isl_int min, isl_int max)
259 struct isl_basic_set *slice = NULL;
260 struct isl_vec *sample = NULL;
263 total = isl_basic_set_total_dim(bset);
266 for (isl_int_set(tmp, min); isl_int_le(tmp, max);
267 isl_int_add_ui(tmp, tmp, 1)) {
270 slice = isl_basic_set_copy(bset);
271 slice = isl_basic_set_cow(slice);
272 slice = isl_basic_set_extend_constraints(slice, 1, 0);
273 k = isl_basic_set_alloc_equality(slice);
276 isl_int_set(slice->eq[k][0], tmp);
277 isl_int_set_si(slice->eq[k][1], -1);
278 isl_seq_clr(slice->eq[k] + 2, total - 1);
279 slice = isl_basic_set_simplify(slice);
280 sample = sample_bounded(slice);
284 if (sample->size > 0)
286 isl_vec_free(sample);
290 sample = empty_sample(bset);
292 isl_basic_set_free(bset);
296 isl_basic_set_free(bset);
297 isl_basic_set_free(slice);
302 /* Given a basic set that is known to be bounded, find and return
303 * an integer point in the basic set, if there is any.
305 * After handling some trivial cases, we check the range of the
306 * first coordinate. If this coordinate can only attain one integer
307 * value, we are happy. Otherwise, we perform basis reduction and
308 * determine the new range.
310 * Then we step through all possible values in the range in sample_scan.
312 * If any basis reduction was performed, the sample value found, if any,
313 * is transformed back to the original space.
315 static struct isl_vec *sample_bounded(struct isl_basic_set *bset)
319 struct isl_vec *sample;
320 struct isl_vec *obj = NULL;
321 struct isl_mat *T = NULL;
323 enum isl_lp_result res;
328 if (isl_basic_set_fast_is_empty(bset))
329 return empty_sample(bset);
332 dim = isl_basic_set_total_dim(bset);
334 return zero_sample(bset);
336 return interval_sample(bset);
338 return sample_eq(bset, sample_bounded);
342 obj = isl_vec_alloc(bset->ctx, 1 + dim);
345 isl_seq_clr(obj->el, 1+ dim);
346 isl_int_set_si(obj->el[1], 1);
348 res = basic_set_range(bset, obj->el, bset->ctx->one, &min, &max);
349 if (res == isl_lp_error)
351 isl_assert(bset->ctx, res != isl_lp_unbounded, goto error);
352 if (res == isl_lp_empty || isl_int_lt(max, min)) {
353 sample = empty_sample(bset);
357 if (isl_int_ne(min, max)) {
358 bset = basic_set_reduced(bset, &T);
362 res = basic_set_range(bset, obj->el, bset->ctx->one, &min, &max);
363 if (res == isl_lp_error)
365 isl_assert(bset->ctx, res != isl_lp_unbounded, goto error);
366 if (res == isl_lp_empty || isl_int_lt(max, min)) {
367 sample = empty_sample(bset);
372 sample = sample_scan(bset, min, max);
375 if (!sample || sample->size == 0)
376 isl_mat_free(ctx, T);
378 sample = isl_mat_vec_product(ctx, T, sample);
385 isl_mat_free(ctx, T);
386 isl_basic_set_free(bset);
393 /* Given a basic set "bset" and a value "sample" for the first coordinates
394 * of bset, plug in these values and drop the corresponding coordinates.
396 * We do this by computing the preimage of the transformation
402 * where [1 s] is the sample value and I is the identity matrix of the
403 * appropriate dimension.
405 static struct isl_basic_set *plug_in(struct isl_basic_set *bset,
406 struct isl_vec *sample)
412 if (!bset || !sample)
415 total = isl_basic_set_total_dim(bset);
416 T = isl_mat_alloc(bset->ctx, 1 + total, 1 + total - (sample->size - 1));
420 for (i = 0; i < sample->size; ++i) {
421 isl_int_set(T->row[i][0], sample->el[i]);
422 isl_seq_clr(T->row[i] + 1, T->n_col - 1);
424 for (i = 0; i < T->n_col - 1; ++i) {
425 isl_seq_clr(T->row[sample->size + i], T->n_col);
426 isl_int_set_si(T->row[sample->size + i][1 + i], 1);
428 isl_vec_free(sample);
430 bset = isl_basic_set_preimage(bset, T);
433 isl_basic_set_free(bset);
434 isl_vec_free(sample);
438 /* Given a basic set "bset", return any (possibly non-integer) point
441 static struct isl_vec *rational_sample(struct isl_basic_set *bset)
444 struct isl_vec *sample;
449 tab = isl_tab_from_basic_set(bset);
450 sample = isl_tab_get_sample_value(bset->ctx, tab);
451 isl_tab_free(bset->ctx, tab);
453 isl_basic_set_free(bset);
458 /* Given a rational vector, with the denominator in the first element
459 * of the vector, round up all coordinates.
461 struct isl_vec *isl_vec_ceil(struct isl_vec *vec)
465 vec = isl_vec_cow(vec);
469 isl_seq_cdiv_q(vec->el + 1, vec->el + 1, vec->el[0], vec->size - 1);
471 isl_int_set_si(vec->el[0], 1);
476 /* Given a linear cone "cone" and a rational point "vec",
477 * construct a polyhedron with shifted copies of the constraints in "cone",
478 * i.e., a polyhedron with "cone" as its recession cone, such that each
479 * point x in this polyhedron is such that the unit box positioned at x
480 * lies entirely inside the affine cone 'vec + cone'.
481 * Any rational point in this polyhedron may therefore be rounded up
482 * to yield an integer point that lies inside said affine cone.
484 * Denote the constraints of cone by "<a_i, x> >= 0" and the rational
485 * point "vec" by v/d.
486 * Let b_i = <a_i, v>. Then the affine cone 'vec + cone' is given
487 * by <a_i, x> - b/d >= 0.
488 * The polyhedron <a_i, x> - ceil{b/d} >= 0 is a subset of this affine cone.
489 * We prefer this polyhedron over the actual affine cone because it doesn't
490 * require a scaling of the constraints.
491 * If each of the vertices of the unit cube positioned at x lies inside
492 * this polyhedron, then the whole unit cube at x lies inside the affine cone.
493 * We therefore impose that x' = x + \sum e_i, for any selection of unit
494 * vectors lies inside the polyhedron, i.e.,
496 * <a_i, x'> - ceil{b/d} = <a_i, x> + sum a_i - ceil{b/d} >= 0
498 * The most stringent of these constraints is the one that selects
499 * all negative a_i, so the polyhedron we are looking for has constraints
501 * <a_i, x> + sum_{a_i < 0} a_i - ceil{b/d} >= 0
503 * Note that if cone were known to have only non-negative rays
504 * (which can be accomplished by a unimodular transformation),
505 * then we would only have to check the points x' = x + e_i
506 * and we only have to add the smallest negative a_i (if any)
507 * instead of the sum of all negative a_i.
509 static struct isl_basic_set *shift_cone(struct isl_basic_set *cone,
515 struct isl_basic_set *shift = NULL;
520 isl_assert(cone->ctx, cone->n_eq == 0, goto error);
522 total = isl_basic_set_total_dim(cone);
524 shift = isl_basic_set_alloc_dim(isl_basic_set_get_dim(cone),
527 for (i = 0; i < cone->n_ineq; ++i) {
528 k = isl_basic_set_alloc_inequality(shift);
531 isl_seq_cpy(shift->ineq[k] + 1, cone->ineq[i] + 1, total);
532 isl_seq_inner_product(shift->ineq[k] + 1, vec->el + 1, total,
534 isl_int_cdiv_q(shift->ineq[k][0],
535 shift->ineq[k][0], vec->el[0]);
536 isl_int_neg(shift->ineq[k][0], shift->ineq[k][0]);
537 for (j = 0; j < total; ++j) {
538 if (isl_int_is_nonneg(shift->ineq[k][1 + j]))
540 isl_int_add(shift->ineq[k][0],
541 shift->ineq[k][0], shift->ineq[k][1 + j]);
545 isl_basic_set_free(cone);
548 return isl_basic_set_finalize(shift);
550 isl_basic_set_free(shift);
551 isl_basic_set_free(cone);
556 /* Given a rational point vec in a (transformed) basic set,
557 * such that cone is the recession cone of the original basic set,
558 * "round up" the rational point to an integer point.
560 * We first check if the rational point just happens to be integer.
561 * If not, we transform the cone in the same way as the basic set,
562 * pick a point x in this cone shifted to the rational point such that
563 * the whole unit cube at x is also inside this affine cone.
564 * Then we simply round up the coordinates of x and return the
565 * resulting integer point.
567 static struct isl_vec *round_up_in_cone(struct isl_vec *vec,
568 struct isl_basic_set *cone, struct isl_mat *U)
572 if (!vec || !cone || !U)
575 isl_assert(vec->ctx, vec->size != 0, goto error);
576 if (isl_int_is_one(vec->el[0])) {
577 isl_mat_free(vec->ctx, U);
578 isl_basic_set_free(cone);
582 total = isl_basic_set_total_dim(cone);
583 cone = isl_basic_set_preimage(cone, U);
584 cone = isl_basic_set_remove_dims(cone, 0, total - (vec->size - 1));
586 cone = shift_cone(cone, vec);
588 vec = rational_sample(cone);
589 vec = isl_vec_ceil(vec);
592 isl_mat_free(vec ? vec->ctx : cone ? cone->ctx : NULL, U);
594 isl_basic_set_free(cone);
598 /* Concatenate two integer vectors, i.e., two vectors with denominator
599 * (stored in element 0) equal to 1.
601 static struct isl_vec *vec_concat(struct isl_vec *vec1, struct isl_vec *vec2)
607 isl_assert(vec1->ctx, vec1->size > 0, goto error);
608 isl_assert(vec2->ctx, vec2->size > 0, goto error);
609 isl_assert(vec1->ctx, isl_int_is_one(vec1->el[0]), goto error);
610 isl_assert(vec2->ctx, isl_int_is_one(vec2->el[0]), goto error);
612 vec = isl_vec_alloc(vec1->ctx, vec1->size + vec2->size - 1);
616 isl_seq_cpy(vec->el, vec1->el, vec1->size);
617 isl_seq_cpy(vec->el + vec1->size, vec2->el + 1, vec2->size - 1);
629 /* Drop all constraints in bset that involve any of the dimensions
630 * first to first+n-1.
632 static struct isl_basic_set *drop_constraints_involving
633 (struct isl_basic_set *bset, unsigned first, unsigned n)
640 bset = isl_basic_set_cow(bset);
642 for (i = bset->n_ineq - 1; i >= 0; --i) {
643 if (isl_seq_first_non_zero(bset->ineq[i] + 1 + first, n) == -1)
645 isl_basic_set_drop_inequality(bset, i);
651 /* Give a basic set "bset" with recession cone "cone", compute and
652 * return an integer point in bset, if any.
654 * If the recession cone is full-dimensional, then we know that
655 * bset contains an infinite number of integer points and it is
656 * fairly easy to pick one of them.
657 * If the recession cone is not full-dimensional, then we first
658 * transform bset such that the bounded directions appear as
659 * the first dimensions of the transformed basic set.
660 * We do this by using a unimodular transformation that transforms
661 * the equalities in the recession cone to equalities on the first
664 * The transformed set is then projected onto its bounded dimensions.
665 * Note that to compute this projection, we can simply drop all constraints
666 * involving any of the unbounded dimensions since these constraints
667 * cannot be combined to produce a constraint on the bounded dimensions.
668 * To see this, assume that there is such a combination of constraints
669 * that produces a constraint on the bounded dimensions. This means
670 * that some combination of the unbounded dimensions has both an upper
671 * bound and a lower bound in terms of the bounded dimensions, but then
672 * this combination would be a bounded direction too and would have been
673 * transformed into a bounded dimensions.
675 * We then compute a sample value in the bounded dimensions.
676 * If no such value can be found, then the original set did not contain
677 * any integer points and we are done.
678 * Otherwise, we plug in the value we found in the bounded dimensions,
679 * project out these bounded dimensions and end up with a set with
680 * a full-dimensional recession cone.
681 * A sample point in this set is computed by "rounding up" any
682 * rational point in the set.
684 * The sample points in the bounded and unbounded dimensions are
685 * then combined into a single sample point and transformed back
686 * to the original space.
688 static struct isl_vec *sample_with_cone(struct isl_basic_set *bset,
689 struct isl_basic_set *cone)
693 struct isl_mat *M, *U;
694 struct isl_vec *sample;
695 struct isl_vec *cone_sample;
697 struct isl_basic_set *bounded;
703 total = isl_basic_set_total_dim(cone);
704 cone_dim = total - cone->n_eq;
706 M = isl_mat_sub_alloc(bset->ctx, cone->eq, 0, cone->n_eq, 1, total);
707 M = isl_mat_left_hermite(bset->ctx, M, 0, &U, NULL);
710 isl_mat_free(bset->ctx, M);
712 U = isl_mat_lin_to_aff(bset->ctx, U);
713 bset = isl_basic_set_preimage(bset, isl_mat_copy(bset->ctx, U));
715 bounded = isl_basic_set_copy(bset);
716 bounded = drop_constraints_involving(bounded, total - cone_dim, cone_dim);
717 bounded = isl_basic_set_drop_dims(bounded, total - cone_dim, cone_dim);
718 sample = sample_bounded(bounded);
719 if (!sample || sample->size == 0) {
720 isl_basic_set_free(bset);
721 isl_basic_set_free(cone);
722 isl_mat_free(ctx, U);
725 bset = plug_in(bset, isl_vec_copy(sample));
726 cone_sample = rational_sample(bset);
727 cone_sample = round_up_in_cone(cone_sample, cone, isl_mat_copy(ctx, U));
728 sample = vec_concat(sample, cone_sample);
729 sample = isl_mat_vec_product(ctx, U, sample);
732 isl_basic_set_free(cone);
733 isl_basic_set_free(bset);
737 /* Compute and return a sample point in bset using generalized basis
738 * reduction. We first check if the input set has a non-trivial
739 * recession cone. If so, we perform some extra preprocessing in
740 * sample_with_cone. Otherwise, we directly perform generalized basis
743 static struct isl_vec *gbr_sample_no_lineality(struct isl_basic_set *bset)
746 struct isl_basic_set *cone;
748 dim = isl_basic_set_total_dim(bset);
750 cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));
752 if (cone->n_eq < dim)
753 return sample_with_cone(bset, cone);
755 isl_basic_set_free(cone);
756 return sample_bounded(bset);
759 static struct isl_vec *sample_no_lineality(struct isl_basic_set *bset)
763 if (isl_basic_set_fast_is_empty(bset))
764 return empty_sample(bset);
766 return sample_eq(bset, sample_no_lineality);
767 dim = isl_basic_set_total_dim(bset);
769 return zero_sample(bset);
771 return interval_sample(bset);
773 switch (bset->ctx->ilp_solver) {
775 return isl_pip_basic_set_sample(bset);
777 return gbr_sample_no_lineality(bset);
779 isl_assert(bset->ctx, 0, );
780 isl_basic_set_free(bset);
784 /* Compute an integer point in "bset" with a lineality space that
785 * is orthogonal to the constraints in "bounds".
787 * We first perform a unimodular transformation on bset that
788 * make the constraints in bounds (and therefore all constraints in bset)
789 * only involve the first dimensions. The remaining dimensions
790 * then do not appear in any constraints and we can select any value
791 * for them, say zero. We therefore project out this final dimensions
792 * and plug in the value zero later. This is accomplished by simply
793 * dropping the final columns of the unimodular transformation.
795 static struct isl_vec *sample_lineality(struct isl_basic_set *bset,
796 struct isl_mat *bounds)
798 struct isl_mat *U = NULL;
799 unsigned old_dim, new_dim;
800 struct isl_vec *sample;
803 if (!bset || !bounds)
807 old_dim = isl_basic_set_n_dim(bset);
808 new_dim = bounds->n_row;
809 bounds = isl_mat_left_hermite(ctx, bounds, 0, &U, NULL);
812 U = isl_mat_lin_to_aff(ctx, U);
813 U = isl_mat_drop_cols(ctx, U, 1 + new_dim, old_dim - new_dim);
814 bset = isl_basic_set_preimage(bset, isl_mat_copy(ctx, U));
817 isl_mat_free(ctx, bounds);
819 sample = sample_no_lineality(bset);
820 if (sample && sample->size != 0)
821 sample = isl_mat_vec_product(ctx, U, sample);
823 isl_mat_free(ctx, U);
826 isl_mat_free(ctx, bounds);
827 isl_mat_free(ctx, U);
828 isl_basic_set_free(bset);
832 struct isl_vec *isl_basic_set_sample(struct isl_basic_set *bset)
835 struct isl_mat *bounds;
841 if (isl_basic_set_fast_is_empty(bset))
842 return empty_sample(bset);
844 dim = isl_basic_set_n_dim(bset);
845 isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
846 isl_assert(ctx, bset->n_div == 0, goto error);
849 return sample_eq(bset, isl_basic_set_sample);
851 return zero_sample(bset);
853 return interval_sample(bset);
854 bounds = independent_bounds(ctx, bset);
858 if (bounds->n_row == 0) {
859 isl_mat_free(ctx, bounds);
860 return zero_sample(bset);
862 if (bounds->n_row < dim)
863 return sample_lineality(bset, bounds);
865 isl_mat_free(ctx, bounds);
866 return sample_no_lineality(bset);
868 isl_basic_set_free(bset);