1 #include "isl_map_private.h"
5 * The implementation of tableaus in this file was inspired by Section 8
6 * of David Detlefs, Greg Nelson and James B. Saxe, "Simplify: a theorem
7 * prover for program checking".
10 struct isl_tab
*isl_tab_alloc(struct isl_ctx
*ctx
,
11 unsigned n_row
, unsigned n_var
)
16 tab
= isl_calloc_type(ctx
, struct isl_tab
);
19 tab
->mat
= isl_mat_alloc(ctx
, n_row
, 2 + n_var
);
22 tab
->var
= isl_alloc_array(ctx
, struct isl_tab_var
, n_var
);
25 tab
->con
= isl_alloc_array(ctx
, struct isl_tab_var
, n_row
);
28 tab
->col_var
= isl_alloc_array(ctx
, int, n_var
);
31 tab
->row_var
= isl_alloc_array(ctx
, int, n_row
);
34 for (i
= 0; i
< n_var
; ++i
) {
35 tab
->var
[i
].index
= i
;
36 tab
->var
[i
].is_row
= 0;
37 tab
->var
[i
].is_nonneg
= 0;
38 tab
->var
[i
].is_zero
= 0;
39 tab
->var
[i
].is_redundant
= 0;
40 tab
->var
[i
].frozen
= 0;
55 tab
->bottom
.type
= isl_tab_undo_bottom
;
56 tab
->bottom
.next
= NULL
;
57 tab
->top
= &tab
->bottom
;
60 isl_tab_free(ctx
, tab
);
64 static int extend_cons(struct isl_ctx
*ctx
, struct isl_tab
*tab
, unsigned n_new
)
66 if (tab
->max_con
< tab
->n_con
+ n_new
) {
67 struct isl_tab_var
*con
;
69 con
= isl_realloc_array(ctx
, tab
->con
,
70 struct isl_tab_var
, tab
->max_con
+ n_new
);
74 tab
->max_con
+= n_new
;
76 if (tab
->mat
->n_row
< tab
->n_row
+ n_new
) {
79 tab
->mat
= isl_mat_extend(ctx
, tab
->mat
,
80 tab
->n_row
+ n_new
, tab
->n_col
);
83 row_var
= isl_realloc_array(ctx
, tab
->row_var
,
84 int, tab
->mat
->n_row
);
87 tab
->row_var
= row_var
;
92 struct isl_tab
*isl_tab_extend(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
95 if (extend_cons(ctx
, tab
, n_new
) >= 0)
98 isl_tab_free(ctx
, tab
);
102 static void free_undo(struct isl_ctx
*ctx
, struct isl_tab
*tab
)
104 struct isl_tab_undo
*undo
, *next
;
106 for (undo
= tab
->top
; undo
&& undo
!= &tab
->bottom
; undo
= next
) {
113 void isl_tab_free(struct isl_ctx
*ctx
, struct isl_tab
*tab
)
118 isl_mat_free(ctx
, tab
->mat
);
119 isl_vec_free(tab
->dual
);
127 static struct isl_tab_var
*var_from_index(struct isl_ctx
*ctx
,
128 struct isl_tab
*tab
, int i
)
133 return &tab
->con
[~i
];
136 static struct isl_tab_var
*var_from_row(struct isl_ctx
*ctx
,
137 struct isl_tab
*tab
, int i
)
139 return var_from_index(ctx
, tab
, tab
->row_var
[i
]);
142 static struct isl_tab_var
*var_from_col(struct isl_ctx
*ctx
,
143 struct isl_tab
*tab
, int i
)
145 return var_from_index(ctx
, tab
, tab
->col_var
[i
]);
148 /* Check if there are any upper bounds on column variable "var",
149 * i.e., non-negative rows where var appears with a negative coefficient.
150 * Return 1 if there are no such bounds.
152 static int max_is_manifestly_unbounded(struct isl_ctx
*ctx
,
153 struct isl_tab
*tab
, struct isl_tab_var
*var
)
159 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
160 if (!isl_int_is_neg(tab
->mat
->row
[i
][2 + var
->index
]))
162 if (var_from_row(ctx
, tab
, i
)->is_nonneg
)
168 /* Check if there are any lower bounds on column variable "var",
169 * i.e., non-negative rows where var appears with a positive coefficient.
170 * Return 1 if there are no such bounds.
172 static int min_is_manifestly_unbounded(struct isl_ctx
*ctx
,
173 struct isl_tab
*tab
, struct isl_tab_var
*var
)
179 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
180 if (!isl_int_is_pos(tab
->mat
->row
[i
][2 + var
->index
]))
182 if (var_from_row(ctx
, tab
, i
)->is_nonneg
)
188 /* Given the index of a column "c", return the index of a row
189 * that can be used to pivot the column in, with either an increase
190 * (sgn > 0) or a decrease (sgn < 0) of the corresponding variable.
191 * If "var" is not NULL, then the row returned will be different from
192 * the one associated with "var".
194 * Each row in the tableau is of the form
196 * x_r = a_r0 + \sum_i a_ri x_i
198 * Only rows with x_r >= 0 and with the sign of a_ri opposite to "sgn"
199 * impose any limit on the increase or decrease in the value of x_c
200 * and this bound is equal to a_r0 / |a_rc|. We are therefore looking
201 * for the row with the smallest (most stringent) such bound.
202 * Note that the common denominator of each row drops out of the fraction.
203 * To check if row j has a smaller bound than row r, i.e.,
204 * a_j0 / |a_jc| < a_r0 / |a_rc| or a_j0 |a_rc| < a_r0 |a_jc|,
205 * we check if -sign(a_jc) (a_j0 a_rc - a_r0 a_jc) < 0,
206 * where -sign(a_jc) is equal to "sgn".
208 static int pivot_row(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
209 struct isl_tab_var
*var
, int sgn
, int c
)
216 for (j
= tab
->n_redundant
; j
< tab
->n_row
; ++j
) {
217 if (var
&& j
== var
->index
)
219 if (!var_from_row(ctx
, tab
, j
)->is_nonneg
)
221 if (sgn
* isl_int_sgn(tab
->mat
->row
[j
][2 + c
]) >= 0)
227 isl_int_mul(t
, tab
->mat
->row
[r
][1], tab
->mat
->row
[j
][2 + c
]);
228 isl_int_submul(t
, tab
->mat
->row
[j
][1], tab
->mat
->row
[r
][2 + c
]);
229 tsgn
= sgn
* isl_int_sgn(t
);
230 if (tsgn
< 0 || (tsgn
== 0 &&
231 tab
->row_var
[j
] < tab
->row_var
[r
]))
238 /* Find a pivot (row and col) that will increase (sgn > 0) or decrease
239 * (sgn < 0) the value of row variable var.
240 * If not NULL, then skip_var is a row variable that should be ignored
241 * while looking for a pivot row. It is usually equal to var.
243 * As the given row in the tableau is of the form
245 * x_r = a_r0 + \sum_i a_ri x_i
247 * we need to find a column such that the sign of a_ri is equal to "sgn"
248 * (such that an increase in x_i will have the desired effect) or a
249 * column with a variable that may attain negative values.
250 * If a_ri is positive, then we need to move x_i in the same direction
251 * to obtain the desired effect. Otherwise, x_i has to move in the
252 * opposite direction.
254 static void find_pivot(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
255 struct isl_tab_var
*var
, struct isl_tab_var
*skip_var
,
256 int sgn
, int *row
, int *col
)
263 isl_assert(ctx
, var
->is_row
, return);
264 tr
= tab
->mat
->row
[var
->index
];
267 for (j
= tab
->n_dead
; j
< tab
->n_col
; ++j
) {
268 if (isl_int_is_zero(tr
[2 + j
]))
270 if (isl_int_sgn(tr
[2 + j
]) != sgn
&&
271 var_from_col(ctx
, tab
, j
)->is_nonneg
)
273 if (c
< 0 || tab
->col_var
[j
] < tab
->col_var
[c
])
279 sgn
*= isl_int_sgn(tr
[2 + c
]);
280 r
= pivot_row(ctx
, tab
, skip_var
, sgn
, c
);
281 *row
= r
< 0 ? var
->index
: r
;
285 /* Return 1 if row "row" represents an obviously redundant inequality.
287 * - it represents an inequality or a variable
288 * - that is the sum of a non-negative sample value and a positive
289 * combination of zero or more non-negative variables.
291 static int is_redundant(struct isl_ctx
*ctx
, struct isl_tab
*tab
, int row
)
295 if (tab
->row_var
[row
] < 0 && !var_from_row(ctx
, tab
, row
)->is_nonneg
)
298 if (isl_int_is_neg(tab
->mat
->row
[row
][1]))
301 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
302 if (isl_int_is_zero(tab
->mat
->row
[row
][2 + i
]))
304 if (isl_int_is_neg(tab
->mat
->row
[row
][2 + i
]))
306 if (!var_from_col(ctx
, tab
, i
)->is_nonneg
)
312 static void swap_rows(struct isl_ctx
*ctx
,
313 struct isl_tab
*tab
, int row1
, int row2
)
316 t
= tab
->row_var
[row1
];
317 tab
->row_var
[row1
] = tab
->row_var
[row2
];
318 tab
->row_var
[row2
] = t
;
319 var_from_row(ctx
, tab
, row1
)->index
= row1
;
320 var_from_row(ctx
, tab
, row2
)->index
= row2
;
321 tab
->mat
= isl_mat_swap_rows(ctx
, tab
->mat
, row1
, row2
);
324 static void push(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
325 enum isl_tab_undo_type type
, struct isl_tab_var
*var
)
327 struct isl_tab_undo
*undo
;
332 undo
= isl_alloc_type(ctx
, struct isl_tab_undo
);
340 undo
->next
= tab
->top
;
344 /* Mark row with index "row" as being redundant.
345 * If we may need to undo the operation or if the row represents
346 * a variable of the original problem, the row is kept,
347 * but no longer considered when looking for a pivot row.
348 * Otherwise, the row is simply removed.
350 * The row may be interchanged with some other row. If it
351 * is interchanged with a later row, return 1. Otherwise return 0.
352 * If the rows are checked in order in the calling function,
353 * then a return value of 1 means that the row with the given
354 * row number may now contain a different row that hasn't been checked yet.
356 static int mark_redundant(struct isl_ctx
*ctx
,
357 struct isl_tab
*tab
, int row
)
359 struct isl_tab_var
*var
= var_from_row(ctx
, tab
, row
);
360 var
->is_redundant
= 1;
361 isl_assert(ctx
, row
>= tab
->n_redundant
, return);
362 if (tab
->need_undo
|| tab
->row_var
[row
] >= 0) {
363 if (tab
->row_var
[row
] >= 0) {
365 push(ctx
, tab
, isl_tab_undo_nonneg
, var
);
367 if (row
!= tab
->n_redundant
)
368 swap_rows(ctx
, tab
, row
, tab
->n_redundant
);
369 push(ctx
, tab
, isl_tab_undo_redundant
, var
);
373 if (row
!= tab
->n_row
- 1)
374 swap_rows(ctx
, tab
, row
, tab
->n_row
- 1);
375 var_from_row(ctx
, tab
, tab
->n_row
- 1)->index
= -1;
381 static void mark_empty(struct isl_ctx
*ctx
, struct isl_tab
*tab
)
383 if (!tab
->empty
&& tab
->need_undo
)
384 push(ctx
, tab
, isl_tab_undo_empty
, NULL
);
388 /* Given a row number "row" and a column number "col", pivot the tableau
389 * such that the associated variable are interchanged.
390 * The given row in the tableau expresses
392 * x_r = a_r0 + \sum_i a_ri x_i
396 * x_c = 1/a_rc x_r - a_r0/a_rc + sum_{i \ne r} -a_ri/a_rc
398 * Substituting this equality into the other rows
400 * x_j = a_j0 + \sum_i a_ji x_i
402 * with a_jc \ne 0, we obtain
404 * x_j = a_jc/a_rc x_r + a_j0 - a_jc a_r0/a_rc + sum a_ji - a_jc a_ri/a_rc
411 * where i is any other column and j is any other row,
412 * is therefore transformed into
414 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
415 * s(n_rc)d_r n_jc/(|n_rc| d_j) (n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j)
417 * The transformation is performed along the following steps
422 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
425 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
426 * n_jc/(|n_rc| d_j) n_ji/(|n_rc| d_j)
428 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
429 * n_jc/(|n_rc| d_j) (n_ji |n_rc|)/(|n_rc| d_j)
431 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
432 * n_jc/(|n_rc| d_j) (n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j)
434 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
435 * s(n_rc)d_r n_jc/(|n_rc| d_j) (n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j)
438 static void pivot(struct isl_ctx
*ctx
,
439 struct isl_tab
*tab
, int row
, int col
)
444 struct isl_mat
*mat
= tab
->mat
;
445 struct isl_tab_var
*var
;
447 isl_int_swap(mat
->row
[row
][0], mat
->row
[row
][2 + col
]);
448 sgn
= isl_int_sgn(mat
->row
[row
][0]);
450 isl_int_neg(mat
->row
[row
][0], mat
->row
[row
][0]);
451 isl_int_neg(mat
->row
[row
][2 + col
], mat
->row
[row
][2 + col
]);
453 for (j
= 0; j
< 1 + tab
->n_col
; ++j
) {
456 isl_int_neg(mat
->row
[row
][1 + j
], mat
->row
[row
][1 + j
]);
458 if (!isl_int_is_one(mat
->row
[row
][0]))
459 isl_seq_normalize(mat
->row
[row
], 2 + tab
->n_col
);
460 for (i
= 0; i
< tab
->n_row
; ++i
) {
463 if (isl_int_is_zero(mat
->row
[i
][2 + col
]))
465 isl_int_mul(mat
->row
[i
][0], mat
->row
[i
][0], mat
->row
[row
][0]);
466 for (j
= 0; j
< 1 + tab
->n_col
; ++j
) {
469 isl_int_mul(mat
->row
[i
][1 + j
],
470 mat
->row
[i
][1 + j
], mat
->row
[row
][0]);
471 isl_int_addmul(mat
->row
[i
][1 + j
],
472 mat
->row
[i
][2 + col
], mat
->row
[row
][1 + j
]);
474 isl_int_mul(mat
->row
[i
][2 + col
],
475 mat
->row
[i
][2 + col
], mat
->row
[row
][2 + col
]);
476 if (!isl_int_is_one(mat
->row
[row
][0]))
477 isl_seq_normalize(mat
->row
[i
], 2 + tab
->n_col
);
479 t
= tab
->row_var
[row
];
480 tab
->row_var
[row
] = tab
->col_var
[col
];
481 tab
->col_var
[col
] = t
;
482 var
= var_from_row(ctx
, tab
, row
);
485 var
= var_from_col(ctx
, tab
, col
);
490 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
491 if (isl_int_is_zero(mat
->row
[i
][2 + col
]))
493 if (!var_from_row(ctx
, tab
, i
)->frozen
&&
494 is_redundant(ctx
, tab
, i
))
495 if (mark_redundant(ctx
, tab
, i
))
500 /* If "var" represents a column variable, then pivot is up (sgn > 0)
501 * or down (sgn < 0) to a row. The variable is assumed not to be
502 * unbounded in the specified direction.
504 static void to_row(struct isl_ctx
*ctx
,
505 struct isl_tab
*tab
, struct isl_tab_var
*var
, int sign
)
512 r
= pivot_row(ctx
, tab
, NULL
, sign
, var
->index
);
513 isl_assert(ctx
, r
>= 0, return);
514 pivot(ctx
, tab
, r
, var
->index
);
517 static void check_table(struct isl_ctx
*ctx
, struct isl_tab
*tab
)
523 for (i
= 0; i
< tab
->n_row
; ++i
) {
524 if (!var_from_row(ctx
, tab
, i
)->is_nonneg
)
526 assert(!isl_int_is_neg(tab
->mat
->row
[i
][1]));
530 /* Return the sign of the maximal value of "var".
531 * If the sign is not negative, then on return from this function,
532 * the sample value will also be non-negative.
534 * If "var" is manifestly unbounded wrt positive values, we are done.
535 * Otherwise, we pivot the variable up to a row if needed
536 * Then we continue pivoting down until either
537 * - no more down pivots can be performed
538 * - the sample value is positive
539 * - the variable is pivoted into a manifestly unbounded column
541 static int sign_of_max(struct isl_ctx
*ctx
,
542 struct isl_tab
*tab
, struct isl_tab_var
*var
)
546 if (max_is_manifestly_unbounded(ctx
, tab
, var
))
548 to_row(ctx
, tab
, var
, 1);
549 while (!isl_int_is_pos(tab
->mat
->row
[var
->index
][1])) {
550 find_pivot(ctx
, tab
, var
, var
, 1, &row
, &col
);
552 return isl_int_sgn(tab
->mat
->row
[var
->index
][1]);
553 pivot(ctx
, tab
, row
, col
);
554 if (!var
->is_row
) /* manifestly unbounded */
560 /* Perform pivots until the row variable "var" has a non-negative
561 * sample value or until no more upward pivots can be performed.
562 * Return the sign of the sample value after the pivots have been
565 static int restore_row(struct isl_ctx
*ctx
,
566 struct isl_tab
*tab
, struct isl_tab_var
*var
)
570 while (isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
571 find_pivot(ctx
, tab
, var
, var
, 1, &row
, &col
);
574 pivot(ctx
, tab
, row
, col
);
575 if (!var
->is_row
) /* manifestly unbounded */
578 return isl_int_sgn(tab
->mat
->row
[var
->index
][1]);
581 /* Perform pivots until we are sure that the row variable "var"
582 * can attain non-negative values. After return from this
583 * function, "var" is still a row variable, but its sample
584 * value may not be non-negative, even if the function returns 1.
586 static int at_least_zero(struct isl_ctx
*ctx
,
587 struct isl_tab
*tab
, struct isl_tab_var
*var
)
591 while (isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
592 find_pivot(ctx
, tab
, var
, var
, 1, &row
, &col
);
595 if (row
== var
->index
) /* manifestly unbounded */
597 pivot(ctx
, tab
, row
, col
);
599 return !isl_int_is_neg(tab
->mat
->row
[var
->index
][1]);
602 /* Return a negative value if "var" can attain negative values.
603 * Return a non-negative value otherwise.
605 * If "var" is manifestly unbounded wrt negative values, we are done.
606 * Otherwise, if var is in a column, we can pivot it down to a row.
607 * Then we continue pivoting down until either
608 * - the pivot would result in a manifestly unbounded column
609 * => we don't perform the pivot, but simply return -1
610 * - no more down pivots can be performed
611 * - the sample value is negative
612 * If the sample value becomes negative and the variable is supposed
613 * to be nonnegative, then we undo the last pivot.
614 * However, if the last pivot has made the pivoting variable
615 * obviously redundant, then it may have moved to another row.
616 * In that case we look for upward pivots until we reach a non-negative
619 static int sign_of_min(struct isl_ctx
*ctx
,
620 struct isl_tab
*tab
, struct isl_tab_var
*var
)
623 struct isl_tab_var
*pivot_var
;
625 if (min_is_manifestly_unbounded(ctx
, tab
, var
))
629 row
= pivot_row(ctx
, tab
, NULL
, -1, col
);
630 pivot_var
= var_from_col(ctx
, tab
, col
);
631 pivot(ctx
, tab
, row
, col
);
632 if (var
->is_redundant
)
634 if (isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
635 if (var
->is_nonneg
) {
636 if (!pivot_var
->is_redundant
&&
637 pivot_var
->index
== row
)
638 pivot(ctx
, tab
, row
, col
);
640 restore_row(ctx
, tab
, var
);
645 if (var
->is_redundant
)
647 while (!isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
648 find_pivot(ctx
, tab
, var
, var
, -1, &row
, &col
);
649 if (row
== var
->index
)
652 return isl_int_sgn(tab
->mat
->row
[var
->index
][1]);
653 pivot_var
= var_from_col(ctx
, tab
, col
);
654 pivot(ctx
, tab
, row
, col
);
655 if (var
->is_redundant
)
658 if (var
->is_nonneg
) {
659 /* pivot back to non-negative value */
660 if (!pivot_var
->is_redundant
&& pivot_var
->index
== row
)
661 pivot(ctx
, tab
, row
, col
);
663 restore_row(ctx
, tab
, var
);
668 /* Return 1 if "var" can attain values <= -1.
669 * Return 0 otherwise.
671 * The sample value of "var" is assumed to be non-negative when the
672 * the function is called and will be made non-negative again before
673 * the function returns.
675 static int min_at_most_neg_one(struct isl_ctx
*ctx
,
676 struct isl_tab
*tab
, struct isl_tab_var
*var
)
679 struct isl_tab_var
*pivot_var
;
681 if (min_is_manifestly_unbounded(ctx
, tab
, var
))
685 row
= pivot_row(ctx
, tab
, NULL
, -1, col
);
686 pivot_var
= var_from_col(ctx
, tab
, col
);
687 pivot(ctx
, tab
, row
, col
);
688 if (var
->is_redundant
)
690 if (isl_int_is_neg(tab
->mat
->row
[var
->index
][1]) &&
691 isl_int_abs_ge(tab
->mat
->row
[var
->index
][1],
692 tab
->mat
->row
[var
->index
][0])) {
693 if (var
->is_nonneg
) {
694 if (!pivot_var
->is_redundant
&&
695 pivot_var
->index
== row
)
696 pivot(ctx
, tab
, row
, col
);
698 restore_row(ctx
, tab
, var
);
703 if (var
->is_redundant
)
706 find_pivot(ctx
, tab
, var
, var
, -1, &row
, &col
);
707 if (row
== var
->index
)
711 pivot_var
= var_from_col(ctx
, tab
, col
);
712 pivot(ctx
, tab
, row
, col
);
713 if (var
->is_redundant
)
715 } while (!isl_int_is_neg(tab
->mat
->row
[var
->index
][1]) ||
716 isl_int_abs_lt(tab
->mat
->row
[var
->index
][1],
717 tab
->mat
->row
[var
->index
][0]));
718 if (var
->is_nonneg
) {
719 /* pivot back to non-negative value */
720 if (!pivot_var
->is_redundant
&& pivot_var
->index
== row
)
721 pivot(ctx
, tab
, row
, col
);
722 restore_row(ctx
, tab
, var
);
727 /* Return 1 if "var" can attain values >= 1.
728 * Return 0 otherwise.
730 static int at_least_one(struct isl_ctx
*ctx
,
731 struct isl_tab
*tab
, struct isl_tab_var
*var
)
736 if (max_is_manifestly_unbounded(ctx
, tab
, var
))
738 to_row(ctx
, tab
, var
, 1);
739 r
= tab
->mat
->row
[var
->index
];
740 while (isl_int_lt(r
[1], r
[0])) {
741 find_pivot(ctx
, tab
, var
, var
, 1, &row
, &col
);
743 return isl_int_ge(r
[1], r
[0]);
744 if (row
== var
->index
) /* manifestly unbounded */
746 pivot(ctx
, tab
, row
, col
);
751 static void swap_cols(struct isl_ctx
*ctx
,
752 struct isl_tab
*tab
, int col1
, int col2
)
755 t
= tab
->col_var
[col1
];
756 tab
->col_var
[col1
] = tab
->col_var
[col2
];
757 tab
->col_var
[col2
] = t
;
758 var_from_col(ctx
, tab
, col1
)->index
= col1
;
759 var_from_col(ctx
, tab
, col2
)->index
= col2
;
760 tab
->mat
= isl_mat_swap_cols(ctx
, tab
->mat
, 2 + col1
, 2 + col2
);
763 /* Mark column with index "col" as representing a zero variable.
764 * If we may need to undo the operation the column is kept,
765 * but no longer considered.
766 * Otherwise, the column is simply removed.
768 * The column may be interchanged with some other column. If it
769 * is interchanged with a later column, return 1. Otherwise return 0.
770 * If the columns are checked in order in the calling function,
771 * then a return value of 1 means that the column with the given
772 * column number may now contain a different column that
773 * hasn't been checked yet.
775 static int kill_col(struct isl_ctx
*ctx
,
776 struct isl_tab
*tab
, int col
)
778 var_from_col(ctx
, tab
, col
)->is_zero
= 1;
779 if (tab
->need_undo
) {
780 push(ctx
, tab
, isl_tab_undo_zero
, var_from_col(ctx
, tab
, col
));
781 if (col
!= tab
->n_dead
)
782 swap_cols(ctx
, tab
, col
, tab
->n_dead
);
786 if (col
!= tab
->n_col
- 1)
787 swap_cols(ctx
, tab
, col
, tab
->n_col
- 1);
788 var_from_col(ctx
, tab
, tab
->n_col
- 1)->index
= -1;
794 /* Row variable "var" is non-negative and cannot attain any values
795 * larger than zero. This means that the coefficients of the unrestricted
796 * column variables are zero and that the coefficients of the non-negative
797 * column variables are zero or negative.
798 * Each of the non-negative variables with a negative coefficient can
799 * then also be written as the negative sum of non-negative variables
800 * and must therefore also be zero.
802 static void close_row(struct isl_ctx
*ctx
,
803 struct isl_tab
*tab
, struct isl_tab_var
*var
)
806 struct isl_mat
*mat
= tab
->mat
;
808 isl_assert(ctx
, var
->is_nonneg
, return);
810 for (j
= tab
->n_dead
; j
< tab
->n_col
; ++j
) {
811 if (isl_int_is_zero(mat
->row
[var
->index
][2 + j
]))
813 isl_assert(ctx
, isl_int_is_neg(mat
->row
[var
->index
][2 + j
]),
815 if (kill_col(ctx
, tab
, j
))
818 mark_redundant(ctx
, tab
, var
->index
);
821 /* Add a row to the tableau. The row is given as an affine combination
822 * of the original variables and needs to be expressed in terms of the
825 * We add each term in turn.
826 * If r = n/d_r is the current sum and we need to add k x, then
827 * if x is a column variable, we increase the numerator of
828 * this column by k d_r
829 * if x = f/d_x is a row variable, then the new representation of r is
831 * n k f d_x/g n + d_r/g k f m/d_r n + m/d_g k f
832 * --- + --- = ------------------- = -------------------
833 * d_r d_r d_r d_x/g m
835 * with g the gcd of d_r and d_x and m the lcm of d_r and d_x.
837 static int add_row(struct isl_ctx
*ctx
, struct isl_tab
*tab
, isl_int
*line
)
844 isl_assert(ctx
, tab
->n_row
< tab
->mat
->n_row
, return -1);
849 tab
->con
[r
].index
= tab
->n_row
;
850 tab
->con
[r
].is_row
= 1;
851 tab
->con
[r
].is_nonneg
= 0;
852 tab
->con
[r
].is_zero
= 0;
853 tab
->con
[r
].is_redundant
= 0;
854 tab
->con
[r
].frozen
= 0;
855 tab
->row_var
[tab
->n_row
] = ~r
;
856 row
= tab
->mat
->row
[tab
->n_row
];
857 isl_int_set_si(row
[0], 1);
858 isl_int_set(row
[1], line
[0]);
859 isl_seq_clr(row
+ 2, tab
->n_col
);
860 for (i
= 0; i
< tab
->n_var
; ++i
) {
861 if (tab
->var
[i
].is_zero
)
863 if (tab
->var
[i
].is_row
) {
865 row
[0], tab
->mat
->row
[tab
->var
[i
].index
][0]);
866 isl_int_swap(a
, row
[0]);
867 isl_int_divexact(a
, row
[0], a
);
869 row
[0], tab
->mat
->row
[tab
->var
[i
].index
][0]);
870 isl_int_mul(b
, b
, line
[1 + i
]);
871 isl_seq_combine(row
+ 1, a
, row
+ 1,
872 b
, tab
->mat
->row
[tab
->var
[i
].index
] + 1,
875 isl_int_addmul(row
[2 + tab
->var
[i
].index
],
876 line
[1 + i
], row
[0]);
878 isl_seq_normalize(row
, 2 + tab
->n_col
);
881 push(ctx
, tab
, isl_tab_undo_allocate
, &tab
->con
[r
]);
888 static int drop_row(struct isl_ctx
*ctx
, struct isl_tab
*tab
, int row
)
890 isl_assert(ctx
, ~tab
->row_var
[row
] == tab
->n_con
- 1, return -1);
891 if (row
!= tab
->n_row
- 1)
892 swap_rows(ctx
, tab
, row
, tab
->n_row
- 1);
898 /* Add inequality "ineq" and check if it conflicts with the
899 * previously added constraints or if it is obviously redundant.
901 struct isl_tab
*isl_tab_add_ineq(struct isl_ctx
*ctx
,
902 struct isl_tab
*tab
, isl_int
*ineq
)
909 r
= add_row(ctx
, tab
, ineq
);
912 tab
->con
[r
].is_nonneg
= 1;
913 push(ctx
, tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
914 if (is_redundant(ctx
, tab
, tab
->con
[r
].index
)) {
915 mark_redundant(ctx
, tab
, tab
->con
[r
].index
);
919 sgn
= restore_row(ctx
, tab
, &tab
->con
[r
]);
921 mark_empty(ctx
, tab
);
922 else if (tab
->con
[r
].is_row
&&
923 is_redundant(ctx
, tab
, tab
->con
[r
].index
))
924 mark_redundant(ctx
, tab
, tab
->con
[r
].index
);
927 isl_tab_free(ctx
, tab
);
931 /* Pivot a non-negative variable down until it reaches the value zero
932 * and then pivot the variable into a column position.
934 static int to_col(struct isl_ctx
*ctx
,
935 struct isl_tab
*tab
, struct isl_tab_var
*var
)
943 while (isl_int_is_pos(tab
->mat
->row
[var
->index
][1])) {
944 find_pivot(ctx
, tab
, var
, NULL
, -1, &row
, &col
);
945 isl_assert(ctx
, row
!= -1, return -1);
946 pivot(ctx
, tab
, row
, col
);
951 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
)
952 if (!isl_int_is_zero(tab
->mat
->row
[var
->index
][2 + i
]))
955 isl_assert(ctx
, i
< tab
->n_col
, return -1);
956 pivot(ctx
, tab
, var
->index
, i
);
961 /* We assume Gaussian elimination has been performed on the equalities.
962 * The equalities can therefore never conflict.
963 * Adding the equalities is currently only really useful for a later call
964 * to isl_tab_ineq_type.
966 static struct isl_tab
*add_eq(struct isl_ctx
*ctx
,
967 struct isl_tab
*tab
, isl_int
*eq
)
974 r
= add_row(ctx
, tab
, eq
);
978 r
= tab
->con
[r
].index
;
979 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
980 if (isl_int_is_zero(tab
->mat
->row
[r
][2 + i
]))
982 pivot(ctx
, tab
, r
, i
);
983 kill_col(ctx
, tab
, i
);
990 isl_tab_free(ctx
, tab
);
994 /* Add an equality that is known to be valid for the given tableau.
996 struct isl_tab
*isl_tab_add_valid_eq(struct isl_ctx
*ctx
,
997 struct isl_tab
*tab
, isl_int
*eq
)
999 struct isl_tab_var
*var
;
1005 r
= add_row(ctx
, tab
, eq
);
1011 if (isl_int_is_neg(tab
->mat
->row
[r
][1]))
1012 isl_seq_neg(tab
->mat
->row
[r
] + 1, tab
->mat
->row
[r
] + 1,
1015 if (to_col(ctx
, tab
, var
) < 0)
1018 kill_col(ctx
, tab
, var
->index
);
1022 isl_tab_free(ctx
, tab
);
1026 struct isl_tab
*isl_tab_from_basic_map(struct isl_basic_map
*bmap
)
1029 struct isl_tab
*tab
;
1033 tab
= isl_tab_alloc(bmap
->ctx
,
1034 isl_basic_map_total_dim(bmap
) + bmap
->n_ineq
+ 1,
1035 isl_basic_map_total_dim(bmap
));
1038 tab
->rational
= ISL_F_ISSET(bmap
, ISL_BASIC_MAP_RATIONAL
);
1039 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_EMPTY
)) {
1040 mark_empty(bmap
->ctx
, tab
);
1043 for (i
= 0; i
< bmap
->n_eq
; ++i
) {
1044 tab
= add_eq(bmap
->ctx
, tab
, bmap
->eq
[i
]);
1048 for (i
= 0; i
< bmap
->n_ineq
; ++i
) {
1049 tab
= isl_tab_add_ineq(bmap
->ctx
, tab
, bmap
->ineq
[i
]);
1050 if (!tab
|| tab
->empty
)
1056 struct isl_tab
*isl_tab_from_basic_set(struct isl_basic_set
*bset
)
1058 return isl_tab_from_basic_map((struct isl_basic_map
*)bset
);
1061 /* Construct a tableau corresponding to the recession cone of "bmap".
1063 struct isl_tab
*isl_tab_from_recession_cone(struct isl_basic_map
*bmap
)
1067 struct isl_tab
*tab
;
1071 tab
= isl_tab_alloc(bmap
->ctx
, bmap
->n_eq
+ bmap
->n_ineq
,
1072 isl_basic_map_total_dim(bmap
));
1075 tab
->rational
= ISL_F_ISSET(bmap
, ISL_BASIC_MAP_RATIONAL
);
1078 for (i
= 0; i
< bmap
->n_eq
; ++i
) {
1079 isl_int_swap(bmap
->eq
[i
][0], cst
);
1080 tab
= add_eq(bmap
->ctx
, tab
, bmap
->eq
[i
]);
1081 isl_int_swap(bmap
->eq
[i
][0], cst
);
1085 for (i
= 0; i
< bmap
->n_ineq
; ++i
) {
1087 isl_int_swap(bmap
->ineq
[i
][0], cst
);
1088 r
= add_row(bmap
->ctx
, tab
, bmap
->ineq
[i
]);
1089 isl_int_swap(bmap
->ineq
[i
][0], cst
);
1092 tab
->con
[r
].is_nonneg
= 1;
1093 push(bmap
->ctx
, tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
1100 isl_tab_free(bmap
->ctx
, tab
);
1104 /* Assuming "tab" is the tableau of a cone, check if the cone is
1105 * bounded, i.e., if it is empty or only contains the origin.
1107 int isl_tab_cone_is_bounded(struct isl_ctx
*ctx
, struct isl_tab
*tab
)
1115 if (tab
->n_dead
== tab
->n_col
)
1118 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1119 struct isl_tab_var
*var
;
1120 var
= var_from_row(ctx
, tab
, i
);
1121 if (!var
->is_nonneg
)
1123 if (sign_of_max(ctx
, tab
, var
) == 0)
1124 close_row(ctx
, tab
, var
);
1127 if (tab
->n_dead
== tab
->n_col
)
1133 static int sample_is_integer(struct isl_ctx
*ctx
, struct isl_tab
*tab
)
1137 for (i
= 0; i
< tab
->n_var
; ++i
) {
1139 if (!tab
->var
[i
].is_row
)
1141 row
= tab
->var
[i
].index
;
1142 if (!isl_int_is_divisible_by(tab
->mat
->row
[row
][1],
1143 tab
->mat
->row
[row
][0]))
1149 static struct isl_vec
*extract_integer_sample(struct isl_ctx
*ctx
,
1150 struct isl_tab
*tab
)
1153 struct isl_vec
*vec
;
1155 vec
= isl_vec_alloc(ctx
, 1 + tab
->n_var
);
1159 isl_int_set_si(vec
->block
.data
[0], 1);
1160 for (i
= 0; i
< tab
->n_var
; ++i
) {
1161 if (!tab
->var
[i
].is_row
)
1162 isl_int_set_si(vec
->block
.data
[1 + i
], 0);
1164 int row
= tab
->var
[i
].index
;
1165 isl_int_divexact(vec
->block
.data
[1 + i
],
1166 tab
->mat
->row
[row
][1], tab
->mat
->row
[row
][0]);
1173 struct isl_vec
*isl_tab_get_sample_value(struct isl_ctx
*ctx
,
1174 struct isl_tab
*tab
)
1177 struct isl_vec
*vec
;
1183 vec
= isl_vec_alloc(ctx
, 1 + tab
->n_var
);
1189 isl_int_set_si(vec
->block
.data
[0], 1);
1190 for (i
= 0; i
< tab
->n_var
; ++i
) {
1192 if (!tab
->var
[i
].is_row
) {
1193 isl_int_set_si(vec
->block
.data
[1 + i
], 0);
1196 row
= tab
->var
[i
].index
;
1197 isl_int_gcd(m
, vec
->block
.data
[0], tab
->mat
->row
[row
][0]);
1198 isl_int_divexact(m
, tab
->mat
->row
[row
][0], m
);
1199 isl_seq_scale(vec
->block
.data
, vec
->block
.data
, m
, 1 + i
);
1200 isl_int_divexact(m
, vec
->block
.data
[0], tab
->mat
->row
[row
][0]);
1201 isl_int_mul(vec
->block
.data
[1 + i
], m
, tab
->mat
->row
[row
][1]);
1203 isl_seq_normalize(vec
->block
.data
, vec
->size
);
1209 /* Update "bmap" based on the results of the tableau "tab".
1210 * In particular, implicit equalities are made explicit, redundant constraints
1211 * are removed and if the sample value happens to be integer, it is stored
1212 * in "bmap" (unless "bmap" already had an integer sample).
1214 * The tableau is assumed to have been created from "bmap" using
1215 * isl_tab_from_basic_map.
1217 struct isl_basic_map
*isl_basic_map_update_from_tab(struct isl_basic_map
*bmap
,
1218 struct isl_tab
*tab
)
1230 bmap
= isl_basic_map_set_to_empty(bmap
);
1232 for (i
= bmap
->n_ineq
- 1; i
>= 0; --i
) {
1233 if (isl_tab_is_equality(bmap
->ctx
, tab
, n_eq
+ i
))
1234 isl_basic_map_inequality_to_equality(bmap
, i
);
1235 else if (isl_tab_is_redundant(bmap
->ctx
, tab
, n_eq
+ i
))
1236 isl_basic_map_drop_inequality(bmap
, i
);
1238 if (!tab
->rational
&&
1239 !bmap
->sample
&& sample_is_integer(bmap
->ctx
, tab
))
1240 bmap
->sample
= extract_integer_sample(bmap
->ctx
, tab
);
1244 struct isl_basic_set
*isl_basic_set_update_from_tab(struct isl_basic_set
*bset
,
1245 struct isl_tab
*tab
)
1247 return (struct isl_basic_set
*)isl_basic_map_update_from_tab(
1248 (struct isl_basic_map
*)bset
, tab
);
1251 /* Given a non-negative variable "var", add a new non-negative variable
1252 * that is the opposite of "var", ensuring that var can only attain the
1254 * If var = n/d is a row variable, then the new variable = -n/d.
1255 * If var is a column variables, then the new variable = -var.
1256 * If the new variable cannot attain non-negative values, then
1257 * the resulting tableau is empty.
1258 * Otherwise, we know the value will be zero and we close the row.
1260 static struct isl_tab
*cut_to_hyperplane(struct isl_ctx
*ctx
,
1261 struct isl_tab
*tab
, struct isl_tab_var
*var
)
1267 if (extend_cons(ctx
, tab
, 1) < 0)
1271 tab
->con
[r
].index
= tab
->n_row
;
1272 tab
->con
[r
].is_row
= 1;
1273 tab
->con
[r
].is_nonneg
= 0;
1274 tab
->con
[r
].is_zero
= 0;
1275 tab
->con
[r
].is_redundant
= 0;
1276 tab
->con
[r
].frozen
= 0;
1277 tab
->row_var
[tab
->n_row
] = ~r
;
1278 row
= tab
->mat
->row
[tab
->n_row
];
1281 isl_int_set(row
[0], tab
->mat
->row
[var
->index
][0]);
1282 isl_seq_neg(row
+ 1,
1283 tab
->mat
->row
[var
->index
] + 1, 1 + tab
->n_col
);
1285 isl_int_set_si(row
[0], 1);
1286 isl_seq_clr(row
+ 1, 1 + tab
->n_col
);
1287 isl_int_set_si(row
[2 + var
->index
], -1);
1292 push(ctx
, tab
, isl_tab_undo_allocate
, &tab
->con
[r
]);
1294 sgn
= sign_of_max(ctx
, tab
, &tab
->con
[r
]);
1296 mark_empty(ctx
, tab
);
1298 tab
->con
[r
].is_nonneg
= 1;
1299 push(ctx
, tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
1301 close_row(ctx
, tab
, &tab
->con
[r
]);
1306 isl_tab_free(ctx
, tab
);
1310 /* Given a tableau "tab" and an inequality constraint "con" of the tableau,
1311 * relax the inequality by one. That is, the inequality r >= 0 is replaced
1312 * by r' = r + 1 >= 0.
1313 * If r is a row variable, we simply increase the constant term by one
1314 * (taking into account the denominator).
1315 * If r is a column variable, then we need to modify each row that
1316 * refers to r = r' - 1 by substituting this equality, effectively
1317 * subtracting the coefficient of the column from the constant.
1319 struct isl_tab
*isl_tab_relax(struct isl_ctx
*ctx
,
1320 struct isl_tab
*tab
, int con
)
1322 struct isl_tab_var
*var
;
1326 var
= &tab
->con
[con
];
1328 if (!var
->is_row
&& !max_is_manifestly_unbounded(ctx
, tab
, var
))
1329 to_row(ctx
, tab
, var
, 1);
1332 isl_int_add(tab
->mat
->row
[var
->index
][1],
1333 tab
->mat
->row
[var
->index
][1], tab
->mat
->row
[var
->index
][0]);
1337 for (i
= 0; i
< tab
->n_row
; ++i
) {
1338 if (isl_int_is_zero(tab
->mat
->row
[i
][2 + var
->index
]))
1340 isl_int_sub(tab
->mat
->row
[i
][1], tab
->mat
->row
[i
][1],
1341 tab
->mat
->row
[i
][2 + var
->index
]);
1346 push(ctx
, tab
, isl_tab_undo_relax
, var
);
1351 struct isl_tab
*isl_tab_select_facet(struct isl_ctx
*ctx
,
1352 struct isl_tab
*tab
, int con
)
1357 return cut_to_hyperplane(ctx
, tab
, &tab
->con
[con
]);
1360 static int may_be_equality(struct isl_tab
*tab
, int row
)
1362 return (tab
->rational
? isl_int_is_zero(tab
->mat
->row
[row
][1])
1363 : isl_int_lt(tab
->mat
->row
[row
][1],
1364 tab
->mat
->row
[row
][0])) &&
1365 isl_seq_first_non_zero(tab
->mat
->row
[row
] + 2 + tab
->n_dead
,
1366 tab
->n_col
- tab
->n_dead
) != -1;
1369 /* Check for (near) equalities among the constraints.
1370 * A constraint is an equality if it is non-negative and if
1371 * its maximal value is either
1372 * - zero (in case of rational tableaus), or
1373 * - strictly less than 1 (in case of integer tableaus)
1375 * We first mark all non-redundant and non-dead variables that
1376 * are not frozen and not obviously not an equality.
1377 * Then we iterate over all marked variables if they can attain
1378 * any values larger than zero or at least one.
1379 * If the maximal value is zero, we mark any column variables
1380 * that appear in the row as being zero and mark the row as being redundant.
1381 * Otherwise, if the maximal value is strictly less than one (and the
1382 * tableau is integer), then we restrict the value to being zero
1383 * by adding an opposite non-negative variable.
1385 struct isl_tab
*isl_tab_detect_equalities(struct isl_ctx
*ctx
,
1386 struct isl_tab
*tab
)
1395 if (tab
->n_dead
== tab
->n_col
)
1399 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1400 struct isl_tab_var
*var
= var_from_row(ctx
, tab
, i
);
1401 var
->marked
= !var
->frozen
&& var
->is_nonneg
&&
1402 may_be_equality(tab
, i
);
1406 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1407 struct isl_tab_var
*var
= var_from_col(ctx
, tab
, i
);
1408 var
->marked
= !var
->frozen
&& var
->is_nonneg
;
1413 struct isl_tab_var
*var
;
1414 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1415 var
= var_from_row(ctx
, tab
, i
);
1419 if (i
== tab
->n_row
) {
1420 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1421 var
= var_from_col(ctx
, tab
, i
);
1425 if (i
== tab
->n_col
)
1430 if (sign_of_max(ctx
, tab
, var
) == 0)
1431 close_row(ctx
, tab
, var
);
1432 else if (!tab
->rational
&& !at_least_one(ctx
, tab
, var
)) {
1433 tab
= cut_to_hyperplane(ctx
, tab
, var
);
1434 return isl_tab_detect_equalities(ctx
, tab
);
1436 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1437 var
= var_from_row(ctx
, tab
, i
);
1440 if (may_be_equality(tab
, i
))
1450 /* Check for (near) redundant constraints.
1451 * A constraint is redundant if it is non-negative and if
1452 * its minimal value (temporarily ignoring the non-negativity) is either
1453 * - zero (in case of rational tableaus), or
1454 * - strictly larger than -1 (in case of integer tableaus)
1456 * We first mark all non-redundant and non-dead variables that
1457 * are not frozen and not obviously negatively unbounded.
1458 * Then we iterate over all marked variables if they can attain
1459 * any values smaller than zero or at most negative one.
1460 * If not, we mark the row as being redundant (assuming it hasn't
1461 * been detected as being obviously redundant in the mean time).
1463 struct isl_tab
*isl_tab_detect_redundant(struct isl_ctx
*ctx
,
1464 struct isl_tab
*tab
)
1473 if (tab
->n_redundant
== tab
->n_row
)
1477 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1478 struct isl_tab_var
*var
= var_from_row(ctx
, tab
, i
);
1479 var
->marked
= !var
->frozen
&& var
->is_nonneg
;
1483 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1484 struct isl_tab_var
*var
= var_from_col(ctx
, tab
, i
);
1485 var
->marked
= !var
->frozen
&& var
->is_nonneg
&&
1486 !min_is_manifestly_unbounded(ctx
, tab
, var
);
1491 struct isl_tab_var
*var
;
1492 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1493 var
= var_from_row(ctx
, tab
, i
);
1497 if (i
== tab
->n_row
) {
1498 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1499 var
= var_from_col(ctx
, tab
, i
);
1503 if (i
== tab
->n_col
)
1508 if ((tab
->rational
? (sign_of_min(ctx
, tab
, var
) >= 0)
1509 : !min_at_most_neg_one(ctx
, tab
, var
)) &&
1511 mark_redundant(ctx
, tab
, var
->index
);
1512 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1513 var
= var_from_col(ctx
, tab
, i
);
1516 if (!min_is_manifestly_unbounded(ctx
, tab
, var
))
1526 int isl_tab_is_equality(struct isl_ctx
*ctx
, struct isl_tab
*tab
, int con
)
1532 if (tab
->con
[con
].is_zero
)
1534 if (tab
->con
[con
].is_redundant
)
1536 if (!tab
->con
[con
].is_row
)
1537 return tab
->con
[con
].index
< tab
->n_dead
;
1539 row
= tab
->con
[con
].index
;
1541 return isl_int_is_zero(tab
->mat
->row
[row
][1]) &&
1542 isl_seq_first_non_zero(tab
->mat
->row
[row
] + 2 + tab
->n_dead
,
1543 tab
->n_col
- tab
->n_dead
) == -1;
1546 /* Return the minimial value of the affine expression "f" with denominator
1547 * "denom" in *opt, *opt_denom, assuming the tableau is not empty and
1548 * the expression cannot attain arbitrarily small values.
1549 * If opt_denom is NULL, then *opt is rounded up to the nearest integer.
1550 * The return value reflects the nature of the result (empty, unbounded,
1551 * minmimal value returned in *opt).
1553 enum isl_lp_result
isl_tab_min(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
1554 isl_int
*f
, isl_int denom
, isl_int
*opt
, isl_int
*opt_denom
,
1558 enum isl_lp_result res
= isl_lp_ok
;
1559 struct isl_tab_var
*var
;
1560 struct isl_tab_undo
*snap
;
1563 return isl_lp_empty
;
1565 snap
= isl_tab_snap(ctx
, tab
);
1566 r
= add_row(ctx
, tab
, f
);
1568 return isl_lp_error
;
1570 isl_int_mul(tab
->mat
->row
[var
->index
][0],
1571 tab
->mat
->row
[var
->index
][0], denom
);
1574 find_pivot(ctx
, tab
, var
, var
, -1, &row
, &col
);
1575 if (row
== var
->index
) {
1576 res
= isl_lp_unbounded
;
1581 pivot(ctx
, tab
, row
, col
);
1583 if (isl_tab_rollback(ctx
, tab
, snap
) < 0)
1584 return isl_lp_error
;
1585 if (ISL_FL_ISSET(flags
, ISL_TAB_SAVE_DUAL
)) {
1588 isl_vec_free(tab
->dual
);
1589 tab
->dual
= isl_vec_alloc(ctx
, 1 + tab
->n_con
);
1591 return isl_lp_error
;
1592 isl_int_set(tab
->dual
->el
[0], tab
->mat
->row
[var
->index
][0]);
1593 for (i
= 0; i
< tab
->n_con
; ++i
) {
1594 if (tab
->con
[i
].is_row
)
1595 isl_int_set_si(tab
->dual
->el
[1 + i
], 0);
1597 int pos
= 2 + tab
->con
[i
].index
;
1598 isl_int_set(tab
->dual
->el
[1 + i
],
1599 tab
->mat
->row
[var
->index
][pos
]);
1603 if (res
== isl_lp_ok
) {
1605 isl_int_set(*opt
, tab
->mat
->row
[var
->index
][1]);
1606 isl_int_set(*opt_denom
, tab
->mat
->row
[var
->index
][0]);
1608 isl_int_cdiv_q(*opt
, tab
->mat
->row
[var
->index
][1],
1609 tab
->mat
->row
[var
->index
][0]);
1614 int isl_tab_is_redundant(struct isl_ctx
*ctx
, struct isl_tab
*tab
, int con
)
1621 if (tab
->con
[con
].is_zero
)
1623 if (tab
->con
[con
].is_redundant
)
1625 return tab
->con
[con
].is_row
&& tab
->con
[con
].index
< tab
->n_redundant
;
1628 /* Take a snapshot of the tableau that can be restored by s call to
1631 struct isl_tab_undo
*isl_tab_snap(struct isl_ctx
*ctx
, struct isl_tab
*tab
)
1639 /* Undo the operation performed by isl_tab_relax.
1641 static void unrelax(struct isl_ctx
*ctx
,
1642 struct isl_tab
*tab
, struct isl_tab_var
*var
)
1644 if (!var
->is_row
&& !max_is_manifestly_unbounded(ctx
, tab
, var
))
1645 to_row(ctx
, tab
, var
, 1);
1648 isl_int_sub(tab
->mat
->row
[var
->index
][1],
1649 tab
->mat
->row
[var
->index
][1], tab
->mat
->row
[var
->index
][0]);
1653 for (i
= 0; i
< tab
->n_row
; ++i
) {
1654 if (isl_int_is_zero(tab
->mat
->row
[i
][2 + var
->index
]))
1656 isl_int_add(tab
->mat
->row
[i
][1], tab
->mat
->row
[i
][1],
1657 tab
->mat
->row
[i
][2 + var
->index
]);
1663 static void perform_undo(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
1664 struct isl_tab_undo
*undo
)
1666 switch(undo
->type
) {
1667 case isl_tab_undo_empty
:
1670 case isl_tab_undo_nonneg
:
1671 undo
->var
->is_nonneg
= 0;
1673 case isl_tab_undo_redundant
:
1674 undo
->var
->is_redundant
= 0;
1677 case isl_tab_undo_zero
:
1678 undo
->var
->is_zero
= 0;
1681 case isl_tab_undo_allocate
:
1682 if (!undo
->var
->is_row
) {
1683 if (max_is_manifestly_unbounded(ctx
, tab
, undo
->var
))
1684 to_row(ctx
, tab
, undo
->var
, -1);
1686 to_row(ctx
, tab
, undo
->var
, 1);
1688 drop_row(ctx
, tab
, undo
->var
->index
);
1690 case isl_tab_undo_relax
:
1691 unrelax(ctx
, tab
, undo
->var
);
1696 /* Return the tableau to the state it was in when the snapshot "snap"
1699 int isl_tab_rollback(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
1700 struct isl_tab_undo
*snap
)
1702 struct isl_tab_undo
*undo
, *next
;
1708 for (undo
= tab
->top
; undo
&& undo
!= &tab
->bottom
; undo
= next
) {
1712 perform_undo(ctx
, tab
, undo
);
1722 /* The given row "row" represents an inequality violated by all
1723 * points in the tableau. Check for some special cases of such
1724 * separating constraints.
1725 * In particular, if the row has been reduced to the constant -1,
1726 * then we know the inequality is adjacent (but opposite) to
1727 * an equality in the tableau.
1728 * If the row has been reduced to r = -1 -r', with r' an inequality
1729 * of the tableau, then the inequality is adjacent (but opposite)
1730 * to the inequality r'.
1732 static enum isl_ineq_type
separation_type(struct isl_ctx
*ctx
,
1733 struct isl_tab
*tab
, unsigned row
)
1738 return isl_ineq_separate
;
1740 if (!isl_int_is_one(tab
->mat
->row
[row
][0]))
1741 return isl_ineq_separate
;
1742 if (!isl_int_is_negone(tab
->mat
->row
[row
][1]))
1743 return isl_ineq_separate
;
1745 pos
= isl_seq_first_non_zero(tab
->mat
->row
[row
] + 2 + tab
->n_dead
,
1746 tab
->n_col
- tab
->n_dead
);
1748 return isl_ineq_adj_eq
;
1750 if (!isl_int_is_negone(tab
->mat
->row
[row
][2 + tab
->n_dead
+ pos
]))
1751 return isl_ineq_separate
;
1753 pos
= isl_seq_first_non_zero(
1754 tab
->mat
->row
[row
] + 2 + tab
->n_dead
+ pos
+ 1,
1755 tab
->n_col
- tab
->n_dead
- pos
- 1);
1757 return pos
== -1 ? isl_ineq_adj_ineq
: isl_ineq_separate
;
1760 /* Check the effect of inequality "ineq" on the tableau "tab".
1762 * isl_ineq_redundant: satisfied by all points in the tableau
1763 * isl_ineq_separate: satisfied by no point in the tableau
1764 * isl_ineq_cut: satisfied by some by not all points
1765 * isl_ineq_adj_eq: adjacent to an equality
1766 * isl_ineq_adj_ineq: adjacent to an inequality.
1768 enum isl_ineq_type
isl_tab_ineq_type(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
1771 enum isl_ineq_type type
= isl_ineq_error
;
1772 struct isl_tab_undo
*snap
= NULL
;
1777 return isl_ineq_error
;
1779 if (extend_cons(ctx
, tab
, 1) < 0)
1780 return isl_ineq_error
;
1782 snap
= isl_tab_snap(ctx
, tab
);
1784 con
= add_row(ctx
, tab
, ineq
);
1788 row
= tab
->con
[con
].index
;
1789 if (is_redundant(ctx
, tab
, row
))
1790 type
= isl_ineq_redundant
;
1791 else if (isl_int_is_neg(tab
->mat
->row
[row
][1]) &&
1793 isl_int_abs_ge(tab
->mat
->row
[row
][1],
1794 tab
->mat
->row
[row
][0]))) {
1795 if (at_least_zero(ctx
, tab
, &tab
->con
[con
]))
1796 type
= isl_ineq_cut
;
1798 type
= separation_type(ctx
, tab
, row
);
1799 } else if (tab
->rational
? (sign_of_min(ctx
, tab
, &tab
->con
[con
]) < 0)
1800 : min_at_most_neg_one(ctx
, tab
, &tab
->con
[con
]))
1801 type
= isl_ineq_cut
;
1803 type
= isl_ineq_redundant
;
1805 if (isl_tab_rollback(ctx
, tab
, snap
))
1806 return isl_ineq_error
;
1809 isl_tab_rollback(ctx
, tab
, snap
);
1810 return isl_ineq_error
;
1813 void isl_tab_dump(struct isl_ctx
*ctx
, struct isl_tab
*tab
,
1814 FILE *out
, int indent
)
1820 fprintf(out
, "%*snull tab\n", indent
, "");
1823 fprintf(out
, "%*sn_redundant: %d, n_dead: %d", indent
, "",
1824 tab
->n_redundant
, tab
->n_dead
);
1826 fprintf(out
, ", rational");
1828 fprintf(out
, ", empty");
1830 fprintf(out
, "%*s[", indent
, "");
1831 for (i
= 0; i
< tab
->n_var
; ++i
) {
1834 fprintf(out
, "%c%d%s", tab
->var
[i
].is_row
? 'r' : 'c',
1836 tab
->var
[i
].is_zero
? " [=0]" :
1837 tab
->var
[i
].is_redundant
? " [R]" : "");
1839 fprintf(out
, "]\n");
1840 fprintf(out
, "%*s[", indent
, "");
1841 for (i
= 0; i
< tab
->n_con
; ++i
) {
1844 fprintf(out
, "%c%d%s", tab
->con
[i
].is_row
? 'r' : 'c',
1846 tab
->con
[i
].is_zero
? " [=0]" :
1847 tab
->con
[i
].is_redundant
? " [R]" : "");
1849 fprintf(out
, "]\n");
1850 fprintf(out
, "%*s[", indent
, "");
1851 for (i
= 0; i
< tab
->n_row
; ++i
) {
1854 fprintf(out
, "r%d: %d%s", i
, tab
->row_var
[i
],
1855 var_from_row(ctx
, tab
, i
)->is_nonneg
? " [>=0]" : "");
1857 fprintf(out
, "]\n");
1858 fprintf(out
, "%*s[", indent
, "");
1859 for (i
= 0; i
< tab
->n_col
; ++i
) {
1862 fprintf(out
, "c%d: %d%s", i
, tab
->col_var
[i
],
1863 var_from_col(ctx
, tab
, i
)->is_nonneg
? " [>=0]" : "");
1865 fprintf(out
, "]\n");
1866 r
= tab
->mat
->n_row
;
1867 tab
->mat
->n_row
= tab
->n_row
;
1868 c
= tab
->mat
->n_col
;
1869 tab
->mat
->n_col
= 2 + tab
->n_col
;
1870 isl_mat_dump(ctx
, tab
->mat
, out
, indent
);
1871 tab
->mat
->n_row
= r
;
1872 tab
->mat
->n_col
= c
;