1 #include "isl_map_private.h"
5 * The implementation of parametric integer linear programming in this file
6 * was inspired by the paper "Parametric Integer Programming" and the
7 * report "Solving systems of affine (in)equalities" by Paul Feautrier
10 * The strategy used for obtaining a feasible solution is different
11 * from the one used in isl_tab.c. In particular, in isl_tab.c,
12 * upon finding a constraint that is not yet satisfied, we pivot
13 * in a row that increases the constant term of row holding the
14 * constraint, making sure the sample solution remains feasible
15 * for all the constraints it already satisfied.
16 * Here, we always pivot in the row holding the constraint,
17 * choosing a column that induces the lexicographically smallest
18 * increment to the sample solution.
20 * By starting out from a sample value that is lexicographically
21 * smaller than any integer point in the problem space, the first
22 * feasible integer sample point we find will also be the lexicographically
23 * smallest. If all variables can be assumed to be non-negative,
24 * then the initial sample value may be chosen equal to zero.
25 * However, we will not make this assumption. Instead, we apply
26 * the "big parameter" trick. Any variable x is then not directly
27 * used in the tableau, but instead it its represented by another
28 * variable x' = M + x, where M is an arbitrarily large (positive)
29 * value. x' is therefore always non-negative, whatever the value of x.
30 * Taking as initial smaple value x' = 0 corresponds to x = -M,
31 * which is always smaller than any possible value of x.
33 * We use the big parameter trick both in the main tableau and
34 * the context tableau, each of course having its own big parameter.
35 * Before doing any real work, we check if all the parameters
36 * happen to be non-negative. If so, we drop the column corresponding
37 * to M from the initial context tableau.
40 /* isl_sol is an interface for constructing a solution to
41 * a parametric integer linear programming problem.
42 * Every time the algorithm reaches a state where a solution
43 * can be read off from the tableau (including cases where the tableau
44 * is empty), the function "add" is called on the isl_sol passed
45 * to find_solutions_main.
47 * The context tableau is owned by isl_sol and is updated incrementally.
49 * There is currently only one implementation of this interface,
50 * isl_sol_map, which simply collects the solutions in an isl_map
51 * and (optionally) the parts of the context where there is no solution
55 struct isl_tab *context_tab;
56 struct isl_sol *(*add)(struct isl_sol *sol, struct isl_tab *tab);
57 void (*free)(struct isl_sol *sol);
60 static void sol_free(struct isl_sol *sol)
70 struct isl_set *empty;
74 static void sol_map_free(struct isl_sol_map *sol_map)
76 isl_tab_free(sol_map->sol.context_tab);
77 isl_map_free(sol_map->map);
78 isl_set_free(sol_map->empty);
82 static void sol_map_free_wrap(struct isl_sol *sol)
84 sol_map_free((struct isl_sol_map *)sol);
87 static struct isl_sol_map *add_empty(struct isl_sol_map *sol)
89 struct isl_basic_set *bset;
93 sol->empty = isl_set_grow(sol->empty, 1);
94 bset = isl_basic_set_copy(sol->sol.context_tab->bset);
95 bset = isl_basic_set_simplify(bset);
96 bset = isl_basic_set_finalize(bset);
97 sol->empty = isl_set_add(sol->empty, bset);
106 /* Add the solution identified by the tableau and the context tableau.
108 * The layout of the variables is as follows.
109 * tab->n_var is equal to the total number of variables in the input
110 * map (including divs that were copied from the context)
111 * + the number of extra divs constructed
112 * Of these, the first tab->n_param and the last tab->n_div variables
113 * correspond to the variables in the context, i.e.,
114 tab->n_param + tab->n_div = context_tab->n_var
115 * tab->n_param is equal to the number of parameters and input
116 * dimensions in the input map
117 * tab->n_div is equal to the number of divs in the context
119 * If there is no solution, then the basic set corresponding to the
120 * context tableau is added to the set "empty".
122 * Otherwise, a basic map is constructed with the same parameters
123 * and divs as the context, the dimensions of the context as input
124 * dimensions and a number of output dimensions that is equal to
125 * the number of output dimensions in the input map.
126 * The divs in the input map (if any) that do not correspond to any
127 * div in the context do not appear in the solution.
128 * The algorithm will make sure that they have an integer value,
129 * but these values themselves are of no interest.
131 * The constraints and divs of the context are simply copied
132 * fron context_tab->bset.
133 * To extract the value of the output variables, it should be noted
134 * that we always use a big parameter M and so the variable stored
135 * in the tableau is not an output variable x itself, but
136 * x' = M + x (in case of minimization)
138 * x' = M - x (in case of maximization)
139 * If x' appears in a column, then its optimal value is zero,
140 * which means that the optimal value of x is an unbounded number
141 * (-M for minimization and M for maximization).
142 * We currently assume that the output dimensions in the original map
143 * are bounded, so this cannot occur.
144 * Similarly, when x' appears in a row, then the coefficient of M in that
145 * row is necessarily 1.
146 * If the row represents
147 * d x' = c + d M + e(y)
148 * then, in case of minimization, an equality
149 * c + e(y) - d x' = 0
150 * is added, and in case of maximization,
151 * c + e(y) + d x' = 0
153 static struct isl_sol_map *sol_map_add(struct isl_sol_map *sol,
157 struct isl_basic_map *bmap = NULL;
158 struct isl_tab *context_tab;
171 return add_empty(sol);
173 context_tab = sol->sol.context_tab;
175 n_out = isl_map_dim(sol->map, isl_dim_out);
176 n_eq = context_tab->bset->n_eq + n_out;
177 n_ineq = context_tab->bset->n_ineq;
178 nparam = tab->n_param;
179 total = isl_map_dim(sol->map, isl_dim_all);
180 bmap = isl_basic_map_alloc_dim(isl_map_get_dim(sol->map),
181 tab->n_div, n_eq, 2 * tab->n_div + n_ineq);
186 ISL_F_SET(bmap, ISL_BASIC_MAP_RATIONAL);
187 for (i = 0; i < context_tab->bset->n_div; ++i) {
188 int k = isl_basic_map_alloc_div(bmap);
191 isl_seq_cpy(bmap->div[k],
192 context_tab->bset->div[i], 1 + 1 + nparam);
193 isl_seq_clr(bmap->div[k] + 1 + 1 + nparam, total - nparam);
194 isl_seq_cpy(bmap->div[k] + 1 + 1 + total,
195 context_tab->bset->div[i] + 1 + 1 + nparam, i);
197 for (i = 0; i < context_tab->bset->n_eq; ++i) {
198 int k = isl_basic_map_alloc_equality(bmap);
201 isl_seq_cpy(bmap->eq[k], context_tab->bset->eq[i], 1 + nparam);
202 isl_seq_clr(bmap->eq[k] + 1 + nparam, total - nparam);
203 isl_seq_cpy(bmap->eq[k] + 1 + total,
204 context_tab->bset->eq[i] + 1 + nparam, n_div);
206 for (i = 0; i < context_tab->bset->n_ineq; ++i) {
207 int k = isl_basic_map_alloc_inequality(bmap);
210 isl_seq_cpy(bmap->ineq[k],
211 context_tab->bset->ineq[i], 1 + nparam);
212 isl_seq_clr(bmap->ineq[k] + 1 + nparam, total - nparam);
213 isl_seq_cpy(bmap->ineq[k] + 1 + total,
214 context_tab->bset->ineq[i] + 1 + nparam, n_div);
216 for (i = tab->n_param; i < total; ++i) {
217 int k = isl_basic_map_alloc_equality(bmap);
220 isl_seq_clr(bmap->eq[k] + 1, isl_basic_map_total_dim(bmap));
221 if (!tab->var[i].is_row) {
223 isl_assert(bmap->ctx, !tab->M, goto error);
224 isl_int_set_si(bmap->eq[k][0], 0);
226 isl_int_set_si(bmap->eq[k][1 + i], 1);
228 isl_int_set_si(bmap->eq[k][1 + i], -1);
231 row = tab->var[i].index;
234 isl_assert(bmap->ctx,
235 isl_int_eq(tab->mat->row[row][2],
236 tab->mat->row[row][0]),
238 isl_int_set(bmap->eq[k][0], tab->mat->row[row][1]);
239 for (j = 0; j < tab->n_param; ++j) {
241 if (tab->var[j].is_row)
243 col = tab->var[j].index;
244 isl_int_set(bmap->eq[k][1 + j],
245 tab->mat->row[row][off + col]);
247 for (j = 0; j < tab->n_div; ++j) {
249 if (tab->var[tab->n_var - tab->n_div+j].is_row)
251 col = tab->var[tab->n_var - tab->n_div+j].index;
252 isl_int_set(bmap->eq[k][1 + total + j],
253 tab->mat->row[row][off + col]);
256 isl_int_set(bmap->eq[k][1 + i],
257 tab->mat->row[row][0]);
259 isl_int_neg(bmap->eq[k][1 + i],
260 tab->mat->row[row][0]);
263 bmap = isl_basic_map_gauss(bmap, NULL);
264 bmap = isl_basic_map_normalize_constraints(bmap);
265 bmap = isl_basic_map_finalize(bmap);
266 sol->map = isl_map_grow(sol->map, 1);
267 sol->map = isl_map_add(sol->map, bmap);
272 isl_basic_map_free(bmap);
277 static struct isl_sol *sol_map_add_wrap(struct isl_sol *sol,
280 return (struct isl_sol *)sol_map_add((struct isl_sol_map *)sol, tab);
284 static struct isl_basic_set *isl_basic_set_add_ineq(struct isl_basic_set *bset,
289 bset = isl_basic_set_extend_constraints(bset, 0, 1);
292 k = isl_basic_set_alloc_inequality(bset);
295 isl_seq_cpy(bset->ineq[k], ineq, 1 + isl_basic_set_total_dim(bset));
298 isl_basic_set_free(bset);
302 static struct isl_basic_set *isl_basic_set_add_eq(struct isl_basic_set *bset,
307 bset = isl_basic_set_extend_constraints(bset, 1, 0);
310 k = isl_basic_set_alloc_equality(bset);
313 isl_seq_cpy(bset->eq[k], eq, 1 + isl_basic_set_total_dim(bset));
316 isl_basic_set_free(bset);
321 /* Store the "parametric constant" of row "row" of tableau "tab" in "line",
322 * i.e., the constant term and the coefficients of all variables that
323 * appear in the context tableau.
324 * Note that the coefficient of the big parameter M is NOT copied.
325 * The context tableau may not have a big parameter and even when it
326 * does, it is a different big parameter.
328 static void get_row_parameter_line(struct isl_tab *tab, int row, isl_int *line)
331 unsigned off = 2 + tab->M;
333 isl_int_set(line[0], tab->mat->row[row][1]);
334 for (i = 0; i < tab->n_param; ++i) {
335 if (tab->var[i].is_row)
336 isl_int_set_si(line[1 + i], 0);
338 int col = tab->var[i].index;
339 isl_int_set(line[1 + i], tab->mat->row[row][off + col]);
342 for (i = 0; i < tab->n_div; ++i) {
343 if (tab->var[tab->n_var - tab->n_div + i].is_row)
344 isl_int_set_si(line[1 + tab->n_param + i], 0);
346 int col = tab->var[tab->n_var - tab->n_div + i].index;
347 isl_int_set(line[1 + tab->n_param + i],
348 tab->mat->row[row][off + col]);
353 /* Check if rows "row1" and "row2" have identical "parametric constants",
354 * as explained above.
355 * In this case, we also insist that the coefficients of the big parameter
356 * be the same as the values of the constants will only be the same
357 * if these coefficients are also the same.
359 static int identical_parameter_line(struct isl_tab *tab, int row1, int row2)
362 unsigned off = 2 + tab->M;
364 if (isl_int_ne(tab->mat->row[row1][1], tab->mat->row[row2][1]))
367 if (tab->M && isl_int_ne(tab->mat->row[row1][2],
368 tab->mat->row[row2][2]))
371 for (i = 0; i < tab->n_param + tab->n_div; ++i) {
372 int pos = i < tab->n_param ? i :
373 tab->n_var - tab->n_div + i - tab->n_param;
376 if (tab->var[pos].is_row)
378 col = tab->var[pos].index;
379 if (isl_int_ne(tab->mat->row[row1][off + col],
380 tab->mat->row[row2][off + col]))
386 /* Return an inequality that expresses that the "parametric constant"
387 * should be non-negative.
388 * This function is only called when the coefficient of the big parameter
391 static struct isl_vec *get_row_parameter_ineq(struct isl_tab *tab, int row)
393 struct isl_vec *ineq;
395 ineq = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_param + tab->n_div);
399 get_row_parameter_line(tab, row, ineq->el);
401 ineq = isl_vec_normalize(ineq);
406 /* Return a integer division for use in a parametric cut based on the given row.
407 * In particular, let the parametric constant of the row be
411 * where y_0 = 1, but none of the y_i corresponds to the big parameter M.
412 * The div returned is equal to
414 * floor(\sum_i {-a_i} y_i) = floor((\sum_i (-a_i mod d) y_i)/d)
416 static struct isl_vec *get_row_parameter_div(struct isl_tab *tab, int row)
420 div = isl_vec_alloc(tab->mat->ctx, 1 + 1 + tab->n_param + tab->n_div);
424 isl_int_set(div->el[0], tab->mat->row[row][0]);
425 get_row_parameter_line(tab, row, div->el + 1);
426 div = isl_vec_normalize(div);
427 isl_seq_neg(div->el + 1, div->el + 1, div->size - 1);
428 isl_seq_fdiv_r(div->el + 1, div->el + 1, div->el[0], div->size - 1);
433 /* Return a integer division for use in transferring an integrality constraint
435 * In particular, let the parametric constant of the row be
439 * where y_0 = 1, but none of the y_i corresponds to the big parameter M.
440 * The the returned div is equal to
442 * floor(\sum_i {a_i} y_i) = floor((\sum_i (a_i mod d) y_i)/d)
444 static struct isl_vec *get_row_split_div(struct isl_tab *tab, int row)
448 div = isl_vec_alloc(tab->mat->ctx, 1 + 1 + tab->n_param + tab->n_div);
452 isl_int_set(div->el[0], tab->mat->row[row][0]);
453 get_row_parameter_line(tab, row, div->el + 1);
454 div = isl_vec_normalize(div);
455 isl_seq_fdiv_r(div->el + 1, div->el + 1, div->el[0], div->size - 1);
460 /* Construct and return an inequality that expresses an upper bound
462 * In particular, if the div is given by
466 * then the inequality expresses
470 static struct isl_vec *ineq_for_div(struct isl_basic_set *bset, unsigned div)
474 struct isl_vec *ineq;
476 total = isl_basic_set_total_dim(bset);
477 div_pos = 1 + total - bset->n_div + div;
479 ineq = isl_vec_alloc(bset->ctx, 1 + total);
483 isl_seq_cpy(ineq->el, bset->div[div] + 1, 1 + total);
484 isl_int_neg(ineq->el[div_pos], bset->div[div][0]);
488 /* Given a row in the tableau and a div that was created
489 * using get_row_split_div and that been constrained to equality, i.e.,
491 * d = floor(\sum_i {a_i} y_i) = \sum_i {a_i} y_i
493 * replace the expression "\sum_i {a_i} y_i" in the row by d,
494 * i.e., we subtract "\sum_i {a_i} y_i" and add 1 d.
495 * The coefficients of the non-parameters in the tableau have been
496 * verified to be integral. We can therefore simply replace coefficient b
497 * by floor(b). For the coefficients of the parameters we have
498 * floor(a_i) = a_i - {a_i}, while for the other coefficients, we have
501 static struct isl_tab *set_row_cst_to_div(struct isl_tab *tab, int row, int div)
505 unsigned off = 2 + tab->M;
507 isl_seq_fdiv_q(tab->mat->row[row] + 1, tab->mat->row[row] + 1,
508 tab->mat->row[row][0], 1 + tab->M + tab->n_col);
510 isl_int_set_si(tab->mat->row[row][0], 1);
512 isl_assert(tab->mat->ctx,
513 !tab->var[tab->n_var - tab->n_div + div].is_row, goto error);
515 col = tab->var[tab->n_var - tab->n_div + div].index;
516 isl_int_set_si(tab->mat->row[row][off + col], 1);
524 /* Check if the (parametric) constant of the given row is obviously
525 * negative, meaning that we don't need to consult the context tableau.
526 * If there is a big parameter and its coefficient is non-zero,
527 * then this coefficient determines the outcome.
528 * Otherwise, we check whether the constant is negative and
529 * all non-zero coefficients of parameters are negative and
530 * belong to non-negative parameters.
532 static int is_obviously_neg(struct isl_tab *tab, int row)
536 unsigned off = 2 + tab->M;
539 if (isl_int_is_pos(tab->mat->row[row][2]))
541 if (isl_int_is_neg(tab->mat->row[row][2]))
545 if (isl_int_is_nonneg(tab->mat->row[row][1]))
547 for (i = 0; i < tab->n_param; ++i) {
548 /* Eliminated parameter */
549 if (tab->var[i].is_row)
551 col = tab->var[i].index;
552 if (isl_int_is_zero(tab->mat->row[row][off + col]))
554 if (!tab->var[i].is_nonneg)
556 if (isl_int_is_pos(tab->mat->row[row][off + col]))
559 for (i = 0; i < tab->n_div; ++i) {
560 if (tab->var[tab->n_var - tab->n_div + i].is_row)
562 col = tab->var[tab->n_var - tab->n_div + i].index;
563 if (isl_int_is_zero(tab->mat->row[row][off + col]))
565 if (!tab->var[tab->n_var - tab->n_div + i].is_nonneg)
567 if (isl_int_is_pos(tab->mat->row[row][off + col]))
573 /* Check if the (parametric) constant of the given row is obviously
574 * non-negative, meaning that we don't need to consult the context tableau.
575 * If there is a big parameter and its coefficient is non-zero,
576 * then this coefficient determines the outcome.
577 * Otherwise, we check whether the constant is non-negative and
578 * all non-zero coefficients of parameters are positive and
579 * belong to non-negative parameters.
581 static int is_obviously_nonneg(struct isl_tab *tab, int row)
585 unsigned off = 2 + tab->M;
588 if (isl_int_is_pos(tab->mat->row[row][2]))
590 if (isl_int_is_neg(tab->mat->row[row][2]))
594 if (isl_int_is_neg(tab->mat->row[row][1]))
596 for (i = 0; i < tab->n_param; ++i) {
597 /* Eliminated parameter */
598 if (tab->var[i].is_row)
600 col = tab->var[i].index;
601 if (isl_int_is_zero(tab->mat->row[row][off + col]))
603 if (!tab->var[i].is_nonneg)
605 if (isl_int_is_neg(tab->mat->row[row][off + col]))
608 for (i = 0; i < tab->n_div; ++i) {
609 if (tab->var[tab->n_var - tab->n_div + i].is_row)
611 col = tab->var[tab->n_var - tab->n_div + i].index;
612 if (isl_int_is_zero(tab->mat->row[row][off + col]))
614 if (!tab->var[tab->n_var - tab->n_div + i].is_nonneg)
616 if (isl_int_is_neg(tab->mat->row[row][off + col]))
622 /* Given a row r and two columns, return the column that would
623 * lead to the lexicographically smallest increment in the sample
624 * solution when leaving the basis in favor of the row.
625 * Pivoting with column c will increment the sample value by a non-negative
626 * constant times a_{V,c}/a_{r,c}, with a_{V,c} the elements of column c
627 * corresponding to the non-parametric variables.
628 * If variable v appears in a column c_v, the a_{v,c} = 1 iff c = c_v,
629 * with all other entries in this virtual row equal to zero.
630 * If variable v appears in a row, then a_{v,c} is the element in column c
633 * Let v be the first variable with a_{v,c1}/a_{r,c1} != a_{v,c2}/a_{r,c2}.
634 * Then if a_{v,c1}/a_{r,c1} < a_{v,c2}/a_{r,c2}, i.e.,
635 * a_{v,c2} a_{r,c1} - a_{v,c1} a_{r,c2} > 0, c1 results in the minimal
636 * increment. Otherwise, it's c2.
638 static int lexmin_col_pair(struct isl_tab *tab,
639 int row, int col1, int col2, isl_int tmp)
644 tr = tab->mat->row[row] + 2 + tab->M;
646 for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
650 if (!tab->var[i].is_row) {
651 if (tab->var[i].index == col1)
653 if (tab->var[i].index == col2)
658 if (tab->var[i].index == row)
661 r = tab->mat->row[tab->var[i].index] + 2 + tab->M;
662 s1 = isl_int_sgn(r[col1]);
663 s2 = isl_int_sgn(r[col2]);
664 if (s1 == 0 && s2 == 0)
671 isl_int_mul(tmp, r[col2], tr[col1]);
672 isl_int_submul(tmp, r[col1], tr[col2]);
673 if (isl_int_is_pos(tmp))
675 if (isl_int_is_neg(tmp))
681 /* Given a row in the tableau, find and return the column that would
682 * result in the lexicographically smallest, but positive, increment
683 * in the sample point.
684 * If there is no such column, then return tab->n_col.
685 * If anything goes wrong, return -1.
687 static int lexmin_pivot_col(struct isl_tab *tab, int row)
690 int col = tab->n_col;
694 tr = tab->mat->row[row] + 2 + tab->M;
698 for (j = tab->n_dead; j < tab->n_col; ++j) {
699 if (tab->col_var[j] >= 0 &&
700 (tab->col_var[j] < tab->n_param ||
701 tab->col_var[j] >= tab->n_var - tab->n_div))
704 if (!isl_int_is_pos(tr[j]))
707 if (col == tab->n_col)
710 col = lexmin_col_pair(tab, row, col, j, tmp);
711 isl_assert(tab->mat->ctx, col >= 0, goto error);
721 /* Return the first known violated constraint, i.e., a non-negative
722 * contraint that currently has an either obviously negative value
723 * or a previously determined to be negative value.
725 * If any constraint has a negative coefficient for the big parameter,
726 * if any, then we return one of these first.
728 static int first_neg(struct isl_tab *tab)
733 for (row = tab->n_redundant; row < tab->n_row; ++row) {
734 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
736 if (isl_int_is_neg(tab->mat->row[row][2]))
739 for (row = tab->n_redundant; row < tab->n_row; ++row) {
740 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
743 if (tab->row_sign[row] == 0 &&
744 is_obviously_neg(tab, row))
745 tab->row_sign[row] = isl_tab_row_neg;
746 if (tab->row_sign[row] != isl_tab_row_neg)
748 } else if (!is_obviously_neg(tab, row))
755 /* Resolve all known or obviously violated constraints through pivoting.
756 * In particular, as long as we can find any violated constraint, we
757 * look for a pivoting column that would result in the lexicographicallly
758 * smallest increment in the sample point. If there is no such column
759 * then the tableau is infeasible.
761 static struct isl_tab *restore_lexmin(struct isl_tab *tab)
769 while ((row = first_neg(tab)) != -1) {
770 col = lexmin_pivot_col(tab, row);
771 if (col >= tab->n_col)
772 return isl_tab_mark_empty(tab);
775 isl_tab_pivot(tab, row, col);
783 /* Given a row that represents an equality, look for an appropriate
785 * In particular, if there are any non-zero coefficients among
786 * the non-parameter variables, then we take the last of these
787 * variables. Eliminating this variable in terms of the other
788 * variables and/or parameters does not influence the property
789 * that all column in the initial tableau are lexicographically
790 * positive. The row corresponding to the eliminated variable
791 * will only have non-zero entries below the diagonal of the
792 * initial tableau. That is, we transform
798 * If there is no such non-parameter variable, then we are dealing with
799 * pure parameter equality and we pick any parameter with coefficient 1 or -1
800 * for elimination. This will ensure that the eliminated parameter
801 * always has an integer value whenever all the other parameters are integral.
802 * If there is no such parameter then we return -1.
804 static int last_var_col_or_int_par_col(struct isl_tab *tab, int row)
806 unsigned off = 2 + tab->M;
809 for (i = tab->n_var - tab->n_div - 1; i >= 0 && i >= tab->n_param; --i) {
811 if (tab->var[i].is_row)
813 col = tab->var[i].index;
814 if (col <= tab->n_dead)
816 if (!isl_int_is_zero(tab->mat->row[row][off + col]))
819 for (i = tab->n_dead; i < tab->n_col; ++i) {
820 if (isl_int_is_one(tab->mat->row[row][off + i]))
822 if (isl_int_is_negone(tab->mat->row[row][off + i]))
828 /* Add an equality that is known to be valid to the tableau.
829 * We first check if we can eliminate a variable or a parameter.
830 * If not, we add the equality as two inequalities.
831 * In this case, the equality was a pure parameter equality and there
832 * is no need to resolve any constraint violations.
834 static struct isl_tab *add_lexmin_valid_eq(struct isl_tab *tab, isl_int *eq)
841 r = isl_tab_add_row(tab, eq);
845 r = tab->con[r].index;
846 i = last_var_col_or_int_par_col(tab, r);
848 tab->con[r].is_nonneg = 1;
849 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]);
850 isl_seq_neg(eq, eq, 1 + tab->n_var);
851 r = isl_tab_add_row(tab, eq);
854 tab->con[r].is_nonneg = 1;
855 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]);
857 isl_tab_pivot(tab, r, i);
858 isl_tab_kill_col(tab, i);
861 tab = restore_lexmin(tab);
870 /* Check if the given row is a pure constant.
872 static int is_constant(struct isl_tab *tab, int row)
874 unsigned off = 2 + tab->M;
876 return isl_seq_first_non_zero(tab->mat->row[row] + off + tab->n_dead,
877 tab->n_col - tab->n_dead) == -1;
880 /* Add an equality that may or may not be valid to the tableau.
881 * If the resulting row is a pure constant, then it must be zero.
882 * Otherwise, the resulting tableau is empty.
884 * If the row is not a pure constant, then we add two inequalities,
885 * each time checking that they can be satisfied.
886 * In the end we try to use one of the two constraints to eliminate
889 static struct isl_tab *add_lexmin_eq(struct isl_tab *tab, isl_int *eq)
898 tab->bset = isl_basic_set_add_eq(tab->bset, eq);
899 isl_tab_push(tab, isl_tab_undo_bset_eq);
903 r1 = isl_tab_add_row(tab, eq);
906 tab->con[r1].is_nonneg = 1;
907 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r1]);
909 row = tab->con[r1].index;
910 if (is_constant(tab, row)) {
911 if (!isl_int_is_zero(tab->mat->row[row][1]) ||
912 (tab->M && !isl_int_is_zero(tab->mat->row[row][2])))
913 return isl_tab_mark_empty(tab);
917 tab = restore_lexmin(tab);
918 if (!tab || tab->empty)
921 isl_seq_neg(eq, eq, 1 + tab->n_var);
923 r2 = isl_tab_add_row(tab, eq);
926 tab->con[r2].is_nonneg = 1;
927 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r2]);
929 tab = restore_lexmin(tab);
930 if (!tab || tab->empty)
933 if (!tab->con[r1].is_row)
934 isl_tab_kill_col(tab, tab->con[r1].index);
935 else if (!tab->con[r2].is_row)
936 isl_tab_kill_col(tab, tab->con[r2].index);
937 else if (isl_int_is_zero(tab->mat->row[tab->con[r1].index][1])) {
938 unsigned off = 2 + tab->M;
940 int row = tab->con[r1].index;
941 i = isl_seq_first_non_zero(tab->mat->row[row]+off+tab->n_dead,
942 tab->n_col - tab->n_dead);
944 isl_tab_pivot(tab, row, tab->n_dead + i);
945 isl_tab_kill_col(tab, tab->n_dead + i);
955 /* Add an inequality to the tableau, resolving violations using
958 static struct isl_tab *add_lexmin_ineq(struct isl_tab *tab, isl_int *ineq)
966 tab->bset = isl_basic_set_add_ineq(tab->bset, ineq);
967 isl_tab_push(tab, isl_tab_undo_bset_ineq);
971 r = isl_tab_add_row(tab, ineq);
974 tab->con[r].is_nonneg = 1;
975 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]);
976 if (isl_tab_row_is_redundant(tab, tab->con[r].index)) {
977 isl_tab_mark_redundant(tab, tab->con[r].index);
981 tab = restore_lexmin(tab);
982 if (tab && !tab->empty && tab->con[r].is_row &&
983 isl_tab_row_is_redundant(tab, tab->con[r].index))
984 isl_tab_mark_redundant(tab, tab->con[r].index);
991 /* Check if the coefficients of the parameters are all integral.
993 static int integer_parameter(struct isl_tab *tab, int row)
997 unsigned off = 2 + tab->M;
999 for (i = 0; i < tab->n_param; ++i) {
1000 /* Eliminated parameter */
1001 if (tab->var[i].is_row)
1003 col = tab->var[i].index;
1004 if (!isl_int_is_divisible_by(tab->mat->row[row][off + col],
1005 tab->mat->row[row][0]))
1008 for (i = 0; i < tab->n_div; ++i) {
1009 if (tab->var[tab->n_var - tab->n_div + i].is_row)
1011 col = tab->var[tab->n_var - tab->n_div + i].index;
1012 if (!isl_int_is_divisible_by(tab->mat->row[row][off + col],
1013 tab->mat->row[row][0]))
1019 /* Check if the coefficients of the non-parameter variables are all integral.
1021 static int integer_variable(struct isl_tab *tab, int row)
1024 unsigned off = 2 + tab->M;
1026 for (i = 0; i < tab->n_col; ++i) {
1027 if (tab->col_var[i] >= 0 &&
1028 (tab->col_var[i] < tab->n_param ||
1029 tab->col_var[i] >= tab->n_var - tab->n_div))
1031 if (!isl_int_is_divisible_by(tab->mat->row[row][off + i],
1032 tab->mat->row[row][0]))
1038 /* Check if the constant term is integral.
1040 static int integer_constant(struct isl_tab *tab, int row)
1042 return isl_int_is_divisible_by(tab->mat->row[row][1],
1043 tab->mat->row[row][0]);
1046 #define I_CST 1 << 0
1047 #define I_PAR 1 << 1
1048 #define I_VAR 1 << 2
1050 /* Check for first (non-parameter) variable that is non-integer and
1051 * therefore requires a cut.
1052 * For parametric tableaus, there are three parts in a row,
1053 * the constant, the coefficients of the parameters and the rest.
1054 * For each part, we check whether the coefficients in that part
1055 * are all integral and if so, set the corresponding flag in *f.
1056 * If the constant and the parameter part are integral, then the
1057 * current sample value is integral and no cut is required
1058 * (irrespective of whether the variable part is integral).
1060 static int first_non_integer(struct isl_tab *tab, int *f)
1064 for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
1067 if (!tab->var[i].is_row)
1069 row = tab->var[i].index;
1070 if (integer_constant(tab, row))
1071 ISL_FL_SET(flags, I_CST);
1072 if (integer_parameter(tab, row))
1073 ISL_FL_SET(flags, I_PAR);
1074 if (ISL_FL_ISSET(flags, I_CST) && ISL_FL_ISSET(flags, I_PAR))
1076 if (integer_variable(tab, row))
1077 ISL_FL_SET(flags, I_VAR);
1084 /* Add a (non-parametric) cut to cut away the non-integral sample
1085 * value of the given row.
1087 * If the row is given by
1089 * m r = f + \sum_i a_i y_i
1093 * c = - {-f/m} + \sum_i {a_i/m} y_i >= 0
1095 * The big parameter, if any, is ignored, since it is assumed to be big
1096 * enough to be divisible by any integer.
1097 * If the tableau is actually a parametric tableau, then this function
1098 * is only called when all coefficients of the parameters are integral.
1099 * The cut therefore has zero coefficients for the parameters.
1101 * The current value is known to be negative, so row_sign, if it
1102 * exists, is set accordingly.
1104 * Return the row of the cut or -1.
1106 static int add_cut(struct isl_tab *tab, int row)
1111 unsigned off = 2 + tab->M;
1113 if (isl_tab_extend_cons(tab, 1) < 0)
1115 r = isl_tab_allocate_con(tab);
1119 r_row = tab->mat->row[tab->con[r].index];
1120 isl_int_set(r_row[0], tab->mat->row[row][0]);
1121 isl_int_neg(r_row[1], tab->mat->row[row][1]);
1122 isl_int_fdiv_r(r_row[1], r_row[1], tab->mat->row[row][0]);
1123 isl_int_neg(r_row[1], r_row[1]);
1125 isl_int_set_si(r_row[2], 0);
1126 for (i = 0; i < tab->n_col; ++i)
1127 isl_int_fdiv_r(r_row[off + i],
1128 tab->mat->row[row][off + i], tab->mat->row[row][0]);
1130 tab->con[r].is_nonneg = 1;
1131 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]);
1133 tab->row_sign[tab->con[r].index] = isl_tab_row_neg;
1135 return tab->con[r].index;
1138 /* Given a non-parametric tableau, add cuts until an integer
1139 * sample point is obtained or until the tableau is determined
1140 * to be integer infeasible.
1141 * As long as there is any non-integer value in the sample point,
1142 * we add an appropriate cut, if possible and resolve the violated
1143 * cut constraint using restore_lexmin.
1144 * If one of the corresponding rows is equal to an integral
1145 * combination of variables/constraints plus a non-integral constant,
1146 * then there is no way to obtain an integer point an we return
1147 * a tableau that is marked empty.
1149 static struct isl_tab *cut_to_integer_lexmin(struct isl_tab *tab)
1159 while ((row = first_non_integer(tab, &flags)) != -1) {
1160 if (ISL_FL_ISSET(flags, I_VAR))
1161 return isl_tab_mark_empty(tab);
1162 row = add_cut(tab, row);
1165 tab = restore_lexmin(tab);
1166 if (!tab || tab->empty)
1175 static struct isl_tab *drop_sample(struct isl_tab *tab, int s)
1177 if (s != tab->n_outside)
1178 isl_mat_swap_rows(tab->samples, tab->n_outside, s);
1180 isl_tab_push(tab, isl_tab_undo_drop_sample);
1185 /* Check whether all the currently active samples also satisfy the inequality
1186 * "ineq" (treated as an equality if eq is set).
1187 * Remove those samples that do not.
1189 static struct isl_tab *check_samples(struct isl_tab *tab, isl_int *ineq, int eq)
1197 isl_assert(tab->mat->ctx, tab->bset, goto error);
1198 isl_assert(tab->mat->ctx, tab->samples, goto error);
1199 isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, goto error);
1202 for (i = tab->n_outside; i < tab->n_sample; ++i) {
1204 isl_seq_inner_product(ineq, tab->samples->row[i],
1205 1 + tab->n_var, &v);
1206 sgn = isl_int_sgn(v);
1207 if (eq ? (sgn == 0) : (sgn >= 0))
1209 tab = drop_sample(tab, i);
1218 /* Check whether the sample value of the tableau is finite,
1219 * i.e., either the tableau does not use a big parameter, or
1220 * all values of the variables are equal to the big parameter plus
1221 * some constant. This constant is the actual sample value.
1223 int sample_is_finite(struct isl_tab *tab)
1230 for (i = 0; i < tab->n_var; ++i) {
1232 if (!tab->var[i].is_row)
1234 row = tab->var[i].index;
1235 if (isl_int_ne(tab->mat->row[row][0], tab->mat->row[row][2]))
1241 /* Check if the context tableau of sol has any integer points.
1242 * Returns -1 if an error occurred.
1243 * If an integer point can be found and if moreover it is finite,
1244 * then it is added to the list of sample values.
1246 * This function is only called when none of the currently active sample
1247 * values satisfies the most recently added constraint.
1249 static int context_is_feasible(struct isl_sol *sol)
1251 struct isl_tab_undo *snap;
1252 struct isl_tab *tab;
1255 if (!sol || !sol->context_tab)
1258 snap = isl_tab_snap(sol->context_tab);
1259 isl_tab_push_basis(sol->context_tab);
1261 sol->context_tab = cut_to_integer_lexmin(sol->context_tab);
1262 if (!sol->context_tab)
1265 tab = sol->context_tab;
1266 if (!tab->empty && sample_is_finite(tab)) {
1267 struct isl_vec *sample;
1269 tab->samples = isl_mat_extend(tab->samples,
1270 tab->n_sample + 1, tab->samples->n_col);
1274 sample = isl_tab_get_sample_value(tab);
1277 isl_seq_cpy(tab->samples->row[tab->n_sample],
1278 sample->el, sample->size);
1279 isl_vec_free(sample);
1283 feasible = !sol->context_tab->empty;
1284 if (isl_tab_rollback(sol->context_tab, snap) < 0)
1289 isl_tab_free(sol->context_tab);
1290 sol->context_tab = NULL;
1294 /* First check if any of the currently active sample values satisfies
1295 * the inequality "ineq" (an equality if eq is set).
1296 * If not, continue with check_integer_feasible.
1298 static int context_valid_sample_or_feasible(struct isl_sol *sol,
1299 isl_int *ineq, int eq)
1303 struct isl_tab *tab;
1305 if (!sol || !sol->context_tab)
1308 tab = sol->context_tab;
1309 isl_assert(tab->mat->ctx, tab->bset, goto error);
1310 isl_assert(tab->mat->ctx, tab->samples, goto error);
1311 isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, goto error);
1314 for (i = tab->n_outside; i < tab->n_sample; ++i) {
1316 isl_seq_inner_product(ineq, tab->samples->row[i],
1317 1 + tab->n_var, &v);
1318 sgn = isl_int_sgn(v);
1319 if (eq ? (sgn == 0) : (sgn >= 0))
1324 if (i < tab->n_sample)
1327 return context_is_feasible(sol);
1330 /* For a div d = floor(f/m), add the constraints
1333 * -(f-(m-1)) + m d >= 0
1335 * Note that the second constraint is the negation of
1339 static struct isl_tab *add_div_constraints(struct isl_tab *tab, unsigned div)
1344 struct isl_vec *ineq;
1349 total = isl_basic_set_total_dim(tab->bset);
1350 div_pos = 1 + total - tab->bset->n_div + div;
1352 ineq = ineq_for_div(tab->bset, div);
1356 tab = add_lexmin_ineq(tab, ineq->el);
1358 isl_seq_neg(ineq->el, tab->bset->div[div] + 1, 1 + total);
1359 isl_int_set(ineq->el[div_pos], tab->bset->div[div][0]);
1360 isl_int_add(ineq->el[0], ineq->el[0], ineq->el[div_pos]);
1361 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
1362 tab = add_lexmin_ineq(tab, ineq->el);
1372 /* Add a div specified by "div" to both the main tableau and
1373 * the context tableau. In case of the main tableau, we only
1374 * need to add an extra div. In the context tableau, we also
1375 * need to express the meaning of the div.
1376 * Return the index of the div or -1 if anything went wrong.
1378 static int add_div(struct isl_tab *tab, struct isl_tab **context_tab,
1379 struct isl_vec *div)
1384 struct isl_mat *samples;
1386 if (isl_tab_extend_vars(*context_tab, 1) < 0)
1388 r = isl_tab_allocate_var(*context_tab);
1391 (*context_tab)->var[r].is_nonneg = 1;
1392 (*context_tab)->var[r].frozen = 1;
1394 samples = isl_mat_extend((*context_tab)->samples,
1395 (*context_tab)->n_sample, 1 + (*context_tab)->n_var);
1396 (*context_tab)->samples = samples;
1399 for (i = (*context_tab)->n_outside; i < samples->n_row; ++i) {
1400 isl_seq_inner_product(div->el + 1, samples->row[i],
1401 div->size - 1, &samples->row[i][samples->n_col - 1]);
1402 isl_int_fdiv_q(samples->row[i][samples->n_col - 1],
1403 samples->row[i][samples->n_col - 1], div->el[0]);
1406 (*context_tab)->bset = isl_basic_set_extend_dim((*context_tab)->bset,
1407 isl_basic_set_get_dim((*context_tab)->bset), 1, 0, 2);
1408 k = isl_basic_set_alloc_div((*context_tab)->bset);
1411 isl_seq_cpy((*context_tab)->bset->div[k], div->el, div->size);
1412 isl_tab_push((*context_tab), isl_tab_undo_bset_div);
1413 *context_tab = add_div_constraints(*context_tab, k);
1417 if (isl_tab_extend_vars(tab, 1) < 0)
1419 r = isl_tab_allocate_var(tab);
1422 if (!(*context_tab)->M)
1423 tab->var[r].is_nonneg = 1;
1424 tab->var[r].frozen = 1;
1427 return tab->n_div - 1;
1429 isl_tab_free(*context_tab);
1430 *context_tab = NULL;
1434 static int find_div(struct isl_tab *tab, isl_int *div, isl_int denom)
1437 unsigned total = isl_basic_set_total_dim(tab->bset);
1439 for (i = 0; i < tab->bset->n_div; ++i) {
1440 if (isl_int_ne(tab->bset->div[i][0], denom))
1442 if (!isl_seq_eq(tab->bset->div[i] + 1, div, total))
1449 /* Return the index of a div that corresponds to "div".
1450 * We first check if we already have such a div and if not, we create one.
1452 static int get_div(struct isl_tab *tab, struct isl_tab **context_tab,
1453 struct isl_vec *div)
1457 d = find_div(*context_tab, div->el + 1, div->el[0]);
1461 return add_div(tab, context_tab, div);
1464 /* Add a parametric cut to cut away the non-integral sample value
1466 * Let a_i be the coefficients of the constant term and the parameters
1467 * and let b_i be the coefficients of the variables or constraints
1468 * in basis of the tableau.
1469 * Let q be the div q = floor(\sum_i {-a_i} y_i).
1471 * The cut is expressed as
1473 * c = \sum_i -{-a_i} y_i + \sum_i {b_i} x_i + q >= 0
1475 * If q did not already exist in the context tableau, then it is added first.
1476 * If q is in a column of the main tableau then the "+ q" can be accomplished
1477 * by setting the corresponding entry to the denominator of the constraint.
1478 * If q happens to be in a row of the main tableau, then the corresponding
1479 * row needs to be added instead (taking care of the denominators).
1480 * Note that this is very unlikely, but perhaps not entirely impossible.
1482 * The current value of the cut is known to be negative (or at least
1483 * non-positive), so row_sign is set accordingly.
1485 * Return the row of the cut or -1.
1487 static int add_parametric_cut(struct isl_tab *tab, int row,
1488 struct isl_tab **context_tab)
1490 struct isl_vec *div;
1496 unsigned off = 2 + tab->M;
1501 if (isl_tab_extend_cons(*context_tab, 3) < 0)
1504 div = get_row_parameter_div(tab, row);
1508 d = get_div(tab, context_tab, div);
1512 if (isl_tab_extend_cons(tab, 1) < 0)
1514 r = isl_tab_allocate_con(tab);
1518 r_row = tab->mat->row[tab->con[r].index];
1519 isl_int_set(r_row[0], tab->mat->row[row][0]);
1520 isl_int_neg(r_row[1], tab->mat->row[row][1]);
1521 isl_int_fdiv_r(r_row[1], r_row[1], tab->mat->row[row][0]);
1522 isl_int_neg(r_row[1], r_row[1]);
1524 isl_int_set_si(r_row[2], 0);
1525 for (i = 0; i < tab->n_param; ++i) {
1526 if (tab->var[i].is_row)
1528 col = tab->var[i].index;
1529 isl_int_neg(r_row[off + col], tab->mat->row[row][off + col]);
1530 isl_int_fdiv_r(r_row[off + col], r_row[off + col],
1531 tab->mat->row[row][0]);
1532 isl_int_neg(r_row[off + col], r_row[off + col]);
1534 for (i = 0; i < tab->n_div; ++i) {
1535 if (tab->var[tab->n_var - tab->n_div + i].is_row)
1537 col = tab->var[tab->n_var - tab->n_div + i].index;
1538 isl_int_neg(r_row[off + col], tab->mat->row[row][off + col]);
1539 isl_int_fdiv_r(r_row[off + col], r_row[off + col],
1540 tab->mat->row[row][0]);
1541 isl_int_neg(r_row[off + col], r_row[off + col]);
1543 for (i = 0; i < tab->n_col; ++i) {
1544 if (tab->col_var[i] >= 0 &&
1545 (tab->col_var[i] < tab->n_param ||
1546 tab->col_var[i] >= tab->n_var - tab->n_div))
1548 isl_int_fdiv_r(r_row[off + i],
1549 tab->mat->row[row][off + i], tab->mat->row[row][0]);
1551 if (tab->var[tab->n_var - tab->n_div + d].is_row) {
1553 int d_row = tab->var[tab->n_var - tab->n_div + d].index;
1555 isl_int_gcd(gcd, tab->mat->row[d_row][0], r_row[0]);
1556 isl_int_divexact(r_row[0], r_row[0], gcd);
1557 isl_int_divexact(gcd, tab->mat->row[d_row][0], gcd);
1558 isl_seq_combine(r_row + 1, gcd, r_row + 1,
1559 r_row[0], tab->mat->row[d_row] + 1,
1560 off - 1 + tab->n_col);
1561 isl_int_mul(r_row[0], r_row[0], tab->mat->row[d_row][0]);
1564 col = tab->var[tab->n_var - tab->n_div + d].index;
1565 isl_int_set(r_row[off + col], tab->mat->row[row][0]);
1568 tab->con[r].is_nonneg = 1;
1569 isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]);
1571 tab->row_sign[tab->con[r].index] = isl_tab_row_neg;
1575 return tab->con[r].index;
1577 isl_tab_free(*context_tab);
1578 *context_tab = NULL;
1582 /* Construct a tableau for bmap that can be used for computing
1583 * the lexicographic minimum (or maximum) of bmap.
1584 * If not NULL, then dom is the domain where the minimum
1585 * should be computed. In this case, we set up a parametric
1586 * tableau with row signs (initialized to "unknown").
1587 * If M is set, then the tableau will use a big parameter.
1588 * If max is set, then a maximum should be computed instead of a minimum.
1589 * This means that for each variable x, the tableau will contain the variable
1590 * x' = M - x, rather than x' = M + x. This in turn means that the coefficient
1591 * of the variables in all constraints are negated prior to adding them
1594 static struct isl_tab *tab_for_lexmin(struct isl_basic_map *bmap,
1595 struct isl_basic_set *dom, unsigned M, int max)
1598 struct isl_tab *tab;
1600 tab = isl_tab_alloc(bmap->ctx, 2 * bmap->n_eq + bmap->n_ineq + 1,
1601 isl_basic_map_total_dim(bmap), M);
1605 tab->rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL);
1607 tab->n_param = isl_basic_set_total_dim(dom) - dom->n_div;
1608 tab->n_div = dom->n_div;
1609 tab->row_sign = isl_calloc_array(bmap->ctx,
1610 enum isl_tab_row_sign, tab->mat->n_row);
1614 if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY))
1615 return isl_tab_mark_empty(tab);
1617 for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) {
1618 tab->var[i].is_nonneg = 1;
1619 tab->var[i].frozen = 1;
1621 for (i = 0; i < bmap->n_eq; ++i) {
1623 isl_seq_neg(bmap->eq[i] + 1 + tab->n_param,
1624 bmap->eq[i] + 1 + tab->n_param,
1625 tab->n_var - tab->n_param - tab->n_div);
1626 tab = add_lexmin_valid_eq(tab, bmap->eq[i]);
1628 isl_seq_neg(bmap->eq[i] + 1 + tab->n_param,
1629 bmap->eq[i] + 1 + tab->n_param,
1630 tab->n_var - tab->n_param - tab->n_div);
1631 if (!tab || tab->empty)
1634 for (i = 0; i < bmap->n_ineq; ++i) {
1636 isl_seq_neg(bmap->ineq[i] + 1 + tab->n_param,
1637 bmap->ineq[i] + 1 + tab->n_param,
1638 tab->n_var - tab->n_param - tab->n_div);
1639 tab = add_lexmin_ineq(tab, bmap->ineq[i]);
1641 isl_seq_neg(bmap->ineq[i] + 1 + tab->n_param,
1642 bmap->ineq[i] + 1 + tab->n_param,
1643 tab->n_var - tab->n_param - tab->n_div);
1644 if (!tab || tab->empty)
1653 static struct isl_tab *context_tab_for_lexmin(struct isl_basic_set *bset)
1655 struct isl_tab *tab;
1657 bset = isl_basic_set_cow(bset);
1660 tab = tab_for_lexmin((struct isl_basic_map *)bset, NULL, 1, 0);
1666 tab->samples = isl_mat_alloc(bset->ctx, 1, 1 + tab->n_var);
1671 isl_basic_set_free(bset);
1675 /* Construct an isl_sol_map structure for accumulating the solution.
1676 * If track_empty is set, then we also keep track of the parts
1677 * of the context where there is no solution.
1678 * If max is set, then we are solving a maximization, rather than
1679 * a minimization problem, which means that the variables in the
1680 * tableau have value "M - x" rather than "M + x".
1682 static struct isl_sol_map *sol_map_init(struct isl_basic_map *bmap,
1683 struct isl_basic_set *dom, int track_empty, int max)
1685 struct isl_sol_map *sol_map;
1686 struct isl_tab *context_tab;
1689 sol_map = isl_calloc_type(bset->ctx, struct isl_sol_map);
1694 sol_map->sol.add = &sol_map_add_wrap;
1695 sol_map->sol.free = &sol_map_free_wrap;
1696 sol_map->map = isl_map_alloc_dim(isl_basic_map_get_dim(bmap), 1,
1701 context_tab = context_tab_for_lexmin(isl_basic_set_copy(dom));
1702 context_tab = restore_lexmin(context_tab);
1703 sol_map->sol.context_tab = context_tab;
1704 f = context_is_feasible(&sol_map->sol);
1709 sol_map->empty = isl_set_alloc_dim(isl_basic_set_get_dim(dom),
1710 1, ISL_SET_DISJOINT);
1711 if (!sol_map->empty)
1715 isl_basic_set_free(dom);
1718 isl_basic_set_free(dom);
1719 sol_map_free(sol_map);
1723 /* For each variable in the context tableau, check if the variable can
1724 * only attain non-negative values. If so, mark the parameter as non-negative
1725 * in the main tableau. This allows for a more direct identification of some
1726 * cases of violated constraints.
1728 static struct isl_tab *tab_detect_nonnegative_parameters(struct isl_tab *tab,
1729 struct isl_tab *context_tab)
1732 struct isl_tab_undo *snap, *snap2;
1733 struct isl_vec *ineq = NULL;
1734 struct isl_tab_var *var;
1737 if (context_tab->n_var == 0)
1740 ineq = isl_vec_alloc(tab->mat->ctx, 1 + context_tab->n_var);
1744 if (isl_tab_extend_cons(context_tab, 1) < 0)
1747 snap = isl_tab_snap(context_tab);
1748 isl_tab_push_basis(context_tab);
1750 snap2 = isl_tab_snap(context_tab);
1753 isl_seq_clr(ineq->el, ineq->size);
1754 for (i = 0; i < context_tab->n_var; ++i) {
1755 isl_int_set_si(ineq->el[1 + i], 1);
1756 context_tab = isl_tab_add_ineq(context_tab, ineq->el);
1757 var = &context_tab->con[context_tab->n_con - 1];
1758 if (!context_tab->empty &&
1759 !isl_tab_min_at_most_neg_one(context_tab, var)) {
1761 if (i >= tab->n_param)
1762 j = i - tab->n_param + tab->n_var - tab->n_div;
1763 tab->var[j].is_nonneg = 1;
1766 isl_int_set_si(ineq->el[1 + i], 0);
1767 if (isl_tab_rollback(context_tab, snap2) < 0)
1771 if (isl_tab_rollback(context_tab, snap) < 0)
1774 if (n == context_tab->n_var) {
1775 context_tab->mat = isl_mat_drop_cols(context_tab->mat, 2, 1);
1787 /* Check whether all coefficients of (non-parameter) variables
1788 * are non-positive, meaning that no pivots can be performed on the row.
1790 static int is_critical(struct isl_tab *tab, int row)
1793 unsigned off = 2 + tab->M;
1795 for (j = tab->n_dead; j < tab->n_col; ++j) {
1796 if (tab->col_var[j] >= 0 &&
1797 (tab->col_var[j] < tab->n_param ||
1798 tab->col_var[j] >= tab->n_var - tab->n_div))
1801 if (isl_int_is_pos(tab->mat->row[row][off + j]))
1808 /* Check whether the inequality represented by vec is strict over the integers,
1809 * i.e., there are no integer values satisfying the constraint with
1810 * equality. This happens if the gcd of the coefficients is not a divisor
1811 * of the constant term. If so, scale the constraint down by the gcd
1812 * of the coefficients.
1814 static int is_strict(struct isl_vec *vec)
1820 isl_seq_gcd(vec->el + 1, vec->size - 1, &gcd);
1821 if (!isl_int_is_one(gcd)) {
1822 strict = !isl_int_is_divisible_by(vec->el[0], gcd);
1823 isl_int_fdiv_q(vec->el[0], vec->el[0], gcd);
1824 isl_seq_scale_down(vec->el + 1, vec->el + 1, gcd, vec->size-1);
1831 /* Determine the sign of the given row of the main tableau.
1832 * The result is one of
1833 * isl_tab_row_pos: always non-negative; no pivot needed
1834 * isl_tab_row_neg: always non-positive; pivot
1835 * isl_tab_row_any: can be both positive and negative; split
1837 * We first handle some simple cases
1838 * - the row sign may be known already
1839 * - the row may be obviously non-negative
1840 * - the parametric constant may be equal to that of another row
1841 * for which we know the sign. This sign will be either "pos" or
1842 * "any". If it had been "neg" then we would have pivoted before.
1844 * If none of these cases hold, we check the value of the row for each
1845 * of the currently active samples. Based on the signs of these values
1846 * we make an initial determination of the sign of the row.
1848 * all zero -> unk(nown)
1849 * all non-negative -> pos
1850 * all non-positive -> neg
1851 * both negative and positive -> all
1853 * If we end up with "all", we are done.
1854 * Otherwise, we perform a check for positive and/or negative
1855 * values as follows.
1857 * samples neg unk pos
1863 * There is no special sign for "zero", because we can usually treat zero
1864 * as either non-negative or non-positive, whatever works out best.
1865 * However, if the row is "critical", meaning that pivoting is impossible
1866 * then we don't want to limp zero with the non-positive case, because
1867 * then we we would lose the solution for those values of the parameters
1868 * where the value of the row is zero. Instead, we treat 0 as non-negative
1869 * ensuring a split if the row can attain both zero and negative values.
1870 * The same happens when the original constraint was one that could not
1871 * be satisfied with equality by any integer values of the parameters.
1872 * In this case, we normalize the constraint, but then a value of zero
1873 * for the normalized constraint is actually a positive value for the
1874 * original constraint, so again we need to treat zero as non-negative.
1875 * In both these cases, we have the following decision tree instead:
1877 * all non-negative -> pos
1878 * all negative -> neg
1879 * both negative and non-negative -> all
1887 static int row_sign(struct isl_tab *tab, struct isl_sol *sol, int row)
1890 struct isl_tab_undo *snap = NULL;
1891 struct isl_vec *ineq = NULL;
1892 int res = isl_tab_row_unknown;
1900 struct isl_tab *context_tab = sol->context_tab;
1902 if (tab->row_sign[row] != isl_tab_row_unknown)
1903 return tab->row_sign[row];
1904 if (is_obviously_nonneg(tab, row))
1905 return isl_tab_row_pos;
1906 for (row2 = tab->n_redundant; row2 < tab->n_row; ++row2) {
1907 if (tab->row_sign[row2] == isl_tab_row_unknown)
1909 if (identical_parameter_line(tab, row, row2))
1910 return tab->row_sign[row2];
1913 critical = is_critical(tab, row);
1915 isl_assert(tab->mat->ctx, context_tab->samples, goto error);
1916 isl_assert(tab->mat->ctx, context_tab->samples->n_col == 1 + context_tab->n_var, goto error);
1918 ineq = get_row_parameter_ineq(tab, row);
1922 strict = is_strict(ineq);
1925 for (i = context_tab->n_outside; i < context_tab->n_sample; ++i) {
1926 isl_seq_inner_product(context_tab->samples->row[i], ineq->el,
1928 sgn = isl_int_sgn(tmp);
1929 if (sgn > 0 || (sgn == 0 && (critical || strict))) {
1930 if (res == isl_tab_row_unknown)
1931 res = isl_tab_row_pos;
1932 if (res == isl_tab_row_neg)
1933 res = isl_tab_row_any;
1936 if (res == isl_tab_row_unknown)
1937 res = isl_tab_row_neg;
1938 if (res == isl_tab_row_pos)
1939 res = isl_tab_row_any;
1941 if (res == isl_tab_row_any)
1946 if (res != isl_tab_row_any) {
1947 if (isl_tab_extend_cons(context_tab, 1) < 0)
1950 snap = isl_tab_snap(context_tab);
1951 isl_tab_push_basis(context_tab);
1954 if (res == isl_tab_row_unknown || res == isl_tab_row_pos) {
1955 /* test for negative values */
1957 isl_seq_neg(ineq->el, ineq->el, ineq->size);
1958 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
1960 isl_tab_push_basis(context_tab);
1961 sol->context_tab = add_lexmin_ineq(sol->context_tab, ineq->el);
1962 feasible = context_is_feasible(sol);
1965 context_tab = sol->context_tab;
1967 res = isl_tab_row_pos;
1969 res = (res == isl_tab_row_unknown) ? isl_tab_row_neg
1971 if (isl_tab_rollback(context_tab, snap) < 0)
1974 if (res == isl_tab_row_neg) {
1975 isl_seq_neg(ineq->el, ineq->el, ineq->size);
1976 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
1980 if (res == isl_tab_row_neg) {
1981 /* test for positive values */
1983 if (!critical && !strict)
1984 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
1986 isl_tab_push_basis(context_tab);
1987 sol->context_tab = add_lexmin_ineq(sol->context_tab, ineq->el);
1988 feasible = context_is_feasible(sol);
1991 context_tab = sol->context_tab;
1993 res = isl_tab_row_any;
1994 if (isl_tab_rollback(context_tab, snap) < 0)
2005 static struct isl_sol *find_solutions(struct isl_sol *sol, struct isl_tab *tab);
2007 /* Find solutions for values of the parameters that satisfy the given
2010 * We currently take a snapshot of the context tableau that is reset
2011 * when we return from this function, while we make a copy of the main
2012 * tableau, leaving the original main tableau untouched.
2013 * These are fairly arbitrary choices. Making a copy also of the context
2014 * tableau would obviate the need to undo any changes made to it later,
2015 * while taking a snapshot of the main tableau could reduce memory usage.
2016 * If we were to switch to taking a snapshot of the main tableau,
2017 * we would have to keep in mind that we need to save the row signs
2018 * and that we need to do this before saving the current basis
2019 * such that the basis has been restore before we restore the row signs.
2021 static struct isl_sol *find_in_pos(struct isl_sol *sol,
2022 struct isl_tab *tab, isl_int *ineq)
2024 struct isl_tab_undo *snap;
2026 snap = isl_tab_snap(sol->context_tab);
2027 isl_tab_push_basis(sol->context_tab);
2028 if (isl_tab_extend_cons(sol->context_tab, 1) < 0)
2031 tab = isl_tab_dup(tab);
2035 sol->context_tab = add_lexmin_ineq(sol->context_tab, ineq);
2036 sol->context_tab = check_samples(sol->context_tab, ineq, 0);
2038 sol = find_solutions(sol, tab);
2040 isl_tab_rollback(sol->context_tab, snap);
2043 isl_tab_rollback(sol->context_tab, snap);
2048 /* Record the absence of solutions for those values of the parameters
2049 * that do not satisfy the given inequality with equality.
2051 static struct isl_sol *no_sol_in_strict(struct isl_sol *sol,
2052 struct isl_tab *tab, struct isl_vec *ineq)
2056 struct isl_tab_undo *snap;
2057 snap = isl_tab_snap(sol->context_tab);
2058 isl_tab_push_basis(sol->context_tab);
2059 if (isl_tab_extend_cons(sol->context_tab, 1) < 0)
2062 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
2064 sol->context_tab = add_lexmin_ineq(sol->context_tab, ineq->el);
2065 f = context_valid_sample_or_feasible(sol, ineq->el, 0);
2071 sol = sol->add(sol, tab);
2074 isl_int_add_ui(ineq->el[0], ineq->el[0], 1);
2076 if (isl_tab_rollback(sol->context_tab, snap) < 0)
2084 /* Given a main tableau where more than one row requires a split,
2085 * determine and return the "best" row to split on.
2087 * Given two rows in the main tableau, if the inequality corresponding
2088 * to the first row is redundant with respect to that of the second row
2089 * in the current tableau, then it is better to split on the second row,
2090 * since in the positive part, both row will be positive.
2091 * (In the negative part a pivot will have to be performed and just about
2092 * anything can happen to the sign of the other row.)
2094 * As a simple heuristic, we therefore select the row that makes the most
2095 * of the other rows redundant.
2097 * Perhaps it would also be useful to look at the number of constraints
2098 * that conflict with any given constraint.
2100 static int best_split(struct isl_tab *tab, struct isl_tab *context_tab)
2102 struct isl_tab_undo *snap, *snap2;
2108 if (isl_tab_extend_cons(context_tab, 2) < 0)
2111 snap = isl_tab_snap(context_tab);
2112 isl_tab_push_basis(context_tab);
2113 snap2 = isl_tab_snap(context_tab);
2115 for (split = tab->n_redundant; split < tab->n_row; ++split) {
2116 struct isl_tab_undo *snap3;
2117 struct isl_vec *ineq = NULL;
2120 if (!isl_tab_var_from_row(tab, split)->is_nonneg)
2122 if (tab->row_sign[split] != isl_tab_row_any)
2125 ineq = get_row_parameter_ineq(tab, split);
2128 context_tab = isl_tab_add_ineq(context_tab, ineq->el);
2131 snap3 = isl_tab_snap(context_tab);
2133 for (row = tab->n_redundant; row < tab->n_row; ++row) {
2134 struct isl_tab_var *var;
2138 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
2140 if (tab->row_sign[row] != isl_tab_row_any)
2143 ineq = get_row_parameter_ineq(tab, row);
2146 context_tab = isl_tab_add_ineq(context_tab, ineq->el);
2148 var = &context_tab->con[context_tab->n_con - 1];
2149 if (!context_tab->empty &&
2150 !isl_tab_min_at_most_neg_one(context_tab, var))
2152 if (isl_tab_rollback(context_tab, snap3) < 0)
2155 if (best == -1 || r > best_r) {
2159 if (isl_tab_rollback(context_tab, snap2) < 0)
2163 if (isl_tab_rollback(context_tab, snap) < 0)
2169 /* Compute the lexicographic minimum of the set represented by the main
2170 * tableau "tab" within the context "sol->context_tab".
2171 * On entry the sample value of the main tableau is lexicographically
2172 * less than or equal to this lexicographic minimum.
2173 * Pivots are performed until a feasible point is found, which is then
2174 * necessarily equal to the minimum, or until the tableau is found to
2175 * be infeasible. Some pivots may need to be performed for only some
2176 * feasible values of the context tableau. If so, the context tableau
2177 * is split into a part where the pivot is needed and a part where it is not.
2179 * Whenever we enter the main loop, the main tableau is such that no
2180 * "obvious" pivots need to be performed on it, where "obvious" means
2181 * that the given row can be seen to be negative without looking at
2182 * the context tableau. In particular, for non-parametric problems,
2183 * no pivots need to be performed on the main tableau.
2184 * The caller of find_solutions is responsible for making this property
2185 * hold prior to the first iteration of the loop, while restore_lexmin
2186 * is called before every other iteration.
2188 * Inside the main loop, we first examine the signs of the rows of
2189 * the main tableau within the context of the context tableau.
2190 * If we find a row that is always non-positive for all values of
2191 * the parameters satisfying the context tableau and negative for at
2192 * least one value of the parameters, we perform the appropriate pivot
2193 * and start over. An exception is the case where no pivot can be
2194 * performed on the row. In this case, we require that the sign of
2195 * the row is negative for all values of the parameters (rather than just
2196 * non-positive). This special case is handled inside row_sign, which
2197 * will say that the row can have any sign if it determines that it can
2198 * attain both negative and zero values.
2200 * If we can't find a row that always requires a pivot, but we can find
2201 * one or more rows that require a pivot for some values of the parameters
2202 * (i.e., the row can attain both positive and negative signs), then we split
2203 * the context tableau into two parts, one where we force the sign to be
2204 * non-negative and one where we force is to be negative.
2205 * The non-negative part is handled by a recursive call (through find_in_pos).
2206 * Upon returning from this call, we continue with the negative part and
2207 * perform the required pivot.
2209 * If no such rows can be found, all rows are non-negative and we have
2210 * found a (rational) feasible point. If we only wanted a rational point
2212 * Otherwise, we check if all values of the sample point of the tableau
2213 * are integral for the variables. If so, we have found the minimal
2214 * integral point and we are done.
2215 * If the sample point is not integral, then we need to make a distinction
2216 * based on whether the constant term is non-integral or the coefficients
2217 * of the parameters. Furthermore, in order to decide how to handle
2218 * the non-integrality, we also need to know whether the coefficients
2219 * of the other columns in the tableau are integral. This leads
2220 * to the following table. The first two rows do not correspond
2221 * to a non-integral sample point and are only mentioned for completeness.
2223 * constant parameters other
2226 * int int rat | -> no problem
2228 * rat int int -> fail
2230 * rat int rat -> cut
2233 * rat rat rat | -> parametric cut
2236 * rat rat int | -> split context
2238 * If the parametric constant is completely integral, then there is nothing
2239 * to be done. If the constant term is non-integral, but all the other
2240 * coefficient are integral, then there is nothing that can be done
2241 * and the tableau has no integral solution.
2242 * If, on the other hand, one or more of the other columns have rational
2243 * coeffcients, but the parameter coefficients are all integral, then
2244 * we can perform a regular (non-parametric) cut.
2245 * Finally, if there is any parameter coefficient that is non-integral,
2246 * then we need to involve the context tableau. There are two cases here.
2247 * If at least one other column has a rational coefficient, then we
2248 * can perform a parametric cut in the main tableau by adding a new
2249 * integer division in the context tableau.
2250 * If all other columns have integral coefficients, then we need to
2251 * enforce that the rational combination of parameters (c + \sum a_i y_i)/m
2252 * is always integral. We do this by introducing an integer division
2253 * q = floor((c + \sum a_i y_i)/m) and stipulating that its argument should
2254 * always be integral in the context tableau, i.e., m q = c + \sum a_i y_i.
2255 * Since q is expressed in the tableau as
2256 * c + \sum a_i y_i - m q >= 0
2257 * -c - \sum a_i y_i + m q + m - 1 >= 0
2258 * it is sufficient to add the inequality
2259 * -c - \sum a_i y_i + m q >= 0
2260 * In the part of the context where this inequality does not hold, the
2261 * main tableau is marked as being empty.
2263 static struct isl_sol *find_solutions(struct isl_sol *sol, struct isl_tab *tab)
2265 struct isl_tab **context_tab;
2270 context_tab = &sol->context_tab;
2274 if ((*context_tab)->empty)
2277 for (; tab && !tab->empty; tab = restore_lexmin(tab)) {
2284 for (row = tab->n_redundant; row < tab->n_row; ++row) {
2285 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
2287 sgn = row_sign(tab, sol, row);
2290 tab->row_sign[row] = sgn;
2291 if (sgn == isl_tab_row_any)
2293 if (sgn == isl_tab_row_any && split == -1)
2295 if (sgn == isl_tab_row_neg)
2298 if (row < tab->n_row)
2301 struct isl_vec *ineq;
2303 split = best_split(tab, *context_tab);
2306 ineq = get_row_parameter_ineq(tab, split);
2310 for (row = tab->n_redundant; row < tab->n_row; ++row) {
2311 if (!isl_tab_var_from_row(tab, row)->is_nonneg)
2313 if (tab->row_sign[row] == isl_tab_row_any)
2314 tab->row_sign[row] = isl_tab_row_unknown;
2316 tab->row_sign[split] = isl_tab_row_pos;
2317 sol = find_in_pos(sol, tab, ineq->el);
2318 tab->row_sign[split] = isl_tab_row_neg;
2320 isl_seq_neg(ineq->el, ineq->el, ineq->size);
2321 isl_int_sub_ui(ineq->el[0], ineq->el[0], 1);
2322 *context_tab = add_lexmin_ineq(*context_tab, ineq->el);
2323 *context_tab = check_samples(*context_tab, ineq->el, 0);
2331 row = first_non_integer(tab, &flags);
2334 if (ISL_FL_ISSET(flags, I_PAR)) {
2335 if (ISL_FL_ISSET(flags, I_VAR)) {
2336 tab = isl_tab_mark_empty(tab);
2339 row = add_cut(tab, row);
2340 } else if (ISL_FL_ISSET(flags, I_VAR)) {
2341 struct isl_vec *div;
2342 struct isl_vec *ineq;
2344 if (isl_tab_extend_cons(*context_tab, 3) < 0)
2346 div = get_row_split_div(tab, row);
2349 d = get_div(tab, context_tab, div);
2353 ineq = ineq_for_div((*context_tab)->bset, d);
2354 sol = no_sol_in_strict(sol, tab, ineq);
2355 isl_seq_neg(ineq->el, ineq->el, ineq->size);
2356 *context_tab = add_lexmin_ineq(*context_tab, ineq->el);
2357 *context_tab = check_samples(*context_tab, ineq->el, 0);
2361 tab = set_row_cst_to_div(tab, row, d);
2363 row = add_parametric_cut(tab, row, context_tab);
2368 sol = sol->add(sol, tab);
2377 /* Compute the lexicographic minimum of the set represented by the main
2378 * tableau "tab" within the context "sol->context_tab".
2380 * As a preprocessing step, we first transfer all the purely parametric
2381 * equalities from the main tableau to the context tableau, i.e.,
2382 * parameters that have been pivoted to a row.
2383 * These equalities are ignored by the main algorithm, because the
2384 * corresponding rows may not be marked as being non-negative.
2385 * In parts of the context where the added equality does not hold,
2386 * the main tableau is marked as being empty.
2388 static struct isl_sol *find_solutions_main(struct isl_sol *sol,
2389 struct isl_tab *tab)
2393 for (row = tab->n_redundant; row < tab->n_row; ++row) {
2397 if (tab->row_var[row] < 0)
2399 if (tab->row_var[row] >= tab->n_param &&
2400 tab->row_var[row] < tab->n_var - tab->n_div)
2402 if (tab->row_var[row] < tab->n_param)
2403 p = tab->row_var[row];
2405 p = tab->row_var[row]
2406 + tab->n_param - (tab->n_var - tab->n_div);
2408 if (isl_tab_extend_cons(sol->context_tab, 2) < 0)
2411 eq = isl_vec_alloc(tab->mat->ctx, 1+tab->n_param+tab->n_div);
2412 get_row_parameter_line(tab, row, eq->el);
2413 isl_int_neg(eq->el[1 + p], tab->mat->row[row][0]);
2414 eq = isl_vec_normalize(eq);
2416 sol = no_sol_in_strict(sol, tab, eq);
2418 isl_seq_neg(eq->el, eq->el, eq->size);
2419 sol = no_sol_in_strict(sol, tab, eq);
2420 isl_seq_neg(eq->el, eq->el, eq->size);
2422 sol->context_tab = add_lexmin_eq(sol->context_tab, eq->el);
2423 context_valid_sample_or_feasible(sol, eq->el, 1);
2424 sol->context_tab = check_samples(sol->context_tab, eq->el, 1);
2428 isl_tab_mark_redundant(tab, row);
2430 if (!sol->context_tab)
2432 if (sol->context_tab->empty)
2435 row = tab->n_redundant - 1;
2438 return find_solutions(sol, tab);
2445 static struct isl_sol_map *sol_map_find_solutions(struct isl_sol_map *sol_map,
2446 struct isl_tab *tab)
2448 return (struct isl_sol_map *)find_solutions_main(&sol_map->sol, tab);
2451 /* Check if integer division "div" of "dom" also occurs in "bmap".
2452 * If so, return its position within the divs.
2453 * If not, return -1.
2455 static int find_context_div(struct isl_basic_map *bmap,
2456 struct isl_basic_set *dom, unsigned div)
2459 unsigned b_dim = isl_dim_total(bmap->dim);
2460 unsigned d_dim = isl_dim_total(dom->dim);
2462 if (isl_int_is_zero(dom->div[div][0]))
2464 if (isl_seq_first_non_zero(dom->div[div] + 2 + d_dim, dom->n_div) != -1)
2467 for (i = 0; i < bmap->n_div; ++i) {
2468 if (isl_int_is_zero(bmap->div[i][0]))
2470 if (isl_seq_first_non_zero(bmap->div[i] + 2 + d_dim,
2471 (b_dim - d_dim) + bmap->n_div) != -1)
2473 if (isl_seq_eq(bmap->div[i], dom->div[div], 2 + d_dim))
2479 /* The correspondence between the variables in the main tableau,
2480 * the context tableau, and the input map and domain is as follows.
2481 * The first n_param and the last n_div variables of the main tableau
2482 * form the variables of the context tableau.
2483 * In the basic map, these n_param variables correspond to the
2484 * parameters and the input dimensions. In the domain, they correspond
2485 * to the parameters and the set dimensions.
2486 * The n_div variables correspond to the integer divisions in the domain.
2487 * To ensure that everything lines up, we may need to copy some of the
2488 * integer divisions of the domain to the map. These have to be placed
2489 * in the same order as those in the context and they have to be placed
2490 * after any other integer divisions that the map may have.
2491 * This function performs the required reordering.
2493 static struct isl_basic_map *align_context_divs(struct isl_basic_map *bmap,
2494 struct isl_basic_set *dom)
2500 for (i = 0; i < dom->n_div; ++i)
2501 if (find_context_div(bmap, dom, i) != -1)
2503 other = bmap->n_div - common;
2504 if (dom->n_div - common > 0) {
2505 bmap = isl_basic_map_extend_dim(bmap, isl_dim_copy(bmap->dim),
2506 dom->n_div - common, 0, 0);
2510 for (i = 0; i < dom->n_div; ++i) {
2511 int pos = find_context_div(bmap, dom, i);
2513 pos = isl_basic_map_alloc_div(bmap);
2516 isl_int_set_si(bmap->div[pos][0], 0);
2518 if (pos != other + i)
2519 isl_basic_map_swap_div(bmap, pos, other + i);
2523 isl_basic_map_free(bmap);
2527 /* Compute the lexicographic minimum (or maximum if "max" is set)
2528 * of "bmap" over the domain "dom" and return the result as a map.
2529 * If "empty" is not NULL, then *empty is assigned a set that
2530 * contains those parts of the domain where there is no solution.
2531 * If "bmap" is marked as rational (ISL_BASIC_MAP_RATIONAL),
2532 * then we compute the rational optimum. Otherwise, we compute
2533 * the integral optimum.
2535 * We perform some preprocessing. As the PILP solver does not
2536 * handle implicit equalities very well, we first make sure all
2537 * the equalities are explicitly available.
2538 * We also make sure the divs in the domain are properly order,
2539 * because they will be added one by one in the given order
2540 * during the construction of the solution map.
2542 struct isl_map *isl_tab_basic_map_partial_lexopt(
2543 struct isl_basic_map *bmap, struct isl_basic_set *dom,
2544 struct isl_set **empty, int max)
2546 struct isl_tab *tab;
2547 struct isl_map *result = NULL;
2548 struct isl_sol_map *sol_map = NULL;
2555 isl_assert(bmap->ctx,
2556 isl_basic_map_compatible_domain(bmap, dom), goto error);
2558 bmap = isl_basic_map_detect_equalities(bmap);
2561 dom = isl_basic_set_order_divs(dom);
2562 bmap = align_context_divs(bmap, dom);
2564 sol_map = sol_map_init(bmap, dom, !!empty, max);
2568 if (isl_basic_set_fast_is_empty(sol_map->sol.context_tab->bset))
2570 else if (isl_basic_map_fast_is_empty(bmap))
2571 sol_map = add_empty(sol_map);
2573 tab = tab_for_lexmin(bmap,
2574 sol_map->sol.context_tab->bset, 1, max);
2575 tab = tab_detect_nonnegative_parameters(tab,
2576 sol_map->sol.context_tab);
2577 sol_map = sol_map_find_solutions(sol_map, tab);
2582 result = isl_map_copy(sol_map->map);
2584 *empty = isl_set_copy(sol_map->empty);
2585 sol_map_free(sol_map);
2586 isl_basic_map_free(bmap);
2589 sol_map_free(sol_map);
2590 isl_basic_map_free(bmap);