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_ctx_private.h>
11 #include <isl_map_private.h>
12 #include "isl_sample.h"
13 #include "isl_sample_piplib.h"
17 #include "isl_equalities.h"
19 #include "isl_basis_reduction.h"
20 #include <isl_factorization.h>
21 #include <isl_point_private.h>
22 #include <isl_options_private.h>
24 static struct isl_vec
*empty_sample(struct isl_basic_set
*bset
)
28 vec
= isl_vec_alloc(bset
->ctx
, 0);
29 isl_basic_set_free(bset
);
33 /* Construct a zero sample of the same dimension as bset.
34 * As a special case, if bset is zero-dimensional, this
35 * function creates a zero-dimensional sample point.
37 static struct isl_vec
*zero_sample(struct isl_basic_set
*bset
)
40 struct isl_vec
*sample
;
42 dim
= isl_basic_set_total_dim(bset
);
43 sample
= isl_vec_alloc(bset
->ctx
, 1 + dim
);
45 isl_int_set_si(sample
->el
[0], 1);
46 isl_seq_clr(sample
->el
+ 1, dim
);
48 isl_basic_set_free(bset
);
52 static struct isl_vec
*interval_sample(struct isl_basic_set
*bset
)
56 struct isl_vec
*sample
;
58 bset
= isl_basic_set_simplify(bset
);
61 if (isl_basic_set_plain_is_empty(bset
))
62 return empty_sample(bset
);
63 if (bset
->n_eq
== 0 && bset
->n_ineq
== 0)
64 return zero_sample(bset
);
66 sample
= isl_vec_alloc(bset
->ctx
, 2);
71 isl_int_set_si(sample
->block
.data
[0], 1);
74 isl_assert(bset
->ctx
, bset
->n_eq
== 1, goto error
);
75 isl_assert(bset
->ctx
, bset
->n_ineq
== 0, goto error
);
76 if (isl_int_is_one(bset
->eq
[0][1]))
77 isl_int_neg(sample
->el
[1], bset
->eq
[0][0]);
79 isl_assert(bset
->ctx
, isl_int_is_negone(bset
->eq
[0][1]),
81 isl_int_set(sample
->el
[1], bset
->eq
[0][0]);
83 isl_basic_set_free(bset
);
88 if (isl_int_is_one(bset
->ineq
[0][1]))
89 isl_int_neg(sample
->block
.data
[1], bset
->ineq
[0][0]);
91 isl_int_set(sample
->block
.data
[1], bset
->ineq
[0][0]);
92 for (i
= 1; i
< bset
->n_ineq
; ++i
) {
93 isl_seq_inner_product(sample
->block
.data
,
94 bset
->ineq
[i
], 2, &t
);
95 if (isl_int_is_neg(t
))
99 if (i
< bset
->n_ineq
) {
100 isl_vec_free(sample
);
101 return empty_sample(bset
);
104 isl_basic_set_free(bset
);
107 isl_basic_set_free(bset
);
108 isl_vec_free(sample
);
112 static struct isl_mat
*independent_bounds(struct isl_basic_set
*bset
)
115 struct isl_mat
*dirs
= NULL
;
116 struct isl_mat
*bounds
= NULL
;
122 dim
= isl_basic_set_n_dim(bset
);
123 bounds
= isl_mat_alloc(bset
->ctx
, 1+dim
, 1+dim
);
127 isl_int_set_si(bounds
->row
[0][0], 1);
128 isl_seq_clr(bounds
->row
[0]+1, dim
);
131 if (bset
->n_ineq
== 0)
134 dirs
= isl_mat_alloc(bset
->ctx
, dim
, dim
);
136 isl_mat_free(bounds
);
139 isl_seq_cpy(dirs
->row
[0], bset
->ineq
[0]+1, dirs
->n_col
);
140 isl_seq_cpy(bounds
->row
[1], bset
->ineq
[0], bounds
->n_col
);
141 for (j
= 1, n
= 1; n
< dim
&& j
< bset
->n_ineq
; ++j
) {
144 isl_seq_cpy(dirs
->row
[n
], bset
->ineq
[j
]+1, dirs
->n_col
);
146 pos
= isl_seq_first_non_zero(dirs
->row
[n
], dirs
->n_col
);
149 for (i
= 0; i
< n
; ++i
) {
151 pos_i
= isl_seq_first_non_zero(dirs
->row
[i
], dirs
->n_col
);
156 isl_seq_elim(dirs
->row
[n
], dirs
->row
[i
], pos
,
158 pos
= isl_seq_first_non_zero(dirs
->row
[n
], dirs
->n_col
);
166 isl_int
*t
= dirs
->row
[n
];
167 for (k
= n
; k
> i
; --k
)
168 dirs
->row
[k
] = dirs
->row
[k
-1];
172 isl_seq_cpy(bounds
->row
[n
], bset
->ineq
[j
], bounds
->n_col
);
179 static void swap_inequality(struct isl_basic_set
*bset
, int a
, int b
)
181 isl_int
*t
= bset
->ineq
[a
];
182 bset
->ineq
[a
] = bset
->ineq
[b
];
186 /* Skew into positive orthant and project out lineality space.
188 * We perform a unimodular transformation that turns a selected
189 * maximal set of linearly independent bounds into constraints
190 * on the first dimensions that impose that these first dimensions
191 * are non-negative. In particular, the constraint matrix is lower
192 * triangular with positive entries on the diagonal and negative
194 * If "bset" has a lineality space then these constraints (and therefore
195 * all constraints in bset) only involve the first dimensions.
196 * The remaining dimensions then do not appear in any constraints and
197 * we can select any value for them, say zero. We therefore project
198 * out this final dimensions and plug in the value zero later. This
199 * is accomplished by simply dropping the final columns of
200 * the unimodular transformation.
202 static struct isl_basic_set
*isl_basic_set_skew_to_positive_orthant(
203 struct isl_basic_set
*bset
, struct isl_mat
**T
)
205 struct isl_mat
*U
= NULL
;
206 struct isl_mat
*bounds
= NULL
;
208 unsigned old_dim
, new_dim
;
214 isl_assert(bset
->ctx
, isl_basic_set_n_param(bset
) == 0, goto error
);
215 isl_assert(bset
->ctx
, bset
->n_div
== 0, goto error
);
216 isl_assert(bset
->ctx
, bset
->n_eq
== 0, goto error
);
218 old_dim
= isl_basic_set_n_dim(bset
);
219 /* Try to move (multiples of) unit rows up. */
220 for (i
= 0, j
= 0; i
< bset
->n_ineq
; ++i
) {
221 int pos
= isl_seq_first_non_zero(bset
->ineq
[i
]+1, old_dim
);
224 if (isl_seq_first_non_zero(bset
->ineq
[i
]+1+pos
+1,
228 swap_inequality(bset
, i
, j
);
231 bounds
= independent_bounds(bset
);
234 new_dim
= bounds
->n_row
- 1;
235 bounds
= isl_mat_left_hermite(bounds
, 1, &U
, NULL
);
238 U
= isl_mat_drop_cols(U
, 1 + new_dim
, old_dim
- new_dim
);
239 bset
= isl_basic_set_preimage(bset
, isl_mat_copy(U
));
243 isl_mat_free(bounds
);
246 isl_mat_free(bounds
);
248 isl_basic_set_free(bset
);
252 /* Find a sample integer point, if any, in bset, which is known
253 * to have equalities. If bset contains no integer points, then
254 * return a zero-length vector.
255 * We simply remove the known equalities, compute a sample
256 * in the resulting bset, using the specified recurse function,
257 * and then transform the sample back to the original space.
259 static struct isl_vec
*sample_eq(struct isl_basic_set
*bset
,
260 struct isl_vec
*(*recurse
)(struct isl_basic_set
*))
263 struct isl_vec
*sample
;
268 bset
= isl_basic_set_remove_equalities(bset
, &T
, NULL
);
269 sample
= recurse(bset
);
270 if (!sample
|| sample
->size
== 0)
273 sample
= isl_mat_vec_product(T
, sample
);
277 /* Return a matrix containing the equalities of the tableau
278 * in constraint form. The tableau is assumed to have
279 * an associated bset that has been kept up-to-date.
281 static struct isl_mat
*tab_equalities(struct isl_tab
*tab
)
286 struct isl_basic_set
*bset
;
291 bset
= isl_tab_peek_bset(tab
);
292 isl_assert(tab
->mat
->ctx
, bset
, return NULL
);
294 n_eq
= tab
->n_var
- tab
->n_col
+ tab
->n_dead
;
295 if (tab
->empty
|| n_eq
== 0)
296 return isl_mat_alloc(tab
->mat
->ctx
, 0, tab
->n_var
);
297 if (n_eq
== tab
->n_var
)
298 return isl_mat_identity(tab
->mat
->ctx
, tab
->n_var
);
300 eq
= isl_mat_alloc(tab
->mat
->ctx
, n_eq
, tab
->n_var
);
303 for (i
= 0, j
= 0; i
< tab
->n_con
; ++i
) {
304 if (tab
->con
[i
].is_row
)
306 if (tab
->con
[i
].index
>= 0 && tab
->con
[i
].index
>= tab
->n_dead
)
309 isl_seq_cpy(eq
->row
[j
], bset
->eq
[i
] + 1, tab
->n_var
);
311 isl_seq_cpy(eq
->row
[j
],
312 bset
->ineq
[i
- bset
->n_eq
] + 1, tab
->n_var
);
315 isl_assert(bset
->ctx
, j
== n_eq
, goto error
);
322 /* Compute and return an initial basis for the bounded tableau "tab".
324 * If the tableau is either full-dimensional or zero-dimensional,
325 * the we simply return an identity matrix.
326 * Otherwise, we construct a basis whose first directions correspond
329 static struct isl_mat
*initial_basis(struct isl_tab
*tab
)
335 tab
->n_unbounded
= 0;
336 tab
->n_zero
= n_eq
= tab
->n_var
- tab
->n_col
+ tab
->n_dead
;
337 if (tab
->empty
|| n_eq
== 0 || n_eq
== tab
->n_var
)
338 return isl_mat_identity(tab
->mat
->ctx
, 1 + tab
->n_var
);
340 eq
= tab_equalities(tab
);
341 eq
= isl_mat_left_hermite(eq
, 0, NULL
, &Q
);
346 Q
= isl_mat_lin_to_aff(Q
);
350 /* Given a tableau representing a set, find and return
351 * an integer point in the set, if there is any.
353 * We perform a depth first search
354 * for an integer point, by scanning all possible values in the range
355 * attained by a basis vector, where an initial basis may have been set
356 * by the calling function. Otherwise an initial basis that exploits
357 * the equalities in the tableau is created.
358 * tab->n_zero is currently ignored and is clobbered by this function.
360 * The tableau is allowed to have unbounded direction, but then
361 * the calling function needs to set an initial basis, with the
362 * unbounded directions last and with tab->n_unbounded set
363 * to the number of unbounded directions.
364 * Furthermore, the calling functions needs to add shifted copies
365 * of all constraints involving unbounded directions to ensure
366 * that any feasible rational value in these directions can be rounded
367 * up to yield a feasible integer value.
368 * In particular, let B define the given basis x' = B x
369 * and let T be the inverse of B, i.e., X = T x'.
370 * Let a x + c >= 0 be a constraint of the set represented by the tableau,
371 * or a T x' + c >= 0 in terms of the given basis. Assume that
372 * the bounded directions have an integer value, then we can safely
373 * round up the values for the unbounded directions if we make sure
374 * that x' not only satisfies the original constraint, but also
375 * the constraint "a T x' + c + s >= 0" with s the sum of all
376 * negative values in the last n_unbounded entries of "a T".
377 * The calling function therefore needs to add the constraint
378 * a x + c + s >= 0. The current function then scans the first
379 * directions for an integer value and once those have been found,
380 * it can compute "T ceil(B x)" to yield an integer point in the set.
381 * Note that during the search, the first rows of B may be changed
382 * by a basis reduction, but the last n_unbounded rows of B remain
383 * unaltered and are also not mixed into the first rows.
385 * The search is implemented iteratively. "level" identifies the current
386 * basis vector. "init" is true if we want the first value at the current
387 * level and false if we want the next value.
389 * The initial basis is the identity matrix. If the range in some direction
390 * contains more than one integer value, we perform basis reduction based
391 * on the value of ctx->opt->gbr
392 * - ISL_GBR_NEVER: never perform basis reduction
393 * - ISL_GBR_ONCE: only perform basis reduction the first
394 * time such a range is encountered
395 * - ISL_GBR_ALWAYS: always perform basis reduction when
396 * such a range is encountered
398 * When ctx->opt->gbr is set to ISL_GBR_ALWAYS, then we allow the basis
399 * reduction computation to return early. That is, as soon as it
400 * finds a reasonable first direction.
402 struct isl_vec
*isl_tab_sample(struct isl_tab
*tab
)
407 struct isl_vec
*sample
;
410 enum isl_lp_result res
;
414 struct isl_tab_undo
**snap
;
419 return isl_vec_alloc(tab
->mat
->ctx
, 0);
422 tab
->basis
= initial_basis(tab
);
425 isl_assert(tab
->mat
->ctx
, tab
->basis
->n_row
== tab
->n_var
+ 1,
427 isl_assert(tab
->mat
->ctx
, tab
->basis
->n_col
== tab
->n_var
+ 1,
434 if (tab
->n_unbounded
== tab
->n_var
) {
435 sample
= isl_tab_get_sample_value(tab
);
436 sample
= isl_mat_vec_product(isl_mat_copy(tab
->basis
), sample
);
437 sample
= isl_vec_ceil(sample
);
438 sample
= isl_mat_vec_inverse_product(isl_mat_copy(tab
->basis
),
443 if (isl_tab_extend_cons(tab
, dim
+ 1) < 0)
446 min
= isl_vec_alloc(ctx
, dim
);
447 max
= isl_vec_alloc(ctx
, dim
);
448 snap
= isl_alloc_array(ctx
, struct isl_tab_undo
*, dim
);
450 if (!min
|| !max
|| !snap
)
460 res
= isl_tab_min(tab
, tab
->basis
->row
[1 + level
],
461 ctx
->one
, &min
->el
[level
], NULL
, 0);
462 if (res
== isl_lp_empty
)
464 isl_assert(ctx
, res
!= isl_lp_unbounded
, goto error
);
465 if (res
== isl_lp_error
)
467 if (!empty
&& isl_tab_sample_is_integer(tab
))
469 isl_seq_neg(tab
->basis
->row
[1 + level
] + 1,
470 tab
->basis
->row
[1 + level
] + 1, dim
);
471 res
= isl_tab_min(tab
, tab
->basis
->row
[1 + level
],
472 ctx
->one
, &max
->el
[level
], NULL
, 0);
473 isl_seq_neg(tab
->basis
->row
[1 + level
] + 1,
474 tab
->basis
->row
[1 + level
] + 1, dim
);
475 isl_int_neg(max
->el
[level
], max
->el
[level
]);
476 if (res
== isl_lp_empty
)
478 isl_assert(ctx
, res
!= isl_lp_unbounded
, goto error
);
479 if (res
== isl_lp_error
)
481 if (!empty
&& isl_tab_sample_is_integer(tab
))
483 if (!empty
&& !reduced
&&
484 ctx
->opt
->gbr
!= ISL_GBR_NEVER
&&
485 isl_int_lt(min
->el
[level
], max
->el
[level
])) {
486 unsigned gbr_only_first
;
487 if (ctx
->opt
->gbr
== ISL_GBR_ONCE
)
488 ctx
->opt
->gbr
= ISL_GBR_NEVER
;
490 gbr_only_first
= ctx
->opt
->gbr_only_first
;
491 ctx
->opt
->gbr_only_first
=
492 ctx
->opt
->gbr
== ISL_GBR_ALWAYS
;
493 tab
= isl_tab_compute_reduced_basis(tab
);
494 ctx
->opt
->gbr_only_first
= gbr_only_first
;
495 if (!tab
|| !tab
->basis
)
501 snap
[level
] = isl_tab_snap(tab
);
503 isl_int_add_ui(min
->el
[level
], min
->el
[level
], 1);
505 if (empty
|| isl_int_gt(min
->el
[level
], max
->el
[level
])) {
509 if (isl_tab_rollback(tab
, snap
[level
]) < 0)
513 isl_int_neg(tab
->basis
->row
[1 + level
][0], min
->el
[level
]);
514 if (isl_tab_add_valid_eq(tab
, tab
->basis
->row
[1 + level
]) < 0)
516 isl_int_set_si(tab
->basis
->row
[1 + level
][0], 0);
517 if (level
+ tab
->n_unbounded
< dim
- 1) {
526 sample
= isl_tab_get_sample_value(tab
);
529 if (tab
->n_unbounded
&& !isl_int_is_one(sample
->el
[0])) {
530 sample
= isl_mat_vec_product(isl_mat_copy(tab
->basis
),
532 sample
= isl_vec_ceil(sample
);
533 sample
= isl_mat_vec_inverse_product(
534 isl_mat_copy(tab
->basis
), sample
);
537 sample
= isl_vec_alloc(ctx
, 0);
552 static struct isl_vec
*sample_bounded(struct isl_basic_set
*bset
);
554 /* Compute a sample point of the given basic set, based on the given,
555 * non-trivial factorization.
557 static __isl_give isl_vec
*factored_sample(__isl_take isl_basic_set
*bset
,
558 __isl_take isl_factorizer
*f
)
561 isl_vec
*sample
= NULL
;
566 ctx
= isl_basic_set_get_ctx(bset
);
570 nparam
= isl_basic_set_dim(bset
, isl_dim_param
);
571 nvar
= isl_basic_set_dim(bset
, isl_dim_set
);
573 sample
= isl_vec_alloc(ctx
, 1 + isl_basic_set_total_dim(bset
));
576 isl_int_set_si(sample
->el
[0], 1);
578 bset
= isl_morph_basic_set(isl_morph_copy(f
->morph
), bset
);
580 for (i
= 0, n
= 0; i
< f
->n_group
; ++i
) {
581 isl_basic_set
*bset_i
;
584 bset_i
= isl_basic_set_copy(bset
);
585 bset_i
= isl_basic_set_drop_constraints_involving(bset_i
,
586 nparam
+ n
+ f
->len
[i
], nvar
- n
- f
->len
[i
]);
587 bset_i
= isl_basic_set_drop_constraints_involving(bset_i
,
589 bset_i
= isl_basic_set_drop(bset_i
, isl_dim_set
,
590 n
+ f
->len
[i
], nvar
- n
- f
->len
[i
]);
591 bset_i
= isl_basic_set_drop(bset_i
, isl_dim_set
, 0, n
);
593 sample_i
= sample_bounded(bset_i
);
596 if (sample_i
->size
== 0) {
597 isl_basic_set_free(bset
);
598 isl_factorizer_free(f
);
599 isl_vec_free(sample
);
602 isl_seq_cpy(sample
->el
+ 1 + nparam
+ n
,
603 sample_i
->el
+ 1, f
->len
[i
]);
604 isl_vec_free(sample_i
);
609 f
->morph
= isl_morph_inverse(f
->morph
);
610 sample
= isl_morph_vec(isl_morph_copy(f
->morph
), sample
);
612 isl_basic_set_free(bset
);
613 isl_factorizer_free(f
);
616 isl_basic_set_free(bset
);
617 isl_factorizer_free(f
);
618 isl_vec_free(sample
);
622 /* Given a basic set that is known to be bounded, find and return
623 * an integer point in the basic set, if there is any.
625 * After handling some trivial cases, we construct a tableau
626 * and then use isl_tab_sample to find a sample, passing it
627 * the identity matrix as initial basis.
629 static struct isl_vec
*sample_bounded(struct isl_basic_set
*bset
)
633 struct isl_vec
*sample
;
634 struct isl_tab
*tab
= NULL
;
640 if (isl_basic_set_plain_is_empty(bset
))
641 return empty_sample(bset
);
643 dim
= isl_basic_set_total_dim(bset
);
645 return zero_sample(bset
);
647 return interval_sample(bset
);
649 return sample_eq(bset
, sample_bounded
);
651 f
= isl_basic_set_factorizer(bset
);
655 return factored_sample(bset
, f
);
656 isl_factorizer_free(f
);
660 tab
= isl_tab_from_basic_set(bset
);
661 if (tab
&& tab
->empty
) {
663 ISL_F_SET(bset
, ISL_BASIC_SET_EMPTY
);
664 sample
= isl_vec_alloc(bset
->ctx
, 0);
665 isl_basic_set_free(bset
);
669 if (isl_tab_track_bset(tab
, isl_basic_set_copy(bset
)) < 0)
671 if (!ISL_F_ISSET(bset
, ISL_BASIC_SET_NO_IMPLICIT
))
672 if (isl_tab_detect_implicit_equalities(tab
) < 0)
675 sample
= isl_tab_sample(tab
);
679 if (sample
->size
> 0) {
680 isl_vec_free(bset
->sample
);
681 bset
->sample
= isl_vec_copy(sample
);
684 isl_basic_set_free(bset
);
688 isl_basic_set_free(bset
);
693 /* Given a basic set "bset" and a value "sample" for the first coordinates
694 * of bset, plug in these values and drop the corresponding coordinates.
696 * We do this by computing the preimage of the transformation
702 * where [1 s] is the sample value and I is the identity matrix of the
703 * appropriate dimension.
705 static struct isl_basic_set
*plug_in(struct isl_basic_set
*bset
,
706 struct isl_vec
*sample
)
712 if (!bset
|| !sample
)
715 total
= isl_basic_set_total_dim(bset
);
716 T
= isl_mat_alloc(bset
->ctx
, 1 + total
, 1 + total
- (sample
->size
- 1));
720 for (i
= 0; i
< sample
->size
; ++i
) {
721 isl_int_set(T
->row
[i
][0], sample
->el
[i
]);
722 isl_seq_clr(T
->row
[i
] + 1, T
->n_col
- 1);
724 for (i
= 0; i
< T
->n_col
- 1; ++i
) {
725 isl_seq_clr(T
->row
[sample
->size
+ i
], T
->n_col
);
726 isl_int_set_si(T
->row
[sample
->size
+ i
][1 + i
], 1);
728 isl_vec_free(sample
);
730 bset
= isl_basic_set_preimage(bset
, T
);
733 isl_basic_set_free(bset
);
734 isl_vec_free(sample
);
738 /* Given a basic set "bset", return any (possibly non-integer) point
741 static struct isl_vec
*rational_sample(struct isl_basic_set
*bset
)
744 struct isl_vec
*sample
;
749 tab
= isl_tab_from_basic_set(bset
);
750 sample
= isl_tab_get_sample_value(tab
);
753 isl_basic_set_free(bset
);
758 /* Given a linear cone "cone" and a rational point "vec",
759 * construct a polyhedron with shifted copies of the constraints in "cone",
760 * i.e., a polyhedron with "cone" as its recession cone, such that each
761 * point x in this polyhedron is such that the unit box positioned at x
762 * lies entirely inside the affine cone 'vec + cone'.
763 * Any rational point in this polyhedron may therefore be rounded up
764 * to yield an integer point that lies inside said affine cone.
766 * Denote the constraints of cone by "<a_i, x> >= 0" and the rational
767 * point "vec" by v/d.
768 * Let b_i = <a_i, v>. Then the affine cone 'vec + cone' is given
769 * by <a_i, x> - b/d >= 0.
770 * The polyhedron <a_i, x> - ceil{b/d} >= 0 is a subset of this affine cone.
771 * We prefer this polyhedron over the actual affine cone because it doesn't
772 * require a scaling of the constraints.
773 * If each of the vertices of the unit cube positioned at x lies inside
774 * this polyhedron, then the whole unit cube at x lies inside the affine cone.
775 * We therefore impose that x' = x + \sum e_i, for any selection of unit
776 * vectors lies inside the polyhedron, i.e.,
778 * <a_i, x'> - ceil{b/d} = <a_i, x> + sum a_i - ceil{b/d} >= 0
780 * The most stringent of these constraints is the one that selects
781 * all negative a_i, so the polyhedron we are looking for has constraints
783 * <a_i, x> + sum_{a_i < 0} a_i - ceil{b/d} >= 0
785 * Note that if cone were known to have only non-negative rays
786 * (which can be accomplished by a unimodular transformation),
787 * then we would only have to check the points x' = x + e_i
788 * and we only have to add the smallest negative a_i (if any)
789 * instead of the sum of all negative a_i.
791 static struct isl_basic_set
*shift_cone(struct isl_basic_set
*cone
,
797 struct isl_basic_set
*shift
= NULL
;
802 isl_assert(cone
->ctx
, cone
->n_eq
== 0, goto error
);
804 total
= isl_basic_set_total_dim(cone
);
806 shift
= isl_basic_set_alloc_space(isl_basic_set_get_space(cone
),
809 for (i
= 0; i
< cone
->n_ineq
; ++i
) {
810 k
= isl_basic_set_alloc_inequality(shift
);
813 isl_seq_cpy(shift
->ineq
[k
] + 1, cone
->ineq
[i
] + 1, total
);
814 isl_seq_inner_product(shift
->ineq
[k
] + 1, vec
->el
+ 1, total
,
816 isl_int_cdiv_q(shift
->ineq
[k
][0],
817 shift
->ineq
[k
][0], vec
->el
[0]);
818 isl_int_neg(shift
->ineq
[k
][0], shift
->ineq
[k
][0]);
819 for (j
= 0; j
< total
; ++j
) {
820 if (isl_int_is_nonneg(shift
->ineq
[k
][1 + j
]))
822 isl_int_add(shift
->ineq
[k
][0],
823 shift
->ineq
[k
][0], shift
->ineq
[k
][1 + j
]);
827 isl_basic_set_free(cone
);
830 return isl_basic_set_finalize(shift
);
832 isl_basic_set_free(shift
);
833 isl_basic_set_free(cone
);
838 /* Given a rational point vec in a (transformed) basic set,
839 * such that cone is the recession cone of the original basic set,
840 * "round up" the rational point to an integer point.
842 * We first check if the rational point just happens to be integer.
843 * If not, we transform the cone in the same way as the basic set,
844 * pick a point x in this cone shifted to the rational point such that
845 * the whole unit cube at x is also inside this affine cone.
846 * Then we simply round up the coordinates of x and return the
847 * resulting integer point.
849 static struct isl_vec
*round_up_in_cone(struct isl_vec
*vec
,
850 struct isl_basic_set
*cone
, struct isl_mat
*U
)
854 if (!vec
|| !cone
|| !U
)
857 isl_assert(vec
->ctx
, vec
->size
!= 0, goto error
);
858 if (isl_int_is_one(vec
->el
[0])) {
860 isl_basic_set_free(cone
);
864 total
= isl_basic_set_total_dim(cone
);
865 cone
= isl_basic_set_preimage(cone
, U
);
866 cone
= isl_basic_set_remove_dims(cone
, isl_dim_set
,
867 0, total
- (vec
->size
- 1));
869 cone
= shift_cone(cone
, vec
);
871 vec
= rational_sample(cone
);
872 vec
= isl_vec_ceil(vec
);
877 isl_basic_set_free(cone
);
881 /* Concatenate two integer vectors, i.e., two vectors with denominator
882 * (stored in element 0) equal to 1.
884 static struct isl_vec
*vec_concat(struct isl_vec
*vec1
, struct isl_vec
*vec2
)
890 isl_assert(vec1
->ctx
, vec1
->size
> 0, goto error
);
891 isl_assert(vec2
->ctx
, vec2
->size
> 0, goto error
);
892 isl_assert(vec1
->ctx
, isl_int_is_one(vec1
->el
[0]), goto error
);
893 isl_assert(vec2
->ctx
, isl_int_is_one(vec2
->el
[0]), goto error
);
895 vec
= isl_vec_alloc(vec1
->ctx
, vec1
->size
+ vec2
->size
- 1);
899 isl_seq_cpy(vec
->el
, vec1
->el
, vec1
->size
);
900 isl_seq_cpy(vec
->el
+ vec1
->size
, vec2
->el
+ 1, vec2
->size
- 1);
912 /* Give a basic set "bset" with recession cone "cone", compute and
913 * return an integer point in bset, if any.
915 * If the recession cone is full-dimensional, then we know that
916 * bset contains an infinite number of integer points and it is
917 * fairly easy to pick one of them.
918 * If the recession cone is not full-dimensional, then we first
919 * transform bset such that the bounded directions appear as
920 * the first dimensions of the transformed basic set.
921 * We do this by using a unimodular transformation that transforms
922 * the equalities in the recession cone to equalities on the first
925 * The transformed set is then projected onto its bounded dimensions.
926 * Note that to compute this projection, we can simply drop all constraints
927 * involving any of the unbounded dimensions since these constraints
928 * cannot be combined to produce a constraint on the bounded dimensions.
929 * To see this, assume that there is such a combination of constraints
930 * that produces a constraint on the bounded dimensions. This means
931 * that some combination of the unbounded dimensions has both an upper
932 * bound and a lower bound in terms of the bounded dimensions, but then
933 * this combination would be a bounded direction too and would have been
934 * transformed into a bounded dimensions.
936 * We then compute a sample value in the bounded dimensions.
937 * If no such value can be found, then the original set did not contain
938 * any integer points and we are done.
939 * Otherwise, we plug in the value we found in the bounded dimensions,
940 * project out these bounded dimensions and end up with a set with
941 * a full-dimensional recession cone.
942 * A sample point in this set is computed by "rounding up" any
943 * rational point in the set.
945 * The sample points in the bounded and unbounded dimensions are
946 * then combined into a single sample point and transformed back
947 * to the original space.
949 __isl_give isl_vec
*isl_basic_set_sample_with_cone(
950 __isl_take isl_basic_set
*bset
, __isl_take isl_basic_set
*cone
)
954 struct isl_mat
*M
, *U
;
955 struct isl_vec
*sample
;
956 struct isl_vec
*cone_sample
;
958 struct isl_basic_set
*bounded
;
964 total
= isl_basic_set_total_dim(cone
);
965 cone_dim
= total
- cone
->n_eq
;
967 M
= isl_mat_sub_alloc6(bset
->ctx
, cone
->eq
, 0, cone
->n_eq
, 1, total
);
968 M
= isl_mat_left_hermite(M
, 0, &U
, NULL
);
973 U
= isl_mat_lin_to_aff(U
);
974 bset
= isl_basic_set_preimage(bset
, isl_mat_copy(U
));
976 bounded
= isl_basic_set_copy(bset
);
977 bounded
= isl_basic_set_drop_constraints_involving(bounded
,
978 total
- cone_dim
, cone_dim
);
979 bounded
= isl_basic_set_drop_dims(bounded
, total
- cone_dim
, cone_dim
);
980 sample
= sample_bounded(bounded
);
981 if (!sample
|| sample
->size
== 0) {
982 isl_basic_set_free(bset
);
983 isl_basic_set_free(cone
);
987 bset
= plug_in(bset
, isl_vec_copy(sample
));
988 cone_sample
= rational_sample(bset
);
989 cone_sample
= round_up_in_cone(cone_sample
, cone
, isl_mat_copy(U
));
990 sample
= vec_concat(sample
, cone_sample
);
991 sample
= isl_mat_vec_product(U
, sample
);
994 isl_basic_set_free(cone
);
995 isl_basic_set_free(bset
);
999 static void vec_sum_of_neg(struct isl_vec
*v
, isl_int
*s
)
1003 isl_int_set_si(*s
, 0);
1005 for (i
= 0; i
< v
->size
; ++i
)
1006 if (isl_int_is_neg(v
->el
[i
]))
1007 isl_int_add(*s
, *s
, v
->el
[i
]);
1010 /* Given a tableau "tab", a tableau "tab_cone" that corresponds
1011 * to the recession cone and the inverse of a new basis U = inv(B),
1012 * with the unbounded directions in B last,
1013 * add constraints to "tab" that ensure any rational value
1014 * in the unbounded directions can be rounded up to an integer value.
1016 * The new basis is given by x' = B x, i.e., x = U x'.
1017 * For any rational value of the last tab->n_unbounded coordinates
1018 * in the update tableau, the value that is obtained by rounding
1019 * up this value should be contained in the original tableau.
1020 * For any constraint "a x + c >= 0", we therefore need to add
1021 * a constraint "a x + c + s >= 0", with s the sum of all negative
1022 * entries in the last elements of "a U".
1024 * Since we are not interested in the first entries of any of the "a U",
1025 * we first drop the columns of U that correpond to bounded directions.
1027 static int tab_shift_cone(struct isl_tab
*tab
,
1028 struct isl_tab
*tab_cone
, struct isl_mat
*U
)
1032 struct isl_basic_set
*bset
= NULL
;
1034 if (tab
&& tab
->n_unbounded
== 0) {
1039 if (!tab
|| !tab_cone
|| !U
)
1041 bset
= isl_tab_peek_bset(tab_cone
);
1042 U
= isl_mat_drop_cols(U
, 0, tab
->n_var
- tab
->n_unbounded
);
1043 for (i
= 0; i
< bset
->n_ineq
; ++i
) {
1045 struct isl_vec
*row
= NULL
;
1046 if (isl_tab_is_equality(tab_cone
, tab_cone
->n_eq
+ i
))
1048 row
= isl_vec_alloc(bset
->ctx
, tab_cone
->n_var
);
1051 isl_seq_cpy(row
->el
, bset
->ineq
[i
] + 1, tab_cone
->n_var
);
1052 row
= isl_vec_mat_product(row
, isl_mat_copy(U
));
1055 vec_sum_of_neg(row
, &v
);
1057 if (isl_int_is_zero(v
))
1059 tab
= isl_tab_extend(tab
, 1);
1060 isl_int_add(bset
->ineq
[i
][0], bset
->ineq
[i
][0], v
);
1061 ok
= isl_tab_add_ineq(tab
, bset
->ineq
[i
]) >= 0;
1062 isl_int_sub(bset
->ineq
[i
][0], bset
->ineq
[i
][0], v
);
1076 /* Compute and return an initial basis for the possibly
1077 * unbounded tableau "tab". "tab_cone" is a tableau
1078 * for the corresponding recession cone.
1079 * Additionally, add constraints to "tab" that ensure
1080 * that any rational value for the unbounded directions
1081 * can be rounded up to an integer value.
1083 * If the tableau is bounded, i.e., if the recession cone
1084 * is zero-dimensional, then we just use inital_basis.
1085 * Otherwise, we construct a basis whose first directions
1086 * correspond to equalities, followed by bounded directions,
1087 * i.e., equalities in the recession cone.
1088 * The remaining directions are then unbounded.
1090 int isl_tab_set_initial_basis_with_cone(struct isl_tab
*tab
,
1091 struct isl_tab
*tab_cone
)
1094 struct isl_mat
*cone_eq
;
1095 struct isl_mat
*U
, *Q
;
1097 if (!tab
|| !tab_cone
)
1100 if (tab_cone
->n_col
== tab_cone
->n_dead
) {
1101 tab
->basis
= initial_basis(tab
);
1102 return tab
->basis
? 0 : -1;
1105 eq
= tab_equalities(tab
);
1108 tab
->n_zero
= eq
->n_row
;
1109 cone_eq
= tab_equalities(tab_cone
);
1110 eq
= isl_mat_concat(eq
, cone_eq
);
1113 tab
->n_unbounded
= tab
->n_var
- (eq
->n_row
- tab
->n_zero
);
1114 eq
= isl_mat_left_hermite(eq
, 0, &U
, &Q
);
1118 tab
->basis
= isl_mat_lin_to_aff(Q
);
1119 if (tab_shift_cone(tab
, tab_cone
, U
) < 0)
1126 /* Compute and return a sample point in bset using generalized basis
1127 * reduction. We first check if the input set has a non-trivial
1128 * recession cone. If so, we perform some extra preprocessing in
1129 * sample_with_cone. Otherwise, we directly perform generalized basis
1132 static struct isl_vec
*gbr_sample(struct isl_basic_set
*bset
)
1135 struct isl_basic_set
*cone
;
1137 dim
= isl_basic_set_total_dim(bset
);
1139 cone
= isl_basic_set_recession_cone(isl_basic_set_copy(bset
));
1143 if (cone
->n_eq
< dim
)
1144 return isl_basic_set_sample_with_cone(bset
, cone
);
1146 isl_basic_set_free(cone
);
1147 return sample_bounded(bset
);
1149 isl_basic_set_free(bset
);
1153 static struct isl_vec
*pip_sample(struct isl_basic_set
*bset
)
1156 struct isl_ctx
*ctx
;
1157 struct isl_vec
*sample
;
1159 bset
= isl_basic_set_skew_to_positive_orthant(bset
, &T
);
1164 sample
= isl_pip_basic_set_sample(bset
);
1166 if (sample
&& sample
->size
!= 0)
1167 sample
= isl_mat_vec_product(T
, sample
);
1174 static struct isl_vec
*basic_set_sample(struct isl_basic_set
*bset
, int bounded
)
1176 struct isl_ctx
*ctx
;
1182 if (isl_basic_set_plain_is_empty(bset
))
1183 return empty_sample(bset
);
1185 dim
= isl_basic_set_n_dim(bset
);
1186 isl_assert(ctx
, isl_basic_set_n_param(bset
) == 0, goto error
);
1187 isl_assert(ctx
, bset
->n_div
== 0, goto error
);
1189 if (bset
->sample
&& bset
->sample
->size
== 1 + dim
) {
1190 int contains
= isl_basic_set_contains(bset
, bset
->sample
);
1194 struct isl_vec
*sample
= isl_vec_copy(bset
->sample
);
1195 isl_basic_set_free(bset
);
1199 isl_vec_free(bset
->sample
);
1200 bset
->sample
= NULL
;
1203 return sample_eq(bset
, bounded
? isl_basic_set_sample_bounded
1204 : isl_basic_set_sample_vec
);
1206 return zero_sample(bset
);
1208 return interval_sample(bset
);
1210 switch (bset
->ctx
->opt
->ilp_solver
) {
1212 return pip_sample(bset
);
1214 return bounded
? sample_bounded(bset
) : gbr_sample(bset
);
1216 isl_assert(bset
->ctx
, 0, );
1218 isl_basic_set_free(bset
);
1222 __isl_give isl_vec
*isl_basic_set_sample_vec(__isl_take isl_basic_set
*bset
)
1224 return basic_set_sample(bset
, 0);
1227 /* Compute an integer sample in "bset", where the caller guarantees
1228 * that "bset" is bounded.
1230 struct isl_vec
*isl_basic_set_sample_bounded(struct isl_basic_set
*bset
)
1232 return basic_set_sample(bset
, 1);
1235 __isl_give isl_basic_set
*isl_basic_set_from_vec(__isl_take isl_vec
*vec
)
1239 struct isl_basic_set
*bset
= NULL
;
1240 struct isl_ctx
*ctx
;
1246 isl_assert(ctx
, vec
->size
!= 0, goto error
);
1248 bset
= isl_basic_set_alloc(ctx
, 0, vec
->size
- 1, 0, vec
->size
- 1, 0);
1251 dim
= isl_basic_set_n_dim(bset
);
1252 for (i
= dim
- 1; i
>= 0; --i
) {
1253 k
= isl_basic_set_alloc_equality(bset
);
1256 isl_seq_clr(bset
->eq
[k
], 1 + dim
);
1257 isl_int_neg(bset
->eq
[k
][0], vec
->el
[1 + i
]);
1258 isl_int_set(bset
->eq
[k
][1 + i
], vec
->el
[0]);
1264 isl_basic_set_free(bset
);
1269 __isl_give isl_basic_map
*isl_basic_map_sample(__isl_take isl_basic_map
*bmap
)
1271 struct isl_basic_set
*bset
;
1272 struct isl_vec
*sample_vec
;
1274 bset
= isl_basic_map_underlying_set(isl_basic_map_copy(bmap
));
1275 sample_vec
= isl_basic_set_sample_vec(bset
);
1278 if (sample_vec
->size
== 0) {
1279 struct isl_basic_map
*sample
;
1280 sample
= isl_basic_map_empty_like(bmap
);
1281 isl_vec_free(sample_vec
);
1282 isl_basic_map_free(bmap
);
1285 bset
= isl_basic_set_from_vec(sample_vec
);
1286 return isl_basic_map_overlying_set(bset
, bmap
);
1288 isl_basic_map_free(bmap
);
1292 __isl_give isl_basic_map
*isl_map_sample(__isl_take isl_map
*map
)
1295 isl_basic_map
*sample
= NULL
;
1300 for (i
= 0; i
< map
->n
; ++i
) {
1301 sample
= isl_basic_map_sample(isl_basic_map_copy(map
->p
[i
]));
1304 if (!ISL_F_ISSET(sample
, ISL_BASIC_MAP_EMPTY
))
1306 isl_basic_map_free(sample
);
1309 sample
= isl_basic_map_empty_like_map(map
);
1317 __isl_give isl_basic_set
*isl_set_sample(__isl_take isl_set
*set
)
1319 return (isl_basic_set
*) isl_map_sample((isl_map
*)set
);
1322 __isl_give isl_point
*isl_basic_set_sample_point(__isl_take isl_basic_set
*bset
)
1327 dim
= isl_basic_set_get_space(bset
);
1328 bset
= isl_basic_set_underlying_set(bset
);
1329 vec
= isl_basic_set_sample_vec(bset
);
1331 return isl_point_alloc(dim
, vec
);
1334 __isl_give isl_point
*isl_set_sample_point(__isl_take isl_set
*set
)
1342 for (i
= 0; i
< set
->n
; ++i
) {
1343 pnt
= isl_basic_set_sample_point(isl_basic_set_copy(set
->p
[i
]));
1346 if (!isl_point_is_void(pnt
))
1348 isl_point_free(pnt
);
1351 pnt
= isl_point_void(isl_set_get_space(set
));