/* * Copyright 2008-2009 Katholieke Universiteit Leuven * Copyright 2010 INRIA Saclay * * Use of this software is governed by the MIT license * * Written by Sven Verdoolaege, K.U.Leuven, Departement * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite, * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France */ #include #include "isl_map_private.h" #include #include "isl_tab.h" #include "isl_sample.h" #include #include #include #include #include /* * The implementation of parametric integer linear programming in this file * was inspired by the paper "Parametric Integer Programming" and the * report "Solving systems of affine (in)equalities" by Paul Feautrier * (and others). * * The strategy used for obtaining a feasible solution is different * from the one used in isl_tab.c. In particular, in isl_tab.c, * upon finding a constraint that is not yet satisfied, we pivot * in a row that increases the constant term of the row holding the * constraint, making sure the sample solution remains feasible * for all the constraints it already satisfied. * Here, we always pivot in the row holding the constraint, * choosing a column that induces the lexicographically smallest * increment to the sample solution. * * By starting out from a sample value that is lexicographically * smaller than any integer point in the problem space, the first * feasible integer sample point we find will also be the lexicographically * smallest. If all variables can be assumed to be non-negative, * then the initial sample value may be chosen equal to zero. * However, we will not make this assumption. Instead, we apply * the "big parameter" trick. Any variable x is then not directly * used in the tableau, but instead it is represented by another * variable x' = M + x, where M is an arbitrarily large (positive) * value. x' is therefore always non-negative, whatever the value of x. * Taking as initial sample value x' = 0 corresponds to x = -M, * which is always smaller than any possible value of x. * * The big parameter trick is used in the main tableau and * also in the context tableau if isl_context_lex is used. * In this case, each tableaus has its own big parameter. * Before doing any real work, we check if all the parameters * happen to be non-negative. If so, we drop the column corresponding * to M from the initial context tableau. * If isl_context_gbr is used, then the big parameter trick is only * used in the main tableau. */ struct isl_context; struct isl_context_op { /* detect nonnegative parameters in context and mark them in tab */ struct isl_tab *(*detect_nonnegative_parameters)( struct isl_context *context, struct isl_tab *tab); /* return temporary reference to basic set representation of context */ struct isl_basic_set *(*peek_basic_set)(struct isl_context *context); /* return temporary reference to tableau representation of context */ struct isl_tab *(*peek_tab)(struct isl_context *context); /* add equality; check is 1 if eq may not be valid; * update is 1 if we may want to call ineq_sign on context later. */ void (*add_eq)(struct isl_context *context, isl_int *eq, int check, int update); /* add inequality; check is 1 if ineq may not be valid; * update is 1 if we may want to call ineq_sign on context later. */ void (*add_ineq)(struct isl_context *context, isl_int *ineq, int check, int update); /* check sign of ineq based on previous information. * strict is 1 if saturation should be treated as a positive sign. */ enum isl_tab_row_sign (*ineq_sign)(struct isl_context *context, isl_int *ineq, int strict); /* check if inequality maintains feasibility */ int (*test_ineq)(struct isl_context *context, isl_int *ineq); /* return index of a div that corresponds to "div" */ int (*get_div)(struct isl_context *context, struct isl_tab *tab, struct isl_vec *div); /* add div "div" to context and return non-negativity */ int (*add_div)(struct isl_context *context, struct isl_vec *div); int (*detect_equalities)(struct isl_context *context, struct isl_tab *tab); /* return row index of "best" split */ int (*best_split)(struct isl_context *context, struct isl_tab *tab); /* check if context has already been determined to be empty */ int (*is_empty)(struct isl_context *context); /* check if context is still usable */ int (*is_ok)(struct isl_context *context); /* save a copy/snapshot of context */ void *(*save)(struct isl_context *context); /* restore saved context */ void (*restore)(struct isl_context *context, void *); /* discard saved context */ void (*discard)(void *); /* invalidate context */ void (*invalidate)(struct isl_context *context); /* free context */ void (*free)(struct isl_context *context); }; struct isl_context { struct isl_context_op *op; }; struct isl_context_lex { struct isl_context context; struct isl_tab *tab; }; /* A stack (linked list) of solutions of subtrees of the search space. * * "M" describes the solution in terms of the dimensions of "dom". * The number of columns of "M" is one more than the total number * of dimensions of "dom". * * If "M" is NULL, then there is no solution on "dom". */ struct isl_partial_sol { int level; struct isl_basic_set *dom; struct isl_mat *M; struct isl_partial_sol *next; }; struct isl_sol; struct isl_sol_callback { struct isl_tab_callback callback; struct isl_sol *sol; }; /* isl_sol is an interface for constructing a solution to * a parametric integer linear programming problem. * Every time the algorithm reaches a state where a solution * can be read off from the tableau (including cases where the tableau * is empty), the function "add" is called on the isl_sol passed * to find_solutions_main. * * The context tableau is owned by isl_sol and is updated incrementally. * * There are currently two implementations of this interface, * isl_sol_map, which simply collects the solutions in an isl_map * and (optionally) the parts of the context where there is no solution * in an isl_set, and * isl_sol_for, which calls a user-defined function for each part of * the solution. */ struct isl_sol { int error; int rational; int level; int max; int n_out; struct isl_context *context; struct isl_partial_sol *partial; void (*add)(struct isl_sol *sol, struct isl_basic_set *dom, struct isl_mat *M); void (*add_empty)(struct isl_sol *sol, struct isl_basic_set *bset); void (*free)(struct isl_sol *sol); struct isl_sol_callback dec_level; }; static void sol_free(struct isl_sol *sol) { struct isl_partial_sol *partial, *next; if (!sol) return; for (partial = sol->partial; partial; partial = next) { next = partial->next; isl_basic_set_free(partial->dom); isl_mat_free(partial->M); free(partial); } sol->free(sol); } /* Push a partial solution represented by a domain and mapping M * onto the stack of partial solutions. */ static void sol_push_sol(struct isl_sol *sol, struct isl_basic_set *dom, struct isl_mat *M) { struct isl_partial_sol *partial; if (sol->error || !dom) goto error; partial = isl_alloc_type(dom->ctx, struct isl_partial_sol); if (!partial) goto error; partial->level = sol->level; partial->dom = dom; partial->M = M; partial->next = sol->partial; sol->partial = partial; return; error: isl_basic_set_free(dom); isl_mat_free(M); sol->error = 1; } /* Pop one partial solution from the partial solution stack and * pass it on to sol->add or sol->add_empty. */ static void sol_pop_one(struct isl_sol *sol) { struct isl_partial_sol *partial; partial = sol->partial; sol->partial = partial->next; if (partial->M) sol->add(sol, partial->dom, partial->M); else sol->add_empty(sol, partial->dom); free(partial); } /* Return a fresh copy of the domain represented by the context tableau. */ static struct isl_basic_set *sol_domain(struct isl_sol *sol) { struct isl_basic_set *bset; if (sol->error) return NULL; bset = isl_basic_set_dup(sol->context->op->peek_basic_set(sol->context)); bset = isl_basic_set_update_from_tab(bset, sol->context->op->peek_tab(sol->context)); return bset; } /* Check whether two partial solutions have the same mapping, where n_div * is the number of divs that the two partial solutions have in common. */ static int same_solution(struct isl_partial_sol *s1, struct isl_partial_sol *s2, unsigned n_div) { int i; unsigned dim; if (!s1->M != !s2->M) return 0; if (!s1->M) return 1; dim = isl_basic_set_total_dim(s1->dom) - s1->dom->n_div; for (i = 0; i < s1->M->n_row; ++i) { if (isl_seq_first_non_zero(s1->M->row[i]+1+dim+n_div, s1->M->n_col-1-dim-n_div) != -1) return 0; if (isl_seq_first_non_zero(s2->M->row[i]+1+dim+n_div, s2->M->n_col-1-dim-n_div) != -1) return 0; if (!isl_seq_eq(s1->M->row[i], s2->M->row[i], 1+dim+n_div)) return 0; } return 1; } /* Pop all solutions from the partial solution stack that were pushed onto * the stack at levels that are deeper than the current level. * If the two topmost elements on the stack have the same level * and represent the same solution, then their domains are combined. * This combined domain is the same as the current context domain * as sol_pop is called each time we move back to a higher level. */ static void sol_pop(struct isl_sol *sol) { struct isl_partial_sol *partial; unsigned n_div; if (sol->error) return; if (sol->level == 0) { for (partial = sol->partial; partial; partial = sol->partial) sol_pop_one(sol); return; } partial = sol->partial; if (!partial) return; if (partial->level <= sol->level) return; if (partial->next && partial->next->level == partial->level) { n_div = isl_basic_set_dim( sol->context->op->peek_basic_set(sol->context), isl_dim_div); if (!same_solution(partial, partial->next, n_div)) { sol_pop_one(sol); sol_pop_one(sol); } else { struct isl_basic_set *bset; isl_mat *M; unsigned n; n = isl_basic_set_dim(partial->next->dom, isl_dim_div); n -= n_div; bset = sol_domain(sol); isl_basic_set_free(partial->next->dom); partial->next->dom = bset; M = partial->next->M; if (M) { M = isl_mat_drop_cols(M, M->n_col - n, n); partial->next->M = M; if (!M) goto error; } partial->next->level = sol->level; if (!bset) goto error; sol->partial = partial->next; isl_basic_set_free(partial->dom); isl_mat_free(partial->M); free(partial); } } else sol_pop_one(sol); if (0) error: sol->error = 1; } static void sol_dec_level(struct isl_sol *sol) { if (sol->error) return; sol->level--; sol_pop(sol); } static int sol_dec_level_wrap(struct isl_tab_callback *cb) { struct isl_sol_callback *callback = (struct isl_sol_callback *)cb; sol_dec_level(callback->sol); return callback->sol->error ? -1 : 0; } /* Move down to next level and push callback onto context tableau * to decrease the level again when it gets rolled back across * the current state. That is, dec_level will be called with * the context tableau in the same state as it is when inc_level * is called. */ static void sol_inc_level(struct isl_sol *sol) { struct isl_tab *tab; if (sol->error) return; sol->level++; tab = sol->context->op->peek_tab(sol->context); if (isl_tab_push_callback(tab, &sol->dec_level.callback) < 0) sol->error = 1; } static void scale_rows(struct isl_mat *mat, isl_int m, int n_row) { int i; if (isl_int_is_one(m)) return; for (i = 0; i < n_row; ++i) isl_seq_scale(mat->row[i], mat->row[i], m, mat->n_col); } /* Add the solution identified by the tableau and the context tableau. * * The layout of the variables is as follows. * tab->n_var is equal to the total number of variables in the input * map (including divs that were copied from the context) * + the number of extra divs constructed * Of these, the first tab->n_param and the last tab->n_div variables * correspond to the variables in the context, i.e., * tab->n_param + tab->n_div = context_tab->n_var * tab->n_param is equal to the number of parameters and input * dimensions in the input map * tab->n_div is equal to the number of divs in the context * * If there is no solution, then call add_empty with a basic set * that corresponds to the context tableau. (If add_empty is NULL, * then do nothing). * * If there is a solution, then first construct a matrix that maps * all dimensions of the context to the output variables, i.e., * the output dimensions in the input map. * The divs in the input map (if any) that do not correspond to any * div in the context do not appear in the solution. * The algorithm will make sure that they have an integer value, * but these values themselves are of no interest. * We have to be careful not to drop or rearrange any divs in the * context because that would change the meaning of the matrix. * * To extract the value of the output variables, it should be noted * that we always use a big parameter M in the main tableau and so * the variable stored in this tableau is not an output variable x itself, but * x' = M + x (in case of minimization) * or * x' = M - x (in case of maximization) * If x' appears in a column, then its optimal value is zero, * which means that the optimal value of x is an unbounded number * (-M for minimization and M for maximization). * We currently assume that the output dimensions in the original map * are bounded, so this cannot occur. * Similarly, when x' appears in a row, then the coefficient of M in that * row is necessarily 1. * If the row in the tableau represents * d x' = c + d M + e(y) * then, in case of minimization, the corresponding row in the matrix * will be * a c + a e(y) * with a d = m, the (updated) common denominator of the matrix. * In case of maximization, the row will be * -a c - a e(y) */ static void sol_add(struct isl_sol *sol, struct isl_tab *tab) { struct isl_basic_set *bset = NULL; struct isl_mat *mat = NULL; unsigned off; int row; isl_int m; if (sol->error || !tab) goto error; if (tab->empty && !sol->add_empty) return; if (sol->context->op->is_empty(sol->context)) return; bset = sol_domain(sol); if (tab->empty) { sol_push_sol(sol, bset, NULL); return; } off = 2 + tab->M; mat = isl_mat_alloc(tab->mat->ctx, 1 + sol->n_out, 1 + tab->n_param + tab->n_div); if (!mat) goto error; isl_int_init(m); isl_seq_clr(mat->row[0] + 1, mat->n_col - 1); isl_int_set_si(mat->row[0][0], 1); for (row = 0; row < sol->n_out; ++row) { int i = tab->n_param + row; int r, j; isl_seq_clr(mat->row[1 + row], mat->n_col); if (!tab->var[i].is_row) { if (tab->M) isl_die(mat->ctx, isl_error_invalid, "unbounded optimum", goto error2); continue; } r = tab->var[i].index; if (tab->M && isl_int_ne(tab->mat->row[r][2], tab->mat->row[r][0])) isl_die(mat->ctx, isl_error_invalid, "unbounded optimum", goto error2); isl_int_gcd(m, mat->row[0][0], tab->mat->row[r][0]); isl_int_divexact(m, tab->mat->row[r][0], m); scale_rows(mat, m, 1 + row); isl_int_divexact(m, mat->row[0][0], tab->mat->row[r][0]); isl_int_mul(mat->row[1 + row][0], m, tab->mat->row[r][1]); for (j = 0; j < tab->n_param; ++j) { int col; if (tab->var[j].is_row) continue; col = tab->var[j].index; isl_int_mul(mat->row[1 + row][1 + j], m, tab->mat->row[r][off + col]); } for (j = 0; j < tab->n_div; ++j) { int col; if (tab->var[tab->n_var - tab->n_div+j].is_row) continue; col = tab->var[tab->n_var - tab->n_div+j].index; isl_int_mul(mat->row[1 + row][1 + tab->n_param + j], m, tab->mat->row[r][off + col]); } if (sol->max) isl_seq_neg(mat->row[1 + row], mat->row[1 + row], mat->n_col); } isl_int_clear(m); sol_push_sol(sol, bset, mat); return; error2: isl_int_clear(m); error: isl_basic_set_free(bset); isl_mat_free(mat); sol->error = 1; } struct isl_sol_map { struct isl_sol sol; struct isl_map *map; struct isl_set *empty; }; static void sol_map_free(struct isl_sol_map *sol_map) { if (!sol_map) return; if (sol_map->sol.context) sol_map->sol.context->op->free(sol_map->sol.context); isl_map_free(sol_map->map); isl_set_free(sol_map->empty); free(sol_map); } static void sol_map_free_wrap(struct isl_sol *sol) { sol_map_free((struct isl_sol_map *)sol); } /* This function is called for parts of the context where there is * no solution, with "bset" corresponding to the context tableau. * Simply add the basic set to the set "empty". */ static void sol_map_add_empty(struct isl_sol_map *sol, struct isl_basic_set *bset) { if (!bset) goto error; isl_assert(bset->ctx, sol->empty, goto error); sol->empty = isl_set_grow(sol->empty, 1); bset = isl_basic_set_simplify(bset); bset = isl_basic_set_finalize(bset); sol->empty = isl_set_add_basic_set(sol->empty, isl_basic_set_copy(bset)); if (!sol->empty) goto error; isl_basic_set_free(bset); return; error: isl_basic_set_free(bset); sol->sol.error = 1; } static void sol_map_add_empty_wrap(struct isl_sol *sol, struct isl_basic_set *bset) { sol_map_add_empty((struct isl_sol_map *)sol, bset); } /* Given a basic map "dom" that represents the context and an affine * matrix "M" that maps the dimensions of the context to the * output variables, construct a basic map with the same parameters * and divs as the context, the dimensions of the context as input * dimensions and a number of output dimensions that is equal to * the number of output dimensions in the input map. * * The constraints and divs of the context are simply copied * from "dom". For each row * x = c + e(y) * an equality * c + e(y) - d x = 0 * is added, with d the common denominator of M. */ static void sol_map_add(struct isl_sol_map *sol, struct isl_basic_set *dom, struct isl_mat *M) { int i; struct isl_basic_map *bmap = NULL; unsigned n_eq; unsigned n_ineq; unsigned nparam; unsigned total; unsigned n_div; unsigned n_out; if (sol->sol.error || !dom || !M) goto error; n_out = sol->sol.n_out; n_eq = dom->n_eq + n_out; n_ineq = dom->n_ineq; n_div = dom->n_div; nparam = isl_basic_set_total_dim(dom) - n_div; total = isl_map_dim(sol->map, isl_dim_all); bmap = isl_basic_map_alloc_space(isl_map_get_space(sol->map), n_div, n_eq, 2 * n_div + n_ineq); if (!bmap) goto error; if (sol->sol.rational) ISL_F_SET(bmap, ISL_BASIC_MAP_RATIONAL); for (i = 0; i < dom->n_div; ++i) { int k = isl_basic_map_alloc_div(bmap); if (k < 0) goto error; isl_seq_cpy(bmap->div[k], dom->div[i], 1 + 1 + nparam); isl_seq_clr(bmap->div[k] + 1 + 1 + nparam, total - nparam); isl_seq_cpy(bmap->div[k] + 1 + 1 + total, dom->div[i] + 1 + 1 + nparam, i); } for (i = 0; i < dom->n_eq; ++i) { int k = isl_basic_map_alloc_equality(bmap); if (k < 0) goto error; isl_seq_cpy(bmap->eq[k], dom->eq[i], 1 + nparam); isl_seq_clr(bmap->eq[k] + 1 + nparam, total - nparam); isl_seq_cpy(bmap->eq[k] + 1 + total, dom->eq[i] + 1 + nparam, n_div); } for (i = 0; i < dom->n_ineq; ++i) { int k = isl_basic_map_alloc_inequality(bmap); if (k < 0) goto error; isl_seq_cpy(bmap->ineq[k], dom->ineq[i], 1 + nparam); isl_seq_clr(bmap->ineq[k] + 1 + nparam, total - nparam); isl_seq_cpy(bmap->ineq[k] + 1 + total, dom->ineq[i] + 1 + nparam, n_div); } for (i = 0; i < M->n_row - 1; ++i) { int k = isl_basic_map_alloc_equality(bmap); if (k < 0) goto error; isl_seq_cpy(bmap->eq[k], M->row[1 + i], 1 + nparam); isl_seq_clr(bmap->eq[k] + 1 + nparam, n_out); isl_int_neg(bmap->eq[k][1 + nparam + i], M->row[0][0]); isl_seq_cpy(bmap->eq[k] + 1 + nparam + n_out, M->row[1 + i] + 1 + nparam, n_div); } bmap = isl_basic_map_simplify(bmap); bmap = isl_basic_map_finalize(bmap); sol->map = isl_map_grow(sol->map, 1); sol->map = isl_map_add_basic_map(sol->map, bmap); isl_basic_set_free(dom); isl_mat_free(M); if (!sol->map) sol->sol.error = 1; return; error: isl_basic_set_free(dom); isl_mat_free(M); isl_basic_map_free(bmap); sol->sol.error = 1; } static void sol_map_add_wrap(struct isl_sol *sol, struct isl_basic_set *dom, struct isl_mat *M) { sol_map_add((struct isl_sol_map *)sol, dom, M); } /* Store the "parametric constant" of row "row" of tableau "tab" in "line", * i.e., the constant term and the coefficients of all variables that * appear in the context tableau. * Note that the coefficient of the big parameter M is NOT copied. * The context tableau may not have a big parameter and even when it * does, it is a different big parameter. */ static void get_row_parameter_line(struct isl_tab *tab, int row, isl_int *line) { int i; unsigned off = 2 + tab->M; isl_int_set(line[0], tab->mat->row[row][1]); for (i = 0; i < tab->n_param; ++i) { if (tab->var[i].is_row) isl_int_set_si(line[1 + i], 0); else { int col = tab->var[i].index; isl_int_set(line[1 + i], tab->mat->row[row][off + col]); } } for (i = 0; i < tab->n_div; ++i) { if (tab->var[tab->n_var - tab->n_div + i].is_row) isl_int_set_si(line[1 + tab->n_param + i], 0); else { int col = tab->var[tab->n_var - tab->n_div + i].index; isl_int_set(line[1 + tab->n_param + i], tab->mat->row[row][off + col]); } } } /* Check if rows "row1" and "row2" have identical "parametric constants", * as explained above. * In this case, we also insist that the coefficients of the big parameter * be the same as the values of the constants will only be the same * if these coefficients are also the same. */ static int identical_parameter_line(struct isl_tab *tab, int row1, int row2) { int i; unsigned off = 2 + tab->M; if (isl_int_ne(tab->mat->row[row1][1], tab->mat->row[row2][1])) return 0; if (tab->M && isl_int_ne(tab->mat->row[row1][2], tab->mat->row[row2][2])) return 0; for (i = 0; i < tab->n_param + tab->n_div; ++i) { int pos = i < tab->n_param ? i : tab->n_var - tab->n_div + i - tab->n_param; int col; if (tab->var[pos].is_row) continue; col = tab->var[pos].index; if (isl_int_ne(tab->mat->row[row1][off + col], tab->mat->row[row2][off + col])) return 0; } return 1; } /* Return an inequality that expresses that the "parametric constant" * should be non-negative. * This function is only called when the coefficient of the big parameter * is equal to zero. */ static struct isl_vec *get_row_parameter_ineq(struct isl_tab *tab, int row) { struct isl_vec *ineq; ineq = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_param + tab->n_div); if (!ineq) return NULL; get_row_parameter_line(tab, row, ineq->el); if (ineq) ineq = isl_vec_normalize(ineq); return ineq; } /* Normalize a div expression of the form * * [(g*f(x) + c)/(g * m)] * * with c the constant term and f(x) the remaining coefficients, to * * [(f(x) + [c/g])/m] */ static void normalize_div(__isl_keep isl_vec *div) { isl_ctx *ctx = isl_vec_get_ctx(div); int len = div->size - 2; isl_seq_gcd(div->el + 2, len, &ctx->normalize_gcd); isl_int_gcd(ctx->normalize_gcd, ctx->normalize_gcd, div->el[0]); if (isl_int_is_one(ctx->normalize_gcd)) return; isl_int_divexact(div->el[0], div->el[0], ctx->normalize_gcd); isl_int_fdiv_q(div->el[1], div->el[1], ctx->normalize_gcd); isl_seq_scale_down(div->el + 2, div->el + 2, ctx->normalize_gcd, len); } /* Return a integer division for use in a parametric cut based on the given row. * In particular, let the parametric constant of the row be * * \sum_i a_i y_i * * where y_0 = 1, but none of the y_i corresponds to the big parameter M. * The div returned is equal to * * floor(\sum_i {-a_i} y_i) = floor((\sum_i (-a_i mod d) y_i)/d) */ static struct isl_vec *get_row_parameter_div(struct isl_tab *tab, int row) { struct isl_vec *div; div = isl_vec_alloc(tab->mat->ctx, 1 + 1 + tab->n_param + tab->n_div); if (!div) return NULL; isl_int_set(div->el[0], tab->mat->row[row][0]); get_row_parameter_line(tab, row, div->el + 1); isl_seq_neg(div->el + 1, div->el + 1, div->size - 1); normalize_div(div); isl_seq_fdiv_r(div->el + 1, div->el + 1, div->el[0], div->size - 1); return div; } /* Return a integer division for use in transferring an integrality constraint * to the context. * In particular, let the parametric constant of the row be * * \sum_i a_i y_i * * where y_0 = 1, but none of the y_i corresponds to the big parameter M. * The the returned div is equal to * * floor(\sum_i {a_i} y_i) = floor((\sum_i (a_i mod d) y_i)/d) */ static struct isl_vec *get_row_split_div(struct isl_tab *tab, int row) { struct isl_vec *div; div = isl_vec_alloc(tab->mat->ctx, 1 + 1 + tab->n_param + tab->n_div); if (!div) return NULL; isl_int_set(div->el[0], tab->mat->row[row][0]); get_row_parameter_line(tab, row, div->el + 1); normalize_div(div); isl_seq_fdiv_r(div->el + 1, div->el + 1, div->el[0], div->size - 1); return div; } /* Construct and return an inequality that expresses an upper bound * on the given div. * In particular, if the div is given by * * d = floor(e/m) * * then the inequality expresses * * m d <= e */ static struct isl_vec *ineq_for_div(struct isl_basic_set *bset, unsigned div) { unsigned total; unsigned div_pos; struct isl_vec *ineq; if (!bset) return NULL; total = isl_basic_set_total_dim(bset); div_pos = 1 + total - bset->n_div + div; ineq = isl_vec_alloc(bset->ctx, 1 + total); if (!ineq) return NULL; isl_seq_cpy(ineq->el, bset->div[div] + 1, 1 + total); isl_int_neg(ineq->el[div_pos], bset->div[div][0]); return ineq; } /* Given a row in the tableau and a div that was created * using get_row_split_div and that has been constrained to equality, i.e., * * d = floor(\sum_i {a_i} y_i) = \sum_i {a_i} y_i * * replace the expression "\sum_i {a_i} y_i" in the row by d, * i.e., we subtract "\sum_i {a_i} y_i" and add 1 d. * The coefficients of the non-parameters in the tableau have been * verified to be integral. We can therefore simply replace coefficient b * by floor(b). For the coefficients of the parameters we have * floor(a_i) = a_i - {a_i}, while for the other coefficients, we have * floor(b) = b. */ static struct isl_tab *set_row_cst_to_div(struct isl_tab *tab, int row, int div) { isl_seq_fdiv_q(tab->mat->row[row] + 1, tab->mat->row[row] + 1, tab->mat->row[row][0], 1 + tab->M + tab->n_col); isl_int_set_si(tab->mat->row[row][0], 1); if (tab->var[tab->n_var - tab->n_div + div].is_row) { int drow = tab->var[tab->n_var - tab->n_div + div].index; isl_assert(tab->mat->ctx, isl_int_is_one(tab->mat->row[drow][0]), goto error); isl_seq_combine(tab->mat->row[row] + 1, tab->mat->ctx->one, tab->mat->row[row] + 1, tab->mat->ctx->one, tab->mat->row[drow] + 1, 1 + tab->M + tab->n_col); } else { int dcol = tab->var[tab->n_var - tab->n_div + div].index; isl_int_add_ui(tab->mat->row[row][2 + tab->M + dcol], tab->mat->row[row][2 + tab->M + dcol], 1); } return tab; error: isl_tab_free(tab); return NULL; } /* Check if the (parametric) constant of the given row is obviously * negative, meaning that we don't need to consult the context tableau. * If there is a big parameter and its coefficient is non-zero, * then this coefficient determines the outcome. * Otherwise, we check whether the constant is negative and * all non-zero coefficients of parameters are negative and * belong to non-negative parameters. */ static int is_obviously_neg(struct isl_tab *tab, int row) { int i; int col; unsigned off = 2 + tab->M; if (tab->M) { if (isl_int_is_pos(tab->mat->row[row][2])) return 0; if (isl_int_is_neg(tab->mat->row[row][2])) return 1; } if (isl_int_is_nonneg(tab->mat->row[row][1])) return 0; for (i = 0; i < tab->n_param; ++i) { /* Eliminated parameter */ if (tab->var[i].is_row) continue; col = tab->var[i].index; if (isl_int_is_zero(tab->mat->row[row][off + col])) continue; if (!tab->var[i].is_nonneg) return 0; if (isl_int_is_pos(tab->mat->row[row][off + col])) return 0; } for (i = 0; i < tab->n_div; ++i) { if (tab->var[tab->n_var - tab->n_div + i].is_row) continue; col = tab->var[tab->n_var - tab->n_div + i].index; if (isl_int_is_zero(tab->mat->row[row][off + col])) continue; if (!tab->var[tab->n_var - tab->n_div + i].is_nonneg) return 0; if (isl_int_is_pos(tab->mat->row[row][off + col])) return 0; } return 1; } /* Check if the (parametric) constant of the given row is obviously * non-negative, meaning that we don't need to consult the context tableau. * If there is a big parameter and its coefficient is non-zero, * then this coefficient determines the outcome. * Otherwise, we check whether the constant is non-negative and * all non-zero coefficients of parameters are positive and * belong to non-negative parameters. */ static int is_obviously_nonneg(struct isl_tab *tab, int row) { int i; int col; unsigned off = 2 + tab->M; if (tab->M) { if (isl_int_is_pos(tab->mat->row[row][2])) return 1; if (isl_int_is_neg(tab->mat->row[row][2])) return 0; } if (isl_int_is_neg(tab->mat->row[row][1])) return 0; for (i = 0; i < tab->n_param; ++i) { /* Eliminated parameter */ if (tab->var[i].is_row) continue; col = tab->var[i].index; if (isl_int_is_zero(tab->mat->row[row][off + col])) continue; if (!tab->var[i].is_nonneg) return 0; if (isl_int_is_neg(tab->mat->row[row][off + col])) return 0; } for (i = 0; i < tab->n_div; ++i) { if (tab->var[tab->n_var - tab->n_div + i].is_row) continue; col = tab->var[tab->n_var - tab->n_div + i].index; if (isl_int_is_zero(tab->mat->row[row][off + col])) continue; if (!tab->var[tab->n_var - tab->n_div + i].is_nonneg) return 0; if (isl_int_is_neg(tab->mat->row[row][off + col])) return 0; } return 1; } /* Given a row r and two columns, return the column that would * lead to the lexicographically smallest increment in the sample * solution when leaving the basis in favor of the row. * Pivoting with column c will increment the sample value by a non-negative * constant times a_{V,c}/a_{r,c}, with a_{V,c} the elements of column c * corresponding to the non-parametric variables. * If variable v appears in a column c_v, the a_{v,c} = 1 iff c = c_v, * with all other entries in this virtual row equal to zero. * If variable v appears in a row, then a_{v,c} is the element in column c * of that row. * * Let v be the first variable with a_{v,c1}/a_{r,c1} != a_{v,c2}/a_{r,c2}. * Then if a_{v,c1}/a_{r,c1} < a_{v,c2}/a_{r,c2}, i.e., * a_{v,c2} a_{r,c1} - a_{v,c1} a_{r,c2} > 0, c1 results in the minimal * increment. Otherwise, it's c2. */ static int lexmin_col_pair(struct isl_tab *tab, int row, int col1, int col2, isl_int tmp) { int i; isl_int *tr; tr = tab->mat->row[row] + 2 + tab->M; for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) { int s1, s2; isl_int *r; if (!tab->var[i].is_row) { if (tab->var[i].index == col1) return col2; if (tab->var[i].index == col2) return col1; continue; } if (tab->var[i].index == row) continue; r = tab->mat->row[tab->var[i].index] + 2 + tab->M; s1 = isl_int_sgn(r[col1]); s2 = isl_int_sgn(r[col2]); if (s1 == 0 && s2 == 0) continue; if (s1 < s2) return col1; if (s2 < s1) return col2; isl_int_mul(tmp, r[col2], tr[col1]); isl_int_submul(tmp, r[col1], tr[col2]); if (isl_int_is_pos(tmp)) return col1; if (isl_int_is_neg(tmp)) return col2; } return -1; } /* Given a row in the tableau, find and return the column that would * result in the lexicographically smallest, but positive, increment * in the sample point. * If there is no such column, then return tab->n_col. * If anything goes wrong, return -1. */ static int lexmin_pivot_col(struct isl_tab *tab, int row) { int j; int col = tab->n_col; isl_int *tr; isl_int tmp; tr = tab->mat->row[row] + 2 + tab->M; isl_int_init(tmp); for (j = tab->n_dead; j < tab->n_col; ++j) { if (tab->col_var[j] >= 0 && (tab->col_var[j] < tab->n_param || tab->col_var[j] >= tab->n_var - tab->n_div)) continue; if (!isl_int_is_pos(tr[j])) continue; if (col == tab->n_col) col = j; else col = lexmin_col_pair(tab, row, col, j, tmp); isl_assert(tab->mat->ctx, col >= 0, goto error); } isl_int_clear(tmp); return col; error: isl_int_clear(tmp); return -1; } /* Return the first known violated constraint, i.e., a non-negative * constraint that currently has an either obviously negative value * or a previously determined to be negative value. * * If any constraint has a negative coefficient for the big parameter, * if any, then we return one of these first. */ static int first_neg(struct isl_tab *tab) { int row; if (tab->M) for (row = tab->n_redundant; row < tab->n_row; ++row) { if (!isl_tab_var_from_row(tab, row)->is_nonneg) continue; if (!isl_int_is_neg(tab->mat->row[row][2])) continue; if (tab->row_sign) tab->row_sign[row] = isl_tab_row_neg; return row; } for (row = tab->n_redundant; row < tab->n_row; ++row) { if (!isl_tab_var_from_row(tab, row)->is_nonneg) continue; if (tab->row_sign) { if (tab->row_sign[row] == 0 && is_obviously_neg(tab, row)) tab->row_sign[row] = isl_tab_row_neg; if (tab->row_sign[row] != isl_tab_row_neg) continue; } else if (!is_obviously_neg(tab, row)) continue; return row; } return -1; } /* Check whether the invariant that all columns are lexico-positive * is satisfied. This function is not called from the current code * but is useful during debugging. */ static void check_lexpos(struct isl_tab *tab) __attribute__ ((unused)); static void check_lexpos(struct isl_tab *tab) { unsigned off = 2 + tab->M; int col; int var; int row; for (col = tab->n_dead; col < tab->n_col; ++col) { if (tab->col_var[col] >= 0 && (tab->col_var[col] < tab->n_param || tab->col_var[col] >= tab->n_var - tab->n_div)) continue; for (var = tab->n_param; var < tab->n_var - tab->n_div; ++var) { if (!tab->var[var].is_row) { if (tab->var[var].index == col) break; else continue; } row = tab->var[var].index; if (isl_int_is_zero(tab->mat->row[row][off + col])) continue; if (isl_int_is_pos(tab->mat->row[row][off + col])) break; fprintf(stderr, "lexneg column %d (row %d)\n", col, row); } if (var >= tab->n_var - tab->n_div) fprintf(stderr, "zero column %d\n", col); } } /* Report to the caller that the given constraint is part of an encountered * conflict. */ static int report_conflicting_constraint(struct isl_tab *tab, int con) { return tab->conflict(con, tab->conflict_user); } /* Given a conflicting row in the tableau, report all constraints * involved in the row to the caller. That is, the row itself * (if it represents a constraint) and all constraint columns with * non-zero (and therefore negative) coefficients. */ static int report_conflict(struct isl_tab *tab, int row) { int j; isl_int *tr; if (!tab->conflict) return 0; if (tab->row_var[row] < 0 && report_conflicting_constraint(tab, ~tab->row_var[row]) < 0) return -1; tr = tab->mat->row[row] + 2 + tab->M; for (j = tab->n_dead; j < tab->n_col; ++j) { if (tab->col_var[j] >= 0 && (tab->col_var[j] < tab->n_param || tab->col_var[j] >= tab->n_var - tab->n_div)) continue; if (!isl_int_is_neg(tr[j])) continue; if (tab->col_var[j] < 0 && report_conflicting_constraint(tab, ~tab->col_var[j]) < 0) return -1; } return 0; } /* Resolve all known or obviously violated constraints through pivoting. * In particular, as long as we can find any violated constraint, we * look for a pivoting column that would result in the lexicographically * smallest increment in the sample point. If there is no such column * then the tableau is infeasible. */ static int restore_lexmin(struct isl_tab *tab) WARN_UNUSED; static int restore_lexmin(struct isl_tab *tab) { int row, col; if (!tab) return -1; if (tab->empty) return 0; while ((row = first_neg(tab)) != -1) { col = lexmin_pivot_col(tab, row); if (col >= tab->n_col) { if (report_conflict(tab, row) < 0) return -1; if (isl_tab_mark_empty(tab) < 0) return -1; return 0; } if (col < 0) return -1; if (isl_tab_pivot(tab, row, col) < 0) return -1; } return 0; } /* Given a row that represents an equality, look for an appropriate * pivoting column. * In particular, if there are any non-zero coefficients among * the non-parameter variables, then we take the last of these * variables. Eliminating this variable in terms of the other * variables and/or parameters does not influence the property * that all column in the initial tableau are lexicographically * positive. The row corresponding to the eliminated variable * will only have non-zero entries below the diagonal of the * initial tableau. That is, we transform * * I I * 1 into a * I I * * If there is no such non-parameter variable, then we are dealing with * pure parameter equality and we pick any parameter with coefficient 1 or -1 * for elimination. This will ensure that the eliminated parameter * always has an integer value whenever all the other parameters are integral. * If there is no such parameter then we return -1. */ static int last_var_col_or_int_par_col(struct isl_tab *tab, int row) { unsigned off = 2 + tab->M; int i; for (i = tab->n_var - tab->n_div - 1; i >= 0 && i >= tab->n_param; --i) { int col; if (tab->var[i].is_row) continue; col = tab->var[i].index; if (col <= tab->n_dead) continue; if (!isl_int_is_zero(tab->mat->row[row][off + col])) return col; } for (i = tab->n_dead; i < tab->n_col; ++i) { if (isl_int_is_one(tab->mat->row[row][off + i])) return i; if (isl_int_is_negone(tab->mat->row[row][off + i])) return i; } return -1; } /* Add an equality that is known to be valid to the tableau. * We first check if we can eliminate a variable or a parameter. * If not, we add the equality as two inequalities. * In this case, the equality was a pure parameter equality and there * is no need to resolve any constraint violations. */ static struct isl_tab *add_lexmin_valid_eq(struct isl_tab *tab, isl_int *eq) { int i; int r; if (!tab) return NULL; r = isl_tab_add_row(tab, eq); if (r < 0) goto error; r = tab->con[r].index; i = last_var_col_or_int_par_col(tab, r); if (i < 0) { tab->con[r].is_nonneg = 1; if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0) goto error; isl_seq_neg(eq, eq, 1 + tab->n_var); r = isl_tab_add_row(tab, eq); if (r < 0) goto error; tab->con[r].is_nonneg = 1; if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0) goto error; } else { if (isl_tab_pivot(tab, r, i) < 0) goto error; if (isl_tab_kill_col(tab, i) < 0) goto error; tab->n_eq++; } return tab; error: isl_tab_free(tab); return NULL; } /* Check if the given row is a pure constant. */ static int is_constant(struct isl_tab *tab, int row) { unsigned off = 2 + tab->M; return isl_seq_first_non_zero(tab->mat->row[row] + off + tab->n_dead, tab->n_col - tab->n_dead) == -1; } /* Add an equality that may or may not be valid to the tableau. * If the resulting row is a pure constant, then it must be zero. * Otherwise, the resulting tableau is empty. * * If the row is not a pure constant, then we add two inequalities, * each time checking that they can be satisfied. * In the end we try to use one of the two constraints to eliminate * a column. */ static int add_lexmin_eq(struct isl_tab *tab, isl_int *eq) WARN_UNUSED; static int add_lexmin_eq(struct isl_tab *tab, isl_int *eq) { int r1, r2; int row; struct isl_tab_undo *snap; if (!tab) return -1; snap = isl_tab_snap(tab); r1 = isl_tab_add_row(tab, eq); if (r1 < 0) return -1; tab->con[r1].is_nonneg = 1; if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r1]) < 0) return -1; row = tab->con[r1].index; if (is_constant(tab, row)) { if (!isl_int_is_zero(tab->mat->row[row][1]) || (tab->M && !isl_int_is_zero(tab->mat->row[row][2]))) { if (isl_tab_mark_empty(tab) < 0) return -1; return 0; } if (isl_tab_rollback(tab, snap) < 0) return -1; return 0; } if (restore_lexmin(tab) < 0) return -1; if (tab->empty) return 0; isl_seq_neg(eq, eq, 1 + tab->n_var); r2 = isl_tab_add_row(tab, eq); if (r2 < 0) return -1; tab->con[r2].is_nonneg = 1; if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r2]) < 0) return -1; if (restore_lexmin(tab) < 0) return -1; if (tab->empty) return 0; if (!tab->con[r1].is_row) { if (isl_tab_kill_col(tab, tab->con[r1].index) < 0) return -1; } else if (!tab->con[r2].is_row) { if (isl_tab_kill_col(tab, tab->con[r2].index) < 0) return -1; } if (tab->bmap) { tab->bmap = isl_basic_map_add_ineq(tab->bmap, eq); if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0) return -1; isl_seq_neg(eq, eq, 1 + tab->n_var); tab->bmap = isl_basic_map_add_ineq(tab->bmap, eq); isl_seq_neg(eq, eq, 1 + tab->n_var); if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0) return -1; if (!tab->bmap) return -1; } return 0; } /* Add an inequality to the tableau, resolving violations using * restore_lexmin. */ static struct isl_tab *add_lexmin_ineq(struct isl_tab *tab, isl_int *ineq) { int r; if (!tab) return NULL; if (tab->bmap) { tab->bmap = isl_basic_map_add_ineq(tab->bmap, ineq); if (isl_tab_push(tab, isl_tab_undo_bmap_ineq) < 0) goto error; if (!tab->bmap) goto error; } r = isl_tab_add_row(tab, ineq); if (r < 0) goto error; tab->con[r].is_nonneg = 1; if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0) goto error; if (isl_tab_row_is_redundant(tab, tab->con[r].index)) { if (isl_tab_mark_redundant(tab, tab->con[r].index) < 0) goto error; return tab; } if (restore_lexmin(tab) < 0) goto error; if (!tab->empty && tab->con[r].is_row && isl_tab_row_is_redundant(tab, tab->con[r].index)) if (isl_tab_mark_redundant(tab, tab->con[r].index) < 0) goto error; return tab; error: isl_tab_free(tab); return NULL; } /* Check if the coefficients of the parameters are all integral. */ static int integer_parameter(struct isl_tab *tab, int row) { int i; int col; unsigned off = 2 + tab->M; for (i = 0; i < tab->n_param; ++i) { /* Eliminated parameter */ if (tab->var[i].is_row) continue; col = tab->var[i].index; if (!isl_int_is_divisible_by(tab->mat->row[row][off + col], tab->mat->row[row][0])) return 0; } for (i = 0; i < tab->n_div; ++i) { if (tab->var[tab->n_var - tab->n_div + i].is_row) continue; col = tab->var[tab->n_var - tab->n_div + i].index; if (!isl_int_is_divisible_by(tab->mat->row[row][off + col], tab->mat->row[row][0])) return 0; } return 1; } /* Check if the coefficients of the non-parameter variables are all integral. */ static int integer_variable(struct isl_tab *tab, int row) { int i; unsigned off = 2 + tab->M; for (i = tab->n_dead; i < tab->n_col; ++i) { if (tab->col_var[i] >= 0 && (tab->col_var[i] < tab->n_param || tab->col_var[i] >= tab->n_var - tab->n_div)) continue; if (!isl_int_is_divisible_by(tab->mat->row[row][off + i], tab->mat->row[row][0])) return 0; } return 1; } /* Check if the constant term is integral. */ static int integer_constant(struct isl_tab *tab, int row) { return isl_int_is_divisible_by(tab->mat->row[row][1], tab->mat->row[row][0]); } #define I_CST 1 << 0 #define I_PAR 1 << 1 #define I_VAR 1 << 2 /* Check for next (non-parameter) variable after "var" (first if var == -1) * that is non-integer and therefore requires a cut and return * the index of the variable. * For parametric tableaus, there are three parts in a row, * the constant, the coefficients of the parameters and the rest. * For each part, we check whether the coefficients in that part * are all integral and if so, set the corresponding flag in *f. * If the constant and the parameter part are integral, then the * current sample value is integral and no cut is required * (irrespective of whether the variable part is integral). */ static int next_non_integer_var(struct isl_tab *tab, int var, int *f) { var = var < 0 ? tab->n_param : var + 1; for (; var < tab->n_var - tab->n_div; ++var) { int flags = 0; int row; if (!tab->var[var].is_row) continue; row = tab->var[var].index; if (integer_constant(tab, row)) ISL_FL_SET(flags, I_CST); if (integer_parameter(tab, row)) ISL_FL_SET(flags, I_PAR); if (ISL_FL_ISSET(flags, I_CST) && ISL_FL_ISSET(flags, I_PAR)) continue; if (integer_variable(tab, row)) ISL_FL_SET(flags, I_VAR); *f = flags; return var; } return -1; } /* Check for first (non-parameter) variable that is non-integer and * therefore requires a cut and return the corresponding row. * For parametric tableaus, there are three parts in a row, * the constant, the coefficients of the parameters and the rest. * For each part, we check whether the coefficients in that part * are all integral and if so, set the corresponding flag in *f. * If the constant and the parameter part are integral, then the * current sample value is integral and no cut is required * (irrespective of whether the variable part is integral). */ static int first_non_integer_row(struct isl_tab *tab, int *f) { int var = next_non_integer_var(tab, -1, f); return var < 0 ? -1 : tab->var[var].index; } /* Add a (non-parametric) cut to cut away the non-integral sample * value of the given row. * * If the row is given by * * m r = f + \sum_i a_i y_i * * then the cut is * * c = - {-f/m} + \sum_i {a_i/m} y_i >= 0 * * The big parameter, if any, is ignored, since it is assumed to be big * enough to be divisible by any integer. * If the tableau is actually a parametric tableau, then this function * is only called when all coefficients of the parameters are integral. * The cut therefore has zero coefficients for the parameters. * * The current value is known to be negative, so row_sign, if it * exists, is set accordingly. * * Return the row of the cut or -1. */ static int add_cut(struct isl_tab *tab, int row) { int i; int r; isl_int *r_row; unsigned off = 2 + tab->M; if (isl_tab_extend_cons(tab, 1) < 0) return -1; r = isl_tab_allocate_con(tab); if (r < 0) return -1; r_row = tab->mat->row[tab->con[r].index]; isl_int_set(r_row[0], tab->mat->row[row][0]); isl_int_neg(r_row[1], tab->mat->row[row][1]); isl_int_fdiv_r(r_row[1], r_row[1], tab->mat->row[row][0]); isl_int_neg(r_row[1], r_row[1]); if (tab->M) isl_int_set_si(r_row[2], 0); for (i = 0; i < tab->n_col; ++i) isl_int_fdiv_r(r_row[off + i], tab->mat->row[row][off + i], tab->mat->row[row][0]); tab->con[r].is_nonneg = 1; if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0) return -1; if (tab->row_sign) tab->row_sign[tab->con[r].index] = isl_tab_row_neg; return tab->con[r].index; } #define CUT_ALL 1 #define CUT_ONE 0 /* Given a non-parametric tableau, add cuts until an integer * sample point is obtained or until the tableau is determined * to be integer infeasible. * As long as there is any non-integer value in the sample point, * we add appropriate cuts, if possible, for each of these * non-integer values and then resolve the violated * cut constraints using restore_lexmin. * If one of the corresponding rows is equal to an integral * combination of variables/constraints plus a non-integral constant, * then there is no way to obtain an integer point and we return * a tableau that is marked empty. * The parameter cutting_strategy controls the strategy used when adding cuts * to remove non-integer points. CUT_ALL adds all possible cuts * before continuing the search. CUT_ONE adds only one cut at a time. */ static struct isl_tab *cut_to_integer_lexmin(struct isl_tab *tab, int cutting_strategy) { int var; int row; int flags; if (!tab) return NULL; if (tab->empty) return tab; while ((var = next_non_integer_var(tab, -1, &flags)) != -1) { do { if (ISL_FL_ISSET(flags, I_VAR)) { if (isl_tab_mark_empty(tab) < 0) goto error; return tab; } row = tab->var[var].index; row = add_cut(tab, row); if (row < 0) goto error; if (cutting_strategy == CUT_ONE) break; } while ((var = next_non_integer_var(tab, var, &flags)) != -1); if (restore_lexmin(tab) < 0) goto error; if (tab->empty) break; } return tab; error: isl_tab_free(tab); return NULL; } /* Check whether all the currently active samples also satisfy the inequality * "ineq" (treated as an equality if eq is set). * Remove those samples that do not. */ static struct isl_tab *check_samples(struct isl_tab *tab, isl_int *ineq, int eq) { int i; isl_int v; if (!tab) return NULL; isl_assert(tab->mat->ctx, tab->bmap, goto error); isl_assert(tab->mat->ctx, tab->samples, goto error); isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, goto error); isl_int_init(v); for (i = tab->n_outside; i < tab->n_sample; ++i) { int sgn; isl_seq_inner_product(ineq, tab->samples->row[i], 1 + tab->n_var, &v); sgn = isl_int_sgn(v); if (eq ? (sgn == 0) : (sgn >= 0)) continue; tab = isl_tab_drop_sample(tab, i); if (!tab) break; } isl_int_clear(v); return tab; error: isl_tab_free(tab); return NULL; } /* Check whether the sample value of the tableau is finite, * i.e., either the tableau does not use a big parameter, or * all values of the variables are equal to the big parameter plus * some constant. This constant is the actual sample value. */ static int sample_is_finite(struct isl_tab *tab) { int i; if (!tab->M) return 1; for (i = 0; i < tab->n_var; ++i) { int row; if (!tab->var[i].is_row) return 0; row = tab->var[i].index; if (isl_int_ne(tab->mat->row[row][0], tab->mat->row[row][2])) return 0; } return 1; } /* Check if the context tableau of sol has any integer points. * Leave tab in empty state if no integer point can be found. * If an integer point can be found and if moreover it is finite, * then it is added to the list of sample values. * * This function is only called when none of the currently active sample * values satisfies the most recently added constraint. */ static struct isl_tab *check_integer_feasible(struct isl_tab *tab) { struct isl_tab_undo *snap; if (!tab) return NULL; snap = isl_tab_snap(tab); if (isl_tab_push_basis(tab) < 0) goto error; tab = cut_to_integer_lexmin(tab, CUT_ALL); if (!tab) goto error; if (!tab->empty && sample_is_finite(tab)) { struct isl_vec *sample; sample = isl_tab_get_sample_value(tab); if (isl_tab_add_sample(tab, sample) < 0) goto error; } if (!tab->empty && isl_tab_rollback(tab, snap) < 0) goto error; return tab; error: isl_tab_free(tab); return NULL; } /* Check if any of the currently active sample values satisfies * the inequality "ineq" (an equality if eq is set). */ static int tab_has_valid_sample(struct isl_tab *tab, isl_int *ineq, int eq) { int i; isl_int v; if (!tab) return -1; isl_assert(tab->mat->ctx, tab->bmap, return -1); isl_assert(tab->mat->ctx, tab->samples, return -1); isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, return -1); isl_int_init(v); for (i = tab->n_outside; i < tab->n_sample; ++i) { int sgn; isl_seq_inner_product(ineq, tab->samples->row[i], 1 + tab->n_var, &v); sgn = isl_int_sgn(v); if (eq ? (sgn == 0) : (sgn >= 0)) break; } isl_int_clear(v); return i < tab->n_sample; } /* Add a div specified by "div" to the tableau "tab" and return * 1 if the div is obviously non-negative. */ static int context_tab_add_div(struct isl_tab *tab, struct isl_vec *div, int (*add_ineq)(void *user, isl_int *), void *user) { int i; int r; struct isl_mat *samples; int nonneg; r = isl_tab_add_div(tab, div, add_ineq, user); if (r < 0) return -1; nonneg = tab->var[r].is_nonneg; tab->var[r].frozen = 1; samples = isl_mat_extend(tab->samples, tab->n_sample, 1 + tab->n_var); tab->samples = samples; if (!samples) return -1; for (i = tab->n_outside; i < samples->n_row; ++i) { isl_seq_inner_product(div->el + 1, samples->row[i], div->size - 1, &samples->row[i][samples->n_col - 1]); isl_int_fdiv_q(samples->row[i][samples->n_col - 1], samples->row[i][samples->n_col - 1], div->el[0]); } return nonneg; } /* Add a div specified by "div" to both the main tableau and * the context tableau. In case of the main tableau, we only * need to add an extra div. In the context tableau, we also * need to express the meaning of the div. * Return the index of the div or -1 if anything went wrong. */ static int add_div(struct isl_tab *tab, struct isl_context *context, struct isl_vec *div) { int r; int nonneg; if ((nonneg = context->op->add_div(context, div)) < 0) goto error; if (!context->op->is_ok(context)) goto error; if (isl_tab_extend_vars(tab, 1) < 0) goto error; r = isl_tab_allocate_var(tab); if (r < 0) goto error; if (nonneg) tab->var[r].is_nonneg = 1; tab->var[r].frozen = 1; tab->n_div++; return tab->n_div - 1; error: context->op->invalidate(context); return -1; } static int find_div(struct isl_tab *tab, isl_int *div, isl_int denom) { int i; unsigned total = isl_basic_map_total_dim(tab->bmap); for (i = 0; i < tab->bmap->n_div; ++i) { if (isl_int_ne(tab->bmap->div[i][0], denom)) continue; if (!isl_seq_eq(tab->bmap->div[i] + 1, div, 1 + total)) continue; return i; } return -1; } /* Return the index of a div that corresponds to "div". * We first check if we already have such a div and if not, we create one. */ static int get_div(struct isl_tab *tab, struct isl_context *context, struct isl_vec *div) { int d; struct isl_tab *context_tab = context->op->peek_tab(context); if (!context_tab) return -1; d = find_div(context_tab, div->el + 1, div->el[0]); if (d != -1) return d; return add_div(tab, context, div); } /* Add a parametric cut to cut away the non-integral sample value * of the give row. * Let a_i be the coefficients of the constant term and the parameters * and let b_i be the coefficients of the variables or constraints * in basis of the tableau. * Let q be the div q = floor(\sum_i {-a_i} y_i). * * The cut is expressed as * * c = \sum_i -{-a_i} y_i + \sum_i {b_i} x_i + q >= 0 * * If q did not already exist in the context tableau, then it is added first. * If q is in a column of the main tableau then the "+ q" can be accomplished * by setting the corresponding entry to the denominator of the constraint. * If q happens to be in a row of the main tableau, then the corresponding * row needs to be added instead (taking care of the denominators). * Note that this is very unlikely, but perhaps not entirely impossible. * * The current value of the cut is known to be negative (or at least * non-positive), so row_sign is set accordingly. * * Return the row of the cut or -1. */ static int add_parametric_cut(struct isl_tab *tab, int row, struct isl_context *context) { struct isl_vec *div; int d; int i; int r; isl_int *r_row; int col; int n; unsigned off = 2 + tab->M; if (!context) return -1; div = get_row_parameter_div(tab, row); if (!div) return -1; n = tab->n_div; d = context->op->get_div(context, tab, div); isl_vec_free(div); if (d < 0) return -1; if (isl_tab_extend_cons(tab, 1) < 0) return -1; r = isl_tab_allocate_con(tab); if (r < 0) return -1; r_row = tab->mat->row[tab->con[r].index]; isl_int_set(r_row[0], tab->mat->row[row][0]); isl_int_neg(r_row[1], tab->mat->row[row][1]); isl_int_fdiv_r(r_row[1], r_row[1], tab->mat->row[row][0]); isl_int_neg(r_row[1], r_row[1]); if (tab->M) isl_int_set_si(r_row[2], 0); for (i = 0; i < tab->n_param; ++i) { if (tab->var[i].is_row) continue; col = tab->var[i].index; isl_int_neg(r_row[off + col], tab->mat->row[row][off + col]); isl_int_fdiv_r(r_row[off + col], r_row[off + col], tab->mat->row[row][0]); isl_int_neg(r_row[off + col], r_row[off + col]); } for (i = 0; i < tab->n_div; ++i) { if (tab->var[tab->n_var - tab->n_div + i].is_row) continue; col = tab->var[tab->n_var - tab->n_div + i].index; isl_int_neg(r_row[off + col], tab->mat->row[row][off + col]); isl_int_fdiv_r(r_row[off + col], r_row[off + col], tab->mat->row[row][0]); isl_int_neg(r_row[off + col], r_row[off + col]); } for (i = 0; i < tab->n_col; ++i) { if (tab->col_var[i] >= 0 && (tab->col_var[i] < tab->n_param || tab->col_var[i] >= tab->n_var - tab->n_div)) continue; isl_int_fdiv_r(r_row[off + i], tab->mat->row[row][off + i], tab->mat->row[row][0]); } if (tab->var[tab->n_var - tab->n_div + d].is_row) { isl_int gcd; int d_row = tab->var[tab->n_var - tab->n_div + d].index; isl_int_init(gcd); isl_int_gcd(gcd, tab->mat->row[d_row][0], r_row[0]); isl_int_divexact(r_row[0], r_row[0], gcd); isl_int_divexact(gcd, tab->mat->row[d_row][0], gcd); isl_seq_combine(r_row + 1, gcd, r_row + 1, r_row[0], tab->mat->row[d_row] + 1, off - 1 + tab->n_col); isl_int_mul(r_row[0], r_row[0], tab->mat->row[d_row][0]); isl_int_clear(gcd); } else { col = tab->var[tab->n_var - tab->n_div + d].index; isl_int_set(r_row[off + col], tab->mat->row[row][0]); } tab->con[r].is_nonneg = 1; if (isl_tab_push_var(tab, isl_tab_undo_nonneg, &tab->con[r]) < 0) return -1; if (tab->row_sign) tab->row_sign[tab->con[r].index] = isl_tab_row_neg; row = tab->con[r].index; if (d >= n && context->op->detect_equalities(context, tab) < 0) return -1; return row; } /* Construct a tableau for bmap that can be used for computing * the lexicographic minimum (or maximum) of bmap. * If not NULL, then dom is the domain where the minimum * should be computed. In this case, we set up a parametric * tableau with row signs (initialized to "unknown"). * If M is set, then the tableau will use a big parameter. * If max is set, then a maximum should be computed instead of a minimum. * This means that for each variable x, the tableau will contain the variable * x' = M - x, rather than x' = M + x. This in turn means that the coefficient * of the variables in all constraints are negated prior to adding them * to the tableau. */ static struct isl_tab *tab_for_lexmin(struct isl_basic_map *bmap, struct isl_basic_set *dom, unsigned M, int max) { int i; struct isl_tab *tab; unsigned n_var; unsigned o_var; tab = isl_tab_alloc(bmap->ctx, 2 * bmap->n_eq + bmap->n_ineq + 1, isl_basic_map_total_dim(bmap), M); if (!tab) return NULL; tab->rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL); if (dom) { tab->n_param = isl_basic_set_total_dim(dom) - dom->n_div; tab->n_div = dom->n_div; tab->row_sign = isl_calloc_array(bmap->ctx, enum isl_tab_row_sign, tab->mat->n_row); if (tab->mat->n_row && !tab->row_sign) goto error; } if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY)) { if (isl_tab_mark_empty(tab) < 0) goto error; return tab; } for (i = tab->n_param; i < tab->n_var - tab->n_div; ++i) { tab->var[i].is_nonneg = 1; tab->var[i].frozen = 1; } o_var = 1 + tab->n_param; n_var = tab->n_var - tab->n_param - tab->n_div; for (i = 0; i < bmap->n_eq; ++i) { if (max) isl_seq_neg(bmap->eq[i] + o_var, bmap->eq[i] + o_var, n_var); tab = add_lexmin_valid_eq(tab, bmap->eq[i]); if (max) isl_seq_neg(bmap->eq[i] + o_var, bmap->eq[i] + o_var, n_var); if (!tab || tab->empty) return tab; } if (bmap->n_eq && restore_lexmin(tab) < 0) goto error; for (i = 0; i < bmap->n_ineq; ++i) { if (max) isl_seq_neg(bmap->ineq[i] + o_var, bmap->ineq[i] + o_var, n_var); tab = add_lexmin_ineq(tab, bmap->ineq[i]); if (max) isl_seq_neg(bmap->ineq[i] + o_var, bmap->ineq[i] + o_var, n_var); if (!tab || tab->empty) return tab; } return tab; error: isl_tab_free(tab); return NULL; } /* Given a main tableau where more than one row requires a split, * determine and return the "best" row to split on. * * Given two rows in the main tableau, if the inequality corresponding * to the first row is redundant with respect to that of the second row * in the current tableau, then it is better to split on the second row, * since in the positive part, both row will be positive. * (In the negative part a pivot will have to be performed and just about * anything can happen to the sign of the other row.) * * As a simple heuristic, we therefore select the row that makes the most * of the other rows redundant. * * Perhaps it would also be useful to look at the number of constraints * that conflict with any given constraint. * * best is the best row so far (-1 when we have not found any row yet). * best_r is the number of other rows made redundant by row best. * When best is still -1, bset_r is meaningless, but it is initialized * to some arbitrary value (0) anyway. Without this redundant initialization * valgrind may warn about uninitialized memory accesses when isl * is compiled with some versions of gcc. */ static int best_split(struct isl_tab *tab, struct isl_tab *context_tab) { struct isl_tab_undo *snap; int split; int row; int best = -1; int best_r = 0; if (isl_tab_extend_cons(context_tab, 2) < 0) return -1; snap = isl_tab_snap(context_tab); for (split = tab->n_redundant; split < tab->n_row; ++split) { struct isl_tab_undo *snap2; struct isl_vec *ineq = NULL; int r = 0; int ok; if (!isl_tab_var_from_row(tab, split)->is_nonneg) continue; if (tab->row_sign[split] != isl_tab_row_any) continue; ineq = get_row_parameter_ineq(tab, split); if (!ineq) return -1; ok = isl_tab_add_ineq(context_tab, ineq->el) >= 0; isl_vec_free(ineq); if (!ok) return -1; snap2 = isl_tab_snap(context_tab); for (row = tab->n_redundant; row < tab->n_row; ++row) { struct isl_tab_var *var; if (row == split) continue; if (!isl_tab_var_from_row(tab, row)->is_nonneg) continue; if (tab->row_sign[row] != isl_tab_row_any) continue; ineq = get_row_parameter_ineq(tab, row); if (!ineq) return -1; ok = isl_tab_add_ineq(context_tab, ineq->el) >= 0; isl_vec_free(ineq); if (!ok) return -1; var = &context_tab->con[context_tab->n_con - 1]; if (!context_tab->empty && !isl_tab_min_at_most_neg_one(context_tab, var)) r++; if (isl_tab_rollback(context_tab, snap2) < 0) return -1; } if (best == -1 || r > best_r) { best = split; best_r = r; } if (isl_tab_rollback(context_tab, snap) < 0) return -1; } return best; } static struct isl_basic_set *context_lex_peek_basic_set( struct isl_context *context) { struct isl_context_lex *clex = (struct isl_context_lex *)context; if (!clex->tab) return NULL; return isl_tab_peek_bset(clex->tab); } static struct isl_tab *context_lex_peek_tab(struct isl_context *context) { struct isl_context_lex *clex = (struct isl_context_lex *)context; return clex->tab; } static void context_lex_add_eq(struct isl_context *context, isl_int *eq, int check, int update) { struct isl_context_lex *clex = (struct isl_context_lex *)context; if (isl_tab_extend_cons(clex->tab, 2) < 0) goto error; if (add_lexmin_eq(clex->tab, eq) < 0) goto error; if (check) { int v = tab_has_valid_sample(clex->tab, eq, 1); if (v < 0) goto error; if (!v) clex->tab = check_integer_feasible(clex->tab); } if (update) clex->tab = check_samples(clex->tab, eq, 1); return; error: isl_tab_free(clex->tab); clex->tab = NULL; } static void context_lex_add_ineq(struct isl_context *context, isl_int *ineq, int check, int update) { struct isl_context_lex *clex = (struct isl_context_lex *)context; if (isl_tab_extend_cons(clex->tab, 1) < 0) goto error; clex->tab = add_lexmin_ineq(clex->tab, ineq); if (check) { int v = tab_has_valid_sample(clex->tab, ineq, 0); if (v < 0) goto error; if (!v) clex->tab = check_integer_feasible(clex->tab); } if (update) clex->tab = check_samples(clex->tab, ineq, 0); return; error: isl_tab_free(clex->tab); clex->tab = NULL; } static int context_lex_add_ineq_wrap(void *user, isl_int *ineq) { struct isl_context *context = (struct isl_context *)user; context_lex_add_ineq(context, ineq, 0, 0); return context->op->is_ok(context) ? 0 : -1; } /* Check which signs can be obtained by "ineq" on all the currently * active sample values. See row_sign for more information. */ static enum isl_tab_row_sign tab_ineq_sign(struct isl_tab *tab, isl_int *ineq, int strict) { int i; int sgn; isl_int tmp; enum isl_tab_row_sign res = isl_tab_row_unknown; isl_assert(tab->mat->ctx, tab->samples, return isl_tab_row_unknown); isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, return isl_tab_row_unknown); isl_int_init(tmp); for (i = tab->n_outside; i < tab->n_sample; ++i) { isl_seq_inner_product(tab->samples->row[i], ineq, 1 + tab->n_var, &tmp); sgn = isl_int_sgn(tmp); if (sgn > 0 || (sgn == 0 && strict)) { if (res == isl_tab_row_unknown) res = isl_tab_row_pos; if (res == isl_tab_row_neg) res = isl_tab_row_any; } if (sgn < 0) { if (res == isl_tab_row_unknown) res = isl_tab_row_neg; if (res == isl_tab_row_pos) res = isl_tab_row_any; } if (res == isl_tab_row_any) break; } isl_int_clear(tmp); return res; } static enum isl_tab_row_sign context_lex_ineq_sign(struct isl_context *context, isl_int *ineq, int strict) { struct isl_context_lex *clex = (struct isl_context_lex *)context; return tab_ineq_sign(clex->tab, ineq, strict); } /* Check whether "ineq" can be added to the tableau without rendering * it infeasible. */ static int context_lex_test_ineq(struct isl_context *context, isl_int *ineq) { struct isl_context_lex *clex = (struct isl_context_lex *)context; struct isl_tab_undo *snap; int feasible; if (!clex->tab) return -1; if (isl_tab_extend_cons(clex->tab, 1) < 0) return -1; snap = isl_tab_snap(clex->tab); if (isl_tab_push_basis(clex->tab) < 0) return -1; clex->tab = add_lexmin_ineq(clex->tab, ineq); clex->tab = check_integer_feasible(clex->tab); if (!clex->tab) return -1; feasible = !clex->tab->empty; if (isl_tab_rollback(clex->tab, snap) < 0) return -1; return feasible; } static int context_lex_get_div(struct isl_context *context, struct isl_tab *tab, struct isl_vec *div) { return get_div(tab, context, div); } /* Add a div specified by "div" to the context tableau and return * 1 if the div is obviously non-negative. * context_tab_add_div will always return 1, because all variables * in a isl_context_lex tableau are non-negative. * However, if we are using a big parameter in the context, then this only * reflects the non-negativity of the variable used to _encode_ the * div, i.e., div' = M + div, so we can't draw any conclusions. */ static int context_lex_add_div(struct isl_context *context, struct isl_vec *div) { struct isl_context_lex *clex = (struct isl_context_lex *)context; int nonneg; nonneg = context_tab_add_div(clex->tab, div, context_lex_add_ineq_wrap, context); if (nonneg < 0) return -1; if (clex->tab->M) return 0; return nonneg; } static int context_lex_detect_equalities(struct isl_context *context, struct isl_tab *tab) { return 0; } static int context_lex_best_split(struct isl_context *context, struct isl_tab *tab) { struct isl_context_lex *clex = (struct isl_context_lex *)context; struct isl_tab_undo *snap; int r; snap = isl_tab_snap(clex->tab); if (isl_tab_push_basis(clex->tab) < 0) return -1; r = best_split(tab, clex->tab); if (r >= 0 && isl_tab_rollback(clex->tab, snap) < 0) return -1; return r; } static int context_lex_is_empty(struct isl_context *context) { struct isl_context_lex *clex = (struct isl_context_lex *)context; if (!clex->tab) return -1; return clex->tab->empty; } static void *context_lex_save(struct isl_context *context) { struct isl_context_lex *clex = (struct isl_context_lex *)context; struct isl_tab_undo *snap; snap = isl_tab_snap(clex->tab); if (isl_tab_push_basis(clex->tab) < 0) return NULL; if (isl_tab_save_samples(clex->tab) < 0) return NULL; return snap; } static void context_lex_restore(struct isl_context *context, void *save) { struct isl_context_lex *clex = (struct isl_context_lex *)context; if (isl_tab_rollback(clex->tab, (struct isl_tab_undo *)save) < 0) { isl_tab_free(clex->tab); clex->tab = NULL; } } static void context_lex_discard(void *save) { } static int context_lex_is_ok(struct isl_context *context) { struct isl_context_lex *clex = (struct isl_context_lex *)context; return !!clex->tab; } /* For each variable in the context tableau, check if the variable can * only attain non-negative values. If so, mark the parameter as non-negative * in the main tableau. This allows for a more direct identification of some * cases of violated constraints. */ static struct isl_tab *tab_detect_nonnegative_parameters(struct isl_tab *tab, struct isl_tab *context_tab) { int i; struct isl_tab_undo *snap; struct isl_vec *ineq = NULL; struct isl_tab_var *var; int n; if (context_tab->n_var == 0) return tab; ineq = isl_vec_alloc(tab->mat->ctx, 1 + context_tab->n_var); if (!ineq) goto error; if (isl_tab_extend_cons(context_tab, 1) < 0) goto error; snap = isl_tab_snap(context_tab); n = 0; isl_seq_clr(ineq->el, ineq->size); for (i = 0; i < context_tab->n_var; ++i) { isl_int_set_si(ineq->el[1 + i], 1); if (isl_tab_add_ineq(context_tab, ineq->el) < 0) goto error; var = &context_tab->con[context_tab->n_con - 1]; if (!context_tab->empty && !isl_tab_min_at_most_neg_one(context_tab, var)) { int j = i; if (i >= tab->n_param) j = i - tab->n_param + tab->n_var - tab->n_div; tab->var[j].is_nonneg = 1; n++; } isl_int_set_si(ineq->el[1 + i], 0); if (isl_tab_rollback(context_tab, snap) < 0) goto error; } if (context_tab->M && n == context_tab->n_var) { context_tab->mat = isl_mat_drop_cols(context_tab->mat, 2, 1); context_tab->M = 0; } isl_vec_free(ineq); return tab; error: isl_vec_free(ineq); isl_tab_free(tab); return NULL; } static struct isl_tab *context_lex_detect_nonnegative_parameters( struct isl_context *context, struct isl_tab *tab) { struct isl_context_lex *clex = (struct isl_context_lex *)context; struct isl_tab_undo *snap; if (!tab) return NULL; snap = isl_tab_snap(clex->tab); if (isl_tab_push_basis(clex->tab) < 0) goto error; tab = tab_detect_nonnegative_parameters(tab, clex->tab); if (isl_tab_rollback(clex->tab, snap) < 0) goto error; return tab; error: isl_tab_free(tab); return NULL; } static void context_lex_invalidate(struct isl_context *context) { struct isl_context_lex *clex = (struct isl_context_lex *)context; isl_tab_free(clex->tab); clex->tab = NULL; } static void context_lex_free(struct isl_context *context) { struct isl_context_lex *clex = (struct isl_context_lex *)context; isl_tab_free(clex->tab); free(clex); } struct isl_context_op isl_context_lex_op = { context_lex_detect_nonnegative_parameters, context_lex_peek_basic_set, context_lex_peek_tab, context_lex_add_eq, context_lex_add_ineq, context_lex_ineq_sign, context_lex_test_ineq, context_lex_get_div, context_lex_add_div, context_lex_detect_equalities, context_lex_best_split, context_lex_is_empty, context_lex_is_ok, context_lex_save, context_lex_restore, context_lex_discard, context_lex_invalidate, context_lex_free, }; static struct isl_tab *context_tab_for_lexmin(struct isl_basic_set *bset) { struct isl_tab *tab; if (!bset) return NULL; tab = tab_for_lexmin((struct isl_basic_map *)bset, NULL, 1, 0); if (!tab) goto error; if (isl_tab_track_bset(tab, bset) < 0) goto error; tab = isl_tab_init_samples(tab); return tab; error: isl_basic_set_free(bset); return NULL; } static struct isl_context *isl_context_lex_alloc(struct isl_basic_set *dom) { struct isl_context_lex *clex; if (!dom) return NULL; clex = isl_alloc_type(dom->ctx, struct isl_context_lex); if (!clex) return NULL; clex->context.op = &isl_context_lex_op; clex->tab = context_tab_for_lexmin(isl_basic_set_copy(dom)); if (restore_lexmin(clex->tab) < 0) goto error; clex->tab = check_integer_feasible(clex->tab); if (!clex->tab) goto error; return &clex->context; error: clex->context.op->free(&clex->context); return NULL; } /* Representation of the context when using generalized basis reduction. * * "shifted" contains the offsets of the unit hypercubes that lie inside the * context. Any rational point in "shifted" can therefore be rounded * up to an integer point in the context. * If the context is constrained by any equality, then "shifted" is not used * as it would be empty. */ struct isl_context_gbr { struct isl_context context; struct isl_tab *tab; struct isl_tab *shifted; struct isl_tab *cone; }; static struct isl_tab *context_gbr_detect_nonnegative_parameters( struct isl_context *context, struct isl_tab *tab) { struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context; if (!tab) return NULL; return tab_detect_nonnegative_parameters(tab, cgbr->tab); } static struct isl_basic_set *context_gbr_peek_basic_set( struct isl_context *context) { struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context; if (!cgbr->tab) return NULL; return isl_tab_peek_bset(cgbr->tab); } static struct isl_tab *context_gbr_peek_tab(struct isl_context *context) { struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context; return cgbr->tab; } /* Initialize the "shifted" tableau of the context, which * contains the constraints of the original tableau shifted * by the sum of all negative coefficients. This ensures * that any rational point in the shifted tableau can * be rounded up to yield an integer point in the original tableau. */ static void gbr_init_shifted(struct isl_context_gbr *cgbr) { int i, j; struct isl_vec *cst; struct isl_basic_set *bset = isl_tab_peek_bset(cgbr->tab); unsigned dim = isl_basic_set_total_dim(bset); cst = isl_vec_alloc(cgbr->tab->mat->ctx, bset->n_ineq); if (!cst) return; for (i = 0; i < bset->n_ineq; ++i) { isl_int_set(cst->el[i], bset->ineq[i][0]); for (j = 0; j < dim; ++j) { if (!isl_int_is_neg(bset->ineq[i][1 + j])) continue; isl_int_add(bset->ineq[i][0], bset->ineq[i][0], bset->ineq[i][1 + j]); } } cgbr->shifted = isl_tab_from_basic_set(bset, 0); for (i = 0; i < bset->n_ineq; ++i) isl_int_set(bset->ineq[i][0], cst->el[i]); isl_vec_free(cst); } /* Check if the shifted tableau is non-empty, and if so * use the sample point to construct an integer point * of the context tableau. */ static struct isl_vec *gbr_get_shifted_sample(struct isl_context_gbr *cgbr) { struct isl_vec *sample; if (!cgbr->shifted) gbr_init_shifted(cgbr); if (!cgbr->shifted) return NULL; if (cgbr->shifted->empty) return isl_vec_alloc(cgbr->tab->mat->ctx, 0); sample = isl_tab_get_sample_value(cgbr->shifted); sample = isl_vec_ceil(sample); return sample; } static struct isl_basic_set *drop_constant_terms(struct isl_basic_set *bset) { int i; if (!bset) return NULL; for (i = 0; i < bset->n_eq; ++i) isl_int_set_si(bset->eq[i][0], 0); for (i = 0; i < bset->n_ineq; ++i) isl_int_set_si(bset->ineq[i][0], 0); return bset; } static int use_shifted(struct isl_context_gbr *cgbr) { if (!cgbr->tab) return 0; return cgbr->tab->bmap->n_eq == 0 && cgbr->tab->bmap->n_div == 0; } static struct isl_vec *gbr_get_sample(struct isl_context_gbr *cgbr) { struct isl_basic_set *bset; struct isl_basic_set *cone; if (isl_tab_sample_is_integer(cgbr->tab)) return isl_tab_get_sample_value(cgbr->tab); if (use_shifted(cgbr)) { struct isl_vec *sample; sample = gbr_get_shifted_sample(cgbr); if (!sample || sample->size > 0) return sample; isl_vec_free(sample); } if (!cgbr->cone) { bset = isl_tab_peek_bset(cgbr->tab); cgbr->cone = isl_tab_from_recession_cone(bset, 0); if (!cgbr->cone) return NULL; if (isl_tab_track_bset(cgbr->cone, isl_basic_set_copy(bset)) < 0) return NULL; } if (isl_tab_detect_implicit_equalities(cgbr->cone) < 0) return NULL; if (cgbr->cone->n_dead == cgbr->cone->n_col) { struct isl_vec *sample; struct isl_tab_undo *snap; if (cgbr->tab->basis) { if (cgbr->tab->basis->n_col != 1 + cgbr->tab->n_var) { isl_mat_free(cgbr->tab->basis); cgbr->tab->basis = NULL; } cgbr->tab->n_zero = 0; cgbr->tab->n_unbounded = 0; } snap = isl_tab_snap(cgbr->tab); sample = isl_tab_sample(cgbr->tab); if (isl_tab_rollback(cgbr->tab, snap) < 0) { isl_vec_free(sample); return NULL; } return sample; } cone = isl_basic_set_dup(isl_tab_peek_bset(cgbr->cone)); cone = drop_constant_terms(cone); cone = isl_basic_set_update_from_tab(cone, cgbr->cone); cone = isl_basic_set_underlying_set(cone); cone = isl_basic_set_gauss(cone, NULL); bset = isl_basic_set_dup(isl_tab_peek_bset(cgbr->tab)); bset = isl_basic_set_update_from_tab(bset, cgbr->tab); bset = isl_basic_set_underlying_set(bset); bset = isl_basic_set_gauss(bset, NULL); return isl_basic_set_sample_with_cone(bset, cone); } static void check_gbr_integer_feasible(struct isl_context_gbr *cgbr) { struct isl_vec *sample; if (!cgbr->tab) return; if (cgbr->tab->empty) return; sample = gbr_get_sample(cgbr); if (!sample) goto error; if (sample->size == 0) { isl_vec_free(sample); if (isl_tab_mark_empty(cgbr->tab) < 0) goto error; return; } if (isl_tab_add_sample(cgbr->tab, sample) < 0) goto error; return; error: isl_tab_free(cgbr->tab); cgbr->tab = NULL; } static struct isl_tab *add_gbr_eq(struct isl_tab *tab, isl_int *eq) { if (!tab) return NULL; if (isl_tab_extend_cons(tab, 2) < 0) goto error; if (isl_tab_add_eq(tab, eq) < 0) goto error; return tab; error: isl_tab_free(tab); return NULL; } /* Add the equality described by "eq" to the context. * If "check" is set, then we check if the context is empty after * adding the equality. * If "update" is set, then we check if the samples are still valid. * * We do not explicitly add shifted copies of the equality to * cgbr->shifted since they would conflict with each other. * Instead, we directly mark cgbr->shifted empty. */ static void context_gbr_add_eq(struct isl_context *context, isl_int *eq, int check, int update) { struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context; cgbr->tab = add_gbr_eq(cgbr->tab, eq); if (cgbr->shifted && !cgbr->shifted->empty && use_shifted(cgbr)) { if (isl_tab_mark_empty(cgbr->shifted) < 0) goto error; } if (cgbr->cone && cgbr->cone->n_col != cgbr->cone->n_dead) { if (isl_tab_extend_cons(cgbr->cone, 2) < 0) goto error; if (isl_tab_add_eq(cgbr->cone, eq) < 0) goto error; } if (check) { int v = tab_has_valid_sample(cgbr->tab, eq, 1); if (v < 0) goto error; if (!v) check_gbr_integer_feasible(cgbr); } if (update) cgbr->tab = check_samples(cgbr->tab, eq, 1); return; error: isl_tab_free(cgbr->tab); cgbr->tab = NULL; } static void add_gbr_ineq(struct isl_context_gbr *cgbr, isl_int *ineq) { if (!cgbr->tab) return; if (isl_tab_extend_cons(cgbr->tab, 1) < 0) goto error; if (isl_tab_add_ineq(cgbr->tab, ineq) < 0) goto error; if (cgbr->shifted && !cgbr->shifted->empty && use_shifted(cgbr)) { int i; unsigned dim; dim = isl_basic_map_total_dim(cgbr->tab->bmap); if (isl_tab_extend_cons(cgbr->shifted, 1) < 0) goto error; for (i = 0; i < dim; ++i) { if (!isl_int_is_neg(ineq[1 + i])) continue; isl_int_add(ineq[0], ineq[0], ineq[1 + i]); } if (isl_tab_add_ineq(cgbr->shifted, ineq) < 0) goto error; for (i = 0; i < dim; ++i) { if (!isl_int_is_neg(ineq[1 + i])) continue; isl_int_sub(ineq[0], ineq[0], ineq[1 + i]); } } if (cgbr->cone && cgbr->cone->n_col != cgbr->cone->n_dead) { if (isl_tab_extend_cons(cgbr->cone, 1) < 0) goto error; if (isl_tab_add_ineq(cgbr->cone, ineq) < 0) goto error; } return; error: isl_tab_free(cgbr->tab); cgbr->tab = NULL; } static void context_gbr_add_ineq(struct isl_context *context, isl_int *ineq, int check, int update) { struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context; add_gbr_ineq(cgbr, ineq); if (!cgbr->tab) return; if (check) { int v = tab_has_valid_sample(cgbr->tab, ineq, 0); if (v < 0) goto error; if (!v) check_gbr_integer_feasible(cgbr); } if (update) cgbr->tab = check_samples(cgbr->tab, ineq, 0); return; error: isl_tab_free(cgbr->tab); cgbr->tab = NULL; } static int context_gbr_add_ineq_wrap(void *user, isl_int *ineq) { struct isl_context *context = (struct isl_context *)user; context_gbr_add_ineq(context, ineq, 0, 0); return context->op->is_ok(context) ? 0 : -1; } static enum isl_tab_row_sign context_gbr_ineq_sign(struct isl_context *context, isl_int *ineq, int strict) { struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context; return tab_ineq_sign(cgbr->tab, ineq, strict); } /* Check whether "ineq" can be added to the tableau without rendering * it infeasible. */ static int context_gbr_test_ineq(struct isl_context *context, isl_int *ineq) { struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context; struct isl_tab_undo *snap; struct isl_tab_undo *shifted_snap = NULL; struct isl_tab_undo *cone_snap = NULL; int feasible; if (!cgbr->tab) return -1; if (isl_tab_extend_cons(cgbr->tab, 1) < 0) return -1; snap = isl_tab_snap(cgbr->tab); if (cgbr->shifted) shifted_snap = isl_tab_snap(cgbr->shifted); if (cgbr->cone) cone_snap = isl_tab_snap(cgbr->cone); add_gbr_ineq(cgbr, ineq); check_gbr_integer_feasible(cgbr); if (!cgbr->tab) return -1; feasible = !cgbr->tab->empty; if (isl_tab_rollback(cgbr->tab, snap) < 0) return -1; if (shifted_snap) { if (isl_tab_rollback(cgbr->shifted, shifted_snap)) return -1; } else if (cgbr->shifted) { isl_tab_free(cgbr->shifted); cgbr->shifted = NULL; } if (cone_snap) { if (isl_tab_rollback(cgbr->cone, cone_snap)) return -1; } else if (cgbr->cone) { isl_tab_free(cgbr->cone); cgbr->cone = NULL; } return feasible; } /* Return the column of the last of the variables associated to * a column that has a non-zero coefficient. * This function is called in a context where only coefficients * of parameters or divs can be non-zero. */ static int last_non_zero_var_col(struct isl_tab *tab, isl_int *p) { int i; int col; if (tab->n_var == 0) return -1; for (i = tab->n_var - 1; i >= 0; --i) { if (i >= tab->n_param && i < tab->n_var - tab->n_div) continue; if (tab->var[i].is_row) continue; col = tab->var[i].index; if (!isl_int_is_zero(p[col])) return col; } return -1; } /* Look through all the recently added equalities in the context * to see if we can propagate any of them to the main tableau. * * The newly added equalities in the context are encoded as pairs * of inequalities starting at inequality "first". * * We tentatively add each of these equalities to the main tableau * and if this happens to result in a row with a final coefficient * that is one or negative one, we use it to kill a column * in the main tableau. Otherwise, we discard the tentatively * added row. * * Return 0 on success and -1 on failure. */ static int propagate_equalities(struct isl_context_gbr *cgbr, struct isl_tab *tab, unsigned first) { int i; struct isl_vec *eq = NULL; eq = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_var); if (!eq) goto error; if (isl_tab_extend_cons(tab, (cgbr->tab->bmap->n_ineq - first)/2) < 0) goto error; isl_seq_clr(eq->el + 1 + tab->n_param, tab->n_var - tab->n_param - tab->n_div); for (i = first; i < cgbr->tab->bmap->n_ineq; i += 2) { int j; int r; struct isl_tab_undo *snap; snap = isl_tab_snap(tab); isl_seq_cpy(eq->el, cgbr->tab->bmap->ineq[i], 1 + tab->n_param); isl_seq_cpy(eq->el + 1 + tab->n_var - tab->n_div, cgbr->tab->bmap->ineq[i] + 1 + tab->n_param, tab->n_div); r = isl_tab_add_row(tab, eq->el); if (r < 0) goto error; r = tab->con[r].index; j = last_non_zero_var_col(tab, tab->mat->row[r] + 2 + tab->M); if (j < 0 || j < tab->n_dead || !isl_int_is_one(tab->mat->row[r][0]) || (!isl_int_is_one(tab->mat->row[r][2 + tab->M + j]) && !isl_int_is_negone(tab->mat->row[r][2 + tab->M + j]))) { if (isl_tab_rollback(tab, snap) < 0) goto error; continue; } if (isl_tab_pivot(tab, r, j) < 0) goto error; if (isl_tab_kill_col(tab, j) < 0) goto error; if (restore_lexmin(tab) < 0) goto error; } isl_vec_free(eq); return 0; error: isl_vec_free(eq); isl_tab_free(cgbr->tab); cgbr->tab = NULL; return -1; } static int context_gbr_detect_equalities(struct isl_context *context, struct isl_tab *tab) { struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context; struct isl_ctx *ctx; unsigned n_ineq; ctx = cgbr->tab->mat->ctx; if (!cgbr->cone) { struct isl_basic_set *bset = isl_tab_peek_bset(cgbr->tab); cgbr->cone = isl_tab_from_recession_cone(bset, 0); if (!cgbr->cone) goto error; if (isl_tab_track_bset(cgbr->cone, isl_basic_set_copy(bset)) < 0) goto error; } if (isl_tab_detect_implicit_equalities(cgbr->cone) < 0) goto error; n_ineq = cgbr->tab->bmap->n_ineq; cgbr->tab = isl_tab_detect_equalities(cgbr->tab, cgbr->cone); if (!cgbr->tab) return -1; if (cgbr->tab->bmap->n_ineq > n_ineq && propagate_equalities(cgbr, tab, n_ineq) < 0) return -1; return 0; error: isl_tab_free(cgbr->tab); cgbr->tab = NULL; return -1; } static int context_gbr_get_div(struct isl_context *context, struct isl_tab *tab, struct isl_vec *div) { return get_div(tab, context, div); } static int context_gbr_add_div(struct isl_context *context, struct isl_vec *div) { struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context; if (cgbr->cone) { int k; if (isl_tab_extend_cons(cgbr->cone, 3) < 0) return -1; if (isl_tab_extend_vars(cgbr->cone, 1) < 0) return -1; if (isl_tab_allocate_var(cgbr->cone) <0) return -1; cgbr->cone->bmap = isl_basic_map_extend_space(cgbr->cone->bmap, isl_basic_map_get_space(cgbr->cone->bmap), 1, 0, 2); k = isl_basic_map_alloc_div(cgbr->cone->bmap); if (k < 0) return -1; isl_seq_cpy(cgbr->cone->bmap->div[k], div->el, div->size); if (isl_tab_push(cgbr->cone, isl_tab_undo_bmap_div) < 0) return -1; } return context_tab_add_div(cgbr->tab, div, context_gbr_add_ineq_wrap, context); } static int context_gbr_best_split(struct isl_context *context, struct isl_tab *tab) { struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context; struct isl_tab_undo *snap; int r; snap = isl_tab_snap(cgbr->tab); r = best_split(tab, cgbr->tab); if (r >= 0 && isl_tab_rollback(cgbr->tab, snap) < 0) return -1; return r; } static int context_gbr_is_empty(struct isl_context *context) { struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context; if (!cgbr->tab) return -1; return cgbr->tab->empty; } struct isl_gbr_tab_undo { struct isl_tab_undo *tab_snap; struct isl_tab_undo *shifted_snap; struct isl_tab_undo *cone_snap; }; static void *context_gbr_save(struct isl_context *context) { struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context; struct isl_gbr_tab_undo *snap; if (!cgbr->tab) return NULL; snap = isl_alloc_type(cgbr->tab->mat->ctx, struct isl_gbr_tab_undo); if (!snap) return NULL; snap->tab_snap = isl_tab_snap(cgbr->tab); if (isl_tab_save_samples(cgbr->tab) < 0) goto error; if (cgbr->shifted) snap->shifted_snap = isl_tab_snap(cgbr->shifted); else snap->shifted_snap = NULL; if (cgbr->cone) snap->cone_snap = isl_tab_snap(cgbr->cone); else snap->cone_snap = NULL; return snap; error: free(snap); return NULL; } static void context_gbr_restore(struct isl_context *context, void *save) { struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context; struct isl_gbr_tab_undo *snap = (struct isl_gbr_tab_undo *)save; if (!snap) goto error; if (isl_tab_rollback(cgbr->tab, snap->tab_snap) < 0) { isl_tab_free(cgbr->tab); cgbr->tab = NULL; } if (snap->shifted_snap) { if (isl_tab_rollback(cgbr->shifted, snap->shifted_snap) < 0) goto error; } else if (cgbr->shifted) { isl_tab_free(cgbr->shifted); cgbr->shifted = NULL; } if (snap->cone_snap) { if (isl_tab_rollback(cgbr->cone, snap->cone_snap) < 0) goto error; } else if (cgbr->cone) { isl_tab_free(cgbr->cone); cgbr->cone = NULL; } free(snap); return; error: free(snap); isl_tab_free(cgbr->tab); cgbr->tab = NULL; } static void context_gbr_discard(void *save) { struct isl_gbr_tab_undo *snap = (struct isl_gbr_tab_undo *)save; free(snap); } static int context_gbr_is_ok(struct isl_context *context) { struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context; return !!cgbr->tab; } static void context_gbr_invalidate(struct isl_context *context) { struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context; isl_tab_free(cgbr->tab); cgbr->tab = NULL; } static void context_gbr_free(struct isl_context *context) { struct isl_context_gbr *cgbr = (struct isl_context_gbr *)context; isl_tab_free(cgbr->tab); isl_tab_free(cgbr->shifted); isl_tab_free(cgbr->cone); free(cgbr); } struct isl_context_op isl_context_gbr_op = { context_gbr_detect_nonnegative_parameters, context_gbr_peek_basic_set, context_gbr_peek_tab, context_gbr_add_eq, context_gbr_add_ineq, context_gbr_ineq_sign, context_gbr_test_ineq, context_gbr_get_div, context_gbr_add_div, context_gbr_detect_equalities, context_gbr_best_split, context_gbr_is_empty, context_gbr_is_ok, context_gbr_save, context_gbr_restore, context_gbr_discard, context_gbr_invalidate, context_gbr_free, }; static struct isl_context *isl_context_gbr_alloc(struct isl_basic_set *dom) { struct isl_context_gbr *cgbr; if (!dom) return NULL; cgbr = isl_calloc_type(dom->ctx, struct isl_context_gbr); if (!cgbr) return NULL; cgbr->context.op = &isl_context_gbr_op; cgbr->shifted = NULL; cgbr->cone = NULL; cgbr->tab = isl_tab_from_basic_set(dom, 1); cgbr->tab = isl_tab_init_samples(cgbr->tab); if (!cgbr->tab) goto error; check_gbr_integer_feasible(cgbr); return &cgbr->context; error: cgbr->context.op->free(&cgbr->context); return NULL; } static struct isl_context *isl_context_alloc(struct isl_basic_set *dom) { if (!dom) return NULL; if (dom->ctx->opt->context == ISL_CONTEXT_LEXMIN) return isl_context_lex_alloc(dom); else return isl_context_gbr_alloc(dom); } /* Construct an isl_sol_map structure for accumulating the solution. * If track_empty is set, then we also keep track of the parts * of the context where there is no solution. * If max is set, then we are solving a maximization, rather than * a minimization problem, which means that the variables in the * tableau have value "M - x" rather than "M + x". */ static struct isl_sol *sol_map_init(struct isl_basic_map *bmap, struct isl_basic_set *dom, int track_empty, int max) { struct isl_sol_map *sol_map = NULL; if (!bmap) goto error; sol_map = isl_calloc_type(bmap->ctx, struct isl_sol_map); if (!sol_map) goto error; sol_map->sol.rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL); sol_map->sol.dec_level.callback.run = &sol_dec_level_wrap; sol_map->sol.dec_level.sol = &sol_map->sol; sol_map->sol.max = max; sol_map->sol.n_out = isl_basic_map_dim(bmap, isl_dim_out); sol_map->sol.add = &sol_map_add_wrap; sol_map->sol.add_empty = track_empty ? &sol_map_add_empty_wrap : NULL; sol_map->sol.free = &sol_map_free_wrap; sol_map->map = isl_map_alloc_space(isl_basic_map_get_space(bmap), 1, ISL_MAP_DISJOINT); if (!sol_map->map) goto error; sol_map->sol.context = isl_context_alloc(dom); if (!sol_map->sol.context) goto error; if (track_empty) { sol_map->empty = isl_set_alloc_space(isl_basic_set_get_space(dom), 1, ISL_SET_DISJOINT); if (!sol_map->empty) goto error; } isl_basic_set_free(dom); return &sol_map->sol; error: isl_basic_set_free(dom); sol_map_free(sol_map); return NULL; } /* Check whether all coefficients of (non-parameter) variables * are non-positive, meaning that no pivots can be performed on the row. */ static int is_critical(struct isl_tab *tab, int row) { int j; unsigned off = 2 + tab->M; for (j = tab->n_dead; j < tab->n_col; ++j) { if (tab->col_var[j] >= 0 && (tab->col_var[j] < tab->n_param || tab->col_var[j] >= tab->n_var - tab->n_div)) continue; if (isl_int_is_pos(tab->mat->row[row][off + j])) return 0; } return 1; } /* Check whether the inequality represented by vec is strict over the integers, * i.e., there are no integer values satisfying the constraint with * equality. This happens if the gcd of the coefficients is not a divisor * of the constant term. If so, scale the constraint down by the gcd * of the coefficients. */ static int is_strict(struct isl_vec *vec) { isl_int gcd; int strict = 0; isl_int_init(gcd); isl_seq_gcd(vec->el + 1, vec->size - 1, &gcd); if (!isl_int_is_one(gcd)) { strict = !isl_int_is_divisible_by(vec->el[0], gcd); isl_int_fdiv_q(vec->el[0], vec->el[0], gcd); isl_seq_scale_down(vec->el + 1, vec->el + 1, gcd, vec->size-1); } isl_int_clear(gcd); return strict; } /* Determine the sign of the given row of the main tableau. * The result is one of * isl_tab_row_pos: always non-negative; no pivot needed * isl_tab_row_neg: always non-positive; pivot * isl_tab_row_any: can be both positive and negative; split * * We first handle some simple cases * - the row sign may be known already * - the row may be obviously non-negative * - the parametric constant may be equal to that of another row * for which we know the sign. This sign will be either "pos" or * "any". If it had been "neg" then we would have pivoted before. * * If none of these cases hold, we check the value of the row for each * of the currently active samples. Based on the signs of these values * we make an initial determination of the sign of the row. * * all zero -> unk(nown) * all non-negative -> pos * all non-positive -> neg * both negative and positive -> all * * If we end up with "all", we are done. * Otherwise, we perform a check for positive and/or negative * values as follows. * * samples neg unk pos * <0 ? Y N Y N * pos any pos * >0 ? Y N Y N * any neg any neg * * There is no special sign for "zero", because we can usually treat zero * as either non-negative or non-positive, whatever works out best. * However, if the row is "critical", meaning that pivoting is impossible * then we don't want to limp zero with the non-positive case, because * then we we would lose the solution for those values of the parameters * where the value of the row is zero. Instead, we treat 0 as non-negative * ensuring a split if the row can attain both zero and negative values. * The same happens when the original constraint was one that could not * be satisfied with equality by any integer values of the parameters. * In this case, we normalize the constraint, but then a value of zero * for the normalized constraint is actually a positive value for the * original constraint, so again we need to treat zero as non-negative. * In both these cases, we have the following decision tree instead: * * all non-negative -> pos * all negative -> neg * both negative and non-negative -> all * * samples neg pos * <0 ? Y N * any pos * >=0 ? Y N * any neg */ static enum isl_tab_row_sign row_sign(struct isl_tab *tab, struct isl_sol *sol, int row) { struct isl_vec *ineq = NULL; enum isl_tab_row_sign res = isl_tab_row_unknown; int critical; int strict; int row2; if (tab->row_sign[row] != isl_tab_row_unknown) return tab->row_sign[row]; if (is_obviously_nonneg(tab, row)) return isl_tab_row_pos; for (row2 = tab->n_redundant; row2 < tab->n_row; ++row2) { if (tab->row_sign[row2] == isl_tab_row_unknown) continue; if (identical_parameter_line(tab, row, row2)) return tab->row_sign[row2]; } critical = is_critical(tab, row); ineq = get_row_parameter_ineq(tab, row); if (!ineq) goto error; strict = is_strict(ineq); res = sol->context->op->ineq_sign(sol->context, ineq->el, critical || strict); if (res == isl_tab_row_unknown || res == isl_tab_row_pos) { /* test for negative values */ int feasible; isl_seq_neg(ineq->el, ineq->el, ineq->size); isl_int_sub_ui(ineq->el[0], ineq->el[0], 1); feasible = sol->context->op->test_ineq(sol->context, ineq->el); if (feasible < 0) goto error; if (!feasible) res = isl_tab_row_pos; else res = (res == isl_tab_row_unknown) ? isl_tab_row_neg : isl_tab_row_any; if (res == isl_tab_row_neg) { isl_seq_neg(ineq->el, ineq->el, ineq->size); isl_int_sub_ui(ineq->el[0], ineq->el[0], 1); } } if (res == isl_tab_row_neg) { /* test for positive values */ int feasible; if (!critical && !strict) isl_int_sub_ui(ineq->el[0], ineq->el[0], 1); feasible = sol->context->op->test_ineq(sol->context, ineq->el); if (feasible < 0) goto error; if (feasible) res = isl_tab_row_any; } isl_vec_free(ineq); return res; error: isl_vec_free(ineq); return isl_tab_row_unknown; } static void find_solutions(struct isl_sol *sol, struct isl_tab *tab); /* Find solutions for values of the parameters that satisfy the given * inequality. * * We currently take a snapshot of the context tableau that is reset * when we return from this function, while we make a copy of the main * tableau, leaving the original main tableau untouched. * These are fairly arbitrary choices. Making a copy also of the context * tableau would obviate the need to undo any changes made to it later, * while taking a snapshot of the main tableau could reduce memory usage. * If we were to switch to taking a snapshot of the main tableau, * we would have to keep in mind that we need to save the row signs * and that we need to do this before saving the current basis * such that the basis has been restore before we restore the row signs. */ static void find_in_pos(struct isl_sol *sol, struct isl_tab *tab, isl_int *ineq) { void *saved; if (!sol->context) goto error; saved = sol->context->op->save(sol->context); tab = isl_tab_dup(tab); if (!tab) goto error; sol->context->op->add_ineq(sol->context, ineq, 0, 1); find_solutions(sol, tab); if (!sol->error) sol->context->op->restore(sol->context, saved); else sol->context->op->discard(saved); return; error: sol->error = 1; } /* Record the absence of solutions for those values of the parameters * that do not satisfy the given inequality with equality. */ static void no_sol_in_strict(struct isl_sol *sol, struct isl_tab *tab, struct isl_vec *ineq) { int empty; void *saved; if (!sol->context || sol->error) goto error; saved = sol->context->op->save(sol->context); isl_int_sub_ui(ineq->el[0], ineq->el[0], 1); sol->context->op->add_ineq(sol->context, ineq->el, 1, 0); if (!sol->context) goto error; empty = tab->empty; tab->empty = 1; sol_add(sol, tab); tab->empty = empty; isl_int_add_ui(ineq->el[0], ineq->el[0], 1); sol->context->op->restore(sol->context, saved); return; error: sol->error = 1; } /* Compute the lexicographic minimum of the set represented by the main * tableau "tab" within the context "sol->context_tab". * On entry the sample value of the main tableau is lexicographically * less than or equal to this lexicographic minimum. * Pivots are performed until a feasible point is found, which is then * necessarily equal to the minimum, or until the tableau is found to * be infeasible. Some pivots may need to be performed for only some * feasible values of the context tableau. If so, the context tableau * is split into a part where the pivot is needed and a part where it is not. * * Whenever we enter the main loop, the main tableau is such that no * "obvious" pivots need to be performed on it, where "obvious" means * that the given row can be seen to be negative without looking at * the context tableau. In particular, for non-parametric problems, * no pivots need to be performed on the main tableau. * The caller of find_solutions is responsible for making this property * hold prior to the first iteration of the loop, while restore_lexmin * is called before every other iteration. * * Inside the main loop, we first examine the signs of the rows of * the main tableau within the context of the context tableau. * If we find a row that is always non-positive for all values of * the parameters satisfying the context tableau and negative for at * least one value of the parameters, we perform the appropriate pivot * and start over. An exception is the case where no pivot can be * performed on the row. In this case, we require that the sign of * the row is negative for all values of the parameters (rather than just * non-positive). This special case is handled inside row_sign, which * will say that the row can have any sign if it determines that it can * attain both negative and zero values. * * If we can't find a row that always requires a pivot, but we can find * one or more rows that require a pivot for some values of the parameters * (i.e., the row can attain both positive and negative signs), then we split * the context tableau into two parts, one where we force the sign to be * non-negative and one where we force is to be negative. * The non-negative part is handled by a recursive call (through find_in_pos). * Upon returning from this call, we continue with the negative part and * perform the required pivot. * * If no such rows can be found, all rows are non-negative and we have * found a (rational) feasible point. If we only wanted a rational point * then we are done. * Otherwise, we check if all values of the sample point of the tableau * are integral for the variables. If so, we have found the minimal * integral point and we are done. * If the sample point is not integral, then we need to make a distinction * based on whether the constant term is non-integral or the coefficients * of the parameters. Furthermore, in order to decide how to handle * the non-integrality, we also need to know whether the coefficients * of the other columns in the tableau are integral. This leads * to the following table. The first two rows do not correspond * to a non-integral sample point and are only mentioned for completeness. * * constant parameters other * * int int int | * int int rat | -> no problem * * rat int int -> fail * * rat int rat -> cut * * int rat rat | * rat rat rat | -> parametric cut * * int rat int | * rat rat int | -> split context * * If the parametric constant is completely integral, then there is nothing * to be done. If the constant term is non-integral, but all the other * coefficient are integral, then there is nothing that can be done * and the tableau has no integral solution. * If, on the other hand, one or more of the other columns have rational * coefficients, but the parameter coefficients are all integral, then * we can perform a regular (non-parametric) cut. * Finally, if there is any parameter coefficient that is non-integral, * then we need to involve the context tableau. There are two cases here. * If at least one other column has a rational coefficient, then we * can perform a parametric cut in the main tableau by adding a new * integer division in the context tableau. * If all other columns have integral coefficients, then we need to * enforce that the rational combination of parameters (c + \sum a_i y_i)/m * is always integral. We do this by introducing an integer division * q = floor((c + \sum a_i y_i)/m) and stipulating that its argument should * always be integral in the context tableau, i.e., m q = c + \sum a_i y_i. * Since q is expressed in the tableau as * c + \sum a_i y_i - m q >= 0 * -c - \sum a_i y_i + m q + m - 1 >= 0 * it is sufficient to add the inequality * -c - \sum a_i y_i + m q >= 0 * In the part of the context where this inequality does not hold, the * main tableau is marked as being empty. */ static void find_solutions(struct isl_sol *sol, struct isl_tab *tab) { struct isl_context *context; int r; if (!tab || sol->error) goto error; context = sol->context; if (tab->empty) goto done; if (context->op->is_empty(context)) goto done; for (r = 0; r >= 0 && tab && !tab->empty; r = restore_lexmin(tab)) { int flags; int row; enum isl_tab_row_sign sgn; int split = -1; int n_split = 0; for (row = tab->n_redundant; row < tab->n_row; ++row) { if (!isl_tab_var_from_row(tab, row)->is_nonneg) continue; sgn = row_sign(tab, sol, row); if (!sgn) goto error; tab->row_sign[row] = sgn; if (sgn == isl_tab_row_any) n_split++; if (sgn == isl_tab_row_any && split == -1) split = row; if (sgn == isl_tab_row_neg) break; } if (row < tab->n_row) continue; if (split != -1) { struct isl_vec *ineq; if (n_split != 1) split = context->op->best_split(context, tab); if (split < 0) goto error; ineq = get_row_parameter_ineq(tab, split); if (!ineq) goto error; is_strict(ineq); for (row = tab->n_redundant; row < tab->n_row; ++row) { if (!isl_tab_var_from_row(tab, row)->is_nonneg) continue; if (tab->row_sign[row] == isl_tab_row_any) tab->row_sign[row] = isl_tab_row_unknown; } tab->row_sign[split] = isl_tab_row_pos; sol_inc_level(sol); find_in_pos(sol, tab, ineq->el); tab->row_sign[split] = isl_tab_row_neg; row = split; isl_seq_neg(ineq->el, ineq->el, ineq->size); isl_int_sub_ui(ineq->el[0], ineq->el[0], 1); if (!sol->error) context->op->add_ineq(context, ineq->el, 0, 1); isl_vec_free(ineq); if (sol->error) goto error; continue; } if (tab->rational) break; row = first_non_integer_row(tab, &flags); if (row < 0) break; if (ISL_FL_ISSET(flags, I_PAR)) { if (ISL_FL_ISSET(flags, I_VAR)) { if (isl_tab_mark_empty(tab) < 0) goto error; break; } row = add_cut(tab, row); } else if (ISL_FL_ISSET(flags, I_VAR)) { struct isl_vec *div; struct isl_vec *ineq; int d; div = get_row_split_div(tab, row); if (!div) goto error; d = context->op->get_div(context, tab, div); isl_vec_free(div); if (d < 0) goto error; ineq = ineq_for_div(context->op->peek_basic_set(context), d); if (!ineq) goto error; sol_inc_level(sol); no_sol_in_strict(sol, tab, ineq); isl_seq_neg(ineq->el, ineq->el, ineq->size); context->op->add_ineq(context, ineq->el, 1, 1); isl_vec_free(ineq); if (sol->error || !context->op->is_ok(context)) goto error; tab = set_row_cst_to_div(tab, row, d); if (context->op->is_empty(context)) break; } else row = add_parametric_cut(tab, row, context); if (row < 0) goto error; } if (r < 0) goto error; done: sol_add(sol, tab); isl_tab_free(tab); return; error: isl_tab_free(tab); sol->error = 1; } /* Does "sol" contain a pair of partial solutions that could potentially * be merged? * * We currently only check that "sol" is not in an error state * and that there are at least two partial solutions of which the final two * are defined at the same level. */ static int sol_has_mergeable_solutions(struct isl_sol *sol) { if (sol->error) return 0; if (!sol->partial) return 0; if (!sol->partial->next) return 0; return sol->partial->level == sol->partial->next->level; } /* Compute the lexicographic minimum of the set represented by the main * tableau "tab" within the context "sol->context_tab". * * As a preprocessing step, we first transfer all the purely parametric * equalities from the main tableau to the context tableau, i.e., * parameters that have been pivoted to a row. * These equalities are ignored by the main algorithm, because the * corresponding rows may not be marked as being non-negative. * In parts of the context where the added equality does not hold, * the main tableau is marked as being empty. * * Before we embark on the actual computation, we save a copy * of the context. When we return, we check if there are any * partial solutions that can potentially be merged. If so, * we perform a rollback to the initial state of the context. * The merging of partial solutions happens inside calls to * sol_dec_level that are pushed onto the undo stack of the context. * If there are no partial solutions that can potentially be merged * then the rollback is skipped as it would just be wasted effort. */ static void find_solutions_main(struct isl_sol *sol, struct isl_tab *tab) { int row; void *saved; if (!tab) goto error; sol->level = 0; for (row = tab->n_redundant; row < tab->n_row; ++row) { int p; struct isl_vec *eq; if (tab->row_var[row] < 0) continue; if (tab->row_var[row] >= tab->n_param && tab->row_var[row] < tab->n_var - tab->n_div) continue; if (tab->row_var[row] < tab->n_param) p = tab->row_var[row]; else p = tab->row_var[row] + tab->n_param - (tab->n_var - tab->n_div); eq = isl_vec_alloc(tab->mat->ctx, 1+tab->n_param+tab->n_div); if (!eq) goto error; get_row_parameter_line(tab, row, eq->el); isl_int_neg(eq->el[1 + p], tab->mat->row[row][0]); eq = isl_vec_normalize(eq); sol_inc_level(sol); no_sol_in_strict(sol, tab, eq); isl_seq_neg(eq->el, eq->el, eq->size); sol_inc_level(sol); no_sol_in_strict(sol, tab, eq); isl_seq_neg(eq->el, eq->el, eq->size); sol->context->op->add_eq(sol->context, eq->el, 1, 1); isl_vec_free(eq); if (isl_tab_mark_redundant(tab, row) < 0) goto error; if (sol->context->op->is_empty(sol->context)) break; row = tab->n_redundant - 1; } saved = sol->context->op->save(sol->context); find_solutions(sol, tab); if (sol_has_mergeable_solutions(sol)) sol->context->op->restore(sol->context, saved); else sol->context->op->discard(saved); sol->level = 0; sol_pop(sol); return; error: isl_tab_free(tab); sol->error = 1; } /* Check if integer division "div" of "dom" also occurs in "bmap". * If so, return its position within the divs. * If not, return -1. */ static int find_context_div(struct isl_basic_map *bmap, struct isl_basic_set *dom, unsigned div) { int i; unsigned b_dim = isl_space_dim(bmap->dim, isl_dim_all); unsigned d_dim = isl_space_dim(dom->dim, isl_dim_all); if (isl_int_is_zero(dom->div[div][0])) return -1; if (isl_seq_first_non_zero(dom->div[div] + 2 + d_dim, dom->n_div) != -1) return -1; for (i = 0; i < bmap->n_div; ++i) { if (isl_int_is_zero(bmap->div[i][0])) continue; if (isl_seq_first_non_zero(bmap->div[i] + 2 + d_dim, (b_dim - d_dim) + bmap->n_div) != -1) continue; if (isl_seq_eq(bmap->div[i], dom->div[div], 2 + d_dim)) return i; } return -1; } /* The correspondence between the variables in the main tableau, * the context tableau, and the input map and domain is as follows. * The first n_param and the last n_div variables of the main tableau * form the variables of the context tableau. * In the basic map, these n_param variables correspond to the * parameters and the input dimensions. In the domain, they correspond * to the parameters and the set dimensions. * The n_div variables correspond to the integer divisions in the domain. * To ensure that everything lines up, we may need to copy some of the * integer divisions of the domain to the map. These have to be placed * in the same order as those in the context and they have to be placed * after any other integer divisions that the map may have. * This function performs the required reordering. */ static struct isl_basic_map *align_context_divs(struct isl_basic_map *bmap, struct isl_basic_set *dom) { int i; int common = 0; int other; for (i = 0; i < dom->n_div; ++i) if (find_context_div(bmap, dom, i) != -1) common++; other = bmap->n_div - common; if (dom->n_div - common > 0) { bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim), dom->n_div - common, 0, 0); if (!bmap) return NULL; } for (i = 0; i < dom->n_div; ++i) { int pos = find_context_div(bmap, dom, i); if (pos < 0) { pos = isl_basic_map_alloc_div(bmap); if (pos < 0) goto error; isl_int_set_si(bmap->div[pos][0], 0); } if (pos != other + i) isl_basic_map_swap_div(bmap, pos, other + i); } return bmap; error: isl_basic_map_free(bmap); return NULL; } /* Base case of isl_tab_basic_map_partial_lexopt, after removing * some obvious symmetries. * * We make sure the divs in the domain are properly ordered, * because they will be added one by one in the given order * during the construction of the solution map. */ static struct isl_sol *basic_map_partial_lexopt_base( __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom, __isl_give isl_set **empty, int max, struct isl_sol *(*init)(__isl_keep isl_basic_map *bmap, __isl_take isl_basic_set *dom, int track_empty, int max)) { struct isl_tab *tab; struct isl_sol *sol = NULL; struct isl_context *context; if (dom->n_div) { dom = isl_basic_set_order_divs(dom); bmap = align_context_divs(bmap, dom); } sol = init(bmap, dom, !!empty, max); if (!sol) goto error; context = sol->context; if (isl_basic_set_plain_is_empty(context->op->peek_basic_set(context))) /* nothing */; else if (isl_basic_map_plain_is_empty(bmap)) { if (sol->add_empty) sol->add_empty(sol, isl_basic_set_copy(context->op->peek_basic_set(context))); } else { tab = tab_for_lexmin(bmap, context->op->peek_basic_set(context), 1, max); tab = context->op->detect_nonnegative_parameters(context, tab); find_solutions_main(sol, tab); } if (sol->error) goto error; isl_basic_map_free(bmap); return sol; error: sol_free(sol); isl_basic_map_free(bmap); return NULL; } /* Base case of isl_tab_basic_map_partial_lexopt, after removing * some obvious symmetries. * * We call basic_map_partial_lexopt_base and extract the results. */ static __isl_give isl_map *basic_map_partial_lexopt_base_map( __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom, __isl_give isl_set **empty, int max) { isl_map *result = NULL; struct isl_sol *sol; struct isl_sol_map *sol_map; sol = basic_map_partial_lexopt_base(bmap, dom, empty, max, &sol_map_init); if (!sol) return NULL; sol_map = (struct isl_sol_map *) sol; result = isl_map_copy(sol_map->map); if (empty) *empty = isl_set_copy(sol_map->empty); sol_free(&sol_map->sol); return result; } /* Structure used during detection of parallel constraints. * n_in: number of "input" variables: isl_dim_param + isl_dim_in * n_out: number of "output" variables: isl_dim_out + isl_dim_div * val: the coefficients of the output variables */ struct isl_constraint_equal_info { isl_basic_map *bmap; unsigned n_in; unsigned n_out; isl_int *val; }; /* Check whether the coefficients of the output variables * of the constraint in "entry" are equal to info->val. */ static int constraint_equal(const void *entry, const void *val) { isl_int **row = (isl_int **)entry; const struct isl_constraint_equal_info *info = val; return isl_seq_eq((*row) + 1 + info->n_in, info->val, info->n_out); } /* Check whether "bmap" has a pair of constraints that have * the same coefficients for the output variables. * Note that the coefficients of the existentially quantified * variables need to be zero since the existentially quantified * of the result are usually not the same as those of the input. * the isl_dim_out and isl_dim_div dimensions. * If so, return 1 and return the row indices of the two constraints * in *first and *second. */ static int parallel_constraints(__isl_keep isl_basic_map *bmap, int *first, int *second) { int i; isl_ctx *ctx = isl_basic_map_get_ctx(bmap); struct isl_hash_table *table = NULL; struct isl_hash_table_entry *entry; struct isl_constraint_equal_info info; unsigned n_out; unsigned n_div; ctx = isl_basic_map_get_ctx(bmap); table = isl_hash_table_alloc(ctx, bmap->n_ineq); if (!table) goto error; info.n_in = isl_basic_map_dim(bmap, isl_dim_param) + isl_basic_map_dim(bmap, isl_dim_in); info.bmap = bmap; n_out = isl_basic_map_dim(bmap, isl_dim_out); n_div = isl_basic_map_dim(bmap, isl_dim_div); info.n_out = n_out + n_div; for (i = 0; i < bmap->n_ineq; ++i) { uint32_t hash; info.val = bmap->ineq[i] + 1 + info.n_in; if (isl_seq_first_non_zero(info.val, n_out) < 0) continue; if (isl_seq_first_non_zero(info.val + n_out, n_div) >= 0) continue; hash = isl_seq_get_hash(info.val, info.n_out); entry = isl_hash_table_find(ctx, table, hash, constraint_equal, &info, 1); if (!entry) goto error; if (entry->data) break; entry->data = &bmap->ineq[i]; } if (i < bmap->n_ineq) { *first = ((isl_int **)entry->data) - bmap->ineq; *second = i; } isl_hash_table_free(ctx, table); return i < bmap->n_ineq; error: isl_hash_table_free(ctx, table); return -1; } /* Given a set of upper bounds in "var", add constraints to "bset" * that make the i-th bound smallest. * * In particular, if there are n bounds b_i, then add the constraints * * b_i <= b_j for j > i * b_i < b_j for j < i */ static __isl_give isl_basic_set *select_minimum(__isl_take isl_basic_set *bset, __isl_keep isl_mat *var, int i) { isl_ctx *ctx; int j, k; ctx = isl_mat_get_ctx(var); for (j = 0; j < var->n_row; ++j) { if (j == i) continue; k = isl_basic_set_alloc_inequality(bset); if (k < 0) goto error; isl_seq_combine(bset->ineq[k], ctx->one, var->row[j], ctx->negone, var->row[i], var->n_col); isl_int_set_si(bset->ineq[k][var->n_col], 0); if (j < i) isl_int_sub_ui(bset->ineq[k][0], bset->ineq[k][0], 1); } bset = isl_basic_set_finalize(bset); return bset; error: isl_basic_set_free(bset); return NULL; } /* Given a set of upper bounds on the last "input" variable m, * construct a set that assigns the minimal upper bound to m, i.e., * construct a set that divides the space into cells where one * of the upper bounds is smaller than all the others and assign * this upper bound to m. * * In particular, if there are n bounds b_i, then the result * consists of n basic sets, each one of the form * * m = b_i * b_i <= b_j for j > i * b_i < b_j for j < i */ static __isl_give isl_set *set_minimum(__isl_take isl_space *dim, __isl_take isl_mat *var) { int i, k; isl_basic_set *bset = NULL; isl_ctx *ctx; isl_set *set = NULL; if (!dim || !var) goto error; ctx = isl_space_get_ctx(dim); set = isl_set_alloc_space(isl_space_copy(dim), var->n_row, ISL_SET_DISJOINT); for (i = 0; i < var->n_row; ++i) { bset = isl_basic_set_alloc_space(isl_space_copy(dim), 0, 1, var->n_row - 1); k = isl_basic_set_alloc_equality(bset); if (k < 0) goto error; isl_seq_cpy(bset->eq[k], var->row[i], var->n_col); isl_int_set_si(bset->eq[k][var->n_col], -1); bset = select_minimum(bset, var, i); set = isl_set_add_basic_set(set, bset); } isl_space_free(dim); isl_mat_free(var); return set; error: isl_basic_set_free(bset); isl_set_free(set); isl_space_free(dim); isl_mat_free(var); return NULL; } /* Given that the last input variable of "bmap" represents the minimum * of the bounds in "cst", check whether we need to split the domain * based on which bound attains the minimum. * * A split is needed when the minimum appears in an integer division * or in an equality. Otherwise, it is only needed if it appears in * an upper bound that is different from the upper bounds on which it * is defined. */ static int need_split_basic_map(__isl_keep isl_basic_map *bmap, __isl_keep isl_mat *cst) { int i, j; unsigned total; unsigned pos; pos = cst->n_col - 1; total = isl_basic_map_dim(bmap, isl_dim_all); for (i = 0; i < bmap->n_div; ++i) if (!isl_int_is_zero(bmap->div[i][2 + pos])) return 1; for (i = 0; i < bmap->n_eq; ++i) if (!isl_int_is_zero(bmap->eq[i][1 + pos])) return 1; for (i = 0; i < bmap->n_ineq; ++i) { if (isl_int_is_nonneg(bmap->ineq[i][1 + pos])) continue; if (!isl_int_is_negone(bmap->ineq[i][1 + pos])) return 1; if (isl_seq_first_non_zero(bmap->ineq[i] + 1 + pos + 1, total - pos - 1) >= 0) return 1; for (j = 0; j < cst->n_row; ++j) if (isl_seq_eq(bmap->ineq[i], cst->row[j], cst->n_col)) break; if (j >= cst->n_row) return 1; } return 0; } /* Given that the last set variable of "bset" represents the minimum * of the bounds in "cst", check whether we need to split the domain * based on which bound attains the minimum. * * We simply call need_split_basic_map here. This is safe because * the position of the minimum is computed from "cst" and not * from "bmap". */ static int need_split_basic_set(__isl_keep isl_basic_set *bset, __isl_keep isl_mat *cst) { return need_split_basic_map((isl_basic_map *)bset, cst); } /* Given that the last set variable of "set" represents the minimum * of the bounds in "cst", check whether we need to split the domain * based on which bound attains the minimum. */ static int need_split_set(__isl_keep isl_set *set, __isl_keep isl_mat *cst) { int i; for (i = 0; i < set->n; ++i) if (need_split_basic_set(set->p[i], cst)) return 1; return 0; } /* Given a set of which the last set variable is the minimum * of the bounds in "cst", split each basic set in the set * in pieces where one of the bounds is (strictly) smaller than the others. * This subdivision is given in "min_expr". * The variable is subsequently projected out. * * We only do the split when it is needed. * For example if the last input variable m = min(a,b) and the only * constraints in the given basic set are lower bounds on m, * i.e., l <= m = min(a,b), then we can simply project out m * to obtain l <= a and l <= b, without having to split on whether * m is equal to a or b. */ static __isl_give isl_set *split(__isl_take isl_set *empty, __isl_take isl_set *min_expr, __isl_take isl_mat *cst) { int n_in; int i; isl_space *dim; isl_set *res; if (!empty || !min_expr || !cst) goto error; n_in = isl_set_dim(empty, isl_dim_set); dim = isl_set_get_space(empty); dim = isl_space_drop_dims(dim, isl_dim_set, n_in - 1, 1); res = isl_set_empty(dim); for (i = 0; i < empty->n; ++i) { isl_set *set; set = isl_set_from_basic_set(isl_basic_set_copy(empty->p[i])); if (need_split_basic_set(empty->p[i], cst)) set = isl_set_intersect(set, isl_set_copy(min_expr)); set = isl_set_remove_dims(set, isl_dim_set, n_in - 1, 1); res = isl_set_union_disjoint(res, set); } isl_set_free(empty); isl_set_free(min_expr); isl_mat_free(cst); return res; error: isl_set_free(empty); isl_set_free(min_expr); isl_mat_free(cst); return NULL; } /* Given a map of which the last input variable is the minimum * of the bounds in "cst", split each basic set in the set * in pieces where one of the bounds is (strictly) smaller than the others. * This subdivision is given in "min_expr". * The variable is subsequently projected out. * * The implementation is essentially the same as that of "split". */ static __isl_give isl_map *split_domain(__isl_take isl_map *opt, __isl_take isl_set *min_expr, __isl_take isl_mat *cst) { int n_in; int i; isl_space *dim; isl_map *res; if (!opt || !min_expr || !cst) goto error; n_in = isl_map_dim(opt, isl_dim_in); dim = isl_map_get_space(opt); dim = isl_space_drop_dims(dim, isl_dim_in, n_in - 1, 1); res = isl_map_empty(dim); for (i = 0; i < opt->n; ++i) { isl_map *map; map = isl_map_from_basic_map(isl_basic_map_copy(opt->p[i])); if (need_split_basic_map(opt->p[i], cst)) map = isl_map_intersect_domain(map, isl_set_copy(min_expr)); map = isl_map_remove_dims(map, isl_dim_in, n_in - 1, 1); res = isl_map_union_disjoint(res, map); } isl_map_free(opt); isl_set_free(min_expr); isl_mat_free(cst); return res; error: isl_map_free(opt); isl_set_free(min_expr); isl_mat_free(cst); return NULL; } static __isl_give isl_map *basic_map_partial_lexopt( __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom, __isl_give isl_set **empty, int max); union isl_lex_res { void *p; isl_map *map; isl_pw_multi_aff *pma; }; /* This function is called from basic_map_partial_lexopt_symm. * The last variable of "bmap" and "dom" corresponds to the minimum * of the bounds in "cst". "map_space" is the space of the original * input relation (of basic_map_partial_lexopt_symm) and "set_space" * is the space of the original domain. * * We recursively call basic_map_partial_lexopt and then plug in * the definition of the minimum in the result. */ static __isl_give union isl_lex_res basic_map_partial_lexopt_symm_map_core( __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom, __isl_give isl_set **empty, int max, __isl_take isl_mat *cst, __isl_take isl_space *map_space, __isl_take isl_space *set_space) { isl_map *opt; isl_set *min_expr; union isl_lex_res res; min_expr = set_minimum(isl_basic_set_get_space(dom), isl_mat_copy(cst)); opt = basic_map_partial_lexopt(bmap, dom, empty, max); if (empty) { *empty = split(*empty, isl_set_copy(min_expr), isl_mat_copy(cst)); *empty = isl_set_reset_space(*empty, set_space); } opt = split_domain(opt, min_expr, cst); opt = isl_map_reset_space(opt, map_space); res.map = opt; return res; } /* Given a basic map with at least two parallel constraints (as found * by the function parallel_constraints), first look for more constraints * parallel to the two constraint and replace the found list of parallel * constraints by a single constraint with as "input" part the minimum * of the input parts of the list of constraints. Then, recursively call * basic_map_partial_lexopt (possibly finding more parallel constraints) * and plug in the definition of the minimum in the result. * * More specifically, given a set of constraints * * a x + b_i(p) >= 0 * * Replace this set by a single constraint * * a x + u >= 0 * * with u a new parameter with constraints * * u <= b_i(p) * * Any solution to the new system is also a solution for the original system * since * * a x >= -u >= -b_i(p) * * Moreover, m = min_i(b_i(p)) satisfies the constraints on u and can * therefore be plugged into the solution. */ static union isl_lex_res basic_map_partial_lexopt_symm( __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom, __isl_give isl_set **empty, int max, int first, int second, __isl_give union isl_lex_res (*core)(__isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom, __isl_give isl_set **empty, int max, __isl_take isl_mat *cst, __isl_take isl_space *map_space, __isl_take isl_space *set_space)) { int i, n, k; int *list = NULL; unsigned n_in, n_out, n_div; isl_ctx *ctx; isl_vec *var = NULL; isl_mat *cst = NULL; isl_space *map_space, *set_space; union isl_lex_res res; map_space = isl_basic_map_get_space(bmap); set_space = empty ? isl_basic_set_get_space(dom) : NULL; n_in = isl_basic_map_dim(bmap, isl_dim_param) + isl_basic_map_dim(bmap, isl_dim_in); n_out = isl_basic_map_dim(bmap, isl_dim_all) - n_in; ctx = isl_basic_map_get_ctx(bmap); list = isl_alloc_array(ctx, int, bmap->n_ineq); var = isl_vec_alloc(ctx, n_out); if ((bmap->n_ineq && !list) || (n_out && !var)) goto error; list[0] = first; list[1] = second; isl_seq_cpy(var->el, bmap->ineq[first] + 1 + n_in, n_out); for (i = second + 1, n = 2; i < bmap->n_ineq; ++i) { if (isl_seq_eq(var->el, bmap->ineq[i] + 1 + n_in, n_out)) list[n++] = i; } cst = isl_mat_alloc(ctx, n, 1 + n_in); if (!cst) goto error; for (i = 0; i < n; ++i) isl_seq_cpy(cst->row[i], bmap->ineq[list[i]], 1 + n_in); bmap = isl_basic_map_cow(bmap); if (!bmap) goto error; for (i = n - 1; i >= 0; --i) if (isl_basic_map_drop_inequality(bmap, list[i]) < 0) goto error; bmap = isl_basic_map_add(bmap, isl_dim_in, 1); bmap = isl_basic_map_extend_constraints(bmap, 0, 1); k = isl_basic_map_alloc_inequality(bmap); if (k < 0) goto error; isl_seq_clr(bmap->ineq[k], 1 + n_in); isl_int_set_si(bmap->ineq[k][1 + n_in], 1); isl_seq_cpy(bmap->ineq[k] + 1 + n_in + 1, var->el, n_out); bmap = isl_basic_map_finalize(bmap); n_div = isl_basic_set_dim(dom, isl_dim_div); dom = isl_basic_set_add_dims(dom, isl_dim_set, 1); dom = isl_basic_set_extend_constraints(dom, 0, n); for (i = 0; i < n; ++i) { k = isl_basic_set_alloc_inequality(dom); if (k < 0) goto error; isl_seq_cpy(dom->ineq[k], cst->row[i], 1 + n_in); isl_int_set_si(dom->ineq[k][1 + n_in], -1); isl_seq_clr(dom->ineq[k] + 1 + n_in + 1, n_div); } isl_vec_free(var); free(list); return core(bmap, dom, empty, max, cst, map_space, set_space); error: isl_space_free(map_space); isl_space_free(set_space); isl_mat_free(cst); isl_vec_free(var); free(list); isl_basic_set_free(dom); isl_basic_map_free(bmap); res.p = NULL; return res; } static __isl_give isl_map *basic_map_partial_lexopt_symm_map( __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom, __isl_give isl_set **empty, int max, int first, int second) { return basic_map_partial_lexopt_symm(bmap, dom, empty, max, first, second, &basic_map_partial_lexopt_symm_map_core).map; } /* Recursive part of isl_tab_basic_map_partial_lexopt, after detecting * equalities and removing redundant constraints. * * We first check if there are any parallel constraints (left). * If not, we are in the base case. * If there are parallel constraints, we replace them by a single * constraint in basic_map_partial_lexopt_symm and then call * this function recursively to look for more parallel constraints. */ static __isl_give isl_map *basic_map_partial_lexopt( __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom, __isl_give isl_set **empty, int max) { int par = 0; int first, second; if (!bmap) goto error; if (bmap->ctx->opt->pip_symmetry) par = parallel_constraints(bmap, &first, &second); if (par < 0) goto error; if (!par) return basic_map_partial_lexopt_base_map(bmap, dom, empty, max); return basic_map_partial_lexopt_symm_map(bmap, dom, empty, max, first, second); error: isl_basic_set_free(dom); isl_basic_map_free(bmap); return NULL; } /* Compute the lexicographic minimum (or maximum if "max" is set) * of "bmap" over the domain "dom" and return the result as a map. * If "empty" is not NULL, then *empty is assigned a set that * contains those parts of the domain where there is no solution. * If "bmap" is marked as rational (ISL_BASIC_MAP_RATIONAL), * then we compute the rational optimum. Otherwise, we compute * the integral optimum. * * We perform some preprocessing. As the PILP solver does not * handle implicit equalities very well, we first make sure all * the equalities are explicitly available. * * We also add context constraints to the basic map and remove * redundant constraints. This is only needed because of the * way we handle simple symmetries. In particular, we currently look * for symmetries on the constraints, before we set up the main tableau. * It is then no good to look for symmetries on possibly redundant constraints. */ struct isl_map *isl_tab_basic_map_partial_lexopt( struct isl_basic_map *bmap, struct isl_basic_set *dom, struct isl_set **empty, int max) { if (empty) *empty = NULL; if (!bmap || !dom) goto error; isl_assert(bmap->ctx, isl_basic_map_compatible_domain(bmap, dom), goto error); if (isl_basic_set_dim(dom, isl_dim_all) == 0) return basic_map_partial_lexopt(bmap, dom, empty, max); bmap = isl_basic_map_intersect_domain(bmap, isl_basic_set_copy(dom)); bmap = isl_basic_map_detect_equalities(bmap); bmap = isl_basic_map_remove_redundancies(bmap); return basic_map_partial_lexopt(bmap, dom, empty, max); error: isl_basic_set_free(dom); isl_basic_map_free(bmap); return NULL; } struct isl_sol_for { struct isl_sol sol; int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_aff_list *list, void *user); void *user; }; static void sol_for_free(struct isl_sol_for *sol_for) { if (!sol_for) return; if (sol_for->sol.context) sol_for->sol.context->op->free(sol_for->sol.context); free(sol_for); } static void sol_for_free_wrap(struct isl_sol *sol) { sol_for_free((struct isl_sol_for *)sol); } /* Add the solution identified by the tableau and the context tableau. * * See documentation of sol_add for more details. * * Instead of constructing a basic map, this function calls a user * defined function with the current context as a basic set and * a list of affine expressions representing the relation between * the input and output. The space over which the affine expressions * are defined is the same as that of the domain. The number of * affine expressions in the list is equal to the number of output variables. */ static void sol_for_add(struct isl_sol_for *sol, struct isl_basic_set *dom, struct isl_mat *M) { int i; isl_ctx *ctx; isl_local_space *ls; isl_aff *aff; isl_aff_list *list; if (sol->sol.error || !dom || !M) goto error; ctx = isl_basic_set_get_ctx(dom); ls = isl_basic_set_get_local_space(dom); list = isl_aff_list_alloc(ctx, M->n_row - 1); for (i = 1; i < M->n_row; ++i) { aff = isl_aff_alloc(isl_local_space_copy(ls)); if (aff) { isl_int_set(aff->v->el[0], M->row[0][0]); isl_seq_cpy(aff->v->el + 1, M->row[i], M->n_col); } aff = isl_aff_normalize(aff); list = isl_aff_list_add(list, aff); } isl_local_space_free(ls); dom = isl_basic_set_finalize(dom); if (sol->fn(isl_basic_set_copy(dom), list, sol->user) < 0) goto error; isl_basic_set_free(dom); isl_mat_free(M); return; error: isl_basic_set_free(dom); isl_mat_free(M); sol->sol.error = 1; } static void sol_for_add_wrap(struct isl_sol *sol, struct isl_basic_set *dom, struct isl_mat *M) { sol_for_add((struct isl_sol_for *)sol, dom, M); } static struct isl_sol_for *sol_for_init(struct isl_basic_map *bmap, int max, int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_aff_list *list, void *user), void *user) { struct isl_sol_for *sol_for = NULL; isl_space *dom_dim; struct isl_basic_set *dom = NULL; sol_for = isl_calloc_type(bmap->ctx, struct isl_sol_for); if (!sol_for) goto error; dom_dim = isl_space_domain(isl_space_copy(bmap->dim)); dom = isl_basic_set_universe(dom_dim); sol_for->sol.rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL); sol_for->sol.dec_level.callback.run = &sol_dec_level_wrap; sol_for->sol.dec_level.sol = &sol_for->sol; sol_for->fn = fn; sol_for->user = user; sol_for->sol.max = max; sol_for->sol.n_out = isl_basic_map_dim(bmap, isl_dim_out); sol_for->sol.add = &sol_for_add_wrap; sol_for->sol.add_empty = NULL; sol_for->sol.free = &sol_for_free_wrap; sol_for->sol.context = isl_context_alloc(dom); if (!sol_for->sol.context) goto error; isl_basic_set_free(dom); return sol_for; error: isl_basic_set_free(dom); sol_for_free(sol_for); return NULL; } static void sol_for_find_solutions(struct isl_sol_for *sol_for, struct isl_tab *tab) { find_solutions_main(&sol_for->sol, tab); } int isl_basic_map_foreach_lexopt(__isl_keep isl_basic_map *bmap, int max, int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_aff_list *list, void *user), void *user) { struct isl_sol_for *sol_for = NULL; bmap = isl_basic_map_copy(bmap); bmap = isl_basic_map_detect_equalities(bmap); if (!bmap) return -1; sol_for = sol_for_init(bmap, max, fn, user); if (!sol_for) goto error; if (isl_basic_map_plain_is_empty(bmap)) /* nothing */; else { struct isl_tab *tab; struct isl_context *context = sol_for->sol.context; tab = tab_for_lexmin(bmap, context->op->peek_basic_set(context), 1, max); tab = context->op->detect_nonnegative_parameters(context, tab); sol_for_find_solutions(sol_for, tab); if (sol_for->sol.error) goto error; } sol_free(&sol_for->sol); isl_basic_map_free(bmap); return 0; error: sol_free(&sol_for->sol); isl_basic_map_free(bmap); return -1; } int isl_basic_set_foreach_lexopt(__isl_keep isl_basic_set *bset, int max, int (*fn)(__isl_take isl_basic_set *dom, __isl_take isl_aff_list *list, void *user), void *user) { return isl_basic_map_foreach_lexopt(bset, max, fn, user); } /* Check if the given sequence of len variables starting at pos * represents a trivial (i.e., zero) solution. * The variables are assumed to be non-negative and to come in pairs, * with each pair representing a variable of unrestricted sign. * The solution is trivial if each such pair in the sequence consists * of two identical values, meaning that the variable being represented * has value zero. */ static int region_is_trivial(struct isl_tab *tab, int pos, int len) { int i; if (len == 0) return 0; for (i = 0; i < len; i += 2) { int neg_row; int pos_row; neg_row = tab->var[pos + i].is_row ? tab->var[pos + i].index : -1; pos_row = tab->var[pos + i + 1].is_row ? tab->var[pos + i + 1].index : -1; if ((neg_row < 0 || isl_int_is_zero(tab->mat->row[neg_row][1])) && (pos_row < 0 || isl_int_is_zero(tab->mat->row[pos_row][1]))) continue; if (neg_row < 0 || pos_row < 0) return 0; if (isl_int_ne(tab->mat->row[neg_row][1], tab->mat->row[pos_row][1])) return 0; } return 1; } /* Return the index of the first trivial region or -1 if all regions * are non-trivial. */ static int first_trivial_region(struct isl_tab *tab, int n_region, struct isl_region *region) { int i; for (i = 0; i < n_region; ++i) { if (region_is_trivial(tab, region[i].pos, region[i].len)) return i; } return -1; } /* Check if the solution is optimal, i.e., whether the first * n_op entries are zero. */ static int is_optimal(__isl_keep isl_vec *sol, int n_op) { int i; for (i = 0; i < n_op; ++i) if (!isl_int_is_zero(sol->el[1 + i])) return 0; return 1; } /* Add constraints to "tab" that ensure that any solution is significantly * better that that represented by "sol". That is, find the first * relevant (within first n_op) non-zero coefficient and force it (along * with all previous coefficients) to be zero. * If the solution is already optimal (all relevant coefficients are zero), * then just mark the table as empty. */ static int force_better_solution(struct isl_tab *tab, __isl_keep isl_vec *sol, int n_op) { int i; isl_ctx *ctx; isl_vec *v = NULL; if (!sol) return -1; for (i = 0; i < n_op; ++i) if (!isl_int_is_zero(sol->el[1 + i])) break; if (i == n_op) { if (isl_tab_mark_empty(tab) < 0) return -1; return 0; } ctx = isl_vec_get_ctx(sol); v = isl_vec_alloc(ctx, 1 + tab->n_var); if (!v) return -1; for (; i >= 0; --i) { v = isl_vec_clr(v); isl_int_set_si(v->el[1 + i], -1); if (add_lexmin_eq(tab, v->el) < 0) goto error; } isl_vec_free(v); return 0; error: isl_vec_free(v); return -1; } struct isl_trivial { int update; int region; int side; struct isl_tab_undo *snap; }; /* Return the lexicographically smallest non-trivial solution of the * given ILP problem. * * All variables are assumed to be non-negative. * * n_op is the number of initial coordinates to optimize. * That is, once a solution has been found, we will only continue looking * for solution that result in significantly better values for those * initial coordinates. That is, we only continue looking for solutions * that increase the number of initial zeros in this sequence. * * A solution is non-trivial, if it is non-trivial on each of the * specified regions. Each region represents a sequence of pairs * of variables. A solution is non-trivial on such a region if * at least one of these pairs consists of different values, i.e., * such that the non-negative variable represented by the pair is non-zero. * * Whenever a conflict is encountered, all constraints involved are * reported to the caller through a call to "conflict". * * We perform a simple branch-and-bound backtracking search. * Each level in the search represents initially trivial region that is forced * to be non-trivial. * At each level we consider n cases, where n is the length of the region. * In terms of the n/2 variables of unrestricted signs being encoded by * the region, we consider the cases * x_0 >= 1 * x_0 <= -1 * x_0 = 0 and x_1 >= 1 * x_0 = 0 and x_1 <= -1 * x_0 = 0 and x_1 = 0 and x_2 >= 1 * x_0 = 0 and x_1 = 0 and x_2 <= -1 * ... * The cases are considered in this order, assuming that each pair * x_i_a x_i_b represents the value x_i_b - x_i_a. * That is, x_0 >= 1 is enforced by adding the constraint * x_0_b - x_0_a >= 1 */ __isl_give isl_vec *isl_tab_basic_set_non_trivial_lexmin( __isl_take isl_basic_set *bset, int n_op, int n_region, struct isl_region *region, int (*conflict)(int con, void *user), void *user) { int i, j; int r; isl_ctx *ctx; isl_vec *v = NULL; isl_vec *sol = NULL; struct isl_tab *tab; struct isl_trivial *triv = NULL; int level, init; if (!bset) return NULL; ctx = isl_basic_set_get_ctx(bset); sol = isl_vec_alloc(ctx, 0); tab = tab_for_lexmin(bset, NULL, 0, 0); if (!tab) goto error; tab->conflict = conflict; tab->conflict_user = user; v = isl_vec_alloc(ctx, 1 + tab->n_var); triv = isl_calloc_array(ctx, struct isl_trivial, n_region); if (!v || (n_region && !triv)) goto error; level = 0; init = 1; while (level >= 0) { int side, base; if (init) { tab = cut_to_integer_lexmin(tab, CUT_ONE); if (!tab) goto error; if (tab->empty) goto backtrack; r = first_trivial_region(tab, n_region, region); if (r < 0) { for (i = 0; i < level; ++i) triv[i].update = 1; isl_vec_free(sol); sol = isl_tab_get_sample_value(tab); if (!sol) goto error; if (is_optimal(sol, n_op)) break; goto backtrack; } if (level >= n_region) isl_die(ctx, isl_error_internal, "nesting level too deep", goto error); if (isl_tab_extend_cons(tab, 2 * region[r].len + 2 * n_op) < 0) goto error; triv[level].region = r; triv[level].side = 0; } r = triv[level].region; side = triv[level].side; base = 2 * (side/2); if (side >= region[r].len) { backtrack: level--; init = 0; if (level >= 0) if (isl_tab_rollback(tab, triv[level].snap) < 0) goto error; continue; } if (triv[level].update) { if (force_better_solution(tab, sol, n_op) < 0) goto error; triv[level].update = 0; } if (side == base && base >= 2) { for (j = base - 2; j < base; ++j) { v = isl_vec_clr(v); isl_int_set_si(v->el[1 + region[r].pos + j], 1); if (add_lexmin_eq(tab, v->el) < 0) goto error; } } triv[level].snap = isl_tab_snap(tab); if (isl_tab_push_basis(tab) < 0) goto error; v = isl_vec_clr(v); isl_int_set_si(v->el[0], -1); isl_int_set_si(v->el[1 + region[r].pos + side], -1); isl_int_set_si(v->el[1 + region[r].pos + (side ^ 1)], 1); tab = add_lexmin_ineq(tab, v->el); triv[level].side++; level++; init = 1; } free(triv); isl_vec_free(v); isl_tab_free(tab); isl_basic_set_free(bset); return sol; error: free(triv); isl_vec_free(v); isl_tab_free(tab); isl_basic_set_free(bset); isl_vec_free(sol); return NULL; } /* Return the lexicographically smallest rational point in "bset", * assuming that all variables are non-negative. * If "bset" is empty, then return a zero-length vector. */ __isl_give isl_vec *isl_tab_basic_set_non_neg_lexmin( __isl_take isl_basic_set *bset) { struct isl_tab *tab; isl_ctx *ctx = isl_basic_set_get_ctx(bset); isl_vec *sol; if (!bset) return NULL; tab = tab_for_lexmin(bset, NULL, 0, 0); if (!tab) goto error; if (tab->empty) sol = isl_vec_alloc(ctx, 0); else sol = isl_tab_get_sample_value(tab); isl_tab_free(tab); isl_basic_set_free(bset); return sol; error: isl_tab_free(tab); isl_basic_set_free(bset); return NULL; } struct isl_sol_pma { struct isl_sol sol; isl_pw_multi_aff *pma; isl_set *empty; }; static void sol_pma_free(struct isl_sol_pma *sol_pma) { if (!sol_pma) return; if (sol_pma->sol.context) sol_pma->sol.context->op->free(sol_pma->sol.context); isl_pw_multi_aff_free(sol_pma->pma); isl_set_free(sol_pma->empty); free(sol_pma); } /* This function is called for parts of the context where there is * no solution, with "bset" corresponding to the context tableau. * Simply add the basic set to the set "empty". */ static void sol_pma_add_empty(struct isl_sol_pma *sol, __isl_take isl_basic_set *bset) { if (!bset) goto error; isl_assert(bset->ctx, sol->empty, goto error); sol->empty = isl_set_grow(sol->empty, 1); bset = isl_basic_set_simplify(bset); bset = isl_basic_set_finalize(bset); sol->empty = isl_set_add_basic_set(sol->empty, bset); if (!sol->empty) sol->sol.error = 1; return; error: isl_basic_set_free(bset); sol->sol.error = 1; } /* Given a basic map "dom" that represents the context and an affine * matrix "M" that maps the dimensions of the context to the * output variables, construct an isl_pw_multi_aff with a single * cell corresponding to "dom" and affine expressions copied from "M". */ static void sol_pma_add(struct isl_sol_pma *sol, __isl_take isl_basic_set *dom, __isl_take isl_mat *M) { int i; isl_local_space *ls; isl_aff *aff; isl_multi_aff *maff; isl_pw_multi_aff *pma; maff = isl_multi_aff_alloc(isl_pw_multi_aff_get_space(sol->pma)); ls = isl_basic_set_get_local_space(dom); for (i = 1; i < M->n_row; ++i) { aff = isl_aff_alloc(isl_local_space_copy(ls)); if (aff) { isl_int_set(aff->v->el[0], M->row[0][0]); isl_seq_cpy(aff->v->el + 1, M->row[i], M->n_col); } aff = isl_aff_normalize(aff); maff = isl_multi_aff_set_aff(maff, i - 1, aff); } isl_local_space_free(ls); isl_mat_free(M); dom = isl_basic_set_simplify(dom); dom = isl_basic_set_finalize(dom); pma = isl_pw_multi_aff_alloc(isl_set_from_basic_set(dom), maff); sol->pma = isl_pw_multi_aff_add_disjoint(sol->pma, pma); if (!sol->pma) sol->sol.error = 1; } static void sol_pma_free_wrap(struct isl_sol *sol) { sol_pma_free((struct isl_sol_pma *)sol); } static void sol_pma_add_empty_wrap(struct isl_sol *sol, __isl_take isl_basic_set *bset) { sol_pma_add_empty((struct isl_sol_pma *)sol, bset); } static void sol_pma_add_wrap(struct isl_sol *sol, __isl_take isl_basic_set *dom, __isl_take isl_mat *M) { sol_pma_add((struct isl_sol_pma *)sol, dom, M); } /* Construct an isl_sol_pma structure for accumulating the solution. * If track_empty is set, then we also keep track of the parts * of the context where there is no solution. * If max is set, then we are solving a maximization, rather than * a minimization problem, which means that the variables in the * tableau have value "M - x" rather than "M + x". */ static struct isl_sol *sol_pma_init(__isl_keep isl_basic_map *bmap, __isl_take isl_basic_set *dom, int track_empty, int max) { struct isl_sol_pma *sol_pma = NULL; if (!bmap) goto error; sol_pma = isl_calloc_type(bmap->ctx, struct isl_sol_pma); if (!sol_pma) goto error; sol_pma->sol.rational = ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL); sol_pma->sol.dec_level.callback.run = &sol_dec_level_wrap; sol_pma->sol.dec_level.sol = &sol_pma->sol; sol_pma->sol.max = max; sol_pma->sol.n_out = isl_basic_map_dim(bmap, isl_dim_out); sol_pma->sol.add = &sol_pma_add_wrap; sol_pma->sol.add_empty = track_empty ? &sol_pma_add_empty_wrap : NULL; sol_pma->sol.free = &sol_pma_free_wrap; sol_pma->pma = isl_pw_multi_aff_empty(isl_basic_map_get_space(bmap)); if (!sol_pma->pma) goto error; sol_pma->sol.context = isl_context_alloc(dom); if (!sol_pma->sol.context) goto error; if (track_empty) { sol_pma->empty = isl_set_alloc_space(isl_basic_set_get_space(dom), 1, ISL_SET_DISJOINT); if (!sol_pma->empty) goto error; } isl_basic_set_free(dom); return &sol_pma->sol; error: isl_basic_set_free(dom); sol_pma_free(sol_pma); return NULL; } /* Base case of isl_tab_basic_map_partial_lexopt, after removing * some obvious symmetries. * * We call basic_map_partial_lexopt_base and extract the results. */ static __isl_give isl_pw_multi_aff *basic_map_partial_lexopt_base_pma( __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom, __isl_give isl_set **empty, int max) { isl_pw_multi_aff *result = NULL; struct isl_sol *sol; struct isl_sol_pma *sol_pma; sol = basic_map_partial_lexopt_base(bmap, dom, empty, max, &sol_pma_init); if (!sol) return NULL; sol_pma = (struct isl_sol_pma *) sol; result = isl_pw_multi_aff_copy(sol_pma->pma); if (empty) *empty = isl_set_copy(sol_pma->empty); sol_free(&sol_pma->sol); return result; } /* Given that the last input variable of "maff" represents the minimum * of some bounds, check whether we need to plug in the expression * of the minimum. * * In particular, check if the last input variable appears in any * of the expressions in "maff". */ static int need_substitution(__isl_keep isl_multi_aff *maff) { int i; unsigned pos; pos = isl_multi_aff_dim(maff, isl_dim_in) - 1; for (i = 0; i < maff->n; ++i) if (isl_aff_involves_dims(maff->p[i], isl_dim_in, pos, 1)) return 1; return 0; } /* Given a set of upper bounds on the last "input" variable m, * construct a piecewise affine expression that selects * the minimal upper bound to m, i.e., * divide the space into cells where one * of the upper bounds is smaller than all the others and select * this upper bound on that cell. * * In particular, if there are n bounds b_i, then the result * consists of n cell, each one of the form * * b_i <= b_j for j > i * b_i < b_j for j < i * * The affine expression on this cell is * * b_i */ static __isl_give isl_pw_aff *set_minimum_pa(__isl_take isl_space *space, __isl_take isl_mat *var) { int i; isl_aff *aff = NULL; isl_basic_set *bset = NULL; isl_ctx *ctx; isl_pw_aff *paff = NULL; isl_space *pw_space; isl_local_space *ls = NULL; if (!space || !var) goto error; ctx = isl_space_get_ctx(space); ls = isl_local_space_from_space(isl_space_copy(space)); pw_space = isl_space_copy(space); pw_space = isl_space_from_domain(pw_space); pw_space = isl_space_add_dims(pw_space, isl_dim_out, 1); paff = isl_pw_aff_alloc_size(pw_space, var->n_row); for (i = 0; i < var->n_row; ++i) { isl_pw_aff *paff_i; aff = isl_aff_alloc(isl_local_space_copy(ls)); bset = isl_basic_set_alloc_space(isl_space_copy(space), 0, 0, var->n_row - 1); if (!aff || !bset) goto error; isl_int_set_si(aff->v->el[0], 1); isl_seq_cpy(aff->v->el + 1, var->row[i], var->n_col); isl_int_set_si(aff->v->el[1 + var->n_col], 0); bset = select_minimum(bset, var, i); paff_i = isl_pw_aff_alloc(isl_set_from_basic_set(bset), aff); paff = isl_pw_aff_add_disjoint(paff, paff_i); } isl_local_space_free(ls); isl_space_free(space); isl_mat_free(var); return paff; error: isl_aff_free(aff); isl_basic_set_free(bset); isl_pw_aff_free(paff); isl_local_space_free(ls); isl_space_free(space); isl_mat_free(var); return NULL; } /* Given a piecewise multi-affine expression of which the last input variable * is the minimum of the bounds in "cst", plug in the value of the minimum. * This minimum expression is given in "min_expr_pa". * The set "min_expr" contains the same information, but in the form of a set. * The variable is subsequently projected out. * * The implementation is similar to those of "split" and "split_domain". * If the variable appears in a given expression, then minimum expression * is plugged in. Otherwise, if the variable appears in the constraints * and a split is required, then the domain is split. Otherwise, no split * is performed. */ static __isl_give isl_pw_multi_aff *split_domain_pma( __isl_take isl_pw_multi_aff *opt, __isl_take isl_pw_aff *min_expr_pa, __isl_take isl_set *min_expr, __isl_take isl_mat *cst) { int n_in; int i; isl_space *space; isl_pw_multi_aff *res; if (!opt || !min_expr || !cst) goto error; n_in = isl_pw_multi_aff_dim(opt, isl_dim_in); space = isl_pw_multi_aff_get_space(opt); space = isl_space_drop_dims(space, isl_dim_in, n_in - 1, 1); res = isl_pw_multi_aff_empty(space); for (i = 0; i < opt->n; ++i) { isl_pw_multi_aff *pma; pma = isl_pw_multi_aff_alloc(isl_set_copy(opt->p[i].set), isl_multi_aff_copy(opt->p[i].maff)); if (need_substitution(opt->p[i].maff)) pma = isl_pw_multi_aff_substitute(pma, isl_dim_in, n_in - 1, min_expr_pa); else if (need_split_set(opt->p[i].set, cst)) pma = isl_pw_multi_aff_intersect_domain(pma, isl_set_copy(min_expr)); pma = isl_pw_multi_aff_project_out(pma, isl_dim_in, n_in - 1, 1); res = isl_pw_multi_aff_add_disjoint(res, pma); } isl_pw_multi_aff_free(opt); isl_pw_aff_free(min_expr_pa); isl_set_free(min_expr); isl_mat_free(cst); return res; error: isl_pw_multi_aff_free(opt); isl_pw_aff_free(min_expr_pa); isl_set_free(min_expr); isl_mat_free(cst); return NULL; } static __isl_give isl_pw_multi_aff *basic_map_partial_lexopt_pma( __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom, __isl_give isl_set **empty, int max); /* This function is called from basic_map_partial_lexopt_symm. * The last variable of "bmap" and "dom" corresponds to the minimum * of the bounds in "cst". "map_space" is the space of the original * input relation (of basic_map_partial_lexopt_symm) and "set_space" * is the space of the original domain. * * We recursively call basic_map_partial_lexopt and then plug in * the definition of the minimum in the result. */ static __isl_give union isl_lex_res basic_map_partial_lexopt_symm_pma_core( __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom, __isl_give isl_set **empty, int max, __isl_take isl_mat *cst, __isl_take isl_space *map_space, __isl_take isl_space *set_space) { isl_pw_multi_aff *opt; isl_pw_aff *min_expr_pa; isl_set *min_expr; union isl_lex_res res; min_expr = set_minimum(isl_basic_set_get_space(dom), isl_mat_copy(cst)); min_expr_pa = set_minimum_pa(isl_basic_set_get_space(dom), isl_mat_copy(cst)); opt = basic_map_partial_lexopt_pma(bmap, dom, empty, max); if (empty) { *empty = split(*empty, isl_set_copy(min_expr), isl_mat_copy(cst)); *empty = isl_set_reset_space(*empty, set_space); } opt = split_domain_pma(opt, min_expr_pa, min_expr, cst); opt = isl_pw_multi_aff_reset_space(opt, map_space); res.pma = opt; return res; } static __isl_give isl_pw_multi_aff *basic_map_partial_lexopt_symm_pma( __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom, __isl_give isl_set **empty, int max, int first, int second) { return basic_map_partial_lexopt_symm(bmap, dom, empty, max, first, second, &basic_map_partial_lexopt_symm_pma_core).pma; } /* Recursive part of isl_basic_map_partial_lexopt_pw_multi_aff, after detecting * equalities and removing redundant constraints. * * We first check if there are any parallel constraints (left). * If not, we are in the base case. * If there are parallel constraints, we replace them by a single * constraint in basic_map_partial_lexopt_symm_pma and then call * this function recursively to look for more parallel constraints. */ static __isl_give isl_pw_multi_aff *basic_map_partial_lexopt_pma( __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom, __isl_give isl_set **empty, int max) { int par = 0; int first, second; if (!bmap) goto error; if (bmap->ctx->opt->pip_symmetry) par = parallel_constraints(bmap, &first, &second); if (par < 0) goto error; if (!par) return basic_map_partial_lexopt_base_pma(bmap, dom, empty, max); return basic_map_partial_lexopt_symm_pma(bmap, dom, empty, max, first, second); error: isl_basic_set_free(dom); isl_basic_map_free(bmap); return NULL; } /* Compute the lexicographic minimum (or maximum if "max" is set) * of "bmap" over the domain "dom" and return the result as a piecewise * multi-affine expression. * If "empty" is not NULL, then *empty is assigned a set that * contains those parts of the domain where there is no solution. * If "bmap" is marked as rational (ISL_BASIC_MAP_RATIONAL), * then we compute the rational optimum. Otherwise, we compute * the integral optimum. * * We perform some preprocessing. As the PILP solver does not * handle implicit equalities very well, we first make sure all * the equalities are explicitly available. * * We also add context constraints to the basic map and remove * redundant constraints. This is only needed because of the * way we handle simple symmetries. In particular, we currently look * for symmetries on the constraints, before we set up the main tableau. * It is then no good to look for symmetries on possibly redundant constraints. */ __isl_give isl_pw_multi_aff *isl_basic_map_partial_lexopt_pw_multi_aff( __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *dom, __isl_give isl_set **empty, int max) { if (empty) *empty = NULL; if (!bmap || !dom) goto error; isl_assert(bmap->ctx, isl_basic_map_compatible_domain(bmap, dom), goto error); if (isl_basic_set_dim(dom, isl_dim_all) == 0) return basic_map_partial_lexopt_pma(bmap, dom, empty, max); bmap = isl_basic_map_intersect_domain(bmap, isl_basic_set_copy(dom)); bmap = isl_basic_map_detect_equalities(bmap); bmap = isl_basic_map_remove_redundancies(bmap); return basic_map_partial_lexopt_pma(bmap, dom, empty, max); error: isl_basic_set_free(dom); isl_basic_map_free(bmap); return NULL; }