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->opt->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->opt->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
&&
466 ctx
->opt
->gbr
!= ISL_GBR_NEVER
&&
467 isl_int_lt(min
->el
[level
], max
->el
[level
])) {
468 unsigned gbr_only_first
;
469 if (ctx
->opt
->gbr
== ISL_GBR_ONCE
)
470 ctx
->opt
->gbr
= ISL_GBR_NEVER
;
472 gbr_only_first
= ctx
->opt
->gbr_only_first
;
473 ctx
->opt
->gbr_only_first
=
474 ctx
->opt
->gbr
== ISL_GBR_ALWAYS
;
475 tab
= isl_tab_compute_reduced_basis(tab
);
476 ctx
->opt
->gbr_only_first
= gbr_only_first
;
477 if (!tab
|| !tab
->basis
)
483 snap
[level
] = isl_tab_snap(tab
);
485 isl_int_add_ui(min
->el
[level
], min
->el
[level
], 1);
487 if (empty
|| isl_int_gt(min
->el
[level
], max
->el
[level
])) {
491 if (isl_tab_rollback(tab
, snap
[level
]) < 0)
495 isl_int_neg(tab
->basis
->row
[1 + level
][0], min
->el
[level
]);
496 tab
= isl_tab_add_valid_eq(tab
, tab
->basis
->row
[1 + level
]);
497 isl_int_set_si(tab
->basis
->row
[1 + level
][0], 0);
498 if (level
+ tab
->n_unbounded
< dim
- 1) {
507 sample
= isl_tab_get_sample_value(tab
);
510 if (tab
->n_unbounded
&& !isl_int_is_one(sample
->el
[0])) {
511 sample
= isl_mat_vec_product(isl_mat_copy(tab
->basis
),
513 sample
= isl_vec_ceil(sample
);
514 sample
= isl_mat_vec_inverse_product(
515 isl_mat_copy(tab
->basis
), sample
);
518 sample
= isl_vec_alloc(ctx
, 0);
533 /* Given a basic set that is known to be bounded, find and return
534 * an integer point in the basic set, if there is any.
536 * After handling some trivial cases, we construct a tableau
537 * and then use isl_tab_sample to find a sample, passing it
538 * the identity matrix as initial basis.
540 static struct isl_vec
*sample_bounded(struct isl_basic_set
*bset
)
544 struct isl_vec
*sample
;
545 struct isl_tab
*tab
= NULL
;
550 if (isl_basic_set_fast_is_empty(bset
))
551 return empty_sample(bset
);
553 dim
= isl_basic_set_total_dim(bset
);
555 return zero_sample(bset
);
557 return interval_sample(bset
);
559 return sample_eq(bset
, sample_bounded
);
563 tab
= isl_tab_from_basic_set(bset
);
564 if (!ISL_F_ISSET(bset
, ISL_BASIC_SET_NO_IMPLICIT
))
565 tab
= isl_tab_detect_implicit_equalities(tab
);
569 tab
->bset
= isl_basic_set_copy(bset
);
571 sample
= isl_tab_sample(tab
);
575 if (sample
->size
> 0) {
576 isl_vec_free(bset
->sample
);
577 bset
->sample
= isl_vec_copy(sample
);
580 isl_basic_set_free(bset
);
584 isl_basic_set_free(bset
);
589 /* Given a basic set "bset" and a value "sample" for the first coordinates
590 * of bset, plug in these values and drop the corresponding coordinates.
592 * We do this by computing the preimage of the transformation
598 * where [1 s] is the sample value and I is the identity matrix of the
599 * appropriate dimension.
601 static struct isl_basic_set
*plug_in(struct isl_basic_set
*bset
,
602 struct isl_vec
*sample
)
608 if (!bset
|| !sample
)
611 total
= isl_basic_set_total_dim(bset
);
612 T
= isl_mat_alloc(bset
->ctx
, 1 + total
, 1 + total
- (sample
->size
- 1));
616 for (i
= 0; i
< sample
->size
; ++i
) {
617 isl_int_set(T
->row
[i
][0], sample
->el
[i
]);
618 isl_seq_clr(T
->row
[i
] + 1, T
->n_col
- 1);
620 for (i
= 0; i
< T
->n_col
- 1; ++i
) {
621 isl_seq_clr(T
->row
[sample
->size
+ i
], T
->n_col
);
622 isl_int_set_si(T
->row
[sample
->size
+ i
][1 + i
], 1);
624 isl_vec_free(sample
);
626 bset
= isl_basic_set_preimage(bset
, T
);
629 isl_basic_set_free(bset
);
630 isl_vec_free(sample
);
634 /* Given a basic set "bset", return any (possibly non-integer) point
637 static struct isl_vec
*rational_sample(struct isl_basic_set
*bset
)
640 struct isl_vec
*sample
;
645 tab
= isl_tab_from_basic_set(bset
);
646 sample
= isl_tab_get_sample_value(tab
);
649 isl_basic_set_free(bset
);
654 /* Given a linear cone "cone" and a rational point "vec",
655 * construct a polyhedron with shifted copies of the constraints in "cone",
656 * i.e., a polyhedron with "cone" as its recession cone, such that each
657 * point x in this polyhedron is such that the unit box positioned at x
658 * lies entirely inside the affine cone 'vec + cone'.
659 * Any rational point in this polyhedron may therefore be rounded up
660 * to yield an integer point that lies inside said affine cone.
662 * Denote the constraints of cone by "<a_i, x> >= 0" and the rational
663 * point "vec" by v/d.
664 * Let b_i = <a_i, v>. Then the affine cone 'vec + cone' is given
665 * by <a_i, x> - b/d >= 0.
666 * The polyhedron <a_i, x> - ceil{b/d} >= 0 is a subset of this affine cone.
667 * We prefer this polyhedron over the actual affine cone because it doesn't
668 * require a scaling of the constraints.
669 * If each of the vertices of the unit cube positioned at x lies inside
670 * this polyhedron, then the whole unit cube at x lies inside the affine cone.
671 * We therefore impose that x' = x + \sum e_i, for any selection of unit
672 * vectors lies inside the polyhedron, i.e.,
674 * <a_i, x'> - ceil{b/d} = <a_i, x> + sum a_i - ceil{b/d} >= 0
676 * The most stringent of these constraints is the one that selects
677 * all negative a_i, so the polyhedron we are looking for has constraints
679 * <a_i, x> + sum_{a_i < 0} a_i - ceil{b/d} >= 0
681 * Note that if cone were known to have only non-negative rays
682 * (which can be accomplished by a unimodular transformation),
683 * then we would only have to check the points x' = x + e_i
684 * and we only have to add the smallest negative a_i (if any)
685 * instead of the sum of all negative a_i.
687 static struct isl_basic_set
*shift_cone(struct isl_basic_set
*cone
,
693 struct isl_basic_set
*shift
= NULL
;
698 isl_assert(cone
->ctx
, cone
->n_eq
== 0, goto error
);
700 total
= isl_basic_set_total_dim(cone
);
702 shift
= isl_basic_set_alloc_dim(isl_basic_set_get_dim(cone
),
705 for (i
= 0; i
< cone
->n_ineq
; ++i
) {
706 k
= isl_basic_set_alloc_inequality(shift
);
709 isl_seq_cpy(shift
->ineq
[k
] + 1, cone
->ineq
[i
] + 1, total
);
710 isl_seq_inner_product(shift
->ineq
[k
] + 1, vec
->el
+ 1, total
,
712 isl_int_cdiv_q(shift
->ineq
[k
][0],
713 shift
->ineq
[k
][0], vec
->el
[0]);
714 isl_int_neg(shift
->ineq
[k
][0], shift
->ineq
[k
][0]);
715 for (j
= 0; j
< total
; ++j
) {
716 if (isl_int_is_nonneg(shift
->ineq
[k
][1 + j
]))
718 isl_int_add(shift
->ineq
[k
][0],
719 shift
->ineq
[k
][0], shift
->ineq
[k
][1 + j
]);
723 isl_basic_set_free(cone
);
726 return isl_basic_set_finalize(shift
);
728 isl_basic_set_free(shift
);
729 isl_basic_set_free(cone
);
734 /* Given a rational point vec in a (transformed) basic set,
735 * such that cone is the recession cone of the original basic set,
736 * "round up" the rational point to an integer point.
738 * We first check if the rational point just happens to be integer.
739 * If not, we transform the cone in the same way as the basic set,
740 * pick a point x in this cone shifted to the rational point such that
741 * the whole unit cube at x is also inside this affine cone.
742 * Then we simply round up the coordinates of x and return the
743 * resulting integer point.
745 static struct isl_vec
*round_up_in_cone(struct isl_vec
*vec
,
746 struct isl_basic_set
*cone
, struct isl_mat
*U
)
750 if (!vec
|| !cone
|| !U
)
753 isl_assert(vec
->ctx
, vec
->size
!= 0, goto error
);
754 if (isl_int_is_one(vec
->el
[0])) {
756 isl_basic_set_free(cone
);
760 total
= isl_basic_set_total_dim(cone
);
761 cone
= isl_basic_set_preimage(cone
, U
);
762 cone
= isl_basic_set_remove_dims(cone
, 0, total
- (vec
->size
- 1));
764 cone
= shift_cone(cone
, vec
);
766 vec
= rational_sample(cone
);
767 vec
= isl_vec_ceil(vec
);
772 isl_basic_set_free(cone
);
776 /* Concatenate two integer vectors, i.e., two vectors with denominator
777 * (stored in element 0) equal to 1.
779 static struct isl_vec
*vec_concat(struct isl_vec
*vec1
, struct isl_vec
*vec2
)
785 isl_assert(vec1
->ctx
, vec1
->size
> 0, goto error
);
786 isl_assert(vec2
->ctx
, vec2
->size
> 0, goto error
);
787 isl_assert(vec1
->ctx
, isl_int_is_one(vec1
->el
[0]), goto error
);
788 isl_assert(vec2
->ctx
, isl_int_is_one(vec2
->el
[0]), goto error
);
790 vec
= isl_vec_alloc(vec1
->ctx
, vec1
->size
+ vec2
->size
- 1);
794 isl_seq_cpy(vec
->el
, vec1
->el
, vec1
->size
);
795 isl_seq_cpy(vec
->el
+ vec1
->size
, vec2
->el
+ 1, vec2
->size
- 1);
807 /* Drop all constraints in bset that involve any of the dimensions
808 * first to first+n-1.
810 static struct isl_basic_set
*drop_constraints_involving
811 (struct isl_basic_set
*bset
, unsigned first
, unsigned n
)
818 bset
= isl_basic_set_cow(bset
);
820 for (i
= bset
->n_ineq
- 1; i
>= 0; --i
) {
821 if (isl_seq_first_non_zero(bset
->ineq
[i
] + 1 + first
, n
) == -1)
823 isl_basic_set_drop_inequality(bset
, i
);
829 /* Give a basic set "bset" with recession cone "cone", compute and
830 * return an integer point in bset, if any.
832 * If the recession cone is full-dimensional, then we know that
833 * bset contains an infinite number of integer points and it is
834 * fairly easy to pick one of them.
835 * If the recession cone is not full-dimensional, then we first
836 * transform bset such that the bounded directions appear as
837 * the first dimensions of the transformed basic set.
838 * We do this by using a unimodular transformation that transforms
839 * the equalities in the recession cone to equalities on the first
842 * The transformed set is then projected onto its bounded dimensions.
843 * Note that to compute this projection, we can simply drop all constraints
844 * involving any of the unbounded dimensions since these constraints
845 * cannot be combined to produce a constraint on the bounded dimensions.
846 * To see this, assume that there is such a combination of constraints
847 * that produces a constraint on the bounded dimensions. This means
848 * that some combination of the unbounded dimensions has both an upper
849 * bound and a lower bound in terms of the bounded dimensions, but then
850 * this combination would be a bounded direction too and would have been
851 * transformed into a bounded dimensions.
853 * We then compute a sample value in the bounded dimensions.
854 * If no such value can be found, then the original set did not contain
855 * any integer points and we are done.
856 * Otherwise, we plug in the value we found in the bounded dimensions,
857 * project out these bounded dimensions and end up with a set with
858 * a full-dimensional recession cone.
859 * A sample point in this set is computed by "rounding up" any
860 * rational point in the set.
862 * The sample points in the bounded and unbounded dimensions are
863 * then combined into a single sample point and transformed back
864 * to the original space.
866 __isl_give isl_vec
*isl_basic_set_sample_with_cone(
867 __isl_take isl_basic_set
*bset
, __isl_take isl_basic_set
*cone
)
871 struct isl_mat
*M
, *U
;
872 struct isl_vec
*sample
;
873 struct isl_vec
*cone_sample
;
875 struct isl_basic_set
*bounded
;
881 total
= isl_basic_set_total_dim(cone
);
882 cone_dim
= total
- cone
->n_eq
;
884 M
= isl_mat_sub_alloc(bset
->ctx
, cone
->eq
, 0, cone
->n_eq
, 1, total
);
885 M
= isl_mat_left_hermite(M
, 0, &U
, NULL
);
890 U
= isl_mat_lin_to_aff(U
);
891 bset
= isl_basic_set_preimage(bset
, isl_mat_copy(U
));
893 bounded
= isl_basic_set_copy(bset
);
894 bounded
= drop_constraints_involving(bounded
, total
- cone_dim
, cone_dim
);
895 bounded
= isl_basic_set_drop_dims(bounded
, total
- cone_dim
, cone_dim
);
896 sample
= sample_bounded(bounded
);
897 if (!sample
|| sample
->size
== 0) {
898 isl_basic_set_free(bset
);
899 isl_basic_set_free(cone
);
903 bset
= plug_in(bset
, isl_vec_copy(sample
));
904 cone_sample
= rational_sample(bset
);
905 cone_sample
= round_up_in_cone(cone_sample
, cone
, isl_mat_copy(U
));
906 sample
= vec_concat(sample
, cone_sample
);
907 sample
= isl_mat_vec_product(U
, sample
);
910 isl_basic_set_free(cone
);
911 isl_basic_set_free(bset
);
915 static void vec_sum_of_neg(struct isl_vec
*v
, isl_int
*s
)
919 isl_int_set_si(*s
, 0);
921 for (i
= 0; i
< v
->size
; ++i
)
922 if (isl_int_is_neg(v
->el
[i
]))
923 isl_int_add(*s
, *s
, v
->el
[i
]);
926 /* Given a tableau "tab", a tableau "tab_cone" that corresponds
927 * to the recession cone and the inverse of a new basis U = inv(B),
928 * with the unbounded directions in B last,
929 * add constraints to "tab" that ensure any rational value
930 * in the unbounded directions can be rounded up to an integer value.
932 * The new basis is given by x' = B x, i.e., x = U x'.
933 * For any rational value of the last tab->n_unbounded coordinates
934 * in the update tableau, the value that is obtained by rounding
935 * up this value should be contained in the original tableau.
936 * For any constraint "a x + c >= 0", we therefore need to add
937 * a constraint "a x + c + s >= 0", with s the sum of all negative
938 * entries in the last elements of "a U".
940 * Since we are not interested in the first entries of any of the "a U",
941 * we first drop the columns of U that correpond to bounded directions.
943 static int tab_shift_cone(struct isl_tab
*tab
,
944 struct isl_tab
*tab_cone
, struct isl_mat
*U
)
948 struct isl_basic_set
*bset
= NULL
;
950 if (tab
&& tab
->n_unbounded
== 0) {
955 if (!tab
|| !tab_cone
|| !U
)
957 bset
= tab_cone
->bset
;
958 U
= isl_mat_drop_cols(U
, 0, tab
->n_var
- tab
->n_unbounded
);
959 for (i
= 0; i
< bset
->n_ineq
; ++i
) {
961 struct isl_vec
*row
= NULL
;
962 if (isl_tab_is_equality(tab_cone
, tab_cone
->n_eq
+ i
))
964 row
= isl_vec_alloc(bset
->ctx
, tab_cone
->n_var
);
967 isl_seq_cpy(row
->el
, bset
->ineq
[i
] + 1, tab_cone
->n_var
);
968 row
= isl_vec_mat_product(row
, isl_mat_copy(U
));
971 vec_sum_of_neg(row
, &v
);
973 if (isl_int_is_zero(v
))
975 tab
= isl_tab_extend(tab
, 1);
976 isl_int_add(bset
->ineq
[i
][0], bset
->ineq
[i
][0], v
);
977 ok
= isl_tab_add_ineq(tab
, bset
->ineq
[i
]) >= 0;
978 isl_int_sub(bset
->ineq
[i
][0], bset
->ineq
[i
][0], v
);
992 /* Compute and return an initial basis for the possibly
993 * unbounded tableau "tab". "tab_cone" is a tableau
994 * for the corresponding recession cone.
995 * Additionally, add constraints to "tab" that ensure
996 * that any rational value for the unbounded directions
997 * can be rounded up to an integer value.
999 * If the tableau is bounded, i.e., if the recession cone
1000 * is zero-dimensional, then we just use inital_basis.
1001 * Otherwise, we construct a basis whose first directions
1002 * correspond to equalities, followed by bounded directions,
1003 * i.e., equalities in the recession cone.
1004 * The remaining directions are then unbounded.
1006 int isl_tab_set_initial_basis_with_cone(struct isl_tab
*tab
,
1007 struct isl_tab
*tab_cone
)
1010 struct isl_mat
*cone_eq
;
1011 struct isl_mat
*U
, *Q
;
1013 if (!tab
|| !tab_cone
)
1016 if (tab_cone
->n_col
== tab_cone
->n_dead
) {
1017 tab
->basis
= initial_basis(tab
);
1018 return tab
->basis
? 0 : -1;
1021 eq
= tab_equalities(tab
);
1024 tab
->n_zero
= eq
->n_row
;
1025 cone_eq
= tab_equalities(tab_cone
);
1026 eq
= isl_mat_concat(eq
, cone_eq
);
1029 tab
->n_unbounded
= tab
->n_var
- (eq
->n_row
- tab
->n_zero
);
1030 eq
= isl_mat_left_hermite(eq
, 0, &U
, &Q
);
1034 tab
->basis
= isl_mat_lin_to_aff(Q
);
1035 if (tab_shift_cone(tab
, tab_cone
, U
) < 0)
1042 /* Compute and return a sample point in bset using generalized basis
1043 * reduction. We first check if the input set has a non-trivial
1044 * recession cone. If so, we perform some extra preprocessing in
1045 * sample_with_cone. Otherwise, we directly perform generalized basis
1048 static struct isl_vec
*gbr_sample(struct isl_basic_set
*bset
)
1051 struct isl_basic_set
*cone
;
1053 dim
= isl_basic_set_total_dim(bset
);
1055 cone
= isl_basic_set_recession_cone(isl_basic_set_copy(bset
));
1057 if (cone
->n_eq
< dim
)
1058 return isl_basic_set_sample_with_cone(bset
, cone
);
1060 isl_basic_set_free(cone
);
1061 return sample_bounded(bset
);
1064 static struct isl_vec
*pip_sample(struct isl_basic_set
*bset
)
1067 struct isl_ctx
*ctx
;
1068 struct isl_vec
*sample
;
1070 bset
= isl_basic_set_skew_to_positive_orthant(bset
, &T
);
1075 sample
= isl_pip_basic_set_sample(bset
);
1077 if (sample
&& sample
->size
!= 0)
1078 sample
= isl_mat_vec_product(T
, sample
);
1085 static struct isl_vec
*basic_set_sample(struct isl_basic_set
*bset
, int bounded
)
1087 struct isl_ctx
*ctx
;
1093 if (isl_basic_set_fast_is_empty(bset
))
1094 return empty_sample(bset
);
1096 dim
= isl_basic_set_n_dim(bset
);
1097 isl_assert(ctx
, isl_basic_set_n_param(bset
) == 0, goto error
);
1098 isl_assert(ctx
, bset
->n_div
== 0, goto error
);
1100 if (bset
->sample
&& bset
->sample
->size
== 1 + dim
) {
1101 int contains
= isl_basic_set_contains(bset
, bset
->sample
);
1105 struct isl_vec
*sample
= isl_vec_copy(bset
->sample
);
1106 isl_basic_set_free(bset
);
1110 isl_vec_free(bset
->sample
);
1111 bset
->sample
= NULL
;
1114 return sample_eq(bset
, bounded
? isl_basic_set_sample_bounded
1115 : isl_basic_set_sample_vec
);
1117 return zero_sample(bset
);
1119 return interval_sample(bset
);
1121 switch (bset
->ctx
->opt
->ilp_solver
) {
1123 return pip_sample(bset
);
1125 return bounded
? sample_bounded(bset
) : gbr_sample(bset
);
1127 isl_assert(bset
->ctx
, 0, );
1129 isl_basic_set_free(bset
);
1133 __isl_give isl_vec
*isl_basic_set_sample_vec(__isl_take isl_basic_set
*bset
)
1135 return basic_set_sample(bset
, 0);
1138 /* Compute an integer sample in "bset", where the caller guarantees
1139 * that "bset" is bounded.
1141 struct isl_vec
*isl_basic_set_sample_bounded(struct isl_basic_set
*bset
)
1143 return basic_set_sample(bset
, 1);
1146 __isl_give isl_basic_set
*isl_basic_set_from_vec(__isl_take isl_vec
*vec
)
1150 struct isl_basic_set
*bset
= NULL
;
1151 struct isl_ctx
*ctx
;
1157 isl_assert(ctx
, vec
->size
!= 0, goto error
);
1159 bset
= isl_basic_set_alloc(ctx
, 0, vec
->size
- 1, 0, vec
->size
- 1, 0);
1162 dim
= isl_basic_set_n_dim(bset
);
1163 for (i
= dim
- 1; i
>= 0; --i
) {
1164 k
= isl_basic_set_alloc_equality(bset
);
1167 isl_seq_clr(bset
->eq
[k
], 1 + dim
);
1168 isl_int_neg(bset
->eq
[k
][0], vec
->el
[1 + i
]);
1169 isl_int_set(bset
->eq
[k
][1 + i
], vec
->el
[0]);
1175 isl_basic_set_free(bset
);
1180 __isl_give isl_basic_map
*isl_basic_map_sample(__isl_take isl_basic_map
*bmap
)
1182 struct isl_basic_set
*bset
;
1183 struct isl_vec
*sample_vec
;
1185 bset
= isl_basic_map_underlying_set(isl_basic_map_copy(bmap
));
1186 sample_vec
= isl_basic_set_sample_vec(bset
);
1189 if (sample_vec
->size
== 0) {
1190 struct isl_basic_map
*sample
;
1191 sample
= isl_basic_map_empty_like(bmap
);
1192 isl_vec_free(sample_vec
);
1193 isl_basic_map_free(bmap
);
1196 bset
= isl_basic_set_from_vec(sample_vec
);
1197 return isl_basic_map_overlying_set(bset
, bmap
);
1199 isl_basic_map_free(bmap
);
1203 __isl_give isl_basic_map
*isl_map_sample(__isl_take isl_map
*map
)
1206 isl_basic_map
*sample
= NULL
;
1211 for (i
= 0; i
< map
->n
; ++i
) {
1212 sample
= isl_basic_map_sample(isl_basic_map_copy(map
->p
[i
]));
1215 if (!ISL_F_ISSET(sample
, ISL_BASIC_MAP_EMPTY
))
1217 isl_basic_map_free(sample
);
1220 sample
= isl_basic_map_empty_like_map(map
);
1228 __isl_give isl_basic_set
*isl_set_sample(__isl_take isl_set
*set
)
1230 return (isl_basic_set
*) isl_map_sample((isl_map
*)set
);