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_union(struct isl_tab
*tab
,
376 enum isl_tab_undo_type type
, union isl_tab_undo_val u
)
378 struct isl_tab_undo
*undo
;
383 undo
= isl_alloc_type(tab
->mat
->ctx
, struct isl_tab_undo
);
391 undo
->next
= tab
->top
;
395 void push_var(struct isl_tab
*tab
,
396 enum isl_tab_undo_type type
, struct isl_tab_var
*var
)
398 union isl_tab_undo_val u
;
400 u
.var_index
= tab
->row_var
[var
->index
];
402 u
.var_index
= tab
->col_var
[var
->index
];
403 push_union(tab
, type
, u
);
406 void push(struct isl_tab
*tab
, enum isl_tab_undo_type type
)
408 union isl_tab_undo_val u
= { 0 };
409 push_union(tab
, type
, u
);
412 /* Mark row with index "row" as being redundant.
413 * If we may need to undo the operation or if the row represents
414 * a variable of the original problem, the row is kept,
415 * but no longer considered when looking for a pivot row.
416 * Otherwise, the row is simply removed.
418 * The row may be interchanged with some other row. If it
419 * is interchanged with a later row, return 1. Otherwise return 0.
420 * If the rows are checked in order in the calling function,
421 * then a return value of 1 means that the row with the given
422 * row number may now contain a different row that hasn't been checked yet.
424 static int mark_redundant(struct isl_tab
*tab
, int row
)
426 struct isl_tab_var
*var
= var_from_row(tab
, row
);
427 var
->is_redundant
= 1;
428 isl_assert(tab
->mat
->ctx
, row
>= tab
->n_redundant
, return);
429 if (tab
->need_undo
|| tab
->row_var
[row
] >= 0) {
430 if (tab
->row_var
[row
] >= 0 && !var
->is_nonneg
) {
432 push_var(tab
, isl_tab_undo_nonneg
, var
);
434 if (row
!= tab
->n_redundant
)
435 swap_rows(tab
, row
, tab
->n_redundant
);
436 push_var(tab
, isl_tab_undo_redundant
, var
);
440 if (row
!= tab
->n_row
- 1)
441 swap_rows(tab
, row
, tab
->n_row
- 1);
442 var_from_row(tab
, tab
->n_row
- 1)->index
= -1;
448 static struct isl_tab
*mark_empty(struct isl_tab
*tab
)
450 if (!tab
->empty
&& tab
->need_undo
)
451 push(tab
, isl_tab_undo_empty
);
456 /* Given a row number "row" and a column number "col", pivot the tableau
457 * such that the associated variables are interchanged.
458 * The given row in the tableau expresses
460 * x_r = a_r0 + \sum_i a_ri x_i
464 * x_c = 1/a_rc x_r - a_r0/a_rc + sum_{i \ne r} -a_ri/a_rc
466 * Substituting this equality into the other rows
468 * x_j = a_j0 + \sum_i a_ji x_i
470 * with a_jc \ne 0, we obtain
472 * 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
479 * where i is any other column and j is any other row,
480 * is therefore transformed into
482 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
483 * 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)
485 * The transformation is performed along the following steps
490 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
493 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
494 * n_jc/(|n_rc| d_j) n_ji/(|n_rc| d_j)
496 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
497 * n_jc/(|n_rc| d_j) (n_ji |n_rc|)/(|n_rc| d_j)
499 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
500 * n_jc/(|n_rc| d_j) (n_ji |n_rc| - s(n_rc)n_jc n_ri)/(|n_rc| d_j)
502 * s(n_rc)d_r/|n_rc| -s(n_rc)n_ri/|n_rc|
503 * 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)
506 static void pivot(struct isl_tab
*tab
, int row
, int col
)
511 struct isl_mat
*mat
= tab
->mat
;
512 struct isl_tab_var
*var
;
514 isl_int_swap(mat
->row
[row
][0], mat
->row
[row
][2 + col
]);
515 sgn
= isl_int_sgn(mat
->row
[row
][0]);
517 isl_int_neg(mat
->row
[row
][0], mat
->row
[row
][0]);
518 isl_int_neg(mat
->row
[row
][2 + col
], mat
->row
[row
][2 + col
]);
520 for (j
= 0; j
< 1 + tab
->n_col
; ++j
) {
523 isl_int_neg(mat
->row
[row
][1 + j
], mat
->row
[row
][1 + j
]);
525 if (!isl_int_is_one(mat
->row
[row
][0]))
526 isl_seq_normalize(mat
->row
[row
], 2 + tab
->n_col
);
527 for (i
= 0; i
< tab
->n_row
; ++i
) {
530 if (isl_int_is_zero(mat
->row
[i
][2 + col
]))
532 isl_int_mul(mat
->row
[i
][0], mat
->row
[i
][0], mat
->row
[row
][0]);
533 for (j
= 0; j
< 1 + tab
->n_col
; ++j
) {
536 isl_int_mul(mat
->row
[i
][1 + j
],
537 mat
->row
[i
][1 + j
], mat
->row
[row
][0]);
538 isl_int_addmul(mat
->row
[i
][1 + j
],
539 mat
->row
[i
][2 + col
], mat
->row
[row
][1 + j
]);
541 isl_int_mul(mat
->row
[i
][2 + col
],
542 mat
->row
[i
][2 + col
], mat
->row
[row
][2 + col
]);
543 if (!isl_int_is_one(mat
->row
[i
][0]))
544 isl_seq_normalize(mat
->row
[i
], 2 + tab
->n_col
);
546 t
= tab
->row_var
[row
];
547 tab
->row_var
[row
] = tab
->col_var
[col
];
548 tab
->col_var
[col
] = t
;
549 var
= var_from_row(tab
, row
);
552 var
= var_from_col(tab
, col
);
557 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
558 if (isl_int_is_zero(mat
->row
[i
][2 + col
]))
560 if (!var_from_row(tab
, i
)->frozen
&&
561 is_redundant(tab
, i
))
562 if (mark_redundant(tab
, i
))
567 /* If "var" represents a column variable, then pivot is up (sgn > 0)
568 * or down (sgn < 0) to a row. The variable is assumed not to be
569 * unbounded in the specified direction.
570 * If sgn = 0, then the variable is unbounded in both directions,
571 * and we pivot with any row we can find.
573 static void to_row(struct isl_tab
*tab
, struct isl_tab_var
*var
, int sign
)
581 for (r
= tab
->n_redundant
; r
< tab
->n_row
; ++r
)
582 if (!isl_int_is_zero(tab
->mat
->row
[r
][2 + var
->index
]))
584 isl_assert(tab
->mat
->ctx
, r
< tab
->n_row
, return);
586 r
= pivot_row(tab
, NULL
, sign
, var
->index
);
587 isl_assert(tab
->mat
->ctx
, r
>= 0, return);
590 pivot(tab
, r
, var
->index
);
593 static void check_table(struct isl_tab
*tab
)
599 for (i
= 0; i
< tab
->n_row
; ++i
) {
600 if (!var_from_row(tab
, i
)->is_nonneg
)
602 assert(!isl_int_is_neg(tab
->mat
->row
[i
][1]));
606 /* Return the sign of the maximal value of "var".
607 * If the sign is not negative, then on return from this function,
608 * the sample value will also be non-negative.
610 * If "var" is manifestly unbounded wrt positive values, we are done.
611 * Otherwise, we pivot the variable up to a row if needed
612 * Then we continue pivoting down until either
613 * - no more down pivots can be performed
614 * - the sample value is positive
615 * - the variable is pivoted into a manifestly unbounded column
617 static int sign_of_max(struct isl_tab
*tab
, struct isl_tab_var
*var
)
621 if (max_is_manifestly_unbounded(tab
, var
))
624 while (!isl_int_is_pos(tab
->mat
->row
[var
->index
][1])) {
625 find_pivot(tab
, var
, var
, 1, &row
, &col
);
627 return isl_int_sgn(tab
->mat
->row
[var
->index
][1]);
628 pivot(tab
, row
, col
);
629 if (!var
->is_row
) /* manifestly unbounded */
635 /* Perform pivots until the row variable "var" has a non-negative
636 * sample value or until no more upward pivots can be performed.
637 * Return the sign of the sample value after the pivots have been
640 static int restore_row(struct isl_tab
*tab
, struct isl_tab_var
*var
)
644 while (isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
645 find_pivot(tab
, var
, var
, 1, &row
, &col
);
648 pivot(tab
, row
, col
);
649 if (!var
->is_row
) /* manifestly unbounded */
652 return isl_int_sgn(tab
->mat
->row
[var
->index
][1]);
655 /* Perform pivots until we are sure that the row variable "var"
656 * can attain non-negative values. After return from this
657 * function, "var" is still a row variable, but its sample
658 * value may not be non-negative, even if the function returns 1.
660 static int at_least_zero(struct isl_tab
*tab
, struct isl_tab_var
*var
)
664 while (isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
665 find_pivot(tab
, var
, var
, 1, &row
, &col
);
668 if (row
== var
->index
) /* manifestly unbounded */
670 pivot(tab
, row
, col
);
672 return !isl_int_is_neg(tab
->mat
->row
[var
->index
][1]);
675 /* Return a negative value if "var" can attain negative values.
676 * Return a non-negative value otherwise.
678 * If "var" is manifestly unbounded wrt negative values, we are done.
679 * Otherwise, if var is in a column, we can pivot it down to a row.
680 * Then we continue pivoting down until either
681 * - the pivot would result in a manifestly unbounded column
682 * => we don't perform the pivot, but simply return -1
683 * - no more down pivots can be performed
684 * - the sample value is negative
685 * If the sample value becomes negative and the variable is supposed
686 * to be nonnegative, then we undo the last pivot.
687 * However, if the last pivot has made the pivoting variable
688 * obviously redundant, then it may have moved to another row.
689 * In that case we look for upward pivots until we reach a non-negative
692 static int sign_of_min(struct isl_tab
*tab
, struct isl_tab_var
*var
)
695 struct isl_tab_var
*pivot_var
;
697 if (min_is_manifestly_unbounded(tab
, var
))
701 row
= pivot_row(tab
, NULL
, -1, col
);
702 pivot_var
= var_from_col(tab
, col
);
703 pivot(tab
, row
, col
);
704 if (var
->is_redundant
)
706 if (isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
707 if (var
->is_nonneg
) {
708 if (!pivot_var
->is_redundant
&&
709 pivot_var
->index
== row
)
710 pivot(tab
, row
, col
);
712 restore_row(tab
, var
);
717 if (var
->is_redundant
)
719 while (!isl_int_is_neg(tab
->mat
->row
[var
->index
][1])) {
720 find_pivot(tab
, var
, var
, -1, &row
, &col
);
721 if (row
== var
->index
)
724 return isl_int_sgn(tab
->mat
->row
[var
->index
][1]);
725 pivot_var
= var_from_col(tab
, col
);
726 pivot(tab
, row
, col
);
727 if (var
->is_redundant
)
730 if (var
->is_nonneg
) {
731 /* pivot back to non-negative value */
732 if (!pivot_var
->is_redundant
&& pivot_var
->index
== row
)
733 pivot(tab
, row
, col
);
735 restore_row(tab
, var
);
740 /* Return 1 if "var" can attain values <= -1.
741 * Return 0 otherwise.
743 * The sample value of "var" is assumed to be non-negative when the
744 * the function is called and will be made non-negative again before
745 * the function returns.
747 static int min_at_most_neg_one(struct isl_tab
*tab
, struct isl_tab_var
*var
)
750 struct isl_tab_var
*pivot_var
;
752 if (min_is_manifestly_unbounded(tab
, var
))
756 row
= pivot_row(tab
, NULL
, -1, col
);
757 pivot_var
= var_from_col(tab
, col
);
758 pivot(tab
, row
, col
);
759 if (var
->is_redundant
)
761 if (isl_int_is_neg(tab
->mat
->row
[var
->index
][1]) &&
762 isl_int_abs_ge(tab
->mat
->row
[var
->index
][1],
763 tab
->mat
->row
[var
->index
][0])) {
764 if (var
->is_nonneg
) {
765 if (!pivot_var
->is_redundant
&&
766 pivot_var
->index
== row
)
767 pivot(tab
, row
, col
);
769 restore_row(tab
, var
);
774 if (var
->is_redundant
)
777 find_pivot(tab
, var
, var
, -1, &row
, &col
);
778 if (row
== var
->index
)
782 pivot_var
= var_from_col(tab
, col
);
783 pivot(tab
, row
, col
);
784 if (var
->is_redundant
)
786 } while (!isl_int_is_neg(tab
->mat
->row
[var
->index
][1]) ||
787 isl_int_abs_lt(tab
->mat
->row
[var
->index
][1],
788 tab
->mat
->row
[var
->index
][0]));
789 if (var
->is_nonneg
) {
790 /* pivot back to non-negative value */
791 if (!pivot_var
->is_redundant
&& pivot_var
->index
== row
)
792 pivot(tab
, row
, col
);
793 restore_row(tab
, var
);
798 /* Return 1 if "var" can attain values >= 1.
799 * Return 0 otherwise.
801 static int at_least_one(struct isl_tab
*tab
, struct isl_tab_var
*var
)
806 if (max_is_manifestly_unbounded(tab
, var
))
809 r
= tab
->mat
->row
[var
->index
];
810 while (isl_int_lt(r
[1], r
[0])) {
811 find_pivot(tab
, var
, var
, 1, &row
, &col
);
813 return isl_int_ge(r
[1], r
[0]);
814 if (row
== var
->index
) /* manifestly unbounded */
816 pivot(tab
, row
, col
);
821 static void swap_cols(struct isl_tab
*tab
, int col1
, int col2
)
824 t
= tab
->col_var
[col1
];
825 tab
->col_var
[col1
] = tab
->col_var
[col2
];
826 tab
->col_var
[col2
] = t
;
827 var_from_col(tab
, col1
)->index
= col1
;
828 var_from_col(tab
, col2
)->index
= col2
;
829 tab
->mat
= isl_mat_swap_cols(tab
->mat
, 2 + col1
, 2 + col2
);
832 /* Mark column with index "col" as representing a zero variable.
833 * If we may need to undo the operation the column is kept,
834 * but no longer considered.
835 * Otherwise, the column is simply removed.
837 * The column may be interchanged with some other column. If it
838 * is interchanged with a later column, return 1. Otherwise return 0.
839 * If the columns are checked in order in the calling function,
840 * then a return value of 1 means that the column with the given
841 * column number may now contain a different column that
842 * hasn't been checked yet.
844 static int kill_col(struct isl_tab
*tab
, int col
)
846 var_from_col(tab
, col
)->is_zero
= 1;
847 if (tab
->need_undo
) {
848 push_var(tab
, isl_tab_undo_zero
, var_from_col(tab
, col
));
849 if (col
!= tab
->n_dead
)
850 swap_cols(tab
, col
, tab
->n_dead
);
854 if (col
!= tab
->n_col
- 1)
855 swap_cols(tab
, col
, tab
->n_col
- 1);
856 var_from_col(tab
, tab
->n_col
- 1)->index
= -1;
862 /* Row variable "var" is non-negative and cannot attain any values
863 * larger than zero. This means that the coefficients of the unrestricted
864 * column variables are zero and that the coefficients of the non-negative
865 * column variables are zero or negative.
866 * Each of the non-negative variables with a negative coefficient can
867 * then also be written as the negative sum of non-negative variables
868 * and must therefore also be zero.
870 static void close_row(struct isl_tab
*tab
, struct isl_tab_var
*var
)
873 struct isl_mat
*mat
= tab
->mat
;
875 isl_assert(tab
->mat
->ctx
, var
->is_nonneg
, return);
877 for (j
= tab
->n_dead
; j
< tab
->n_col
; ++j
) {
878 if (isl_int_is_zero(mat
->row
[var
->index
][2 + j
]))
880 isl_assert(tab
->mat
->ctx
,
881 isl_int_is_neg(mat
->row
[var
->index
][2 + j
]), return);
882 if (kill_col(tab
, j
))
885 mark_redundant(tab
, var
->index
);
888 /* Add a constraint to the tableau and allocate a row for it.
889 * Return the index into the constraint array "con".
891 static int allocate_con(struct isl_tab
*tab
)
895 isl_assert(tab
->mat
->ctx
, tab
->n_row
< tab
->mat
->n_row
, return -1);
898 tab
->con
[r
].index
= tab
->n_row
;
899 tab
->con
[r
].is_row
= 1;
900 tab
->con
[r
].is_nonneg
= 0;
901 tab
->con
[r
].is_zero
= 0;
902 tab
->con
[r
].is_redundant
= 0;
903 tab
->con
[r
].frozen
= 0;
904 tab
->row_var
[tab
->n_row
] = ~r
;
908 push_var(tab
, isl_tab_undo_allocate
, &tab
->con
[r
]);
913 /* Add a row to the tableau. The row is given as an affine combination
914 * of the original variables and needs to be expressed in terms of the
917 * We add each term in turn.
918 * If r = n/d_r is the current sum and we need to add k x, then
919 * if x is a column variable, we increase the numerator of
920 * this column by k d_r
921 * if x = f/d_x is a row variable, then the new representation of r is
923 * n k f d_x/g n + d_r/g k f m/d_r n + m/d_g k f
924 * --- + --- = ------------------- = -------------------
925 * d_r d_r d_r d_x/g m
927 * with g the gcd of d_r and d_x and m the lcm of d_r and d_x.
929 static int add_row(struct isl_tab
*tab
, isl_int
*line
)
936 r
= allocate_con(tab
);
942 row
= tab
->mat
->row
[tab
->con
[r
].index
];
943 isl_int_set_si(row
[0], 1);
944 isl_int_set(row
[1], line
[0]);
945 isl_seq_clr(row
+ 2, tab
->n_col
);
946 for (i
= 0; i
< tab
->n_var
; ++i
) {
947 if (tab
->var
[i
].is_zero
)
949 if (tab
->var
[i
].is_row
) {
951 row
[0], tab
->mat
->row
[tab
->var
[i
].index
][0]);
952 isl_int_swap(a
, row
[0]);
953 isl_int_divexact(a
, row
[0], a
);
955 row
[0], tab
->mat
->row
[tab
->var
[i
].index
][0]);
956 isl_int_mul(b
, b
, line
[1 + i
]);
957 isl_seq_combine(row
+ 1, a
, row
+ 1,
958 b
, tab
->mat
->row
[tab
->var
[i
].index
] + 1,
961 isl_int_addmul(row
[2 + tab
->var
[i
].index
],
962 line
[1 + i
], row
[0]);
964 isl_seq_normalize(row
, 2 + tab
->n_col
);
971 static int drop_row(struct isl_tab
*tab
, int row
)
973 isl_assert(tab
->mat
->ctx
, ~tab
->row_var
[row
] == tab
->n_con
- 1, return -1);
974 if (row
!= tab
->n_row
- 1)
975 swap_rows(tab
, row
, tab
->n_row
- 1);
981 /* Add inequality "ineq" and check if it conflicts with the
982 * previously added constraints or if it is obviously redundant.
984 struct isl_tab
*isl_tab_add_ineq(struct isl_tab
*tab
, isl_int
*ineq
)
991 r
= add_row(tab
, ineq
);
994 tab
->con
[r
].is_nonneg
= 1;
995 push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
996 if (is_redundant(tab
, tab
->con
[r
].index
)) {
997 mark_redundant(tab
, tab
->con
[r
].index
);
1001 sgn
= restore_row(tab
, &tab
->con
[r
]);
1003 return mark_empty(tab
);
1004 if (tab
->con
[r
].is_row
&& is_redundant(tab
, tab
->con
[r
].index
))
1005 mark_redundant(tab
, tab
->con
[r
].index
);
1012 /* Pivot a non-negative variable down until it reaches the value zero
1013 * and then pivot the variable into a column position.
1015 static int to_col(struct isl_tab
*tab
, struct isl_tab_var
*var
)
1023 while (isl_int_is_pos(tab
->mat
->row
[var
->index
][1])) {
1024 find_pivot(tab
, var
, NULL
, -1, &row
, &col
);
1025 isl_assert(tab
->mat
->ctx
, row
!= -1, return -1);
1026 pivot(tab
, row
, col
);
1031 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
)
1032 if (!isl_int_is_zero(tab
->mat
->row
[var
->index
][2 + i
]))
1035 isl_assert(tab
->mat
->ctx
, i
< tab
->n_col
, return -1);
1036 pivot(tab
, var
->index
, i
);
1041 /* We assume Gaussian elimination has been performed on the equalities.
1042 * The equalities can therefore never conflict.
1043 * Adding the equalities is currently only really useful for a later call
1044 * to isl_tab_ineq_type.
1046 static struct isl_tab
*add_eq(struct isl_tab
*tab
, isl_int
*eq
)
1053 r
= add_row(tab
, eq
);
1057 r
= tab
->con
[r
].index
;
1058 i
= isl_seq_first_non_zero(tab
->mat
->row
[r
] + 2 + tab
->n_dead
,
1059 tab
->n_col
- tab
->n_dead
);
1060 isl_assert(tab
->mat
->ctx
, i
>= 0, goto error
);
1072 /* Add an equality that is known to be valid for the given tableau.
1074 struct isl_tab
*isl_tab_add_valid_eq(struct isl_tab
*tab
, isl_int
*eq
)
1076 struct isl_tab_var
*var
;
1082 r
= add_row(tab
, eq
);
1088 if (isl_int_is_neg(tab
->mat
->row
[r
][1]))
1089 isl_seq_neg(tab
->mat
->row
[r
] + 1, tab
->mat
->row
[r
] + 1,
1092 if (to_col(tab
, var
) < 0)
1095 kill_col(tab
, var
->index
);
1103 struct isl_tab
*isl_tab_from_basic_map(struct isl_basic_map
*bmap
)
1106 struct isl_tab
*tab
;
1110 tab
= isl_tab_alloc(bmap
->ctx
,
1111 isl_basic_map_total_dim(bmap
) + bmap
->n_ineq
+ 1,
1112 isl_basic_map_total_dim(bmap
));
1115 tab
->rational
= ISL_F_ISSET(bmap
, ISL_BASIC_MAP_RATIONAL
);
1116 if (ISL_F_ISSET(bmap
, ISL_BASIC_MAP_EMPTY
))
1117 return mark_empty(tab
);
1118 for (i
= 0; i
< bmap
->n_eq
; ++i
) {
1119 tab
= add_eq(tab
, bmap
->eq
[i
]);
1123 for (i
= 0; i
< bmap
->n_ineq
; ++i
) {
1124 tab
= isl_tab_add_ineq(tab
, bmap
->ineq
[i
]);
1125 if (!tab
|| tab
->empty
)
1131 struct isl_tab
*isl_tab_from_basic_set(struct isl_basic_set
*bset
)
1133 return isl_tab_from_basic_map((struct isl_basic_map
*)bset
);
1136 /* Construct a tableau corresponding to the recession cone of "bmap".
1138 struct isl_tab
*isl_tab_from_recession_cone(struct isl_basic_map
*bmap
)
1142 struct isl_tab
*tab
;
1146 tab
= isl_tab_alloc(bmap
->ctx
, bmap
->n_eq
+ bmap
->n_ineq
,
1147 isl_basic_map_total_dim(bmap
));
1150 tab
->rational
= ISL_F_ISSET(bmap
, ISL_BASIC_MAP_RATIONAL
);
1153 for (i
= 0; i
< bmap
->n_eq
; ++i
) {
1154 isl_int_swap(bmap
->eq
[i
][0], cst
);
1155 tab
= add_eq(tab
, bmap
->eq
[i
]);
1156 isl_int_swap(bmap
->eq
[i
][0], cst
);
1160 for (i
= 0; i
< bmap
->n_ineq
; ++i
) {
1162 isl_int_swap(bmap
->ineq
[i
][0], cst
);
1163 r
= add_row(tab
, bmap
->ineq
[i
]);
1164 isl_int_swap(bmap
->ineq
[i
][0], cst
);
1167 tab
->con
[r
].is_nonneg
= 1;
1168 push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
1179 /* Assuming "tab" is the tableau of a cone, check if the cone is
1180 * bounded, i.e., if it is empty or only contains the origin.
1182 int isl_tab_cone_is_bounded(struct isl_tab
*tab
)
1190 if (tab
->n_dead
== tab
->n_col
)
1194 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1195 struct isl_tab_var
*var
;
1196 var
= var_from_row(tab
, i
);
1197 if (!var
->is_nonneg
)
1199 if (sign_of_max(tab
, var
) != 0)
1201 close_row(tab
, var
);
1204 if (tab
->n_dead
== tab
->n_col
)
1206 if (i
== tab
->n_row
)
1211 int isl_tab_sample_is_integer(struct isl_tab
*tab
)
1218 for (i
= 0; i
< tab
->n_var
; ++i
) {
1220 if (!tab
->var
[i
].is_row
)
1222 row
= tab
->var
[i
].index
;
1223 if (!isl_int_is_divisible_by(tab
->mat
->row
[row
][1],
1224 tab
->mat
->row
[row
][0]))
1230 static struct isl_vec
*extract_integer_sample(struct isl_tab
*tab
)
1233 struct isl_vec
*vec
;
1235 vec
= isl_vec_alloc(tab
->mat
->ctx
, 1 + tab
->n_var
);
1239 isl_int_set_si(vec
->block
.data
[0], 1);
1240 for (i
= 0; i
< tab
->n_var
; ++i
) {
1241 if (!tab
->var
[i
].is_row
)
1242 isl_int_set_si(vec
->block
.data
[1 + i
], 0);
1244 int row
= tab
->var
[i
].index
;
1245 isl_int_divexact(vec
->block
.data
[1 + i
],
1246 tab
->mat
->row
[row
][1], tab
->mat
->row
[row
][0]);
1253 struct isl_vec
*isl_tab_get_sample_value(struct isl_tab
*tab
)
1256 struct isl_vec
*vec
;
1262 vec
= isl_vec_alloc(tab
->mat
->ctx
, 1 + tab
->n_var
);
1268 isl_int_set_si(vec
->block
.data
[0], 1);
1269 for (i
= 0; i
< tab
->n_var
; ++i
) {
1271 if (!tab
->var
[i
].is_row
) {
1272 isl_int_set_si(vec
->block
.data
[1 + i
], 0);
1275 row
= tab
->var
[i
].index
;
1276 isl_int_gcd(m
, vec
->block
.data
[0], tab
->mat
->row
[row
][0]);
1277 isl_int_divexact(m
, tab
->mat
->row
[row
][0], m
);
1278 isl_seq_scale(vec
->block
.data
, vec
->block
.data
, m
, 1 + i
);
1279 isl_int_divexact(m
, vec
->block
.data
[0], tab
->mat
->row
[row
][0]);
1280 isl_int_mul(vec
->block
.data
[1 + i
], m
, tab
->mat
->row
[row
][1]);
1282 isl_seq_normalize(vec
->block
.data
, vec
->size
);
1288 /* Update "bmap" based on the results of the tableau "tab".
1289 * In particular, implicit equalities are made explicit, redundant constraints
1290 * are removed and if the sample value happens to be integer, it is stored
1291 * in "bmap" (unless "bmap" already had an integer sample).
1293 * The tableau is assumed to have been created from "bmap" using
1294 * isl_tab_from_basic_map.
1296 struct isl_basic_map
*isl_basic_map_update_from_tab(struct isl_basic_map
*bmap
,
1297 struct isl_tab
*tab
)
1309 bmap
= isl_basic_map_set_to_empty(bmap
);
1311 for (i
= bmap
->n_ineq
- 1; i
>= 0; --i
) {
1312 if (isl_tab_is_equality(tab
, n_eq
+ i
))
1313 isl_basic_map_inequality_to_equality(bmap
, i
);
1314 else if (isl_tab_is_redundant(tab
, n_eq
+ i
))
1315 isl_basic_map_drop_inequality(bmap
, i
);
1317 if (!tab
->rational
&&
1318 !bmap
->sample
&& isl_tab_sample_is_integer(tab
))
1319 bmap
->sample
= extract_integer_sample(tab
);
1323 struct isl_basic_set
*isl_basic_set_update_from_tab(struct isl_basic_set
*bset
,
1324 struct isl_tab
*tab
)
1326 return (struct isl_basic_set
*)isl_basic_map_update_from_tab(
1327 (struct isl_basic_map
*)bset
, tab
);
1330 /* Given a non-negative variable "var", add a new non-negative variable
1331 * that is the opposite of "var", ensuring that var can only attain the
1333 * If var = n/d is a row variable, then the new variable = -n/d.
1334 * If var is a column variables, then the new variable = -var.
1335 * If the new variable cannot attain non-negative values, then
1336 * the resulting tableau is empty.
1337 * Otherwise, we know the value will be zero and we close the row.
1339 static struct isl_tab
*cut_to_hyperplane(struct isl_tab
*tab
,
1340 struct isl_tab_var
*var
)
1346 if (extend_cons(tab
, 1) < 0)
1350 tab
->con
[r
].index
= tab
->n_row
;
1351 tab
->con
[r
].is_row
= 1;
1352 tab
->con
[r
].is_nonneg
= 0;
1353 tab
->con
[r
].is_zero
= 0;
1354 tab
->con
[r
].is_redundant
= 0;
1355 tab
->con
[r
].frozen
= 0;
1356 tab
->row_var
[tab
->n_row
] = ~r
;
1357 row
= tab
->mat
->row
[tab
->n_row
];
1360 isl_int_set(row
[0], tab
->mat
->row
[var
->index
][0]);
1361 isl_seq_neg(row
+ 1,
1362 tab
->mat
->row
[var
->index
] + 1, 1 + tab
->n_col
);
1364 isl_int_set_si(row
[0], 1);
1365 isl_seq_clr(row
+ 1, 1 + tab
->n_col
);
1366 isl_int_set_si(row
[2 + var
->index
], -1);
1371 push_var(tab
, isl_tab_undo_allocate
, &tab
->con
[r
]);
1373 sgn
= sign_of_max(tab
, &tab
->con
[r
]);
1375 return mark_empty(tab
);
1376 tab
->con
[r
].is_nonneg
= 1;
1377 push_var(tab
, isl_tab_undo_nonneg
, &tab
->con
[r
]);
1379 close_row(tab
, &tab
->con
[r
]);
1387 /* Given a tableau "tab" and an inequality constraint "con" of the tableau,
1388 * relax the inequality by one. That is, the inequality r >= 0 is replaced
1389 * by r' = r + 1 >= 0.
1390 * If r is a row variable, we simply increase the constant term by one
1391 * (taking into account the denominator).
1392 * If r is a column variable, then we need to modify each row that
1393 * refers to r = r' - 1 by substituting this equality, effectively
1394 * subtracting the coefficient of the column from the constant.
1396 struct isl_tab
*isl_tab_relax(struct isl_tab
*tab
, int con
)
1398 struct isl_tab_var
*var
;
1402 var
= &tab
->con
[con
];
1404 if (!var
->is_row
&& !max_is_manifestly_unbounded(tab
, var
))
1405 to_row(tab
, var
, 1);
1408 isl_int_add(tab
->mat
->row
[var
->index
][1],
1409 tab
->mat
->row
[var
->index
][1], tab
->mat
->row
[var
->index
][0]);
1413 for (i
= 0; i
< tab
->n_row
; ++i
) {
1414 if (isl_int_is_zero(tab
->mat
->row
[i
][2 + var
->index
]))
1416 isl_int_sub(tab
->mat
->row
[i
][1], tab
->mat
->row
[i
][1],
1417 tab
->mat
->row
[i
][2 + var
->index
]);
1422 push_var(tab
, isl_tab_undo_relax
, var
);
1427 struct isl_tab
*isl_tab_select_facet(struct isl_tab
*tab
, int con
)
1432 return cut_to_hyperplane(tab
, &tab
->con
[con
]);
1435 static int may_be_equality(struct isl_tab
*tab
, int row
)
1437 return (tab
->rational
? isl_int_is_zero(tab
->mat
->row
[row
][1])
1438 : isl_int_lt(tab
->mat
->row
[row
][1],
1439 tab
->mat
->row
[row
][0])) &&
1440 isl_seq_first_non_zero(tab
->mat
->row
[row
] + 2 + tab
->n_dead
,
1441 tab
->n_col
- tab
->n_dead
) != -1;
1444 /* Check for (near) equalities among the constraints.
1445 * A constraint is an equality if it is non-negative and if
1446 * its maximal value is either
1447 * - zero (in case of rational tableaus), or
1448 * - strictly less than 1 (in case of integer tableaus)
1450 * We first mark all non-redundant and non-dead variables that
1451 * are not frozen and not obviously not an equality.
1452 * Then we iterate over all marked variables if they can attain
1453 * any values larger than zero or at least one.
1454 * If the maximal value is zero, we mark any column variables
1455 * that appear in the row as being zero and mark the row as being redundant.
1456 * Otherwise, if the maximal value is strictly less than one (and the
1457 * tableau is integer), then we restrict the value to being zero
1458 * by adding an opposite non-negative variable.
1460 struct isl_tab
*isl_tab_detect_equalities(struct isl_tab
*tab
)
1469 if (tab
->n_dead
== tab
->n_col
)
1473 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1474 struct isl_tab_var
*var
= var_from_row(tab
, i
);
1475 var
->marked
= !var
->frozen
&& var
->is_nonneg
&&
1476 may_be_equality(tab
, i
);
1480 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1481 struct isl_tab_var
*var
= var_from_col(tab
, i
);
1482 var
->marked
= !var
->frozen
&& var
->is_nonneg
;
1487 struct isl_tab_var
*var
;
1488 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1489 var
= var_from_row(tab
, i
);
1493 if (i
== tab
->n_row
) {
1494 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1495 var
= var_from_col(tab
, i
);
1499 if (i
== tab
->n_col
)
1504 if (sign_of_max(tab
, var
) == 0)
1505 close_row(tab
, var
);
1506 else if (!tab
->rational
&& !at_least_one(tab
, var
)) {
1507 tab
= cut_to_hyperplane(tab
, var
);
1508 return isl_tab_detect_equalities(tab
);
1510 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1511 var
= var_from_row(tab
, i
);
1514 if (may_be_equality(tab
, i
))
1524 /* Check for (near) redundant constraints.
1525 * A constraint is redundant if it is non-negative and if
1526 * its minimal value (temporarily ignoring the non-negativity) is either
1527 * - zero (in case of rational tableaus), or
1528 * - strictly larger than -1 (in case of integer tableaus)
1530 * We first mark all non-redundant and non-dead variables that
1531 * are not frozen and not obviously negatively unbounded.
1532 * Then we iterate over all marked variables if they can attain
1533 * any values smaller than zero or at most negative one.
1534 * If not, we mark the row as being redundant (assuming it hasn't
1535 * been detected as being obviously redundant in the mean time).
1537 struct isl_tab
*isl_tab_detect_redundant(struct isl_tab
*tab
)
1546 if (tab
->n_redundant
== tab
->n_row
)
1550 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1551 struct isl_tab_var
*var
= var_from_row(tab
, i
);
1552 var
->marked
= !var
->frozen
&& var
->is_nonneg
;
1556 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1557 struct isl_tab_var
*var
= var_from_col(tab
, i
);
1558 var
->marked
= !var
->frozen
&& var
->is_nonneg
&&
1559 !min_is_manifestly_unbounded(tab
, var
);
1564 struct isl_tab_var
*var
;
1565 for (i
= tab
->n_redundant
; i
< tab
->n_row
; ++i
) {
1566 var
= var_from_row(tab
, i
);
1570 if (i
== tab
->n_row
) {
1571 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1572 var
= var_from_col(tab
, i
);
1576 if (i
== tab
->n_col
)
1581 if ((tab
->rational
? (sign_of_min(tab
, var
) >= 0)
1582 : !min_at_most_neg_one(tab
, var
)) &&
1584 mark_redundant(tab
, var
->index
);
1585 for (i
= tab
->n_dead
; i
< tab
->n_col
; ++i
) {
1586 var
= var_from_col(tab
, i
);
1589 if (!min_is_manifestly_unbounded(tab
, var
))
1599 int isl_tab_is_equality(struct isl_tab
*tab
, int con
)
1605 if (tab
->con
[con
].is_zero
)
1607 if (tab
->con
[con
].is_redundant
)
1609 if (!tab
->con
[con
].is_row
)
1610 return tab
->con
[con
].index
< tab
->n_dead
;
1612 row
= tab
->con
[con
].index
;
1614 return isl_int_is_zero(tab
->mat
->row
[row
][1]) &&
1615 isl_seq_first_non_zero(tab
->mat
->row
[row
] + 2 + tab
->n_dead
,
1616 tab
->n_col
- tab
->n_dead
) == -1;
1619 /* Return the minimial value of the affine expression "f" with denominator
1620 * "denom" in *opt, *opt_denom, assuming the tableau is not empty and
1621 * the expression cannot attain arbitrarily small values.
1622 * If opt_denom is NULL, then *opt is rounded up to the nearest integer.
1623 * The return value reflects the nature of the result (empty, unbounded,
1624 * minmimal value returned in *opt).
1626 enum isl_lp_result
isl_tab_min(struct isl_tab
*tab
,
1627 isl_int
*f
, isl_int denom
, isl_int
*opt
, isl_int
*opt_denom
,
1631 enum isl_lp_result res
= isl_lp_ok
;
1632 struct isl_tab_var
*var
;
1633 struct isl_tab_undo
*snap
;
1636 return isl_lp_empty
;
1638 snap
= isl_tab_snap(tab
);
1639 r
= add_row(tab
, f
);
1641 return isl_lp_error
;
1643 isl_int_mul(tab
->mat
->row
[var
->index
][0],
1644 tab
->mat
->row
[var
->index
][0], denom
);
1647 find_pivot(tab
, var
, var
, -1, &row
, &col
);
1648 if (row
== var
->index
) {
1649 res
= isl_lp_unbounded
;
1654 pivot(tab
, row
, col
);
1656 if (isl_tab_rollback(tab
, snap
) < 0)
1657 return isl_lp_error
;
1658 if (ISL_FL_ISSET(flags
, ISL_TAB_SAVE_DUAL
)) {
1661 isl_vec_free(tab
->dual
);
1662 tab
->dual
= isl_vec_alloc(tab
->mat
->ctx
, 1 + tab
->n_con
);
1664 return isl_lp_error
;
1665 isl_int_set(tab
->dual
->el
[0], tab
->mat
->row
[var
->index
][0]);
1666 for (i
= 0; i
< tab
->n_con
; ++i
) {
1667 if (tab
->con
[i
].is_row
)
1668 isl_int_set_si(tab
->dual
->el
[1 + i
], 0);
1670 int pos
= 2 + tab
->con
[i
].index
;
1671 isl_int_set(tab
->dual
->el
[1 + i
],
1672 tab
->mat
->row
[var
->index
][pos
]);
1676 if (res
== isl_lp_ok
) {
1678 isl_int_set(*opt
, tab
->mat
->row
[var
->index
][1]);
1679 isl_int_set(*opt_denom
, tab
->mat
->row
[var
->index
][0]);
1681 isl_int_cdiv_q(*opt
, tab
->mat
->row
[var
->index
][1],
1682 tab
->mat
->row
[var
->index
][0]);
1687 int isl_tab_is_redundant(struct isl_tab
*tab
, int con
)
1694 if (tab
->con
[con
].is_zero
)
1696 if (tab
->con
[con
].is_redundant
)
1698 return tab
->con
[con
].is_row
&& tab
->con
[con
].index
< tab
->n_redundant
;
1701 /* Take a snapshot of the tableau that can be restored by s call to
1704 struct isl_tab_undo
*isl_tab_snap(struct isl_tab
*tab
)
1712 /* Undo the operation performed by isl_tab_relax.
1714 static void unrelax(struct isl_tab
*tab
, struct isl_tab_var
*var
)
1716 if (!var
->is_row
&& !max_is_manifestly_unbounded(tab
, var
))
1717 to_row(tab
, var
, 1);
1720 isl_int_sub(tab
->mat
->row
[var
->index
][1],
1721 tab
->mat
->row
[var
->index
][1], tab
->mat
->row
[var
->index
][0]);
1725 for (i
= 0; i
< tab
->n_row
; ++i
) {
1726 if (isl_int_is_zero(tab
->mat
->row
[i
][2 + var
->index
]))
1728 isl_int_add(tab
->mat
->row
[i
][1], tab
->mat
->row
[i
][1],
1729 tab
->mat
->row
[i
][2 + var
->index
]);
1735 static void perform_undo_var(struct isl_tab
*tab
, struct isl_tab_undo
*undo
)
1737 struct isl_tab_var
*var
= var_from_index(tab
, undo
->u
.var_index
);
1738 switch(undo
->type
) {
1739 case isl_tab_undo_nonneg
:
1742 case isl_tab_undo_redundant
:
1743 var
->is_redundant
= 0;
1746 case isl_tab_undo_zero
:
1750 case isl_tab_undo_allocate
:
1752 if (!max_is_manifestly_unbounded(tab
, var
))
1753 to_row(tab
, var
, 1);
1754 else if (!min_is_manifestly_unbounded(tab
, var
))
1755 to_row(tab
, var
, -1);
1757 to_row(tab
, var
, 0);
1759 drop_row(tab
, var
->index
);
1761 case isl_tab_undo_relax
:
1767 static int perform_undo(struct isl_tab
*tab
, struct isl_tab_undo
*undo
)
1769 switch (undo
->type
) {
1770 case isl_tab_undo_empty
:
1773 case isl_tab_undo_nonneg
:
1774 case isl_tab_undo_redundant
:
1775 case isl_tab_undo_zero
:
1776 case isl_tab_undo_allocate
:
1777 case isl_tab_undo_relax
:
1778 perform_undo_var(tab
, undo
);
1781 isl_assert(tab
->mat
->ctx
, 0, return -1);
1786 /* Return the tableau to the state it was in when the snapshot "snap"
1789 int isl_tab_rollback(struct isl_tab
*tab
, struct isl_tab_undo
*snap
)
1791 struct isl_tab_undo
*undo
, *next
;
1797 for (undo
= tab
->top
; undo
&& undo
!= &tab
->bottom
; undo
= next
) {
1801 if (perform_undo(tab
, undo
) < 0) {
1815 /* The given row "row" represents an inequality violated by all
1816 * points in the tableau. Check for some special cases of such
1817 * separating constraints.
1818 * In particular, if the row has been reduced to the constant -1,
1819 * then we know the inequality is adjacent (but opposite) to
1820 * an equality in the tableau.
1821 * If the row has been reduced to r = -1 -r', with r' an inequality
1822 * of the tableau, then the inequality is adjacent (but opposite)
1823 * to the inequality r'.
1825 static enum isl_ineq_type
separation_type(struct isl_tab
*tab
, unsigned row
)
1830 return isl_ineq_separate
;
1832 if (!isl_int_is_one(tab
->mat
->row
[row
][0]))
1833 return isl_ineq_separate
;
1834 if (!isl_int_is_negone(tab
->mat
->row
[row
][1]))
1835 return isl_ineq_separate
;
1837 pos
= isl_seq_first_non_zero(tab
->mat
->row
[row
] + 2 + tab
->n_dead
,
1838 tab
->n_col
- tab
->n_dead
);
1840 return isl_ineq_adj_eq
;
1842 if (!isl_int_is_negone(tab
->mat
->row
[row
][2 + tab
->n_dead
+ pos
]))
1843 return isl_ineq_separate
;
1845 pos
= isl_seq_first_non_zero(
1846 tab
->mat
->row
[row
] + 2 + tab
->n_dead
+ pos
+ 1,
1847 tab
->n_col
- tab
->n_dead
- pos
- 1);
1849 return pos
== -1 ? isl_ineq_adj_ineq
: isl_ineq_separate
;
1852 /* Check the effect of inequality "ineq" on the tableau "tab".
1854 * isl_ineq_redundant: satisfied by all points in the tableau
1855 * isl_ineq_separate: satisfied by no point in the tableau
1856 * isl_ineq_cut: satisfied by some by not all points
1857 * isl_ineq_adj_eq: adjacent to an equality
1858 * isl_ineq_adj_ineq: adjacent to an inequality.
1860 enum isl_ineq_type
isl_tab_ineq_type(struct isl_tab
*tab
, isl_int
*ineq
)
1862 enum isl_ineq_type type
= isl_ineq_error
;
1863 struct isl_tab_undo
*snap
= NULL
;
1868 return isl_ineq_error
;
1870 if (extend_cons(tab
, 1) < 0)
1871 return isl_ineq_error
;
1873 snap
= isl_tab_snap(tab
);
1875 con
= add_row(tab
, ineq
);
1879 row
= tab
->con
[con
].index
;
1880 if (is_redundant(tab
, row
))
1881 type
= isl_ineq_redundant
;
1882 else if (isl_int_is_neg(tab
->mat
->row
[row
][1]) &&
1884 isl_int_abs_ge(tab
->mat
->row
[row
][1],
1885 tab
->mat
->row
[row
][0]))) {
1886 if (at_least_zero(tab
, &tab
->con
[con
]))
1887 type
= isl_ineq_cut
;
1889 type
= separation_type(tab
, row
);
1890 } else if (tab
->rational
? (sign_of_min(tab
, &tab
->con
[con
]) < 0)
1891 : min_at_most_neg_one(tab
, &tab
->con
[con
]))
1892 type
= isl_ineq_cut
;
1894 type
= isl_ineq_redundant
;
1896 if (isl_tab_rollback(tab
, snap
))
1897 return isl_ineq_error
;
1900 isl_tab_rollback(tab
, snap
);
1901 return isl_ineq_error
;
1904 void isl_tab_dump(struct isl_tab
*tab
, FILE *out
, int indent
)
1910 fprintf(out
, "%*snull tab\n", indent
, "");
1913 fprintf(out
, "%*sn_redundant: %d, n_dead: %d", indent
, "",
1914 tab
->n_redundant
, tab
->n_dead
);
1916 fprintf(out
, ", rational");
1918 fprintf(out
, ", empty");
1920 fprintf(out
, "%*s[", indent
, "");
1921 for (i
= 0; i
< tab
->n_var
; ++i
) {
1924 fprintf(out
, "%c%d%s", tab
->var
[i
].is_row
? 'r' : 'c',
1926 tab
->var
[i
].is_zero
? " [=0]" :
1927 tab
->var
[i
].is_redundant
? " [R]" : "");
1929 fprintf(out
, "]\n");
1930 fprintf(out
, "%*s[", indent
, "");
1931 for (i
= 0; i
< tab
->n_con
; ++i
) {
1934 fprintf(out
, "%c%d%s", tab
->con
[i
].is_row
? 'r' : 'c',
1936 tab
->con
[i
].is_zero
? " [=0]" :
1937 tab
->con
[i
].is_redundant
? " [R]" : "");
1939 fprintf(out
, "]\n");
1940 fprintf(out
, "%*s[", indent
, "");
1941 for (i
= 0; i
< tab
->n_row
; ++i
) {
1944 fprintf(out
, "r%d: %d%s", i
, tab
->row_var
[i
],
1945 var_from_row(tab
, i
)->is_nonneg
? " [>=0]" : "");
1947 fprintf(out
, "]\n");
1948 fprintf(out
, "%*s[", indent
, "");
1949 for (i
= 0; i
< tab
->n_col
; ++i
) {
1952 fprintf(out
, "c%d: %d%s", i
, tab
->col_var
[i
],
1953 var_from_col(tab
, i
)->is_nonneg
? " [>=0]" : "");
1955 fprintf(out
, "]\n");
1956 r
= tab
->mat
->n_row
;
1957 tab
->mat
->n_row
= tab
->n_row
;
1958 c
= tab
->mat
->n_col
;
1959 tab
->mat
->n_col
= 2 + tab
->n_col
;
1960 isl_mat_dump(tab
->mat
, out
, indent
);
1961 tab
->mat
->n_row
= r
;
1962 tab
->mat
->n_col
= c
;