2 #include "isl_map_private.h"
6 * The implementation of tableaus in this file was inspired by Section 8
7 * of David Detlefs, Greg Nelson and James B. Saxe, "Simplify: a theorem
8 * prover for program checking".
11 struct isl_tab
*isl_tab_alloc(struct isl_ctx
*ctx
,
12 unsigned n_row
, unsigned n_var
)
17 tab
= isl_calloc_type(ctx
, struct isl_tab
);
20 tab
->mat
= isl_mat_alloc(ctx
, n_row
, 2 + n_var
);
23 tab
->var
= isl_alloc_array(ctx
, struct isl_tab_var
, n_var
);
26 tab
->con
= isl_alloc_array(ctx
, struct isl_tab_var
, n_row
);
29 tab
->col_var
= isl_alloc_array(ctx
, int, n_var
);
32 tab
->row_var
= isl_alloc_array(ctx
, int, n_row
);
35 for (i
= 0; i
< n_var
; ++i
) {
36 tab
->var
[i
].index
= i
;
37 tab
->var
[i
].is_row
= 0;
38 tab
->var
[i
].is_nonneg
= 0;
39 tab
->var
[i
].is_zero
= 0;
40 tab
->var
[i
].is_redundant
= 0;
41 tab
->var
[i
].frozen
= 0;
56 tab
->bottom
.type
= isl_tab_undo_bottom
;
57 tab
->bottom
.next
= NULL
;
58 tab
->top
= &tab
->bottom
;
65 static int extend_cons(struct isl_tab
*tab
, unsigned n_new
)
67 if (tab
->max_con
< tab
->n_con
+ n_new
) {
68 struct isl_tab_var
*con
;
70 con
= isl_realloc_array(tab
->mat
->ctx
, tab
->con
,
71 struct isl_tab_var
, tab
->max_con
+ n_new
);
75 tab
->max_con
+= n_new
;
77 if (tab
->mat
->n_row
< tab
->n_row
+ n_new
) {
80 tab
->mat
= isl_mat_extend(tab
->mat
,
81 tab
->n_row
+ n_new
, tab
->n_col
);
84 row_var
= isl_realloc_array(tab
->mat
->ctx
, tab
->row_var
,
85 int, tab
->mat
->n_row
);
88 tab
->row_var
= row_var
;
93 struct isl_tab
*isl_tab_extend(struct isl_tab
*tab
, unsigned n_new
)
95 if (extend_cons(tab
, n_new
) >= 0)
102 static void free_undo(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_tab
*tab
)
118 isl_mat_free(tab
->mat
);
119 isl_vec_free(tab
->dual
);
127 struct isl_tab
*isl_tab_dup(struct isl_tab
*tab
)
135 dup
= isl_calloc_type(tab
->ctx
, struct isl_tab
);
138 dup
->mat
= isl_mat_dup(tab
->mat
);
141 dup
->var
= isl_alloc_array(tab
->ctx
, struct isl_tab_var
, tab
->n_var
);
144 for (i
= 0; i
< tab
->n_var
; ++i
)
145 dup
->var
[i
] = tab
->var
[i
];
146 dup
->con
= isl_alloc_array(tab
->ctx
, struct isl_tab_var
, tab
->max_con
);
149 for (i
= 0; i
< tab
->n_con
; ++i
)
150 dup
->con
[i
] = tab
->con
[i
];
151 dup
->col_var
= isl_alloc_array(tab
->ctx
, int, tab
->mat
->n_col
);
154 for (i
= 0; i
< tab
->n_var
; ++i
)
155 dup
->col_var
[i
] = tab
->col_var
[i
];
156 dup
->row_var
= isl_alloc_array(tab
->ctx
, int, tab
->mat
->n_row
);
159 for (i
= 0; i
< tab
->n_row
; ++i
)
160 dup
->row_var
[i
] = tab
->row_var
[i
];
161 dup
->n_row
= tab
->n_row
;
162 dup
->n_con
= tab
->n_con
;
163 dup
->n_eq
= tab
->n_eq
;
164 dup
->max_con
= tab
->max_con
;
165 dup
->n_col
= tab
->n_col
;
166 dup
->n_var
= tab
->n_var
;
167 dup
->n_dead
= tab
->n_dead
;
168 dup
->n_redundant
= tab
->n_redundant
;
169 dup
->rational
= tab
->rational
;
170 dup
->empty
= tab
->empty
;
173 dup
->bottom
.type
= isl_tab_undo_bottom
;
174 dup
->bottom
.next
= NULL
;
175 dup
->top
= &dup
->bottom
;
182 static struct isl_tab_var
*var_from_index(struct isl_tab
*tab
, int i
)
187 return &tab
->con
[~i
];
190 static struct isl_tab_var
*var_from_row(struct isl_tab
*tab
, int i
)
192 return var_from_index(tab
, tab
->row_var
[i
]);
195 static struct isl_tab_var
*var_from_col(struct isl_tab
*tab
, int i
)
197 return var_from_index(tab
, tab
->col_var
[i
]);
200 /* Check if there are any upper bounds on column variable "var",
201 * i.e., non-negative rows where var appears with a negative coefficient.
202 * Return 1 if there are no such bounds.
204 static int max_is_manifestly_unbounded(struct isl_tab
*tab
,
205 struct isl_tab_var
*var
)
211 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
212 if (!isl_int_is_neg(tab
->mat
->row
[i
][2 + var
->index
]))
214 if (var_from_row(tab
, i
)->is_nonneg
)
220 /* Check if there are any lower bounds on column variable "var",
221 * i.e., non-negative rows where var appears with a positive coefficient.
222 * Return 1 if there are no such bounds.
224 static int min_is_manifestly_unbounded(struct isl_tab
*tab
,
225 struct isl_tab_var
*var
)
231 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
232 if (!isl_int_is_pos(tab
->mat
->row
[i
][2 + var
->index
]))
234 if (var_from_row(tab
, i
)->is_nonneg
)
240 /* Given the index of a column "c", return the index of a row
241 * that can be used to pivot the column in, with either an increase
242 * (sgn > 0) or a decrease (sgn < 0) of the corresponding variable.
243 * If "var" is not NULL, then the row returned will be different from
244 * the one associated with "var".
246 * Each row in the tableau is of the form
248 * x_r = a_r0 + \sum_i a_ri x_i
250 * Only rows with x_r >= 0 and with the sign of a_ri opposite to "sgn"
251 * impose any limit on the increase or decrease in the value of x_c
252 * and this bound is equal to a_r0 / |a_rc|. We are therefore looking
253 * for the row with the smallest (most stringent) such bound.
254 * Note that the common denominator of each row drops out of the fraction.
255 * To check if row j has a smaller bound than row r, i.e.,
256 * a_j0 / |a_jc| < a_r0 / |a_rc| or a_j0 |a_rc| < a_r0 |a_jc|,
257 * we check if -sign(a_jc) (a_j0 a_rc - a_r0 a_jc) < 0,
258 * where -sign(a_jc) is equal to "sgn".
260 static int pivot_row(struct isl_tab
*tab
,
261 struct isl_tab_var
*var
, int sgn
, int c
)
268 for (j
= tab
->n_redundant
; j
< tab
->n_row
; ++j
) {
269 if (var
&& j
== var
->index
)
271 if (!var_from_row(tab
, j
)->is_nonneg
)
273 if (sgn
* isl_int_sgn(tab
->mat
->row
[j
][2 + c
]) >= 0)
279 isl_int_mul(t
, tab
->mat
->row
[r
][1], tab
->mat
->row
[j
][2 + c
]);
280 isl_int_submul(t
, tab
->mat
->row
[j
][1], tab
->mat
->row
[r
][2 + c
]);
281 tsgn
= sgn
* isl_int_sgn(t
);
282 if (tsgn
< 0 || (tsgn
== 0 &&
283 tab
->row_var
[j
] < tab
->row_var
[r
]))
290 /* Find a pivot (row and col) that will increase (sgn > 0) or decrease
291 * (sgn < 0) the value of row variable var.
292 * If not NULL, then skip_var is a row variable that should be ignored
293 * while looking for a pivot row. It is usually equal to var.
295 * As the given row in the tableau is of the form
297 * x_r = a_r0 + \sum_i a_ri x_i
299 * we need to find a column such that the sign of a_ri is equal to "sgn"
300 * (such that an increase in x_i will have the desired effect) or a
301 * column with a variable that may attain negative values.
302 * If a_ri is positive, then we need to move x_i in the same direction
303 * to obtain the desired effect. Otherwise, x_i has to move in the
304 * opposite direction.
306 static void find_pivot(struct isl_tab
*tab
,
307 struct isl_tab_var
*var
, struct isl_tab_var
*skip_var
,
308 int sgn
, int *row
, int *col
)
315 isl_assert(tab
->mat
->ctx
, var
->is_row
, return);
316 tr
= tab
->mat
->row
[var
->index
];
319 for (j
= tab
->n_dead
; j
< tab
->n_col
; ++j
) {
320 if (isl_int_is_zero(tr
[2 + j
]))
322 if (isl_int_sgn(tr
[2 + j
]) != sgn
&&
323 var_from_col(tab
, j
)->is_nonneg
)
325 if (c
< 0 || tab
->col_var
[j
] < tab
->col_var
[c
])
331 sgn
*= isl_int_sgn(tr
[2 + c
]);
332 r
= pivot_row(tab
, skip_var
, sgn
, c
);
333 *row
= r
< 0 ? var
->index
: r
;
337 /* Return 1 if row "row" represents an obviously redundant inequality.
339 * - it represents an inequality or a variable
340 * - that is the sum of a non-negative sample value and a positive
341 * combination of zero or more non-negative variables.
343 static int is_redundant(struct isl_tab
*tab
, int row
)
347 if (tab
->row_var
[row
] < 0 && !var_from_row(tab
, row
)->is_nonneg
)
350 if (isl_int_is_neg(tab
->mat
->row
[row
][1]))
353 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
354 if (isl_int_is_zero(tab
->mat
->row
[row
][2 + i
]))
356 if (isl_int_is_neg(tab
->mat
->row
[row
][2 + i
]))
358 if (!var_from_col(tab
, i
)->is_nonneg
)
364 static void swap_rows(struct isl_tab
*tab
, int row1
, int row2
)
367 t
= tab
->row_var
[row1
];
368 tab
->row_var
[row1
] = tab
->row_var
[row2
];
369 tab
->row_var
[row2
] = t
;
370 var_from_row(tab
, row1
)->index
= row1
;
371 var_from_row(tab
, row2
)->index
= row2
;
372 tab
->mat
= isl_mat_swap_rows(tab
->mat
, row1
, row2
);
375 static void push(struct isl_tab
*tab
,
376 enum isl_tab_undo_type type
, struct isl_tab_var
*var
)
378 struct isl_tab_undo
*undo
;
383 undo
= isl_alloc_type(tab
->mat
->ctx
, struct isl_tab_undo
);
392 else if (var
->is_row
)
393 undo
->var_index
= tab
->row_var
[var
->index
];
395 undo
->var_index
= tab
->col_var
[var
->index
];
396 undo
->next
= tab
->top
;
400 /* Mark row with index "row" as being redundant.
401 * If we may need to undo the operation or if the row represents
402 * a variable of the original problem, the row is kept,
403 * but no longer considered when looking for a pivot row.
404 * Otherwise, the row is simply removed.
406 * The row may be interchanged with some other row. If it
407 * is interchanged with a later row, return 1. Otherwise return 0.
408 * If the rows are checked in order in the calling function,
409 * then a return value of 1 means that the row with the given
410 * row number may now contain a different row that hasn't been checked yet.
412 static int mark_redundant(struct isl_tab
*tab
, int row
)
414 struct isl_tab_var
*var
= var_from_row(tab
, row
);
415 var
->is_redundant
= 1;
416 isl_assert(tab
->mat
->ctx
, row
>= tab
->n_redundant
, return);
417 if (tab
->need_undo
|| tab
->row_var
[row
] >= 0) {
418 if (tab
->row_var
[row
] >= 0 && !var
->is_nonneg
) {
420 push(tab
, isl_tab_undo_nonneg
, var
);
422 if (row
!= tab
->n_redundant
)
423 swap_rows(tab
, row
, tab
->n_redundant
);
424 push(tab
, isl_tab_undo_redundant
, var
);
428 if (row
!= tab
->n_row
- 1)
429 swap_rows(tab
, row
, tab
->n_row
- 1);
430 var_from_row(tab
, tab
->n_row
- 1)->index
= -1;
436 static struct isl_tab
*mark_empty(struct isl_tab
*tab
)
438 if (!tab
->empty
&& tab
->need_undo
)
439 push(tab
, isl_tab_undo_empty
, NULL
);
444 /* Given a row number "row" and a column number "col", pivot the tableau
445 * such that the associated variables are interchanged.
446 * The given row in the tableau expresses
448 * x_r = a_r0 + \sum_i a_ri x_i
452 * x_c = 1/a_rc x_r - a_r0/a_rc + sum_{i \ne r} -a_ri/a_rc
454 * Substituting this equality into the other rows
456 * x_j = a_j0 + \sum_i a_ji x_i
458 * with a_jc \ne 0, we obtain
460 * 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
467 * where i is any other column and j is any other row,
468 * is therefore transformed into
470 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
471 * 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)
473 * The transformation is performed along the following steps
478 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
481 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
482 * n_jc/(|n_rc| d_j) n_ji/(|n_rc| d_j)
484 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
485 * n_jc/(|n_rc| d_j) (n_ji |n_rc|)/(|n_rc| d_j)
487 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
488 * n_jc/(|n_rc| d_j) (n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j)
490 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
491 * 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)
494 static void pivot(struct isl_tab
*tab
, int row
, int col
)
499 struct isl_mat
*mat
= tab
->mat
;
500 struct isl_tab_var
*var
;
502 isl_int_swap(mat
->row
[row
][0], mat
->row
[row
][2 + col
]);
503 sgn
= isl_int_sgn(mat
->row
[row
][0]);
505 isl_int_neg(mat
->row
[row
][0], mat
->row
[row
][0]);
506 isl_int_neg(mat
->row
[row
][2 + col
], mat
->row
[row
][2 + col
]);
508 for (j
= 0; j
< 1 + tab
->n_col
; ++j
) {
511 isl_int_neg(mat
->row
[row
][1 + j
], mat
->row
[row
][1 + j
]);
513 if (!isl_int_is_one(mat
->row
[row
][0]))
514 isl_seq_normalize(mat
->row
[row
], 2 + tab
->n_col
);
515 for (i
= 0; i
< tab
->n_row
; ++i
) {
518 if (isl_int_is_zero(mat
->row
[i
][2 + col
]))
520 isl_int_mul(mat
->row
[i
][0], mat
->row
[i
][0], mat
->row
[row
][0]);
521 for (j
= 0; j
< 1 + tab
->n_col
; ++j
) {
524 isl_int_mul(mat
->row
[i
][1 + j
],
525 mat
->row
[i
][1 + j
], mat
->row
[row
][0]);
526 isl_int_addmul(mat
->row
[i
][1 + j
],
527 mat
->row
[i
][2 + col
], mat
->row
[row
][1 + j
]);
529 isl_int_mul(mat
->row
[i
][2 + col
],
530 mat
->row
[i
][2 + col
], mat
->row
[row
][2 + col
]);
531 if (!isl_int_is_one(mat
->row
[i
][0]))
532 isl_seq_normalize(mat
->row
[i
], 2 + tab
->n_col
);
534 t
= tab
->row_var
[row
];
535 tab
->row_var
[row
] = tab
->col_var
[col
];
536 tab
->col_var
[col
] = t
;
537 var
= var_from_row(tab
, row
);
540 var
= var_from_col(tab
, col
);
545 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
546 if (isl_int_is_zero(mat
->row
[i
][2 + col
]))
548 if (!var_from_row(tab
, i
)->frozen
&&
549 is_redundant(tab
, i
))
550 if (mark_redundant(tab
, i
))
555 /* If "var" represents a column variable, then pivot is up (sgn > 0)
556 * or down (sgn < 0) to a row. The variable is assumed not to be
557 * unbounded in the specified direction.
558 * If sgn = 0, then the variable is unbounded in both directions,
559 * and we pivot with any row we can find.
561 static void to_row(struct isl_tab
*tab
, struct isl_tab_var
*var
, int sign
)
569 for (r
= tab
->n_redundant
; r
< tab
->n_row
; ++r
)
570 if (!isl_int_is_zero(tab
->mat
->row
[r
][2 + var
->index
]))
572 isl_assert(tab
->mat
->ctx
, r
< tab
->n_row
, return);
574 r
= pivot_row(tab
, NULL
, sign
, var
->index
);
575 isl_assert(tab
->mat
->ctx
, r
>= 0, return);
578 pivot(tab
, r
, var
->index
);
581 static void check_table(struct isl_tab
*tab
)
587 for (i
= 0; i
< tab
->n_row
; ++i
) {
588 if (!var_from_row(tab
, i
)->is_nonneg
)
590 assert(!isl_int_is_neg(tab
->mat
->row
[i
][1]));
594 /* Return the sign of the maximal value of "var".
595 * If the sign is not negative, then on return from this function,
596 * the sample value will also be non-negative.
598 * If "var" is manifestly unbounded wrt positive values, we are done.
599 * Otherwise, we pivot the variable up to a row if needed
600 * Then we continue pivoting down until either
601 * - no more down pivots can be performed
602 * - the sample value is positive
603 * - the variable is pivoted into a manifestly unbounded column
605 static int sign_of_max(struct isl_tab
*tab
, struct isl_tab_var
*var
)
609 if (max_is_manifestly_unbounded(tab
, var
))
612 while (!isl_int_is_pos(tab
->mat
->row
[var
->index
][1])) {
613 find_pivot(tab
, var
, var
, 1, &row
, &col
);
615 return isl_int_sgn(tab
->mat
->row
[var
->index
][1]);
616 pivot(tab
, row
, col
);
617 if (!var
->is_row
) /* manifestly unbounded */
623 /* Perform pivots until the row variable "var" has a non-negative
624 * sample value or until no more upward pivots can be performed.
625 * Return the sign of the sample value after the pivots have been
628 static int restore_row(struct isl_tab
*tab
, struct isl_tab_var
*var
)
632 while (isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
633 find_pivot(tab
, var
, var
, 1, &row
, &col
);
636 pivot(tab
, row
, col
);
637 if (!var
->is_row
) /* manifestly unbounded */
640 return isl_int_sgn(tab
->mat
->row
[var
->index
][1]);
643 /* Perform pivots until we are sure that the row variable "var"
644 * can attain non-negative values. After return from this
645 * function, "var" is still a row variable, but its sample
646 * value may not be non-negative, even if the function returns 1.
648 static int at_least_zero(struct isl_tab
*tab
, struct isl_tab_var
*var
)
652 while (isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
653 find_pivot(tab
, var
, var
, 1, &row
, &col
);
656 if (row
== var
->index
) /* manifestly unbounded */
658 pivot(tab
, row
, col
);
660 return !isl_int_is_neg(tab
->mat
->row
[var
->index
][1]);
663 /* Return a negative value if "var" can attain negative values.
664 * Return a non-negative value otherwise.
666 * If "var" is manifestly unbounded wrt negative values, we are done.
667 * Otherwise, if var is in a column, we can pivot it down to a row.
668 * Then we continue pivoting down until either
669 * - the pivot would result in a manifestly unbounded column
670 * => we don't perform the pivot, but simply return -1
671 * - no more down pivots can be performed
672 * - the sample value is negative
673 * If the sample value becomes negative and the variable is supposed
674 * to be nonnegative, then we undo the last pivot.
675 * However, if the last pivot has made the pivoting variable
676 * obviously redundant, then it may have moved to another row.
677 * In that case we look for upward pivots until we reach a non-negative
680 static int sign_of_min(struct isl_tab
*tab
, struct isl_tab_var
*var
)
683 struct isl_tab_var
*pivot_var
;
685 if (min_is_manifestly_unbounded(tab
, var
))
689 row
= pivot_row(tab
, NULL
, -1, col
);
690 pivot_var
= var_from_col(tab
, col
);
691 pivot(tab
, row
, col
);
692 if (var
->is_redundant
)
694 if (isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
695 if (var
->is_nonneg
) {
696 if (!pivot_var
->is_redundant
&&
697 pivot_var
->index
== row
)
698 pivot(tab
, row
, col
);
700 restore_row(tab
, var
);
705 if (var
->is_redundant
)
707 while (!isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
708 find_pivot(tab
, var
, var
, -1, &row
, &col
);
709 if (row
== var
->index
)
712 return isl_int_sgn(tab
->mat
->row
[var
->index
][1]);
713 pivot_var
= var_from_col(tab
, col
);
714 pivot(tab
, row
, col
);
715 if (var
->is_redundant
)
718 if (var
->is_nonneg
) {
719 /* pivot back to non-negative value */
720 if (!pivot_var
->is_redundant
&& pivot_var
->index
== row
)
721 pivot(tab
, row
, col
);
723 restore_row(tab
, var
);
728 /* Return 1 if "var" can attain values <= -1.
729 * Return 0 otherwise.
731 * The sample value of "var" is assumed to be non-negative when the
732 * the function is called and will be made non-negative again before
733 * the function returns.
735 static int min_at_most_neg_one(struct isl_tab
*tab
, struct isl_tab_var
*var
)
738 struct isl_tab_var
*pivot_var
;
740 if (min_is_manifestly_unbounded(tab
, var
))
744 row
= pivot_row(tab
, NULL
, -1, col
);
745 pivot_var
= var_from_col(tab
, col
);
746 pivot(tab
, row
, col
);
747 if (var
->is_redundant
)
749 if (isl_int_is_neg(tab
->mat
->row
[var
->index
][1]) &&
750 isl_int_abs_ge(tab
->mat
->row
[var
->index
][1],
751 tab
->mat
->row
[var
->index
][0])) {
752 if (var
->is_nonneg
) {
753 if (!pivot_var
->is_redundant
&&
754 pivot_var
->index
== row
)
755 pivot(tab
, row
, col
);
757 restore_row(tab
, var
);
762 if (var
->is_redundant
)
765 find_pivot(tab
, var
, var
, -1, &row
, &col
);
766 if (row
== var
->index
)
770 pivot_var
= var_from_col(tab
, col
);
771 pivot(tab
, row
, col
);
772 if (var
->is_redundant
)
774 } while (!isl_int_is_neg(tab
->mat
->row
[var
->index
][1]) ||
775 isl_int_abs_lt(tab
->mat
->row
[var
->index
][1],
776 tab
->mat
->row
[var
->index
][0]));
777 if (var
->is_nonneg
) {
778 /* pivot back to non-negative value */
779 if (!pivot_var
->is_redundant
&& pivot_var
->index
== row
)
780 pivot(tab
, row
, col
);
781 restore_row(tab
, var
);
786 /* Return 1 if "var" can attain values >= 1.
787 * Return 0 otherwise.
789 static int at_least_one(struct isl_tab
*tab
, struct isl_tab_var
*var
)
794 if (max_is_manifestly_unbounded(tab
, var
))
797 r
= tab
->mat
->row
[var
->index
];
798 while (isl_int_lt(r
[1], r
[0])) {
799 find_pivot(tab
, var
, var
, 1, &row
, &col
);
801 return isl_int_ge(r
[1], r
[0]);
802 if (row
== var
->index
) /* manifestly unbounded */
804 pivot(tab
, row
, col
);
809 static void swap_cols(struct isl_tab
*tab
, int col1
, int col2
)
812 t
= tab
->col_var
[col1
];
813 tab
->col_var
[col1
] = tab
->col_var
[col2
];
814 tab
->col_var
[col2
] = t
;
815 var_from_col(tab
, col1
)->index
= col1
;
816 var_from_col(tab
, col2
)->index
= col2
;
817 tab
->mat
= isl_mat_swap_cols(tab
->mat
, 2 + col1
, 2 + col2
);
820 /* Mark column with index "col" as representing a zero variable.
821 * If we may need to undo the operation the column is kept,
822 * but no longer considered.
823 * Otherwise, the column is simply removed.
825 * The column may be interchanged with some other column. If it
826 * is interchanged with a later column, return 1. Otherwise return 0.
827 * If the columns are checked in order in the calling function,
828 * then a return value of 1 means that the column with the given
829 * column number may now contain a different column that
830 * hasn't been checked yet.
832 static int kill_col(struct isl_tab
*tab
, int col
)
834 var_from_col(tab
, col
)->is_zero
= 1;
835 if (tab
->need_undo
) {
836 push(tab
, isl_tab_undo_zero
, var_from_col(tab
, col
));
837 if (col
!= tab
->n_dead
)
838 swap_cols(tab
, col
, tab
->n_dead
);
842 if (col
!= tab
->n_col
- 1)
843 swap_cols(tab
, col
, tab
->n_col
- 1);
844 var_from_col(tab
, tab
->n_col
- 1)->index
= -1;
850 /* Row variable "var" is non-negative and cannot attain any values
851 * larger than zero. This means that the coefficients of the unrestricted
852 * column variables are zero and that the coefficients of the non-negative
853 * column variables are zero or negative.
854 * Each of the non-negative variables with a negative coefficient can
855 * then also be written as the negative sum of non-negative variables
856 * and must therefore also be zero.
858 static void close_row(struct isl_tab
*tab
, struct isl_tab_var
*var
)
861 struct isl_mat
*mat
= tab
->mat
;
863 isl_assert(tab
->mat
->ctx
, var
->is_nonneg
, return);
865 for (j
= tab
->n_dead
; j
< tab
->n_col
; ++j
) {
866 if (isl_int_is_zero(mat
->row
[var
->index
][2 + j
]))
868 isl_assert(tab
->mat
->ctx
,
869 isl_int_is_neg(mat
->row
[var
->index
][2 + j
]), return);
870 if (kill_col(tab
, j
))
873 mark_redundant(tab
, var
->index
);
876 /* Add a constraint to the tableau and allocate a row for it.
877 * Return the index into the constraint array "con".
879 static int allocate_con(struct isl_tab
*tab
)
883 isl_assert(tab
->mat
->ctx
, tab
->n_row
< tab
->mat
->n_row
, return -1);
886 tab
->con
[r
].index
= tab
->n_row
;
887 tab
->con
[r
].is_row
= 1;
888 tab
->con
[r
].is_nonneg
= 0;
889 tab
->con
[r
].is_zero
= 0;
890 tab
->con
[r
].is_redundant
= 0;
891 tab
->con
[r
].frozen
= 0;
892 tab
->row_var
[tab
->n_row
] = ~r
;
896 push(tab
, isl_tab_undo_allocate
, &tab
->con
[r
]);
901 /* Add a row to the tableau. The row is given as an affine combination
902 * of the original variables and needs to be expressed in terms of the
905 * We add each term in turn.
906 * If r = n/d_r is the current sum and we need to add k x, then
907 * if x is a column variable, we increase the numerator of
908 * this column by k d_r
909 * if x = f/d_x is a row variable, then the new representation of r is
911 * n k f d_x/g n + d_r/g k f m/d_r n + m/d_g k f
912 * --- + --- = ------------------- = -------------------
913 * d_r d_r d_r d_x/g m
915 * with g the gcd of d_r and d_x and m the lcm of d_r and d_x.
917 static int add_row(struct isl_tab
*tab
, isl_int
*line
)
924 r
= allocate_con(tab
);
930 row
= tab
->mat
->row
[tab
->con
[r
].index
];
931 isl_int_set_si(row
[0], 1);
932 isl_int_set(row
[1], line
[0]);
933 isl_seq_clr(row
+ 2, tab
->n_col
);
934 for (i
= 0; i
< tab
->n_var
; ++i
) {
935 if (tab
->var
[i
].is_zero
)
937 if (tab
->var
[i
].is_row
) {
939 row
[0], tab
->mat
->row
[tab
->var
[i
].index
][0]);
940 isl_int_swap(a
, row
[0]);
941 isl_int_divexact(a
, row
[0], a
);
943 row
[0], tab
->mat
->row
[tab
->var
[i
].index
][0]);
944 isl_int_mul(b
, b
, line
[1 + i
]);
945 isl_seq_combine(row
+ 1, a
, row
+ 1,
946 b
, tab
->mat
->row
[tab
->var
[i
].index
] + 1,
949 isl_int_addmul(row
[2 + tab
->var
[i
].index
],
950 line
[1 + i
], row
[0]);
952 isl_seq_normalize(row
, 2 + tab
->n_col
);
959 static int drop_row(struct isl_tab
*tab
, int row
)
961 isl_assert(tab
->mat
->ctx
, ~tab
->row_var
[row
] == tab
->n_con
- 1, return -1);
962 if (row
!= tab
->n_row
- 1)
963 swap_rows(tab
, row
, tab
->n_row
- 1);
969 /* Add inequality "ineq" and check if it conflicts with the
970 * previously added constraints or if it is obviously redundant.
972 struct isl_tab
*isl_tab_add_ineq(struct isl_tab
*tab
, isl_int
*ineq
)
979 r
= add_row(tab
, ineq
);
982 tab
->con
[r
].is_nonneg
= 1;
983 push(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
984 if (is_redundant(tab
, tab
->con
[r
].index
)) {
985 mark_redundant(tab
, tab
->con
[r
].index
);
989 sgn
= restore_row(tab
, &tab
->con
[r
]);
991 return mark_empty(tab
);
992 if (tab
->con
[r
].is_row
&& is_redundant(tab
, tab
->con
[r
].index
))
993 mark_redundant(tab
, tab
->con
[r
].index
);
1000 /* Pivot a non-negative variable down until it reaches the value zero
1001 * and then pivot the variable into a column position.
1003 static int to_col(struct isl_tab
*tab
, struct isl_tab_var
*var
)
1011 while (isl_int_is_pos(tab
->mat
->row
[var
->index
][1])) {
1012 find_pivot(tab
, var
, NULL
, -1, &row
, &col
);
1013 isl_assert(tab
->mat
->ctx
, row
!= -1, return -1);
1014 pivot(tab
, row
, col
);
1019 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
)
1020 if (!isl_int_is_zero(tab
->mat
->row
[var
->index
][2 + i
]))
1023 isl_assert(tab
->mat
->ctx
, i
< tab
->n_col
, return -1);
1024 pivot(tab
, var
->index
, i
);
1029 /* We assume Gaussian elimination has been performed on the equalities.
1030 * The equalities can therefore never conflict.
1031 * Adding the equalities is currently only really useful for a later call
1032 * to isl_tab_ineq_type.
1034 static struct isl_tab
*add_eq(struct isl_tab
*tab
, isl_int
*eq
)
1041 r
= add_row(tab
, eq
);
1045 r
= tab
->con
[r
].index
;
1046 i
= isl_seq_first_non_zero(tab
->mat
->row
[r
] + 2 + tab
->n_dead
,
1047 tab
->n_col
- tab
->n_dead
);
1048 isl_assert(tab
->mat
->ctx
, i
>= 0, goto error
);
1060 /* Add an equality that is known to be valid for the given tableau.
1062 struct isl_tab
*isl_tab_add_valid_eq(struct isl_tab
*tab
, isl_int
*eq
)
1064 struct isl_tab_var
*var
;
1070 r
= add_row(tab
, eq
);
1076 if (isl_int_is_neg(tab
->mat
->row
[r
][1]))
1077 isl_seq_neg(tab
->mat
->row
[r
] + 1, tab
->mat
->row
[r
] + 1,
1080 if (to_col(tab
, var
) < 0)
1083 kill_col(tab
, var
->index
);
1091 struct isl_tab
*isl_tab_from_basic_map(struct isl_basic_map
*bmap
)
1094 struct isl_tab
*tab
;
1098 tab
= isl_tab_alloc(bmap
->ctx
,
1099 isl_basic_map_total_dim(bmap
) + bmap
->n_ineq
+ 1,
1100 isl_basic_map_total_dim(bmap
));
1103 tab
->rational
= ISL_F_ISSET(bmap
, ISL_BASIC_MAP_RATIONAL
);
1104 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_EMPTY
))
1105 return mark_empty(tab
);
1106 for (i
= 0; i
< bmap
->n_eq
; ++i
) {
1107 tab
= add_eq(tab
, bmap
->eq
[i
]);
1111 for (i
= 0; i
< bmap
->n_ineq
; ++i
) {
1112 tab
= isl_tab_add_ineq(tab
, bmap
->ineq
[i
]);
1113 if (!tab
|| tab
->empty
)
1119 struct isl_tab
*isl_tab_from_basic_set(struct isl_basic_set
*bset
)
1121 return isl_tab_from_basic_map((struct isl_basic_map
*)bset
);
1124 /* Construct a tableau corresponding to the recession cone of "bmap".
1126 struct isl_tab
*isl_tab_from_recession_cone(struct isl_basic_map
*bmap
)
1130 struct isl_tab
*tab
;
1134 tab
= isl_tab_alloc(bmap
->ctx
, bmap
->n_eq
+ bmap
->n_ineq
,
1135 isl_basic_map_total_dim(bmap
));
1138 tab
->rational
= ISL_F_ISSET(bmap
, ISL_BASIC_MAP_RATIONAL
);
1141 for (i
= 0; i
< bmap
->n_eq
; ++i
) {
1142 isl_int_swap(bmap
->eq
[i
][0], cst
);
1143 tab
= add_eq(tab
, bmap
->eq
[i
]);
1144 isl_int_swap(bmap
->eq
[i
][0], cst
);
1148 for (i
= 0; i
< bmap
->n_ineq
; ++i
) {
1150 isl_int_swap(bmap
->ineq
[i
][0], cst
);
1151 r
= add_row(tab
, bmap
->ineq
[i
]);
1152 isl_int_swap(bmap
->ineq
[i
][0], cst
);
1155 tab
->con
[r
].is_nonneg
= 1;
1156 push(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
1167 /* Assuming "tab" is the tableau of a cone, check if the cone is
1168 * bounded, i.e., if it is empty or only contains the origin.
1170 int isl_tab_cone_is_bounded(struct isl_tab
*tab
)
1178 if (tab
->n_dead
== tab
->n_col
)
1182 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1183 struct isl_tab_var
*var
;
1184 var
= var_from_row(tab
, i
);
1185 if (!var
->is_nonneg
)
1187 if (sign_of_max(tab
, var
) != 0)
1189 close_row(tab
, var
);
1192 if (tab
->n_dead
== tab
->n_col
)
1194 if (i
== tab
->n_row
)
1199 int isl_tab_sample_is_integer(struct isl_tab
*tab
)
1206 for (i
= 0; i
< tab
->n_var
; ++i
) {
1208 if (!tab
->var
[i
].is_row
)
1210 row
= tab
->var
[i
].index
;
1211 if (!isl_int_is_divisible_by(tab
->mat
->row
[row
][1],
1212 tab
->mat
->row
[row
][0]))
1218 static struct isl_vec
*extract_integer_sample(struct isl_tab
*tab
)
1221 struct isl_vec
*vec
;
1223 vec
= isl_vec_alloc(tab
->mat
->ctx
, 1 + tab
->n_var
);
1227 isl_int_set_si(vec
->block
.data
[0], 1);
1228 for (i
= 0; i
< tab
->n_var
; ++i
) {
1229 if (!tab
->var
[i
].is_row
)
1230 isl_int_set_si(vec
->block
.data
[1 + i
], 0);
1232 int row
= tab
->var
[i
].index
;
1233 isl_int_divexact(vec
->block
.data
[1 + i
],
1234 tab
->mat
->row
[row
][1], tab
->mat
->row
[row
][0]);
1241 struct isl_vec
*isl_tab_get_sample_value(struct isl_tab
*tab
)
1244 struct isl_vec
*vec
;
1250 vec
= isl_vec_alloc(tab
->mat
->ctx
, 1 + tab
->n_var
);
1256 isl_int_set_si(vec
->block
.data
[0], 1);
1257 for (i
= 0; i
< tab
->n_var
; ++i
) {
1259 if (!tab
->var
[i
].is_row
) {
1260 isl_int_set_si(vec
->block
.data
[1 + i
], 0);
1263 row
= tab
->var
[i
].index
;
1264 isl_int_gcd(m
, vec
->block
.data
[0], tab
->mat
->row
[row
][0]);
1265 isl_int_divexact(m
, tab
->mat
->row
[row
][0], m
);
1266 isl_seq_scale(vec
->block
.data
, vec
->block
.data
, m
, 1 + i
);
1267 isl_int_divexact(m
, vec
->block
.data
[0], tab
->mat
->row
[row
][0]);
1268 isl_int_mul(vec
->block
.data
[1 + i
], m
, tab
->mat
->row
[row
][1]);
1270 isl_seq_normalize(vec
->block
.data
, vec
->size
);
1276 /* Update "bmap" based on the results of the tableau "tab".
1277 * In particular, implicit equalities are made explicit, redundant constraints
1278 * are removed and if the sample value happens to be integer, it is stored
1279 * in "bmap" (unless "bmap" already had an integer sample).
1281 * The tableau is assumed to have been created from "bmap" using
1282 * isl_tab_from_basic_map.
1284 struct isl_basic_map
*isl_basic_map_update_from_tab(struct isl_basic_map
*bmap
,
1285 struct isl_tab
*tab
)
1297 bmap
= isl_basic_map_set_to_empty(bmap
);
1299 for (i
= bmap
->n_ineq
- 1; i
>= 0; --i
) {
1300 if (isl_tab_is_equality(tab
, n_eq
+ i
))
1301 isl_basic_map_inequality_to_equality(bmap
, i
);
1302 else if (isl_tab_is_redundant(tab
, n_eq
+ i
))
1303 isl_basic_map_drop_inequality(bmap
, i
);
1305 if (!tab
->rational
&&
1306 !bmap
->sample
&& isl_tab_sample_is_integer(tab
))
1307 bmap
->sample
= extract_integer_sample(tab
);
1311 struct isl_basic_set
*isl_basic_set_update_from_tab(struct isl_basic_set
*bset
,
1312 struct isl_tab
*tab
)
1314 return (struct isl_basic_set
*)isl_basic_map_update_from_tab(
1315 (struct isl_basic_map
*)bset
, tab
);
1318 /* Given a non-negative variable "var", add a new non-negative variable
1319 * that is the opposite of "var", ensuring that var can only attain the
1321 * If var = n/d is a row variable, then the new variable = -n/d.
1322 * If var is a column variables, then the new variable = -var.
1323 * If the new variable cannot attain non-negative values, then
1324 * the resulting tableau is empty.
1325 * Otherwise, we know the value will be zero and we close the row.
1327 static struct isl_tab
*cut_to_hyperplane(struct isl_tab
*tab
,
1328 struct isl_tab_var
*var
)
1334 if (extend_cons(tab
, 1) < 0)
1338 tab
->con
[r
].index
= tab
->n_row
;
1339 tab
->con
[r
].is_row
= 1;
1340 tab
->con
[r
].is_nonneg
= 0;
1341 tab
->con
[r
].is_zero
= 0;
1342 tab
->con
[r
].is_redundant
= 0;
1343 tab
->con
[r
].frozen
= 0;
1344 tab
->row_var
[tab
->n_row
] = ~r
;
1345 row
= tab
->mat
->row
[tab
->n_row
];
1348 isl_int_set(row
[0], tab
->mat
->row
[var
->index
][0]);
1349 isl_seq_neg(row
+ 1,
1350 tab
->mat
->row
[var
->index
] + 1, 1 + tab
->n_col
);
1352 isl_int_set_si(row
[0], 1);
1353 isl_seq_clr(row
+ 1, 1 + tab
->n_col
);
1354 isl_int_set_si(row
[2 + var
->index
], -1);
1359 push(tab
, isl_tab_undo_allocate
, &tab
->con
[r
]);
1361 sgn
= sign_of_max(tab
, &tab
->con
[r
]);
1363 return mark_empty(tab
);
1364 tab
->con
[r
].is_nonneg
= 1;
1365 push(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
1367 close_row(tab
, &tab
->con
[r
]);
1375 /* Given a tableau "tab" and an inequality constraint "con" of the tableau,
1376 * relax the inequality by one. That is, the inequality r >= 0 is replaced
1377 * by r' = r + 1 >= 0.
1378 * If r is a row variable, we simply increase the constant term by one
1379 * (taking into account the denominator).
1380 * If r is a column variable, then we need to modify each row that
1381 * refers to r = r' - 1 by substituting this equality, effectively
1382 * subtracting the coefficient of the column from the constant.
1384 struct isl_tab
*isl_tab_relax(struct isl_tab
*tab
, int con
)
1386 struct isl_tab_var
*var
;
1390 var
= &tab
->con
[con
];
1392 if (!var
->is_row
&& !max_is_manifestly_unbounded(tab
, var
))
1393 to_row(tab
, var
, 1);
1396 isl_int_add(tab
->mat
->row
[var
->index
][1],
1397 tab
->mat
->row
[var
->index
][1], tab
->mat
->row
[var
->index
][0]);
1401 for (i
= 0; i
< tab
->n_row
; ++i
) {
1402 if (isl_int_is_zero(tab
->mat
->row
[i
][2 + var
->index
]))
1404 isl_int_sub(tab
->mat
->row
[i
][1], tab
->mat
->row
[i
][1],
1405 tab
->mat
->row
[i
][2 + var
->index
]);
1410 push(tab
, isl_tab_undo_relax
, var
);
1415 struct isl_tab
*isl_tab_select_facet(struct isl_tab
*tab
, int con
)
1420 return cut_to_hyperplane(tab
, &tab
->con
[con
]);
1423 static int may_be_equality(struct isl_tab
*tab
, int row
)
1425 return (tab
->rational
? isl_int_is_zero(tab
->mat
->row
[row
][1])
1426 : isl_int_lt(tab
->mat
->row
[row
][1],
1427 tab
->mat
->row
[row
][0])) &&
1428 isl_seq_first_non_zero(tab
->mat
->row
[row
] + 2 + tab
->n_dead
,
1429 tab
->n_col
- tab
->n_dead
) != -1;
1432 /* Check for (near) equalities among the constraints.
1433 * A constraint is an equality if it is non-negative and if
1434 * its maximal value is either
1435 * - zero (in case of rational tableaus), or
1436 * - strictly less than 1 (in case of integer tableaus)
1438 * We first mark all non-redundant and non-dead variables that
1439 * are not frozen and not obviously not an equality.
1440 * Then we iterate over all marked variables if they can attain
1441 * any values larger than zero or at least one.
1442 * If the maximal value is zero, we mark any column variables
1443 * that appear in the row as being zero and mark the row as being redundant.
1444 * Otherwise, if the maximal value is strictly less than one (and the
1445 * tableau is integer), then we restrict the value to being zero
1446 * by adding an opposite non-negative variable.
1448 struct isl_tab
*isl_tab_detect_equalities(struct isl_tab
*tab
)
1457 if (tab
->n_dead
== tab
->n_col
)
1461 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1462 struct isl_tab_var
*var
= var_from_row(tab
, i
);
1463 var
->marked
= !var
->frozen
&& var
->is_nonneg
&&
1464 may_be_equality(tab
, i
);
1468 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1469 struct isl_tab_var
*var
= var_from_col(tab
, i
);
1470 var
->marked
= !var
->frozen
&& var
->is_nonneg
;
1475 struct isl_tab_var
*var
;
1476 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1477 var
= var_from_row(tab
, i
);
1481 if (i
== tab
->n_row
) {
1482 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1483 var
= var_from_col(tab
, i
);
1487 if (i
== tab
->n_col
)
1492 if (sign_of_max(tab
, var
) == 0)
1493 close_row(tab
, var
);
1494 else if (!tab
->rational
&& !at_least_one(tab
, var
)) {
1495 tab
= cut_to_hyperplane(tab
, var
);
1496 return isl_tab_detect_equalities(tab
);
1498 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1499 var
= var_from_row(tab
, i
);
1502 if (may_be_equality(tab
, i
))
1512 /* Check for (near) redundant constraints.
1513 * A constraint is redundant if it is non-negative and if
1514 * its minimal value (temporarily ignoring the non-negativity) is either
1515 * - zero (in case of rational tableaus), or
1516 * - strictly larger than -1 (in case of integer tableaus)
1518 * We first mark all non-redundant and non-dead variables that
1519 * are not frozen and not obviously negatively unbounded.
1520 * Then we iterate over all marked variables if they can attain
1521 * any values smaller than zero or at most negative one.
1522 * If not, we mark the row as being redundant (assuming it hasn't
1523 * been detected as being obviously redundant in the mean time).
1525 struct isl_tab
*isl_tab_detect_redundant(struct isl_tab
*tab
)
1534 if (tab
->n_redundant
== tab
->n_row
)
1538 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1539 struct isl_tab_var
*var
= var_from_row(tab
, i
);
1540 var
->marked
= !var
->frozen
&& var
->is_nonneg
;
1544 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1545 struct isl_tab_var
*var
= var_from_col(tab
, i
);
1546 var
->marked
= !var
->frozen
&& var
->is_nonneg
&&
1547 !min_is_manifestly_unbounded(tab
, var
);
1552 struct isl_tab_var
*var
;
1553 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1554 var
= var_from_row(tab
, i
);
1558 if (i
== tab
->n_row
) {
1559 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1560 var
= var_from_col(tab
, i
);
1564 if (i
== tab
->n_col
)
1569 if ((tab
->rational
? (sign_of_min(tab
, var
) >= 0)
1570 : !min_at_most_neg_one(tab
, var
)) &&
1572 mark_redundant(tab
, var
->index
);
1573 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1574 var
= var_from_col(tab
, i
);
1577 if (!min_is_manifestly_unbounded(tab
, var
))
1587 int isl_tab_is_equality(struct isl_tab
*tab
, int con
)
1593 if (tab
->con
[con
].is_zero
)
1595 if (tab
->con
[con
].is_redundant
)
1597 if (!tab
->con
[con
].is_row
)
1598 return tab
->con
[con
].index
< tab
->n_dead
;
1600 row
= tab
->con
[con
].index
;
1602 return isl_int_is_zero(tab
->mat
->row
[row
][1]) &&
1603 isl_seq_first_non_zero(tab
->mat
->row
[row
] + 2 + tab
->n_dead
,
1604 tab
->n_col
- tab
->n_dead
) == -1;
1607 /* Return the minimial value of the affine expression "f" with denominator
1608 * "denom" in *opt, *opt_denom, assuming the tableau is not empty and
1609 * the expression cannot attain arbitrarily small values.
1610 * If opt_denom is NULL, then *opt is rounded up to the nearest integer.
1611 * The return value reflects the nature of the result (empty, unbounded,
1612 * minmimal value returned in *opt).
1614 enum isl_lp_result
isl_tab_min(struct isl_tab
*tab
,
1615 isl_int
*f
, isl_int denom
, isl_int
*opt
, isl_int
*opt_denom
,
1619 enum isl_lp_result res
= isl_lp_ok
;
1620 struct isl_tab_var
*var
;
1621 struct isl_tab_undo
*snap
;
1624 return isl_lp_empty
;
1626 snap
= isl_tab_snap(tab
);
1627 r
= add_row(tab
, f
);
1629 return isl_lp_error
;
1631 isl_int_mul(tab
->mat
->row
[var
->index
][0],
1632 tab
->mat
->row
[var
->index
][0], denom
);
1635 find_pivot(tab
, var
, var
, -1, &row
, &col
);
1636 if (row
== var
->index
) {
1637 res
= isl_lp_unbounded
;
1642 pivot(tab
, row
, col
);
1644 if (isl_tab_rollback(tab
, snap
) < 0)
1645 return isl_lp_error
;
1646 if (ISL_FL_ISSET(flags
, ISL_TAB_SAVE_DUAL
)) {
1649 isl_vec_free(tab
->dual
);
1650 tab
->dual
= isl_vec_alloc(tab
->mat
->ctx
, 1 + tab
->n_con
);
1652 return isl_lp_error
;
1653 isl_int_set(tab
->dual
->el
[0], tab
->mat
->row
[var
->index
][0]);
1654 for (i
= 0; i
< tab
->n_con
; ++i
) {
1655 if (tab
->con
[i
].is_row
)
1656 isl_int_set_si(tab
->dual
->el
[1 + i
], 0);
1658 int pos
= 2 + tab
->con
[i
].index
;
1659 isl_int_set(tab
->dual
->el
[1 + i
],
1660 tab
->mat
->row
[var
->index
][pos
]);
1664 if (res
== isl_lp_ok
) {
1666 isl_int_set(*opt
, tab
->mat
->row
[var
->index
][1]);
1667 isl_int_set(*opt_denom
, tab
->mat
->row
[var
->index
][0]);
1669 isl_int_cdiv_q(*opt
, tab
->mat
->row
[var
->index
][1],
1670 tab
->mat
->row
[var
->index
][0]);
1675 int isl_tab_is_redundant(struct isl_tab
*tab
, int con
)
1682 if (tab
->con
[con
].is_zero
)
1684 if (tab
->con
[con
].is_redundant
)
1686 return tab
->con
[con
].is_row
&& tab
->con
[con
].index
< tab
->n_redundant
;
1689 /* Take a snapshot of the tableau that can be restored by s call to
1692 struct isl_tab_undo
*isl_tab_snap(struct isl_tab
*tab
)
1700 /* Undo the operation performed by isl_tab_relax.
1702 static void unrelax(struct isl_tab
*tab
, struct isl_tab_var
*var
)
1704 if (!var
->is_row
&& !max_is_manifestly_unbounded(tab
, var
))
1705 to_row(tab
, var
, 1);
1708 isl_int_sub(tab
->mat
->row
[var
->index
][1],
1709 tab
->mat
->row
[var
->index
][1], tab
->mat
->row
[var
->index
][0]);
1713 for (i
= 0; i
< tab
->n_row
; ++i
) {
1714 if (isl_int_is_zero(tab
->mat
->row
[i
][2 + var
->index
]))
1716 isl_int_add(tab
->mat
->row
[i
][1], tab
->mat
->row
[i
][1],
1717 tab
->mat
->row
[i
][2 + var
->index
]);
1723 static void perform_undo(struct isl_tab
*tab
, struct isl_tab_undo
*undo
)
1725 struct isl_tab_var
*var
= var_from_index(tab
, undo
->var_index
);
1726 switch(undo
->type
) {
1727 case isl_tab_undo_empty
:
1730 case isl_tab_undo_nonneg
:
1733 case isl_tab_undo_redundant
:
1734 var
->is_redundant
= 0;
1737 case isl_tab_undo_zero
:
1741 case isl_tab_undo_allocate
:
1743 if (!max_is_manifestly_unbounded(tab
, var
))
1744 to_row(tab
, var
, 1);
1745 else if (!min_is_manifestly_unbounded(tab
, var
))
1746 to_row(tab
, var
, -1);
1748 to_row(tab
, var
, 0);
1750 drop_row(tab
, var
->index
);
1752 case isl_tab_undo_relax
:
1758 /* Return the tableau to the state it was in when the snapshot "snap"
1761 int isl_tab_rollback(struct isl_tab
*tab
, struct isl_tab_undo
*snap
)
1763 struct isl_tab_undo
*undo
, *next
;
1769 for (undo
= tab
->top
; undo
&& undo
!= &tab
->bottom
; undo
= next
) {
1773 perform_undo(tab
, undo
);
1783 /* The given row "row" represents an inequality violated by all
1784 * points in the tableau. Check for some special cases of such
1785 * separating constraints.
1786 * In particular, if the row has been reduced to the constant -1,
1787 * then we know the inequality is adjacent (but opposite) to
1788 * an equality in the tableau.
1789 * If the row has been reduced to r = -1 -r', with r' an inequality
1790 * of the tableau, then the inequality is adjacent (but opposite)
1791 * to the inequality r'.
1793 static enum isl_ineq_type
separation_type(struct isl_tab
*tab
, unsigned row
)
1798 return isl_ineq_separate
;
1800 if (!isl_int_is_one(tab
->mat
->row
[row
][0]))
1801 return isl_ineq_separate
;
1802 if (!isl_int_is_negone(tab
->mat
->row
[row
][1]))
1803 return isl_ineq_separate
;
1805 pos
= isl_seq_first_non_zero(tab
->mat
->row
[row
] + 2 + tab
->n_dead
,
1806 tab
->n_col
- tab
->n_dead
);
1808 return isl_ineq_adj_eq
;
1810 if (!isl_int_is_negone(tab
->mat
->row
[row
][2 + tab
->n_dead
+ pos
]))
1811 return isl_ineq_separate
;
1813 pos
= isl_seq_first_non_zero(
1814 tab
->mat
->row
[row
] + 2 + tab
->n_dead
+ pos
+ 1,
1815 tab
->n_col
- tab
->n_dead
- pos
- 1);
1817 return pos
== -1 ? isl_ineq_adj_ineq
: isl_ineq_separate
;
1820 /* Check the effect of inequality "ineq" on the tableau "tab".
1822 * isl_ineq_redundant: satisfied by all points in the tableau
1823 * isl_ineq_separate: satisfied by no point in the tableau
1824 * isl_ineq_cut: satisfied by some by not all points
1825 * isl_ineq_adj_eq: adjacent to an equality
1826 * isl_ineq_adj_ineq: adjacent to an inequality.
1828 enum isl_ineq_type
isl_tab_ineq_type(struct isl_tab
*tab
, isl_int
*ineq
)
1830 enum isl_ineq_type type
= isl_ineq_error
;
1831 struct isl_tab_undo
*snap
= NULL
;
1836 return isl_ineq_error
;
1838 if (extend_cons(tab
, 1) < 0)
1839 return isl_ineq_error
;
1841 snap
= isl_tab_snap(tab
);
1843 con
= add_row(tab
, ineq
);
1847 row
= tab
->con
[con
].index
;
1848 if (is_redundant(tab
, row
))
1849 type
= isl_ineq_redundant
;
1850 else if (isl_int_is_neg(tab
->mat
->row
[row
][1]) &&
1852 isl_int_abs_ge(tab
->mat
->row
[row
][1],
1853 tab
->mat
->row
[row
][0]))) {
1854 if (at_least_zero(tab
, &tab
->con
[con
]))
1855 type
= isl_ineq_cut
;
1857 type
= separation_type(tab
, row
);
1858 } else if (tab
->rational
? (sign_of_min(tab
, &tab
->con
[con
]) < 0)
1859 : min_at_most_neg_one(tab
, &tab
->con
[con
]))
1860 type
= isl_ineq_cut
;
1862 type
= isl_ineq_redundant
;
1864 if (isl_tab_rollback(tab
, snap
))
1865 return isl_ineq_error
;
1868 isl_tab_rollback(tab
, snap
);
1869 return isl_ineq_error
;
1872 void isl_tab_dump(struct isl_tab
*tab
, FILE *out
, int indent
)
1878 fprintf(out
, "%*snull tab\n", indent
, "");
1881 fprintf(out
, "%*sn_redundant: %d, n_dead: %d", indent
, "",
1882 tab
->n_redundant
, tab
->n_dead
);
1884 fprintf(out
, ", rational");
1886 fprintf(out
, ", empty");
1888 fprintf(out
, "%*s[", indent
, "");
1889 for (i
= 0; i
< tab
->n_var
; ++i
) {
1892 fprintf(out
, "%c%d%s", tab
->var
[i
].is_row
? 'r' : 'c',
1894 tab
->var
[i
].is_zero
? " [=0]" :
1895 tab
->var
[i
].is_redundant
? " [R]" : "");
1897 fprintf(out
, "]\n");
1898 fprintf(out
, "%*s[", indent
, "");
1899 for (i
= 0; i
< tab
->n_con
; ++i
) {
1902 fprintf(out
, "%c%d%s", tab
->con
[i
].is_row
? 'r' : 'c',
1904 tab
->con
[i
].is_zero
? " [=0]" :
1905 tab
->con
[i
].is_redundant
? " [R]" : "");
1907 fprintf(out
, "]\n");
1908 fprintf(out
, "%*s[", indent
, "");
1909 for (i
= 0; i
< tab
->n_row
; ++i
) {
1912 fprintf(out
, "r%d: %d%s", i
, tab
->row_var
[i
],
1913 var_from_row(tab
, i
)->is_nonneg
? " [>=0]" : "");
1915 fprintf(out
, "]\n");
1916 fprintf(out
, "%*s[", indent
, "");
1917 for (i
= 0; i
< tab
->n_col
; ++i
) {
1920 fprintf(out
, "c%d: %d%s", i
, tab
->col_var
[i
],
1921 var_from_col(tab
, i
)->is_nonneg
? " [>=0]" : "");
1923 fprintf(out
, "]\n");
1924 r
= tab
->mat
->n_row
;
1925 tab
->mat
->n_row
= tab
->n_row
;
1926 c
= tab
->mat
->n_col
;
1927 tab
->mat
->n_col
= 2 + tab
->n_col
;
1928 isl_mat_dump(tab
->mat
, out
, indent
);
1929 tab
->mat
->n_row
= r
;
1930 tab
->mat
->n_col
= c
;