2 * Copyright 2008-2009 Katholieke Universiteit Leuven
3 * Copyright 2010 INRIA Saclay
5 * Use of this software is governed by the MIT license
7 * Written by Sven Verdoolaege, K.U.Leuven, Departement
8 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
9 * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite,
10 * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France
13 #include <isl_ctx_private.h>
14 #include "isl_map_private.h"
17 #include "isl_sample.h"
18 #include <isl_mat_private.h>
19 #include <isl_aff_private.h>
20 #include <isl_options_private.h>
21 #include <isl_config.h>
24 * The implementation of parametric integer linear programming in this file
25 * was inspired by the paper "Parametric Integer Programming" and the
26 * report "Solving systems of affine (in)equalities" by Paul Feautrier
29 * The strategy used for obtaining a feasible solution is different
30 * from the one used in isl_tab.c. In particular, in isl_tab.c,
31 * upon finding a constraint that is not yet satisfied, we pivot
32 * in a row that increases the constant term of the row holding the
33 * constraint, making sure the sample solution remains feasible
34 * for all the constraints it already satisfied.
35 * Here, we always pivot in the row holding the constraint,
36 * choosing a column that induces the lexicographically smallest
37 * increment to the sample solution.
39 * By starting out from a sample value that is lexicographically
40 * smaller than any integer point in the problem space, the first
41 * feasible integer sample point we find will also be the lexicographically
42 * smallest. If all variables can be assumed to be non-negative,
43 * then the initial sample value may be chosen equal to zero.
44 * However, we will not make this assumption. Instead, we apply
45 * the "big parameter" trick. Any variable x is then not directly
46 * used in the tableau, but instead it is represented by another
47 * variable x' = M + x, where M is an arbitrarily large (positive)
48 * value. x' is therefore always non-negative, whatever the value of x.
49 * Taking as initial sample value x' = 0 corresponds to x = -M,
50 * which is always smaller than any possible value of x.
52 * The big parameter trick is used in the main tableau and
53 * also in the context tableau if isl_context_lex is used.
54 * In this case, each tableaus has its own big parameter.
55 * Before doing any real work, we check if all the parameters
56 * happen to be non-negative. If so, we drop the column corresponding
57 * to M from the initial context tableau.
58 * If isl_context_gbr is used, then the big parameter trick is only
59 * used in the main tableau.
63 struct isl_context_op {
64 /* detect nonnegative parameters in context and mark them in tab */
65 struct isl_tab *(*detect_nonnegative_parameters)(
66 struct isl_context *context, struct isl_tab *tab);
67 /* return temporary reference to basic set representation of context */
68 struct isl_basic_set *(*peek_basic_set)(struct isl_context *context);
69 /* return temporary reference to tableau representation of context */
70 struct isl_tab *(*peek_tab)(struct isl_context *context);
71 /* add equality; check is 1 if eq may not be valid;
72 * update is 1 if we may want to call ineq_sign on context later.
74 void (*add_eq)(struct isl_context *context, isl_int *eq,
75 int check, int update);
76 /* add inequality; check is 1 if ineq may not be valid;
77 * update is 1 if we may want to call ineq_sign on context later.
79 void (*add_ineq)(struct isl_context *context, isl_int *ineq,
80 int check, int update);
81 /* check sign of ineq based on previous information.
82 * strict is 1 if saturation should be treated as a positive sign.
84 enum isl_tab_row_sign (*ineq_sign)(struct isl_context *context,
85 isl_int *ineq, int strict);
86 /* check if inequality maintains feasibility */
87 int (*test_ineq)(struct isl_context *context, isl_int *ineq);
88 /* return index of a div that corresponds to "div" */
89 int (*get_div)(struct isl_context *context, struct isl_tab *tab,
91 /* add div "div" to context and return non-negativity */
92 int (*add_div)(struct isl_context *context, struct isl_vec *div);
93 int (*detect_equalities)(struct isl_context *context,
95 /* return row index of "best" split */
96 int (*best_split)(struct isl_context *context, struct isl_tab *tab);
97 /* check if context has already been determined to be empty */
98 int (*is_empty)(struct isl_context *context);
99 /* check if context is still usable */
100 int (*is_ok)(struct isl_context *context);
101 /* save a copy/snapshot of context */
102 void *(*save)(struct isl_context *context);
103 /* restore saved context */
104 void (*restore)(struct isl_context *context, void *);
105 /* discard saved context */
106 void (*discard)(void *);
107 /* invalidate context */
108 void (*invalidate)(struct isl_context *context);
110 void (*free)(struct isl_context *context);
114 struct isl_context_op *op;
117 struct isl_context_lex {
118 struct isl_context context;
122 struct isl_partial_sol {
124 struct isl_basic_set *dom;
127 struct isl_partial_sol *next;
131 struct isl_sol_callback {
132 struct isl_tab_callback callback;
136 /* isl_sol is an interface for constructing a solution to
137 * a parametric integer linear programming problem.
138 * Every time the algorithm reaches a state where a solution
139 * can be read off from the tableau (including cases where the tableau
140 * is empty), the function "add" is called on the isl_sol passed
141 * to find_solutions_main.
143 * The context tableau is owned by isl_sol and is updated incrementally.
145 * There are currently two implementations of this interface,
146 * isl_sol_map, which simply collects the solutions in an isl_map
147 * and (optionally) the parts of the context where there is no solution
149 * isl_sol_for, which calls a user-defined function for each part of
158 struct isl_context *context;
159 struct isl_partial_sol *partial;
160 void (*add)(struct isl_sol *sol,
161 struct isl_basic_set *dom, struct isl_mat *M);
162 void (*add_empty)(struct isl_sol *sol, struct isl_basic_set *bset);
163 void (*free)(struct isl_sol *sol);
164 struct isl_sol_callback dec_level;
167 static void sol_free(struct isl_sol *sol)
169 struct isl_partial_sol *partial, *next;
172 for (partial = sol->partial; partial; partial = next) {
173 next = partial->next;
174 isl_basic_set_free(partial->dom);
175 isl_mat_free(partial->M);
181 /* Push a partial solution represented by a domain and mapping M
182 * onto the stack of partial solutions.
184 static void sol_push_sol(struct isl_sol *sol,
185 struct isl_basic_set *dom, struct isl_mat *M)
187 struct isl_partial_sol *partial;
189 if (sol->error || !dom)
192 partial = isl_alloc_type(dom->ctx, struct isl_partial_sol);
196 partial->level = sol->level;
199 partial->next = sol->partial;
201 sol->partial = partial;
205 isl_basic_set_free(dom);
210 /* Pop one partial solution from the partial solution stack and
211 * pass it on to sol->add or sol->add_empty.
213 static void sol_pop_one(struct isl_sol *sol)
215 struct isl_partial_sol *partial;
217 partial = sol->partial;
218 sol->partial = partial->next;
221 sol->add(sol, partial->dom, partial->M);
223 sol->add_empty(sol, partial->dom);
227 /* Return a fresh copy of the domain represented by the context tableau.
229 static struct isl_basic_set *sol_domain(struct isl_sol *sol)
231 struct isl_basic_set *bset;
236 bset = isl_basic_set_dup(sol->context->op->peek_basic_set(sol->context));
237 bset = isl_basic_set_update_from_tab(bset,
238 sol->context->op->peek_tab(sol->context));
243 /* Check whether two partial solutions have the same mapping, where n_div
244 * is the number of divs that the two partial solutions have in common.
246 static int same_solution(struct isl_partial_sol *s1, struct isl_partial_sol *s2,
252 if (!s1->M != !s2->M)
257 dim = isl_basic_set_total_dim(s1->dom) - s1->dom->n_div;
259 for (i = 0; i < s1->M->n_row; ++i) {
260 if (isl_seq_first_non_zero(s1->M->row[i]+1+dim+n_div,
261 s1->M->n_col-1-dim-n_div) != -1)
263 if (isl_seq_first_non_zero(s2->M->row[i]+1+dim+n_div,
264 s2->M->n_col-1-dim-n_div) != -1)
266 if (!isl_seq_eq(s1->M->row[i], s2->M->row[i], 1+dim+n_div))
272 /* Pop all solutions from the partial solution stack that were pushed onto
273 * the stack at levels that are deeper than the current level.
274 * If the two topmost elements on the stack have the same level
275 * and represent the same solution, then their domains are combined.
276 * This combined domain is the same as the current context domain
277 * as sol_pop is called each time we move back to a higher level.
279 static void sol_pop(struct isl_sol *sol)
281 struct isl_partial_sol *partial;
287 if (sol->level == 0) {
288 for (partial = sol->partial; partial; partial = sol->partial)
293 partial = sol->partial;
297 if (partial->level <= sol->level)
300 if (partial->next && partial->next->level == partial->level) {
301 n_div = isl_basic_set_dim(
302 sol->context->op->peek_basic_set(sol->context),
305 if (!same_solution(partial, partial->next, n_div)) {
309 struct isl_basic_set *bset;
311 bset = sol_domain(sol);
315 isl_basic_set_free(partial->next->dom);
316 partial->next->dom = bset;
317 partial->next->level = sol->level;
319 sol->partial = partial->next;
320 isl_basic_set_free(partial->dom);
321 isl_mat_free(partial->M);
328 error: sol->error = 1;
331 static void sol_dec_level(struct isl_sol *sol)
341 static int sol_dec_level_wrap(struct isl_tab_callback *cb)
343 struct isl_sol_callback *callback = (struct isl_sol_callback *)cb;
345 sol_dec_level(callback->sol);
347 return callback->sol->error ? -1 : 0;
350 /* Move down to next level and push callback onto context tableau
351 * to decrease the level again when it gets rolled back across
352 * the current state. That is, dec_level will be called with
353 * the context tableau in the same state as it is when inc_level
356 static void sol_inc_level(struct isl_sol *sol)
364 tab = sol->context->op->peek_tab(sol->context);
365 if (isl_tab_push_callback(tab, &sol->dec_level.callback) < 0)
369 static void scale_rows(struct isl_mat *mat, isl_int m, int n_row)
373 if (isl_int_is_one(m))
376 for (i = 0; i < n_row; ++i)
377 isl_seq_scale(mat->row[i], mat->row[i], m, mat->n_col);
380 /* Add the solution identified by the tableau and the context tableau.
382 * The layout of the variables is as follows.
383 * tab->n_var is equal to the total number of variables in the input
384 * map (including divs that were copied from the context)
385 * + the number of extra divs constructed
386 * Of these, the first tab->n_param and the last tab->n_div variables
387 * correspond to the variables in the context, i.e.,
388 * tab->n_param + tab->n_div = context_tab->n_var
389 * tab->n_param is equal to the number of parameters and input
390 * dimensions in the input map
391 * tab->n_div is equal to the number of divs in the context
393 * If there is no solution, then call add_empty with a basic set
394 * that corresponds to the context tableau. (If add_empty is NULL,
397 * If there is a solution, then first construct a matrix that maps
398 * all dimensions of the context to the output variables, i.e.,
399 * the output dimensions in the input map.
400 * The divs in the input map (if any) that do not correspond to any
401 * div in the context do not appear in the solution.
402 * The algorithm will make sure that they have an integer value,
403 * but these values themselves are of no interest.
404 * We have to be careful not to drop or rearrange any divs in the
405 * context because that would change the meaning of the matrix.
407 * To extract the value of the output variables, it should be noted
408 * that we always use a big parameter M in the main tableau and so
409 * the variable stored in this tableau is not an output variable x itself, but
410 * x' = M + x (in case of minimization)
412 * x' = M - x (in case of maximization)
413 * If x' appears in a column, then its optimal value is zero,
414 * which means that the optimal value of x is an unbounded number
415 * (-M for minimization and M for maximization).
416 * We currently assume that the output dimensions in the original map
417 * are bounded, so this cannot occur.
418 * Similarly, when x' appears in a row, then the coefficient of M in that
419 * row is necessarily 1.
420 * If the row in the tableau represents
421 * d x' = c + d M + e(y)
422 * then, in case of minimization, the corresponding row in the matrix
425 * with a d = m, the (updated) common denominator of the matrix.
426 * In case of maximization, the row will be
429 static void sol_add(struct isl_sol *sol, struct isl_tab *tab)
431 struct isl_basic_set *bset = NULL;
432 struct isl_mat *mat = NULL;
437 if (sol->error || !tab)
440 if (tab->empty && !sol->add_empty)
442 if (sol->context->op->is_empty(sol->context))
445 bset = sol_domain(sol);
448 sol_push_sol(sol, bset, NULL);
454 mat = isl_mat_alloc(tab->mat->ctx, 1 + sol->n_out,
455 1 + tab->n_param + tab->n_div);
461 isl_seq_clr(mat->row[0] + 1, mat->n_col - 1);
462 isl_int_set_si(mat->row[0][0], 1);
463 for (row = 0; row < sol->n_out; ++row) {
464 int i = tab->n_param + row;
467 isl_seq_clr(mat->row[1 + row], mat->n_col);
468 if (!tab->var[i].is_row) {
470 isl_die(mat->ctx, isl_error_invalid,
471 "unbounded optimum", goto error2);
475 r = tab->var[i].index;
477 isl_int_ne(tab->mat->row[r][2], tab->mat->row[r][0]))
478 isl_die(mat->ctx, isl_error_invalid,
479 "unbounded optimum", goto error2);
480 isl_int_gcd(m, mat->row[0][0], tab->mat->row[r][0]);
481 isl_int_divexact(m, tab->mat->row[r][0], m);
482 scale_rows(mat, m, 1 + row);
483 isl_int_divexact(m, mat->row[0][0], tab->mat->row[r][0]);
484 isl_int_mul(mat->row[1 + row][0], m, tab->mat->row[r][1]);
485 for (j = 0; j < tab->n_param; ++j) {
487 if (tab->var[j].is_row)
489 col = tab->var[j].index;
490 isl_int_mul(mat->row[1 + row][1 + j], m,
491 tab->mat->row[r][off + col]);
493 for (j = 0; j < tab->n_div; ++j) {
495 if (tab->var[tab->n_var - tab->n_div+j].is_row)
497 col = tab->var[tab->n_var - tab->n_div+j].index;
498 isl_int_mul(mat->row[1 + row][1 + tab->n_param + j], m,
499 tab->mat->row[r][off + col]);
502 isl_seq_neg(mat->row[1 + row], mat->row[1 + row],
508 sol_push_sol(sol, bset, mat);
513 isl_basic_set_free(bset);
521 struct isl_set *empty;
524 static void sol_map_free(struct isl_sol_map *sol_map)
528 if (sol_map->sol.context)
529 sol_map->sol.context->op->free(sol_map->sol.context);
530 isl_map_free(sol_map->map);
531 isl_set_free(sol_map->empty);
535 static void sol_map_free_wrap(struct isl_sol *sol)
537 sol_map_free((struct isl_sol_map *)sol);
540 /* This function is called for parts of the context where there is
541 * no solution, with "bset" corresponding to the context tableau.
542 * Simply add the basic set to the set "empty".
544 static void sol_map_add_empty(struct isl_sol_map *sol,
545 struct isl_basic_set *bset)
549 isl_assert(bset->ctx, sol->empty, goto error);
551 sol->empty = isl_set_grow(sol->empty, 1);
552 bset = isl_basic_set_simplify(bset);
553 bset = isl_basic_set_finalize(bset);
554 sol->empty = isl_set_add_basic_set(sol->empty, isl_basic_set_copy(bset));
557 isl_basic_set_free(bset);
560 isl_basic_set_free(bset);
564 static void sol_map_add_empty_wrap(struct isl_sol *sol,
565 struct isl_basic_set *bset)
567 sol_map_add_empty((struct isl_sol_map *)sol, bset);
570 /* Given a basic map "dom" that represents the context and an affine
571 * matrix "M" that maps the dimensions of the context to the
572 * output variables, construct a basic map with the same parameters
573 * and divs as the context, the dimensions of the context as input
574 * dimensions and a number of output dimensions that is equal to
575 * the number of output dimensions in the input map.
577 * The constraints and divs of the context are simply copied
578 * from "dom". For each row
582 * is added, with d the common denominator of M.
584 static void sol_map_add(struct isl_sol_map *sol,
585 struct isl_basic_set *dom, struct isl_mat *M)
588 struct isl_basic_map *bmap = NULL;
596 if (sol->sol.error || !dom || !M)
599 n_out = sol->sol.n_out;
600 n_eq = dom->n_eq + n_out;
601 n_ineq = dom->n_ineq;
603 nparam = isl_basic_set_total_dim(dom) - n_div;
604 total = isl_map_dim(sol->map, isl_dim_all);
605 bmap = isl_basic_map_alloc_space(isl_map_get_space(sol->map),
606 n_div, n_eq, 2 * n_div + n_ineq);
609 if (sol->sol.rational)
610 ISL_F_SET(bmap, ISL_BASIC_MAP_RATIONAL);
611 for (i = 0; i < dom->n_div; ++i) {
612 int k = isl_basic_map_alloc_div(bmap);
615 isl_seq_cpy(bmap->div[k], dom->div[i], 1 + 1 + nparam);
616 isl_seq_clr(bmap->div[k] + 1 + 1 + nparam, total - nparam);
617 isl_seq_cpy(bmap->div[k] + 1 + 1 + total,
618 dom->div[i] + 1 + 1 + nparam, i);
620 for (i = 0; i < dom->n_eq; ++i) {
621 int k = isl_basic_map_alloc_equality(bmap);
624 isl_seq_cpy(bmap->eq[k], dom->eq[i], 1 + nparam);
625 isl_seq_clr(bmap->eq[k] + 1 + nparam, total - nparam);
626 isl_seq_cpy(bmap->eq[k] + 1 + total,
627 dom->eq[i] + 1 + nparam, n_div);
629 for (i = 0; i < dom->n_ineq; ++i) {
630 int k = isl_basic_map_alloc_inequality(bmap);
633 isl_seq_cpy(bmap->ineq[k], dom->ineq[i], 1 + nparam);
634 isl_seq_clr(bmap->ineq[k] + 1 + nparam, total - nparam);
635 isl_seq_cpy(bmap->ineq[k] + 1 + total,
636 dom->ineq[i] + 1 + nparam, n_div);
638 for (i = 0; i < M->n_row - 1; ++i) {
639 int k = isl_basic_map_alloc_equality(bmap);
642 isl_seq_cpy(bmap->eq[k], M->row[1 + i], 1 + nparam);
643 isl_seq_clr(bmap->eq[k] + 1 + nparam, n_out);
644 isl_int_neg(bmap->eq[k][1 + nparam + i], M->row[0][0]);
645 isl_seq_cpy(bmap->eq[k] + 1 + nparam + n_out,
646 M->row[1 + i] + 1 + nparam, n_div);
648 bmap = isl_basic_map_simplify(bmap);
649 bmap = isl_basic_map_finalize(bmap);
650 sol->map = isl_map_grow(sol->map, 1);
651 sol->map = isl_map_add_basic_map(sol->map, bmap);
652 isl_basic_set_free(dom);
658 isl_basic_set_free(dom);
660 isl_basic_map_free(bmap);
664 static void sol_map_add_wrap(struct isl_sol *sol,
665 struct isl_basic_set *dom, struct isl_mat *M)
667 sol_map_add((struct isl_sol_map *)sol, dom, M);
671 /* Store the "parametric constant" of row "row" of tableau "tab" in "line",
672 * i.e., the constant term and the coefficients of all variables that
673 * appear in the context tableau.
674 * Note that the coefficient of the big parameter M is NOT copied.
675 * The context tableau may not have a big parameter and even when it
676 * does, it is a different big parameter.
678 static void get_row_parameter_line(struct isl_tab *tab, int row, isl_int *line)
681 unsigned off = 2 + tab->M;
683 isl_int_set(line[0], tab->mat->row[row][1]);
684 for (i = 0; i < tab->n_param; ++i) {
685 if (tab->var[i].is_row)
686 isl_int_set_si(line[1 + i], 0);
688 int col = tab->var[i].index;
689 isl_int_set(line[1 + i], tab->mat->row[row][off + col]);
692 for (i = 0; i < tab->n_div; ++i) {
693 if (tab->var[tab->n_var - tab->n_div + i].is_row)
694 isl_int_set_si(line[1 + tab->n_param + i], 0);
696 int col = tab->var[tab->n_var - tab->n_div + i].index;
697 isl_int_set(line[1 + tab->n_param + i],
698 tab->mat->row[row][off + col]);
703 /* Check if rows "row1" and "row2" have identical "parametric constants",
704 * as explained above.
705 * In this case, we also insist that the coefficients of the big parameter
706 * be the same as the values of the constants will only be the same
707 * if these coefficients are also the same.
709 static int identical_parameter_line(struct isl_tab *tab, int row1, int row2)
712 unsigned off = 2 + tab->M;
714 if (isl_int_ne(tab->mat->row[row1][1], tab->mat->row[row2][1]))
717 if (tab->M && isl_int_ne(tab->mat->row[row1][2],
718 tab->mat->row[row2][2]))
721 for (i = 0; i < tab->n_param + tab->n_div; ++i) {
722 int pos = i < tab->n_param ? i :
723 tab->n_var - tab->n_div + i - tab->n_param;
726 if (tab->var[pos].is_row)
728 col = tab->var[pos].index;
729 if (isl_int_ne(tab->mat->row[row1][off + col],
730 tab->mat->row[row2][off + col]))
736 /* Return an inequality that expresses that the "parametric constant"
737 * should be non-negative.
738 * This function is only called when the coefficient of the big parameter
741 static struct isl_vec *get_row_parameter_ineq(struct isl_tab *tab, int row)
743 struct isl_vec *ineq;
745 ineq = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_param + tab->n_div);
749 get_row_parameter_line(tab, row, ineq->el);
751 ineq = isl_vec_normalize(ineq);
756 /* Normalize a div expression of the form
758 * [(g*f(x) + c)/(g * m)]
760 * with c the constant term and f(x) the remaining coefficients, to
764 static void normalize_div(__isl_keep isl_vec *div)
766 isl_ctx *ctx = isl_vec_get_ctx(div);
767 int len = div->size - 2;
769 isl_seq_gcd(div->el + 2, len, &ctx->normalize_gcd);
770 isl_int_gcd(ctx->normalize_gcd, ctx->normalize_gcd, div->el[0]);
772 if (isl_int_is_one(ctx->normalize_gcd))
775 isl_int_divexact(div->el[0], div->el[0], ctx->normalize_gcd);
776 isl_int_fdiv_q(div->el[1], div->el[1], ctx->normalize_gcd);
777 isl_seq_scale_down(div->el + 2, div->el + 2, ctx->normalize_gcd, len);
780 /* Return a integer division for use in a parametric cut based on the given row.
781 * In particular, let the parametric constant of the row be
785 * where y_0 = 1, but none of the y_i corresponds to the big parameter M.
786 * The div returned is equal to
788 * floor(\sum_i {-a_i} y_i) = floor((\sum_i (-a_i mod d) y_i)/d)
790 static struct isl_vec *get_row_parameter_div(struct isl_tab *tab, int row)
794 div = isl_vec_alloc(tab->mat->ctx, 1 + 1 + tab->n_param + tab->n_div);
798 isl_int_set(div->el[0], tab->mat->row[row][0]);
799 get_row_parameter_line(tab, row, div->el + 1);
800 isl_seq_neg(div->el + 1, div->el + 1, div->size - 1);
802 isl_seq_fdiv_r(div->el + 1, div->el + 1, div->el[0], div->size - 1);
807 /* Return a integer division for use in transferring an integrality constraint
809 * In particular, let the parametric constant of the row be
813 * where y_0 = 1, but none of the y_i corresponds to the big parameter M.
814 * The the returned div is equal to
816 * floor(\sum_i {a_i} y_i) = floor((\sum_i (a_i mod d) y_i)/d)
818 static struct isl_vec *get_row_split_div(struct isl_tab *tab, int row)
822 div = isl_vec_alloc(tab->mat->ctx, 1 + 1 + tab->n_param + tab->n_div);
826 isl_int_set(div->el[0], tab->mat->row[row][0]);
827 get_row_parameter_line(tab, row, div->el + 1);
829 isl_seq_fdiv_r(div->el + 1, div->el + 1, div->el[0], div->size - 1);
834 /* Construct and return an inequality that expresses an upper bound
836 * In particular, if the div is given by
840 * then the inequality expresses
844 static struct isl_vec *ineq_for_div(struct isl_basic_set *bset, unsigned div)
848 struct isl_vec *ineq;
853 total = isl_basic_set_total_dim(bset);
854 div_pos = 1 + total - bset->n_div + div;
856 ineq = isl_vec_alloc(bset->ctx, 1 + total);
860 isl_seq_cpy(ineq->el, bset->div[div] + 1, 1 + total);
861 isl_int_neg(ineq->el[div_pos], bset->div[div][0]);
865 /* Given a row in the tableau and a div that was created
866 * using get_row_split_div and that has been constrained to equality, i.e.,
868 * d = floor(\sum_i {a_i} y_i) = \sum_i {a_i} y_i
870 * replace the expression "\sum_i {a_i} y_i" in the row by d,
871 * i.e., we subtract "\sum_i {a_i} y_i" and add 1 d.
872 * The coefficients of the non-parameters in the tableau have been
873 * verified to be integral. We can therefore simply replace coefficient b
874 * by floor(b). For the coefficients of the parameters we have
875 * floor(a_i) = a_i - {a_i}, while for the other coefficients, we have
878 static struct isl_tab *set_row_cst_to_div(struct isl_tab *tab, int row, int div)
880 isl_seq_fdiv_q(tab->mat->row[row] + 1, tab->mat->row[row] + 1,
881 tab->mat->row[row][0], 1 + tab->M + tab->n_col);
883 isl_int_set_si(tab->mat->row[row][0], 1);
885 if (tab->var[tab->n_var - tab->n_div + div].is_row) {
886 int drow = tab->var[tab->n_var - tab->n_div + div].index;
888 isl_assert(tab->mat->ctx,
889 isl_int_is_one(tab->mat->row[drow][0]), goto error);
890 isl_seq_combine(tab->mat->row[row] + 1,
891 tab->mat->ctx->one, tab->mat->row[row] + 1,
892 tab->mat->ctx->one, tab->mat->row[drow] + 1,
893 1 + tab->M + tab->n_col);
895 int dcol = tab->var[tab->n_var - tab->n_div + div].index;
897 isl_int_add_ui(tab->mat->row[row][2 + tab->M + dcol],
898 tab->mat->row[row][2 + tab->M + dcol], 1);
907 /* Check if the (parametric) constant of the given row is obviously
908 * negative, meaning that we don't need to consult the context tableau.
909 * If there is a big parameter and its coefficient is non-zero,
910 * then this coefficient determines the outcome.
911 * Otherwise, we check whether the constant is negative and
912 * all non-zero coefficients of parameters are negative and
913 * belong to non-negative parameters.
915 static int is_obviously_neg(struct isl_tab *tab, int row)
919 unsigned off = 2 + tab->M;
922 if (isl_int_is_pos(tab->mat->row[row][2]))
924 if (isl_int_is_neg(tab->mat->row[row][2]))
928 if (isl_int_is_nonneg(tab->mat->row[row][1]))
930 for (i = 0; i < tab->n_param; ++i) {
931 /* Eliminated parameter */
932 if (tab->var[i].is_row)
934 col = tab->var[i].index;
935 if (isl_int_is_zero(tab->mat->row[row][off + col]))
937 if (!tab->var[i].is_nonneg)
939 if (isl_int_is_pos(tab->mat->row[row][off + col]))
942 for (i = 0; i < tab->n_div; ++i) {
943 if (tab->var[tab->n_var - tab->n_div + i].is_row)
945 col = tab->var[tab->n_var - tab->n_div + i].index;
946 if (isl_int_is_zero(tab->mat->row[row][off + col]))
948 if (!tab->var[tab->n_var - tab->n_div + i].is_nonneg)
950 if (isl_int_is_pos(tab->mat->row[row][off + col]))
956 /* Check if the (parametric) constant of the given row is obviously
957 * non-negative, meaning that we don't need to consult the context tableau.
958 * If there is a big parameter and its coefficient is non-zero,
959 * then this coefficient determines the outcome.
960 * Otherwise, we check whether the constant is non-negative and
961 * all non-zero coefficients of parameters are positive and
962 * belong to non-negative parameters.
964 static int is_obviously_nonneg(struct isl_tab *tab, int row)
968 unsigned off = 2 + tab->M;
971 if (isl_int_is_pos(tab->mat->row[row][2]))
973 if (isl_int_is_neg(tab->mat->row[row][2]))
977 if (isl_int_is_neg(tab->mat->row[row][1]))
979 for (i = 0; i < tab->n_param; ++i) {
980 /* Eliminated parameter */
981 if (tab->var[i].is_row)
983 col = tab->var[i].index;
984 if (isl_int_is_zero(tab->mat->row[row][off + col]))
986 if (!tab->var[i].is_nonneg)
988 if (isl_int_is_neg(tab->mat->row[row][off + col]))
991 for (i = 0; i < tab->n_div; ++i) {
992 if (tab->var[tab->n_var - tab->n_div + i].is_row)
994 col = tab->var[tab->n_var - tab->n_div + i].index;
995 if (isl_int_is_zero(tab->mat->row[row][off + col]))
997 if (!tab->var[tab->n_var - tab->n_div + i].is_nonneg)
999 if (isl_int_is_neg(tab->mat->row[row][off + col]))
1005 /* Given a row r and two columns, return the column that would
1006 * lead to the lexicographically smallest increment in the sample
1007 * solution when leaving the basis in favor of the row.
1008 * Pivoting with column c will increment the sample value by a non-negative
1009 * constant times a_{V,c}/a_{r,c}, with a_{V,c} the elements of column c
1010 * corresponding to the non-parametric variables.
1011 * If variable v appears in a column c_v, the a_{v,c} = 1 iff c = c_v,
1012 * with all other entries in this virtual row equal to zero.
1013 * If variable v appears in a row, then a_{v,c} is the element in column c
1016 * Let v be the first variable with a_{v,c1}/a_{r,c1} != a_{v,c2}/a_{r,c2}.
1017 * Then if a_{v,c1}/a_{r,c1} < a_{v,c2}/a_{r,c2}, i.e.,
1018 * a_{v,c2} a_{r,c1} - a_{v,c1} a_{r,c2} > 0, c1 results in the minimal
1019 * increment. Otherwise, it's c2.
1021 static int lexmin_col_pair(struct isl_tab *tab,
1022 int row, int col1, int col2, isl_int tmp)
1027 tr = tab->mat->row[row] + 2 + tab->M;
1029 for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
1033 if (!tab->var[i].is_row) {
1034 if (tab->var[i].index == col1)
1036 if (tab->var[i].index == col2)
1041 if (tab->var[i].index == row)
1044 r = tab->mat->row[tab->var[i].index] + 2 + tab->M;
1045 s1 = isl_int_sgn(r[col1]);
1046 s2 = isl_int_sgn(r[col2]);
1047 if (s1 == 0 && s2 == 0)
1054 isl_int_mul(tmp, r[col2], tr[col1]);
1055 isl_int_submul(tmp, r[col1], tr[col2]);
1056 if (isl_int_is_pos(tmp))
1058 if (isl_int_is_neg(tmp))
1064 /* Given a row in the tableau, find and return the column that would
1065 * result in the lexicographically smallest, but positive, increment
1066 * in the sample point.
1067 * If there is no such column, then return tab->n_col.
1068 * If anything goes wrong, return -1.
1070 static int lexmin_pivot_col(struct isl_tab *tab, int row)
1073 int col = tab->n_col;
1077 tr = tab->mat->row[row] + 2 + tab->M;
1081 for (j = tab->n_dead; j < tab->n_col; ++j) {
1082 if (tab->col_var[j] >= 0 &&
1083 (tab->col_var[j] < tab->n_param ||
1084 tab->col_var[j] >= tab->n_var - tab->n_div))
1087 if (!isl_int_is_pos(tr[j]))
1090 if (col == tab->n_col)
1093 col = lexmin_col_pair(tab, row, col, j, tmp);
1094 isl_assert(tab->mat->ctx, col >= 0, goto error);
1104 /* Return the first known violated constraint, i.e., a non-negative
1105 * constraint that currently has an either obviously negative value
1106 * or a previously determined to be negative value.
1108 * If any constraint has a negative coefficient for the big parameter,
1109 * if any, then we return one of these first.
1111 static int first_neg(struct isl_tab *tab)
1116 for (row = tab->n_redundant; row < tab->n_row; ++row) {
1117 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
1119 if (!isl_int_is_neg(tab->mat->row[row][2]))
1122 tab->row_sign[row] = isl_tab_row_neg;
1125 for (row = tab->n_redundant; row < tab->n_row; ++row) {
1126 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
1128 if (tab->row_sign) {
1129 if (tab->row_sign[row] == 0 &&
1130 is_obviously_neg(tab, row))
1131 tab->row_sign[row] = isl_tab_row_neg;
1132 if (tab->row_sign[row] != isl_tab_row_neg)
1134 } else if (!is_obviously_neg(tab, row))
1141 /* Check whether the invariant that all columns are lexico-positive
1142 * is satisfied. This function is not called from the current code
1143 * but is useful during debugging.
1145 static void check_lexpos(struct isl_tab *tab) __attribute__ ((unused));
1146 static void check_lexpos(struct isl_tab *tab)
1148 unsigned off = 2 + tab->M;
1153 for (col = tab->n_dead; col < tab->n_col; ++col) {
1154 if (tab->col_var[col] >= 0 &&
1155 (tab->col_var[col] < tab->n_param ||
1156 tab->col_var[col] >= tab->n_var - tab->n_div))
1158 for (var = tab->n_param; var < tab->n_var - tab->n_div; ++var) {
1159 if (!tab->var[var].is_row) {
1160 if (tab->var[var].index == col)
1165 row = tab->var[var].index;
1166 if (isl_int_is_zero(tab->mat->row[row][off + col]))
1168 if (isl_int_is_pos(tab->mat->row[row][off + col]))
1170 fprintf(stderr, "lexneg column %d (row %d)\n",
1173 if (var >= tab->n_var - tab->n_div)
1174 fprintf(stderr, "zero column %d\n", col);
1178 /* Report to the caller that the given constraint is part of an encountered
1181 static int report_conflicting_constraint(struct isl_tab *tab, int con)
1183 return tab->conflict(con, tab->conflict_user);
1186 /* Given a conflicting row in the tableau, report all constraints
1187 * involved in the row to the caller. That is, the row itself
1188 * (if it represents a constraint) and all constraint columns with
1189 * non-zero (and therefore negative) coefficients.
1191 static int report_conflict(struct isl_tab *tab, int row)
1199 if (tab->row_var[row] < 0 &&
1200 report_conflicting_constraint(tab, ~tab->row_var[row]) < 0)
1203 tr = tab->mat->row[row] + 2 + tab->M;
1205 for (j = tab->n_dead; j < tab->n_col; ++j) {
1206 if (tab->col_var[j] >= 0 &&
1207 (tab->col_var[j] < tab->n_param ||
1208 tab->col_var[j] >= tab->n_var - tab->n_div))
1211 if (!isl_int_is_neg(tr[j]))
1214 if (tab->col_var[j] < 0 &&
1215 report_conflicting_constraint(tab, ~tab->col_var[j]) < 0)
1222 /* Resolve all known or obviously violated constraints through pivoting.
1223 * In particular, as long as we can find any violated constraint, we
1224 * look for a pivoting column that would result in the lexicographically
1225 * smallest increment in the sample point. If there is no such column
1226 * then the tableau is infeasible.
1228 static int restore_lexmin(struct isl_tab *tab) WARN_UNUSED;
1229 static int restore_lexmin(struct isl_tab *tab)
1237 while ((row = first_neg(tab)) != -1) {
1238 col = lexmin_pivot_col(tab, row);
1239 if (col >= tab->n_col) {
1240 if (report_conflict(tab, row) < 0)
1242 if (isl_tab_mark_empty(tab) < 0)
1248 if (isl_tab_pivot(tab, row, col) < 0)
1254 /* Given a row that represents an equality, look for an appropriate
1256 * In particular, if there are any non-zero coefficients among
1257 * the non-parameter variables, then we take the last of these
1258 * variables. Eliminating this variable in terms of the other
1259 * variables and/or parameters does not influence the property
1260 * that all column in the initial tableau are lexicographically
1261 * positive. The row corresponding to the eliminated variable
1262 * will only have non-zero entries below the diagonal of the
1263 * initial tableau. That is, we transform
1269 * If there is no such non-parameter variable, then we are dealing with
1270 * pure parameter equality and we pick any parameter with coefficient 1 or -1
1271 * for elimination. This will ensure that the eliminated parameter
1272 * always has an integer value whenever all the other parameters are integral.
1273 * If there is no such parameter then we return -1.
1275 static int last_var_col_or_int_par_col(struct isl_tab *tab, int row)
1277 unsigned off = 2 + tab->M;
1280 for (i = tab->n_var - tab->n_div - 1; i >= 0 && i >= tab->n_param; --i) {
1282 if (tab->var[i].is_row)
1284 col = tab->var[i].index;
1285 if (col <= tab->n_dead)
1287 if (!isl_int_is_zero(tab->mat->row[row][off + col]))
1290 for (i = tab->n_dead; i < tab->n_col; ++i) {
1291 if (isl_int_is_one(tab->mat->row[row][off + i]))
1293 if (isl_int_is_negone(tab->mat->row[row][off + i]))
1299 /* Add an equality that is known to be valid to the tableau.
1300 * We first check if we can eliminate a variable or a parameter.
1301 * If not, we add the equality as two inequalities.
1302 * In this case, the equality was a pure parameter equality and there
1303 * is no need to resolve any constraint violations.
1305 static struct isl_tab *add_lexmin_valid_eq(struct isl_tab *tab, isl_int *eq)
1312 r = isl_tab_add_row(tab, eq);
1316 r = tab->con[r].index;
1317 i = last_var_col_or_int_par_col(tab, r);
1319 tab->con[r].is_nonneg = 1;
1320 if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
1322 isl_seq_neg(eq, eq, 1 + tab->n_var);
1323 r = isl_tab_add_row(tab, eq);
1326 tab->con[r].is_nonneg = 1;
1327 if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
1330 if (isl_tab_pivot(tab, r, i) < 0)
1332 if (isl_tab_kill_col(tab, i) < 0)
1343 /* Check if the given row is a pure constant.
1345 static int is_constant(struct isl_tab *tab, int row)
1347 unsigned off = 2 + tab->M;
1349 return isl_seq_first_non_zero(tab->mat->row[row] + off + tab->n_dead,
1350 tab->n_col - tab->n_dead) == -1;
1353 /* Add an equality that may or may not be valid to the tableau.
1354 * If the resulting row is a pure constant, then it must be zero.
1355 * Otherwise, the resulting tableau is empty.
1357 * If the row is not a pure constant, then we add two inequalities,
1358 * each time checking that they can be satisfied.
1359 * In the end we try to use one of the two constraints to eliminate
1362 static int add_lexmin_eq(struct isl_tab *tab, isl_int *eq) WARN_UNUSED;
1363 static int add_lexmin_eq(struct isl_tab *tab, isl_int *eq)
1367 struct isl_tab_undo *snap;
1371 snap = isl_tab_snap(tab);
1372 r1 = isl_tab_add_row(tab, eq);
1375 tab->con[r1].is_nonneg = 1;
1376 if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r1]) < 0)
1379 row = tab->con[r1].index;
1380 if (is_constant(tab, row)) {
1381 if (!isl_int_is_zero(tab->mat->row[row][1]) ||
1382 (tab->M && !isl_int_is_zero(tab->mat->row[row][2]))) {
1383 if (isl_tab_mark_empty(tab) < 0)
1387 if (isl_tab_rollback(tab, snap) < 0)
1392 if (restore_lexmin(tab) < 0)
1397 isl_seq_neg(eq, eq, 1 + tab->n_var);
1399 r2 = isl_tab_add_row(tab, eq);
1402 tab->con[r2].is_nonneg = 1;
1403 if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r2]) < 0)
1406 if (restore_lexmin(tab) < 0)
1411 if (!tab->con[r1].is_row) {
1412 if (isl_tab_kill_col(tab, tab->con[r1].index) < 0)
1414 } else if (!tab->con[r2].is_row) {
1415 if (isl_tab_kill_col(tab, tab->con[r2].index) < 0)
1420 tab->bmap = isl_basic_map_add_ineq(tab->bmap, eq);
1421 if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0)
1423 isl_seq_neg(eq, eq, 1 + tab->n_var);
1424 tab->bmap = isl_basic_map_add_ineq(tab->bmap, eq);
1425 isl_seq_neg(eq, eq, 1 + tab->n_var);
1426 if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0)
1435 /* Add an inequality to the tableau, resolving violations using
1438 static struct isl_tab *add_lexmin_ineq(struct isl_tab *tab, isl_int *ineq)
1445 tab->bmap = isl_basic_map_add_ineq(tab->bmap, ineq);
1446 if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0)
1451 r = isl_tab_add_row(tab, ineq);
1454 tab->con[r].is_nonneg = 1;
1455 if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
1457 if (isl_tab_row_is_redundant(tab, tab->con[r].index)) {
1458 if (isl_tab_mark_redundant(tab, tab->con[r].index) < 0)
1463 if (restore_lexmin(tab) < 0)
1465 if (!tab->empty && tab->con[r].is_row &&
1466 isl_tab_row_is_redundant(tab, tab->con[r].index))
1467 if (isl_tab_mark_redundant(tab, tab->con[r].index) < 0)
1475 /* Check if the coefficients of the parameters are all integral.
1477 static int integer_parameter(struct isl_tab *tab, int row)
1481 unsigned off = 2 + tab->M;
1483 for (i = 0; i < tab->n_param; ++i) {
1484 /* Eliminated parameter */
1485 if (tab->var[i].is_row)
1487 col = tab->var[i].index;
1488 if (!isl_int_is_divisible_by(tab->mat->row[row][off + col],
1489 tab->mat->row[row][0]))
1492 for (i = 0; i < tab->n_div; ++i) {
1493 if (tab->var[tab->n_var - tab->n_div + i].is_row)
1495 col = tab->var[tab->n_var - tab->n_div + i].index;
1496 if (!isl_int_is_divisible_by(tab->mat->row[row][off + col],
1497 tab->mat->row[row][0]))
1503 /* Check if the coefficients of the non-parameter variables are all integral.
1505 static int integer_variable(struct isl_tab *tab, int row)
1508 unsigned off = 2 + tab->M;
1510 for (i = tab->n_dead; i < tab->n_col; ++i) {
1511 if (tab->col_var[i] >= 0 &&
1512 (tab->col_var[i] < tab->n_param ||
1513 tab->col_var[i] >= tab->n_var - tab->n_div))
1515 if (!isl_int_is_divisible_by(tab->mat->row[row][off + i],
1516 tab->mat->row[row][0]))
1522 /* Check if the constant term is integral.
1524 static int integer_constant(struct isl_tab *tab, int row)
1526 return isl_int_is_divisible_by(tab->mat->row[row][1],
1527 tab->mat->row[row][0]);
1530 #define I_CST 1 << 0
1531 #define I_PAR 1 << 1
1532 #define I_VAR 1 << 2
1534 /* Check for next (non-parameter) variable after "var" (first if var == -1)
1535 * that is non-integer and therefore requires a cut and return
1536 * the index of the variable.
1537 * For parametric tableaus, there are three parts in a row,
1538 * the constant, the coefficients of the parameters and the rest.
1539 * For each part, we check whether the coefficients in that part
1540 * are all integral and if so, set the corresponding flag in *f.
1541 * If the constant and the parameter part are integral, then the
1542 * current sample value is integral and no cut is required
1543 * (irrespective of whether the variable part is integral).
1545 static int next_non_integer_var(struct isl_tab *tab, int var, int *f)
1547 var = var < 0 ? tab->n_param : var + 1;
1549 for (; var < tab->n_var - tab->n_div; ++var) {
1552 if (!tab->var[var].is_row)
1554 row = tab->var[var].index;
1555 if (integer_constant(tab, row))
1556 ISL_FL_SET(flags, I_CST);
1557 if (integer_parameter(tab, row))
1558 ISL_FL_SET(flags, I_PAR);
1559 if (ISL_FL_ISSET(flags, I_CST) && ISL_FL_ISSET(flags, I_PAR))
1561 if (integer_variable(tab, row))
1562 ISL_FL_SET(flags, I_VAR);
1569 /* Check for first (non-parameter) variable that is non-integer and
1570 * therefore requires a cut and return the corresponding row.
1571 * For parametric tableaus, there are three parts in a row,
1572 * the constant, the coefficients of the parameters and the rest.
1573 * For each part, we check whether the coefficients in that part
1574 * are all integral and if so, set the corresponding flag in *f.
1575 * If the constant and the parameter part are integral, then the
1576 * current sample value is integral and no cut is required
1577 * (irrespective of whether the variable part is integral).
1579 static int first_non_integer_row(struct isl_tab *tab, int *f)
1581 int var = next_non_integer_var(tab, -1, f);
1583 return var < 0 ? -1 : tab->var[var].index;
1586 /* Add a (non-parametric) cut to cut away the non-integral sample
1587 * value of the given row.
1589 * If the row is given by
1591 * m r = f + \sum_i a_i y_i
1595 * c = - {-f/m} + \sum_i {a_i/m} y_i >= 0
1597 * The big parameter, if any, is ignored, since it is assumed to be big
1598 * enough to be divisible by any integer.
1599 * If the tableau is actually a parametric tableau, then this function
1600 * is only called when all coefficients of the parameters are integral.
1601 * The cut therefore has zero coefficients for the parameters.
1603 * The current value is known to be negative, so row_sign, if it
1604 * exists, is set accordingly.
1606 * Return the row of the cut or -1.
1608 static int add_cut(struct isl_tab *tab, int row)
1613 unsigned off = 2 + tab->M;
1615 if (isl_tab_extend_cons(tab, 1) < 0)
1617 r = isl_tab_allocate_con(tab);
1621 r_row = tab->mat->row[tab->con[r].index];
1622 isl_int_set(r_row[0], tab->mat->row[row][0]);
1623 isl_int_neg(r_row[1], tab->mat->row[row][1]);
1624 isl_int_fdiv_r(r_row[1], r_row[1], tab->mat->row[row][0]);
1625 isl_int_neg(r_row[1], r_row[1]);
1627 isl_int_set_si(r_row[2], 0);
1628 for (i = 0; i < tab->n_col; ++i)
1629 isl_int_fdiv_r(r_row[off + i],
1630 tab->mat->row[row][off + i], tab->mat->row[row][0]);
1632 tab->con[r].is_nonneg = 1;
1633 if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
1636 tab->row_sign[tab->con[r].index] = isl_tab_row_neg;
1638 return tab->con[r].index;
1644 /* Given a non-parametric tableau, add cuts until an integer
1645 * sample point is obtained or until the tableau is determined
1646 * to be integer infeasible.
1647 * As long as there is any non-integer value in the sample point,
1648 * we add appropriate cuts, if possible, for each of these
1649 * non-integer values and then resolve the violated
1650 * cut constraints using restore_lexmin.
1651 * If one of the corresponding rows is equal to an integral
1652 * combination of variables/constraints plus a non-integral constant,
1653 * then there is no way to obtain an integer point and we return
1654 * a tableau that is marked empty.
1655 * The parameter cutting_strategy controls the strategy used when adding cuts
1656 * to remove non-integer points. CUT_ALL adds all possible cuts
1657 * before continuing the search. CUT_ONE adds only one cut at a time.
1659 static struct isl_tab *cut_to_integer_lexmin(struct isl_tab *tab,
1660 int cutting_strategy)
1671 while ((var = next_non_integer_var(tab, -1, &flags)) != -1) {
1673 if (ISL_FL_ISSET(flags, I_VAR)) {
1674 if (isl_tab_mark_empty(tab) < 0)
1678 row = tab->var[var].index;
1679 row = add_cut(tab, row);
1682 if (cutting_strategy == CUT_ONE)
1684 } while ((var = next_non_integer_var(tab, var, &flags)) != -1);
1685 if (restore_lexmin(tab) < 0)
1696 /* Check whether all the currently active samples also satisfy the inequality
1697 * "ineq" (treated as an equality if eq is set).
1698 * Remove those samples that do not.
1700 static struct isl_tab *check_samples(struct isl_tab *tab, isl_int *ineq, int eq)
1708 isl_assert(tab->mat->ctx, tab->bmap, goto error);
1709 isl_assert(tab->mat->ctx, tab->samples, goto error);
1710 isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, goto error);
1713 for (i = tab->n_outside; i < tab->n_sample; ++i) {
1715 isl_seq_inner_product(ineq, tab->samples->row[i],
1716 1 + tab->n_var, &v);
1717 sgn = isl_int_sgn(v);
1718 if (eq ? (sgn == 0) : (sgn >= 0))
1720 tab = isl_tab_drop_sample(tab, i);
1732 /* Check whether the sample value of the tableau is finite,
1733 * i.e., either the tableau does not use a big parameter, or
1734 * all values of the variables are equal to the big parameter plus
1735 * some constant. This constant is the actual sample value.
1737 static int sample_is_finite(struct isl_tab *tab)
1744 for (i = 0; i < tab->n_var; ++i) {
1746 if (!tab->var[i].is_row)
1748 row = tab->var[i].index;
1749 if (isl_int_ne(tab->mat->row[row][0], tab->mat->row[row][2]))
1755 /* Check if the context tableau of sol has any integer points.
1756 * Leave tab in empty state if no integer point can be found.
1757 * If an integer point can be found and if moreover it is finite,
1758 * then it is added to the list of sample values.
1760 * This function is only called when none of the currently active sample
1761 * values satisfies the most recently added constraint.
1763 static struct isl_tab *check_integer_feasible(struct isl_tab *tab)
1765 struct isl_tab_undo *snap;
1770 snap = isl_tab_snap(tab);
1771 if (isl_tab_push_basis(tab) < 0)
1774 tab = cut_to_integer_lexmin(tab, CUT_ALL);
1778 if (!tab->empty && sample_is_finite(tab)) {
1779 struct isl_vec *sample;
1781 sample = isl_tab_get_sample_value(tab);
1783 tab = isl_tab_add_sample(tab, sample);
1786 if (!tab->empty && isl_tab_rollback(tab, snap) < 0)
1795 /* Check if any of the currently active sample values satisfies
1796 * the inequality "ineq" (an equality if eq is set).
1798 static int tab_has_valid_sample(struct isl_tab *tab, isl_int *ineq, int eq)
1806 isl_assert(tab->mat->ctx, tab->bmap, return -1);
1807 isl_assert(tab->mat->ctx, tab->samples, return -1);
1808 isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, return -1);
1811 for (i = tab->n_outside; i < tab->n_sample; ++i) {
1813 isl_seq_inner_product(ineq, tab->samples->row[i],
1814 1 + tab->n_var, &v);
1815 sgn = isl_int_sgn(v);
1816 if (eq ? (sgn == 0) : (sgn >= 0))
1821 return i < tab->n_sample;
1824 /* Add a div specified by "div" to the tableau "tab" and return
1825 * 1 if the div is obviously non-negative.
1827 static int context_tab_add_div(struct isl_tab *tab, struct isl_vec *div,
1828 int (*add_ineq)(void *user, isl_int *), void *user)
1832 struct isl_mat *samples;
1835 r = isl_tab_add_div(tab, div, add_ineq, user);
1838 nonneg = tab->var[r].is_nonneg;
1839 tab->var[r].frozen = 1;
1841 samples = isl_mat_extend(tab->samples,
1842 tab->n_sample, 1 + tab->n_var);
1843 tab->samples = samples;
1846 for (i = tab->n_outside; i < samples->n_row; ++i) {
1847 isl_seq_inner_product(div->el + 1, samples->row[i],
1848 div->size - 1, &samples->row[i][samples->n_col - 1]);
1849 isl_int_fdiv_q(samples->row[i][samples->n_col - 1],
1850 samples->row[i][samples->n_col - 1], div->el[0]);
1856 /* Add a div specified by "div" to both the main tableau and
1857 * the context tableau. In case of the main tableau, we only
1858 * need to add an extra div. In the context tableau, we also
1859 * need to express the meaning of the div.
1860 * Return the index of the div or -1 if anything went wrong.
1862 static int add_div(struct isl_tab *tab, struct isl_context *context,
1863 struct isl_vec *div)
1868 if ((nonneg = context->op->add_div(context, div)) < 0)
1871 if (!context->op->is_ok(context))
1874 if (isl_tab_extend_vars(tab, 1) < 0)
1876 r = isl_tab_allocate_var(tab);
1880 tab->var[r].is_nonneg = 1;
1881 tab->var[r].frozen = 1;
1884 return tab->n_div - 1;
1886 context->op->invalidate(context);
1890 static int find_div(struct isl_tab *tab, isl_int *div, isl_int denom)
1893 unsigned total = isl_basic_map_total_dim(tab->bmap);
1895 for (i = 0; i < tab->bmap->n_div; ++i) {
1896 if (isl_int_ne(tab->bmap->div[i][0], denom))
1898 if (!isl_seq_eq(tab->bmap->div[i] + 1, div, 1 + total))
1905 /* Return the index of a div that corresponds to "div".
1906 * We first check if we already have such a div and if not, we create one.
1908 static int get_div(struct isl_tab *tab, struct isl_context *context,
1909 struct isl_vec *div)
1912 struct isl_tab *context_tab = context->op->peek_tab(context);
1917 d = find_div(context_tab, div->el + 1, div->el[0]);
1921 return add_div(tab, context, div);
1924 /* Add a parametric cut to cut away the non-integral sample value
1926 * Let a_i be the coefficients of the constant term and the parameters
1927 * and let b_i be the coefficients of the variables or constraints
1928 * in basis of the tableau.
1929 * Let q be the div q = floor(\sum_i {-a_i} y_i).
1931 * The cut is expressed as
1933 * c = \sum_i -{-a_i} y_i + \sum_i {b_i} x_i + q >= 0
1935 * If q did not already exist in the context tableau, then it is added first.
1936 * If q is in a column of the main tableau then the "+ q" can be accomplished
1937 * by setting the corresponding entry to the denominator of the constraint.
1938 * If q happens to be in a row of the main tableau, then the corresponding
1939 * row needs to be added instead (taking care of the denominators).
1940 * Note that this is very unlikely, but perhaps not entirely impossible.
1942 * The current value of the cut is known to be negative (or at least
1943 * non-positive), so row_sign is set accordingly.
1945 * Return the row of the cut or -1.
1947 static int add_parametric_cut(struct isl_tab *tab, int row,
1948 struct isl_context *context)
1950 struct isl_vec *div;
1957 unsigned off = 2 + tab->M;
1962 div = get_row_parameter_div(tab, row);
1967 d = context->op->get_div(context, tab, div);
1972 if (isl_tab_extend_cons(tab, 1) < 0)
1974 r = isl_tab_allocate_con(tab);
1978 r_row = tab->mat->row[tab->con[r].index];
1979 isl_int_set(r_row[0], tab->mat->row[row][0]);
1980 isl_int_neg(r_row[1], tab->mat->row[row][1]);
1981 isl_int_fdiv_r(r_row[1], r_row[1], tab->mat->row[row][0]);
1982 isl_int_neg(r_row[1], r_row[1]);
1984 isl_int_set_si(r_row[2], 0);
1985 for (i = 0; i < tab->n_param; ++i) {
1986 if (tab->var[i].is_row)
1988 col = tab->var[i].index;
1989 isl_int_neg(r_row[off + col], tab->mat->row[row][off + col]);
1990 isl_int_fdiv_r(r_row[off + col], r_row[off + col],
1991 tab->mat->row[row][0]);
1992 isl_int_neg(r_row[off + col], r_row[off + col]);
1994 for (i = 0; i < tab->n_div; ++i) {
1995 if (tab->var[tab->n_var - tab->n_div + i].is_row)
1997 col = tab->var[tab->n_var - tab->n_div + i].index;
1998 isl_int_neg(r_row[off + col], tab->mat->row[row][off + col]);
1999 isl_int_fdiv_r(r_row[off + col], r_row[off + col],
2000 tab->mat->row[row][0]);
2001 isl_int_neg(r_row[off + col], r_row[off + col]);
2003 for (i = 0; i < tab->n_col; ++i) {
2004 if (tab->col_var[i] >= 0 &&
2005 (tab->col_var[i] < tab->n_param ||
2006 tab->col_var[i] >= tab->n_var - tab->n_div))
2008 isl_int_fdiv_r(r_row[off + i],
2009 tab->mat->row[row][off + i], tab->mat->row[row][0]);
2011 if (tab->var[tab->n_var - tab->n_div + d].is_row) {
2013 int d_row = tab->var[tab->n_var - tab->n_div + d].index;
2015 isl_int_gcd(gcd, tab->mat->row[d_row][0], r_row[0]);
2016 isl_int_divexact(r_row[0], r_row[0], gcd);
2017 isl_int_divexact(gcd, tab->mat->row[d_row][0], gcd);
2018 isl_seq_combine(r_row + 1, gcd, r_row + 1,
2019 r_row[0], tab->mat->row[d_row] + 1,
2020 off - 1 + tab->n_col);
2021 isl_int_mul(r_row[0], r_row[0], tab->mat->row[d_row][0]);
2024 col = tab->var[tab->n_var - tab->n_div + d].index;
2025 isl_int_set(r_row[off + col], tab->mat->row[row][0]);
2028 tab->con[r].is_nonneg = 1;
2029 if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0)
2032 tab->row_sign[tab->con[r].index] = isl_tab_row_neg;
2034 row = tab->con[r].index;
2036 if (d >= n && context->op->detect_equalities(context, tab) < 0)
2042 /* Construct a tableau for bmap that can be used for computing
2043 * the lexicographic minimum (or maximum) of bmap.
2044 * If not NULL, then dom is the domain where the minimum
2045 * should be computed. In this case, we set up a parametric
2046 * tableau with row signs (initialized to "unknown").
2047 * If M is set, then the tableau will use a big parameter.
2048 * If max is set, then a maximum should be computed instead of a minimum.
2049 * This means that for each variable x, the tableau will contain the variable
2050 * x' = M - x, rather than x' = M + x. This in turn means that the coefficient
2051 * of the variables in all constraints are negated prior to adding them
2054 static struct isl_tab *tab_for_lexmin(struct isl_basic_map *bmap,
2055 struct isl_basic_set *dom, unsigned M, int max)
2058 struct isl_tab *tab;
2060 tab = isl_tab_alloc(bmap->ctx, 2 * bmap->n_eq + bmap->n_ineq + 1,
2061 isl_basic_map_total_dim(bmap), M);
2065 tab->rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
2067 tab->n_param = isl_basic_set_total_dim(dom) - dom->n_div;
2068 tab->n_div = dom->n_div;
2069 tab->row_sign = isl_calloc_array(bmap->ctx,
2070 enum isl_tab_row_sign, tab->mat->n_row);
2074 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY)) {
2075 if (isl_tab_mark_empty(tab) < 0)
2080 for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
2081 tab->var[i].is_nonneg = 1;
2082 tab->var[i].frozen = 1;
2084 for (i = 0; i < bmap->n_eq; ++i) {
2086 isl_seq_neg(bmap->eq[i] + 1 + tab->n_param,
2087 bmap->eq[i] + 1 + tab->n_param,
2088 tab->n_var - tab->n_param - tab->n_div);
2089 tab = add_lexmin_valid_eq(tab, bmap->eq[i]);
2091 isl_seq_neg(bmap->eq[i] + 1 + tab->n_param,
2092 bmap->eq[i] + 1 + tab->n_param,
2093 tab->n_var - tab->n_param - tab->n_div);
2094 if (!tab || tab->empty)
2097 if (bmap->n_eq && restore_lexmin(tab) < 0)
2099 for (i = 0; i < bmap->n_ineq; ++i) {
2101 isl_seq_neg(bmap->ineq[i] + 1 + tab->n_param,
2102 bmap->ineq[i] + 1 + tab->n_param,
2103 tab->n_var - tab->n_param - tab->n_div);
2104 tab = add_lexmin_ineq(tab, bmap->ineq[i]);
2106 isl_seq_neg(bmap->ineq[i] + 1 + tab->n_param,
2107 bmap->ineq[i] + 1 + tab->n_param,
2108 tab->n_var - tab->n_param - tab->n_div);
2109 if (!tab || tab->empty)
2118 /* Given a main tableau where more than one row requires a split,
2119 * determine and return the "best" row to split on.
2121 * Given two rows in the main tableau, if the inequality corresponding
2122 * to the first row is redundant with respect to that of the second row
2123 * in the current tableau, then it is better to split on the second row,
2124 * since in the positive part, both row will be positive.
2125 * (In the negative part a pivot will have to be performed and just about
2126 * anything can happen to the sign of the other row.)
2128 * As a simple heuristic, we therefore select the row that makes the most
2129 * of the other rows redundant.
2131 * Perhaps it would also be useful to look at the number of constraints
2132 * that conflict with any given constraint.
2134 static int best_split(struct isl_tab *tab, struct isl_tab *context_tab)
2136 struct isl_tab_undo *snap;
2142 if (isl_tab_extend_cons(context_tab, 2) < 0)
2145 snap = isl_tab_snap(context_tab);
2147 for (split = tab->n_redundant; split < tab->n_row; ++split) {
2148 struct isl_tab_undo *snap2;
2149 struct isl_vec *ineq = NULL;
2153 if (!isl_tab_var_from_row(tab, split)->is_nonneg)
2155 if (tab->row_sign[split] != isl_tab_row_any)
2158 ineq = get_row_parameter_ineq(tab, split);
2161 ok = isl_tab_add_ineq(context_tab, ineq->el) >= 0;
2166 snap2 = isl_tab_snap(context_tab);
2168 for (row = tab->n_redundant; row < tab->n_row; ++row) {
2169 struct isl_tab_var *var;
2173 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
2175 if (tab->row_sign[row] != isl_tab_row_any)
2178 ineq = get_row_parameter_ineq(tab, row);
2181 ok = isl_tab_add_ineq(context_tab, ineq->el) >= 0;
2185 var = &context_tab->con[context_tab->n_con - 1];
2186 if (!context_tab->empty &&
2187 !isl_tab_min_at_most_neg_one(context_tab, var))
2189 if (isl_tab_rollback(context_tab, snap2) < 0)
2192 if (best == -1 || r > best_r) {
2196 if (isl_tab_rollback(context_tab, snap) < 0)
2203 static struct isl_basic_set *context_lex_peek_basic_set(
2204 struct isl_context *context)
2206 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2209 return isl_tab_peek_bset(clex->tab);
2212 static struct isl_tab *context_lex_peek_tab(struct isl_context *context)
2214 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2218 static void context_lex_add_eq(struct isl_context *context, isl_int *eq,
2219 int check, int update)
2221 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2222 if (isl_tab_extend_cons(clex->tab, 2) < 0)
2224 if (add_lexmin_eq(clex->tab, eq) < 0)
2227 int v = tab_has_valid_sample(clex->tab, eq, 1);
2231 clex->tab = check_integer_feasible(clex->tab);
2234 clex->tab = check_samples(clex->tab, eq, 1);
2237 isl_tab_free(clex->tab);
2241 static void context_lex_add_ineq(struct isl_context *context, isl_int *ineq,
2242 int check, int update)
2244 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2245 if (isl_tab_extend_cons(clex->tab, 1) < 0)
2247 clex->tab = add_lexmin_ineq(clex->tab, ineq);
2249 int v = tab_has_valid_sample(clex->tab, ineq, 0);
2253 clex->tab = check_integer_feasible(clex->tab);
2256 clex->tab = check_samples(clex->tab, ineq, 0);
2259 isl_tab_free(clex->tab);
2263 static int context_lex_add_ineq_wrap(void *user, isl_int *ineq)
2265 struct isl_context *context = (struct isl_context *)user;
2266 context_lex_add_ineq(context, ineq, 0, 0);
2267 return context->op->is_ok(context) ? 0 : -1;
2270 /* Check which signs can be obtained by "ineq" on all the currently
2271 * active sample values. See row_sign for more information.
2273 static enum isl_tab_row_sign tab_ineq_sign(struct isl_tab *tab, isl_int *ineq,
2279 enum isl_tab_row_sign res = isl_tab_row_unknown;
2281 isl_assert(tab->mat->ctx, tab->samples, return isl_tab_row_unknown);
2282 isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var,
2283 return isl_tab_row_unknown);
2286 for (i = tab->n_outside; i < tab->n_sample; ++i) {
2287 isl_seq_inner_product(tab->samples->row[i], ineq,
2288 1 + tab->n_var, &tmp);
2289 sgn = isl_int_sgn(tmp);
2290 if (sgn > 0 || (sgn == 0 && strict)) {
2291 if (res == isl_tab_row_unknown)
2292 res = isl_tab_row_pos;
2293 if (res == isl_tab_row_neg)
2294 res = isl_tab_row_any;
2297 if (res == isl_tab_row_unknown)
2298 res = isl_tab_row_neg;
2299 if (res == isl_tab_row_pos)
2300 res = isl_tab_row_any;
2302 if (res == isl_tab_row_any)
2310 static enum isl_tab_row_sign context_lex_ineq_sign(struct isl_context *context,
2311 isl_int *ineq, int strict)
2313 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2314 return tab_ineq_sign(clex->tab, ineq, strict);
2317 /* Check whether "ineq" can be added to the tableau without rendering
2320 static int context_lex_test_ineq(struct isl_context *context, isl_int *ineq)
2322 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2323 struct isl_tab_undo *snap;
2329 if (isl_tab_extend_cons(clex->tab, 1) < 0)
2332 snap = isl_tab_snap(clex->tab);
2333 if (isl_tab_push_basis(clex->tab) < 0)
2335 clex->tab = add_lexmin_ineq(clex->tab, ineq);
2336 clex->tab = check_integer_feasible(clex->tab);
2339 feasible = !clex->tab->empty;
2340 if (isl_tab_rollback(clex->tab, snap) < 0)
2346 static int context_lex_get_div(struct isl_context *context, struct isl_tab *tab,
2347 struct isl_vec *div)
2349 return get_div(tab, context, div);
2352 /* Add a div specified by "div" to the context tableau and return
2353 * 1 if the div is obviously non-negative.
2354 * context_tab_add_div will always return 1, because all variables
2355 * in a isl_context_lex tableau are non-negative.
2356 * However, if we are using a big parameter in the context, then this only
2357 * reflects the non-negativity of the variable used to _encode_ the
2358 * div, i.e., div' = M + div, so we can't draw any conclusions.
2360 static int context_lex_add_div(struct isl_context *context, struct isl_vec *div)
2362 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2364 nonneg = context_tab_add_div(clex->tab, div,
2365 context_lex_add_ineq_wrap, context);
2373 static int context_lex_detect_equalities(struct isl_context *context,
2374 struct isl_tab *tab)
2379 static int context_lex_best_split(struct isl_context *context,
2380 struct isl_tab *tab)
2382 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2383 struct isl_tab_undo *snap;
2386 snap = isl_tab_snap(clex->tab);
2387 if (isl_tab_push_basis(clex->tab) < 0)
2389 r = best_split(tab, clex->tab);
2391 if (r >= 0 && isl_tab_rollback(clex->tab, snap) < 0)
2397 static int context_lex_is_empty(struct isl_context *context)
2399 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2402 return clex->tab->empty;
2405 static void *context_lex_save(struct isl_context *context)
2407 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2408 struct isl_tab_undo *snap;
2410 snap = isl_tab_snap(clex->tab);
2411 if (isl_tab_push_basis(clex->tab) < 0)
2413 if (isl_tab_save_samples(clex->tab) < 0)
2419 static void context_lex_restore(struct isl_context *context, void *save)
2421 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2422 if (isl_tab_rollback(clex->tab, (struct isl_tab_undo *)save) < 0) {
2423 isl_tab_free(clex->tab);
2428 static void context_lex_discard(void *save)
2432 static int context_lex_is_ok(struct isl_context *context)
2434 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2438 /* For each variable in the context tableau, check if the variable can
2439 * only attain non-negative values. If so, mark the parameter as non-negative
2440 * in the main tableau. This allows for a more direct identification of some
2441 * cases of violated constraints.
2443 static struct isl_tab *tab_detect_nonnegative_parameters(struct isl_tab *tab,
2444 struct isl_tab *context_tab)
2447 struct isl_tab_undo *snap;
2448 struct isl_vec *ineq = NULL;
2449 struct isl_tab_var *var;
2452 if (context_tab->n_var == 0)
2455 ineq = isl_vec_alloc(tab->mat->ctx, 1 + context_tab->n_var);
2459 if (isl_tab_extend_cons(context_tab, 1) < 0)
2462 snap = isl_tab_snap(context_tab);
2465 isl_seq_clr(ineq->el, ineq->size);
2466 for (i = 0; i < context_tab->n_var; ++i) {
2467 isl_int_set_si(ineq->el[1 + i], 1);
2468 if (isl_tab_add_ineq(context_tab, ineq->el) < 0)
2470 var = &context_tab->con[context_tab->n_con - 1];
2471 if (!context_tab->empty &&
2472 !isl_tab_min_at_most_neg_one(context_tab, var)) {
2474 if (i >= tab->n_param)
2475 j = i - tab->n_param + tab->n_var - tab->n_div;
2476 tab->var[j].is_nonneg = 1;
2479 isl_int_set_si(ineq->el[1 + i], 0);
2480 if (isl_tab_rollback(context_tab, snap) < 0)
2484 if (context_tab->M && n == context_tab->n_var) {
2485 context_tab->mat = isl_mat_drop_cols(context_tab->mat, 2, 1);
2497 static struct isl_tab *context_lex_detect_nonnegative_parameters(
2498 struct isl_context *context, struct isl_tab *tab)
2500 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2501 struct isl_tab_undo *snap;
2506 snap = isl_tab_snap(clex->tab);
2507 if (isl_tab_push_basis(clex->tab) < 0)
2510 tab = tab_detect_nonnegative_parameters(tab, clex->tab);
2512 if (isl_tab_rollback(clex->tab, snap) < 0)
2521 static void context_lex_invalidate(struct isl_context *context)
2523 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2524 isl_tab_free(clex->tab);
2528 static void context_lex_free(struct isl_context *context)
2530 struct isl_context_lex *clex = (struct isl_context_lex *)context;
2531 isl_tab_free(clex->tab);
2535 struct isl_context_op isl_context_lex_op = {
2536 context_lex_detect_nonnegative_parameters,
2537 context_lex_peek_basic_set,
2538 context_lex_peek_tab,
2540 context_lex_add_ineq,
2541 context_lex_ineq_sign,
2542 context_lex_test_ineq,
2543 context_lex_get_div,
2544 context_lex_add_div,
2545 context_lex_detect_equalities,
2546 context_lex_best_split,
2547 context_lex_is_empty,
2550 context_lex_restore,
2551 context_lex_discard,
2552 context_lex_invalidate,
2556 static struct isl_tab *context_tab_for_lexmin(struct isl_basic_set *bset)
2558 struct isl_tab *tab;
2562 tab = tab_for_lexmin((struct isl_basic_map *)bset, NULL, 1, 0);
2565 if (isl_tab_track_bset(tab, bset) < 0)
2567 tab = isl_tab_init_samples(tab);
2570 isl_basic_set_free(bset);
2574 static struct isl_context *isl_context_lex_alloc(struct isl_basic_set *dom)
2576 struct isl_context_lex *clex;
2581 clex = isl_alloc_type(dom->ctx, struct isl_context_lex);
2585 clex->context.op = &isl_context_lex_op;
2587 clex->tab = context_tab_for_lexmin(isl_basic_set_copy(dom));
2588 if (restore_lexmin(clex->tab) < 0)
2590 clex->tab = check_integer_feasible(clex->tab);
2594 return &clex->context;
2596 clex->context.op->free(&clex->context);
2600 struct isl_context_gbr {
2601 struct isl_context context;
2602 struct isl_tab *tab;
2603 struct isl_tab *shifted;
2604 struct isl_tab *cone;
2607 static struct isl_tab *context_gbr_detect_nonnegative_parameters(
2608 struct isl_context *context, struct isl_tab *tab)
2610 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
2613 return tab_detect_nonnegative_parameters(tab, cgbr->tab);
2616 static struct isl_basic_set *context_gbr_peek_basic_set(
2617 struct isl_context *context)
2619 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
2622 return isl_tab_peek_bset(cgbr->tab);
2625 static struct isl_tab *context_gbr_peek_tab(struct isl_context *context)
2627 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
2631 /* Initialize the "shifted" tableau of the context, which
2632 * contains the constraints of the original tableau shifted
2633 * by the sum of all negative coefficients. This ensures
2634 * that any rational point in the shifted tableau can
2635 * be rounded up to yield an integer point in the original tableau.
2637 static void gbr_init_shifted(struct isl_context_gbr *cgbr)
2640 struct isl_vec *cst;
2641 struct isl_basic_set *bset = isl_tab_peek_bset(cgbr->tab);
2642 unsigned dim = isl_basic_set_total_dim(bset);
2644 cst = isl_vec_alloc(cgbr->tab->mat->ctx, bset->n_ineq);
2648 for (i = 0; i < bset->n_ineq; ++i) {
2649 isl_int_set(cst->el[i], bset->ineq[i][0]);
2650 for (j = 0; j < dim; ++j) {
2651 if (!isl_int_is_neg(bset->ineq[i][1 + j]))
2653 isl_int_add(bset->ineq[i][0], bset->ineq[i][0],
2654 bset->ineq[i][1 + j]);
2658 cgbr->shifted = isl_tab_from_basic_set(bset, 0);
2660 for (i = 0; i < bset->n_ineq; ++i)
2661 isl_int_set(bset->ineq[i][0], cst->el[i]);
2666 /* Check if the shifted tableau is non-empty, and if so
2667 * use the sample point to construct an integer point
2668 * of the context tableau.
2670 static struct isl_vec *gbr_get_shifted_sample(struct isl_context_gbr *cgbr)
2672 struct isl_vec *sample;
2675 gbr_init_shifted(cgbr);
2678 if (cgbr->shifted->empty)
2679 return isl_vec_alloc(cgbr->tab->mat->ctx, 0);
2681 sample = isl_tab_get_sample_value(cgbr->shifted);
2682 sample = isl_vec_ceil(sample);
2687 static struct isl_basic_set *drop_constant_terms(struct isl_basic_set *bset)
2694 for (i = 0; i < bset->n_eq; ++i)
2695 isl_int_set_si(bset->eq[i][0], 0);
2697 for (i = 0; i < bset->n_ineq; ++i)
2698 isl_int_set_si(bset->ineq[i][0], 0);
2703 static int use_shifted(struct isl_context_gbr *cgbr)
2705 return cgbr->tab->bmap->n_eq == 0 && cgbr->tab->bmap->n_div == 0;
2708 static struct isl_vec *gbr_get_sample(struct isl_context_gbr *cgbr)
2710 struct isl_basic_set *bset;
2711 struct isl_basic_set *cone;
2713 if (isl_tab_sample_is_integer(cgbr->tab))
2714 return isl_tab_get_sample_value(cgbr->tab);
2716 if (use_shifted(cgbr)) {
2717 struct isl_vec *sample;
2719 sample = gbr_get_shifted_sample(cgbr);
2720 if (!sample || sample->size > 0)
2723 isl_vec_free(sample);
2727 bset = isl_tab_peek_bset(cgbr->tab);
2728 cgbr->cone = isl_tab_from_recession_cone(bset, 0);
2731 if (isl_tab_track_bset(cgbr->cone,
2732 isl_basic_set_copy(bset)) < 0)
2735 if (isl_tab_detect_implicit_equalities(cgbr->cone) < 0)
2738 if (cgbr->cone->n_dead == cgbr->cone->n_col) {
2739 struct isl_vec *sample;
2740 struct isl_tab_undo *snap;
2742 if (cgbr->tab->basis) {
2743 if (cgbr->tab->basis->n_col != 1 + cgbr->tab->n_var) {
2744 isl_mat_free(cgbr->tab->basis);
2745 cgbr->tab->basis = NULL;
2747 cgbr->tab->n_zero = 0;
2748 cgbr->tab->n_unbounded = 0;
2751 snap = isl_tab_snap(cgbr->tab);
2753 sample = isl_tab_sample(cgbr->tab);
2755 if (isl_tab_rollback(cgbr->tab, snap) < 0) {
2756 isl_vec_free(sample);
2763 cone = isl_basic_set_dup(isl_tab_peek_bset(cgbr->cone));
2764 cone = drop_constant_terms(cone);
2765 cone = isl_basic_set_update_from_tab(cone, cgbr->cone);
2766 cone = isl_basic_set_underlying_set(cone);
2767 cone = isl_basic_set_gauss(cone, NULL);
2769 bset = isl_basic_set_dup(isl_tab_peek_bset(cgbr->tab));
2770 bset = isl_basic_set_update_from_tab(bset, cgbr->tab);
2771 bset = isl_basic_set_underlying_set(bset);
2772 bset = isl_basic_set_gauss(bset, NULL);
2774 return isl_basic_set_sample_with_cone(bset, cone);
2777 static void check_gbr_integer_feasible(struct isl_context_gbr *cgbr)
2779 struct isl_vec *sample;
2784 if (cgbr->tab->empty)
2787 sample = gbr_get_sample(cgbr);
2791 if (sample->size == 0) {
2792 isl_vec_free(sample);
2793 if (isl_tab_mark_empty(cgbr->tab) < 0)
2798 cgbr->tab = isl_tab_add_sample(cgbr->tab, sample);
2802 isl_tab_free(cgbr->tab);
2806 static struct isl_tab *add_gbr_eq(struct isl_tab *tab, isl_int *eq)
2811 if (isl_tab_extend_cons(tab, 2) < 0)
2814 if (isl_tab_add_eq(tab, eq) < 0)
2823 static void context_gbr_add_eq(struct isl_context *context, isl_int *eq,
2824 int check, int update)
2826 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
2828 cgbr->tab = add_gbr_eq(cgbr->tab, eq);
2830 if (cgbr->cone && cgbr->cone->n_col != cgbr->cone->n_dead) {
2831 if (isl_tab_extend_cons(cgbr->cone, 2) < 0)
2833 if (isl_tab_add_eq(cgbr->cone, eq) < 0)
2838 int v = tab_has_valid_sample(cgbr->tab, eq, 1);
2842 check_gbr_integer_feasible(cgbr);
2845 cgbr->tab = check_samples(cgbr->tab, eq, 1);
2848 isl_tab_free(cgbr->tab);
2852 static void add_gbr_ineq(struct isl_context_gbr *cgbr, isl_int *ineq)
2857 if (isl_tab_extend_cons(cgbr->tab, 1) < 0)
2860 if (isl_tab_add_ineq(cgbr->tab, ineq) < 0)
2863 if (cgbr->shifted && !cgbr->shifted->empty && use_shifted(cgbr)) {
2866 dim = isl_basic_map_total_dim(cgbr->tab->bmap);
2868 if (isl_tab_extend_cons(cgbr->shifted, 1) < 0)
2871 for (i = 0; i < dim; ++i) {
2872 if (!isl_int_is_neg(ineq[1 + i]))
2874 isl_int_add(ineq[0], ineq[0], ineq[1 + i]);
2877 if (isl_tab_add_ineq(cgbr->shifted, ineq) < 0)
2880 for (i = 0; i < dim; ++i) {
2881 if (!isl_int_is_neg(ineq[1 + i]))
2883 isl_int_sub(ineq[0], ineq[0], ineq[1 + i]);
2887 if (cgbr->cone && cgbr->cone->n_col != cgbr->cone->n_dead) {
2888 if (isl_tab_extend_cons(cgbr->cone, 1) < 0)
2890 if (isl_tab_add_ineq(cgbr->cone, ineq) < 0)
2896 isl_tab_free(cgbr->tab);
2900 static void context_gbr_add_ineq(struct isl_context *context, isl_int *ineq,
2901 int check, int update)
2903 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
2905 add_gbr_ineq(cgbr, ineq);
2910 int v = tab_has_valid_sample(cgbr->tab, ineq, 0);
2914 check_gbr_integer_feasible(cgbr);
2917 cgbr->tab = check_samples(cgbr->tab, ineq, 0);
2920 isl_tab_free(cgbr->tab);
2924 static int context_gbr_add_ineq_wrap(void *user, isl_int *ineq)
2926 struct isl_context *context = (struct isl_context *)user;
2927 context_gbr_add_ineq(context, ineq, 0, 0);
2928 return context->op->is_ok(context) ? 0 : -1;
2931 static enum isl_tab_row_sign context_gbr_ineq_sign(struct isl_context *context,
2932 isl_int *ineq, int strict)
2934 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
2935 return tab_ineq_sign(cgbr->tab, ineq, strict);
2938 /* Check whether "ineq" can be added to the tableau without rendering
2941 static int context_gbr_test_ineq(struct isl_context *context, isl_int *ineq)
2943 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
2944 struct isl_tab_undo *snap;
2945 struct isl_tab_undo *shifted_snap = NULL;
2946 struct isl_tab_undo *cone_snap = NULL;
2952 if (isl_tab_extend_cons(cgbr->tab, 1) < 0)
2955 snap = isl_tab_snap(cgbr->tab);
2957 shifted_snap = isl_tab_snap(cgbr->shifted);
2959 cone_snap = isl_tab_snap(cgbr->cone);
2960 add_gbr_ineq(cgbr, ineq);
2961 check_gbr_integer_feasible(cgbr);
2964 feasible = !cgbr->tab->empty;
2965 if (isl_tab_rollback(cgbr->tab, snap) < 0)
2968 if (isl_tab_rollback(cgbr->shifted, shifted_snap))
2970 } else if (cgbr->shifted) {
2971 isl_tab_free(cgbr->shifted);
2972 cgbr->shifted = NULL;
2975 if (isl_tab_rollback(cgbr->cone, cone_snap))
2977 } else if (cgbr->cone) {
2978 isl_tab_free(cgbr->cone);
2985 /* Return the column of the last of the variables associated to
2986 * a column that has a non-zero coefficient.
2987 * This function is called in a context where only coefficients
2988 * of parameters or divs can be non-zero.
2990 static int last_non_zero_var_col(struct isl_tab *tab, isl_int *p)
2995 if (tab->n_var == 0)
2998 for (i = tab->n_var - 1; i >= 0; --i) {
2999 if (i >= tab->n_param && i < tab->n_var - tab->n_div)
3001 if (tab->var[i].is_row)
3003 col = tab->var[i].index;
3004 if (!isl_int_is_zero(p[col]))
3011 /* Look through all the recently added equalities in the context
3012 * to see if we can propagate any of them to the main tableau.
3014 * The newly added equalities in the context are encoded as pairs
3015 * of inequalities starting at inequality "first".
3017 * We tentatively add each of these equalities to the main tableau
3018 * and if this happens to result in a row with a final coefficient
3019 * that is one or negative one, we use it to kill a column
3020 * in the main tableau. Otherwise, we discard the tentatively
3023 static void propagate_equalities(struct isl_context_gbr *cgbr,
3024 struct isl_tab *tab, unsigned first)
3027 struct isl_vec *eq = NULL;
3029 eq = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_var);
3033 if (isl_tab_extend_cons(tab, (cgbr->tab->bmap->n_ineq - first)/2) < 0)
3036 isl_seq_clr(eq->el + 1 + tab->n_param,
3037 tab->n_var - tab->n_param - tab->n_div);
3038 for (i = first; i < cgbr->tab->bmap->n_ineq; i += 2) {
3041 struct isl_tab_undo *snap;
3042 snap = isl_tab_snap(tab);
3044 isl_seq_cpy(eq->el, cgbr->tab->bmap->ineq[i], 1 + tab->n_param);
3045 isl_seq_cpy(eq->el + 1 + tab->n_var - tab->n_div,
3046 cgbr->tab->bmap->ineq[i] + 1 + tab->n_param,
3049 r = isl_tab_add_row(tab, eq->el);
3052 r = tab->con[r].index;
3053 j = last_non_zero_var_col(tab, tab->mat->row[r] + 2 + tab->M);
3054 if (j < 0 || j < tab->n_dead ||
3055 !isl_int_is_one(tab->mat->row[r][0]) ||
3056 (!isl_int_is_one(tab->mat->row[r][2 + tab->M + j]) &&
3057 !isl_int_is_negone(tab->mat->row[r][2 + tab->M + j]))) {
3058 if (isl_tab_rollback(tab, snap) < 0)
3062 if (isl_tab_pivot(tab, r, j) < 0)
3064 if (isl_tab_kill_col(tab, j) < 0)
3067 if (restore_lexmin(tab) < 0)
3076 isl_tab_free(cgbr->tab);
3080 static int context_gbr_detect_equalities(struct isl_context *context,
3081 struct isl_tab *tab)
3083 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
3084 struct isl_ctx *ctx;
3087 ctx = cgbr->tab->mat->ctx;
3090 struct isl_basic_set *bset = isl_tab_peek_bset(cgbr->tab);
3091 cgbr->cone = isl_tab_from_recession_cone(bset, 0);
3094 if (isl_tab_track_bset(cgbr->cone,
3095 isl_basic_set_copy(bset)) < 0)
3098 if (isl_tab_detect_implicit_equalities(cgbr->cone) < 0)
3101 n_ineq = cgbr->tab->bmap->n_ineq;
3102 cgbr->tab = isl_tab_detect_equalities(cgbr->tab, cgbr->cone);
3103 if (cgbr->tab && cgbr->tab->bmap->n_ineq > n_ineq)
3104 propagate_equalities(cgbr, tab, n_ineq);
3108 isl_tab_free(cgbr->tab);
3113 static int context_gbr_get_div(struct isl_context *context, struct isl_tab *tab,
3114 struct isl_vec *div)
3116 return get_div(tab, context, div);
3119 static int context_gbr_add_div(struct isl_context *context, struct isl_vec *div)
3121 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
3125 if (isl_tab_extend_cons(cgbr->cone, 3) < 0)
3127 if (isl_tab_extend_vars(cgbr->cone, 1) < 0)
3129 if (isl_tab_allocate_var(cgbr->cone) <0)
3132 cgbr->cone->bmap = isl_basic_map_extend_space(cgbr->cone->bmap,
3133 isl_basic_map_get_space(cgbr->cone->bmap), 1, 0, 2);
3134 k = isl_basic_map_alloc_div(cgbr->cone->bmap);
3137 isl_seq_cpy(cgbr->cone->bmap->div[k], div->el, div->size);
3138 if (isl_tab_push(cgbr->cone, isl_tab_undo_bmap_div) < 0)
3141 return context_tab_add_div(cgbr->tab, div,
3142 context_gbr_add_ineq_wrap, context);
3145 static int context_gbr_best_split(struct isl_context *context,
3146 struct isl_tab *tab)
3148 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
3149 struct isl_tab_undo *snap;
3152 snap = isl_tab_snap(cgbr->tab);
3153 r = best_split(tab, cgbr->tab);
3155 if (r >= 0 && isl_tab_rollback(cgbr->tab, snap) < 0)
3161 static int context_gbr_is_empty(struct isl_context *context)
3163 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
3166 return cgbr->tab->empty;
3169 struct isl_gbr_tab_undo {
3170 struct isl_tab_undo *tab_snap;
3171 struct isl_tab_undo *shifted_snap;
3172 struct isl_tab_undo *cone_snap;
3175 static void *context_gbr_save(struct isl_context *context)
3177 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
3178 struct isl_gbr_tab_undo *snap;
3180 snap = isl_alloc_type(cgbr->tab->mat->ctx, struct isl_gbr_tab_undo);
3184 snap->tab_snap = isl_tab_snap(cgbr->tab);
3185 if (isl_tab_save_samples(cgbr->tab) < 0)
3189 snap->shifted_snap = isl_tab_snap(cgbr->shifted);
3191 snap->shifted_snap = NULL;
3194 snap->cone_snap = isl_tab_snap(cgbr->cone);
3196 snap->cone_snap = NULL;
3204 static void context_gbr_restore(struct isl_context *context, void *save)
3206 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
3207 struct isl_gbr_tab_undo *snap = (struct isl_gbr_tab_undo *)save;
3210 if (isl_tab_rollback(cgbr->tab, snap->tab_snap) < 0) {
3211 isl_tab_free(cgbr->tab);
3215 if (snap->shifted_snap) {
3216 if (isl_tab_rollback(cgbr->shifted, snap->shifted_snap) < 0)
3218 } else if (cgbr->shifted) {
3219 isl_tab_free(cgbr->shifted);
3220 cgbr->shifted = NULL;
3223 if (snap->cone_snap) {
3224 if (isl_tab_rollback(cgbr->cone, snap->cone_snap) < 0)
3226 } else if (cgbr->cone) {
3227 isl_tab_free(cgbr->cone);
3236 isl_tab_free(cgbr->tab);
3240 static void context_gbr_discard(void *save)
3242 struct isl_gbr_tab_undo *snap = (struct isl_gbr_tab_undo *)save;
3246 static int context_gbr_is_ok(struct isl_context *context)
3248 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
3252 static void context_gbr_invalidate(struct isl_context *context)
3254 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
3255 isl_tab_free(cgbr->tab);
3259 static void context_gbr_free(struct isl_context *context)
3261 struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context;
3262 isl_tab_free(cgbr->tab);
3263 isl_tab_free(cgbr->shifted);
3264 isl_tab_free(cgbr->cone);
3268 struct isl_context_op isl_context_gbr_op = {
3269 context_gbr_detect_nonnegative_parameters,
3270 context_gbr_peek_basic_set,
3271 context_gbr_peek_tab,
3273 context_gbr_add_ineq,
3274 context_gbr_ineq_sign,
3275 context_gbr_test_ineq,
3276 context_gbr_get_div,
3277 context_gbr_add_div,
3278 context_gbr_detect_equalities,
3279 context_gbr_best_split,
3280 context_gbr_is_empty,
3283 context_gbr_restore,
3284 context_gbr_discard,
3285 context_gbr_invalidate,
3289 static struct isl_context *isl_context_gbr_alloc(struct isl_basic_set *dom)
3291 struct isl_context_gbr *cgbr;
3296 cgbr = isl_calloc_type(dom->ctx, struct isl_context_gbr);
3300 cgbr->context.op = &isl_context_gbr_op;
3302 cgbr->shifted = NULL;
3304 cgbr->tab = isl_tab_from_basic_set(dom, 1);
3305 cgbr->tab = isl_tab_init_samples(cgbr->tab);
3308 check_gbr_integer_feasible(cgbr);
3310 return &cgbr->context;
3312 cgbr->context.op->free(&cgbr->context);
3316 static struct isl_context *isl_context_alloc(struct isl_basic_set *dom)
3321 if (dom->ctx->opt->context == ISL_CONTEXT_LEXMIN)
3322 return isl_context_lex_alloc(dom);
3324 return isl_context_gbr_alloc(dom);
3327 /* Construct an isl_sol_map structure for accumulating the solution.
3328 * If track_empty is set, then we also keep track of the parts
3329 * of the context where there is no solution.
3330 * If max is set, then we are solving a maximization, rather than
3331 * a minimization problem, which means that the variables in the
3332 * tableau have value "M - x" rather than "M + x".
3334 static struct isl_sol *sol_map_init(struct isl_basic_map *bmap,
3335 struct isl_basic_set *dom, int track_empty, int max)
3337 struct isl_sol_map *sol_map = NULL;
3342 sol_map = isl_calloc_type(bmap->ctx, struct isl_sol_map);
3346 sol_map->sol.rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
3347 sol_map->sol.dec_level.callback.run = &sol_dec_level_wrap;
3348 sol_map->sol.dec_level.sol = &sol_map->sol;
3349 sol_map->sol.max = max;
3350 sol_map->sol.n_out = isl_basic_map_dim(bmap, isl_dim_out);
3351 sol_map->sol.add = &sol_map_add_wrap;
3352 sol_map->sol.add_empty = track_empty ? &sol_map_add_empty_wrap : NULL;
3353 sol_map->sol.free = &sol_map_free_wrap;
3354 sol_map->map = isl_map_alloc_space(isl_basic_map_get_space(bmap), 1,
3359 sol_map->sol.context = isl_context_alloc(dom);
3360 if (!sol_map->sol.context)
3364 sol_map->empty = isl_set_alloc_space(isl_basic_set_get_space(dom),
3365 1, ISL_SET_DISJOINT);
3366 if (!sol_map->empty)
3370 isl_basic_set_free(dom);
3371 return &sol_map->sol;
3373 isl_basic_set_free(dom);
3374 sol_map_free(sol_map);
3378 /* Check whether all coefficients of (non-parameter) variables
3379 * are non-positive, meaning that no pivots can be performed on the row.
3381 static int is_critical(struct isl_tab *tab, int row)
3384 unsigned off = 2 + tab->M;
3386 for (j = tab->n_dead; j < tab->n_col; ++j) {
3387 if (tab->col_var[j] >= 0 &&
3388 (tab->col_var[j] < tab->n_param ||
3389 tab->col_var[j] >= tab->n_var - tab->n_div))
3392 if (isl_int_is_pos(tab->mat->row[row][off + j]))
3399 /* Check whether the inequality represented by vec is strict over the integers,
3400 * i.e., there are no integer values satisfying the constraint with
3401 * equality. This happens if the gcd of the coefficients is not a divisor
3402 * of the constant term. If so, scale the constraint down by the gcd
3403 * of the coefficients.
3405 static int is_strict(struct isl_vec *vec)
3411 isl_seq_gcd(vec->el + 1, vec->size - 1, &gcd);
3412 if (!isl_int_is_one(gcd)) {
3413 strict = !isl_int_is_divisible_by(vec->el[0], gcd);
3414 isl_int_fdiv_q(vec->el[0], vec->el[0], gcd);
3415 isl_seq_scale_down(vec->el + 1, vec->el + 1, gcd, vec->size-1);
3422 /* Determine the sign of the given row of the main tableau.
3423 * The result is one of
3424 * isl_tab_row_pos: always non-negative; no pivot needed
3425 * isl_tab_row_neg: always non-positive; pivot
3426 * isl_tab_row_any: can be both positive and negative; split
3428 * We first handle some simple cases
3429 * - the row sign may be known already
3430 * - the row may be obviously non-negative
3431 * - the parametric constant may be equal to that of another row
3432 * for which we know the sign. This sign will be either "pos" or
3433 * "any". If it had been "neg" then we would have pivoted before.
3435 * If none of these cases hold, we check the value of the row for each
3436 * of the currently active samples. Based on the signs of these values
3437 * we make an initial determination of the sign of the row.
3439 * all zero -> unk(nown)
3440 * all non-negative -> pos
3441 * all non-positive -> neg
3442 * both negative and positive -> all
3444 * If we end up with "all", we are done.
3445 * Otherwise, we perform a check for positive and/or negative
3446 * values as follows.
3448 * samples neg unk pos
3454 * There is no special sign for "zero", because we can usually treat zero
3455 * as either non-negative or non-positive, whatever works out best.
3456 * However, if the row is "critical", meaning that pivoting is impossible
3457 * then we don't want to limp zero with the non-positive case, because
3458 * then we we would lose the solution for those values of the parameters
3459 * where the value of the row is zero. Instead, we treat 0 as non-negative
3460 * ensuring a split if the row can attain both zero and negative values.
3461 * The same happens when the original constraint was one that could not
3462 * be satisfied with equality by any integer values of the parameters.
3463 * In this case, we normalize the constraint, but then a value of zero
3464 * for the normalized constraint is actually a positive value for the
3465 * original constraint, so again we need to treat zero as non-negative.
3466 * In both these cases, we have the following decision tree instead:
3468 * all non-negative -> pos
3469 * all negative -> neg
3470 * both negative and non-negative -> all
3478 static enum isl_tab_row_sign row_sign(struct isl_tab *tab,
3479 struct isl_sol *sol, int row)
3481 struct isl_vec *ineq = NULL;
3482 enum isl_tab_row_sign res = isl_tab_row_unknown;
3487 if (tab->row_sign[row] != isl_tab_row_unknown)
3488 return tab->row_sign[row];
3489 if (is_obviously_nonneg(tab, row))
3490 return isl_tab_row_pos;
3491 for (row2 = tab->n_redundant; row2 < tab->n_row; ++row2) {
3492 if (tab->row_sign[row2] == isl_tab_row_unknown)
3494 if (identical_parameter_line(tab, row, row2))
3495 return tab->row_sign[row2];
3498 critical = is_critical(tab, row);
3500 ineq = get_row_parameter_ineq(tab, row);
3504 strict = is_strict(ineq);
3506 res = sol->context->op->ineq_sign(sol->context, ineq->el,
3507 critical || strict);
3509 if (res == isl_tab_row_unknown || res == isl_tab_row_pos) {
3510 /* test for negative values */
3512 isl_seq_neg(ineq->el, ineq->el, ineq->size);
3513 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
3515 feasible = sol->context->op->test_ineq(sol->context, ineq->el);
3519 res = isl_tab_row_pos;
3521 res = (res == isl_tab_row_unknown) ? isl_tab_row_neg
3523 if (res == isl_tab_row_neg) {
3524 isl_seq_neg(ineq->el, ineq->el, ineq->size);
3525 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
3529 if (res == isl_tab_row_neg) {
3530 /* test for positive values */
3532 if (!critical && !strict)
3533 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
3535 feasible = sol->context->op->test_ineq(sol->context, ineq->el);
3539 res = isl_tab_row_any;
3546 return isl_tab_row_unknown;
3549 static void find_solutions(struct isl_sol *sol, struct isl_tab *tab);
3551 /* Find solutions for values of the parameters that satisfy the given
3554 * We currently take a snapshot of the context tableau that is reset
3555 * when we return from this function, while we make a copy of the main
3556 * tableau, leaving the original main tableau untouched.
3557 * These are fairly arbitrary choices. Making a copy also of the context
3558 * tableau would obviate the need to undo any changes made to it later,
3559 * while taking a snapshot of the main tableau could reduce memory usage.
3560 * If we were to switch to taking a snapshot of the main tableau,
3561 * we would have to keep in mind that we need to save the row signs
3562 * and that we need to do this before saving the current basis
3563 * such that the basis has been restore before we restore the row signs.
3565 static void find_in_pos(struct isl_sol *sol, struct isl_tab *tab, isl_int *ineq)
3571 saved = sol->context->op->save(sol->context);
3573 tab = isl_tab_dup(tab);
3577 sol->context->op->add_ineq(sol->context, ineq, 0, 1);
3579 find_solutions(sol, tab);
3582 sol->context->op->restore(sol->context, saved);
3584 sol->context->op->discard(saved);
3590 /* Record the absence of solutions for those values of the parameters
3591 * that do not satisfy the given inequality with equality.
3593 static void no_sol_in_strict(struct isl_sol *sol,
3594 struct isl_tab *tab, struct isl_vec *ineq)
3599 if (!sol->context || sol->error)
3601 saved = sol->context->op->save(sol->context);
3603 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
3605 sol->context->op->add_ineq(sol->context, ineq->el, 1, 0);
3614 isl_int_add_ui(ineq->el[0], ineq->el[0], 1);
3616 sol->context->op->restore(sol->context, saved);
3622 /* Compute the lexicographic minimum of the set represented by the main
3623 * tableau "tab" within the context "sol->context_tab".
3624 * On entry the sample value of the main tableau is lexicographically
3625 * less than or equal to this lexicographic minimum.
3626 * Pivots are performed until a feasible point is found, which is then
3627 * necessarily equal to the minimum, or until the tableau is found to
3628 * be infeasible. Some pivots may need to be performed for only some
3629 * feasible values of the context tableau. If so, the context tableau
3630 * is split into a part where the pivot is needed and a part where it is not.
3632 * Whenever we enter the main loop, the main tableau is such that no
3633 * "obvious" pivots need to be performed on it, where "obvious" means
3634 * that the given row can be seen to be negative without looking at
3635 * the context tableau. In particular, for non-parametric problems,
3636 * no pivots need to be performed on the main tableau.
3637 * The caller of find_solutions is responsible for making this property
3638 * hold prior to the first iteration of the loop, while restore_lexmin
3639 * is called before every other iteration.
3641 * Inside the main loop, we first examine the signs of the rows of
3642 * the main tableau within the context of the context tableau.
3643 * If we find a row that is always non-positive for all values of
3644 * the parameters satisfying the context tableau and negative for at
3645 * least one value of the parameters, we perform the appropriate pivot
3646 * and start over. An exception is the case where no pivot can be
3647 * performed on the row. In this case, we require that the sign of
3648 * the row is negative for all values of the parameters (rather than just
3649 * non-positive). This special case is handled inside row_sign, which
3650 * will say that the row can have any sign if it determines that it can
3651 * attain both negative and zero values.
3653 * If we can't find a row that always requires a pivot, but we can find
3654 * one or more rows that require a pivot for some values of the parameters
3655 * (i.e., the row can attain both positive and negative signs), then we split
3656 * the context tableau into two parts, one where we force the sign to be
3657 * non-negative and one where we force is to be negative.
3658 * The non-negative part is handled by a recursive call (through find_in_pos).
3659 * Upon returning from this call, we continue with the negative part and
3660 * perform the required pivot.
3662 * If no such rows can be found, all rows are non-negative and we have
3663 * found a (rational) feasible point. If we only wanted a rational point
3665 * Otherwise, we check if all values of the sample point of the tableau
3666 * are integral for the variables. If so, we have found the minimal
3667 * integral point and we are done.
3668 * If the sample point is not integral, then we need to make a distinction
3669 * based on whether the constant term is non-integral or the coefficients
3670 * of the parameters. Furthermore, in order to decide how to handle
3671 * the non-integrality, we also need to know whether the coefficients
3672 * of the other columns in the tableau are integral. This leads
3673 * to the following table. The first two rows do not correspond
3674 * to a non-integral sample point and are only mentioned for completeness.
3676 * constant parameters other
3679 * int int rat | -> no problem
3681 * rat int int -> fail
3683 * rat int rat -> cut
3686 * rat rat rat | -> parametric cut
3689 * rat rat int | -> split context
3691 * If the parametric constant is completely integral, then there is nothing
3692 * to be done. If the constant term is non-integral, but all the other
3693 * coefficient are integral, then there is nothing that can be done
3694 * and the tableau has no integral solution.
3695 * If, on the other hand, one or more of the other columns have rational
3696 * coefficients, but the parameter coefficients are all integral, then
3697 * we can perform a regular (non-parametric) cut.
3698 * Finally, if there is any parameter coefficient that is non-integral,
3699 * then we need to involve the context tableau. There are two cases here.
3700 * If at least one other column has a rational coefficient, then we
3701 * can perform a parametric cut in the main tableau by adding a new
3702 * integer division in the context tableau.
3703 * If all other columns have integral coefficients, then we need to
3704 * enforce that the rational combination of parameters (c + \sum a_i y_i)/m
3705 * is always integral. We do this by introducing an integer division
3706 * q = floor((c + \sum a_i y_i)/m) and stipulating that its argument should
3707 * always be integral in the context tableau, i.e., m q = c + \sum a_i y_i.
3708 * Since q is expressed in the tableau as
3709 * c + \sum a_i y_i - m q >= 0
3710 * -c - \sum a_i y_i + m q + m - 1 >= 0
3711 * it is sufficient to add the inequality
3712 * -c - \sum a_i y_i + m q >= 0
3713 * In the part of the context where this inequality does not hold, the
3714 * main tableau is marked as being empty.
3716 static void find_solutions(struct isl_sol *sol, struct isl_tab *tab)
3718 struct isl_context *context;
3721 if (!tab || sol->error)
3724 context = sol->context;
3728 if (context->op->is_empty(context))
3731 for (r = 0; r >= 0 && tab && !tab->empty; r = restore_lexmin(tab)) {
3734 enum isl_tab_row_sign sgn;
3738 for (row = tab->n_redundant; row < tab->n_row; ++row) {
3739 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
3741 sgn = row_sign(tab, sol, row);
3744 tab->row_sign[row] = sgn;
3745 if (sgn == isl_tab_row_any)
3747 if (sgn == isl_tab_row_any && split == -1)
3749 if (sgn == isl_tab_row_neg)
3752 if (row < tab->n_row)
3755 struct isl_vec *ineq;
3757 split = context->op->best_split(context, tab);
3760 ineq = get_row_parameter_ineq(tab, split);
3764 for (row = tab->n_redundant; row < tab->n_row; ++row) {
3765 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
3767 if (tab->row_sign[row] == isl_tab_row_any)
3768 tab->row_sign[row] = isl_tab_row_unknown;
3770 tab->row_sign[split] = isl_tab_row_pos;
3772 find_in_pos(sol, tab, ineq->el);
3773 tab->row_sign[split] = isl_tab_row_neg;
3775 isl_seq_neg(ineq->el, ineq->el, ineq->size);
3776 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
3778 context->op->add_ineq(context, ineq->el, 0, 1);
3786 row = first_non_integer_row(tab, &flags);
3789 if (ISL_FL_ISSET(flags, I_PAR)) {
3790 if (ISL_FL_ISSET(flags, I_VAR)) {
3791 if (isl_tab_mark_empty(tab) < 0)
3795 row = add_cut(tab, row);
3796 } else if (ISL_FL_ISSET(flags, I_VAR)) {
3797 struct isl_vec *div;
3798 struct isl_vec *ineq;
3800 div = get_row_split_div(tab, row);
3803 d = context->op->get_div(context, tab, div);
3807 ineq = ineq_for_div(context->op->peek_basic_set(context), d);
3811 no_sol_in_strict(sol, tab, ineq);
3812 isl_seq_neg(ineq->el, ineq->el, ineq->size);
3813 context->op->add_ineq(context, ineq->el, 1, 1);
3815 if (sol->error || !context->op->is_ok(context))
3817 tab = set_row_cst_to_div(tab, row, d);
3818 if (context->op->is_empty(context))
3821 row = add_parametric_cut(tab, row, context);
3836 /* Compute the lexicographic minimum of the set represented by the main
3837 * tableau "tab" within the context "sol->context_tab".
3839 * As a preprocessing step, we first transfer all the purely parametric
3840 * equalities from the main tableau to the context tableau, i.e.,
3841 * parameters that have been pivoted to a row.
3842 * These equalities are ignored by the main algorithm, because the
3843 * corresponding rows may not be marked as being non-negative.
3844 * In parts of the context where the added equality does not hold,
3845 * the main tableau is marked as being empty.
3847 static void find_solutions_main(struct isl_sol *sol, struct isl_tab *tab)
3856 for (row = tab->n_redundant; row < tab->n_row; ++row) {
3860 if (tab->row_var[row] < 0)
3862 if (tab->row_var[row] >= tab->n_param &&
3863 tab->row_var[row] < tab->n_var - tab->n_div)
3865 if (tab->row_var[row] < tab->n_param)
3866 p = tab->row_var[row];
3868 p = tab->row_var[row]
3869 + tab->n_param - (tab->n_var - tab->n_div);
3871 eq = isl_vec_alloc(tab->mat->ctx, 1+tab->n_param+tab->n_div);
3874 get_row_parameter_line(tab, row, eq->el);
3875 isl_int_neg(eq->el[1 + p], tab->mat->row[row][0]);
3876 eq = isl_vec_normalize(eq);
3879 no_sol_in_strict(sol, tab, eq);
3881 isl_seq_neg(eq->el, eq->el, eq->size);
3883 no_sol_in_strict(sol, tab, eq);
3884 isl_seq_neg(eq->el, eq->el, eq->size);
3886 sol->context->op->add_eq(sol->context, eq->el, 1, 1);
3890 if (isl_tab_mark_redundant(tab, row) < 0)
3893 if (sol->context->op->is_empty(sol->context))
3896 row = tab->n_redundant - 1;
3899 find_solutions(sol, tab);
3910 /* Check if integer division "div" of "dom" also occurs in "bmap".
3911 * If so, return its position within the divs.
3912 * If not, return -1.
3914 static int find_context_div(struct isl_basic_map *bmap,
3915 struct isl_basic_set *dom, unsigned div)
3918 unsigned b_dim = isl_space_dim(bmap->dim, isl_dim_all);
3919 unsigned d_dim = isl_space_dim(dom->dim, isl_dim_all);
3921 if (isl_int_is_zero(dom->div[div][0]))
3923 if (isl_seq_first_non_zero(dom->div[div] + 2 + d_dim, dom->n_div) != -1)
3926 for (i = 0; i < bmap->n_div; ++i) {
3927 if (isl_int_is_zero(bmap->div[i][0]))
3929 if (isl_seq_first_non_zero(bmap->div[i] + 2 + d_dim,
3930 (b_dim - d_dim) + bmap->n_div) != -1)
3932 if (isl_seq_eq(bmap->div[i], dom->div[div], 2 + d_dim))
3938 /* The correspondence between the variables in the main tableau,
3939 * the context tableau, and the input map and domain is as follows.
3940 * The first n_param and the last n_div variables of the main tableau
3941 * form the variables of the context tableau.
3942 * In the basic map, these n_param variables correspond to the
3943 * parameters and the input dimensions. In the domain, they correspond
3944 * to the parameters and the set dimensions.
3945 * The n_div variables correspond to the integer divisions in the domain.
3946 * To ensure that everything lines up, we may need to copy some of the
3947 * integer divisions of the domain to the map. These have to be placed
3948 * in the same order as those in the context and they have to be placed
3949 * after any other integer divisions that the map may have.
3950 * This function performs the required reordering.
3952 static struct isl_basic_map *align_context_divs(struct isl_basic_map *bmap,
3953 struct isl_basic_set *dom)
3959 for (i = 0; i < dom->n_div; ++i)
3960 if (find_context_div(bmap, dom, i) != -1)
3962 other = bmap->n_div - common;
3963 if (dom->n_div - common > 0) {
3964 bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim),
3965 dom->n_div - common, 0, 0);
3969 for (i = 0; i < dom->n_div; ++i) {
3970 int pos = find_context_div(bmap, dom, i);
3972 pos = isl_basic_map_alloc_div(bmap);
3975 isl_int_set_si(bmap->div[pos][0], 0);
3977 if (pos != other + i)
3978 isl_basic_map_swap_div(bmap, pos, other + i);
3982 isl_basic_map_free(bmap);
3986 /* Base case of isl_tab_basic_map_partial_lexopt, after removing
3987 * some obvious symmetries.
3989 * We make sure the divs in the domain are properly ordered,
3990 * because they will be added one by one in the given order
3991 * during the construction of the solution map.
3993 static struct isl_sol *basic_map_partial_lexopt_base(
3994 __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
3995 __isl_give isl_set **empty, int max,
3996 struct isl_sol *(*init)(__isl_keep isl_basic_map *bmap,
3997 __isl_take isl_basic_set *dom, int track_empty, int max))
3999 struct isl_tab *tab;
4000 struct isl_sol *sol = NULL;
4001 struct isl_context *context;
4004 dom = isl_basic_set_order_divs(dom);
4005 bmap = align_context_divs(bmap, dom);
4007 sol = init(bmap, dom, !!empty, max);
4011 context = sol->context;
4012 if (isl_basic_set_plain_is_empty(context->op->peek_basic_set(context)))
4014 else if (isl_basic_map_plain_is_empty(bmap)) {
4017 isl_basic_set_copy(context->op->peek_basic_set(context)));
4019 tab = tab_for_lexmin(bmap,
4020 context->op->peek_basic_set(context), 1, max);
4021 tab = context->op->detect_nonnegative_parameters(context, tab);
4022 find_solutions_main(sol, tab);
4027 isl_basic_map_free(bmap);
4031 isl_basic_map_free(bmap);
4035 /* Base case of isl_tab_basic_map_partial_lexopt, after removing
4036 * some obvious symmetries.
4038 * We call basic_map_partial_lexopt_base and extract the results.
4040 static __isl_give isl_map *basic_map_partial_lexopt_base_map(
4041 __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
4042 __isl_give isl_set **empty, int max)
4044 isl_map *result = NULL;
4045 struct isl_sol *sol;
4046 struct isl_sol_map *sol_map;
4048 sol = basic_map_partial_lexopt_base(bmap, dom, empty, max,
4052 sol_map = (struct isl_sol_map *) sol;
4054 result = isl_map_copy(sol_map->map);
4056 *empty = isl_set_copy(sol_map->empty);
4057 sol_free(&sol_map->sol);
4061 /* Structure used during detection of parallel constraints.
4062 * n_in: number of "input" variables: isl_dim_param + isl_dim_in
4063 * n_out: number of "output" variables: isl_dim_out + isl_dim_div
4064 * val: the coefficients of the output variables
4066 struct isl_constraint_equal_info {
4067 isl_basic_map *bmap;
4073 /* Check whether the coefficients of the output variables
4074 * of the constraint in "entry" are equal to info->val.
4076 static int constraint_equal(const void *entry, const void *val)
4078 isl_int **row = (isl_int **)entry;
4079 const struct isl_constraint_equal_info *info = val;
4081 return isl_seq_eq((*row) + 1 + info->n_in, info->val, info->n_out);
4084 /* Check whether "bmap" has a pair of constraints that have
4085 * the same coefficients for the output variables.
4086 * Note that the coefficients of the existentially quantified
4087 * variables need to be zero since the existentially quantified
4088 * of the result are usually not the same as those of the input.
4089 * the isl_dim_out and isl_dim_div dimensions.
4090 * If so, return 1 and return the row indices of the two constraints
4091 * in *first and *second.
4093 static int parallel_constraints(__isl_keep isl_basic_map *bmap,
4094 int *first, int *second)
4097 isl_ctx *ctx = isl_basic_map_get_ctx(bmap);
4098 struct isl_hash_table *table = NULL;
4099 struct isl_hash_table_entry *entry;
4100 struct isl_constraint_equal_info info;
4104 ctx = isl_basic_map_get_ctx(bmap);
4105 table = isl_hash_table_alloc(ctx, bmap->n_ineq);
4109 info.n_in = isl_basic_map_dim(bmap, isl_dim_param) +
4110 isl_basic_map_dim(bmap, isl_dim_in);
4112 n_out = isl_basic_map_dim(bmap, isl_dim_out);
4113 n_div = isl_basic_map_dim(bmap, isl_dim_div);
4114 info.n_out = n_out + n_div;
4115 for (i = 0; i < bmap->n_ineq; ++i) {
4118 info.val = bmap->ineq[i] + 1 + info.n_in;
4119 if (isl_seq_first_non_zero(info.val, n_out) < 0)
4121 if (isl_seq_first_non_zero(info.val + n_out, n_div) >= 0)
4123 hash = isl_seq_get_hash(info.val, info.n_out);
4124 entry = isl_hash_table_find(ctx, table, hash,
4125 constraint_equal, &info, 1);
4130 entry->data = &bmap->ineq[i];
4133 if (i < bmap->n_ineq) {
4134 *first = ((isl_int **)entry->data) - bmap->ineq;
4138 isl_hash_table_free(ctx, table);
4140 return i < bmap->n_ineq;
4142 isl_hash_table_free(ctx, table);
4146 /* Given a set of upper bounds in "var", add constraints to "bset"
4147 * that make the i-th bound smallest.
4149 * In particular, if there are n bounds b_i, then add the constraints
4151 * b_i <= b_j for j > i
4152 * b_i < b_j for j < i
4154 static __isl_give isl_basic_set *select_minimum(__isl_take isl_basic_set *bset,
4155 __isl_keep isl_mat *var, int i)
4160 ctx = isl_mat_get_ctx(var);
4162 for (j = 0; j < var->n_row; ++j) {
4165 k = isl_basic_set_alloc_inequality(bset);
4168 isl_seq_combine(bset->ineq[k], ctx->one, var->row[j],
4169 ctx->negone, var->row[i], var->n_col);
4170 isl_int_set_si(bset->ineq[k][var->n_col], 0);
4172 isl_int_sub_ui(bset->ineq[k][0], bset->ineq[k][0], 1);
4175 bset = isl_basic_set_finalize(bset);
4179 isl_basic_set_free(bset);
4183 /* Given a set of upper bounds on the last "input" variable m,
4184 * construct a set that assigns the minimal upper bound to m, i.e.,
4185 * construct a set that divides the space into cells where one
4186 * of the upper bounds is smaller than all the others and assign
4187 * this upper bound to m.
4189 * In particular, if there are n bounds b_i, then the result
4190 * consists of n basic sets, each one of the form
4193 * b_i <= b_j for j > i
4194 * b_i < b_j for j < i
4196 static __isl_give isl_set *set_minimum(__isl_take isl_space *dim,
4197 __isl_take isl_mat *var)
4200 isl_basic_set *bset = NULL;
4202 isl_set *set = NULL;
4207 ctx = isl_space_get_ctx(dim);
4208 set = isl_set_alloc_space(isl_space_copy(dim),
4209 var->n_row, ISL_SET_DISJOINT);
4211 for (i = 0; i < var->n_row; ++i) {
4212 bset = isl_basic_set_alloc_space(isl_space_copy(dim), 0,
4214 k = isl_basic_set_alloc_equality(bset);
4217 isl_seq_cpy(bset->eq[k], var->row[i], var->n_col);
4218 isl_int_set_si(bset->eq[k][var->n_col], -1);
4219 bset = select_minimum(bset, var, i);
4220 set = isl_set_add_basic_set(set, bset);
4223 isl_space_free(dim);
4227 isl_basic_set_free(bset);
4229 isl_space_free(dim);
4234 /* Given that the last input variable of "bmap" represents the minimum
4235 * of the bounds in "cst", check whether we need to split the domain
4236 * based on which bound attains the minimum.
4238 * A split is needed when the minimum appears in an integer division
4239 * or in an equality. Otherwise, it is only needed if it appears in
4240 * an upper bound that is different from the upper bounds on which it
4243 static int need_split_basic_map(__isl_keep isl_basic_map *bmap,
4244 __isl_keep isl_mat *cst)
4250 pos = cst->n_col - 1;
4251 total = isl_basic_map_dim(bmap, isl_dim_all);
4253 for (i = 0; i < bmap->n_div; ++i)
4254 if (!isl_int_is_zero(bmap->div[i][2 + pos]))
4257 for (i = 0; i < bmap->n_eq; ++i)
4258 if (!isl_int_is_zero(bmap->eq[i][1 + pos]))
4261 for (i = 0; i < bmap->n_ineq; ++i) {
4262 if (isl_int_is_nonneg(bmap->ineq[i][1 + pos]))
4264 if (!isl_int_is_negone(bmap->ineq[i][1 + pos]))
4266 if (isl_seq_first_non_zero(bmap->ineq[i] + 1 + pos + 1,
4267 total - pos - 1) >= 0)
4270 for (j = 0; j < cst->n_row; ++j)
4271 if (isl_seq_eq(bmap->ineq[i], cst->row[j], cst->n_col))
4273 if (j >= cst->n_row)
4280 /* Given that the last set variable of "bset" represents the minimum
4281 * of the bounds in "cst", check whether we need to split the domain
4282 * based on which bound attains the minimum.
4284 * We simply call need_split_basic_map here. This is safe because
4285 * the position of the minimum is computed from "cst" and not
4288 static int need_split_basic_set(__isl_keep isl_basic_set *bset,
4289 __isl_keep isl_mat *cst)
4291 return need_split_basic_map((isl_basic_map *)bset, cst);
4294 /* Given that the last set variable of "set" represents the minimum
4295 * of the bounds in "cst", check whether we need to split the domain
4296 * based on which bound attains the minimum.
4298 static int need_split_set(__isl_keep isl_set *set, __isl_keep isl_mat *cst)
4302 for (i = 0; i < set->n; ++i)
4303 if (need_split_basic_set(set->p[i], cst))
4309 /* Given a set of which the last set variable is the minimum
4310 * of the bounds in "cst", split each basic set in the set
4311 * in pieces where one of the bounds is (strictly) smaller than the others.
4312 * This subdivision is given in "min_expr".
4313 * The variable is subsequently projected out.
4315 * We only do the split when it is needed.
4316 * For example if the last input variable m = min(a,b) and the only
4317 * constraints in the given basic set are lower bounds on m,
4318 * i.e., l <= m = min(a,b), then we can simply project out m
4319 * to obtain l <= a and l <= b, without having to split on whether
4320 * m is equal to a or b.
4322 static __isl_give isl_set *split(__isl_take isl_set *empty,
4323 __isl_take isl_set *min_expr, __isl_take isl_mat *cst)
4330 if (!empty || !min_expr || !cst)
4333 n_in = isl_set_dim(empty, isl_dim_set);
4334 dim = isl_set_get_space(empty);
4335 dim = isl_space_drop_dims(dim, isl_dim_set, n_in - 1, 1);
4336 res = isl_set_empty(dim);
4338 for (i = 0; i < empty->n; ++i) {
4341 set = isl_set_from_basic_set(isl_basic_set_copy(empty->p[i]));
4342 if (need_split_basic_set(empty->p[i], cst))
4343 set = isl_set_intersect(set, isl_set_copy(min_expr));
4344 set = isl_set_remove_dims(set, isl_dim_set, n_in - 1, 1);
4346 res = isl_set_union_disjoint(res, set);
4349 isl_set_free(empty);
4350 isl_set_free(min_expr);
4354 isl_set_free(empty);
4355 isl_set_free(min_expr);
4360 /* Given a map of which the last input variable is the minimum
4361 * of the bounds in "cst", split each basic set in the set
4362 * in pieces where one of the bounds is (strictly) smaller than the others.
4363 * This subdivision is given in "min_expr".
4364 * The variable is subsequently projected out.
4366 * The implementation is essentially the same as that of "split".
4368 static __isl_give isl_map *split_domain(__isl_take isl_map *opt,
4369 __isl_take isl_set *min_expr, __isl_take isl_mat *cst)
4376 if (!opt || !min_expr || !cst)
4379 n_in = isl_map_dim(opt, isl_dim_in);
4380 dim = isl_map_get_space(opt);
4381 dim = isl_space_drop_dims(dim, isl_dim_in, n_in - 1, 1);
4382 res = isl_map_empty(dim);
4384 for (i = 0; i < opt->n; ++i) {
4387 map = isl_map_from_basic_map(isl_basic_map_copy(opt->p[i]));
4388 if (need_split_basic_map(opt->p[i], cst))
4389 map = isl_map_intersect_domain(map,
4390 isl_set_copy(min_expr));
4391 map = isl_map_remove_dims(map, isl_dim_in, n_in - 1, 1);
4393 res = isl_map_union_disjoint(res, map);
4397 isl_set_free(min_expr);
4402 isl_set_free(min_expr);
4407 static __isl_give isl_map *basic_map_partial_lexopt(
4408 __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
4409 __isl_give isl_set **empty, int max);
4414 isl_pw_multi_aff *pma;
4417 /* This function is called from basic_map_partial_lexopt_symm.
4418 * The last variable of "bmap" and "dom" corresponds to the minimum
4419 * of the bounds in "cst". "map_space" is the space of the original
4420 * input relation (of basic_map_partial_lexopt_symm) and "set_space"
4421 * is the space of the original domain.
4423 * We recursively call basic_map_partial_lexopt and then plug in
4424 * the definition of the minimum in the result.
4426 static __isl_give union isl_lex_res basic_map_partial_lexopt_symm_map_core(
4427 __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
4428 __isl_give isl_set **empty, int max, __isl_take isl_mat *cst,
4429 __isl_take isl_space *map_space, __isl_take isl_space *set_space)
4433 union isl_lex_res res;
4435 min_expr = set_minimum(isl_basic_set_get_space(dom), isl_mat_copy(cst));
4437 opt = basic_map_partial_lexopt(bmap, dom, empty, max);
4440 *empty = split(*empty,
4441 isl_set_copy(min_expr), isl_mat_copy(cst));
4442 *empty = isl_set_reset_space(*empty, set_space);
4445 opt = split_domain(opt, min_expr, cst);
4446 opt = isl_map_reset_space(opt, map_space);
4452 /* Given a basic map with at least two parallel constraints (as found
4453 * by the function parallel_constraints), first look for more constraints
4454 * parallel to the two constraint and replace the found list of parallel
4455 * constraints by a single constraint with as "input" part the minimum
4456 * of the input parts of the list of constraints. Then, recursively call
4457 * basic_map_partial_lexopt (possibly finding more parallel constraints)
4458 * and plug in the definition of the minimum in the result.
4460 * More specifically, given a set of constraints
4464 * Replace this set by a single constraint
4468 * with u a new parameter with constraints
4472 * Any solution to the new system is also a solution for the original system
4475 * a x >= -u >= -b_i(p)
4477 * Moreover, m = min_i(b_i(p)) satisfies the constraints on u and can
4478 * therefore be plugged into the solution.
4480 static union isl_lex_res basic_map_partial_lexopt_symm(
4481 __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
4482 __isl_give isl_set **empty, int max, int first, int second,
4483 __isl_give union isl_lex_res (*core)(__isl_take isl_basic_map *bmap,
4484 __isl_take isl_basic_set *dom,
4485 __isl_give isl_set **empty,
4486 int max, __isl_take isl_mat *cst,
4487 __isl_take isl_space *map_space,
4488 __isl_take isl_space *set_space))
4492 unsigned n_in, n_out, n_div;
4494 isl_vec *var = NULL;
4495 isl_mat *cst = NULL;
4496 isl_space *map_space, *set_space;
4497 union isl_lex_res res;
4499 map_space = isl_basic_map_get_space(bmap);
4500 set_space = empty ? isl_basic_set_get_space(dom) : NULL;
4502 n_in = isl_basic_map_dim(bmap, isl_dim_param) +
4503 isl_basic_map_dim(bmap, isl_dim_in);
4504 n_out = isl_basic_map_dim(bmap, isl_dim_all) - n_in;
4506 ctx = isl_basic_map_get_ctx(bmap);
4507 list = isl_alloc_array(ctx, int, bmap->n_ineq);
4508 var = isl_vec_alloc(ctx, n_out);
4514 isl_seq_cpy(var->el, bmap->ineq[first] + 1 + n_in, n_out);
4515 for (i = second + 1, n = 2; i < bmap->n_ineq; ++i) {
4516 if (isl_seq_eq(var->el, bmap->ineq[i] + 1 + n_in, n_out))
4520 cst = isl_mat_alloc(ctx, n, 1 + n_in);
4524 for (i = 0; i < n; ++i)
4525 isl_seq_cpy(cst->row[i], bmap->ineq[list[i]], 1 + n_in);
4527 bmap = isl_basic_map_cow(bmap);
4530 for (i = n - 1; i >= 0; --i)
4531 if (isl_basic_map_drop_inequality(bmap, list[i]) < 0)
4534 bmap = isl_basic_map_add(bmap, isl_dim_in, 1);
4535 bmap = isl_basic_map_extend_constraints(bmap, 0, 1);
4536 k = isl_basic_map_alloc_inequality(bmap);
4539 isl_seq_clr(bmap->ineq[k], 1 + n_in);
4540 isl_int_set_si(bmap->ineq[k][1 + n_in], 1);
4541 isl_seq_cpy(bmap->ineq[k] + 1 + n_in + 1, var->el, n_out);
4542 bmap = isl_basic_map_finalize(bmap);
4544 n_div = isl_basic_set_dim(dom, isl_dim_div);
4545 dom = isl_basic_set_add_dims(dom, isl_dim_set, 1);
4546 dom = isl_basic_set_extend_constraints(dom, 0, n);
4547 for (i = 0; i < n; ++i) {
4548 k = isl_basic_set_alloc_inequality(dom);
4551 isl_seq_cpy(dom->ineq[k], cst->row[i], 1 + n_in);
4552 isl_int_set_si(dom->ineq[k][1 + n_in], -1);
4553 isl_seq_clr(dom->ineq[k] + 1 + n_in + 1, n_div);
4559 return core(bmap, dom, empty, max, cst, map_space, set_space);
4561 isl_space_free(map_space);
4562 isl_space_free(set_space);
4566 isl_basic_set_free(dom);
4567 isl_basic_map_free(bmap);
4572 static __isl_give isl_map *basic_map_partial_lexopt_symm_map(
4573 __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
4574 __isl_give isl_set **empty, int max, int first, int second)
4576 return basic_map_partial_lexopt_symm(bmap, dom, empty, max,
4577 first, second, &basic_map_partial_lexopt_symm_map_core).map;
4580 /* Recursive part of isl_tab_basic_map_partial_lexopt, after detecting
4581 * equalities and removing redundant constraints.
4583 * We first check if there are any parallel constraints (left).
4584 * If not, we are in the base case.
4585 * If there are parallel constraints, we replace them by a single
4586 * constraint in basic_map_partial_lexopt_symm and then call
4587 * this function recursively to look for more parallel constraints.
4589 static __isl_give isl_map *basic_map_partial_lexopt(
4590 __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
4591 __isl_give isl_set **empty, int max)
4599 if (bmap->ctx->opt->pip_symmetry)
4600 par = parallel_constraints(bmap, &first, &second);
4604 return basic_map_partial_lexopt_base_map(bmap, dom, empty, max);
4606 return basic_map_partial_lexopt_symm_map(bmap, dom, empty, max,
4609 isl_basic_set_free(dom);
4610 isl_basic_map_free(bmap);
4614 /* Compute the lexicographic minimum (or maximum if "max" is set)
4615 * of "bmap" over the domain "dom" and return the result as a map.
4616 * If "empty" is not NULL, then *empty is assigned a set that
4617 * contains those parts of the domain where there is no solution.
4618 * If "bmap" is marked as rational (ISL_BASIC_MAP_RATIONAL),
4619 * then we compute the rational optimum. Otherwise, we compute
4620 * the integral optimum.
4622 * We perform some preprocessing. As the PILP solver does not
4623 * handle implicit equalities very well, we first make sure all
4624 * the equalities are explicitly available.
4626 * We also add context constraints to the basic map and remove
4627 * redundant constraints. This is only needed because of the
4628 * way we handle simple symmetries. In particular, we currently look
4629 * for symmetries on the constraints, before we set up the main tableau.
4630 * It is then no good to look for symmetries on possibly redundant constraints.
4632 struct isl_map *isl_tab_basic_map_partial_lexopt(
4633 struct isl_basic_map *bmap, struct isl_basic_set *dom,
4634 struct isl_set **empty, int max)
4641 isl_assert(bmap->ctx,
4642 isl_basic_map_compatible_domain(bmap, dom), goto error);
4644 if (isl_basic_set_dim(dom, isl_dim_all) == 0)
4645 return basic_map_partial_lexopt(bmap, dom, empty, max);
4647 bmap = isl_basic_map_intersect_domain(bmap, isl_basic_set_copy(dom));
4648 bmap = isl_basic_map_detect_equalities(bmap);
4649 bmap = isl_basic_map_remove_redundancies(bmap);
4651 return basic_map_partial_lexopt(bmap, dom, empty, max);
4653 isl_basic_set_free(dom);
4654 isl_basic_map_free(bmap);
4658 struct isl_sol_for {
4660 int (*fn)(__isl_take isl_basic_set *dom,
4661 __isl_take isl_aff_list *list, void *user);
4665 static void sol_for_free(struct isl_sol_for *sol_for)
4667 if (sol_for->sol.context)
4668 sol_for->sol.context->op->free(sol_for->sol.context);
4672 static void sol_for_free_wrap(struct isl_sol *sol)
4674 sol_for_free((struct isl_sol_for *)sol);
4677 /* Add the solution identified by the tableau and the context tableau.
4679 * See documentation of sol_add for more details.
4681 * Instead of constructing a basic map, this function calls a user
4682 * defined function with the current context as a basic set and
4683 * a list of affine expressions representing the relation between
4684 * the input and output. The space over which the affine expressions
4685 * are defined is the same as that of the domain. The number of
4686 * affine expressions in the list is equal to the number of output variables.
4688 static void sol_for_add(struct isl_sol_for *sol,
4689 struct isl_basic_set *dom, struct isl_mat *M)
4693 isl_local_space *ls;
4697 if (sol->sol.error || !dom || !M)
4700 ctx = isl_basic_set_get_ctx(dom);
4701 ls = isl_basic_set_get_local_space(dom);
4702 list = isl_aff_list_alloc(ctx, M->n_row - 1);
4703 for (i = 1; i < M->n_row; ++i) {
4704 aff = isl_aff_alloc(isl_local_space_copy(ls));
4706 isl_int_set(aff->v->el[0], M->row[0][0]);
4707 isl_seq_cpy(aff->v->el + 1, M->row[i], M->n_col);
4709 aff = isl_aff_normalize(aff);
4710 list = isl_aff_list_add(list, aff);
4712 isl_local_space_free(ls);
4714 dom = isl_basic_set_finalize(dom);
4716 if (sol->fn(isl_basic_set_copy(dom), list, sol->user) < 0)
4719 isl_basic_set_free(dom);
4723 isl_basic_set_free(dom);
4728 static void sol_for_add_wrap(struct isl_sol *sol,
4729 struct isl_basic_set *dom, struct isl_mat *M)
4731 sol_for_add((struct isl_sol_for *)sol, dom, M);
4734 static struct isl_sol_for *sol_for_init(struct isl_basic_map *bmap, int max,
4735 int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_aff_list *list,
4739 struct isl_sol_for *sol_for = NULL;
4741 struct isl_basic_set *dom = NULL;
4743 sol_for = isl_calloc_type(bmap->ctx, struct isl_sol_for);
4747 dom_dim = isl_space_domain(isl_space_copy(bmap->dim));
4748 dom = isl_basic_set_universe(dom_dim);
4750 sol_for->sol.rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
4751 sol_for->sol.dec_level.callback.run = &sol_dec_level_wrap;
4752 sol_for->sol.dec_level.sol = &sol_for->sol;
4754 sol_for->user = user;
4755 sol_for->sol.max = max;
4756 sol_for->sol.n_out = isl_basic_map_dim(bmap, isl_dim_out);
4757 sol_for->sol.add = &sol_for_add_wrap;
4758 sol_for->sol.add_empty = NULL;
4759 sol_for->sol.free = &sol_for_free_wrap;
4761 sol_for->sol.context = isl_context_alloc(dom);
4762 if (!sol_for->sol.context)
4765 isl_basic_set_free(dom);
4768 isl_basic_set_free(dom);
4769 sol_for_free(sol_for);
4773 static void sol_for_find_solutions(struct isl_sol_for *sol_for,
4774 struct isl_tab *tab)
4776 find_solutions_main(&sol_for->sol, tab);
4779 int isl_basic_map_foreach_lexopt(__isl_keep isl_basic_map *bmap, int max,
4780 int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_aff_list *list,
4784 struct isl_sol_for *sol_for = NULL;
4786 bmap = isl_basic_map_copy(bmap);
4787 bmap = isl_basic_map_detect_equalities(bmap);
4791 sol_for = sol_for_init(bmap, max, fn, user);
4795 if (isl_basic_map_plain_is_empty(bmap))
4798 struct isl_tab *tab;
4799 struct isl_context *context = sol_for->sol.context;
4800 tab = tab_for_lexmin(bmap,
4801 context->op->peek_basic_set(context), 1, max);
4802 tab = context->op->detect_nonnegative_parameters(context, tab);
4803 sol_for_find_solutions(sol_for, tab);
4804 if (sol_for->sol.error)
4808 sol_free(&sol_for->sol);
4809 isl_basic_map_free(bmap);
4812 sol_free(&sol_for->sol);
4813 isl_basic_map_free(bmap);
4817 int isl_basic_set_foreach_lexopt(__isl_keep isl_basic_set *bset, int max,
4818 int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_aff_list *list,
4822 return isl_basic_map_foreach_lexopt(bset, max, fn, user);
4825 /* Check if the given sequence of len variables starting at pos
4826 * represents a trivial (i.e., zero) solution.
4827 * The variables are assumed to be non-negative and to come in pairs,
4828 * with each pair representing a variable of unrestricted sign.
4829 * The solution is trivial if each such pair in the sequence consists
4830 * of two identical values, meaning that the variable being represented
4833 static int region_is_trivial(struct isl_tab *tab, int pos, int len)
4840 for (i = 0; i < len; i += 2) {
4844 neg_row = tab->var[pos + i].is_row ?
4845 tab->var[pos + i].index : -1;
4846 pos_row = tab->var[pos + i + 1].is_row ?
4847 tab->var[pos + i + 1].index : -1;
4850 isl_int_is_zero(tab->mat->row[neg_row][1])) &&
4852 isl_int_is_zero(tab->mat->row[pos_row][1])))
4855 if (neg_row < 0 || pos_row < 0)
4857 if (isl_int_ne(tab->mat->row[neg_row][1],
4858 tab->mat->row[pos_row][1]))
4865 /* Return the index of the first trivial region or -1 if all regions
4868 static int first_trivial_region(struct isl_tab *tab,
4869 int n_region, struct isl_region *region)
4873 for (i = 0; i < n_region; ++i) {
4874 if (region_is_trivial(tab, region[i].pos, region[i].len))
4881 /* Check if the solution is optimal, i.e., whether the first
4882 * n_op entries are zero.
4884 static int is_optimal(__isl_keep isl_vec *sol, int n_op)
4888 for (i = 0; i < n_op; ++i)
4889 if (!isl_int_is_zero(sol->el[1 + i]))
4894 /* Add constraints to "tab" that ensure that any solution is significantly
4895 * better that that represented by "sol". That is, find the first
4896 * relevant (within first n_op) non-zero coefficient and force it (along
4897 * with all previous coefficients) to be zero.
4898 * If the solution is already optimal (all relevant coefficients are zero),
4899 * then just mark the table as empty.
4901 static int force_better_solution(struct isl_tab *tab,
4902 __isl_keep isl_vec *sol, int n_op)
4911 for (i = 0; i < n_op; ++i)
4912 if (!isl_int_is_zero(sol->el[1 + i]))
4916 if (isl_tab_mark_empty(tab) < 0)
4921 ctx = isl_vec_get_ctx(sol);
4922 v = isl_vec_alloc(ctx, 1 + tab->n_var);
4926 for (; i >= 0; --i) {
4928 isl_int_set_si(v->el[1 + i], -1);
4929 if (add_lexmin_eq(tab, v->el) < 0)
4940 struct isl_trivial {
4944 struct isl_tab_undo *snap;
4947 /* Return the lexicographically smallest non-trivial solution of the
4948 * given ILP problem.
4950 * All variables are assumed to be non-negative.
4952 * n_op is the number of initial coordinates to optimize.
4953 * That is, once a solution has been found, we will only continue looking
4954 * for solution that result in significantly better values for those
4955 * initial coordinates. That is, we only continue looking for solutions
4956 * that increase the number of initial zeros in this sequence.
4958 * A solution is non-trivial, if it is non-trivial on each of the
4959 * specified regions. Each region represents a sequence of pairs
4960 * of variables. A solution is non-trivial on such a region if
4961 * at least one of these pairs consists of different values, i.e.,
4962 * such that the non-negative variable represented by the pair is non-zero.
4964 * Whenever a conflict is encountered, all constraints involved are
4965 * reported to the caller through a call to "conflict".
4967 * We perform a simple branch-and-bound backtracking search.
4968 * Each level in the search represents initially trivial region that is forced
4969 * to be non-trivial.
4970 * At each level we consider n cases, where n is the length of the region.
4971 * In terms of the n/2 variables of unrestricted signs being encoded by
4972 * the region, we consider the cases
4975 * x_0 = 0 and x_1 >= 1
4976 * x_0 = 0 and x_1 <= -1
4977 * x_0 = 0 and x_1 = 0 and x_2 >= 1
4978 * x_0 = 0 and x_1 = 0 and x_2 <= -1
4980 * The cases are considered in this order, assuming that each pair
4981 * x_i_a x_i_b represents the value x_i_b - x_i_a.
4982 * That is, x_0 >= 1 is enforced by adding the constraint
4983 * x_0_b - x_0_a >= 1
4985 __isl_give isl_vec *isl_tab_basic_set_non_trivial_lexmin(
4986 __isl_take isl_basic_set *bset, int n_op, int n_region,
4987 struct isl_region *region,
4988 int (*conflict)(int con, void *user), void *user)
4994 isl_vec *sol = NULL;
4995 struct isl_tab *tab;
4996 struct isl_trivial *triv = NULL;
5002 ctx = isl_basic_set_get_ctx(bset);
5003 sol = isl_vec_alloc(ctx, 0);
5005 tab = tab_for_lexmin(bset, NULL, 0, 0);
5008 tab->conflict = conflict;
5009 tab->conflict_user = user;
5011 v = isl_vec_alloc(ctx, 1 + tab->n_var);
5012 triv = isl_calloc_array(ctx, struct isl_trivial, n_region);
5019 while (level >= 0) {
5023 tab = cut_to_integer_lexmin(tab, CUT_ONE);
5028 r = first_trivial_region(tab, n_region, region);
5030 for (i = 0; i < level; ++i)
5033 sol = isl_tab_get_sample_value(tab);
5036 if (is_optimal(sol, n_op))
5040 if (level >= n_region)
5041 isl_die(ctx, isl_error_internal,
5042 "nesting level too deep", goto error);
5043 if (isl_tab_extend_cons(tab,
5044 2 * region[r].len + 2 * n_op) < 0)
5046 triv[level].region = r;
5047 triv[level].side = 0;
5050 r = triv[level].region;
5051 side = triv[level].side;
5052 base = 2 * (side/2);
5054 if (side >= region[r].len) {
5059 if (isl_tab_rollback(tab, triv[level].snap) < 0)
5064 if (triv[level].update) {
5065 if (force_better_solution(tab, sol, n_op) < 0)
5067 triv[level].update = 0;
5070 if (side == base && base >= 2) {
5071 for (j = base - 2; j < base; ++j) {
5073 isl_int_set_si(v->el[1 + region[r].pos + j], 1);
5074 if (add_lexmin_eq(tab, v->el) < 0)
5079 triv[level].snap = isl_tab_snap(tab);
5080 if (isl_tab_push_basis(tab) < 0)
5084 isl_int_set_si(v->el[0], -1);
5085 isl_int_set_si(v->el[1 + region[r].pos + side], -1);
5086 isl_int_set_si(v->el[1 + region[r].pos + (side ^ 1)], 1);
5087 tab = add_lexmin_ineq(tab, v->el);
5097 isl_basic_set_free(bset);
5104 isl_basic_set_free(bset);
5109 /* Return the lexicographically smallest rational point in "bset",
5110 * assuming that all variables are non-negative.
5111 * If "bset" is empty, then return a zero-length vector.
5113 __isl_give isl_vec *isl_tab_basic_set_non_neg_lexmin(
5114 __isl_take isl_basic_set *bset)
5116 struct isl_tab *tab;
5117 isl_ctx *ctx = isl_basic_set_get_ctx(bset);
5123 tab = tab_for_lexmin(bset, NULL, 0, 0);
5127 sol = isl_vec_alloc(ctx, 0);
5129 sol = isl_tab_get_sample_value(tab);
5131 isl_basic_set_free(bset);
5135 isl_basic_set_free(bset);
5139 struct isl_sol_pma {
5141 isl_pw_multi_aff *pma;
5145 static void sol_pma_free(struct isl_sol_pma *sol_pma)
5149 if (sol_pma->sol.context)
5150 sol_pma->sol.context->op->free(sol_pma->sol.context);
5151 isl_pw_multi_aff_free(sol_pma->pma);
5152 isl_set_free(sol_pma->empty);
5156 /* This function is called for parts of the context where there is
5157 * no solution, with "bset" corresponding to the context tableau.
5158 * Simply add the basic set to the set "empty".
5160 static void sol_pma_add_empty(struct isl_sol_pma *sol,
5161 __isl_take isl_basic_set *bset)
5165 isl_assert(bset->ctx, sol->empty, goto error);
5167 sol->empty = isl_set_grow(sol->empty, 1);
5168 bset = isl_basic_set_simplify(bset);
5169 bset = isl_basic_set_finalize(bset);
5170 sol->empty = isl_set_add_basic_set(sol->empty, bset);
5175 isl_basic_set_free(bset);
5179 /* Given a basic map "dom" that represents the context and an affine
5180 * matrix "M" that maps the dimensions of the context to the
5181 * output variables, construct an isl_pw_multi_aff with a single
5182 * cell corresponding to "dom" and affine expressions copied from "M".
5184 static void sol_pma_add(struct isl_sol_pma *sol,
5185 __isl_take isl_basic_set *dom, __isl_take isl_mat *M)
5188 isl_local_space *ls;
5190 isl_multi_aff *maff;
5191 isl_pw_multi_aff *pma;
5193 maff = isl_multi_aff_alloc(isl_pw_multi_aff_get_space(sol->pma));
5194 ls = isl_basic_set_get_local_space(dom);
5195 for (i = 1; i < M->n_row; ++i) {
5196 aff = isl_aff_alloc(isl_local_space_copy(ls));
5198 isl_int_set(aff->v->el[0], M->row[0][0]);
5199 isl_seq_cpy(aff->v->el + 1, M->row[i], M->n_col);
5201 aff = isl_aff_normalize(aff);
5202 maff = isl_multi_aff_set_aff(maff, i - 1, aff);
5204 isl_local_space_free(ls);
5206 dom = isl_basic_set_simplify(dom);
5207 dom = isl_basic_set_finalize(dom);
5208 pma = isl_pw_multi_aff_alloc(isl_set_from_basic_set(dom), maff);
5209 sol->pma = isl_pw_multi_aff_add_disjoint(sol->pma, pma);
5214 static void sol_pma_free_wrap(struct isl_sol *sol)
5216 sol_pma_free((struct isl_sol_pma *)sol);
5219 static void sol_pma_add_empty_wrap(struct isl_sol *sol,
5220 __isl_take isl_basic_set *bset)
5222 sol_pma_add_empty((struct isl_sol_pma *)sol, bset);
5225 static void sol_pma_add_wrap(struct isl_sol *sol,
5226 __isl_take isl_basic_set *dom, __isl_take isl_mat *M)
5228 sol_pma_add((struct isl_sol_pma *)sol, dom, M);
5231 /* Construct an isl_sol_pma structure for accumulating the solution.
5232 * If track_empty is set, then we also keep track of the parts
5233 * of the context where there is no solution.
5234 * If max is set, then we are solving a maximization, rather than
5235 * a minimization problem, which means that the variables in the
5236 * tableau have value "M - x" rather than "M + x".
5238 static struct isl_sol *sol_pma_init(__isl_keep isl_basic_map *bmap,
5239 __isl_take isl_basic_set *dom, int track_empty, int max)
5241 struct isl_sol_pma *sol_pma = NULL;
5246 sol_pma = isl_calloc_type(bmap->ctx, struct isl_sol_pma);
5250 sol_pma->sol.rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
5251 sol_pma->sol.dec_level.callback.run = &sol_dec_level_wrap;
5252 sol_pma->sol.dec_level.sol = &sol_pma->sol;
5253 sol_pma->sol.max = max;
5254 sol_pma->sol.n_out = isl_basic_map_dim(bmap, isl_dim_out);
5255 sol_pma->sol.add = &sol_pma_add_wrap;
5256 sol_pma->sol.add_empty = track_empty ? &sol_pma_add_empty_wrap : NULL;
5257 sol_pma->sol.free = &sol_pma_free_wrap;
5258 sol_pma->pma = isl_pw_multi_aff_empty(isl_basic_map_get_space(bmap));
5262 sol_pma->sol.context = isl_context_alloc(dom);
5263 if (!sol_pma->sol.context)
5267 sol_pma->empty = isl_set_alloc_space(isl_basic_set_get_space(dom),
5268 1, ISL_SET_DISJOINT);
5269 if (!sol_pma->empty)
5273 isl_basic_set_free(dom);
5274 return &sol_pma->sol;
5276 isl_basic_set_free(dom);
5277 sol_pma_free(sol_pma);
5281 /* Base case of isl_tab_basic_map_partial_lexopt, after removing
5282 * some obvious symmetries.
5284 * We call basic_map_partial_lexopt_base and extract the results.
5286 static __isl_give isl_pw_multi_aff *basic_map_partial_lexopt_base_pma(
5287 __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
5288 __isl_give isl_set **empty, int max)
5290 isl_pw_multi_aff *result = NULL;
5291 struct isl_sol *sol;
5292 struct isl_sol_pma *sol_pma;
5294 sol = basic_map_partial_lexopt_base(bmap, dom, empty, max,
5298 sol_pma = (struct isl_sol_pma *) sol;
5300 result = isl_pw_multi_aff_copy(sol_pma->pma);
5302 *empty = isl_set_copy(sol_pma->empty);
5303 sol_free(&sol_pma->sol);
5307 /* Given that the last input variable of "maff" represents the minimum
5308 * of some bounds, check whether we need to plug in the expression
5311 * In particular, check if the last input variable appears in any
5312 * of the expressions in "maff".
5314 static int need_substitution(__isl_keep isl_multi_aff *maff)
5319 pos = isl_multi_aff_dim(maff, isl_dim_in) - 1;
5321 for (i = 0; i < maff->n; ++i)
5322 if (isl_aff_involves_dims(maff->p[i], isl_dim_in, pos, 1))
5328 /* Given a set of upper bounds on the last "input" variable m,
5329 * construct a piecewise affine expression that selects
5330 * the minimal upper bound to m, i.e.,
5331 * divide the space into cells where one
5332 * of the upper bounds is smaller than all the others and select
5333 * this upper bound on that cell.
5335 * In particular, if there are n bounds b_i, then the result
5336 * consists of n cell, each one of the form
5338 * b_i <= b_j for j > i
5339 * b_i < b_j for j < i
5341 * The affine expression on this cell is
5345 static __isl_give isl_pw_aff *set_minimum_pa(__isl_take isl_space *space,
5346 __isl_take isl_mat *var)
5349 isl_aff *aff = NULL;
5350 isl_basic_set *bset = NULL;
5352 isl_pw_aff *paff = NULL;
5353 isl_space *pw_space;
5354 isl_local_space *ls = NULL;
5359 ctx = isl_space_get_ctx(space);
5360 ls = isl_local_space_from_space(isl_space_copy(space));
5361 pw_space = isl_space_copy(space);
5362 pw_space = isl_space_from_domain(pw_space);
5363 pw_space = isl_space_add_dims(pw_space, isl_dim_out, 1);
5364 paff = isl_pw_aff_alloc_size(pw_space, var->n_row);
5366 for (i = 0; i < var->n_row; ++i) {
5369 aff = isl_aff_alloc(isl_local_space_copy(ls));
5370 bset = isl_basic_set_alloc_space(isl_space_copy(space), 0,
5374 isl_int_set_si(aff->v->el[0], 1);
5375 isl_seq_cpy(aff->v->el + 1, var->row[i], var->n_col);
5376 isl_int_set_si(aff->v->el[1 + var->n_col], 0);
5377 bset = select_minimum(bset, var, i);
5378 paff_i = isl_pw_aff_alloc(isl_set_from_basic_set(bset), aff);
5379 paff = isl_pw_aff_add_disjoint(paff, paff_i);
5382 isl_local_space_free(ls);
5383 isl_space_free(space);
5388 isl_basic_set_free(bset);
5389 isl_pw_aff_free(paff);
5390 isl_local_space_free(ls);
5391 isl_space_free(space);
5396 /* Given a piecewise multi-affine expression of which the last input variable
5397 * is the minimum of the bounds in "cst", plug in the value of the minimum.
5398 * This minimum expression is given in "min_expr_pa".
5399 * The set "min_expr" contains the same information, but in the form of a set.
5400 * The variable is subsequently projected out.
5402 * The implementation is similar to those of "split" and "split_domain".
5403 * If the variable appears in a given expression, then minimum expression
5404 * is plugged in. Otherwise, if the variable appears in the constraints
5405 * and a split is required, then the domain is split. Otherwise, no split
5408 static __isl_give isl_pw_multi_aff *split_domain_pma(
5409 __isl_take isl_pw_multi_aff *opt, __isl_take isl_pw_aff *min_expr_pa,
5410 __isl_take isl_set *min_expr, __isl_take isl_mat *cst)
5415 isl_pw_multi_aff *res;
5417 if (!opt || !min_expr || !cst)
5420 n_in = isl_pw_multi_aff_dim(opt, isl_dim_in);
5421 space = isl_pw_multi_aff_get_space(opt);
5422 space = isl_space_drop_dims(space, isl_dim_in, n_in - 1, 1);
5423 res = isl_pw_multi_aff_empty(space);
5425 for (i = 0; i < opt->n; ++i) {
5426 isl_pw_multi_aff *pma;
5428 pma = isl_pw_multi_aff_alloc(isl_set_copy(opt->p[i].set),
5429 isl_multi_aff_copy(opt->p[i].maff));
5430 if (need_substitution(opt->p[i].maff))
5431 pma = isl_pw_multi_aff_substitute(pma,
5432 isl_dim_in, n_in - 1, min_expr_pa);
5433 else if (need_split_set(opt->p[i].set, cst))
5434 pma = isl_pw_multi_aff_intersect_domain(pma,
5435 isl_set_copy(min_expr));
5436 pma = isl_pw_multi_aff_project_out(pma,
5437 isl_dim_in, n_in - 1, 1);
5439 res = isl_pw_multi_aff_add_disjoint(res, pma);
5442 isl_pw_multi_aff_free(opt);
5443 isl_pw_aff_free(min_expr_pa);
5444 isl_set_free(min_expr);
5448 isl_pw_multi_aff_free(opt);
5449 isl_pw_aff_free(min_expr_pa);
5450 isl_set_free(min_expr);
5455 static __isl_give isl_pw_multi_aff *basic_map_partial_lexopt_pma(
5456 __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
5457 __isl_give isl_set **empty, int max);
5459 /* This function is called from basic_map_partial_lexopt_symm.
5460 * The last variable of "bmap" and "dom" corresponds to the minimum
5461 * of the bounds in "cst". "map_space" is the space of the original
5462 * input relation (of basic_map_partial_lexopt_symm) and "set_space"
5463 * is the space of the original domain.
5465 * We recursively call basic_map_partial_lexopt and then plug in
5466 * the definition of the minimum in the result.
5468 static __isl_give union isl_lex_res basic_map_partial_lexopt_symm_pma_core(
5469 __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
5470 __isl_give isl_set **empty, int max, __isl_take isl_mat *cst,
5471 __isl_take isl_space *map_space, __isl_take isl_space *set_space)
5473 isl_pw_multi_aff *opt;
5474 isl_pw_aff *min_expr_pa;
5476 union isl_lex_res res;
5478 min_expr = set_minimum(isl_basic_set_get_space(dom), isl_mat_copy(cst));
5479 min_expr_pa = set_minimum_pa(isl_basic_set_get_space(dom),
5482 opt = basic_map_partial_lexopt_pma(bmap, dom, empty, max);
5485 *empty = split(*empty,
5486 isl_set_copy(min_expr), isl_mat_copy(cst));
5487 *empty = isl_set_reset_space(*empty, set_space);
5490 opt = split_domain_pma(opt, min_expr_pa, min_expr, cst);
5491 opt = isl_pw_multi_aff_reset_space(opt, map_space);
5497 static __isl_give isl_pw_multi_aff *basic_map_partial_lexopt_symm_pma(
5498 __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
5499 __isl_give isl_set **empty, int max, int first, int second)
5501 return basic_map_partial_lexopt_symm(bmap, dom, empty, max,
5502 first, second, &basic_map_partial_lexopt_symm_pma_core).pma;
5505 /* Recursive part of isl_basic_map_partial_lexopt_pw_multi_aff, after detecting
5506 * equalities and removing redundant constraints.
5508 * We first check if there are any parallel constraints (left).
5509 * If not, we are in the base case.
5510 * If there are parallel constraints, we replace them by a single
5511 * constraint in basic_map_partial_lexopt_symm_pma and then call
5512 * this function recursively to look for more parallel constraints.
5514 static __isl_give isl_pw_multi_aff *basic_map_partial_lexopt_pma(
5515 __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
5516 __isl_give isl_set **empty, int max)
5524 if (bmap->ctx->opt->pip_symmetry)
5525 par = parallel_constraints(bmap, &first, &second);
5529 return basic_map_partial_lexopt_base_pma(bmap, dom, empty, max);
5531 return basic_map_partial_lexopt_symm_pma(bmap, dom, empty, max,
5534 isl_basic_set_free(dom);
5535 isl_basic_map_free(bmap);
5539 /* Compute the lexicographic minimum (or maximum if "max" is set)
5540 * of "bmap" over the domain "dom" and return the result as a piecewise
5541 * multi-affine expression.
5542 * If "empty" is not NULL, then *empty is assigned a set that
5543 * contains those parts of the domain where there is no solution.
5544 * If "bmap" is marked as rational (ISL_BASIC_MAP_RATIONAL),
5545 * then we compute the rational optimum. Otherwise, we compute
5546 * the integral optimum.
5548 * We perform some preprocessing. As the PILP solver does not
5549 * handle implicit equalities very well, we first make sure all
5550 * the equalities are explicitly available.
5552 * We also add context constraints to the basic map and remove
5553 * redundant constraints. This is only needed because of the
5554 * way we handle simple symmetries. In particular, we currently look
5555 * for symmetries on the constraints, before we set up the main tableau.
5556 * It is then no good to look for symmetries on possibly redundant constraints.
5558 __isl_give isl_pw_multi_aff *isl_basic_map_partial_lexopt_pw_multi_aff(
5559 __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom,
5560 __isl_give isl_set **empty, int max)
5567 isl_assert(bmap->ctx,
5568 isl_basic_map_compatible_domain(bmap, dom), goto error);
5570 if (isl_basic_set_dim(dom, isl_dim_all) == 0)
5571 return basic_map_partial_lexopt_pma(bmap, dom, empty, max);
5573 bmap = isl_basic_map_intersect_domain(bmap, isl_basic_set_copy(dom));
5574 bmap = isl_basic_map_detect_equalities(bmap);
5575 bmap = isl_basic_map_remove_redundancies(bmap);
5577 return basic_map_partial_lexopt_pma(bmap, dom, empty, max);
5579 isl_basic_set_free(dom);
5580 isl_basic_map_free(bmap);