1 #include "isl_map_private.h"
4 #include "isl_sample.h"
7 * The implementation of parametric integer linear programming in this file
8 * was inspired by the paper "Parametric Integer Programming" and the
9 * report "Solving systems of affine (in)equalities" by Paul Feautrier
12 * The strategy used for obtaining a feasible solution is different
13 * from the one used in isl_tab.c. In particular, in isl_tab.c,
14 * upon finding a constraint that is not yet satisfied, we pivot
15 * in a row that increases the constant term of row holding the
16 * constraint, making sure the sample solution remains feasible
17 * for all the constraints it already satisfied.
18 * Here, we always pivot in the row holding the constraint,
19 * choosing a column that induces the lexicographically smallest
20 * increment to the sample solution.
22 * By starting out from a sample value that is lexicographically
23 * smaller than any integer point in the problem space, the first
24 * feasible integer sample point we find will also be the lexicographically
25 * smallest. If all variables can be assumed to be non-negative,
26 * then the initial sample value may be chosen equal to zero.
27 * However, we will not make this assumption. Instead, we apply
28 * the "big parameter" trick. Any variable x is then not directly
29 * used in the tableau, but instead it its represented by another
30 * variable x' = M + x, where M is an arbitrarily large (positive)
31 * value. x' is therefore always non-negative, whatever the value of x.
32 * Taking as initial smaple value x' = 0 corresponds to x = -M,
33 * which is always smaller than any possible value of x.
35 * The big parameter trick is used in the main tableau and
36 * also in the context tableau if isl_context_lex is used.
37 * In this case, each tableaus has its own big parameter.
38 * Before doing any real work, we check if all the parameters
39 * happen to be non-negative. If so, we drop the column corresponding
40 * to M from the initial context tableau.
41 * If isl_context_gbr is used, then the big parameter trick is only
42 * used in the main tableau.
46 struct isl_context_op {
47 /* detect nonnegative parameters in context and mark them in tab */
48 struct isl_tab *(*detect_nonnegative_parameters)(
49 struct isl_context *context, struct isl_tab *tab);
50 /* return temporary reference to basic set representation of context */
51 struct isl_basic_set *(*peek_basic_set)(struct isl_context *context);
52 /* return temporary reference to tableau representation of context */
53 struct isl_tab *(*peek_tab)(struct isl_context *context);
54 /* add equality; check is 1 if eq may not be valid;
55 * update is 1 if we may want to call ineq_sign on context later.
57 void (*add_eq)(struct isl_context *context, isl_int *eq,
58 int check, int update);
59 /* add inequality; check is 1 if ineq may not be valid;
60 * update is 1 if we may want to call ineq_sign on context later.
62 void (*add_ineq)(struct isl_context *context, isl_int *ineq,
63 int check, int update);
64 /* check sign of ineq based on previous information.
65 * strict is 1 if saturation should be treated as a positive sign.
67 enum isl_tab_row_sign (*ineq_sign)(struct isl_context *context,
68 isl_int *ineq, int strict);
69 /* check if inequality maintains feasibility */
70 int (*test_ineq)(struct isl_context *context, isl_int *ineq);
71 /* return index of a div that corresponds to "div" */
72 int (*get_div)(struct isl_context *context, struct isl_tab *tab,
74 /* add div "div" to context and return index and non-negativity */
75 int (*add_div)(struct isl_context *context, struct isl_vec *div,
77 int (*detect_equalities)(struct isl_context *context,
79 /* return row index of "best" split */
80 int (*best_split)(struct isl_context *context, struct isl_tab *tab);
81 /* check if context has already been determined to be empty */
82 int (*is_empty)(struct isl_context *context);
83 /* check if context is still usable */
84 int (*is_ok)(struct isl_context *context);
85 /* save a copy/snapshot of context */
86 void *(*save)(struct isl_context *context);
87 /* restore saved context */
88 void (*restore)(struct isl_context *context, void *);
89 /* invalidate context */
90 void (*invalidate)(struct isl_context *context);
92 void (*free)(struct isl_context *context);
96 struct isl_context_op *op;
99 struct isl_context_lex {
100 struct isl_context context;
104 struct isl_partial_sol {
106 struct isl_basic_set *dom;
109 struct isl_partial_sol *next;
113 struct isl_sol_callback {
114 struct isl_tab_callback callback;
118 /* isl_sol is an interface for constructing a solution to
119 * a parametric integer linear programming problem.
120 * Every time the algorithm reaches a state where a solution
121 * can be read off from the tableau (including cases where the tableau
122 * is empty), the function "add" is called on the isl_sol passed
123 * to find_solutions_main.
125 * The context tableau is owned by isl_sol and is updated incrementally.
127 * There are currently two implementations of this interface,
128 * isl_sol_map, which simply collects the solutions in an isl_map
129 * and (optionally) the parts of the context where there is no solution
131 * isl_sol_for, which calls a user-defined function for each part of
140 struct isl_context *context;
141 struct isl_partial_sol *partial;
142 void (*add)(struct isl_sol *sol,
143 struct isl_basic_set *dom, struct isl_mat *M);
144 void (*add_empty)(struct isl_sol *sol, struct isl_basic_set *bset);
145 void (*free)(struct isl_sol *sol);
146 struct isl_sol_callback dec_level;
149 static void sol_free(struct isl_sol *sol)
151 struct isl_partial_sol *partial, *next;
154 for (partial = sol->partial; partial; partial = next) {
155 next = partial->next;
156 isl_basic_set_free(partial->dom);
157 isl_mat_free(partial->M);
163 /* Push a partial solution represented by a domain and mapping M
164 * onto the stack of partial solutions.
166 static void sol_push_sol(struct isl_sol *sol,
167 struct isl_basic_set *dom, struct isl_mat *M)
169 struct isl_partial_sol *partial;
171 if (sol->error || !dom)
174 partial = isl_alloc_type(dom->ctx, struct isl_partial_sol);
178 partial->level = sol->level;
181 partial->next = sol->partial;
183 sol->partial = partial;
187 isl_basic_set_free(dom);
191 /* Pop one partial solution from the partial solution stack and
192 * pass it on to sol->add or sol->add_empty.
194 static void sol_pop_one(struct isl_sol *sol)
196 struct isl_partial_sol *partial;
198 partial = sol->partial;
199 sol->partial = partial->next;
202 sol->add(sol, partial->dom, partial->M);
204 sol->add_empty(sol, partial->dom);
208 /* Return a fresh copy of the domain represented by the context tableau.
210 static struct isl_basic_set *sol_domain(struct isl_sol *sol)
212 struct isl_basic_set *bset;
217 bset = isl_basic_set_dup(sol->context->op->peek_basic_set(sol->context));
218 bset = isl_basic_set_update_from_tab(bset,
219 sol->context->op->peek_tab(sol->context));
224 /* Check whether two partial solutions have the same mapping, where n_div
225 * is the number of divs that the two partial solutions have in common.
227 static int same_solution(struct isl_partial_sol *s1, struct isl_partial_sol *s2,
233 if (!s1->M != !s2->M)
238 dim = isl_basic_set_total_dim(s1->dom) - s1->dom->n_div;
240 for (i = 0; i < s1->M->n_row; ++i) {
241 if (isl_seq_first_non_zero(s1->M->row[i]+1+dim+n_div,
242 s1->M->n_col-1-dim-n_div) != -1)
244 if (isl_seq_first_non_zero(s2->M->row[i]+1+dim+n_div,
245 s2->M->n_col-1-dim-n_div) != -1)
247 if (!isl_seq_eq(s1->M->row[i], s2->M->row[i], 1+dim+n_div))
253 /* Pop all solutions from the partial solution stack that were pushed onto
254 * the stack at levels that are deeper than the current level.
255 * If the two topmost elements on the stack have the same level
256 * and represent the same solution, then their domains are combined.
257 * This combined domain is the same as the current context domain
258 * as sol_pop is called each time we move back to a higher level.
260 static void sol_pop(struct isl_sol *sol)
262 struct isl_partial_sol *partial;
268 if (sol->level == 0) {
269 for (partial = sol->partial; partial; partial = sol->partial)
274 partial = sol->partial;
278 if (partial->level <= sol->level)
281 if (partial->next && partial->next->level == partial->level) {
282 n_div = isl_basic_set_dim(
283 sol->context->op->peek_basic_set(sol->context),
286 if (!same_solution(partial, partial->next, n_div)) {
290 struct isl_basic_set *bset;
292 bset = sol_domain(sol);
294 isl_basic_set_free(partial->next->dom);
295 partial->next->dom = bset;
296 partial->next->level = sol->level;
298 sol->partial = partial->next;
299 isl_basic_set_free(partial->dom);
300 isl_mat_free(partial->M);
307 static void sol_dec_level(struct isl_sol *sol)
317 static int sol_dec_level_wrap(struct isl_tab_callback *cb)
319 struct isl_sol_callback *callback = (struct isl_sol_callback *)cb;
321 sol_dec_level(callback->sol);
323 return callback->sol->error ? -1 : 0;
326 /* Move down to next level and push callback onto context tableau
327 * to decrease the level again when it gets rolled back across
328 * the current state. That is, dec_level will be called with
329 * the context tableau in the same state as it is when inc_level
332 static void sol_inc_level(struct isl_sol *sol)
340 tab = sol->context->op->peek_tab(sol->context);
341 if (isl_tab_push_callback(tab, &sol->dec_level.callback) < 0)
345 static void scale_rows(struct isl_mat *mat, isl_int m, int n_row)
349 if (isl_int_is_one(m))
352 for (i = 0; i < n_row; ++i)
353 isl_seq_scale(mat->row[i], mat->row[i], m, mat->n_col);
356 /* Add the solution identified by the tableau and the context tableau.
358 * The layout of the variables is as follows.
359 * tab->n_var is equal to the total number of variables in the input
360 * map (including divs that were copied from the context)
361 * + the number of extra divs constructed
362 * Of these, the first tab->n_param and the last tab->n_div variables
363 * correspond to the variables in the context, i.e.,
364 * tab->n_param + tab->n_div = context_tab->n_var
365 * tab->n_param is equal to the number of parameters and input
366 * dimensions in the input map
367 * tab->n_div is equal to the number of divs in the context
369 * If there is no solution, then call add_empty with a basic set
370 * that corresponds to the context tableau. (If add_empty is NULL,
373 * If there is a solution, then first construct a matrix that maps
374 * all dimensions of the context to the output variables, i.e.,
375 * the output dimensions in the input map.
376 * The divs in the input map (if any) that do not correspond to any
377 * div in the context do not appear in the solution.
378 * The algorithm will make sure that they have an integer value,
379 * but these values themselves are of no interest.
380 * We have to be careful not to drop or rearrange any divs in the
381 * context because that would change the meaning of the matrix.
383 * To extract the value of the output variables, it should be noted
384 * that we always use a big parameter M in the main tableau and so
385 * the variable stored in this tableau is not an output variable x itself, but
386 * x' = M + x (in case of minimization)
388 * x' = M - x (in case of maximization)
389 * If x' appears in a column, then its optimal value is zero,
390 * which means that the optimal value of x is an unbounded number
391 * (-M for minimization and M for maximization).
392 * We currently assume that the output dimensions in the original map
393 * are bounded, so this cannot occur.
394 * Similarly, when x' appears in a row, then the coefficient of M in that
395 * row is necessarily 1.
396 * If the row in the tableau represents
397 * d x' = c + d M + e(y)
398 * then, in case of minimization, the corresponding row in the matrix
401 * with a d = m, the (updated) common denominator of the matrix.
402 * In case of maximization, the row will be
405 static void sol_add(struct isl_sol *sol, struct isl_tab *tab)
407 struct isl_basic_set *bset = NULL;
408 struct isl_mat *mat = NULL;
413 if (sol->error || !tab)
416 if (tab->empty && !sol->add_empty)
419 bset = sol_domain(sol);
422 sol_push_sol(sol, bset, NULL);
428 mat = isl_mat_alloc(tab->mat->ctx, 1 + sol->n_out,
429 1 + tab->n_param + tab->n_div);
435 isl_seq_clr(mat->row[0] + 1, mat->n_col - 1);
436 isl_int_set_si(mat->row[0][0], 1);
437 for (row = 0; row < sol->n_out; ++row) {
438 int i = tab->n_param + row;
441 isl_seq_clr(mat->row[1 + row], mat->n_col);
442 if (!tab->var[i].is_row) {
444 isl_assert(mat->ctx, !tab->M, goto error2);
448 r = tab->var[i].index;
451 isl_assert(mat->ctx, isl_int_eq(tab->mat->row[r][2],
452 tab->mat->row[r][0]),
454 isl_int_gcd(m, mat->row[0][0], tab->mat->row[r][0]);
455 isl_int_divexact(m, tab->mat->row[r][0], m);
456 scale_rows(mat, m, 1 + row);
457 isl_int_divexact(m, mat->row[0][0], tab->mat->row[r][0]);
458 isl_int_mul(mat->row[1 + row][0], m, tab->mat->row[r][1]);
459 for (j = 0; j < tab->n_param; ++j) {
461 if (tab->var[j].is_row)
463 col = tab->var[j].index;
464 isl_int_mul(mat->row[1 + row][1 + j], m,
465 tab->mat->row[r][off + col]);
467 for (j = 0; j < tab->n_div; ++j) {
469 if (tab->var[tab->n_var - tab->n_div+j].is_row)
471 col = tab->var[tab->n_var - tab->n_div+j].index;
472 isl_int_mul(mat->row[1 + row][1 + tab->n_param + j], m,
473 tab->mat->row[r][off + col]);
476 isl_seq_neg(mat->row[1 + row], mat->row[1 + row],
482 sol_push_sol(sol, bset, mat);
487 isl_basic_set_free(bset);
495 struct isl_set *empty;
498 static void sol_map_free(struct isl_sol_map *sol_map)
500 if (sol_map->sol.context)
501 sol_map->sol.context->op->free(sol_map->sol.context);
502 isl_map_free(sol_map->map);
503 isl_set_free(sol_map->empty);
507 static void sol_map_free_wrap(struct isl_sol *sol)
509 sol_map_free((struct isl_sol_map *)sol);
512 /* This function is called for parts of the context where there is
513 * no solution, with "bset" corresponding to the context tableau.
514 * Simply add the basic set to the set "empty".
516 static void sol_map_add_empty(struct isl_sol_map *sol,
517 struct isl_basic_set *bset)
521 isl_assert(bset->ctx, sol->empty, goto error);
523 sol->empty = isl_set_grow(sol->empty, 1);
524 bset = isl_basic_set_simplify(bset);
525 bset = isl_basic_set_finalize(bset);
526 sol->empty = isl_set_add(sol->empty, isl_basic_set_copy(bset));
529 isl_basic_set_free(bset);
532 isl_basic_set_free(bset);
536 static void sol_map_add_empty_wrap(struct isl_sol *sol,
537 struct isl_basic_set *bset)
539 sol_map_add_empty((struct isl_sol_map *)sol, bset);
542 /* Given a basic map "dom" that represents the context and an affine
543 * matrix "M" that maps the dimensions of the context to the
544 * output variables, construct a basic map with the same parameters
545 * and divs as the context, the dimensions of the context as input
546 * dimensions and a number of output dimensions that is equal to
547 * the number of output dimensions in the input map.
549 * The constraints and divs of the context are simply copied
550 * from "dom". For each row
554 * is added, with d the common denominator of M.
556 static void sol_map_add(struct isl_sol_map *sol,
557 struct isl_basic_set *dom, struct isl_mat *M)
560 struct isl_basic_map *bmap = NULL;
561 isl_basic_set *context_bset;
569 if (sol->sol.error || !dom || !M)
572 n_out = sol->sol.n_out;
573 n_eq = dom->n_eq + n_out;
574 n_ineq = dom->n_ineq;
576 nparam = isl_basic_set_total_dim(dom) - n_div;
577 total = isl_map_dim(sol->map, isl_dim_all);
578 bmap = isl_basic_map_alloc_dim(isl_map_get_dim(sol->map),
579 n_div, n_eq, 2 * n_div + n_ineq);
582 if (sol->sol.rational)
583 ISL_F_SET(bmap, ISL_BASIC_MAP_RATIONAL);
584 for (i = 0; i < dom->n_div; ++i) {
585 int k = isl_basic_map_alloc_div(bmap);
588 isl_seq_cpy(bmap->div[k], dom->div[i], 1 + 1 + nparam);
589 isl_seq_clr(bmap->div[k] + 1 + 1 + nparam, total - nparam);
590 isl_seq_cpy(bmap->div[k] + 1 + 1 + total,
591 dom->div[i] + 1 + 1 + nparam, i);
593 for (i = 0; i < dom->n_eq; ++i) {
594 int k = isl_basic_map_alloc_equality(bmap);
597 isl_seq_cpy(bmap->eq[k], dom->eq[i], 1 + nparam);
598 isl_seq_clr(bmap->eq[k] + 1 + nparam, total - nparam);
599 isl_seq_cpy(bmap->eq[k] + 1 + total,
600 dom->eq[i] + 1 + nparam, n_div);
602 for (i = 0; i < dom->n_ineq; ++i) {
603 int k = isl_basic_map_alloc_inequality(bmap);
606 isl_seq_cpy(bmap->ineq[k], dom->ineq[i], 1 + nparam);
607 isl_seq_clr(bmap->ineq[k] + 1 + nparam, total - nparam);
608 isl_seq_cpy(bmap->ineq[k] + 1 + total,
609 dom->ineq[i] + 1 + nparam, n_div);
611 for (i = 0; i < M->n_row - 1; ++i) {
612 int k = isl_basic_map_alloc_equality(bmap);
615 isl_seq_cpy(bmap->eq[k], M->row[1 + i], 1 + nparam);
616 isl_seq_clr(bmap->eq[k] + 1 + nparam, n_out);
617 isl_int_neg(bmap->eq[k][1 + nparam + i], M->row[0][0]);
618 isl_seq_cpy(bmap->eq[k] + 1 + nparam + n_out,
619 M->row[1 + i] + 1 + nparam, n_div);
621 bmap = isl_basic_map_simplify(bmap);
622 bmap = isl_basic_map_finalize(bmap);
623 sol->map = isl_map_grow(sol->map, 1);
624 sol->map = isl_map_add(sol->map, bmap);
627 isl_basic_set_free(dom);
631 isl_basic_set_free(dom);
633 isl_basic_map_free(bmap);
637 static void sol_map_add_wrap(struct isl_sol *sol,
638 struct isl_basic_set *dom, struct isl_mat *M)
640 sol_map_add((struct isl_sol_map *)sol, dom, M);
644 /* Store the "parametric constant" of row "row" of tableau "tab" in "line",
645 * i.e., the constant term and the coefficients of all variables that
646 * appear in the context tableau.
647 * Note that the coefficient of the big parameter M is NOT copied.
648 * The context tableau may not have a big parameter and even when it
649 * does, it is a different big parameter.
651 static void get_row_parameter_line(struct isl_tab *tab, int row, isl_int *line)
654 unsigned off = 2 + tab->M;
656 isl_int_set(line[0], tab->mat->row[row][1]);
657 for (i = 0; i < tab->n_param; ++i) {
658 if (tab->var[i].is_row)
659 isl_int_set_si(line[1 + i], 0);
661 int col = tab->var[i].index;
662 isl_int_set(line[1 + i], tab->mat->row[row][off + col]);
665 for (i = 0; i < tab->n_div; ++i) {
666 if (tab->var[tab->n_var - tab->n_div + i].is_row)
667 isl_int_set_si(line[1 + tab->n_param + i], 0);
669 int col = tab->var[tab->n_var - tab->n_div + i].index;
670 isl_int_set(line[1 + tab->n_param + i],
671 tab->mat->row[row][off + col]);
676 /* Check if rows "row1" and "row2" have identical "parametric constants",
677 * as explained above.
678 * In this case, we also insist that the coefficients of the big parameter
679 * be the same as the values of the constants will only be the same
680 * if these coefficients are also the same.
682 static int identical_parameter_line(struct isl_tab *tab, int row1, int row2)
685 unsigned off = 2 + tab->M;
687 if (isl_int_ne(tab->mat->row[row1][1], tab->mat->row[row2][1]))
690 if (tab->M && isl_int_ne(tab->mat->row[row1][2],
691 tab->mat->row[row2][2]))
694 for (i = 0; i < tab->n_param + tab->n_div; ++i) {
695 int pos = i < tab->n_param ? i :
696 tab->n_var - tab->n_div + i - tab->n_param;
699 if (tab->var[pos].is_row)
701 col = tab->var[pos].index;
702 if (isl_int_ne(tab->mat->row[row1][off + col],
703 tab->mat->row[row2][off + col]))
709 /* Return an inequality that expresses that the "parametric constant"
710 * should be non-negative.
711 * This function is only called when the coefficient of the big parameter
714 static struct isl_vec *get_row_parameter_ineq(struct isl_tab *tab, int row)
716 struct isl_vec *ineq;
718 ineq = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_param + tab->n_div);
722 get_row_parameter_line(tab, row, ineq->el);
724 ineq = isl_vec_normalize(ineq);
729 /* Return a integer division for use in a parametric cut based on the given row.
730 * In particular, let the parametric constant of the row be
734 * where y_0 = 1, but none of the y_i corresponds to the big parameter M.
735 * The div returned is equal to
737 * floor(\sum_i {-a_i} y_i) = floor((\sum_i (-a_i mod d) y_i)/d)
739 static struct isl_vec *get_row_parameter_div(struct isl_tab *tab, int row)
743 div = isl_vec_alloc(tab->mat->ctx, 1 + 1 + tab->n_param + tab->n_div);
747 isl_int_set(div->el[0], tab->mat->row[row][0]);
748 get_row_parameter_line(tab, row, div->el + 1);
749 div = isl_vec_normalize(div);
750 isl_seq_neg(div->el + 1, div->el + 1, div->size - 1);
751 isl_seq_fdiv_r(div->el + 1, div->el + 1, div->el[0], div->size - 1);
756 /* Return a integer division for use in transferring an integrality constraint
758 * In particular, let the parametric constant of the row be
762 * where y_0 = 1, but none of the y_i corresponds to the big parameter M.
763 * The the returned div is equal to
765 * floor(\sum_i {a_i} y_i) = floor((\sum_i (a_i mod d) y_i)/d)
767 static struct isl_vec *get_row_split_div(struct isl_tab *tab, int row)
771 div = isl_vec_alloc(tab->mat->ctx, 1 + 1 + tab->n_param + tab->n_div);
775 isl_int_set(div->el[0], tab->mat->row[row][0]);
776 get_row_parameter_line(tab, row, div->el + 1);
777 div = isl_vec_normalize(div);
778 isl_seq_fdiv_r(div->el + 1, div->el + 1, div->el[0], div->size - 1);
783 /* Construct and return an inequality that expresses an upper bound
785 * In particular, if the div is given by
789 * then the inequality expresses
793 static struct isl_vec *ineq_for_div(struct isl_basic_set *bset, unsigned div)
797 struct isl_vec *ineq;
802 total = isl_basic_set_total_dim(bset);
803 div_pos = 1 + total - bset->n_div + div;
805 ineq = isl_vec_alloc(bset->ctx, 1 + total);
809 isl_seq_cpy(ineq->el, bset->div[div] + 1, 1 + total);
810 isl_int_neg(ineq->el[div_pos], bset->div[div][0]);
814 /* Given a row in the tableau and a div that was created
815 * using get_row_split_div and that been constrained to equality, i.e.,
817 * d = floor(\sum_i {a_i} y_i) = \sum_i {a_i} y_i
819 * replace the expression "\sum_i {a_i} y_i" in the row by d,
820 * i.e., we subtract "\sum_i {a_i} y_i" and add 1 d.
821 * The coefficients of the non-parameters in the tableau have been
822 * verified to be integral. We can therefore simply replace coefficient b
823 * by floor(b). For the coefficients of the parameters we have
824 * floor(a_i) = a_i - {a_i}, while for the other coefficients, we have
827 static struct isl_tab *set_row_cst_to_div(struct isl_tab *tab, int row, int div)
829 isl_seq_fdiv_q(tab->mat->row[row] + 1, tab->mat->row[row] + 1,
830 tab->mat->row[row][0], 1 + tab->M + tab->n_col);
832 isl_int_set_si(tab->mat->row[row][0], 1);
834 if (tab->var[tab->n_var - tab->n_div + div].is_row) {
835 int drow = tab->var[tab->n_var - tab->n_div + div].index;
837 isl_assert(tab->mat->ctx,
838 isl_int_is_one(tab->mat->row[drow][0]), goto error);
839 isl_seq_combine(tab->mat->row[row] + 1,
840 tab->mat->ctx->one, tab->mat->row[row] + 1,
841 tab->mat->ctx->one, tab->mat->row[drow] + 1,
842 1 + tab->M + tab->n_col);
844 int dcol = tab->var[tab->n_var - tab->n_div + div].index;
846 isl_int_set_si(tab->mat->row[row][2 + tab->M + dcol], 1);
855 /* Check if the (parametric) constant of the given row is obviously
856 * negative, meaning that we don't need to consult the context tableau.
857 * If there is a big parameter and its coefficient is non-zero,
858 * then this coefficient determines the outcome.
859 * Otherwise, we check whether the constant is negative and
860 * all non-zero coefficients of parameters are negative and
861 * belong to non-negative parameters.
863 static int is_obviously_neg(struct isl_tab *tab, int row)
867 unsigned off = 2 + tab->M;
870 if (isl_int_is_pos(tab->mat->row[row][2]))
872 if (isl_int_is_neg(tab->mat->row[row][2]))
876 if (isl_int_is_nonneg(tab->mat->row[row][1]))
878 for (i = 0; i < tab->n_param; ++i) {
879 /* Eliminated parameter */
880 if (tab->var[i].is_row)
882 col = tab->var[i].index;
883 if (isl_int_is_zero(tab->mat->row[row][off + col]))
885 if (!tab->var[i].is_nonneg)
887 if (isl_int_is_pos(tab->mat->row[row][off + col]))
890 for (i = 0; i < tab->n_div; ++i) {
891 if (tab->var[tab->n_var - tab->n_div + i].is_row)
893 col = tab->var[tab->n_var - tab->n_div + i].index;
894 if (isl_int_is_zero(tab->mat->row[row][off + col]))
896 if (!tab->var[tab->n_var - tab->n_div + i].is_nonneg)
898 if (isl_int_is_pos(tab->mat->row[row][off + col]))
904 /* Check if the (parametric) constant of the given row is obviously
905 * non-negative, meaning that we don't need to consult the context tableau.
906 * If there is a big parameter and its coefficient is non-zero,
907 * then this coefficient determines the outcome.
908 * Otherwise, we check whether the constant is non-negative and
909 * all non-zero coefficients of parameters are positive and
910 * belong to non-negative parameters.
912 static int is_obviously_nonneg(struct isl_tab *tab, int row)
916 unsigned off = 2 + tab->M;
919 if (isl_int_is_pos(tab->mat->row[row][2]))
921 if (isl_int_is_neg(tab->mat->row[row][2]))
925 if (isl_int_is_neg(tab->mat->row[row][1]))
927 for (i = 0; i < tab->n_param; ++i) {
928 /* Eliminated parameter */
929 if (tab->var[i].is_row)
931 col = tab->var[i].index;
932 if (isl_int_is_zero(tab->mat->row[row][off + col]))
934 if (!tab->var[i].is_nonneg)
936 if (isl_int_is_neg(tab->mat->row[row][off + col]))
939 for (i = 0; i < tab->n_div; ++i) {
940 if (tab->var[tab->n_var - tab->n_div + i].is_row)
942 col = tab->var[tab->n_var - tab->n_div + i].index;
943 if (isl_int_is_zero(tab->mat->row[row][off + col]))
945 if (!tab->var[tab->n_var - tab->n_div + i].is_nonneg)
947 if (isl_int_is_neg(tab->mat->row[row][off + col]))
953 /* Given a row r and two columns, return the column that would
954 * lead to the lexicographically smallest increment in the sample
955 * solution when leaving the basis in favor of the row.
956 * Pivoting with column c will increment the sample value by a non-negative
957 * constant times a_{V,c}/a_{r,c}, with a_{V,c} the elements of column c
958 * corresponding to the non-parametric variables.
959 * If variable v appears in a column c_v, the a_{v,c} = 1 iff c = c_v,
960 * with all other entries in this virtual row equal to zero.
961 * If variable v appears in a row, then a_{v,c} is the element in column c
964 * Let v be the first variable with a_{v,c1}/a_{r,c1} != a_{v,c2}/a_{r,c2}.
965 * Then if a_{v,c1}/a_{r,c1} < a_{v,c2}/a_{r,c2}, i.e.,
966 * a_{v,c2} a_{r,c1} - a_{v,c1} a_{r,c2} > 0, c1 results in the minimal
967 * increment. Otherwise, it's c2.
969 static int lexmin_col_pair(struct isl_tab *tab,
970 int row, int col1, int col2, isl_int tmp)
975 tr = tab->mat->row[row] + 2 + tab->M;
977 for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
981 if (!tab->var[i].is_row) {
982 if (tab->var[i].index == col1)
984 if (tab->var[i].index == col2)
989 if (tab->var[i].index == row)
992 r = tab->mat->row[tab->var[i].index] + 2 + tab->M;
993 s1 = isl_int_sgn(r[col1]);
994 s2 = isl_int_sgn(r[col2]);
995 if (s1 == 0 && s2 == 0)
1002 isl_int_mul(tmp, r[col2], tr[col1]);
1003 isl_int_submul(tmp, r[col1], tr[col2]);
1004 if (isl_int_is_pos(tmp))
1006 if (isl_int_is_neg(tmp))
1012 /* Given a row in the tableau, find and return the column that would
1013 * result in the lexicographically smallest, but positive, increment
1014 * in the sample point.
1015 * If there is no such column, then return tab->n_col.
1016 * If anything goes wrong, return -1.
1018 static int lexmin_pivot_col(struct isl_tab *tab, int row)
1021 int col = tab->n_col;
1025 tr = tab->mat->row[row] + 2 + tab->M;
1029 for (j = tab->n_dead; j < tab->n_col; ++j) {
1030 if (tab->col_var[j] >= 0 &&
1031 (tab->col_var[j] < tab->n_param ||
1032 tab->col_var[j] >= tab->n_var - tab->n_div))
1035 if (!isl_int_is_pos(tr[j]))
1038 if (col == tab->n_col)
1041 col = lexmin_col_pair(tab, row, col, j, tmp);
1042 isl_assert(tab->mat->ctx, col >= 0, goto error);
1052 /* Return the first known violated constraint, i.e., a non-negative
1053 * contraint that currently has an either obviously negative value
1054 * or a previously determined to be negative value.
1056 * If any constraint has a negative coefficient for the big parameter,
1057 * if any, then we return one of these first.
1059 static int first_neg(struct isl_tab *tab)
1064 for (row = tab->n_redundant; row < tab->n_row; ++row) {
1065 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
1067 if (isl_int_is_neg(tab->mat->row[row][2]))
1070 for (row = tab->n_redundant; row < tab->n_row; ++row) {
1071 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
1073 if (tab->row_sign) {
1074 if (tab->row_sign[row] == 0 &&
1075 is_obviously_neg(tab, row))
1076 tab->row_sign[row] = isl_tab_row_neg;
1077 if (tab->row_sign[row] != isl_tab_row_neg)
1079 } else if (!is_obviously_neg(tab, row))
1086 /* Resolve all known or obviously violated constraints through pivoting.
1087 * In particular, as long as we can find any violated constraint, we
1088 * look for a pivoting column that would result in the lexicographicallly
1089 * smallest increment in the sample point. If there is no such column
1090 * then the tableau is infeasible.
1092 static struct isl_tab *restore_lexmin(struct isl_tab *tab) WARN_UNUSED;
1093 static struct isl_tab *restore_lexmin(struct isl_tab *tab)
1101 while ((row = first_neg(tab)) != -1) {
1102 col = lexmin_pivot_col(tab, row);
1103 if (col >= tab->n_col)
1104 return isl_tab_mark_empty(tab);
1107 if (isl_tab_pivot(tab, row, col) < 0)
1116 /* Given a row that represents an equality, look for an appropriate
1118 * In particular, if there are any non-zero coefficients among
1119 * the non-parameter variables, then we take the last of these
1120 * variables. Eliminating this variable in terms of the other
1121 * variables and/or parameters does not influence the property
1122 * that all column in the initial tableau are lexicographically
1123 * positive. The row corresponding to the eliminated variable
1124 * will only have non-zero entries below the diagonal of the
1125 * initial tableau. That is, we transform
1131 * If there is no such non-parameter variable, then we are dealing with
1132 * pure parameter equality and we pick any parameter with coefficient 1 or -1
1133 * for elimination. This will ensure that the eliminated parameter
1134 * always has an integer value whenever all the other parameters are integral.
1135 * If there is no such parameter then we return -1.
1137 static int last_var_col_or_int_par_col(struct isl_tab *tab, int row)
1139 unsigned off = 2 + tab->M;
1142 for (i = tab->n_var - tab->n_div - 1; i >= 0 && i >= tab->n_param; --i) {
1144 if (tab->var[i].is_row)
1146 col = tab->var[i].index;
1147 if (col <= tab->n_dead)
1149 if (!isl_int_is_zero(tab->mat->row[row][off + col]))
1152 for (i = tab->n_dead; i < tab->n_col; ++i) {
1153 if (isl_int_is_one(tab->mat->row[row][off + i]))
1155 if (isl_int_is_negone(tab->mat->row[row][off + i]))
1161 /* Add an equality that is known to be valid to the tableau.
1162 * We first check if we can eliminate a variable or a parameter.
1163 * If not, we add the equality as two inequalities.
1164 * In this case, the equality was a pure parameter equality and there
1165 * is no need to resolve any constraint violations.
1167 static struct isl_tab *add_lexmin_valid_eq(struct isl_tab *tab, isl_int *eq)
1174 r = isl_tab_add_row(tab, eq);
1178 r = tab->con[r].index;
1179 i = last_var_col_or_int_par_col(tab, r);
1181 tab->con[r].is_nonneg = 1;
1182 if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
1184 isl_seq_neg(eq, eq, 1 + tab->n_var);
1185 r = isl_tab_add_row(tab, eq);
1188 tab->con[r].is_nonneg = 1;
1189 if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
1192 if (isl_tab_pivot(tab, r, i) < 0)
1194 if (isl_tab_kill_col(tab, i) < 0)
1198 tab = restore_lexmin(tab);
1207 /* Check if the given row is a pure constant.
1209 static int is_constant(struct isl_tab *tab, int row)
1211 unsigned off = 2 + tab->M;
1213 return isl_seq_first_non_zero(tab->mat->row[row] + off + tab->n_dead,
1214 tab->n_col - tab->n_dead) == -1;
1217 /* Add an equality that may or may not be valid to the tableau.
1218 * If the resulting row is a pure constant, then it must be zero.
1219 * Otherwise, the resulting tableau is empty.
1221 * If the row is not a pure constant, then we add two inequalities,
1222 * each time checking that they can be satisfied.
1223 * In the end we try to use one of the two constraints to eliminate
1226 static struct isl_tab *add_lexmin_eq(struct isl_tab *tab, isl_int *eq) WARN_UNUSED;
1227 static struct isl_tab *add_lexmin_eq(struct isl_tab *tab, isl_int *eq)
1231 struct isl_tab_undo *snap;
1235 snap = isl_tab_snap(tab);
1236 r1 = isl_tab_add_row(tab, eq);
1239 tab->con[r1].is_nonneg = 1;
1240 if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r1]) < 0)
1243 row = tab->con[r1].index;
1244 if (is_constant(tab, row)) {
1245 if (!isl_int_is_zero(tab->mat->row[row][1]) ||
1246 (tab->M && !isl_int_is_zero(tab->mat->row[row][2])))
1247 return isl_tab_mark_empty(tab);
1248 if (isl_tab_rollback(tab, snap) < 0)
1253 tab = restore_lexmin(tab);
1254 if (!tab || tab->empty)
1257 isl_seq_neg(eq, eq, 1 + tab->n_var);
1259 r2 = isl_tab_add_row(tab, eq);
1262 tab->con[r2].is_nonneg = 1;
1263 if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r2]) < 0)
1266 tab = restore_lexmin(tab);
1267 if (!tab || tab->empty)
1270 if (!tab->con[r1].is_row) {
1271 if (isl_tab_kill_col(tab, tab->con[r1].index) < 0)
1273 } else if (!tab->con[r2].is_row) {
1274 if (isl_tab_kill_col(tab, tab->con[r2].index) < 0)
1276 } else if (isl_int_is_zero(tab->mat->row[tab->con[r1].index][1])) {
1277 unsigned off = 2 + tab->M;
1279 int row = tab->con[r1].index;
1280 i = isl_seq_first_non_zero(tab->mat->row[row]+off+tab->n_dead,
1281 tab->n_col - tab->n_dead);
1283 if (isl_tab_pivot(tab, row, tab->n_dead + i) < 0)
1285 if (isl_tab_kill_col(tab, tab->n_dead + i) < 0)
1291 tab->bset = isl_basic_set_add_ineq(tab->bset, eq);
1292 if (isl_tab_push(tab, isl_tab_undo_bset_ineq) < 0)
1294 isl_seq_neg(eq, eq, 1 + tab->n_var);
1295 tab->bset = isl_basic_set_add_ineq(tab->bset, eq);
1296 isl_seq_neg(eq, eq, 1 + tab->n_var);
1297 if (isl_tab_push(tab, isl_tab_undo_bset_ineq) < 0)
1309 /* Add an inequality to the tableau, resolving violations using
1312 static struct isl_tab *add_lexmin_ineq(struct isl_tab *tab, isl_int *ineq)
1319 tab->bset = isl_basic_set_add_ineq(tab->bset, ineq);
1320 if (isl_tab_push(tab, isl_tab_undo_bset_ineq) < 0)
1325 r = isl_tab_add_row(tab, ineq);
1328 tab->con[r].is_nonneg = 1;
1329 if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
1331 if (isl_tab_row_is_redundant(tab, tab->con[r].index)) {
1332 if (isl_tab_mark_redundant(tab, tab->con[r].index) < 0)
1337 tab = restore_lexmin(tab);
1338 if (tab && !tab->empty && tab->con[r].is_row &&
1339 isl_tab_row_is_redundant(tab, tab->con[r].index))
1340 if (isl_tab_mark_redundant(tab, tab->con[r].index) < 0)
1348 /* Check if the coefficients of the parameters are all integral.
1350 static int integer_parameter(struct isl_tab *tab, int row)
1354 unsigned off = 2 + tab->M;
1356 for (i = 0; i < tab->n_param; ++i) {
1357 /* Eliminated parameter */
1358 if (tab->var[i].is_row)
1360 col = tab->var[i].index;
1361 if (!isl_int_is_divisible_by(tab->mat->row[row][off + col],
1362 tab->mat->row[row][0]))
1365 for (i = 0; i < tab->n_div; ++i) {
1366 if (tab->var[tab->n_var - tab->n_div + i].is_row)
1368 col = tab->var[tab->n_var - tab->n_div + i].index;
1369 if (!isl_int_is_divisible_by(tab->mat->row[row][off + col],
1370 tab->mat->row[row][0]))
1376 /* Check if the coefficients of the non-parameter variables are all integral.
1378 static int integer_variable(struct isl_tab *tab, int row)
1381 unsigned off = 2 + tab->M;
1383 for (i = tab->n_dead; i < tab->n_col; ++i) {
1384 if (tab->col_var[i] >= 0 &&
1385 (tab->col_var[i] < tab->n_param ||
1386 tab->col_var[i] >= tab->n_var - tab->n_div))
1388 if (!isl_int_is_divisible_by(tab->mat->row[row][off + i],
1389 tab->mat->row[row][0]))
1395 /* Check if the constant term is integral.
1397 static int integer_constant(struct isl_tab *tab, int row)
1399 return isl_int_is_divisible_by(tab->mat->row[row][1],
1400 tab->mat->row[row][0]);
1403 #define I_CST 1 << 0
1404 #define I_PAR 1 << 1
1405 #define I_VAR 1 << 2
1407 /* Check for first (non-parameter) variable that is non-integer and
1408 * therefore requires a cut.
1409 * For parametric tableaus, there are three parts in a row,
1410 * the constant, the coefficients of the parameters and the rest.
1411 * For each part, we check whether the coefficients in that part
1412 * are all integral and if so, set the corresponding flag in *f.
1413 * If the constant and the parameter part are integral, then the
1414 * current sample value is integral and no cut is required
1415 * (irrespective of whether the variable part is integral).
1417 static int first_non_integer(struct isl_tab *tab, int *f)
1421 for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
1424 if (!tab->var[i].is_row)
1426 row = tab->var[i].index;
1427 if (integer_constant(tab, row))
1428 ISL_FL_SET(flags, I_CST);
1429 if (integer_parameter(tab, row))
1430 ISL_FL_SET(flags, I_PAR);
1431 if (ISL_FL_ISSET(flags, I_CST) && ISL_FL_ISSET(flags, I_PAR))
1433 if (integer_variable(tab, row))
1434 ISL_FL_SET(flags, I_VAR);
1441 /* Add a (non-parametric) cut to cut away the non-integral sample
1442 * value of the given row.
1444 * If the row is given by
1446 * m r = f + \sum_i a_i y_i
1450 * c = - {-f/m} + \sum_i {a_i/m} y_i >= 0
1452 * The big parameter, if any, is ignored, since it is assumed to be big
1453 * enough to be divisible by any integer.
1454 * If the tableau is actually a parametric tableau, then this function
1455 * is only called when all coefficients of the parameters are integral.
1456 * The cut therefore has zero coefficients for the parameters.
1458 * The current value is known to be negative, so row_sign, if it
1459 * exists, is set accordingly.
1461 * Return the row of the cut or -1.
1463 static int add_cut(struct isl_tab *tab, int row)
1468 unsigned off = 2 + tab->M;
1470 if (isl_tab_extend_cons(tab, 1) < 0)
1472 r = isl_tab_allocate_con(tab);
1476 r_row = tab->mat->row[tab->con[r].index];
1477 isl_int_set(r_row[0], tab->mat->row[row][0]);
1478 isl_int_neg(r_row[1], tab->mat->row[row][1]);
1479 isl_int_fdiv_r(r_row[1], r_row[1], tab->mat->row[row][0]);
1480 isl_int_neg(r_row[1], r_row[1]);
1482 isl_int_set_si(r_row[2], 0);
1483 for (i = 0; i < tab->n_col; ++i)
1484 isl_int_fdiv_r(r_row[off + i],
1485 tab->mat->row[row][off + i], tab->mat->row[row][0]);
1487 tab->con[r].is_nonneg = 1;
1488 if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
1491 tab->row_sign[tab->con[r].index] = isl_tab_row_neg;
1493 return tab->con[r].index;
1496 /* Given a non-parametric tableau, add cuts until an integer
1497 * sample point is obtained or until the tableau is determined
1498 * to be integer infeasible.
1499 * As long as there is any non-integer value in the sample point,
1500 * we add an appropriate cut, if possible and resolve the violated
1501 * cut constraint using restore_lexmin.
1502 * If one of the corresponding rows is equal to an integral
1503 * combination of variables/constraints plus a non-integral constant,
1504 * then there is no way to obtain an integer point an we return
1505 * a tableau that is marked empty.
1507 static struct isl_tab *cut_to_integer_lexmin(struct isl_tab *tab)
1517 while ((row = first_non_integer(tab, &flags)) != -1) {
1518 if (ISL_FL_ISSET(flags, I_VAR))
1519 return isl_tab_mark_empty(tab);
1520 row = add_cut(tab, row);
1523 tab = restore_lexmin(tab);
1524 if (!tab || tab->empty)
1533 /* Check whether all the currently active samples also satisfy the inequality
1534 * "ineq" (treated as an equality if eq is set).
1535 * Remove those samples that do not.
1537 static struct isl_tab *check_samples(struct isl_tab *tab, isl_int *ineq, int eq)
1545 isl_assert(tab->mat->ctx, tab->bset, goto error);
1546 isl_assert(tab->mat->ctx, tab->samples, goto error);
1547 isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, goto error);
1550 for (i = tab->n_outside; i < tab->n_sample; ++i) {
1552 isl_seq_inner_product(ineq, tab->samples->row[i],
1553 1 + tab->n_var, &v);
1554 sgn = isl_int_sgn(v);
1555 if (eq ? (sgn == 0) : (sgn >= 0))
1557 tab = isl_tab_drop_sample(tab, i);
1569 /* Check whether the sample value of the tableau is finite,
1570 * i.e., either the tableau does not use a big parameter, or
1571 * all values of the variables are equal to the big parameter plus
1572 * some constant. This constant is the actual sample value.
1574 static int sample_is_finite(struct isl_tab *tab)
1581 for (i = 0; i < tab->n_var; ++i) {
1583 if (!tab->var[i].is_row)
1585 row = tab->var[i].index;
1586 if (isl_int_ne(tab->mat->row[row][0], tab->mat->row[row][2]))
1592 /* Check if the context tableau of sol has any integer points.
1593 * Leave tab in empty state if no integer point can be found.
1594 * If an integer point can be found and if moreover it is finite,
1595 * then it is added to the list of sample values.
1597 * This function is only called when none of the currently active sample
1598 * values satisfies the most recently added constraint.
1600 static struct isl_tab *check_integer_feasible(struct isl_tab *tab)
1602 struct isl_tab_undo *snap;
1608 snap = isl_tab_snap(tab);
1609 if (isl_tab_push_basis(tab) < 0)
1612 tab = cut_to_integer_lexmin(tab);
1616 if (!tab->empty && sample_is_finite(tab)) {
1617 struct isl_vec *sample;
1619 sample = isl_tab_get_sample_value(tab);
1621 tab = isl_tab_add_sample(tab, sample);
1624 if (!tab->empty && isl_tab_rollback(tab, snap) < 0)
1633 /* Check if any of the currently active sample values satisfies
1634 * the inequality "ineq" (an equality if eq is set).
1636 static int tab_has_valid_sample(struct isl_tab *tab, isl_int *ineq, int eq)
1644 isl_assert(tab->mat->ctx, tab->bset, return -1);
1645 isl_assert(tab->mat->ctx, tab->samples, return -1);
1646 isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, return -1);
1649 for (i = tab->n_outside; i < tab->n_sample; ++i) {
1651 isl_seq_inner_product(ineq, tab->samples->row[i],
1652 1 + tab->n_var, &v);
1653 sgn = isl_int_sgn(v);
1654 if (eq ? (sgn == 0) : (sgn >= 0))
1659 return i < tab->n_sample;
1662 /* For a div d = floor(f/m), add the constraints
1665 * -(f-(m-1)) + m d >= 0
1667 * Note that the second constraint is the negation of
1671 static void add_div_constraints(struct isl_context *context, unsigned div)
1675 struct isl_vec *ineq;
1676 struct isl_basic_set *bset;
1678 bset = context->op->peek_basic_set(context);
1682 total = isl_basic_set_total_dim(bset);
1683 div_pos = 1 + total - bset->n_div + div;
1685 ineq = ineq_for_div(bset, div);
1689 context->op->add_ineq(context, ineq->el, 0, 0);
1691 isl_seq_neg(ineq->el, bset->div[div] + 1, 1 + total);
1692 isl_int_set(ineq->el[div_pos], bset->div[div][0]);
1693 isl_int_add(ineq->el[0], ineq->el[0], ineq->el[div_pos]);
1694 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
1696 context->op->add_ineq(context, ineq->el, 0, 0);
1702 context->op->invalidate(context);
1705 /* Add a div specifed by "div" to the tableau "tab" and return
1706 * the index of the new div. *nonneg is set to 1 if the div
1707 * is obviously non-negative.
1709 static int context_tab_add_div(struct isl_tab *tab, struct isl_vec *div,
1715 struct isl_mat *samples;
1717 for (i = 0; i < tab->n_var; ++i) {
1718 if (isl_int_is_zero(div->el[2 + i]))
1720 if (!tab->var[i].is_nonneg)
1723 *nonneg = i == tab->n_var;
1725 if (isl_tab_extend_cons(tab, 3) < 0)
1727 if (isl_tab_extend_vars(tab, 1) < 0)
1729 r = isl_tab_allocate_var(tab);
1733 tab->var[r].is_nonneg = 1;
1734 tab->var[r].frozen = 1;
1736 samples = isl_mat_extend(tab->samples,
1737 tab->n_sample, 1 + tab->n_var);
1738 tab->samples = samples;
1741 for (i = tab->n_outside; i < samples->n_row; ++i) {
1742 isl_seq_inner_product(div->el + 1, samples->row[i],
1743 div->size - 1, &samples->row[i][samples->n_col - 1]);
1744 isl_int_fdiv_q(samples->row[i][samples->n_col - 1],
1745 samples->row[i][samples->n_col - 1], div->el[0]);
1748 tab->bset = isl_basic_set_extend_dim(tab->bset,
1749 isl_basic_set_get_dim(tab->bset), 1, 0, 2);
1750 k = isl_basic_set_alloc_div(tab->bset);
1753 isl_seq_cpy(tab->bset->div[k], div->el, div->size);
1754 if (isl_tab_push(tab, isl_tab_undo_bset_div) < 0)
1760 /* Add a div specified by "div" to both the main tableau and
1761 * the context tableau. In case of the main tableau, we only
1762 * need to add an extra div. In the context tableau, we also
1763 * need to express the meaning of the div.
1764 * Return the index of the div or -1 if anything went wrong.
1766 static int add_div(struct isl_tab *tab, struct isl_context *context,
1767 struct isl_vec *div)
1773 k = context->op->add_div(context, div, &nonneg);
1777 add_div_constraints(context, k);
1778 if (!context->op->is_ok(context))
1781 if (isl_tab_extend_vars(tab, 1) < 0)
1783 r = isl_tab_allocate_var(tab);
1787 tab->var[r].is_nonneg = 1;
1788 tab->var[r].frozen = 1;
1791 return tab->n_div - 1;
1793 context->op->invalidate(context);
1797 static int find_div(struct isl_tab *tab, isl_int *div, isl_int denom)
1800 unsigned total = isl_basic_set_total_dim(tab->bset);
1802 for (i = 0; i < tab->bset->n_div; ++i) {
1803 if (isl_int_ne(tab->bset->div[i][0], denom))
1805 if (!isl_seq_eq(tab->bset->div[i] + 1, div, total))
1812 /* Return the index of a div that corresponds to "div".
1813 * We first check if we already have such a div and if not, we create one.
1815 static int get_div(struct isl_tab *tab, struct isl_context *context,
1816 struct isl_vec *div)
1819 struct isl_tab *context_tab = context->op->peek_tab(context);
1824 d = find_div(context_tab, div->el + 1, div->el[0]);
1828 return add_div(tab, context, div);
1831 /* Add a parametric cut to cut away the non-integral sample value
1833 * Let a_i be the coefficients of the constant term and the parameters
1834 * and let b_i be the coefficients of the variables or constraints
1835 * in basis of the tableau.
1836 * Let q be the div q = floor(\sum_i {-a_i} y_i).
1838 * The cut is expressed as
1840 * c = \sum_i -{-a_i} y_i + \sum_i {b_i} x_i + q >= 0
1842 * If q did not already exist in the context tableau, then it is added first.
1843 * If q is in a column of the main tableau then the "+ q" can be accomplished
1844 * by setting the corresponding entry to the denominator of the constraint.
1845 * If q happens to be in a row of the main tableau, then the corresponding
1846 * row needs to be added instead (taking care of the denominators).
1847 * Note that this is very unlikely, but perhaps not entirely impossible.
1849 * The current value of the cut is known to be negative (or at least
1850 * non-positive), so row_sign is set accordingly.
1852 * Return the row of the cut or -1.
1854 static int add_parametric_cut(struct isl_tab *tab, int row,
1855 struct isl_context *context)
1857 struct isl_vec *div;
1864 unsigned off = 2 + tab->M;
1869 div = get_row_parameter_div(tab, row);
1874 d = context->op->get_div(context, tab, div);
1878 if (isl_tab_extend_cons(tab, 1) < 0)
1880 r = isl_tab_allocate_con(tab);
1884 r_row = tab->mat->row[tab->con[r].index];
1885 isl_int_set(r_row[0], tab->mat->row[row][0]);
1886 isl_int_neg(r_row[1], tab->mat->row[row][1]);
1887 isl_int_fdiv_r(r_row[1], r_row[1], tab->mat->row[row][0]);
1888 isl_int_neg(r_row[1], r_row[1]);
1890 isl_int_set_si(r_row[2], 0);
1891 for (i = 0; i < tab->n_param; ++i) {
1892 if (tab->var[i].is_row)
1894 col = tab->var[i].index;
1895 isl_int_neg(r_row[off + col], tab->mat->row[row][off + col]);
1896 isl_int_fdiv_r(r_row[off + col], r_row[off + col],
1897 tab->mat->row[row][0]);
1898 isl_int_neg(r_row[off + col], r_row[off + col]);
1900 for (i = 0; i < tab->n_div; ++i) {
1901 if (tab->var[tab->n_var - tab->n_div + i].is_row)
1903 col = tab->var[tab->n_var - tab->n_div + i].index;
1904 isl_int_neg(r_row[off + col], tab->mat->row[row][off + col]);
1905 isl_int_fdiv_r(r_row[off + col], r_row[off + col],
1906 tab->mat->row[row][0]);
1907 isl_int_neg(r_row[off + col], r_row[off + col]);
1909 for (i = 0; i < tab->n_col; ++i) {
1910 if (tab->col_var[i] >= 0 &&
1911 (tab->col_var[i] < tab->n_param ||
1912 tab->col_var[i] >= tab->n_var - tab->n_div))
1914 isl_int_fdiv_r(r_row[off + i],
1915 tab->mat->row[row][off + i], tab->mat->row[row][0]);
1917 if (tab->var[tab->n_var - tab->n_div + d].is_row) {
1919 int d_row = tab->var[tab->n_var - tab->n_div + d].index;
1921 isl_int_gcd(gcd, tab->mat->row[d_row][0], r_row[0]);
1922 isl_int_divexact(r_row[0], r_row[0], gcd);
1923 isl_int_divexact(gcd, tab->mat->row[d_row][0], gcd);
1924 isl_seq_combine(r_row + 1, gcd, r_row + 1,
1925 r_row[0], tab->mat->row[d_row] + 1,
1926 off - 1 + tab->n_col);
1927 isl_int_mul(r_row[0], r_row[0], tab->mat->row[d_row][0]);
1930 col = tab->var[tab->n_var - tab->n_div + d].index;
1931 isl_int_set(r_row[off + col], tab->mat->row[row][0]);
1934 tab->con[r].is_nonneg = 1;
1935 if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
1938 tab->row_sign[tab->con[r].index] = isl_tab_row_neg;
1942 row = tab->con[r].index;
1944 if (d >= n && context->op->detect_equalities(context, tab) < 0)
1950 /* Construct a tableau for bmap that can be used for computing
1951 * the lexicographic minimum (or maximum) of bmap.
1952 * If not NULL, then dom is the domain where the minimum
1953 * should be computed. In this case, we set up a parametric
1954 * tableau with row signs (initialized to "unknown").
1955 * If M is set, then the tableau will use a big parameter.
1956 * If max is set, then a maximum should be computed instead of a minimum.
1957 * This means that for each variable x, the tableau will contain the variable
1958 * x' = M - x, rather than x' = M + x. This in turn means that the coefficient
1959 * of the variables in all constraints are negated prior to adding them
1962 static struct isl_tab *tab_for_lexmin(struct isl_basic_map *bmap,
1963 struct isl_basic_set *dom, unsigned M, int max)
1966 struct isl_tab *tab;
1968 tab = isl_tab_alloc(bmap->ctx, 2 * bmap->n_eq + bmap->n_ineq + 1,
1969 isl_basic_map_total_dim(bmap), M);
1973 tab->rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
1975 tab->n_param = isl_basic_set_total_dim(dom) - dom->n_div;
1976 tab->n_div = dom->n_div;
1977 tab->row_sign = isl_calloc_array(bmap->ctx,
1978 enum isl_tab_row_sign, tab->mat->n_row);
1982 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
1983 return isl_tab_mark_empty(tab);
1985 for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
1986 tab->var[i].is_nonneg = 1;
1987 tab->var[i].frozen = 1;
1989 for (i = 0; i < bmap->n_eq; ++i) {
1991 isl_seq_neg(bmap->eq[i] + 1 + tab->n_param,
1992 bmap->eq[i] + 1 + tab->n_param,
1993 tab->n_var - tab->n_param - tab->n_div);
1994 tab = add_lexmin_valid_eq(tab, bmap->eq[i]);
1996 isl_seq_neg(bmap->eq[i] + 1 + tab->n_param,
1997 bmap->eq[i] + 1 + tab->n_param,
1998 tab->n_var - tab->n_param - tab->n_div);
1999 if (!tab || tab->empty)
2002 for (i = 0; i < bmap->n_ineq; ++i) {
2004 isl_seq_neg(bmap->ineq[i] + 1 + tab->n_param,
2005 bmap->ineq[i] + 1 + tab->n_param,
2006 tab->n_var - tab->n_param - tab->n_div);
2007 tab = add_lexmin_ineq(tab, bmap->ineq[i]);
2009 isl_seq_neg(bmap->ineq[i] + 1 + tab->n_param,
2010 bmap->ineq[i] + 1 + tab->n_param,
2011 tab->n_var - tab->n_param - tab->n_div);
2012 if (!tab || tab->empty)
2021 /* Given a main tableau where more than one row requires a split,
2022 * determine and return the "best" row to split on.
2024 * Given two rows in the main tableau, if the inequality corresponding
2025 * to the first row is redundant with respect to that of the second row
2026 * in the current tableau, then it is better to split on the second row,
2027 * since in the positive part, both row will be positive.
2028 * (In the negative part a pivot will have to be performed and just about
2029 * anything can happen to the sign of the other row.)
2031 * As a simple heuristic, we therefore select the row that makes the most
2032 * of the other rows redundant.
2034 * Perhaps it would also be useful to look at the number of constraints
2035 * that conflict with any given constraint.
2037 static int best_split(struct isl_tab *tab, struct isl_tab *context_tab)
2039 struct isl_tab_undo *snap;
2045 if (isl_tab_extend_cons(context_tab, 2) < 0)
2048 snap = isl_tab_snap(context_tab);
2050 for (split = tab->n_redundant; split < tab->n_row; ++split) {
2051 struct isl_tab_undo *snap2;
2052 struct isl_vec *ineq = NULL;
2055 if (!isl_tab_var_from_row(tab, split)->is_nonneg)
2057 if (tab->row_sign[split] != isl_tab_row_any)
2060 ineq = get_row_parameter_ineq(tab, split);
2063 context_tab = isl_tab_add_ineq(context_tab, ineq->el);
2066 snap2 = isl_tab_snap(context_tab);
2068 for (row = tab->n_redundant; row < tab->n_row; ++row) {
2069 struct isl_tab_var *var;
2073 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
2075 if (tab->row_sign[row] != isl_tab_row_any)
2078 ineq = get_row_parameter_ineq(tab, row);
2081 context_tab = isl_tab_add_ineq(context_tab, ineq->el);
2083 var = &context_tab->con[context_tab->n_con - 1];
2084 if (!context_tab->empty &&
2085 !isl_tab_min_at_most_neg_one(context_tab, var))
2087 if (isl_tab_rollback(context_tab, snap2) < 0)
2090 if (best == -1 || r > best_r) {
2094 if (isl_tab_rollback(context_tab, snap) < 0)
2101 static struct isl_basic_set *context_lex_peek_basic_set(
2102 struct isl_context *context)
2104 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2107 return clex->tab->bset;
2110 static struct isl_tab *context_lex_peek_tab(struct isl_context *context)
2112 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2116 static void context_lex_extend(struct isl_context *context, int n)
2118 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2121 if (isl_tab_extend_cons(clex->tab, n) >= 0)
2123 isl_tab_free(clex->tab);
2127 static void context_lex_add_eq(struct isl_context *context, isl_int *eq,
2128 int check, int update)
2130 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2131 if (isl_tab_extend_cons(clex->tab, 2) < 0)
2133 clex->tab = add_lexmin_eq(clex->tab, eq);
2135 int v = tab_has_valid_sample(clex->tab, eq, 1);
2139 clex->tab = check_integer_feasible(clex->tab);
2142 clex->tab = check_samples(clex->tab, eq, 1);
2145 isl_tab_free(clex->tab);
2149 static void context_lex_add_ineq(struct isl_context *context, isl_int *ineq,
2150 int check, int update)
2152 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2153 if (isl_tab_extend_cons(clex->tab, 1) < 0)
2155 clex->tab = add_lexmin_ineq(clex->tab, ineq);
2157 int v = tab_has_valid_sample(clex->tab, ineq, 0);
2161 clex->tab = check_integer_feasible(clex->tab);
2164 clex->tab = check_samples(clex->tab, ineq, 0);
2167 isl_tab_free(clex->tab);
2171 /* Check which signs can be obtained by "ineq" on all the currently
2172 * active sample values. See row_sign for more information.
2174 static enum isl_tab_row_sign tab_ineq_sign(struct isl_tab *tab, isl_int *ineq,
2180 int res = isl_tab_row_unknown;
2182 isl_assert(tab->mat->ctx, tab->samples, return 0);
2183 isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, return 0);
2186 for (i = tab->n_outside; i < tab->n_sample; ++i) {
2187 isl_seq_inner_product(tab->samples->row[i], ineq,
2188 1 + tab->n_var, &tmp);
2189 sgn = isl_int_sgn(tmp);
2190 if (sgn > 0 || (sgn == 0 && strict)) {
2191 if (res == isl_tab_row_unknown)
2192 res = isl_tab_row_pos;
2193 if (res == isl_tab_row_neg)
2194 res = isl_tab_row_any;
2197 if (res == isl_tab_row_unknown)
2198 res = isl_tab_row_neg;
2199 if (res == isl_tab_row_pos)
2200 res = isl_tab_row_any;
2202 if (res == isl_tab_row_any)
2210 static enum isl_tab_row_sign context_lex_ineq_sign(struct isl_context *context,
2211 isl_int *ineq, int strict)
2213 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2214 return tab_ineq_sign(clex->tab, ineq, strict);
2217 /* Check whether "ineq" can be added to the tableau without rendering
2220 static int context_lex_test_ineq(struct isl_context *context, isl_int *ineq)
2222 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2223 struct isl_tab_undo *snap;
2229 if (isl_tab_extend_cons(clex->tab, 1) < 0)
2232 snap = isl_tab_snap(clex->tab);
2233 if (isl_tab_push_basis(clex->tab) < 0)
2235 clex->tab = add_lexmin_ineq(clex->tab, ineq);
2236 clex->tab = check_integer_feasible(clex->tab);
2239 feasible = !clex->tab->empty;
2240 if (isl_tab_rollback(clex->tab, snap) < 0)
2246 static int context_lex_get_div(struct isl_context *context, struct isl_tab *tab,
2247 struct isl_vec *div)
2249 return get_div(tab, context, div);
2252 static int context_lex_add_div(struct isl_context *context, struct isl_vec *div,
2255 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2256 return context_tab_add_div(clex->tab, div, nonneg);
2259 static int context_lex_detect_equalities(struct isl_context *context,
2260 struct isl_tab *tab)
2265 static int context_lex_best_split(struct isl_context *context,
2266 struct isl_tab *tab)
2268 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2269 struct isl_tab_undo *snap;
2272 snap = isl_tab_snap(clex->tab);
2273 if (isl_tab_push_basis(clex->tab) < 0)
2275 r = best_split(tab, clex->tab);
2277 if (isl_tab_rollback(clex->tab, snap) < 0)
2283 static int context_lex_is_empty(struct isl_context *context)
2285 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2288 return clex->tab->empty;
2291 static void *context_lex_save(struct isl_context *context)
2293 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2294 struct isl_tab_undo *snap;
2296 snap = isl_tab_snap(clex->tab);
2297 if (isl_tab_push_basis(clex->tab) < 0)
2299 if (isl_tab_save_samples(clex->tab) < 0)
2305 static void context_lex_restore(struct isl_context *context, void *save)
2307 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2308 if (isl_tab_rollback(clex->tab, (struct isl_tab_undo *)save) < 0) {
2309 isl_tab_free(clex->tab);
2314 static int context_lex_is_ok(struct isl_context *context)
2316 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2320 /* For each variable in the context tableau, check if the variable can
2321 * only attain non-negative values. If so, mark the parameter as non-negative
2322 * in the main tableau. This allows for a more direct identification of some
2323 * cases of violated constraints.
2325 static struct isl_tab *tab_detect_nonnegative_parameters(struct isl_tab *tab,
2326 struct isl_tab *context_tab)
2329 struct isl_tab_undo *snap;
2330 struct isl_vec *ineq = NULL;
2331 struct isl_tab_var *var;
2334 if (context_tab->n_var == 0)
2337 ineq = isl_vec_alloc(tab->mat->ctx, 1 + context_tab->n_var);
2341 if (isl_tab_extend_cons(context_tab, 1) < 0)
2344 snap = isl_tab_snap(context_tab);
2347 isl_seq_clr(ineq->el, ineq->size);
2348 for (i = 0; i < context_tab->n_var; ++i) {
2349 isl_int_set_si(ineq->el[1 + i], 1);
2350 context_tab = isl_tab_add_ineq(context_tab, ineq->el);
2351 var = &context_tab->con[context_tab->n_con - 1];
2352 if (!context_tab->empty &&
2353 !isl_tab_min_at_most_neg_one(context_tab, var)) {
2355 if (i >= tab->n_param)
2356 j = i - tab->n_param + tab->n_var - tab->n_div;
2357 tab->var[j].is_nonneg = 1;
2360 isl_int_set_si(ineq->el[1 + i], 0);
2361 if (isl_tab_rollback(context_tab, snap) < 0)
2365 if (context_tab->M && n == context_tab->n_var) {
2366 context_tab->mat = isl_mat_drop_cols(context_tab->mat, 2, 1);
2378 static struct isl_tab *context_lex_detect_nonnegative_parameters(
2379 struct isl_context *context, struct isl_tab *tab)
2381 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2382 struct isl_tab_undo *snap;
2384 snap = isl_tab_snap(clex->tab);
2385 if (isl_tab_push_basis(clex->tab) < 0)
2388 tab = tab_detect_nonnegative_parameters(tab, clex->tab);
2390 if (isl_tab_rollback(clex->tab, snap) < 0)
2399 static void context_lex_invalidate(struct isl_context *context)
2401 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2402 isl_tab_free(clex->tab);
2406 static void context_lex_free(struct isl_context *context)
2408 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2409 isl_tab_free(clex->tab);
2413 struct isl_context_op isl_context_lex_op = {
2414 context_lex_detect_nonnegative_parameters,
2415 context_lex_peek_basic_set,
2416 context_lex_peek_tab,
2418 context_lex_add_ineq,
2419 context_lex_ineq_sign,
2420 context_lex_test_ineq,
2421 context_lex_get_div,
2422 context_lex_add_div,
2423 context_lex_detect_equalities,
2424 context_lex_best_split,
2425 context_lex_is_empty,
2428 context_lex_restore,
2429 context_lex_invalidate,
2433 static struct isl_tab *context_tab_for_lexmin(struct isl_basic_set *bset)
2435 struct isl_tab *tab;
2437 bset = isl_basic_set_cow(bset);
2440 tab = tab_for_lexmin((struct isl_basic_map *)bset, NULL, 1, 0);
2444 tab = isl_tab_init_samples(tab);
2447 isl_basic_set_free(bset);
2451 static struct isl_context *isl_context_lex_alloc(struct isl_basic_set *dom)
2453 struct isl_context_lex *clex;
2458 clex = isl_alloc_type(dom->ctx, struct isl_context_lex);
2462 clex->context.op = &isl_context_lex_op;
2464 clex->tab = context_tab_for_lexmin(isl_basic_set_copy(dom));
2465 clex->tab = restore_lexmin(clex->tab);
2466 clex->tab = check_integer_feasible(clex->tab);
2470 return &clex->context;
2472 clex->context.op->free(&clex->context);
2476 struct isl_context_gbr {
2477 struct isl_context context;
2478 struct isl_tab *tab;
2479 struct isl_tab *shifted;
2480 struct isl_tab *cone;
2483 static struct isl_tab *context_gbr_detect_nonnegative_parameters(
2484 struct isl_context *context, struct isl_tab *tab)
2486 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
2487 return tab_detect_nonnegative_parameters(tab, cgbr->tab);
2490 static struct isl_basic_set *context_gbr_peek_basic_set(
2491 struct isl_context *context)
2493 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
2496 return cgbr->tab->bset;
2499 static struct isl_tab *context_gbr_peek_tab(struct isl_context *context)
2501 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
2505 /* Initialize the "shifted" tableau of the context, which
2506 * contains the constraints of the original tableau shifted
2507 * by the sum of all negative coefficients. This ensures
2508 * that any rational point in the shifted tableau can
2509 * be rounded up to yield an integer point in the original tableau.
2511 static void gbr_init_shifted(struct isl_context_gbr *cgbr)
2514 struct isl_vec *cst;
2515 struct isl_basic_set *bset = cgbr->tab->bset;
2516 unsigned dim = isl_basic_set_total_dim(bset);
2518 cst = isl_vec_alloc(cgbr->tab->mat->ctx, bset->n_ineq);
2522 for (i = 0; i < bset->n_ineq; ++i) {
2523 isl_int_set(cst->el[i], bset->ineq[i][0]);
2524 for (j = 0; j < dim; ++j) {
2525 if (!isl_int_is_neg(bset->ineq[i][1 + j]))
2527 isl_int_add(bset->ineq[i][0], bset->ineq[i][0],
2528 bset->ineq[i][1 + j]);
2532 cgbr->shifted = isl_tab_from_basic_set(bset);
2534 for (i = 0; i < bset->n_ineq; ++i)
2535 isl_int_set(bset->ineq[i][0], cst->el[i]);
2540 /* Check if the shifted tableau is non-empty, and if so
2541 * use the sample point to construct an integer point
2542 * of the context tableau.
2544 static struct isl_vec *gbr_get_shifted_sample(struct isl_context_gbr *cgbr)
2546 struct isl_vec *sample;
2549 gbr_init_shifted(cgbr);
2552 if (cgbr->shifted->empty)
2553 return isl_vec_alloc(cgbr->tab->mat->ctx, 0);
2555 sample = isl_tab_get_sample_value(cgbr->shifted);
2556 sample = isl_vec_ceil(sample);
2561 static struct isl_basic_set *drop_constant_terms(struct isl_basic_set *bset)
2568 for (i = 0; i < bset->n_eq; ++i)
2569 isl_int_set_si(bset->eq[i][0], 0);
2571 for (i = 0; i < bset->n_ineq; ++i)
2572 isl_int_set_si(bset->ineq[i][0], 0);
2577 static int use_shifted(struct isl_context_gbr *cgbr)
2579 return cgbr->tab->bset->n_eq == 0 && cgbr->tab->bset->n_div == 0;
2582 static struct isl_vec *gbr_get_sample(struct isl_context_gbr *cgbr)
2584 struct isl_basic_set *bset;
2585 struct isl_basic_set *cone;
2587 if (isl_tab_sample_is_integer(cgbr->tab))
2588 return isl_tab_get_sample_value(cgbr->tab);
2590 if (use_shifted(cgbr)) {
2591 struct isl_vec *sample;
2593 sample = gbr_get_shifted_sample(cgbr);
2594 if (!sample || sample->size > 0)
2597 isl_vec_free(sample);
2601 cgbr->cone = isl_tab_from_recession_cone(cgbr->tab->bset);
2604 cgbr->cone->bset = isl_basic_set_dup(cgbr->tab->bset);
2606 cgbr->cone = isl_tab_detect_implicit_equalities(cgbr->cone);
2610 if (cgbr->cone->n_dead == cgbr->cone->n_col) {
2611 struct isl_vec *sample;
2612 struct isl_tab_undo *snap;
2614 if (cgbr->tab->basis) {
2615 if (cgbr->tab->basis->n_col != 1 + cgbr->tab->n_var) {
2616 isl_mat_free(cgbr->tab->basis);
2617 cgbr->tab->basis = NULL;
2619 cgbr->tab->n_zero = 0;
2620 cgbr->tab->n_unbounded = 0;
2624 snap = isl_tab_snap(cgbr->tab);
2626 sample = isl_tab_sample(cgbr->tab);
2628 if (isl_tab_rollback(cgbr->tab, snap) < 0) {
2629 isl_vec_free(sample);
2636 cone = isl_basic_set_dup(cgbr->cone->bset);
2637 cone = drop_constant_terms(cone);
2638 cone = isl_basic_set_update_from_tab(cone, cgbr->cone);
2639 cone = isl_basic_set_underlying_set(cone);
2640 cone = isl_basic_set_gauss(cone, NULL);
2642 bset = isl_basic_set_dup(cgbr->tab->bset);
2643 bset = isl_basic_set_update_from_tab(bset, cgbr->tab);
2644 bset = isl_basic_set_underlying_set(bset);
2645 bset = isl_basic_set_gauss(bset, NULL);
2647 return isl_basic_set_sample_with_cone(bset, cone);
2650 static void check_gbr_integer_feasible(struct isl_context_gbr *cgbr)
2652 struct isl_vec *sample;
2657 if (cgbr->tab->empty)
2660 sample = gbr_get_sample(cgbr);
2664 if (sample->size == 0) {
2665 isl_vec_free(sample);
2666 cgbr->tab = isl_tab_mark_empty(cgbr->tab);
2670 cgbr->tab = isl_tab_add_sample(cgbr->tab, sample);
2674 isl_tab_free(cgbr->tab);
2678 static struct isl_tab *add_gbr_eq(struct isl_tab *tab, isl_int *eq)
2685 if (isl_tab_extend_cons(tab, 2) < 0)
2688 tab = isl_tab_add_eq(tab, eq);
2696 static void context_gbr_add_eq(struct isl_context *context, isl_int *eq,
2697 int check, int update)
2699 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
2701 cgbr->tab = add_gbr_eq(cgbr->tab, eq);
2703 if (cgbr->cone && cgbr->cone->n_col != cgbr->cone->n_dead) {
2704 if (isl_tab_extend_cons(cgbr->cone, 2) < 0)
2706 cgbr->cone = isl_tab_add_eq(cgbr->cone, eq);
2710 int v = tab_has_valid_sample(cgbr->tab, eq, 1);
2714 check_gbr_integer_feasible(cgbr);
2717 cgbr->tab = check_samples(cgbr->tab, eq, 1);
2720 isl_tab_free(cgbr->tab);
2724 static void add_gbr_ineq(struct isl_context_gbr *cgbr, isl_int *ineq)
2729 if (isl_tab_extend_cons(cgbr->tab, 1) < 0)
2732 cgbr->tab = isl_tab_add_ineq(cgbr->tab, ineq);
2734 if (cgbr->shifted && !cgbr->shifted->empty && use_shifted(cgbr)) {
2737 dim = isl_basic_set_total_dim(cgbr->tab->bset);
2739 if (isl_tab_extend_cons(cgbr->shifted, 1) < 0)
2742 for (i = 0; i < dim; ++i) {
2743 if (!isl_int_is_neg(ineq[1 + i]))
2745 isl_int_add(ineq[0], ineq[0], ineq[1 + i]);
2748 cgbr->shifted = isl_tab_add_ineq(cgbr->shifted, ineq);
2750 for (i = 0; i < dim; ++i) {
2751 if (!isl_int_is_neg(ineq[1 + i]))
2753 isl_int_sub(ineq[0], ineq[0], ineq[1 + i]);
2757 if (cgbr->cone && cgbr->cone->n_col != cgbr->cone->n_dead) {
2758 if (isl_tab_extend_cons(cgbr->cone, 1) < 0)
2760 cgbr->cone = isl_tab_add_ineq(cgbr->cone, ineq);
2765 isl_tab_free(cgbr->tab);
2769 static void context_gbr_add_ineq(struct isl_context *context, isl_int *ineq,
2770 int check, int update)
2772 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
2774 add_gbr_ineq(cgbr, ineq);
2779 int v = tab_has_valid_sample(cgbr->tab, ineq, 0);
2783 check_gbr_integer_feasible(cgbr);
2786 cgbr->tab = check_samples(cgbr->tab, ineq, 0);
2789 isl_tab_free(cgbr->tab);
2793 static enum isl_tab_row_sign context_gbr_ineq_sign(struct isl_context *context,
2794 isl_int *ineq, int strict)
2796 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
2797 return tab_ineq_sign(cgbr->tab, ineq, strict);
2800 /* Check whether "ineq" can be added to the tableau without rendering
2803 static int context_gbr_test_ineq(struct isl_context *context, isl_int *ineq)
2805 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
2806 struct isl_tab_undo *snap;
2807 struct isl_tab_undo *shifted_snap = NULL;
2808 struct isl_tab_undo *cone_snap = NULL;
2814 if (isl_tab_extend_cons(cgbr->tab, 1) < 0)
2817 snap = isl_tab_snap(cgbr->tab);
2819 shifted_snap = isl_tab_snap(cgbr->shifted);
2821 cone_snap = isl_tab_snap(cgbr->cone);
2822 add_gbr_ineq(cgbr, ineq);
2823 check_gbr_integer_feasible(cgbr);
2826 feasible = !cgbr->tab->empty;
2827 if (isl_tab_rollback(cgbr->tab, snap) < 0)
2830 if (isl_tab_rollback(cgbr->shifted, shifted_snap))
2832 } else if (cgbr->shifted) {
2833 isl_tab_free(cgbr->shifted);
2834 cgbr->shifted = NULL;
2837 if (isl_tab_rollback(cgbr->cone, cone_snap))
2839 } else if (cgbr->cone) {
2840 isl_tab_free(cgbr->cone);
2847 /* Return the column of the last of the variables associated to
2848 * a column that has a non-zero coefficient.
2849 * This function is called in a context where only coefficients
2850 * of parameters or divs can be non-zero.
2852 static int last_non_zero_var_col(struct isl_tab *tab, isl_int *p)
2856 unsigned dim = tab->n_var - tab->n_param - tab->n_div;
2858 if (tab->n_var == 0)
2861 for (i = tab->n_var - 1; i >= 0; --i) {
2862 if (i >= tab->n_param && i < tab->n_var - tab->n_div)
2864 if (tab->var[i].is_row)
2866 col = tab->var[i].index;
2867 if (!isl_int_is_zero(p[col]))
2874 /* Look through all the recently added equalities in the context
2875 * to see if we can propagate any of them to the main tableau.
2877 * The newly added equalities in the context are encoded as pairs
2878 * of inequalities starting at inequality "first".
2880 * We tentatively add each of these equalities to the main tableau
2881 * and if this happens to result in a row with a final coefficient
2882 * that is one or negative one, we use it to kill a column
2883 * in the main tableau. Otherwise, we discard the tentatively
2886 static void propagate_equalities(struct isl_context_gbr *cgbr,
2887 struct isl_tab *tab, unsigned first)
2890 struct isl_vec *eq = NULL;
2892 eq = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_var);
2896 if (isl_tab_extend_cons(tab, (cgbr->tab->bset->n_ineq - first)/2) < 0)
2899 isl_seq_clr(eq->el + 1 + tab->n_param,
2900 tab->n_var - tab->n_param - tab->n_div);
2901 for (i = first; i < cgbr->tab->bset->n_ineq; i += 2) {
2904 struct isl_tab_undo *snap;
2905 snap = isl_tab_snap(tab);
2907 isl_seq_cpy(eq->el, cgbr->tab->bset->ineq[i], 1 + tab->n_param);
2908 isl_seq_cpy(eq->el + 1 + tab->n_var - tab->n_div,
2909 cgbr->tab->bset->ineq[i] + 1 + tab->n_param,
2912 r = isl_tab_add_row(tab, eq->el);
2915 r = tab->con[r].index;
2916 j = last_non_zero_var_col(tab, tab->mat->row[r] + 2 + tab->M);
2917 if (j < 0 || j < tab->n_dead ||
2918 !isl_int_is_one(tab->mat->row[r][0]) ||
2919 (!isl_int_is_one(tab->mat->row[r][2 + tab->M + j]) &&
2920 !isl_int_is_negone(tab->mat->row[r][2 + tab->M + j]))) {
2921 if (isl_tab_rollback(tab, snap) < 0)
2925 if (isl_tab_pivot(tab, r, j) < 0)
2927 if (isl_tab_kill_col(tab, j) < 0)
2930 tab = restore_lexmin(tab);
2938 isl_tab_free(cgbr->tab);
2942 static int context_gbr_detect_equalities(struct isl_context *context,
2943 struct isl_tab *tab)
2945 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
2946 struct isl_ctx *ctx;
2948 enum isl_lp_result res;
2951 ctx = cgbr->tab->mat->ctx;
2954 cgbr->cone = isl_tab_from_recession_cone(cgbr->tab->bset);
2957 cgbr->cone->bset = isl_basic_set_dup(cgbr->tab->bset);
2959 cgbr->cone = isl_tab_detect_implicit_equalities(cgbr->cone);
2961 n_ineq = cgbr->tab->bset->n_ineq;
2962 cgbr->tab = isl_tab_detect_equalities(cgbr->tab, cgbr->cone);
2963 if (cgbr->tab && cgbr->tab->bset->n_ineq > n_ineq)
2964 propagate_equalities(cgbr, tab, n_ineq);
2968 isl_tab_free(cgbr->tab);
2973 static int context_gbr_get_div(struct isl_context *context, struct isl_tab *tab,
2974 struct isl_vec *div)
2976 return get_div(tab, context, div);
2979 static int context_gbr_add_div(struct isl_context *context, struct isl_vec *div,
2982 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
2986 if (isl_tab_extend_cons(cgbr->cone, 3) < 0)
2988 if (isl_tab_extend_vars(cgbr->cone, 1) < 0)
2990 if (isl_tab_allocate_var(cgbr->cone) <0)
2993 cgbr->cone->bset = isl_basic_set_extend_dim(cgbr->cone->bset,
2994 isl_basic_set_get_dim(cgbr->cone->bset), 1, 0, 2);
2995 k = isl_basic_set_alloc_div(cgbr->cone->bset);
2998 isl_seq_cpy(cgbr->cone->bset->div[k], div->el, div->size);
2999 if (isl_tab_push(cgbr->cone, isl_tab_undo_bset_div) < 0)
3002 return context_tab_add_div(cgbr->tab, div, nonneg);
3005 static int context_gbr_best_split(struct isl_context *context,
3006 struct isl_tab *tab)
3008 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
3009 struct isl_tab_undo *snap;
3012 snap = isl_tab_snap(cgbr->tab);
3013 r = best_split(tab, cgbr->tab);
3015 if (isl_tab_rollback(cgbr->tab, snap) < 0)
3021 static int context_gbr_is_empty(struct isl_context *context)
3023 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
3026 return cgbr->tab->empty;
3029 struct isl_gbr_tab_undo {
3030 struct isl_tab_undo *tab_snap;
3031 struct isl_tab_undo *shifted_snap;
3032 struct isl_tab_undo *cone_snap;
3035 static void *context_gbr_save(struct isl_context *context)
3037 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
3038 struct isl_gbr_tab_undo *snap;
3040 snap = isl_alloc_type(cgbr->tab->mat->ctx, struct isl_gbr_tab_undo);
3044 snap->tab_snap = isl_tab_snap(cgbr->tab);
3045 if (isl_tab_save_samples(cgbr->tab) < 0)
3049 snap->shifted_snap = isl_tab_snap(cgbr->shifted);
3051 snap->shifted_snap = NULL;
3054 snap->cone_snap = isl_tab_snap(cgbr->cone);
3056 snap->cone_snap = NULL;
3064 static void context_gbr_restore(struct isl_context *context, void *save)
3066 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
3067 struct isl_gbr_tab_undo *snap = (struct isl_gbr_tab_undo *)save;
3070 if (isl_tab_rollback(cgbr->tab, snap->tab_snap) < 0) {
3071 isl_tab_free(cgbr->tab);
3075 if (snap->shifted_snap) {
3076 if (isl_tab_rollback(cgbr->shifted, snap->shifted_snap) < 0)
3078 } else if (cgbr->shifted) {
3079 isl_tab_free(cgbr->shifted);
3080 cgbr->shifted = NULL;
3083 if (snap->cone_snap) {
3084 if (isl_tab_rollback(cgbr->cone, snap->cone_snap) < 0)
3086 } else if (cgbr->cone) {
3087 isl_tab_free(cgbr->cone);
3096 isl_tab_free(cgbr->tab);
3100 static int context_gbr_is_ok(struct isl_context *context)
3102 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
3106 static void context_gbr_invalidate(struct isl_context *context)
3108 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
3109 isl_tab_free(cgbr->tab);
3113 static void context_gbr_free(struct isl_context *context)
3115 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
3116 isl_tab_free(cgbr->tab);
3117 isl_tab_free(cgbr->shifted);
3118 isl_tab_free(cgbr->cone);
3122 struct isl_context_op isl_context_gbr_op = {
3123 context_gbr_detect_nonnegative_parameters,
3124 context_gbr_peek_basic_set,
3125 context_gbr_peek_tab,
3127 context_gbr_add_ineq,
3128 context_gbr_ineq_sign,
3129 context_gbr_test_ineq,
3130 context_gbr_get_div,
3131 context_gbr_add_div,
3132 context_gbr_detect_equalities,
3133 context_gbr_best_split,
3134 context_gbr_is_empty,
3137 context_gbr_restore,
3138 context_gbr_invalidate,
3142 static struct isl_context *isl_context_gbr_alloc(struct isl_basic_set *dom)
3144 struct isl_context_gbr *cgbr;
3149 cgbr = isl_calloc_type(dom->ctx, struct isl_context_gbr);
3153 cgbr->context.op = &isl_context_gbr_op;
3155 cgbr->shifted = NULL;
3157 cgbr->tab = isl_tab_from_basic_set(dom);
3158 cgbr->tab = isl_tab_init_samples(cgbr->tab);
3161 cgbr->tab->bset = isl_basic_set_cow(isl_basic_set_copy(dom));
3162 if (!cgbr->tab->bset)
3164 check_gbr_integer_feasible(cgbr);
3166 return &cgbr->context;
3168 cgbr->context.op->free(&cgbr->context);
3172 static struct isl_context *isl_context_alloc(struct isl_basic_set *dom)
3177 if (dom->ctx->opt->context == ISL_CONTEXT_LEXMIN)
3178 return isl_context_lex_alloc(dom);
3180 return isl_context_gbr_alloc(dom);
3183 /* Construct an isl_sol_map structure for accumulating the solution.
3184 * If track_empty is set, then we also keep track of the parts
3185 * of the context where there is no solution.
3186 * If max is set, then we are solving a maximization, rather than
3187 * a minimization problem, which means that the variables in the
3188 * tableau have value "M - x" rather than "M + x".
3190 static struct isl_sol_map *sol_map_init(struct isl_basic_map *bmap,
3191 struct isl_basic_set *dom, int track_empty, int max)
3193 struct isl_sol_map *sol_map;
3195 sol_map = isl_calloc_type(bset->ctx, struct isl_sol_map);
3199 sol_map->sol.rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
3200 sol_map->sol.dec_level.callback.run = &sol_dec_level_wrap;
3201 sol_map->sol.dec_level.sol = &sol_map->sol;
3202 sol_map->sol.max = max;
3203 sol_map->sol.n_out = isl_basic_map_dim(bmap, isl_dim_out);
3204 sol_map->sol.add = &sol_map_add_wrap;
3205 sol_map->sol.add_empty = track_empty ? &sol_map_add_empty_wrap : NULL;
3206 sol_map->sol.free = &sol_map_free_wrap;
3207 sol_map->map = isl_map_alloc_dim(isl_basic_map_get_dim(bmap), 1,
3212 sol_map->sol.context = isl_context_alloc(dom);
3213 if (!sol_map->sol.context)
3217 sol_map->empty = isl_set_alloc_dim(isl_basic_set_get_dim(dom),
3218 1, ISL_SET_DISJOINT);
3219 if (!sol_map->empty)
3223 isl_basic_set_free(dom);
3226 isl_basic_set_free(dom);
3227 sol_map_free(sol_map);
3231 /* Check whether all coefficients of (non-parameter) variables
3232 * are non-positive, meaning that no pivots can be performed on the row.
3234 static int is_critical(struct isl_tab *tab, int row)
3237 unsigned off = 2 + tab->M;
3239 for (j = tab->n_dead; j < tab->n_col; ++j) {
3240 if (tab->col_var[j] >= 0 &&
3241 (tab->col_var[j] < tab->n_param ||
3242 tab->col_var[j] >= tab->n_var - tab->n_div))
3245 if (isl_int_is_pos(tab->mat->row[row][off + j]))
3252 /* Check whether the inequality represented by vec is strict over the integers,
3253 * i.e., there are no integer values satisfying the constraint with
3254 * equality. This happens if the gcd of the coefficients is not a divisor
3255 * of the constant term. If so, scale the constraint down by the gcd
3256 * of the coefficients.
3258 static int is_strict(struct isl_vec *vec)
3264 isl_seq_gcd(vec->el + 1, vec->size - 1, &gcd);
3265 if (!isl_int_is_one(gcd)) {
3266 strict = !isl_int_is_divisible_by(vec->el[0], gcd);
3267 isl_int_fdiv_q(vec->el[0], vec->el[0], gcd);
3268 isl_seq_scale_down(vec->el + 1, vec->el + 1, gcd, vec->size-1);
3275 /* Determine the sign of the given row of the main tableau.
3276 * The result is one of
3277 * isl_tab_row_pos: always non-negative; no pivot needed
3278 * isl_tab_row_neg: always non-positive; pivot
3279 * isl_tab_row_any: can be both positive and negative; split
3281 * We first handle some simple cases
3282 * - the row sign may be known already
3283 * - the row may be obviously non-negative
3284 * - the parametric constant may be equal to that of another row
3285 * for which we know the sign. This sign will be either "pos" or
3286 * "any". If it had been "neg" then we would have pivoted before.
3288 * If none of these cases hold, we check the value of the row for each
3289 * of the currently active samples. Based on the signs of these values
3290 * we make an initial determination of the sign of the row.
3292 * all zero -> unk(nown)
3293 * all non-negative -> pos
3294 * all non-positive -> neg
3295 * both negative and positive -> all
3297 * If we end up with "all", we are done.
3298 * Otherwise, we perform a check for positive and/or negative
3299 * values as follows.
3301 * samples neg unk pos
3307 * There is no special sign for "zero", because we can usually treat zero
3308 * as either non-negative or non-positive, whatever works out best.
3309 * However, if the row is "critical", meaning that pivoting is impossible
3310 * then we don't want to limp zero with the non-positive case, because
3311 * then we we would lose the solution for those values of the parameters
3312 * where the value of the row is zero. Instead, we treat 0 as non-negative
3313 * ensuring a split if the row can attain both zero and negative values.
3314 * The same happens when the original constraint was one that could not
3315 * be satisfied with equality by any integer values of the parameters.
3316 * In this case, we normalize the constraint, but then a value of zero
3317 * for the normalized constraint is actually a positive value for the
3318 * original constraint, so again we need to treat zero as non-negative.
3319 * In both these cases, we have the following decision tree instead:
3321 * all non-negative -> pos
3322 * all negative -> neg
3323 * both negative and non-negative -> all
3331 static enum isl_tab_row_sign row_sign(struct isl_tab *tab,
3332 struct isl_sol *sol, int row)
3334 struct isl_vec *ineq = NULL;
3335 int res = isl_tab_row_unknown;
3340 if (tab->row_sign[row] != isl_tab_row_unknown)
3341 return tab->row_sign[row];
3342 if (is_obviously_nonneg(tab, row))
3343 return isl_tab_row_pos;
3344 for (row2 = tab->n_redundant; row2 < tab->n_row; ++row2) {
3345 if (tab->row_sign[row2] == isl_tab_row_unknown)
3347 if (identical_parameter_line(tab, row, row2))
3348 return tab->row_sign[row2];
3351 critical = is_critical(tab, row);
3353 ineq = get_row_parameter_ineq(tab, row);
3357 strict = is_strict(ineq);
3359 res = sol->context->op->ineq_sign(sol->context, ineq->el,
3360 critical || strict);
3362 if (res == isl_tab_row_unknown || res == isl_tab_row_pos) {
3363 /* test for negative values */
3365 isl_seq_neg(ineq->el, ineq->el, ineq->size);
3366 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
3368 feasible = sol->context->op->test_ineq(sol->context, ineq->el);
3372 res = isl_tab_row_pos;
3374 res = (res == isl_tab_row_unknown) ? isl_tab_row_neg
3376 if (res == isl_tab_row_neg) {
3377 isl_seq_neg(ineq->el, ineq->el, ineq->size);
3378 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
3382 if (res == isl_tab_row_neg) {
3383 /* test for positive values */
3385 if (!critical && !strict)
3386 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
3388 feasible = sol->context->op->test_ineq(sol->context, ineq->el);
3392 res = isl_tab_row_any;
3402 static void find_solutions(struct isl_sol *sol, struct isl_tab *tab);
3404 /* Find solutions for values of the parameters that satisfy the given
3407 * We currently take a snapshot of the context tableau that is reset
3408 * when we return from this function, while we make a copy of the main
3409 * tableau, leaving the original main tableau untouched.
3410 * These are fairly arbitrary choices. Making a copy also of the context
3411 * tableau would obviate the need to undo any changes made to it later,
3412 * while taking a snapshot of the main tableau could reduce memory usage.
3413 * If we were to switch to taking a snapshot of the main tableau,
3414 * we would have to keep in mind that we need to save the row signs
3415 * and that we need to do this before saving the current basis
3416 * such that the basis has been restore before we restore the row signs.
3418 static void find_in_pos(struct isl_sol *sol, struct isl_tab *tab, isl_int *ineq)
3424 saved = sol->context->op->save(sol->context);
3426 tab = isl_tab_dup(tab);
3430 sol->context->op->add_ineq(sol->context, ineq, 0, 1);
3432 find_solutions(sol, tab);
3434 sol->context->op->restore(sol->context, saved);
3440 /* Record the absence of solutions for those values of the parameters
3441 * that do not satisfy the given inequality with equality.
3443 static void no_sol_in_strict(struct isl_sol *sol,
3444 struct isl_tab *tab, struct isl_vec *ineq)
3451 saved = sol->context->op->save(sol->context);
3453 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
3455 sol->context->op->add_ineq(sol->context, ineq->el, 1, 0);
3464 isl_int_add_ui(ineq->el[0], ineq->el[0], 1);
3466 sol->context->op->restore(sol->context, saved);
3472 /* Compute the lexicographic minimum of the set represented by the main
3473 * tableau "tab" within the context "sol->context_tab".
3474 * On entry the sample value of the main tableau is lexicographically
3475 * less than or equal to this lexicographic minimum.
3476 * Pivots are performed until a feasible point is found, which is then
3477 * necessarily equal to the minimum, or until the tableau is found to
3478 * be infeasible. Some pivots may need to be performed for only some
3479 * feasible values of the context tableau. If so, the context tableau
3480 * is split into a part where the pivot is needed and a part where it is not.
3482 * Whenever we enter the main loop, the main tableau is such that no
3483 * "obvious" pivots need to be performed on it, where "obvious" means
3484 * that the given row can be seen to be negative without looking at
3485 * the context tableau. In particular, for non-parametric problems,
3486 * no pivots need to be performed on the main tableau.
3487 * The caller of find_solutions is responsible for making this property
3488 * hold prior to the first iteration of the loop, while restore_lexmin
3489 * is called before every other iteration.
3491 * Inside the main loop, we first examine the signs of the rows of
3492 * the main tableau within the context of the context tableau.
3493 * If we find a row that is always non-positive for all values of
3494 * the parameters satisfying the context tableau and negative for at
3495 * least one value of the parameters, we perform the appropriate pivot
3496 * and start over. An exception is the case where no pivot can be
3497 * performed on the row. In this case, we require that the sign of
3498 * the row is negative for all values of the parameters (rather than just
3499 * non-positive). This special case is handled inside row_sign, which
3500 * will say that the row can have any sign if it determines that it can
3501 * attain both negative and zero values.
3503 * If we can't find a row that always requires a pivot, but we can find
3504 * one or more rows that require a pivot for some values of the parameters
3505 * (i.e., the row can attain both positive and negative signs), then we split
3506 * the context tableau into two parts, one where we force the sign to be
3507 * non-negative and one where we force is to be negative.
3508 * The non-negative part is handled by a recursive call (through find_in_pos).
3509 * Upon returning from this call, we continue with the negative part and
3510 * perform the required pivot.
3512 * If no such rows can be found, all rows are non-negative and we have
3513 * found a (rational) feasible point. If we only wanted a rational point
3515 * Otherwise, we check if all values of the sample point of the tableau
3516 * are integral for the variables. If so, we have found the minimal
3517 * integral point and we are done.
3518 * If the sample point is not integral, then we need to make a distinction
3519 * based on whether the constant term is non-integral or the coefficients
3520 * of the parameters. Furthermore, in order to decide how to handle
3521 * the non-integrality, we also need to know whether the coefficients
3522 * of the other columns in the tableau are integral. This leads
3523 * to the following table. The first two rows do not correspond
3524 * to a non-integral sample point and are only mentioned for completeness.
3526 * constant parameters other
3529 * int int rat | -> no problem
3531 * rat int int -> fail
3533 * rat int rat -> cut
3536 * rat rat rat | -> parametric cut
3539 * rat rat int | -> split context
3541 * If the parametric constant is completely integral, then there is nothing
3542 * to be done. If the constant term is non-integral, but all the other
3543 * coefficient are integral, then there is nothing that can be done
3544 * and the tableau has no integral solution.
3545 * If, on the other hand, one or more of the other columns have rational
3546 * coeffcients, but the parameter coefficients are all integral, then
3547 * we can perform a regular (non-parametric) cut.
3548 * Finally, if there is any parameter coefficient that is non-integral,
3549 * then we need to involve the context tableau. There are two cases here.
3550 * If at least one other column has a rational coefficient, then we
3551 * can perform a parametric cut in the main tableau by adding a new
3552 * integer division in the context tableau.
3553 * If all other columns have integral coefficients, then we need to
3554 * enforce that the rational combination of parameters (c + \sum a_i y_i)/m
3555 * is always integral. We do this by introducing an integer division
3556 * q = floor((c + \sum a_i y_i)/m) and stipulating that its argument should
3557 * always be integral in the context tableau, i.e., m q = c + \sum a_i y_i.
3558 * Since q is expressed in the tableau as
3559 * c + \sum a_i y_i - m q >= 0
3560 * -c - \sum a_i y_i + m q + m - 1 >= 0
3561 * it is sufficient to add the inequality
3562 * -c - \sum a_i y_i + m q >= 0
3563 * In the part of the context where this inequality does not hold, the
3564 * main tableau is marked as being empty.
3566 static void find_solutions(struct isl_sol *sol, struct isl_tab *tab)
3568 struct isl_context *context;
3570 if (!tab || sol->error)
3573 context = sol->context;
3577 if (context->op->is_empty(context))
3580 for (; tab && !tab->empty; tab = restore_lexmin(tab)) {
3587 for (row = tab->n_redundant; row < tab->n_row; ++row) {
3588 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
3590 sgn = row_sign(tab, sol, row);
3593 tab->row_sign[row] = sgn;
3594 if (sgn == isl_tab_row_any)
3596 if (sgn == isl_tab_row_any && split == -1)
3598 if (sgn == isl_tab_row_neg)
3601 if (row < tab->n_row)
3604 struct isl_vec *ineq;
3606 split = context->op->best_split(context, tab);
3609 ineq = get_row_parameter_ineq(tab, split);
3613 for (row = tab->n_redundant; row < tab->n_row; ++row) {
3614 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
3616 if (tab->row_sign[row] == isl_tab_row_any)
3617 tab->row_sign[row] = isl_tab_row_unknown;
3619 tab->row_sign[split] = isl_tab_row_pos;
3621 find_in_pos(sol, tab, ineq->el);
3622 tab->row_sign[split] = isl_tab_row_neg;
3624 isl_seq_neg(ineq->el, ineq->el, ineq->size);
3625 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
3626 context->op->add_ineq(context, ineq->el, 0, 1);
3634 row = first_non_integer(tab, &flags);
3637 if (ISL_FL_ISSET(flags, I_PAR)) {
3638 if (ISL_FL_ISSET(flags, I_VAR)) {
3639 tab = isl_tab_mark_empty(tab);
3642 row = add_cut(tab, row);
3643 } else if (ISL_FL_ISSET(flags, I_VAR)) {
3644 struct isl_vec *div;
3645 struct isl_vec *ineq;
3647 div = get_row_split_div(tab, row);
3650 d = context->op->get_div(context, tab, div);
3654 ineq = ineq_for_div(context->op->peek_basic_set(context), d);
3656 no_sol_in_strict(sol, tab, ineq);
3657 isl_seq_neg(ineq->el, ineq->el, ineq->size);
3658 context->op->add_ineq(context, ineq->el, 1, 1);
3660 if (sol->error || !context->op->is_ok(context))
3662 tab = set_row_cst_to_div(tab, row, d);
3664 row = add_parametric_cut(tab, row, context);
3677 /* Compute the lexicographic minimum of the set represented by the main
3678 * tableau "tab" within the context "sol->context_tab".
3680 * As a preprocessing step, we first transfer all the purely parametric
3681 * equalities from the main tableau to the context tableau, i.e.,
3682 * parameters that have been pivoted to a row.
3683 * These equalities are ignored by the main algorithm, because the
3684 * corresponding rows may not be marked as being non-negative.
3685 * In parts of the context where the added equality does not hold,
3686 * the main tableau is marked as being empty.
3688 static void find_solutions_main(struct isl_sol *sol, struct isl_tab *tab)
3694 for (row = tab->n_redundant; row < tab->n_row; ++row) {
3698 if (tab->row_var[row] < 0)
3700 if (tab->row_var[row] >= tab->n_param &&
3701 tab->row_var[row] < tab->n_var - tab->n_div)
3703 if (tab->row_var[row] < tab->n_param)
3704 p = tab->row_var[row];
3706 p = tab->row_var[row]
3707 + tab->n_param - (tab->n_var - tab->n_div);
3709 eq = isl_vec_alloc(tab->mat->ctx, 1+tab->n_param+tab->n_div);
3710 get_row_parameter_line(tab, row, eq->el);
3711 isl_int_neg(eq->el[1 + p], tab->mat->row[row][0]);
3712 eq = isl_vec_normalize(eq);
3715 no_sol_in_strict(sol, tab, eq);
3717 isl_seq_neg(eq->el, eq->el, eq->size);
3719 no_sol_in_strict(sol, tab, eq);
3720 isl_seq_neg(eq->el, eq->el, eq->size);
3722 sol->context->op->add_eq(sol->context, eq->el, 1, 1);
3726 if (isl_tab_mark_redundant(tab, row) < 0)
3729 if (sol->context->op->is_empty(sol->context))
3732 row = tab->n_redundant - 1;
3735 find_solutions(sol, tab);
3746 static void sol_map_find_solutions(struct isl_sol_map *sol_map,
3747 struct isl_tab *tab)
3749 find_solutions_main(&sol_map->sol, tab);
3752 /* Check if integer division "div" of "dom" also occurs in "bmap".
3753 * If so, return its position within the divs.
3754 * If not, return -1.
3756 static int find_context_div(struct isl_basic_map *bmap,
3757 struct isl_basic_set *dom, unsigned div)
3760 unsigned b_dim = isl_dim_total(bmap->dim);
3761 unsigned d_dim = isl_dim_total(dom->dim);
3763 if (isl_int_is_zero(dom->div[div][0]))
3765 if (isl_seq_first_non_zero(dom->div[div] + 2 + d_dim, dom->n_div) != -1)
3768 for (i = 0; i < bmap->n_div; ++i) {
3769 if (isl_int_is_zero(bmap->div[i][0]))
3771 if (isl_seq_first_non_zero(bmap->div[i] + 2 + d_dim,
3772 (b_dim - d_dim) + bmap->n_div) != -1)
3774 if (isl_seq_eq(bmap->div[i], dom->div[div], 2 + d_dim))
3780 /* The correspondence between the variables in the main tableau,
3781 * the context tableau, and the input map and domain is as follows.
3782 * The first n_param and the last n_div variables of the main tableau
3783 * form the variables of the context tableau.
3784 * In the basic map, these n_param variables correspond to the
3785 * parameters and the input dimensions. In the domain, they correspond
3786 * to the parameters and the set dimensions.
3787 * The n_div variables correspond to the integer divisions in the domain.
3788 * To ensure that everything lines up, we may need to copy some of the
3789 * integer divisions of the domain to the map. These have to be placed
3790 * in the same order as those in the context and they have to be placed
3791 * after any other integer divisions that the map may have.
3792 * This function performs the required reordering.
3794 static struct isl_basic_map *align_context_divs(struct isl_basic_map *bmap,
3795 struct isl_basic_set *dom)
3801 for (i = 0; i < dom->n_div; ++i)
3802 if (find_context_div(bmap, dom, i) != -1)
3804 other = bmap->n_div - common;
3805 if (dom->n_div - common > 0) {
3806 bmap = isl_basic_map_extend_dim(bmap, isl_dim_copy(bmap->dim),
3807 dom->n_div - common, 0, 0);
3811 for (i = 0; i < dom->n_div; ++i) {
3812 int pos = find_context_div(bmap, dom, i);
3814 pos = isl_basic_map_alloc_div(bmap);
3817 isl_int_set_si(bmap->div[pos][0], 0);
3819 if (pos != other + i)
3820 isl_basic_map_swap_div(bmap, pos, other + i);
3824 isl_basic_map_free(bmap);
3828 /* Compute the lexicographic minimum (or maximum if "max" is set)
3829 * of "bmap" over the domain "dom" and return the result as a map.
3830 * If "empty" is not NULL, then *empty is assigned a set that
3831 * contains those parts of the domain where there is no solution.
3832 * If "bmap" is marked as rational (ISL_BASIC_MAP_RATIONAL),
3833 * then we compute the rational optimum. Otherwise, we compute
3834 * the integral optimum.
3836 * We perform some preprocessing. As the PILP solver does not
3837 * handle implicit equalities very well, we first make sure all
3838 * the equalities are explicitly available.
3839 * We also make sure the divs in the domain are properly order,
3840 * because they will be added one by one in the given order
3841 * during the construction of the solution map.
3843 struct isl_map *isl_tab_basic_map_partial_lexopt(
3844 struct isl_basic_map *bmap, struct isl_basic_set *dom,
3845 struct isl_set **empty, int max)
3847 struct isl_tab *tab;
3848 struct isl_map *result = NULL;
3849 struct isl_sol_map *sol_map = NULL;
3850 struct isl_context *context;
3857 isl_assert(bmap->ctx,
3858 isl_basic_map_compatible_domain(bmap, dom), goto error);
3860 bmap = isl_basic_map_detect_equalities(bmap);
3863 dom = isl_basic_set_order_divs(dom);
3864 bmap = align_context_divs(bmap, dom);
3866 sol_map = sol_map_init(bmap, dom, !!empty, max);
3870 context = sol_map->sol.context;
3871 if (isl_basic_set_fast_is_empty(context->op->peek_basic_set(context)))
3873 else if (isl_basic_map_fast_is_empty(bmap))
3874 sol_map_add_empty(sol_map,
3875 isl_basic_set_dup(context->op->peek_basic_set(context)));
3877 tab = tab_for_lexmin(bmap,
3878 context->op->peek_basic_set(context), 1, max);
3879 tab = context->op->detect_nonnegative_parameters(context, tab);
3880 sol_map_find_solutions(sol_map, tab);
3882 if (sol_map->sol.error)
3885 result = isl_map_copy(sol_map->map);
3887 *empty = isl_set_copy(sol_map->empty);
3888 sol_free(&sol_map->sol);
3889 isl_basic_map_free(bmap);
3892 sol_free(&sol_map->sol);
3893 isl_basic_map_free(bmap);
3897 struct isl_sol_for {
3899 int (*fn)(__isl_take isl_basic_set *dom,
3900 __isl_take isl_mat *map, void *user);
3904 static void sol_for_free(struct isl_sol_for *sol_for)
3906 if (sol_for->sol.context)
3907 sol_for->sol.context->op->free(sol_for->sol.context);
3911 static void sol_for_free_wrap(struct isl_sol *sol)
3913 sol_for_free((struct isl_sol_for *)sol);
3916 /* Add the solution identified by the tableau and the context tableau.
3918 * See documentation of sol_add for more details.
3920 * Instead of constructing a basic map, this function calls a user
3921 * defined function with the current context as a basic set and
3922 * an affine matrix reprenting the relation between the input and output.
3923 * The number of rows in this matrix is equal to one plus the number
3924 * of output variables. The number of columns is equal to one plus
3925 * the total dimension of the context, i.e., the number of parameters,
3926 * input variables and divs. Since some of the columns in the matrix
3927 * may refer to the divs, the basic set is not simplified.
3928 * (Simplification may reorder or remove divs.)
3930 static void sol_for_add(struct isl_sol_for *sol,
3931 struct isl_basic_set *dom, struct isl_mat *M)
3933 if (sol->sol.error || !dom || !M)
3936 dom = isl_basic_set_simplify(dom);
3937 dom = isl_basic_set_finalize(dom);
3939 if (sol->fn(isl_basic_set_copy(dom), isl_mat_copy(M), sol->user) < 0)
3942 isl_basic_set_free(dom);
3946 isl_basic_set_free(dom);
3951 static void sol_for_add_wrap(struct isl_sol *sol,
3952 struct isl_basic_set *dom, struct isl_mat *M)
3954 sol_for_add((struct isl_sol_for *)sol, dom, M);
3957 static struct isl_sol_for *sol_for_init(struct isl_basic_map *bmap, int max,
3958 int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_mat *map,
3962 struct isl_sol_for *sol_for = NULL;
3963 struct isl_dim *dom_dim;
3964 struct isl_basic_set *dom = NULL;
3966 sol_for = isl_calloc_type(bset->ctx, struct isl_sol_for);
3970 dom_dim = isl_dim_domain(isl_dim_copy(bmap->dim));
3971 dom = isl_basic_set_universe(dom_dim);
3973 sol_for->sol.rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
3974 sol_for->sol.dec_level.callback.run = &sol_dec_level_wrap;
3975 sol_for->sol.dec_level.sol = &sol_for->sol;
3977 sol_for->user = user;
3978 sol_for->sol.max = max;
3979 sol_for->sol.n_out = isl_basic_map_dim(bmap, isl_dim_out);
3980 sol_for->sol.add = &sol_for_add_wrap;
3981 sol_for->sol.add_empty = NULL;
3982 sol_for->sol.free = &sol_for_free_wrap;
3984 sol_for->sol.context = isl_context_alloc(dom);
3985 if (!sol_for->sol.context)
3988 isl_basic_set_free(dom);
3991 isl_basic_set_free(dom);
3992 sol_for_free(sol_for);
3996 static void sol_for_find_solutions(struct isl_sol_for *sol_for,
3997 struct isl_tab *tab)
3999 find_solutions_main(&sol_for->sol, tab);
4002 int isl_basic_map_foreach_lexopt(__isl_keep isl_basic_map *bmap, int max,
4003 int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_mat *map,
4007 struct isl_sol_for *sol_for = NULL;
4009 bmap = isl_basic_map_copy(bmap);
4013 bmap = isl_basic_map_detect_equalities(bmap);
4014 sol_for = sol_for_init(bmap, max, fn, user);
4016 if (isl_basic_map_fast_is_empty(bmap))
4019 struct isl_tab *tab;
4020 struct isl_context *context = sol_for->sol.context;
4021 tab = tab_for_lexmin(bmap,
4022 context->op->peek_basic_set(context), 1, max);
4023 tab = context->op->detect_nonnegative_parameters(context, tab);
4024 sol_for_find_solutions(sol_for, tab);
4025 if (sol_for->sol.error)
4029 sol_free(&sol_for->sol);
4030 isl_basic_map_free(bmap);
4033 sol_free(&sol_for->sol);
4034 isl_basic_map_free(bmap);
4038 int isl_basic_map_foreach_lexmin(__isl_keep isl_basic_map *bmap,
4039 int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_mat *map,
4043 return isl_basic_map_foreach_lexopt(bmap, 0, fn, user);
4046 int isl_basic_map_foreach_lexmax(__isl_keep isl_basic_map *bmap,
4047 int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_mat *map,
4051 return isl_basic_map_foreach_lexopt(bmap, 1, fn, user);