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 tab
= isl_tab_add_valid_eq(tab
, tab
->basis
->row
[1 + level
]);
512 isl_int_set_si(tab
->basis
->row
[1 + level
][0], 0);
513 if (level
+ tab
->n_unbounded
< dim
- 1) {
522 sample
= isl_tab_get_sample_value(tab
);
525 if (tab
->n_unbounded
&& !isl_int_is_one(sample
->el
[0])) {
526 sample
= isl_mat_vec_product(isl_mat_copy(tab
->basis
),
528 sample
= isl_vec_ceil(sample
);
529 sample
= isl_mat_vec_inverse_product(
530 isl_mat_copy(tab
->basis
), sample
);
533 sample
= isl_vec_alloc(ctx
, 0);
548 /* Given a basic set that is known to be bounded, find and return
549 * an integer point in the basic set, if there is any.
551 * After handling some trivial cases, we construct a tableau
552 * and then use isl_tab_sample to find a sample, passing it
553 * the identity matrix as initial basis.
555 static struct isl_vec
*sample_bounded(struct isl_basic_set
*bset
)
559 struct isl_vec
*sample
;
560 struct isl_tab
*tab
= NULL
;
565 if (isl_basic_set_fast_is_empty(bset
))
566 return empty_sample(bset
);
568 dim
= isl_basic_set_total_dim(bset
);
570 return zero_sample(bset
);
572 return interval_sample(bset
);
574 return sample_eq(bset
, sample_bounded
);
578 tab
= isl_tab_from_basic_set(bset
);
579 if (tab
&& tab
->empty
) {
581 ISL_F_SET(bset
, ISL_BASIC_SET_EMPTY
);
582 sample
= isl_vec_alloc(bset
->ctx
, 0);
583 isl_basic_set_free(bset
);
587 if (isl_tab_track_bset(tab
, isl_basic_set_copy(bset
)) < 0)
589 if (!ISL_F_ISSET(bset
, ISL_BASIC_SET_NO_IMPLICIT
))
590 if (isl_tab_detect_implicit_equalities(tab
) < 0)
593 sample
= isl_tab_sample(tab
);
597 if (sample
->size
> 0) {
598 isl_vec_free(bset
->sample
);
599 bset
->sample
= isl_vec_copy(sample
);
602 isl_basic_set_free(bset
);
606 isl_basic_set_free(bset
);
611 /* Given a basic set "bset" and a value "sample" for the first coordinates
612 * of bset, plug in these values and drop the corresponding coordinates.
614 * We do this by computing the preimage of the transformation
620 * where [1 s] is the sample value and I is the identity matrix of the
621 * appropriate dimension.
623 static struct isl_basic_set
*plug_in(struct isl_basic_set
*bset
,
624 struct isl_vec
*sample
)
630 if (!bset
|| !sample
)
633 total
= isl_basic_set_total_dim(bset
);
634 T
= isl_mat_alloc(bset
->ctx
, 1 + total
, 1 + total
- (sample
->size
- 1));
638 for (i
= 0; i
< sample
->size
; ++i
) {
639 isl_int_set(T
->row
[i
][0], sample
->el
[i
]);
640 isl_seq_clr(T
->row
[i
] + 1, T
->n_col
- 1);
642 for (i
= 0; i
< T
->n_col
- 1; ++i
) {
643 isl_seq_clr(T
->row
[sample
->size
+ i
], T
->n_col
);
644 isl_int_set_si(T
->row
[sample
->size
+ i
][1 + i
], 1);
646 isl_vec_free(sample
);
648 bset
= isl_basic_set_preimage(bset
, T
);
651 isl_basic_set_free(bset
);
652 isl_vec_free(sample
);
656 /* Given a basic set "bset", return any (possibly non-integer) point
659 static struct isl_vec
*rational_sample(struct isl_basic_set
*bset
)
662 struct isl_vec
*sample
;
667 tab
= isl_tab_from_basic_set(bset
);
668 sample
= isl_tab_get_sample_value(tab
);
671 isl_basic_set_free(bset
);
676 /* Given a linear cone "cone" and a rational point "vec",
677 * construct a polyhedron with shifted copies of the constraints in "cone",
678 * i.e., a polyhedron with "cone" as its recession cone, such that each
679 * point x in this polyhedron is such that the unit box positioned at x
680 * lies entirely inside the affine cone 'vec + cone'.
681 * Any rational point in this polyhedron may therefore be rounded up
682 * to yield an integer point that lies inside said affine cone.
684 * Denote the constraints of cone by "<a_i, x> >= 0" and the rational
685 * point "vec" by v/d.
686 * Let b_i = <a_i, v>. Then the affine cone 'vec + cone' is given
687 * by <a_i, x> - b/d >= 0.
688 * The polyhedron <a_i, x> - ceil{b/d} >= 0 is a subset of this affine cone.
689 * We prefer this polyhedron over the actual affine cone because it doesn't
690 * require a scaling of the constraints.
691 * If each of the vertices of the unit cube positioned at x lies inside
692 * this polyhedron, then the whole unit cube at x lies inside the affine cone.
693 * We therefore impose that x' = x + \sum e_i, for any selection of unit
694 * vectors lies inside the polyhedron, i.e.,
696 * <a_i, x'> - ceil{b/d} = <a_i, x> + sum a_i - ceil{b/d} >= 0
698 * The most stringent of these constraints is the one that selects
699 * all negative a_i, so the polyhedron we are looking for has constraints
701 * <a_i, x> + sum_{a_i < 0} a_i - ceil{b/d} >= 0
703 * Note that if cone were known to have only non-negative rays
704 * (which can be accomplished by a unimodular transformation),
705 * then we would only have to check the points x' = x + e_i
706 * and we only have to add the smallest negative a_i (if any)
707 * instead of the sum of all negative a_i.
709 static struct isl_basic_set
*shift_cone(struct isl_basic_set
*cone
,
715 struct isl_basic_set
*shift
= NULL
;
720 isl_assert(cone
->ctx
, cone
->n_eq
== 0, goto error
);
722 total
= isl_basic_set_total_dim(cone
);
724 shift
= isl_basic_set_alloc_dim(isl_basic_set_get_dim(cone
),
727 for (i
= 0; i
< cone
->n_ineq
; ++i
) {
728 k
= isl_basic_set_alloc_inequality(shift
);
731 isl_seq_cpy(shift
->ineq
[k
] + 1, cone
->ineq
[i
] + 1, total
);
732 isl_seq_inner_product(shift
->ineq
[k
] + 1, vec
->el
+ 1, total
,
734 isl_int_cdiv_q(shift
->ineq
[k
][0],
735 shift
->ineq
[k
][0], vec
->el
[0]);
736 isl_int_neg(shift
->ineq
[k
][0], shift
->ineq
[k
][0]);
737 for (j
= 0; j
< total
; ++j
) {
738 if (isl_int_is_nonneg(shift
->ineq
[k
][1 + j
]))
740 isl_int_add(shift
->ineq
[k
][0],
741 shift
->ineq
[k
][0], shift
->ineq
[k
][1 + j
]);
745 isl_basic_set_free(cone
);
748 return isl_basic_set_finalize(shift
);
750 isl_basic_set_free(shift
);
751 isl_basic_set_free(cone
);
756 /* Given a rational point vec in a (transformed) basic set,
757 * such that cone is the recession cone of the original basic set,
758 * "round up" the rational point to an integer point.
760 * We first check if the rational point just happens to be integer.
761 * If not, we transform the cone in the same way as the basic set,
762 * pick a point x in this cone shifted to the rational point such that
763 * the whole unit cube at x is also inside this affine cone.
764 * Then we simply round up the coordinates of x and return the
765 * resulting integer point.
767 static struct isl_vec
*round_up_in_cone(struct isl_vec
*vec
,
768 struct isl_basic_set
*cone
, struct isl_mat
*U
)
772 if (!vec
|| !cone
|| !U
)
775 isl_assert(vec
->ctx
, vec
->size
!= 0, goto error
);
776 if (isl_int_is_one(vec
->el
[0])) {
778 isl_basic_set_free(cone
);
782 total
= isl_basic_set_total_dim(cone
);
783 cone
= isl_basic_set_preimage(cone
, U
);
784 cone
= isl_basic_set_remove_dims(cone
, 0, total
- (vec
->size
- 1));
786 cone
= shift_cone(cone
, vec
);
788 vec
= rational_sample(cone
);
789 vec
= isl_vec_ceil(vec
);
794 isl_basic_set_free(cone
);
798 /* Concatenate two integer vectors, i.e., two vectors with denominator
799 * (stored in element 0) equal to 1.
801 static struct isl_vec
*vec_concat(struct isl_vec
*vec1
, struct isl_vec
*vec2
)
807 isl_assert(vec1
->ctx
, vec1
->size
> 0, goto error
);
808 isl_assert(vec2
->ctx
, vec2
->size
> 0, goto error
);
809 isl_assert(vec1
->ctx
, isl_int_is_one(vec1
->el
[0]), goto error
);
810 isl_assert(vec2
->ctx
, isl_int_is_one(vec2
->el
[0]), goto error
);
812 vec
= isl_vec_alloc(vec1
->ctx
, vec1
->size
+ vec2
->size
- 1);
816 isl_seq_cpy(vec
->el
, vec1
->el
, vec1
->size
);
817 isl_seq_cpy(vec
->el
+ vec1
->size
, vec2
->el
+ 1, vec2
->size
- 1);
829 /* Drop all constraints in bset that involve any of the dimensions
830 * first to first+n-1.
832 static struct isl_basic_set
*drop_constraints_involving
833 (struct isl_basic_set
*bset
, unsigned first
, unsigned n
)
837 bset
= isl_basic_set_cow(bset
);
842 for (i
= bset
->n_ineq
- 1; i
>= 0; --i
) {
843 if (isl_seq_first_non_zero(bset
->ineq
[i
] + 1 + first
, n
) == -1)
845 isl_basic_set_drop_inequality(bset
, i
);
851 /* Give a basic set "bset" with recession cone "cone", compute and
852 * return an integer point in bset, if any.
854 * If the recession cone is full-dimensional, then we know that
855 * bset contains an infinite number of integer points and it is
856 * fairly easy to pick one of them.
857 * If the recession cone is not full-dimensional, then we first
858 * transform bset such that the bounded directions appear as
859 * the first dimensions of the transformed basic set.
860 * We do this by using a unimodular transformation that transforms
861 * the equalities in the recession cone to equalities on the first
864 * The transformed set is then projected onto its bounded dimensions.
865 * Note that to compute this projection, we can simply drop all constraints
866 * involving any of the unbounded dimensions since these constraints
867 * cannot be combined to produce a constraint on the bounded dimensions.
868 * To see this, assume that there is such a combination of constraints
869 * that produces a constraint on the bounded dimensions. This means
870 * that some combination of the unbounded dimensions has both an upper
871 * bound and a lower bound in terms of the bounded dimensions, but then
872 * this combination would be a bounded direction too and would have been
873 * transformed into a bounded dimensions.
875 * We then compute a sample value in the bounded dimensions.
876 * If no such value can be found, then the original set did not contain
877 * any integer points and we are done.
878 * Otherwise, we plug in the value we found in the bounded dimensions,
879 * project out these bounded dimensions and end up with a set with
880 * a full-dimensional recession cone.
881 * A sample point in this set is computed by "rounding up" any
882 * rational point in the set.
884 * The sample points in the bounded and unbounded dimensions are
885 * then combined into a single sample point and transformed back
886 * to the original space.
888 __isl_give isl_vec
*isl_basic_set_sample_with_cone(
889 __isl_take isl_basic_set
*bset
, __isl_take isl_basic_set
*cone
)
893 struct isl_mat
*M
, *U
;
894 struct isl_vec
*sample
;
895 struct isl_vec
*cone_sample
;
897 struct isl_basic_set
*bounded
;
903 total
= isl_basic_set_total_dim(cone
);
904 cone_dim
= total
- cone
->n_eq
;
906 M
= isl_mat_sub_alloc(bset
->ctx
, cone
->eq
, 0, cone
->n_eq
, 1, total
);
907 M
= isl_mat_left_hermite(M
, 0, &U
, NULL
);
912 U
= isl_mat_lin_to_aff(U
);
913 bset
= isl_basic_set_preimage(bset
, isl_mat_copy(U
));
915 bounded
= isl_basic_set_copy(bset
);
916 bounded
= drop_constraints_involving(bounded
, total
- cone_dim
, cone_dim
);
917 bounded
= isl_basic_set_drop_dims(bounded
, total
- cone_dim
, cone_dim
);
918 sample
= sample_bounded(bounded
);
919 if (!sample
|| sample
->size
== 0) {
920 isl_basic_set_free(bset
);
921 isl_basic_set_free(cone
);
925 bset
= plug_in(bset
, isl_vec_copy(sample
));
926 cone_sample
= rational_sample(bset
);
927 cone_sample
= round_up_in_cone(cone_sample
, cone
, isl_mat_copy(U
));
928 sample
= vec_concat(sample
, cone_sample
);
929 sample
= isl_mat_vec_product(U
, sample
);
932 isl_basic_set_free(cone
);
933 isl_basic_set_free(bset
);
937 static void vec_sum_of_neg(struct isl_vec
*v
, isl_int
*s
)
941 isl_int_set_si(*s
, 0);
943 for (i
= 0; i
< v
->size
; ++i
)
944 if (isl_int_is_neg(v
->el
[i
]))
945 isl_int_add(*s
, *s
, v
->el
[i
]);
948 /* Given a tableau "tab", a tableau "tab_cone" that corresponds
949 * to the recession cone and the inverse of a new basis U = inv(B),
950 * with the unbounded directions in B last,
951 * add constraints to "tab" that ensure any rational value
952 * in the unbounded directions can be rounded up to an integer value.
954 * The new basis is given by x' = B x, i.e., x = U x'.
955 * For any rational value of the last tab->n_unbounded coordinates
956 * in the update tableau, the value that is obtained by rounding
957 * up this value should be contained in the original tableau.
958 * For any constraint "a x + c >= 0", we therefore need to add
959 * a constraint "a x + c + s >= 0", with s the sum of all negative
960 * entries in the last elements of "a U".
962 * Since we are not interested in the first entries of any of the "a U",
963 * we first drop the columns of U that correpond to bounded directions.
965 static int tab_shift_cone(struct isl_tab
*tab
,
966 struct isl_tab
*tab_cone
, struct isl_mat
*U
)
970 struct isl_basic_set
*bset
= NULL
;
972 if (tab
&& tab
->n_unbounded
== 0) {
977 if (!tab
|| !tab_cone
|| !U
)
979 bset
= isl_tab_peek_bset(tab_cone
);
980 U
= isl_mat_drop_cols(U
, 0, tab
->n_var
- tab
->n_unbounded
);
981 for (i
= 0; i
< bset
->n_ineq
; ++i
) {
983 struct isl_vec
*row
= NULL
;
984 if (isl_tab_is_equality(tab_cone
, tab_cone
->n_eq
+ i
))
986 row
= isl_vec_alloc(bset
->ctx
, tab_cone
->n_var
);
989 isl_seq_cpy(row
->el
, bset
->ineq
[i
] + 1, tab_cone
->n_var
);
990 row
= isl_vec_mat_product(row
, isl_mat_copy(U
));
993 vec_sum_of_neg(row
, &v
);
995 if (isl_int_is_zero(v
))
997 tab
= isl_tab_extend(tab
, 1);
998 isl_int_add(bset
->ineq
[i
][0], bset
->ineq
[i
][0], v
);
999 ok
= isl_tab_add_ineq(tab
, bset
->ineq
[i
]) >= 0;
1000 isl_int_sub(bset
->ineq
[i
][0], bset
->ineq
[i
][0], v
);
1014 /* Compute and return an initial basis for the possibly
1015 * unbounded tableau "tab". "tab_cone" is a tableau
1016 * for the corresponding recession cone.
1017 * Additionally, add constraints to "tab" that ensure
1018 * that any rational value for the unbounded directions
1019 * can be rounded up to an integer value.
1021 * If the tableau is bounded, i.e., if the recession cone
1022 * is zero-dimensional, then we just use inital_basis.
1023 * Otherwise, we construct a basis whose first directions
1024 * correspond to equalities, followed by bounded directions,
1025 * i.e., equalities in the recession cone.
1026 * The remaining directions are then unbounded.
1028 int isl_tab_set_initial_basis_with_cone(struct isl_tab
*tab
,
1029 struct isl_tab
*tab_cone
)
1032 struct isl_mat
*cone_eq
;
1033 struct isl_mat
*U
, *Q
;
1035 if (!tab
|| !tab_cone
)
1038 if (tab_cone
->n_col
== tab_cone
->n_dead
) {
1039 tab
->basis
= initial_basis(tab
);
1040 return tab
->basis
? 0 : -1;
1043 eq
= tab_equalities(tab
);
1046 tab
->n_zero
= eq
->n_row
;
1047 cone_eq
= tab_equalities(tab_cone
);
1048 eq
= isl_mat_concat(eq
, cone_eq
);
1051 tab
->n_unbounded
= tab
->n_var
- (eq
->n_row
- tab
->n_zero
);
1052 eq
= isl_mat_left_hermite(eq
, 0, &U
, &Q
);
1056 tab
->basis
= isl_mat_lin_to_aff(Q
);
1057 if (tab_shift_cone(tab
, tab_cone
, U
) < 0)
1064 /* Compute and return a sample point in bset using generalized basis
1065 * reduction. We first check if the input set has a non-trivial
1066 * recession cone. If so, we perform some extra preprocessing in
1067 * sample_with_cone. Otherwise, we directly perform generalized basis
1070 static struct isl_vec
*gbr_sample(struct isl_basic_set
*bset
)
1073 struct isl_basic_set
*cone
;
1075 dim
= isl_basic_set_total_dim(bset
);
1077 cone
= isl_basic_set_recession_cone(isl_basic_set_copy(bset
));
1081 if (cone
->n_eq
< dim
)
1082 return isl_basic_set_sample_with_cone(bset
, cone
);
1084 isl_basic_set_free(cone
);
1085 return sample_bounded(bset
);
1087 isl_basic_set_free(bset
);
1091 static struct isl_vec
*pip_sample(struct isl_basic_set
*bset
)
1094 struct isl_ctx
*ctx
;
1095 struct isl_vec
*sample
;
1097 bset
= isl_basic_set_skew_to_positive_orthant(bset
, &T
);
1102 sample
= isl_pip_basic_set_sample(bset
);
1104 if (sample
&& sample
->size
!= 0)
1105 sample
= isl_mat_vec_product(T
, sample
);
1112 static struct isl_vec
*basic_set_sample(struct isl_basic_set
*bset
, int bounded
)
1114 struct isl_ctx
*ctx
;
1120 if (isl_basic_set_fast_is_empty(bset
))
1121 return empty_sample(bset
);
1123 dim
= isl_basic_set_n_dim(bset
);
1124 isl_assert(ctx
, isl_basic_set_n_param(bset
) == 0, goto error
);
1125 isl_assert(ctx
, bset
->n_div
== 0, goto error
);
1127 if (bset
->sample
&& bset
->sample
->size
== 1 + dim
) {
1128 int contains
= isl_basic_set_contains(bset
, bset
->sample
);
1132 struct isl_vec
*sample
= isl_vec_copy(bset
->sample
);
1133 isl_basic_set_free(bset
);
1137 isl_vec_free(bset
->sample
);
1138 bset
->sample
= NULL
;
1141 return sample_eq(bset
, bounded
? isl_basic_set_sample_bounded
1142 : isl_basic_set_sample_vec
);
1144 return zero_sample(bset
);
1146 return interval_sample(bset
);
1148 switch (bset
->ctx
->opt
->ilp_solver
) {
1150 return pip_sample(bset
);
1152 return bounded
? sample_bounded(bset
) : gbr_sample(bset
);
1154 isl_assert(bset
->ctx
, 0, );
1156 isl_basic_set_free(bset
);
1160 __isl_give isl_vec
*isl_basic_set_sample_vec(__isl_take isl_basic_set
*bset
)
1162 return basic_set_sample(bset
, 0);
1165 /* Compute an integer sample in "bset", where the caller guarantees
1166 * that "bset" is bounded.
1168 struct isl_vec
*isl_basic_set_sample_bounded(struct isl_basic_set
*bset
)
1170 return basic_set_sample(bset
, 1);
1173 __isl_give isl_basic_set
*isl_basic_set_from_vec(__isl_take isl_vec
*vec
)
1177 struct isl_basic_set
*bset
= NULL
;
1178 struct isl_ctx
*ctx
;
1184 isl_assert(ctx
, vec
->size
!= 0, goto error
);
1186 bset
= isl_basic_set_alloc(ctx
, 0, vec
->size
- 1, 0, vec
->size
- 1, 0);
1189 dim
= isl_basic_set_n_dim(bset
);
1190 for (i
= dim
- 1; i
>= 0; --i
) {
1191 k
= isl_basic_set_alloc_equality(bset
);
1194 isl_seq_clr(bset
->eq
[k
], 1 + dim
);
1195 isl_int_neg(bset
->eq
[k
][0], vec
->el
[1 + i
]);
1196 isl_int_set(bset
->eq
[k
][1 + i
], vec
->el
[0]);
1202 isl_basic_set_free(bset
);
1207 __isl_give isl_basic_map
*isl_basic_map_sample(__isl_take isl_basic_map
*bmap
)
1209 struct isl_basic_set
*bset
;
1210 struct isl_vec
*sample_vec
;
1212 bset
= isl_basic_map_underlying_set(isl_basic_map_copy(bmap
));
1213 sample_vec
= isl_basic_set_sample_vec(bset
);
1216 if (sample_vec
->size
== 0) {
1217 struct isl_basic_map
*sample
;
1218 sample
= isl_basic_map_empty_like(bmap
);
1219 isl_vec_free(sample_vec
);
1220 isl_basic_map_free(bmap
);
1223 bset
= isl_basic_set_from_vec(sample_vec
);
1224 return isl_basic_map_overlying_set(bset
, bmap
);
1226 isl_basic_map_free(bmap
);
1230 __isl_give isl_basic_map
*isl_map_sample(__isl_take isl_map
*map
)
1233 isl_basic_map
*sample
= NULL
;
1238 for (i
= 0; i
< map
->n
; ++i
) {
1239 sample
= isl_basic_map_sample(isl_basic_map_copy(map
->p
[i
]));
1242 if (!ISL_F_ISSET(sample
, ISL_BASIC_MAP_EMPTY
))
1244 isl_basic_map_free(sample
);
1247 sample
= isl_basic_map_empty_like_map(map
);
1255 __isl_give isl_basic_set
*isl_set_sample(__isl_take isl_set
*set
)
1257 return (isl_basic_set
*) isl_map_sample((isl_map
*)set
);
1260 __isl_give isl_point
*isl_basic_set_sample_point(__isl_take isl_basic_set
*bset
)
1265 dim
= isl_basic_set_get_dim(bset
);
1266 bset
= isl_basic_set_underlying_set(bset
);
1267 vec
= isl_basic_set_sample_vec(bset
);
1269 return isl_point_alloc(dim
, vec
);
1272 __isl_give isl_point
*isl_set_sample_point(__isl_take isl_set
*set
)
1280 for (i
= 0; i
< set
->n
; ++i
) {
1281 pnt
= isl_basic_set_sample_point(isl_basic_set_copy(set
->p
[i
]));
1284 if (!isl_point_is_void(pnt
))
1286 isl_point_free(pnt
);
1289 pnt
= isl_point_void(isl_set_get_dim(set
));