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"
20 static struct isl_vec *empty_sample(struct isl_basic_set *bset)
24 vec = isl_vec_alloc(bset->ctx, 0);
25 isl_basic_set_free(bset);
29 /* Construct a zero sample of the same dimension as bset.
30 * As a special case, if bset is zero-dimensional, this
31 * function creates a zero-dimensional sample point.
33 static struct isl_vec *zero_sample(struct isl_basic_set *bset)
36 struct isl_vec *sample;
38 dim = isl_basic_set_total_dim(bset);
39 sample = isl_vec_alloc(bset->ctx, 1 + dim);
41 isl_int_set_si(sample->el[0], 1);
42 isl_seq_clr(sample->el + 1, dim);
44 isl_basic_set_free(bset);
48 static struct isl_vec *interval_sample(struct isl_basic_set *bset)
52 struct isl_vec *sample;
54 bset = isl_basic_set_simplify(bset);
57 if (isl_basic_set_fast_is_empty(bset))
58 return empty_sample(bset);
59 if (bset->n_eq == 0 && bset->n_ineq == 0)
60 return zero_sample(bset);
62 sample = isl_vec_alloc(bset->ctx, 2);
63 isl_int_set_si(sample->block.data[0], 1);
66 isl_assert(bset->ctx, bset->n_eq == 1, goto error);
67 isl_assert(bset->ctx, bset->n_ineq == 0, goto error);
68 if (isl_int_is_one(bset->eq[0][1]))
69 isl_int_neg(sample->el[1], bset->eq[0][0]);
71 isl_assert(bset->ctx, isl_int_is_negone(bset->eq[0][1]),
73 isl_int_set(sample->el[1], bset->eq[0][0]);
75 isl_basic_set_free(bset);
80 if (isl_int_is_one(bset->ineq[0][1]))
81 isl_int_neg(sample->block.data[1], bset->ineq[0][0]);
83 isl_int_set(sample->block.data[1], bset->ineq[0][0]);
84 for (i = 1; i < bset->n_ineq; ++i) {
85 isl_seq_inner_product(sample->block.data,
86 bset->ineq[i], 2, &t);
87 if (isl_int_is_neg(t))
91 if (i < bset->n_ineq) {
93 return empty_sample(bset);
96 isl_basic_set_free(bset);
99 isl_basic_set_free(bset);
100 isl_vec_free(sample);
104 static struct isl_mat *independent_bounds(struct isl_basic_set *bset)
107 struct isl_mat *dirs = NULL;
108 struct isl_mat *bounds = NULL;
114 dim = isl_basic_set_n_dim(bset);
115 bounds = isl_mat_alloc(bset->ctx, 1+dim, 1+dim);
119 isl_int_set_si(bounds->row[0][0], 1);
120 isl_seq_clr(bounds->row[0]+1, dim);
123 if (bset->n_ineq == 0)
126 dirs = isl_mat_alloc(bset->ctx, dim, dim);
128 isl_mat_free(bounds);
131 isl_seq_cpy(dirs->row[0], bset->ineq[0]+1, dirs->n_col);
132 isl_seq_cpy(bounds->row[1], bset->ineq[0], bounds->n_col);
133 for (j = 1, n = 1; n < dim && j < bset->n_ineq; ++j) {
136 isl_seq_cpy(dirs->row[n], bset->ineq[j]+1, dirs->n_col);
138 pos = isl_seq_first_non_zero(dirs->row[n], dirs->n_col);
141 for (i = 0; i < n; ++i) {
143 pos_i = isl_seq_first_non_zero(dirs->row[i], dirs->n_col);
148 isl_seq_elim(dirs->row[n], dirs->row[i], pos,
150 pos = isl_seq_first_non_zero(dirs->row[n], dirs->n_col);
158 isl_int *t = dirs->row[n];
159 for (k = n; k > i; --k)
160 dirs->row[k] = dirs->row[k-1];
164 isl_seq_cpy(bounds->row[n], bset->ineq[j], bounds->n_col);
171 static void swap_inequality(struct isl_basic_set *bset, int a, int b)
173 isl_int *t = bset->ineq[a];
174 bset->ineq[a] = bset->ineq[b];
178 /* Skew into positive orthant and project out lineality space.
180 * We perform a unimodular transformation that turns a selected
181 * maximal set of linearly independent bounds into constraints
182 * on the first dimensions that impose that these first dimensions
183 * are non-negative. In particular, the constraint matrix is lower
184 * triangular with positive entries on the diagonal and negative
186 * If "bset" has a lineality space then these constraints (and therefore
187 * all constraints in bset) only involve the first dimensions.
188 * The remaining dimensions then do not appear in any constraints and
189 * we can select any value for them, say zero. We therefore project
190 * out this final dimensions and plug in the value zero later. This
191 * is accomplished by simply dropping the final columns of
192 * the unimodular transformation.
194 static struct isl_basic_set *isl_basic_set_skew_to_positive_orthant(
195 struct isl_basic_set *bset, struct isl_mat **T)
197 struct isl_mat *U = NULL;
198 struct isl_mat *bounds = NULL;
200 unsigned old_dim, new_dim;
206 isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
207 isl_assert(bset->ctx, bset->n_div == 0, goto error);
208 isl_assert(bset->ctx, bset->n_eq == 0, goto error);
210 old_dim = isl_basic_set_n_dim(bset);
211 /* Try to move (multiples of) unit rows up. */
212 for (i = 0, j = 0; i < bset->n_ineq; ++i) {
213 int pos = isl_seq_first_non_zero(bset->ineq[i]+1, old_dim);
216 if (isl_seq_first_non_zero(bset->ineq[i]+1+pos+1,
220 swap_inequality(bset, i, j);
223 bounds = independent_bounds(bset);
226 new_dim = bounds->n_row - 1;
227 bounds = isl_mat_left_hermite(bounds, 1, &U, NULL);
230 U = isl_mat_drop_cols(U, 1 + new_dim, old_dim - new_dim);
231 bset = isl_basic_set_preimage(bset, isl_mat_copy(U));
235 isl_mat_free(bounds);
238 isl_mat_free(bounds);
240 isl_basic_set_free(bset);
244 /* Find a sample integer point, if any, in bset, which is known
245 * to have equalities. If bset contains no integer points, then
246 * return a zero-length vector.
247 * We simply remove the known equalities, compute a sample
248 * in the resulting bset, using the specified recurse function,
249 * and then transform the sample back to the original space.
251 static struct isl_vec *sample_eq(struct isl_basic_set *bset,
252 struct isl_vec *(*recurse)(struct isl_basic_set *))
255 struct isl_vec *sample;
260 bset = isl_basic_set_remove_equalities(bset, &T, NULL);
261 sample = recurse(bset);
262 if (!sample || sample->size == 0)
265 sample = isl_mat_vec_product(T, sample);
269 /* Return a matrix containing the equalities of the tableau
270 * in constraint form. The tableau is assumed to have
271 * an associated bset that has been kept up-to-date.
273 static struct isl_mat *tab_equalities(struct isl_tab *tab)
278 struct isl_basic_set *bset;
283 bset = isl_tab_peek_bset(tab);
284 isl_assert(tab->mat->ctx, bset, return NULL);
286 n_eq = tab->n_var - tab->n_col + tab->n_dead;
287 if (tab->empty || n_eq == 0)
288 return isl_mat_alloc(tab->mat->ctx, 0, tab->n_var);
289 if (n_eq == tab->n_var)
290 return isl_mat_identity(tab->mat->ctx, tab->n_var);
292 eq = isl_mat_alloc(tab->mat->ctx, n_eq, tab->n_var);
295 for (i = 0, j = 0; i < tab->n_con; ++i) {
296 if (tab->con[i].is_row)
298 if (tab->con[i].index >= 0 && tab->con[i].index >= tab->n_dead)
301 isl_seq_cpy(eq->row[j], bset->eq[i] + 1, tab->n_var);
303 isl_seq_cpy(eq->row[j],
304 bset->ineq[i - bset->n_eq] + 1, tab->n_var);
307 isl_assert(bset->ctx, j == n_eq, goto error);
314 /* Compute and return an initial basis for the bounded tableau "tab".
316 * If the tableau is either full-dimensional or zero-dimensional,
317 * the we simply return an identity matrix.
318 * Otherwise, we construct a basis whose first directions correspond
321 static struct isl_mat *initial_basis(struct isl_tab *tab)
327 n_eq = tab->n_var - tab->n_col + tab->n_dead;
328 if (tab->empty || n_eq == 0 || n_eq == tab->n_var)
329 return isl_mat_identity(tab->mat->ctx, 1 + tab->n_var);
331 eq = tab_equalities(tab);
332 eq = isl_mat_left_hermite(eq, 0, NULL, &Q);
337 Q = isl_mat_lin_to_aff(Q);
341 /* Given a tableau representing a set, find and return
342 * an integer point in the set, if there is any.
344 * We perform a depth first search
345 * for an integer point, by scanning all possible values in the range
346 * attained by a basis vector, where an initial basis may have been set
347 * by the calling function. Otherwise an initial basis that exploits
348 * the equalities in the tableau is created.
349 * tab->n_zero is currently ignored and is clobbered by this function.
351 * The tableau is allowed to have unbounded direction, but then
352 * the calling function needs to set an initial basis, with the
353 * unbounded directions last and with tab->n_unbounded set
354 * to the number of unbounded directions.
355 * Furthermore, the calling functions needs to add shifted copies
356 * of all constraints involving unbounded directions to ensure
357 * that any feasible rational value in these directions can be rounded
358 * up to yield a feasible integer value.
359 * In particular, let B define the given basis x' = B x
360 * and let T be the inverse of B, i.e., X = T x'.
361 * Let a x + c >= 0 be a constraint of the set represented by the tableau,
362 * or a T x' + c >= 0 in terms of the given basis. Assume that
363 * the bounded directions have an integer value, then we can safely
364 * round up the values for the unbounded directions if we make sure
365 * that x' not only satisfies the original constraint, but also
366 * the constraint "a T x' + c + s >= 0" with s the sum of all
367 * negative values in the last n_unbounded entries of "a T".
368 * The calling function therefore needs to add the constraint
369 * a x + c + s >= 0. The current function then scans the first
370 * directions for an integer value and once those have been found,
371 * it can compute "T ceil(B x)" to yield an integer point in the set.
372 * Note that during the search, the first rows of B may be changed
373 * by a basis reduction, but the last n_unbounded rows of B remain
374 * unaltered and are also not mixed into the first rows.
376 * The search is implemented iteratively. "level" identifies the current
377 * basis vector. "init" is true if we want the first value at the current
378 * level and false if we want the next value.
380 * The initial basis is the identity matrix. If the range in some direction
381 * contains more than one integer value, we perform basis reduction based
382 * on the value of ctx->opt->gbr
383 * - ISL_GBR_NEVER: never perform basis reduction
384 * - ISL_GBR_ONCE: only perform basis reduction the first
385 * time such a range is encountered
386 * - ISL_GBR_ALWAYS: always perform basis reduction when
387 * such a range is encountered
389 * When ctx->opt->gbr is set to ISL_GBR_ALWAYS, then we allow the basis
390 * reduction computation to return early. That is, as soon as it
391 * finds a reasonable first direction.
393 struct isl_vec *isl_tab_sample(struct isl_tab *tab)
398 struct isl_vec *sample;
401 enum isl_lp_result res;
405 struct isl_tab_undo **snap;
410 return isl_vec_alloc(tab->mat->ctx, 0);
413 tab->basis = initial_basis(tab);
416 isl_assert(tab->mat->ctx, tab->basis->n_row == tab->n_var + 1,
418 isl_assert(tab->mat->ctx, tab->basis->n_col == tab->n_var + 1,
425 if (tab->n_unbounded == tab->n_var) {
426 sample = isl_tab_get_sample_value(tab);
427 sample = isl_mat_vec_product(isl_mat_copy(tab->basis), sample);
428 sample = isl_vec_ceil(sample);
429 sample = isl_mat_vec_inverse_product(isl_mat_copy(tab->basis),
434 if (isl_tab_extend_cons(tab, dim + 1) < 0)
437 min = isl_vec_alloc(ctx, dim);
438 max = isl_vec_alloc(ctx, dim);
439 snap = isl_alloc_array(ctx, struct isl_tab_undo *, dim);
441 if (!min || !max || !snap)
451 res = isl_tab_min(tab, tab->basis->row[1 + level],
452 ctx->one, &min->el[level], NULL, 0);
453 if (res == isl_lp_empty)
455 isl_assert(ctx, res != isl_lp_unbounded, goto error);
456 if (res == isl_lp_error)
458 if (!empty && isl_tab_sample_is_integer(tab))
460 isl_seq_neg(tab->basis->row[1 + level] + 1,
461 tab->basis->row[1 + level] + 1, dim);
462 res = isl_tab_min(tab, tab->basis->row[1 + level],
463 ctx->one, &max->el[level], NULL, 0);
464 isl_seq_neg(tab->basis->row[1 + level] + 1,
465 tab->basis->row[1 + level] + 1, dim);
466 isl_int_neg(max->el[level], max->el[level]);
467 if (res == isl_lp_empty)
469 isl_assert(ctx, res != isl_lp_unbounded, goto error);
470 if (res == isl_lp_error)
472 if (!empty && isl_tab_sample_is_integer(tab))
474 if (!empty && !reduced &&
475 ctx->opt->gbr != ISL_GBR_NEVER &&
476 isl_int_lt(min->el[level], max->el[level])) {
477 unsigned gbr_only_first;
478 if (ctx->opt->gbr == ISL_GBR_ONCE)
479 ctx->opt->gbr = ISL_GBR_NEVER;
481 gbr_only_first = ctx->opt->gbr_only_first;
482 ctx->opt->gbr_only_first =
483 ctx->opt->gbr == ISL_GBR_ALWAYS;
484 tab = isl_tab_compute_reduced_basis(tab);
485 ctx->opt->gbr_only_first = gbr_only_first;
486 if (!tab || !tab->basis)
492 snap[level] = isl_tab_snap(tab);
494 isl_int_add_ui(min->el[level], min->el[level], 1);
496 if (empty || isl_int_gt(min->el[level], max->el[level])) {
500 if (isl_tab_rollback(tab, snap[level]) < 0)
504 isl_int_neg(tab->basis->row[1 + level][0], min->el[level]);
505 tab = isl_tab_add_valid_eq(tab, tab->basis->row[1 + level]);
506 isl_int_set_si(tab->basis->row[1 + level][0], 0);
507 if (level + tab->n_unbounded < dim - 1) {
516 sample = isl_tab_get_sample_value(tab);
519 if (tab->n_unbounded && !isl_int_is_one(sample->el[0])) {
520 sample = isl_mat_vec_product(isl_mat_copy(tab->basis),
522 sample = isl_vec_ceil(sample);
523 sample = isl_mat_vec_inverse_product(
524 isl_mat_copy(tab->basis), sample);
527 sample = isl_vec_alloc(ctx, 0);
542 /* Given a basic set that is known to be bounded, find and return
543 * an integer point in the basic set, if there is any.
545 * After handling some trivial cases, we construct a tableau
546 * and then use isl_tab_sample to find a sample, passing it
547 * the identity matrix as initial basis.
549 static struct isl_vec *sample_bounded(struct isl_basic_set *bset)
553 struct isl_vec *sample;
554 struct isl_tab *tab = NULL;
559 if (isl_basic_set_fast_is_empty(bset))
560 return empty_sample(bset);
562 dim = isl_basic_set_total_dim(bset);
564 return zero_sample(bset);
566 return interval_sample(bset);
568 return sample_eq(bset, sample_bounded);
572 tab = isl_tab_from_basic_set(bset);
573 if (tab && tab->empty) {
575 ISL_F_SET(bset, ISL_BASIC_SET_EMPTY);
576 sample = isl_vec_alloc(bset->ctx, 0);
577 isl_basic_set_free(bset);
581 if (isl_tab_track_bset(tab, isl_basic_set_copy(bset)) < 0)
583 if (!ISL_F_ISSET(bset, ISL_BASIC_SET_NO_IMPLICIT))
584 tab = isl_tab_detect_implicit_equalities(tab);
588 sample = isl_tab_sample(tab);
592 if (sample->size > 0) {
593 isl_vec_free(bset->sample);
594 bset->sample = isl_vec_copy(sample);
597 isl_basic_set_free(bset);
601 isl_basic_set_free(bset);
606 /* Given a basic set "bset" and a value "sample" for the first coordinates
607 * of bset, plug in these values and drop the corresponding coordinates.
609 * We do this by computing the preimage of the transformation
615 * where [1 s] is the sample value and I is the identity matrix of the
616 * appropriate dimension.
618 static struct isl_basic_set *plug_in(struct isl_basic_set *bset,
619 struct isl_vec *sample)
625 if (!bset || !sample)
628 total = isl_basic_set_total_dim(bset);
629 T = isl_mat_alloc(bset->ctx, 1 + total, 1 + total - (sample->size - 1));
633 for (i = 0; i < sample->size; ++i) {
634 isl_int_set(T->row[i][0], sample->el[i]);
635 isl_seq_clr(T->row[i] + 1, T->n_col - 1);
637 for (i = 0; i < T->n_col - 1; ++i) {
638 isl_seq_clr(T->row[sample->size + i], T->n_col);
639 isl_int_set_si(T->row[sample->size + i][1 + i], 1);
641 isl_vec_free(sample);
643 bset = isl_basic_set_preimage(bset, T);
646 isl_basic_set_free(bset);
647 isl_vec_free(sample);
651 /* Given a basic set "bset", return any (possibly non-integer) point
654 static struct isl_vec *rational_sample(struct isl_basic_set *bset)
657 struct isl_vec *sample;
662 tab = isl_tab_from_basic_set(bset);
663 sample = isl_tab_get_sample_value(tab);
666 isl_basic_set_free(bset);
671 /* Given a linear cone "cone" and a rational point "vec",
672 * construct a polyhedron with shifted copies of the constraints in "cone",
673 * i.e., a polyhedron with "cone" as its recession cone, such that each
674 * point x in this polyhedron is such that the unit box positioned at x
675 * lies entirely inside the affine cone 'vec + cone'.
676 * Any rational point in this polyhedron may therefore be rounded up
677 * to yield an integer point that lies inside said affine cone.
679 * Denote the constraints of cone by "<a_i, x> >= 0" and the rational
680 * point "vec" by v/d.
681 * Let b_i = <a_i, v>. Then the affine cone 'vec + cone' is given
682 * by <a_i, x> - b/d >= 0.
683 * The polyhedron <a_i, x> - ceil{b/d} >= 0 is a subset of this affine cone.
684 * We prefer this polyhedron over the actual affine cone because it doesn't
685 * require a scaling of the constraints.
686 * If each of the vertices of the unit cube positioned at x lies inside
687 * this polyhedron, then the whole unit cube at x lies inside the affine cone.
688 * We therefore impose that x' = x + \sum e_i, for any selection of unit
689 * vectors lies inside the polyhedron, i.e.,
691 * <a_i, x'> - ceil{b/d} = <a_i, x> + sum a_i - ceil{b/d} >= 0
693 * The most stringent of these constraints is the one that selects
694 * all negative a_i, so the polyhedron we are looking for has constraints
696 * <a_i, x> + sum_{a_i < 0} a_i - ceil{b/d} >= 0
698 * Note that if cone were known to have only non-negative rays
699 * (which can be accomplished by a unimodular transformation),
700 * then we would only have to check the points x' = x + e_i
701 * and we only have to add the smallest negative a_i (if any)
702 * instead of the sum of all negative a_i.
704 static struct isl_basic_set *shift_cone(struct isl_basic_set *cone,
710 struct isl_basic_set *shift = NULL;
715 isl_assert(cone->ctx, cone->n_eq == 0, goto error);
717 total = isl_basic_set_total_dim(cone);
719 shift = isl_basic_set_alloc_dim(isl_basic_set_get_dim(cone),
722 for (i = 0; i < cone->n_ineq; ++i) {
723 k = isl_basic_set_alloc_inequality(shift);
726 isl_seq_cpy(shift->ineq[k] + 1, cone->ineq[i] + 1, total);
727 isl_seq_inner_product(shift->ineq[k] + 1, vec->el + 1, total,
729 isl_int_cdiv_q(shift->ineq[k][0],
730 shift->ineq[k][0], vec->el[0]);
731 isl_int_neg(shift->ineq[k][0], shift->ineq[k][0]);
732 for (j = 0; j < total; ++j) {
733 if (isl_int_is_nonneg(shift->ineq[k][1 + j]))
735 isl_int_add(shift->ineq[k][0],
736 shift->ineq[k][0], shift->ineq[k][1 + j]);
740 isl_basic_set_free(cone);
743 return isl_basic_set_finalize(shift);
745 isl_basic_set_free(shift);
746 isl_basic_set_free(cone);
751 /* Given a rational point vec in a (transformed) basic set,
752 * such that cone is the recession cone of the original basic set,
753 * "round up" the rational point to an integer point.
755 * We first check if the rational point just happens to be integer.
756 * If not, we transform the cone in the same way as the basic set,
757 * pick a point x in this cone shifted to the rational point such that
758 * the whole unit cube at x is also inside this affine cone.
759 * Then we simply round up the coordinates of x and return the
760 * resulting integer point.
762 static struct isl_vec *round_up_in_cone(struct isl_vec *vec,
763 struct isl_basic_set *cone, struct isl_mat *U)
767 if (!vec || !cone || !U)
770 isl_assert(vec->ctx, vec->size != 0, goto error);
771 if (isl_int_is_one(vec->el[0])) {
773 isl_basic_set_free(cone);
777 total = isl_basic_set_total_dim(cone);
778 cone = isl_basic_set_preimage(cone, U);
779 cone = isl_basic_set_remove_dims(cone, 0, total - (vec->size - 1));
781 cone = shift_cone(cone, vec);
783 vec = rational_sample(cone);
784 vec = isl_vec_ceil(vec);
789 isl_basic_set_free(cone);
793 /* Concatenate two integer vectors, i.e., two vectors with denominator
794 * (stored in element 0) equal to 1.
796 static struct isl_vec *vec_concat(struct isl_vec *vec1, struct isl_vec *vec2)
802 isl_assert(vec1->ctx, vec1->size > 0, goto error);
803 isl_assert(vec2->ctx, vec2->size > 0, goto error);
804 isl_assert(vec1->ctx, isl_int_is_one(vec1->el[0]), goto error);
805 isl_assert(vec2->ctx, isl_int_is_one(vec2->el[0]), goto error);
807 vec = isl_vec_alloc(vec1->ctx, vec1->size + vec2->size - 1);
811 isl_seq_cpy(vec->el, vec1->el, vec1->size);
812 isl_seq_cpy(vec->el + vec1->size, vec2->el + 1, vec2->size - 1);
824 /* Drop all constraints in bset that involve any of the dimensions
825 * first to first+n-1.
827 static struct isl_basic_set *drop_constraints_involving
828 (struct isl_basic_set *bset, unsigned first, unsigned n)
835 bset = isl_basic_set_cow(bset);
837 for (i = bset->n_ineq - 1; i >= 0; --i) {
838 if (isl_seq_first_non_zero(bset->ineq[i] + 1 + first, n) == -1)
840 isl_basic_set_drop_inequality(bset, i);
846 /* Give a basic set "bset" with recession cone "cone", compute and
847 * return an integer point in bset, if any.
849 * If the recession cone is full-dimensional, then we know that
850 * bset contains an infinite number of integer points and it is
851 * fairly easy to pick one of them.
852 * If the recession cone is not full-dimensional, then we first
853 * transform bset such that the bounded directions appear as
854 * the first dimensions of the transformed basic set.
855 * We do this by using a unimodular transformation that transforms
856 * the equalities in the recession cone to equalities on the first
859 * The transformed set is then projected onto its bounded dimensions.
860 * Note that to compute this projection, we can simply drop all constraints
861 * involving any of the unbounded dimensions since these constraints
862 * cannot be combined to produce a constraint on the bounded dimensions.
863 * To see this, assume that there is such a combination of constraints
864 * that produces a constraint on the bounded dimensions. This means
865 * that some combination of the unbounded dimensions has both an upper
866 * bound and a lower bound in terms of the bounded dimensions, but then
867 * this combination would be a bounded direction too and would have been
868 * transformed into a bounded dimensions.
870 * We then compute a sample value in the bounded dimensions.
871 * If no such value can be found, then the original set did not contain
872 * any integer points and we are done.
873 * Otherwise, we plug in the value we found in the bounded dimensions,
874 * project out these bounded dimensions and end up with a set with
875 * a full-dimensional recession cone.
876 * A sample point in this set is computed by "rounding up" any
877 * rational point in the set.
879 * The sample points in the bounded and unbounded dimensions are
880 * then combined into a single sample point and transformed back
881 * to the original space.
883 __isl_give isl_vec *isl_basic_set_sample_with_cone(
884 __isl_take isl_basic_set *bset, __isl_take isl_basic_set *cone)
888 struct isl_mat *M, *U;
889 struct isl_vec *sample;
890 struct isl_vec *cone_sample;
892 struct isl_basic_set *bounded;
898 total = isl_basic_set_total_dim(cone);
899 cone_dim = total - cone->n_eq;
901 M = isl_mat_sub_alloc(bset->ctx, cone->eq, 0, cone->n_eq, 1, total);
902 M = isl_mat_left_hermite(M, 0, &U, NULL);
907 U = isl_mat_lin_to_aff(U);
908 bset = isl_basic_set_preimage(bset, isl_mat_copy(U));
910 bounded = isl_basic_set_copy(bset);
911 bounded = drop_constraints_involving(bounded, total - cone_dim, cone_dim);
912 bounded = isl_basic_set_drop_dims(bounded, total - cone_dim, cone_dim);
913 sample = sample_bounded(bounded);
914 if (!sample || sample->size == 0) {
915 isl_basic_set_free(bset);
916 isl_basic_set_free(cone);
920 bset = plug_in(bset, isl_vec_copy(sample));
921 cone_sample = rational_sample(bset);
922 cone_sample = round_up_in_cone(cone_sample, cone, isl_mat_copy(U));
923 sample = vec_concat(sample, cone_sample);
924 sample = isl_mat_vec_product(U, sample);
927 isl_basic_set_free(cone);
928 isl_basic_set_free(bset);
932 static void vec_sum_of_neg(struct isl_vec *v, isl_int *s)
936 isl_int_set_si(*s, 0);
938 for (i = 0; i < v->size; ++i)
939 if (isl_int_is_neg(v->el[i]))
940 isl_int_add(*s, *s, v->el[i]);
943 /* Given a tableau "tab", a tableau "tab_cone" that corresponds
944 * to the recession cone and the inverse of a new basis U = inv(B),
945 * with the unbounded directions in B last,
946 * add constraints to "tab" that ensure any rational value
947 * in the unbounded directions can be rounded up to an integer value.
949 * The new basis is given by x' = B x, i.e., x = U x'.
950 * For any rational value of the last tab->n_unbounded coordinates
951 * in the update tableau, the value that is obtained by rounding
952 * up this value should be contained in the original tableau.
953 * For any constraint "a x + c >= 0", we therefore need to add
954 * a constraint "a x + c + s >= 0", with s the sum of all negative
955 * entries in the last elements of "a U".
957 * Since we are not interested in the first entries of any of the "a U",
958 * we first drop the columns of U that correpond to bounded directions.
960 static int tab_shift_cone(struct isl_tab *tab,
961 struct isl_tab *tab_cone, struct isl_mat *U)
965 struct isl_basic_set *bset = NULL;
967 if (tab && tab->n_unbounded == 0) {
972 if (!tab || !tab_cone || !U)
974 bset = isl_tab_peek_bset(tab_cone);
975 U = isl_mat_drop_cols(U, 0, tab->n_var - tab->n_unbounded);
976 for (i = 0; i < bset->n_ineq; ++i) {
978 struct isl_vec *row = NULL;
979 if (isl_tab_is_equality(tab_cone, tab_cone->n_eq + i))
981 row = isl_vec_alloc(bset->ctx, tab_cone->n_var);
984 isl_seq_cpy(row->el, bset->ineq[i] + 1, tab_cone->n_var);
985 row = isl_vec_mat_product(row, isl_mat_copy(U));
988 vec_sum_of_neg(row, &v);
990 if (isl_int_is_zero(v))
992 tab = isl_tab_extend(tab, 1);
993 isl_int_add(bset->ineq[i][0], bset->ineq[i][0], v);
994 ok = isl_tab_add_ineq(tab, bset->ineq[i]) >= 0;
995 isl_int_sub(bset->ineq[i][0], bset->ineq[i][0], v);
1009 /* Compute and return an initial basis for the possibly
1010 * unbounded tableau "tab". "tab_cone" is a tableau
1011 * for the corresponding recession cone.
1012 * Additionally, add constraints to "tab" that ensure
1013 * that any rational value for the unbounded directions
1014 * can be rounded up to an integer value.
1016 * If the tableau is bounded, i.e., if the recession cone
1017 * is zero-dimensional, then we just use inital_basis.
1018 * Otherwise, we construct a basis whose first directions
1019 * correspond to equalities, followed by bounded directions,
1020 * i.e., equalities in the recession cone.
1021 * The remaining directions are then unbounded.
1023 int isl_tab_set_initial_basis_with_cone(struct isl_tab *tab,
1024 struct isl_tab *tab_cone)
1027 struct isl_mat *cone_eq;
1028 struct isl_mat *U, *Q;
1030 if (!tab || !tab_cone)
1033 if (tab_cone->n_col == tab_cone->n_dead) {
1034 tab->basis = initial_basis(tab);
1035 return tab->basis ? 0 : -1;
1038 eq = tab_equalities(tab);
1041 tab->n_zero = eq->n_row;
1042 cone_eq = tab_equalities(tab_cone);
1043 eq = isl_mat_concat(eq, cone_eq);
1046 tab->n_unbounded = tab->n_var - (eq->n_row - tab->n_zero);
1047 eq = isl_mat_left_hermite(eq, 0, &U, &Q);
1051 tab->basis = isl_mat_lin_to_aff(Q);
1052 if (tab_shift_cone(tab, tab_cone, U) < 0)
1059 /* Compute and return a sample point in bset using generalized basis
1060 * reduction. We first check if the input set has a non-trivial
1061 * recession cone. If so, we perform some extra preprocessing in
1062 * sample_with_cone. Otherwise, we directly perform generalized basis
1065 static struct isl_vec *gbr_sample(struct isl_basic_set *bset)
1068 struct isl_basic_set *cone;
1070 dim = isl_basic_set_total_dim(bset);
1072 cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));
1074 if (cone->n_eq < dim)
1075 return isl_basic_set_sample_with_cone(bset, cone);
1077 isl_basic_set_free(cone);
1078 return sample_bounded(bset);
1081 static struct isl_vec *pip_sample(struct isl_basic_set *bset)
1084 struct isl_ctx *ctx;
1085 struct isl_vec *sample;
1087 bset = isl_basic_set_skew_to_positive_orthant(bset, &T);
1092 sample = isl_pip_basic_set_sample(bset);
1094 if (sample && sample->size != 0)
1095 sample = isl_mat_vec_product(T, sample);
1102 static struct isl_vec *basic_set_sample(struct isl_basic_set *bset, int bounded)
1104 struct isl_ctx *ctx;
1110 if (isl_basic_set_fast_is_empty(bset))
1111 return empty_sample(bset);
1113 dim = isl_basic_set_n_dim(bset);
1114 isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
1115 isl_assert(ctx, bset->n_div == 0, goto error);
1117 if (bset->sample && bset->sample->size == 1 + dim) {
1118 int contains = isl_basic_set_contains(bset, bset->sample);
1122 struct isl_vec *sample = isl_vec_copy(bset->sample);
1123 isl_basic_set_free(bset);
1127 isl_vec_free(bset->sample);
1128 bset->sample = NULL;
1131 return sample_eq(bset, bounded ? isl_basic_set_sample_bounded
1132 : isl_basic_set_sample_vec);
1134 return zero_sample(bset);
1136 return interval_sample(bset);
1138 switch (bset->ctx->opt->ilp_solver) {
1140 return pip_sample(bset);
1142 return bounded ? sample_bounded(bset) : gbr_sample(bset);
1144 isl_assert(bset->ctx, 0, );
1146 isl_basic_set_free(bset);
1150 __isl_give isl_vec *isl_basic_set_sample_vec(__isl_take isl_basic_set *bset)
1152 return basic_set_sample(bset, 0);
1155 /* Compute an integer sample in "bset", where the caller guarantees
1156 * that "bset" is bounded.
1158 struct isl_vec *isl_basic_set_sample_bounded(struct isl_basic_set *bset)
1160 return basic_set_sample(bset, 1);
1163 __isl_give isl_basic_set *isl_basic_set_from_vec(__isl_take isl_vec *vec)
1167 struct isl_basic_set *bset = NULL;
1168 struct isl_ctx *ctx;
1174 isl_assert(ctx, vec->size != 0, goto error);
1176 bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
1179 dim = isl_basic_set_n_dim(bset);
1180 for (i = dim - 1; i >= 0; --i) {
1181 k = isl_basic_set_alloc_equality(bset);
1184 isl_seq_clr(bset->eq[k], 1 + dim);
1185 isl_int_neg(bset->eq[k][0], vec->el[1 + i]);
1186 isl_int_set(bset->eq[k][1 + i], vec->el[0]);
1192 isl_basic_set_free(bset);
1197 __isl_give isl_basic_map *isl_basic_map_sample(__isl_take isl_basic_map *bmap)
1199 struct isl_basic_set *bset;
1200 struct isl_vec *sample_vec;
1202 bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
1203 sample_vec = isl_basic_set_sample_vec(bset);
1206 if (sample_vec->size == 0) {
1207 struct isl_basic_map *sample;
1208 sample = isl_basic_map_empty_like(bmap);
1209 isl_vec_free(sample_vec);
1210 isl_basic_map_free(bmap);
1213 bset = isl_basic_set_from_vec(sample_vec);
1214 return isl_basic_map_overlying_set(bset, bmap);
1216 isl_basic_map_free(bmap);
1220 __isl_give isl_basic_map *isl_map_sample(__isl_take isl_map *map)
1223 isl_basic_map *sample = NULL;
1228 for (i = 0; i < map->n; ++i) {
1229 sample = isl_basic_map_sample(isl_basic_map_copy(map->p[i]));
1232 if (!ISL_F_ISSET(sample, ISL_BASIC_MAP_EMPTY))
1234 isl_basic_map_free(sample);
1237 sample = isl_basic_map_empty_like_map(map);
1245 __isl_give isl_basic_set *isl_set_sample(__isl_take isl_set *set)
1247 return (isl_basic_set *) isl_map_sample((isl_map *)set);