/* * Copyright 2008-2009 Katholieke Universiteit Leuven * Copyright 2012-2013 Ecole Normale Superieure * Copyright 2014-2015 INRIA Rocquencourt * Copyright 2016 Sven Verdoolaege * * 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 Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt, * B.P. 105 - 78153 Le Chesnay, France */ #include #include #include "isl_equalities.h" #include #include #include "isl_tab.h" #include #include #include #include #include #include #include static void swap_equality(__isl_keep isl_basic_map *bmap, int a, int b) { isl_int *t = bmap->eq[a]; bmap->eq[a] = bmap->eq[b]; bmap->eq[b] = t; } static void swap_inequality(__isl_keep isl_basic_map *bmap, int a, int b) { if (a != b) { isl_int *t = bmap->ineq[a]; bmap->ineq[a] = bmap->ineq[b]; bmap->ineq[b] = t; } } __isl_give isl_basic_map *isl_basic_map_normalize_constraints( __isl_take isl_basic_map *bmap) { int i; isl_int gcd; isl_size total = isl_basic_map_dim(bmap, isl_dim_all); if (total < 0) return isl_basic_map_free(bmap); isl_int_init(gcd); for (i = bmap->n_eq - 1; i >= 0; --i) { isl_seq_gcd(bmap->eq[i]+1, total, &gcd); if (isl_int_is_zero(gcd)) { if (!isl_int_is_zero(bmap->eq[i][0])) { bmap = isl_basic_map_set_to_empty(bmap); break; } if (isl_basic_map_drop_equality(bmap, i) < 0) goto error; continue; } if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL)) isl_int_gcd(gcd, gcd, bmap->eq[i][0]); if (isl_int_is_one(gcd)) continue; if (!isl_int_is_divisible_by(bmap->eq[i][0], gcd)) { bmap = isl_basic_map_set_to_empty(bmap); break; } isl_seq_scale_down(bmap->eq[i], bmap->eq[i], gcd, 1+total); } for (i = bmap->n_ineq - 1; i >= 0; --i) { isl_seq_gcd(bmap->ineq[i]+1, total, &gcd); if (isl_int_is_zero(gcd)) { if (isl_int_is_neg(bmap->ineq[i][0])) { bmap = isl_basic_map_set_to_empty(bmap); break; } if (isl_basic_map_drop_inequality(bmap, i) < 0) goto error; continue; } if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL)) isl_int_gcd(gcd, gcd, bmap->ineq[i][0]); if (isl_int_is_one(gcd)) continue; isl_int_fdiv_q(bmap->ineq[i][0], bmap->ineq[i][0], gcd); isl_seq_scale_down(bmap->ineq[i]+1, bmap->ineq[i]+1, gcd, total); } isl_int_clear(gcd); return bmap; error: isl_int_clear(gcd); isl_basic_map_free(bmap); return NULL; } __isl_give isl_basic_set *isl_basic_set_normalize_constraints( __isl_take isl_basic_set *bset) { isl_basic_map *bmap = bset_to_bmap(bset); return bset_from_bmap(isl_basic_map_normalize_constraints(bmap)); } /* Reduce the coefficient of the variable at position "pos" * in integer division "div", such that it lies in the half-open * interval (1/2,1/2], extracting any excess value from this integer division. * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0 * corresponds to the constant term. * * That is, the integer division is of the form * * floor((... + (c * d + r) * x_pos + ...)/d) * * with -d < 2 * r <= d. * Replace it by * * floor((... + r * x_pos + ...)/d) + c * x_pos * * If 2 * ((c * d + r) % d) <= d, then c = floor((c * d + r)/d). * Otherwise, c = floor((c * d + r)/d) + 1. * * This is the same normalization that is performed by isl_aff_floor. */ static __isl_give isl_basic_map *reduce_coefficient_in_div( __isl_take isl_basic_map *bmap, int div, int pos) { isl_int shift; int add_one; isl_int_init(shift); isl_int_fdiv_r(shift, bmap->div[div][1 + pos], bmap->div[div][0]); isl_int_mul_ui(shift, shift, 2); add_one = isl_int_gt(shift, bmap->div[div][0]); isl_int_fdiv_q(shift, bmap->div[div][1 + pos], bmap->div[div][0]); if (add_one) isl_int_add_ui(shift, shift, 1); isl_int_neg(shift, shift); bmap = isl_basic_map_shift_div(bmap, div, pos, shift); isl_int_clear(shift); return bmap; } /* Does the coefficient of the variable at position "pos" * in integer division "div" need to be reduced? * That is, does it lie outside the half-open interval (1/2,1/2]? * The coefficient c/d lies outside this interval if abs(2 * c) >= d and * 2 * c != d. */ static isl_bool needs_reduction(__isl_keep isl_basic_map *bmap, int div, int pos) { isl_bool r; if (isl_int_is_zero(bmap->div[div][1 + pos])) return isl_bool_false; isl_int_mul_ui(bmap->div[div][1 + pos], bmap->div[div][1 + pos], 2); r = isl_int_abs_ge(bmap->div[div][1 + pos], bmap->div[div][0]) && !isl_int_eq(bmap->div[div][1 + pos], bmap->div[div][0]); isl_int_divexact_ui(bmap->div[div][1 + pos], bmap->div[div][1 + pos], 2); return r; } /* Reduce the coefficients (including the constant term) of * integer division "div", if needed. * In particular, make sure all coefficients lie in * the half-open interval (1/2,1/2]. */ static __isl_give isl_basic_map *reduce_div_coefficients_of_div( __isl_take isl_basic_map *bmap, int div) { int i; isl_size total; total = isl_basic_map_dim(bmap, isl_dim_all); if (total < 0) return isl_basic_map_free(bmap); for (i = 0; i < 1 + total; ++i) { isl_bool reduce; reduce = needs_reduction(bmap, div, i); if (reduce < 0) return isl_basic_map_free(bmap); if (!reduce) continue; bmap = reduce_coefficient_in_div(bmap, div, i); if (!bmap) break; } return bmap; } /* Reduce the coefficients (including the constant term) of * the known integer divisions, if needed * In particular, make sure all coefficients lie in * the half-open interval (1/2,1/2]. */ static __isl_give isl_basic_map *reduce_div_coefficients( __isl_take isl_basic_map *bmap) { int i; if (!bmap) return NULL; if (bmap->n_div == 0) return bmap; for (i = 0; i < bmap->n_div; ++i) { if (isl_int_is_zero(bmap->div[i][0])) continue; bmap = reduce_div_coefficients_of_div(bmap, i); if (!bmap) break; } return bmap; } /* Remove any common factor in numerator and denominator of the div expression, * not taking into account the constant term. * That is, if the div is of the form * * floor((a + m f(x))/(m d)) * * then replace it by * * floor((floor(a/m) + f(x))/d) * * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d * and can therefore not influence the result of the floor. */ static __isl_give isl_basic_map *normalize_div_expression( __isl_take isl_basic_map *bmap, int div) { isl_size total = isl_basic_map_dim(bmap, isl_dim_all); isl_ctx *ctx = bmap->ctx; if (total < 0) return isl_basic_map_free(bmap); if (isl_int_is_zero(bmap->div[div][0])) return bmap; isl_seq_gcd(bmap->div[div] + 2, total, &ctx->normalize_gcd); isl_int_gcd(ctx->normalize_gcd, ctx->normalize_gcd, bmap->div[div][0]); if (isl_int_is_one(ctx->normalize_gcd)) return bmap; isl_int_fdiv_q(bmap->div[div][1], bmap->div[div][1], ctx->normalize_gcd); isl_int_divexact(bmap->div[div][0], bmap->div[div][0], ctx->normalize_gcd); isl_seq_scale_down(bmap->div[div] + 2, bmap->div[div] + 2, ctx->normalize_gcd, total); return bmap; } /* Remove any common factor in numerator and denominator of a div expression, * not taking into account the constant term. * That is, look for any div of the form * * floor((a + m f(x))/(m d)) * * and replace it by * * floor((floor(a/m) + f(x))/d) * * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d * and can therefore not influence the result of the floor. */ static __isl_give isl_basic_map *normalize_div_expressions( __isl_take isl_basic_map *bmap) { int i; if (!bmap) return NULL; if (bmap->n_div == 0) return bmap; for (i = 0; i < bmap->n_div; ++i) bmap = normalize_div_expression(bmap, i); return bmap; } /* Assumes divs have been ordered if keep_divs is set. */ static __isl_give isl_basic_map *eliminate_var_using_equality( __isl_take isl_basic_map *bmap, unsigned pos, isl_int *eq, int keep_divs, int *progress) { isl_size total; isl_size v_div; int k; int last_div; total = isl_basic_map_dim(bmap, isl_dim_all); v_div = isl_basic_map_var_offset(bmap, isl_dim_div); if (total < 0 || v_div < 0) return isl_basic_map_free(bmap); last_div = isl_seq_last_non_zero(eq + 1 + v_div, bmap->n_div); for (k = 0; k < bmap->n_eq; ++k) { if (bmap->eq[k] == eq) continue; if (isl_int_is_zero(bmap->eq[k][1+pos])) continue; if (progress) *progress = 1; isl_seq_elim(bmap->eq[k], eq, 1+pos, 1+total, NULL); isl_seq_normalize(bmap->ctx, bmap->eq[k], 1 + total); } for (k = 0; k < bmap->n_ineq; ++k) { if (isl_int_is_zero(bmap->ineq[k][1+pos])) continue; if (progress) *progress = 1; isl_seq_elim(bmap->ineq[k], eq, 1+pos, 1+total, NULL); isl_seq_normalize(bmap->ctx, bmap->ineq[k], 1 + total); ISL_F_CLR(bmap, ISL_BASIC_MAP_NO_REDUNDANT); ISL_F_CLR(bmap, ISL_BASIC_MAP_SORTED); } for (k = 0; k < bmap->n_div; ++k) { if (isl_int_is_zero(bmap->div[k][0])) continue; if (isl_int_is_zero(bmap->div[k][1+1+pos])) continue; if (progress) *progress = 1; /* We need to be careful about circular definitions, * so for now we just remove the definition of div k * if the equality contains any divs. * If keep_divs is set, then the divs have been ordered * and we can keep the definition as long as the result * is still ordered. */ if (last_div == -1 || (keep_divs && last_div < k)) { isl_seq_elim(bmap->div[k]+1, eq, 1+pos, 1+total, &bmap->div[k][0]); bmap = normalize_div_expression(bmap, k); if (!bmap) return NULL; } else isl_seq_clr(bmap->div[k], 1 + total); } return bmap; } /* Assumes divs have been ordered if keep_divs is set. */ static __isl_give isl_basic_map *eliminate_div(__isl_take isl_basic_map *bmap, isl_int *eq, unsigned div, int keep_divs) { isl_size v_div; unsigned pos; v_div = isl_basic_map_var_offset(bmap, isl_dim_div); if (v_div < 0) return isl_basic_map_free(bmap); pos = v_div + div; bmap = eliminate_var_using_equality(bmap, pos, eq, keep_divs, NULL); bmap = isl_basic_map_drop_div(bmap, div); return bmap; } /* Check if elimination of div "div" using equality "eq" would not * result in a div depending on a later div. */ static isl_bool ok_to_eliminate_div(__isl_keep isl_basic_map *bmap, isl_int *eq, unsigned div) { int k; int last_div; isl_size v_div; unsigned pos; v_div = isl_basic_map_var_offset(bmap, isl_dim_div); if (v_div < 0) return isl_bool_error; pos = v_div + div; last_div = isl_seq_last_non_zero(eq + 1 + v_div, bmap->n_div); if (last_div < 0 || last_div <= div) return isl_bool_true; for (k = 0; k <= last_div; ++k) { if (isl_int_is_zero(bmap->div[k][0])) continue; if (!isl_int_is_zero(bmap->div[k][1 + 1 + pos])) return isl_bool_false; } return isl_bool_true; } /* Eliminate divs based on equalities */ static __isl_give isl_basic_map *eliminate_divs_eq( __isl_take isl_basic_map *bmap, int *progress) { int d; int i; int modified = 0; unsigned off; bmap = isl_basic_map_order_divs(bmap); if (!bmap) return NULL; off = isl_basic_map_offset(bmap, isl_dim_div); for (d = bmap->n_div - 1; d >= 0 ; --d) { for (i = 0; i < bmap->n_eq; ++i) { isl_bool ok; if (!isl_int_is_one(bmap->eq[i][off + d]) && !isl_int_is_negone(bmap->eq[i][off + d])) continue; ok = ok_to_eliminate_div(bmap, bmap->eq[i], d); if (ok < 0) return isl_basic_map_free(bmap); if (!ok) continue; modified = 1; *progress = 1; bmap = eliminate_div(bmap, bmap->eq[i], d, 1); if (isl_basic_map_drop_equality(bmap, i) < 0) return isl_basic_map_free(bmap); break; } } if (modified) return eliminate_divs_eq(bmap, progress); return bmap; } /* Eliminate divs based on inequalities */ static __isl_give isl_basic_map *eliminate_divs_ineq( __isl_take isl_basic_map *bmap, int *progress) { int d; int i; unsigned off; struct isl_ctx *ctx; if (!bmap) return NULL; ctx = bmap->ctx; off = isl_basic_map_offset(bmap, isl_dim_div); for (d = bmap->n_div - 1; d >= 0 ; --d) { for (i = 0; i < bmap->n_eq; ++i) if (!isl_int_is_zero(bmap->eq[i][off + d])) break; if (i < bmap->n_eq) continue; for (i = 0; i < bmap->n_ineq; ++i) if (isl_int_abs_gt(bmap->ineq[i][off + d], ctx->one)) break; if (i < bmap->n_ineq) continue; *progress = 1; bmap = isl_basic_map_eliminate_vars(bmap, (off-1)+d, 1); if (!bmap || ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY)) break; bmap = isl_basic_map_drop_div(bmap, d); if (!bmap) break; } return bmap; } /* Does the equality constraint at position "eq" in "bmap" involve * any local variables in the range [first, first + n) * that are not marked as having an explicit representation? */ static isl_bool bmap_eq_involves_unknown_divs(__isl_keep isl_basic_map *bmap, int eq, unsigned first, unsigned n) { unsigned o_div; int i; if (!bmap) return isl_bool_error; o_div = isl_basic_map_offset(bmap, isl_dim_div); for (i = 0; i < n; ++i) { isl_bool unknown; if (isl_int_is_zero(bmap->eq[eq][o_div + first + i])) continue; unknown = isl_basic_map_div_is_marked_unknown(bmap, first + i); if (unknown < 0) return isl_bool_error; if (unknown) return isl_bool_true; } return isl_bool_false; } /* The last local variable involved in the equality constraint * at position "eq" in "bmap" is the local variable at position "div". * It can therefore be used to extract an explicit representation * for that variable. * Do so unless the local variable already has an explicit representation or * the explicit representation would involve any other local variables * that in turn do not have an explicit representation. * An equality constraint involving local variables without an explicit * representation can be used in isl_basic_map_drop_redundant_divs * to separate out an independent local variable. Introducing * an explicit representation here would block this transformation, * while the partial explicit representation in itself is not very useful. * Set *progress if anything is changed. * * The equality constraint is of the form * * f(x) + n e >= 0 * * with n a positive number. The explicit representation derived from * this constraint is * * floor((-f(x))/n) */ static __isl_give isl_basic_map *set_div_from_eq(__isl_take isl_basic_map *bmap, int div, int eq, int *progress) { isl_size total; unsigned o_div; isl_bool involves; if (!bmap) return NULL; if (!isl_int_is_zero(bmap->div[div][0])) return bmap; involves = bmap_eq_involves_unknown_divs(bmap, eq, 0, div); if (involves < 0) return isl_basic_map_free(bmap); if (involves) return bmap; total = isl_basic_map_dim(bmap, isl_dim_all); if (total < 0) return isl_basic_map_free(bmap); o_div = isl_basic_map_offset(bmap, isl_dim_div); isl_seq_neg(bmap->div[div] + 1, bmap->eq[eq], 1 + total); isl_int_set_si(bmap->div[div][1 + o_div + div], 0); isl_int_set(bmap->div[div][0], bmap->eq[eq][o_div + div]); if (progress) *progress = 1; return bmap; } /* Perform fangcheng (Gaussian elimination) on the equality * constraints of "bmap". * That is, put them into row-echelon form, starting from the last column * backward and use them to eliminate the corresponding coefficients * from all constraints. * * If "progress" is not NULL, then it gets set if the elimination * results in any changes. * The elimination process may result in some equality constraints * getting interchanged or removed. * If "swap" or "drop" are not NULL, then they get called when * two equality constraints get interchanged or * when a number of final equality constraints get removed. * As a special case, if the input turns out to be empty, * then drop gets called with the number of removed equality * constraints set to the total number of equality constraints. * If "swap" or "drop" are not NULL, then the local variables (if any) * are assumed to be in a valid order. */ __isl_give isl_basic_map *isl_basic_map_gauss5(__isl_take isl_basic_map *bmap, int *progress, isl_stat (*swap)(unsigned a, unsigned b, void *user), isl_stat (*drop)(unsigned n, void *user), void *user) { int k; int done; int last_var; unsigned total_var; isl_size total; unsigned n_drop; if (!swap && !drop) bmap = isl_basic_map_order_divs(bmap); total = isl_basic_map_dim(bmap, isl_dim_all); if (total < 0) return isl_basic_map_free(bmap); total_var = total - bmap->n_div; last_var = total - 1; for (done = 0; done < bmap->n_eq; ++done) { for (; last_var >= 0; --last_var) { for (k = done; k < bmap->n_eq; ++k) if (!isl_int_is_zero(bmap->eq[k][1+last_var])) break; if (k < bmap->n_eq) break; } if (last_var < 0) break; if (k != done) { swap_equality(bmap, k, done); if (swap && swap(k, done, user) < 0) return isl_basic_map_free(bmap); } if (isl_int_is_neg(bmap->eq[done][1+last_var])) isl_seq_neg(bmap->eq[done], bmap->eq[done], 1+total); bmap = eliminate_var_using_equality(bmap, last_var, bmap->eq[done], 1, progress); if (last_var >= total_var) bmap = set_div_from_eq(bmap, last_var - total_var, done, progress); if (!bmap) return NULL; } if (done == bmap->n_eq) return bmap; for (k = done; k < bmap->n_eq; ++k) { if (isl_int_is_zero(bmap->eq[k][0])) continue; if (drop && drop(bmap->n_eq, user) < 0) return isl_basic_map_free(bmap); return isl_basic_map_set_to_empty(bmap); } n_drop = bmap->n_eq - done; bmap = isl_basic_map_free_equality(bmap, n_drop); if (drop && drop(n_drop, user) < 0) return isl_basic_map_free(bmap); return bmap; } __isl_give isl_basic_map *isl_basic_map_gauss(__isl_take isl_basic_map *bmap, int *progress) { return isl_basic_map_gauss5(bmap, progress, NULL, NULL, NULL); } __isl_give isl_basic_set *isl_basic_set_gauss( __isl_take isl_basic_set *bset, int *progress) { return bset_from_bmap(isl_basic_map_gauss(bset_to_bmap(bset), progress)); } static unsigned int round_up(unsigned int v) { int old_v = v; while (v) { old_v = v; v ^= v & -v; } return old_v << 1; } /* Hash table of inequalities in a basic map. * "index" is an array of addresses of inequalities in the basic map, some * of which are NULL. The inequalities are hashed on the coefficients * except the constant term. * "size" is the number of elements in the array and is always a power of two * "bits" is the number of bits need to represent an index into the array. * "total" is the total dimension of the basic map. */ struct isl_constraint_index { unsigned int size; int bits; isl_int ***index; isl_size total; }; /* Fill in the "ci" data structure for holding the inequalities of "bmap". */ static isl_stat create_constraint_index(struct isl_constraint_index *ci, __isl_keep isl_basic_map *bmap) { isl_ctx *ctx; ci->index = NULL; if (!bmap) return isl_stat_error; ci->total = isl_basic_map_dim(bmap, isl_dim_all); if (ci->total < 0) return isl_stat_error; if (bmap->n_ineq == 0) return isl_stat_ok; ci->size = round_up(4 * (bmap->n_ineq + 1) / 3 - 1); ci->bits = ffs(ci->size) - 1; ctx = isl_basic_map_get_ctx(bmap); ci->index = isl_calloc_array(ctx, isl_int **, ci->size); if (!ci->index) return isl_stat_error; return isl_stat_ok; } /* Free the memory allocated by create_constraint_index. */ static void constraint_index_free(struct isl_constraint_index *ci) { free(ci->index); } /* Return the position in ci->index that contains the address of * an inequality that is equal to *ineq up to the constant term, * provided this address is not identical to "ineq". * If there is no such inequality, then return the position where * such an inequality should be inserted. */ static int hash_index_ineq(struct isl_constraint_index *ci, isl_int **ineq) { int h; uint32_t hash = isl_seq_get_hash_bits((*ineq) + 1, ci->total, ci->bits); for (h = hash; ci->index[h]; h = (h+1) % ci->size) if (ineq != ci->index[h] && isl_seq_eq((*ineq) + 1, ci->index[h][0]+1, ci->total)) break; return h; } /* Return the position in ci->index that contains the address of * an inequality that is equal to the k'th inequality of "bmap" * up to the constant term, provided it does not point to the very * same inequality. * If there is no such inequality, then return the position where * such an inequality should be inserted. */ static int hash_index(struct isl_constraint_index *ci, __isl_keep isl_basic_map *bmap, int k) { return hash_index_ineq(ci, &bmap->ineq[k]); } static int set_hash_index(struct isl_constraint_index *ci, __isl_keep isl_basic_set *bset, int k) { return hash_index(ci, bset, k); } /* Fill in the "ci" data structure with the inequalities of "bset". */ static isl_stat setup_constraint_index(struct isl_constraint_index *ci, __isl_keep isl_basic_set *bset) { int k, h; if (create_constraint_index(ci, bset) < 0) return isl_stat_error; for (k = 0; k < bset->n_ineq; ++k) { h = set_hash_index(ci, bset, k); ci->index[h] = &bset->ineq[k]; } return isl_stat_ok; } /* Is the inequality ineq (obviously) redundant with respect * to the constraints in "ci"? * * Look for an inequality in "ci" with the same coefficients and then * check if the contant term of "ineq" is greater than or equal * to the constant term of that inequality. If so, "ineq" is clearly * redundant. * * Note that hash_index_ineq ignores a stored constraint if it has * the same address as the passed inequality. It is ok to pass * the address of a local variable here since it will never be * the same as the address of a constraint in "ci". */ static isl_bool constraint_index_is_redundant(struct isl_constraint_index *ci, isl_int *ineq) { int h; h = hash_index_ineq(ci, &ineq); if (!ci->index[h]) return isl_bool_false; return isl_int_ge(ineq[0], (*ci->index[h])[0]); } /* If we can eliminate more than one div, then we need to make * sure we do it from last div to first div, in order not to * change the position of the other divs that still need to * be removed. */ static __isl_give isl_basic_map *remove_duplicate_divs( __isl_take isl_basic_map *bmap, int *progress) { unsigned int size; int *index; int *elim_for; int k, l, h; int bits; struct isl_blk eq; isl_size v_div; unsigned total; struct isl_ctx *ctx; bmap = isl_basic_map_order_divs(bmap); if (!bmap || bmap->n_div <= 1) return bmap; v_div = isl_basic_map_var_offset(bmap, isl_dim_div); if (v_div < 0) return isl_basic_map_free(bmap); total = v_div + bmap->n_div; ctx = bmap->ctx; for (k = bmap->n_div - 1; k >= 0; --k) if (!isl_int_is_zero(bmap->div[k][0])) break; if (k <= 0) return bmap; size = round_up(4 * bmap->n_div / 3 - 1); if (size == 0) return bmap; elim_for = isl_calloc_array(ctx, int, bmap->n_div); bits = ffs(size) - 1; index = isl_calloc_array(ctx, int, size); if (!elim_for || !index) goto out; eq = isl_blk_alloc(ctx, 1+total); if (isl_blk_is_error(eq)) goto out; isl_seq_clr(eq.data, 1+total); index[isl_seq_get_hash_bits(bmap->div[k], 2+total, bits)] = k + 1; for (--k; k >= 0; --k) { uint32_t hash; if (isl_int_is_zero(bmap->div[k][0])) continue; hash = isl_seq_get_hash_bits(bmap->div[k], 2+total, bits); for (h = hash; index[h]; h = (h+1) % size) if (isl_seq_eq(bmap->div[k], bmap->div[index[h]-1], 2+total)) break; if (index[h]) { *progress = 1; l = index[h] - 1; elim_for[l] = k + 1; } index[h] = k+1; } for (l = bmap->n_div - 1; l >= 0; --l) { if (!elim_for[l]) continue; k = elim_for[l] - 1; isl_int_set_si(eq.data[1 + v_div + k], -1); isl_int_set_si(eq.data[1 + v_div + l], 1); bmap = eliminate_div(bmap, eq.data, l, 1); if (!bmap) break; isl_int_set_si(eq.data[1 + v_div + k], 0); isl_int_set_si(eq.data[1 + v_div + l], 0); } isl_blk_free(ctx, eq); out: free(index); free(elim_for); return bmap; } static int n_pure_div_eq(__isl_keep isl_basic_map *bmap) { int i, j; isl_size v_div; v_div = isl_basic_map_var_offset(bmap, isl_dim_div); if (v_div < 0) return -1; for (i = 0, j = bmap->n_div-1; i < bmap->n_eq; ++i) { while (j >= 0 && isl_int_is_zero(bmap->eq[i][1 + v_div + j])) --j; if (j < 0) break; if (isl_seq_first_non_zero(bmap->eq[i] + 1 + v_div, j) != -1) return 0; } return i; } /* Normalize divs that appear in equalities. * * In particular, we assume that bmap contains some equalities * of the form * * a x = m * e_i * * and we want to replace the set of e_i by a minimal set and * such that the new e_i have a canonical representation in terms * of the vector x. * If any of the equalities involves more than one divs, then * we currently simply bail out. * * Let us first additionally assume that all equalities involve * a div. The equalities then express modulo constraints on the * remaining variables and we can use "parameter compression" * to find a minimal set of constraints. The result is a transformation * * x = T(x') = x_0 + G x' * * with G a lower-triangular matrix with all elements below the diagonal * non-negative and smaller than the diagonal element on the same row. * We first normalize x_0 by making the same property hold in the affine * T matrix. * The rows i of G with a 1 on the diagonal do not impose any modulo * constraint and simply express x_i = x'_i. * For each of the remaining rows i, we introduce a div and a corresponding * equality. In particular * * g_ii e_j = x_i - g_i(x') * * where each x'_k is replaced either by x_k (if g_kk = 1) or the * corresponding div (if g_kk != 1). * * If there are any equalities not involving any div, then we * first apply a variable compression on the variables x: * * x = C x'' x'' = C_2 x * * and perform the above parameter compression on A C instead of on A. * The resulting compression is then of the form * * x'' = T(x') = x_0 + G x' * * and in constructing the new divs and the corresponding equalities, * we have to replace each x'', i.e., the x'_k with (g_kk = 1), * by the corresponding row from C_2. */ static __isl_give isl_basic_map *normalize_divs(__isl_take isl_basic_map *bmap, int *progress) { int i, j, k; isl_size v_div; int div_eq; struct isl_mat *B; struct isl_vec *d; struct isl_mat *T = NULL; struct isl_mat *C = NULL; struct isl_mat *C2 = NULL; isl_int v; int *pos = NULL; int dropped, needed; if (!bmap) return NULL; if (bmap->n_div == 0) return bmap; if (bmap->n_eq == 0) return bmap; if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS)) return bmap; v_div = isl_basic_map_var_offset(bmap, isl_dim_div); div_eq = n_pure_div_eq(bmap); if (v_div < 0 || div_eq < 0) return isl_basic_map_free(bmap); if (div_eq == 0) return bmap; if (div_eq < bmap->n_eq) { B = isl_mat_sub_alloc6(bmap->ctx, bmap->eq, div_eq, bmap->n_eq - div_eq, 0, 1 + v_div); C = isl_mat_variable_compression(B, &C2); if (!C || !C2) goto error; if (C->n_col == 0) { bmap = isl_basic_map_set_to_empty(bmap); isl_mat_free(C); isl_mat_free(C2); goto done; } } d = isl_vec_alloc(bmap->ctx, div_eq); if (!d) goto error; for (i = 0, j = bmap->n_div-1; i < div_eq; ++i) { while (j >= 0 && isl_int_is_zero(bmap->eq[i][1 + v_div + j])) --j; isl_int_set(d->block.data[i], bmap->eq[i][1 + v_div + j]); } B = isl_mat_sub_alloc6(bmap->ctx, bmap->eq, 0, div_eq, 0, 1 + v_div); if (C) { B = isl_mat_product(B, C); C = NULL; } T = isl_mat_parameter_compression(B, d); if (!T) goto error; if (T->n_col == 0) { bmap = isl_basic_map_set_to_empty(bmap); isl_mat_free(C2); isl_mat_free(T); goto done; } isl_int_init(v); for (i = 0; i < T->n_row - 1; ++i) { isl_int_fdiv_q(v, T->row[1 + i][0], T->row[1 + i][1 + i]); if (isl_int_is_zero(v)) continue; isl_mat_col_submul(T, 0, v, 1 + i); } isl_int_clear(v); pos = isl_alloc_array(bmap->ctx, int, T->n_row); if (!pos) goto error; /* We have to be careful because dropping equalities may reorder them */ dropped = 0; for (j = bmap->n_div - 1; j >= 0; --j) { for (i = 0; i < bmap->n_eq; ++i) if (!isl_int_is_zero(bmap->eq[i][1 + v_div + j])) break; if (i < bmap->n_eq) { bmap = isl_basic_map_drop_div(bmap, j); if (isl_basic_map_drop_equality(bmap, i) < 0) goto error; ++dropped; } } pos[0] = 0; needed = 0; for (i = 1; i < T->n_row; ++i) { if (isl_int_is_one(T->row[i][i])) pos[i] = i; else needed++; } if (needed > dropped) { bmap = isl_basic_map_extend(bmap, needed, needed, 0); if (!bmap) goto error; } for (i = 1; i < T->n_row; ++i) { if (isl_int_is_one(T->row[i][i])) continue; k = isl_basic_map_alloc_div(bmap); pos[i] = 1 + v_div + k; isl_seq_clr(bmap->div[k] + 1, 1 + v_div + bmap->n_div); isl_int_set(bmap->div[k][0], T->row[i][i]); if (C2) isl_seq_cpy(bmap->div[k] + 1, C2->row[i], 1 + v_div); else isl_int_set_si(bmap->div[k][1 + i], 1); for (j = 0; j < i; ++j) { if (isl_int_is_zero(T->row[i][j])) continue; if (pos[j] < T->n_row && C2) isl_seq_submul(bmap->div[k] + 1, T->row[i][j], C2->row[pos[j]], 1 + v_div); else isl_int_neg(bmap->div[k][1 + pos[j]], T->row[i][j]); } j = isl_basic_map_alloc_equality(bmap); isl_seq_neg(bmap->eq[j], bmap->div[k]+1, 1+v_div+bmap->n_div); isl_int_set(bmap->eq[j][pos[i]], bmap->div[k][0]); } free(pos); isl_mat_free(C2); isl_mat_free(T); if (progress) *progress = 1; done: ISL_F_SET(bmap, ISL_BASIC_MAP_NORMALIZED_DIVS); return bmap; error: free(pos); isl_mat_free(C); isl_mat_free(C2); isl_mat_free(T); isl_basic_map_free(bmap); return NULL; } static __isl_give isl_basic_map *set_div_from_lower_bound( __isl_take isl_basic_map *bmap, int div, int ineq) { unsigned total = isl_basic_map_offset(bmap, isl_dim_div); isl_seq_neg(bmap->div[div] + 1, bmap->ineq[ineq], total + bmap->n_div); isl_int_set(bmap->div[div][0], bmap->ineq[ineq][total + div]); isl_int_add(bmap->div[div][1], bmap->div[div][1], bmap->div[div][0]); isl_int_sub_ui(bmap->div[div][1], bmap->div[div][1], 1); isl_int_set_si(bmap->div[div][1 + total + div], 0); return bmap; } /* Check whether it is ok to define a div based on an inequality. * To avoid the introduction of circular definitions of divs, we * do not allow such a definition if the resulting expression would refer to * any other undefined divs or if any known div is defined in * terms of the unknown div. */ static isl_bool ok_to_set_div_from_bound(__isl_keep isl_basic_map *bmap, int div, int ineq) { int j; unsigned total = isl_basic_map_offset(bmap, isl_dim_div); /* Not defined in terms of unknown divs */ for (j = 0; j < bmap->n_div; ++j) { if (div == j) continue; if (isl_int_is_zero(bmap->ineq[ineq][total + j])) continue; if (isl_int_is_zero(bmap->div[j][0])) return isl_bool_false; } /* No other div defined in terms of this one => avoid loops */ for (j = 0; j < bmap->n_div; ++j) { if (div == j) continue; if (isl_int_is_zero(bmap->div[j][0])) continue; if (!isl_int_is_zero(bmap->div[j][1 + total + div])) return isl_bool_false; } return isl_bool_true; } /* Would an expression for div "div" based on inequality "ineq" of "bmap" * be a better expression than the current one? * * If we do not have any expression yet, then any expression would be better. * Otherwise we check if the last variable involved in the inequality * (disregarding the div that it would define) is in an earlier position * than the last variable involved in the current div expression. */ static isl_bool better_div_constraint(__isl_keep isl_basic_map *bmap, int div, int ineq) { unsigned total = isl_basic_map_offset(bmap, isl_dim_div); int last_div; int last_ineq; if (isl_int_is_zero(bmap->div[div][0])) return isl_bool_true; if (isl_seq_last_non_zero(bmap->ineq[ineq] + total + div + 1, bmap->n_div - (div + 1)) >= 0) return isl_bool_false; last_ineq = isl_seq_last_non_zero(bmap->ineq[ineq], total + div); last_div = isl_seq_last_non_zero(bmap->div[div] + 1, total + bmap->n_div); return last_ineq < last_div; } /* Given two constraints "k" and "l" that are opposite to each other, * except for the constant term, check if we can use them * to obtain an expression for one of the hitherto unknown divs or * a "better" expression for a div for which we already have an expression. * "sum" is the sum of the constant terms of the constraints. * If this sum is strictly smaller than the coefficient of one * of the divs, then this pair can be used to define the div. * To avoid the introduction of circular definitions of divs, we * do not use the pair if the resulting expression would refer to * any other undefined divs or if any known div is defined in * terms of the unknown div. */ static __isl_give isl_basic_map *check_for_div_constraints( __isl_take isl_basic_map *bmap, int k, int l, isl_int sum, int *progress) { int i; unsigned total = isl_basic_map_offset(bmap, isl_dim_div); for (i = 0; i < bmap->n_div; ++i) { isl_bool set_div; if (isl_int_is_zero(bmap->ineq[k][total + i])) continue; if (isl_int_abs_ge(sum, bmap->ineq[k][total + i])) continue; set_div = better_div_constraint(bmap, i, k); if (set_div >= 0 && set_div) set_div = ok_to_set_div_from_bound(bmap, i, k); if (set_div < 0) return isl_basic_map_free(bmap); if (!set_div) break; if (isl_int_is_pos(bmap->ineq[k][total + i])) bmap = set_div_from_lower_bound(bmap, i, k); else bmap = set_div_from_lower_bound(bmap, i, l); if (progress) *progress = 1; break; } return bmap; } __isl_give isl_basic_map *isl_basic_map_remove_duplicate_constraints( __isl_take isl_basic_map *bmap, int *progress, int detect_divs) { struct isl_constraint_index ci; int k, l, h; isl_size total = isl_basic_map_dim(bmap, isl_dim_all); isl_int sum; if (total < 0 || bmap->n_ineq <= 1) return bmap; if (create_constraint_index(&ci, bmap) < 0) return bmap; h = isl_seq_get_hash_bits(bmap->ineq[0] + 1, total, ci.bits); ci.index[h] = &bmap->ineq[0]; for (k = 1; k < bmap->n_ineq; ++k) { h = hash_index(&ci, bmap, k); if (!ci.index[h]) { ci.index[h] = &bmap->ineq[k]; continue; } if (progress) *progress = 1; l = ci.index[h] - &bmap->ineq[0]; if (isl_int_lt(bmap->ineq[k][0], bmap->ineq[l][0])) swap_inequality(bmap, k, l); isl_basic_map_drop_inequality(bmap, k); --k; } isl_int_init(sum); for (k = 0; bmap && k < bmap->n_ineq-1; ++k) { isl_seq_neg(bmap->ineq[k]+1, bmap->ineq[k]+1, total); h = hash_index(&ci, bmap, k); isl_seq_neg(bmap->ineq[k]+1, bmap->ineq[k]+1, total); if (!ci.index[h]) continue; l = ci.index[h] - &bmap->ineq[0]; isl_int_add(sum, bmap->ineq[k][0], bmap->ineq[l][0]); if (isl_int_is_pos(sum)) { if (detect_divs) bmap = check_for_div_constraints(bmap, k, l, sum, progress); continue; } if (isl_int_is_zero(sum)) { /* We need to break out of the loop after these * changes since the contents of the hash * will no longer be valid. * Plus, we probably we want to regauss first. */ if (progress) *progress = 1; isl_basic_map_drop_inequality(bmap, l); isl_basic_map_inequality_to_equality(bmap, k); } else bmap = isl_basic_map_set_to_empty(bmap); break; } isl_int_clear(sum); constraint_index_free(&ci); return bmap; } /* Detect all pairs of inequalities that form an equality. * * isl_basic_map_remove_duplicate_constraints detects at most one such pair. * Call it repeatedly while it is making progress. */ __isl_give isl_basic_map *isl_basic_map_detect_inequality_pairs( __isl_take isl_basic_map *bmap, int *progress) { int duplicate; do { duplicate = 0; bmap = isl_basic_map_remove_duplicate_constraints(bmap, &duplicate, 0); if (progress && duplicate) *progress = 1; } while (duplicate); return bmap; } /* Given a known integer division "div" that is not integral * (with denominator 1), eliminate it from the constraints in "bmap" * where it appears with a (positive or negative) unit coefficient. * If "progress" is not NULL, then it gets set if the elimination * results in any changes. * * That is, replace * * floor(e/m) + f >= 0 * * by * * e + m f >= 0 * * and * * -floor(e/m) + f >= 0 * * by * * -e + m f + m - 1 >= 0 * * The first conversion is valid because floor(e/m) >= -f is equivalent * to e/m >= -f because -f is an integral expression. * The second conversion follows from the fact that * * -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m) * * * Note that one of the div constraints may have been eliminated * due to being redundant with respect to the constraint that is * being modified by this function. The modified constraint may * no longer imply this div constraint, so we add it back to make * sure we do not lose any information. */ static __isl_give isl_basic_map *eliminate_unit_div( __isl_take isl_basic_map *bmap, int div, int *progress) { int j; isl_size v_div, dim; isl_ctx *ctx; v_div = isl_basic_map_var_offset(bmap, isl_dim_div); dim = isl_basic_map_dim(bmap, isl_dim_all); if (v_div < 0 || dim < 0) return isl_basic_map_free(bmap); ctx = isl_basic_map_get_ctx(bmap); for (j = 0; j < bmap->n_ineq; ++j) { int s; if (!isl_int_is_one(bmap->ineq[j][1 + v_div + div]) && !isl_int_is_negone(bmap->ineq[j][1 + v_div + div])) continue; if (progress) *progress = 1; s = isl_int_sgn(bmap->ineq[j][1 + v_div + div]); isl_int_set_si(bmap->ineq[j][1 + v_div + div], 0); if (s < 0) isl_seq_combine(bmap->ineq[j], ctx->negone, bmap->div[div] + 1, bmap->div[div][0], bmap->ineq[j], 1 + dim); else isl_seq_combine(bmap->ineq[j], ctx->one, bmap->div[div] + 1, bmap->div[div][0], bmap->ineq[j], 1 + dim); if (s < 0) { isl_int_add(bmap->ineq[j][0], bmap->ineq[j][0], bmap->div[div][0]); isl_int_sub_ui(bmap->ineq[j][0], bmap->ineq[j][0], 1); } bmap = isl_basic_map_extend_constraints(bmap, 0, 1); bmap = isl_basic_map_add_div_constraint(bmap, div, s); if (!bmap) return NULL; } return bmap; } /* Eliminate selected known divs from constraints where they appear with * a (positive or negative) unit coefficient. * In particular, only handle those for which "select" returns isl_bool_true. * If "progress" is not NULL, then it gets set if the elimination * results in any changes. * * We skip integral divs, i.e., those with denominator 1, as we would * risk eliminating the div from the div constraints. We do not need * to handle those divs here anyway since the div constraints will turn * out to form an equality and this equality can then be used to eliminate * the div from all constraints. */ static __isl_give isl_basic_map *eliminate_selected_unit_divs( __isl_take isl_basic_map *bmap, isl_bool (*select)(__isl_keep isl_basic_map *bmap, int div), int *progress) { int i; if (!bmap) return NULL; for (i = 0; i < bmap->n_div; ++i) { isl_bool selected; if (isl_int_is_zero(bmap->div[i][0])) continue; if (isl_int_is_one(bmap->div[i][0])) continue; selected = select(bmap, i); if (selected < 0) return isl_basic_map_free(bmap); if (!selected) continue; bmap = eliminate_unit_div(bmap, i, progress); if (!bmap) return NULL; } return bmap; } /* eliminate_selected_unit_divs callback that selects every * integer division. */ static isl_bool is_any_div(__isl_keep isl_basic_map *bmap, int div) { return isl_bool_true; } /* Eliminate known divs from constraints where they appear with * a (positive or negative) unit coefficient. * If "progress" is not NULL, then it gets set if the elimination * results in any changes. */ static __isl_give isl_basic_map *eliminate_unit_divs( __isl_take isl_basic_map *bmap, int *progress) { return eliminate_selected_unit_divs(bmap, &is_any_div, progress); } /* eliminate_selected_unit_divs callback that selects * integer divisions that only appear with * a (positive or negative) unit coefficient * (outside their div constraints). */ static isl_bool is_pure_unit_div(__isl_keep isl_basic_map *bmap, int div) { int i; isl_size v_div, n_ineq; v_div = isl_basic_map_var_offset(bmap, isl_dim_div); n_ineq = isl_basic_map_n_inequality(bmap); if (v_div < 0 || n_ineq < 0) return isl_bool_error; for (i = 0; i < n_ineq; ++i) { isl_bool skip; if (isl_int_is_zero(bmap->ineq[i][1 + v_div + div])) continue; skip = isl_basic_map_is_div_constraint(bmap, bmap->ineq[i], div); if (skip < 0) return isl_bool_error; if (skip) continue; if (!isl_int_is_one(bmap->ineq[i][1 + v_div + div]) && !isl_int_is_negone(bmap->ineq[i][1 + v_div + div])) return isl_bool_false; } return isl_bool_true; } /* Eliminate known divs from constraints where they appear with * a (positive or negative) unit coefficient, * but only if they do not appear in any other constraints * (other than the div constraints). */ __isl_give isl_basic_map *isl_basic_map_eliminate_pure_unit_divs( __isl_take isl_basic_map *bmap) { return eliminate_selected_unit_divs(bmap, &is_pure_unit_div, NULL); } __isl_give isl_basic_map *isl_basic_map_simplify(__isl_take isl_basic_map *bmap) { int progress = 1; if (!bmap) return NULL; while (progress) { isl_bool empty; progress = 0; empty = isl_basic_map_plain_is_empty(bmap); if (empty < 0) return isl_basic_map_free(bmap); if (empty) break; bmap = isl_basic_map_normalize_constraints(bmap); bmap = reduce_div_coefficients(bmap); bmap = normalize_div_expressions(bmap); bmap = remove_duplicate_divs(bmap, &progress); bmap = eliminate_unit_divs(bmap, &progress); bmap = eliminate_divs_eq(bmap, &progress); bmap = eliminate_divs_ineq(bmap, &progress); bmap = isl_basic_map_gauss(bmap, &progress); /* requires equalities in normal form */ bmap = normalize_divs(bmap, &progress); bmap = isl_basic_map_remove_duplicate_constraints(bmap, &progress, 1); if (bmap && progress) ISL_F_CLR(bmap, ISL_BASIC_MAP_REDUCED_COEFFICIENTS); } return bmap; } __isl_give isl_basic_set *isl_basic_set_simplify( __isl_take isl_basic_set *bset) { return bset_from_bmap(isl_basic_map_simplify(bset_to_bmap(bset))); } isl_bool isl_basic_map_is_div_constraint(__isl_keep isl_basic_map *bmap, isl_int *constraint, unsigned div) { unsigned pos; if (!bmap) return isl_bool_error; pos = isl_basic_map_offset(bmap, isl_dim_div) + div; if (isl_int_eq(constraint[pos], bmap->div[div][0])) { int neg; isl_int_sub(bmap->div[div][1], bmap->div[div][1], bmap->div[div][0]); isl_int_add_ui(bmap->div[div][1], bmap->div[div][1], 1); neg = isl_seq_is_neg(constraint, bmap->div[div]+1, pos); isl_int_sub_ui(bmap->div[div][1], bmap->div[div][1], 1); isl_int_add(bmap->div[div][1], bmap->div[div][1], bmap->div[div][0]); if (!neg) return isl_bool_false; if (isl_seq_first_non_zero(constraint+pos+1, bmap->n_div-div-1) != -1) return isl_bool_false; } else if (isl_int_abs_eq(constraint[pos], bmap->div[div][0])) { if (!isl_seq_eq(constraint, bmap->div[div]+1, pos)) return isl_bool_false; if (isl_seq_first_non_zero(constraint+pos+1, bmap->n_div-div-1) != -1) return isl_bool_false; } else return isl_bool_false; return isl_bool_true; } /* If the only constraints a div d=floor(f/m) * appears in are its two defining constraints * * f - m d >=0 * -(f - (m - 1)) + m d >= 0 * * then it can safely be removed. */ static isl_bool div_is_redundant(__isl_keep isl_basic_map *bmap, int div) { int i; isl_size v_div = isl_basic_map_var_offset(bmap, isl_dim_div); unsigned pos = 1 + v_div + div; if (v_div < 0) return isl_bool_error; for (i = 0; i < bmap->n_eq; ++i) if (!isl_int_is_zero(bmap->eq[i][pos])) return isl_bool_false; for (i = 0; i < bmap->n_ineq; ++i) { isl_bool red; if (isl_int_is_zero(bmap->ineq[i][pos])) continue; red = isl_basic_map_is_div_constraint(bmap, bmap->ineq[i], div); if (red < 0 || !red) return red; } for (i = 0; i < bmap->n_div; ++i) { if (isl_int_is_zero(bmap->div[i][0])) continue; if (!isl_int_is_zero(bmap->div[i][1+pos])) return isl_bool_false; } return isl_bool_true; } /* * Remove divs that don't occur in any of the constraints or other divs. * These can arise when dropping constraints from a basic map or * when the divs of a basic map have been temporarily aligned * with the divs of another basic map. */ static __isl_give isl_basic_map *remove_redundant_divs( __isl_take isl_basic_map *bmap) { int i; isl_size v_div; v_div = isl_basic_map_var_offset(bmap, isl_dim_div); if (v_div < 0) return isl_basic_map_free(bmap); for (i = bmap->n_div-1; i >= 0; --i) { isl_bool redundant; redundant = div_is_redundant(bmap, i); if (redundant < 0) return isl_basic_map_free(bmap); if (!redundant) continue; bmap = isl_basic_map_drop_constraints_involving(bmap, v_div + i, 1); bmap = isl_basic_map_drop_div(bmap, i); } return bmap; } /* Mark "bmap" as final, without checking for obviously redundant * integer divisions. This function should be used when "bmap" * is known not to involve any such integer divisions. */ __isl_give isl_basic_map *isl_basic_map_mark_final( __isl_take isl_basic_map *bmap) { if (!bmap) return NULL; ISL_F_SET(bmap, ISL_BASIC_SET_FINAL); return bmap; } /* Mark "bmap" as final, after removing obviously redundant integer divisions. */ __isl_give isl_basic_map *isl_basic_map_finalize(__isl_take isl_basic_map *bmap) { bmap = remove_redundant_divs(bmap); bmap = isl_basic_map_mark_final(bmap); return bmap; } __isl_give isl_basic_set *isl_basic_set_finalize( __isl_take isl_basic_set *bset) { return bset_from_bmap(isl_basic_map_finalize(bset_to_bmap(bset))); } /* Remove definition of any div that is defined in terms of the given variable. * The div itself is not removed. Functions such as * eliminate_divs_ineq depend on the other divs remaining in place. */ static __isl_give isl_basic_map *remove_dependent_vars( __isl_take isl_basic_map *bmap, int pos) { int i; if (!bmap) return NULL; for (i = 0; i < bmap->n_div; ++i) { if (isl_int_is_zero(bmap->div[i][0])) continue; if (isl_int_is_zero(bmap->div[i][1+1+pos])) continue; bmap = isl_basic_map_mark_div_unknown(bmap, i); if (!bmap) return NULL; } return bmap; } /* Eliminate the specified variables from the constraints using * Fourier-Motzkin. The variables themselves are not removed. */ __isl_give isl_basic_map *isl_basic_map_eliminate_vars( __isl_take isl_basic_map *bmap, unsigned pos, unsigned n) { int d; int i, j, k; isl_size total; int need_gauss = 0; if (n == 0) return bmap; total = isl_basic_map_dim(bmap, isl_dim_all); if (total < 0) return isl_basic_map_free(bmap); bmap = isl_basic_map_cow(bmap); for (d = pos + n - 1; d >= 0 && d >= pos; --d) bmap = remove_dependent_vars(bmap, d); if (!bmap) return NULL; for (d = pos + n - 1; d >= 0 && d >= total - bmap->n_div && d >= pos; --d) isl_seq_clr(bmap->div[d-(total-bmap->n_div)], 2+total); for (d = pos + n - 1; d >= 0 && d >= pos; --d) { int n_lower, n_upper; if (!bmap) return NULL; for (i = 0; i < bmap->n_eq; ++i) { if (isl_int_is_zero(bmap->eq[i][1+d])) continue; bmap = eliminate_var_using_equality(bmap, d, bmap->eq[i], 0, NULL); if (isl_basic_map_drop_equality(bmap, i) < 0) return isl_basic_map_free(bmap); need_gauss = 1; break; } if (i < bmap->n_eq) continue; n_lower = 0; n_upper = 0; for (i = 0; i < bmap->n_ineq; ++i) { if (isl_int_is_pos(bmap->ineq[i][1+d])) n_lower++; else if (isl_int_is_neg(bmap->ineq[i][1+d])) n_upper++; } bmap = isl_basic_map_extend_constraints(bmap, 0, n_lower * n_upper); if (!bmap) goto error; for (i = bmap->n_ineq - 1; i >= 0; --i) { int last; if (isl_int_is_zero(bmap->ineq[i][1+d])) continue; last = -1; for (j = 0; j < i; ++j) { if (isl_int_is_zero(bmap->ineq[j][1+d])) continue; last = j; if (isl_int_sgn(bmap->ineq[i][1+d]) == isl_int_sgn(bmap->ineq[j][1+d])) continue; k = isl_basic_map_alloc_inequality(bmap); if (k < 0) goto error; isl_seq_cpy(bmap->ineq[k], bmap->ineq[i], 1+total); isl_seq_elim(bmap->ineq[k], bmap->ineq[j], 1+d, 1+total, NULL); } isl_basic_map_drop_inequality(bmap, i); i = last + 1; } if (n_lower > 0 && n_upper > 0) { bmap = isl_basic_map_normalize_constraints(bmap); bmap = isl_basic_map_remove_duplicate_constraints(bmap, NULL, 0); bmap = isl_basic_map_gauss(bmap, NULL); bmap = isl_basic_map_remove_redundancies(bmap); need_gauss = 0; if (!bmap) goto error; if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY)) break; } } if (need_gauss) bmap = isl_basic_map_gauss(bmap, NULL); return bmap; error: isl_basic_map_free(bmap); return NULL; } __isl_give isl_basic_set *isl_basic_set_eliminate_vars( __isl_take isl_basic_set *bset, unsigned pos, unsigned n) { return bset_from_bmap(isl_basic_map_eliminate_vars(bset_to_bmap(bset), pos, n)); } /* Eliminate the specified n dimensions starting at first from the * constraints, without removing the dimensions from the space. * If the set is rational, the dimensions are eliminated using Fourier-Motzkin. * Otherwise, they are projected out and the original space is restored. */ __isl_give isl_basic_map *isl_basic_map_eliminate( __isl_take isl_basic_map *bmap, enum isl_dim_type type, unsigned first, unsigned n) { isl_space *space; if (!bmap) return NULL; if (n == 0) return bmap; if (isl_basic_map_check_range(bmap, type, first, n) < 0) return isl_basic_map_free(bmap); if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL)) { first += isl_basic_map_offset(bmap, type) - 1; bmap = isl_basic_map_eliminate_vars(bmap, first, n); return isl_basic_map_finalize(bmap); } space = isl_basic_map_get_space(bmap); bmap = isl_basic_map_project_out(bmap, type, first, n); bmap = isl_basic_map_insert_dims(bmap, type, first, n); bmap = isl_basic_map_reset_space(bmap, space); return bmap; } __isl_give isl_basic_set *isl_basic_set_eliminate( __isl_take isl_basic_set *bset, enum isl_dim_type type, unsigned first, unsigned n) { return isl_basic_map_eliminate(bset, type, first, n); } /* Remove all constraints from "bmap" that reference any unknown local * variables (directly or indirectly). * * Dropping all constraints on a local variable will make it redundant, * so it will get removed implicitly by * isl_basic_map_drop_constraints_involving_dims. Some other local * variables may also end up becoming redundant if they only appear * in constraints together with the unknown local variable. * Therefore, start over after calling * isl_basic_map_drop_constraints_involving_dims. */ __isl_give isl_basic_map *isl_basic_map_drop_constraints_involving_unknown_divs( __isl_take isl_basic_map *bmap) { isl_bool known; isl_size n_div; int i, o_div; known = isl_basic_map_divs_known(bmap); if (known < 0) return isl_basic_map_free(bmap); if (known) return bmap; n_div = isl_basic_map_dim(bmap, isl_dim_div); if (n_div < 0) return isl_basic_map_free(bmap); o_div = isl_basic_map_offset(bmap, isl_dim_div) - 1; for (i = 0; i < n_div; ++i) { known = isl_basic_map_div_is_known(bmap, i); if (known < 0) return isl_basic_map_free(bmap); if (known) continue; bmap = remove_dependent_vars(bmap, o_div + i); bmap = isl_basic_map_drop_constraints_involving_dims(bmap, isl_dim_div, i, 1); n_div = isl_basic_map_dim(bmap, isl_dim_div); if (n_div < 0) return isl_basic_map_free(bmap); i = -1; } return bmap; } /* Remove all constraints from "bset" that reference any unknown local * variables (directly or indirectly). */ __isl_give isl_basic_set *isl_basic_set_drop_constraints_involving_unknown_divs( __isl_take isl_basic_set *bset) { isl_basic_map *bmap; bmap = bset_to_bmap(bset); bmap = isl_basic_map_drop_constraints_involving_unknown_divs(bmap); return bset_from_bmap(bmap); } /* Remove all constraints from "map" that reference any unknown local * variables (directly or indirectly). * * Since constraints may get dropped from the basic maps, * they may no longer be disjoint from each other. */ __isl_give isl_map *isl_map_drop_constraints_involving_unknown_divs( __isl_take isl_map *map) { int i; isl_bool known; known = isl_map_divs_known(map); if (known < 0) return isl_map_free(map); if (known) return map; map = isl_map_cow(map); if (!map) return NULL; for (i = 0; i < map->n; ++i) { map->p[i] = isl_basic_map_drop_constraints_involving_unknown_divs( map->p[i]); if (!map->p[i]) return isl_map_free(map); } if (map->n > 1) ISL_F_CLR(map, ISL_MAP_DISJOINT); return map; } /* Don't assume equalities are in order, because align_divs * may have changed the order of the divs. */ static void compute_elimination_index(__isl_keep isl_basic_map *bmap, int *elim, unsigned len) { int d, i; for (d = 0; d < len; ++d) elim[d] = -1; for (i = 0; i < bmap->n_eq; ++i) { for (d = len - 1; d >= 0; --d) { if (isl_int_is_zero(bmap->eq[i][1+d])) continue; elim[d] = i; break; } } } static void set_compute_elimination_index(__isl_keep isl_basic_set *bset, int *elim, unsigned len) { compute_elimination_index(bset_to_bmap(bset), elim, len); } static int reduced_using_equalities(isl_int *dst, isl_int *src, __isl_keep isl_basic_map *bmap, int *elim, unsigned total) { int d; int copied = 0; for (d = total - 1; d >= 0; --d) { if (isl_int_is_zero(src[1+d])) continue; if (elim[d] == -1) continue; if (!copied) { isl_seq_cpy(dst, src, 1 + total); copied = 1; } isl_seq_elim(dst, bmap->eq[elim[d]], 1 + d, 1 + total, NULL); } return copied; } static int set_reduced_using_equalities(isl_int *dst, isl_int *src, __isl_keep isl_basic_set *bset, int *elim, unsigned total) { return reduced_using_equalities(dst, src, bset_to_bmap(bset), elim, total); } static __isl_give isl_basic_set *isl_basic_set_reduce_using_equalities( __isl_take isl_basic_set *bset, __isl_take isl_basic_set *context) { int i; int *elim; isl_size dim; if (!bset || !context) goto error; if (context->n_eq == 0) { isl_basic_set_free(context); return bset; } bset = isl_basic_set_cow(bset); dim = isl_basic_set_dim(bset, isl_dim_set); if (dim < 0) goto error; elim = isl_alloc_array(bset->ctx, int, dim); if (!elim) goto error; set_compute_elimination_index(context, elim, dim); for (i = 0; i < bset->n_eq; ++i) set_reduced_using_equalities(bset->eq[i], bset->eq[i], context, elim, dim); for (i = 0; i < bset->n_ineq; ++i) set_reduced_using_equalities(bset->ineq[i], bset->ineq[i], context, elim, dim); isl_basic_set_free(context); free(elim); bset = isl_basic_set_simplify(bset); bset = isl_basic_set_finalize(bset); return bset; error: isl_basic_set_free(bset); isl_basic_set_free(context); return NULL; } /* For each inequality in "ineq" that is a shifted (more relaxed) * copy of an inequality in "context", mark the corresponding entry * in "row" with -1. * If an inequality only has a non-negative constant term, then * mark it as well. */ static isl_stat mark_shifted_constraints(__isl_keep isl_mat *ineq, __isl_keep isl_basic_set *context, int *row) { struct isl_constraint_index ci; isl_size n_ineq, cols; unsigned total; int k; if (!ineq || !context) return isl_stat_error; if (context->n_ineq == 0) return isl_stat_ok; if (setup_constraint_index(&ci, context) < 0) return isl_stat_error; n_ineq = isl_mat_rows(ineq); cols = isl_mat_cols(ineq); if (n_ineq < 0 || cols < 0) return isl_stat_error; total = cols - 1; for (k = 0; k < n_ineq; ++k) { int l; isl_bool redundant; l = isl_seq_first_non_zero(ineq->row[k] + 1, total); if (l < 0 && isl_int_is_nonneg(ineq->row[k][0])) { row[k] = -1; continue; } redundant = constraint_index_is_redundant(&ci, ineq->row[k]); if (redundant < 0) goto error; if (!redundant) continue; row[k] = -1; } constraint_index_free(&ci); return isl_stat_ok; error: constraint_index_free(&ci); return isl_stat_error; } static __isl_give isl_basic_set *remove_shifted_constraints( __isl_take isl_basic_set *bset, __isl_keep isl_basic_set *context) { struct isl_constraint_index ci; int k; if (!bset || !context) return bset; if (context->n_ineq == 0) return bset; if (setup_constraint_index(&ci, context) < 0) return bset; for (k = 0; k < bset->n_ineq; ++k) { isl_bool redundant; redundant = constraint_index_is_redundant(&ci, bset->ineq[k]); if (redundant < 0) goto error; if (!redundant) continue; bset = isl_basic_set_cow(bset); if (!bset) goto error; isl_basic_set_drop_inequality(bset, k); --k; } constraint_index_free(&ci); return bset; error: constraint_index_free(&ci); return bset; } /* Remove constraints from "bmap" that are identical to constraints * in "context" or that are more relaxed (greater constant term). * * We perform the test for shifted copies on the pure constraints * in remove_shifted_constraints. */ static __isl_give isl_basic_map *isl_basic_map_remove_shifted_constraints( __isl_take isl_basic_map *bmap, __isl_take isl_basic_map *context) { isl_basic_set *bset, *bset_context; if (!bmap || !context) goto error; if (bmap->n_ineq == 0 || context->n_ineq == 0) { isl_basic_map_free(context); return bmap; } bmap = isl_basic_map_order_divs(bmap); context = isl_basic_map_align_divs(context, bmap); bmap = isl_basic_map_align_divs(bmap, context); bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap)); bset_context = isl_basic_map_underlying_set(context); bset = remove_shifted_constraints(bset, bset_context); isl_basic_set_free(bset_context); bmap = isl_basic_map_overlying_set(bset, bmap); return bmap; error: isl_basic_map_free(bmap); isl_basic_map_free(context); return NULL; } /* Does the (linear part of a) constraint "c" involve any of the "len" * "relevant" dimensions? */ static int is_related(isl_int *c, int len, int *relevant) { int i; for (i = 0; i < len; ++i) { if (!relevant[i]) continue; if (!isl_int_is_zero(c[i])) return 1; } return 0; } /* Drop constraints from "bmap" that do not involve any of * the dimensions marked "relevant". */ static __isl_give isl_basic_map *drop_unrelated_constraints( __isl_take isl_basic_map *bmap, int *relevant) { int i; isl_size dim; dim = isl_basic_map_dim(bmap, isl_dim_all); if (dim < 0) return isl_basic_map_free(bmap); for (i = 0; i < dim; ++i) if (!relevant[i]) break; if (i >= dim) return bmap; for (i = bmap->n_eq - 1; i >= 0; --i) if (!is_related(bmap->eq[i] + 1, dim, relevant)) { bmap = isl_basic_map_cow(bmap); if (isl_basic_map_drop_equality(bmap, i) < 0) return isl_basic_map_free(bmap); } for (i = bmap->n_ineq - 1; i >= 0; --i) if (!is_related(bmap->ineq[i] + 1, dim, relevant)) { bmap = isl_basic_map_cow(bmap); if (isl_basic_map_drop_inequality(bmap, i) < 0) return isl_basic_map_free(bmap); } return bmap; } /* Update the groups in "group" based on the (linear part of a) constraint "c". * * In particular, for any variable involved in the constraint, * find the actual group id from before and replace the group * of the corresponding variable by the minimal group of all * the variables involved in the constraint considered so far * (if this minimum is smaller) or replace the minimum by this group * (if the minimum is larger). * * At the end, all the variables in "c" will (indirectly) point * to the minimal of the groups that they referred to originally. */ static void update_groups(int dim, int *group, isl_int *c) { int j; int min = dim; for (j = 0; j < dim; ++j) { if (isl_int_is_zero(c[j])) continue; while (group[j] >= 0 && group[group[j]] != group[j]) group[j] = group[group[j]]; if (group[j] == min) continue; if (group[j] < min) { if (min >= 0 && min < dim) group[min] = group[j]; min = group[j]; } else group[group[j]] = min; } } /* Allocate an array of groups of variables, one for each variable * in "context", initialized to zero. */ static int *alloc_groups(__isl_keep isl_basic_set *context) { isl_ctx *ctx; isl_size dim; dim = isl_basic_set_dim(context, isl_dim_set); if (dim < 0) return NULL; ctx = isl_basic_set_get_ctx(context); return isl_calloc_array(ctx, int, dim); } /* Drop constraints from "bmap" that only involve variables that are * not related to any of the variables marked with a "-1" in "group". * * We construct groups of variables that collect variables that * (indirectly) appear in some common constraint of "bmap". * Each group is identified by the first variable in the group, * except for the special group of variables that was already identified * in the input as -1 (or are related to those variables). * If group[i] is equal to i (or -1), then the group of i is i (or -1), * otherwise the group of i is the group of group[i]. * * We first initialize groups for the remaining variables. * Then we iterate over the constraints of "bmap" and update the * group of the variables in the constraint by the smallest group. * Finally, we resolve indirect references to groups by running over * the variables. * * After computing the groups, we drop constraints that do not involve * any variables in the -1 group. */ __isl_give isl_basic_map *isl_basic_map_drop_unrelated_constraints( __isl_take isl_basic_map *bmap, __isl_take int *group) { isl_size dim; int i; int last; dim = isl_basic_map_dim(bmap, isl_dim_all); if (dim < 0) return isl_basic_map_free(bmap); last = -1; for (i = 0; i < dim; ++i) if (group[i] >= 0) last = group[i] = i; if (last < 0) { free(group); return bmap; } for (i = 0; i < bmap->n_eq; ++i) update_groups(dim, group, bmap->eq[i] + 1); for (i = 0; i < bmap->n_ineq; ++i) update_groups(dim, group, bmap->ineq[i] + 1); for (i = 0; i < dim; ++i) if (group[i] >= 0) group[i] = group[group[i]]; for (i = 0; i < dim; ++i) group[i] = group[i] == -1; bmap = drop_unrelated_constraints(bmap, group); free(group); return bmap; } /* Drop constraints from "context" that are irrelevant for computing * the gist of "bset". * * In particular, drop constraints in variables that are not related * to any of the variables involved in the constraints of "bset" * in the sense that there is no sequence of constraints that connects them. * * We first mark all variables that appear in "bset" as belonging * to a "-1" group and then continue with group_and_drop_irrelevant_constraints. */ static __isl_give isl_basic_set *drop_irrelevant_constraints( __isl_take isl_basic_set *context, __isl_keep isl_basic_set *bset) { int *group; isl_size dim; int i, j; dim = isl_basic_set_dim(bset, isl_dim_set); if (!context || dim < 0) return isl_basic_set_free(context); group = alloc_groups(context); if (!group) return isl_basic_set_free(context); for (i = 0; i < dim; ++i) { for (j = 0; j < bset->n_eq; ++j) if (!isl_int_is_zero(bset->eq[j][1 + i])) break; if (j < bset->n_eq) { group[i] = -1; continue; } for (j = 0; j < bset->n_ineq; ++j) if (!isl_int_is_zero(bset->ineq[j][1 + i])) break; if (j < bset->n_ineq) group[i] = -1; } return isl_basic_map_drop_unrelated_constraints(context, group); } /* Drop constraints from "context" that are irrelevant for computing * the gist of the inequalities "ineq". * Inequalities in "ineq" for which the corresponding element of row * is set to -1 have already been marked for removal and should be ignored. * * In particular, drop constraints in variables that are not related * to any of the variables involved in "ineq" * in the sense that there is no sequence of constraints that connects them. * * We first mark all variables that appear in "bset" as belonging * to a "-1" group and then continue with group_and_drop_irrelevant_constraints. */ static __isl_give isl_basic_set *drop_irrelevant_constraints_marked( __isl_take isl_basic_set *context, __isl_keep isl_mat *ineq, int *row) { int *group; isl_size dim; int i, j; isl_size n; dim = isl_basic_set_dim(context, isl_dim_set); n = isl_mat_rows(ineq); if (dim < 0 || n < 0) return isl_basic_set_free(context); group = alloc_groups(context); if (!group) return isl_basic_set_free(context); for (i = 0; i < dim; ++i) { for (j = 0; j < n; ++j) { if (row[j] < 0) continue; if (!isl_int_is_zero(ineq->row[j][1 + i])) break; } if (j < n) group[i] = -1; } return isl_basic_map_drop_unrelated_constraints(context, group); } /* Do all "n" entries of "row" contain a negative value? */ static int all_neg(int *row, int n) { int i; for (i = 0; i < n; ++i) if (row[i] >= 0) return 0; return 1; } /* Update the inequalities in "bset" based on the information in "row" * and "tab". * * In particular, the array "row" contains either -1, meaning that * the corresponding inequality of "bset" is redundant, or the index * of an inequality in "tab". * * If the row entry is -1, then drop the inequality. * Otherwise, if the constraint is marked redundant in the tableau, * then drop the inequality. Similarly, if it is marked as an equality * in the tableau, then turn the inequality into an equality and * perform Gaussian elimination. */ static __isl_give isl_basic_set *update_ineq(__isl_take isl_basic_set *bset, __isl_keep int *row, struct isl_tab *tab) { int i; unsigned n_ineq; unsigned n_eq; int found_equality = 0; if (!bset) return NULL; if (tab && tab->empty) return isl_basic_set_set_to_empty(bset); n_ineq = bset->n_ineq; for (i = n_ineq - 1; i >= 0; --i) { if (row[i] < 0) { if (isl_basic_set_drop_inequality(bset, i) < 0) return isl_basic_set_free(bset); continue; } if (!tab) continue; n_eq = tab->n_eq; if (isl_tab_is_equality(tab, n_eq + row[i])) { isl_basic_map_inequality_to_equality(bset, i); found_equality = 1; } else if (isl_tab_is_redundant(tab, n_eq + row[i])) { if (isl_basic_set_drop_inequality(bset, i) < 0) return isl_basic_set_free(bset); } } if (found_equality) bset = isl_basic_set_gauss(bset, NULL); bset = isl_basic_set_finalize(bset); return bset; } /* Update the inequalities in "bset" based on the information in "row" * and "tab" and free all arguments (other than "bset"). */ static __isl_give isl_basic_set *update_ineq_free( __isl_take isl_basic_set *bset, __isl_take isl_mat *ineq, __isl_take isl_basic_set *context, __isl_take int *row, struct isl_tab *tab) { isl_mat_free(ineq); isl_basic_set_free(context); bset = update_ineq(bset, row, tab); free(row); isl_tab_free(tab); return bset; } /* Remove all information from bset that is redundant in the context * of context. * "ineq" contains the (possibly transformed) inequalities of "bset", * in the same order. * The (explicit) equalities of "bset" are assumed to have been taken * into account by the transformation such that only the inequalities * are relevant. * "context" is assumed not to be empty. * * "row" keeps track of the constraint index of a "bset" inequality in "tab". * A value of -1 means that the inequality is obviously redundant and may * not even appear in "tab". * * We first mark the inequalities of "bset" * that are obviously redundant with respect to some inequality in "context". * Then we remove those constraints from "context" that have become * irrelevant for computing the gist of "bset". * Note that this removal of constraints cannot be replaced by * a factorization because factors in "bset" may still be connected * to each other through constraints in "context". * * If there are any inequalities left, we construct a tableau for * the context and then add the inequalities of "bset". * Before adding these inequalities, we freeze all constraints such that * they won't be considered redundant in terms of the constraints of "bset". * Then we detect all redundant constraints (among the * constraints that weren't frozen), first by checking for redundancy in the * the tableau and then by checking if replacing a constraint by its negation * would lead to an empty set. This last step is fairly expensive * and could be optimized by more reuse of the tableau. * Finally, we update bset according to the results. */ static __isl_give isl_basic_set *uset_gist_full(__isl_take isl_basic_set *bset, __isl_take isl_mat *ineq, __isl_take isl_basic_set *context) { int i, r; int *row = NULL; isl_ctx *ctx; isl_basic_set *combined = NULL; struct isl_tab *tab = NULL; unsigned n_eq, context_ineq; if (!bset || !ineq || !context) goto error; if (bset->n_ineq == 0 || isl_basic_set_plain_is_universe(context)) { isl_basic_set_free(context); isl_mat_free(ineq); return bset; } ctx = isl_basic_set_get_ctx(context); row = isl_calloc_array(ctx, int, bset->n_ineq); if (!row) goto error; if (mark_shifted_constraints(ineq, context, row) < 0) goto error; if (all_neg(row, bset->n_ineq)) return update_ineq_free(bset, ineq, context, row, NULL); context = drop_irrelevant_constraints_marked(context, ineq, row); if (!context) goto error; if (isl_basic_set_plain_is_universe(context)) return update_ineq_free(bset, ineq, context, row, NULL); n_eq = context->n_eq; context_ineq = context->n_ineq; combined = isl_basic_set_cow(isl_basic_set_copy(context)); combined = isl_basic_set_extend_constraints(combined, 0, bset->n_ineq); tab = isl_tab_from_basic_set(combined, 0); for (i = 0; i < context_ineq; ++i) if (isl_tab_freeze_constraint(tab, n_eq + i) < 0) goto error; if (isl_tab_extend_cons(tab, bset->n_ineq) < 0) goto error; r = context_ineq; for (i = 0; i < bset->n_ineq; ++i) { if (row[i] < 0) continue; combined = isl_basic_set_add_ineq(combined, ineq->row[i]); if (isl_tab_add_ineq(tab, ineq->row[i]) < 0) goto error; row[i] = r++; } if (isl_tab_detect_implicit_equalities(tab) < 0) goto error; if (isl_tab_detect_redundant(tab) < 0) goto error; for (i = bset->n_ineq - 1; i >= 0; --i) { isl_basic_set *test; int is_empty; if (row[i] < 0) continue; r = row[i]; if (tab->con[n_eq + r].is_redundant) continue; test = isl_basic_set_dup(combined); test = isl_inequality_negate(test, r); test = isl_basic_set_update_from_tab(test, tab); is_empty = isl_basic_set_is_empty(test); isl_basic_set_free(test); if (is_empty < 0) goto error; if (is_empty) tab->con[n_eq + r].is_redundant = 1; } bset = update_ineq_free(bset, ineq, context, row, tab); if (bset) { ISL_F_SET(bset, ISL_BASIC_SET_NO_IMPLICIT); ISL_F_SET(bset, ISL_BASIC_SET_NO_REDUNDANT); } isl_basic_set_free(combined); return bset; error: free(row); isl_mat_free(ineq); isl_tab_free(tab); isl_basic_set_free(combined); isl_basic_set_free(context); isl_basic_set_free(bset); return NULL; } /* Extract the inequalities of "bset" as an isl_mat. */ static __isl_give isl_mat *extract_ineq(__isl_keep isl_basic_set *bset) { isl_size total; isl_ctx *ctx; isl_mat *ineq; total = isl_basic_set_dim(bset, isl_dim_all); if (total < 0) return NULL; ctx = isl_basic_set_get_ctx(bset); ineq = isl_mat_sub_alloc6(ctx, bset->ineq, 0, bset->n_ineq, 0, 1 + total); return ineq; } /* Remove all information from "bset" that is redundant in the context * of "context", for the case where both "bset" and "context" are * full-dimensional. */ static __isl_give isl_basic_set *uset_gist_uncompressed( __isl_take isl_basic_set *bset, __isl_take isl_basic_set *context) { isl_mat *ineq; ineq = extract_ineq(bset); return uset_gist_full(bset, ineq, context); } /* Replace "bset" by an empty basic set in the same space. */ static __isl_give isl_basic_set *replace_by_empty( __isl_take isl_basic_set *bset) { isl_space *space; space = isl_basic_set_get_space(bset); isl_basic_set_free(bset); return isl_basic_set_empty(space); } /* Remove all information from "bset" that is redundant in the context * of "context", for the case where the combined equalities of * "bset" and "context" allow for a compression that can be obtained * by preapplication of "T". * If the compression of "context" is empty, meaning that "bset" and * "context" do not intersect, then return the empty set. * * "bset" itself is not transformed by "T". Instead, the inequalities * are extracted from "bset" and those are transformed by "T". * uset_gist_full then determines which of the transformed inequalities * are redundant with respect to the transformed "context" and removes * the corresponding inequalities from "bset". * * After preapplying "T" to the inequalities, any common factor is * removed from the coefficients. If this results in a tightening * of the constant term, then the same tightening is applied to * the corresponding untransformed inequality in "bset". * That is, if after plugging in T, a constraint f(x) >= 0 is of the form * * g f'(x) + r >= 0 * * with 0 <= r < g, then it is equivalent to * * f'(x) >= 0 * * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine * subspace compressed by T since the latter would be transformed to * * g f'(x) >= 0 */ static __isl_give isl_basic_set *uset_gist_compressed( __isl_take isl_basic_set *bset, __isl_take isl_basic_set *context, __isl_take isl_mat *T) { isl_ctx *ctx; isl_mat *ineq; int i; isl_size n_row, n_col; isl_int rem; ineq = extract_ineq(bset); ineq = isl_mat_product(ineq, isl_mat_copy(T)); context = isl_basic_set_preimage(context, T); if (!ineq || !context) goto error; if (isl_basic_set_plain_is_empty(context)) { isl_mat_free(ineq); isl_basic_set_free(context); return replace_by_empty(bset); } ctx = isl_mat_get_ctx(ineq); n_row = isl_mat_rows(ineq); n_col = isl_mat_cols(ineq); if (n_row < 0 || n_col < 0) goto error; isl_int_init(rem); for (i = 0; i < n_row; ++i) { isl_seq_gcd(ineq->row[i] + 1, n_col - 1, &ctx->normalize_gcd); if (isl_int_is_zero(ctx->normalize_gcd)) continue; if (isl_int_is_one(ctx->normalize_gcd)) continue; isl_seq_scale_down(ineq->row[i] + 1, ineq->row[i] + 1, ctx->normalize_gcd, n_col - 1); isl_int_fdiv_r(rem, ineq->row[i][0], ctx->normalize_gcd); isl_int_fdiv_q(ineq->row[i][0], ineq->row[i][0], ctx->normalize_gcd); if (isl_int_is_zero(rem)) continue; bset = isl_basic_set_cow(bset); if (!bset) break; isl_int_sub(bset->ineq[i][0], bset->ineq[i][0], rem); } isl_int_clear(rem); return uset_gist_full(bset, ineq, context); error: isl_mat_free(ineq); isl_basic_set_free(context); isl_basic_set_free(bset); return NULL; } /* Project "bset" onto the variables that are involved in "template". */ static __isl_give isl_basic_set *project_onto_involved( __isl_take isl_basic_set *bset, __isl_keep isl_basic_set *template) { int i; isl_size n; n = isl_basic_set_dim(template, isl_dim_set); if (n < 0 || !template) return isl_basic_set_free(bset); for (i = 0; i < n; ++i) { isl_bool involved; involved = isl_basic_set_involves_dims(template, isl_dim_set, i, 1); if (involved < 0) return isl_basic_set_free(bset); if (involved) continue; bset = isl_basic_set_eliminate_vars(bset, i, 1); } return bset; } /* Remove all information from bset that is redundant in the context * of context. In particular, equalities that are linear combinations * of those in context are removed. Then the inequalities that are * redundant in the context of the equalities and inequalities of * context are removed. * * First of all, we drop those constraints from "context" * that are irrelevant for computing the gist of "bset". * Alternatively, we could factorize the intersection of "context" and "bset". * * We first compute the intersection of the integer affine hulls * of "bset" and "context", * compute the gist inside this intersection and then reduce * the constraints with respect to the equalities of the context * that only involve variables already involved in the input. * If the intersection of the affine hulls turns out to be empty, * then return the empty set. * * If two constraints are mutually redundant, then uset_gist_full * will remove the second of those constraints. We therefore first * sort the constraints so that constraints not involving existentially * quantified variables are given precedence over those that do. * We have to perform this sorting before the variable compression, * because that may effect the order of the variables. */ static __isl_give isl_basic_set *uset_gist(__isl_take isl_basic_set *bset, __isl_take isl_basic_set *context) { isl_mat *eq; isl_mat *T; isl_basic_set *aff; isl_basic_set *aff_context; isl_size total; total = isl_basic_set_dim(bset, isl_dim_all); if (total < 0 || !context) goto error; context = drop_irrelevant_constraints(context, bset); bset = isl_basic_set_detect_equalities(bset); aff = isl_basic_set_copy(bset); aff = isl_basic_set_plain_affine_hull(aff); context = isl_basic_set_detect_equalities(context); aff_context = isl_basic_set_copy(context); aff_context = isl_basic_set_plain_affine_hull(aff_context); aff = isl_basic_set_intersect(aff, aff_context); if (!aff) goto error; if (isl_basic_set_plain_is_empty(aff)) { isl_basic_set_free(bset); isl_basic_set_free(context); return aff; } bset = isl_basic_set_sort_constraints(bset); if (aff->n_eq == 0) { isl_basic_set_free(aff); return uset_gist_uncompressed(bset, context); } eq = isl_mat_sub_alloc6(bset->ctx, aff->eq, 0, aff->n_eq, 0, 1 + total); eq = isl_mat_cow(eq); T = isl_mat_variable_compression(eq, NULL); isl_basic_set_free(aff); if (T && T->n_col == 0) { isl_mat_free(T); isl_basic_set_free(context); return replace_by_empty(bset); } aff_context = isl_basic_set_affine_hull(isl_basic_set_copy(context)); aff_context = project_onto_involved(aff_context, bset); bset = uset_gist_compressed(bset, context, T); bset = isl_basic_set_reduce_using_equalities(bset, aff_context); if (bset) { ISL_F_SET(bset, ISL_BASIC_SET_NO_IMPLICIT); ISL_F_SET(bset, ISL_BASIC_SET_NO_REDUNDANT); } return bset; error: isl_basic_set_free(bset); isl_basic_set_free(context); return NULL; } /* Return the number of equality constraints in "bmap" that involve * local variables. This function assumes that Gaussian elimination * has been applied to the equality constraints. */ static int n_div_eq(__isl_keep isl_basic_map *bmap) { int i; isl_size total, n_div; if (!bmap) return -1; if (bmap->n_eq == 0) return 0; total = isl_basic_map_dim(bmap, isl_dim_all); n_div = isl_basic_map_dim(bmap, isl_dim_div); if (total < 0 || n_div < 0) return -1; total -= n_div; for (i = 0; i < bmap->n_eq; ++i) if (isl_seq_first_non_zero(bmap->eq[i] + 1 + total, n_div) == -1) return i; return bmap->n_eq; } /* Construct a basic map in "space" defined by the equality constraints in "eq". * The constraints are assumed not to involve any local variables. */ static __isl_give isl_basic_map *basic_map_from_equalities( __isl_take isl_space *space, __isl_take isl_mat *eq) { int i, k; isl_size total; isl_basic_map *bmap = NULL; total = isl_space_dim(space, isl_dim_all); if (total < 0 || !eq) goto error; if (1 + total != eq->n_col) isl_die(isl_space_get_ctx(space), isl_error_internal, "unexpected number of columns", goto error); bmap = isl_basic_map_alloc_space(isl_space_copy(space), 0, eq->n_row, 0); for (i = 0; i < eq->n_row; ++i) { k = isl_basic_map_alloc_equality(bmap); if (k < 0) goto error; isl_seq_cpy(bmap->eq[k], eq->row[i], eq->n_col); } isl_space_free(space); isl_mat_free(eq); return bmap; error: isl_space_free(space); isl_mat_free(eq); isl_basic_map_free(bmap); return NULL; } /* Construct and return a variable compression based on the equality * constraints in "bmap1" and "bmap2" that do not involve the local variables. * "n1" is the number of (initial) equality constraints in "bmap1" * that do involve local variables. * "n2" is the number of (initial) equality constraints in "bmap2" * that do involve local variables. * "total" is the total number of other variables. * This function assumes that Gaussian elimination * has been applied to the equality constraints in both "bmap1" and "bmap2" * such that the equality constraints not involving local variables * are those that start at "n1" or "n2". * * If either of "bmap1" and "bmap2" does not have such equality constraints, * then simply compute the compression based on the equality constraints * in the other basic map. * Otherwise, combine the equality constraints from both into a new * basic map such that Gaussian elimination can be applied to this combination * and then construct a variable compression from the resulting * equality constraints. */ static __isl_give isl_mat *combined_variable_compression( __isl_keep isl_basic_map *bmap1, int n1, __isl_keep isl_basic_map *bmap2, int n2, int total) { isl_ctx *ctx; isl_mat *E1, *E2, *V; isl_basic_map *bmap; ctx = isl_basic_map_get_ctx(bmap1); if (bmap1->n_eq == n1) { E2 = isl_mat_sub_alloc6(ctx, bmap2->eq, n2, bmap2->n_eq - n2, 0, 1 + total); return isl_mat_variable_compression(E2, NULL); } if (bmap2->n_eq == n2) { E1 = isl_mat_sub_alloc6(ctx, bmap1->eq, n1, bmap1->n_eq - n1, 0, 1 + total); return isl_mat_variable_compression(E1, NULL); } E1 = isl_mat_sub_alloc6(ctx, bmap1->eq, n1, bmap1->n_eq - n1, 0, 1 + total); E2 = isl_mat_sub_alloc6(ctx, bmap2->eq, n2, bmap2->n_eq - n2, 0, 1 + total); E1 = isl_mat_concat(E1, E2); bmap = basic_map_from_equalities(isl_basic_map_get_space(bmap1), E1); bmap = isl_basic_map_gauss(bmap, NULL); if (!bmap) return NULL; E1 = isl_mat_sub_alloc6(ctx, bmap->eq, 0, bmap->n_eq, 0, 1 + total); V = isl_mat_variable_compression(E1, NULL); isl_basic_map_free(bmap); return V; } /* Extract the stride constraints from "bmap", compressed * with respect to both the stride constraints in "context" and * the remaining equality constraints in both "bmap" and "context". * "bmap_n_eq" is the number of (initial) stride constraints in "bmap". * "context_n_eq" is the number of (initial) stride constraints in "context". * * Let x be all variables in "bmap" (and "context") other than the local * variables. First compute a variable compression * * x = V x' * * based on the non-stride equality constraints in "bmap" and "context". * Consider the stride constraints of "context", * * A(x) + B(y) = 0 * * with y the local variables and plug in the variable compression, * resulting in * * A(V x') + B(y) = 0 * * Use these constraints to compute a parameter compression on x' * * x' = T x'' * * Now consider the stride constraints of "bmap" * * C(x) + D(y) = 0 * * and plug in x = V*T x''. * That is, return A = [C*V*T D]. */ static __isl_give isl_mat *extract_compressed_stride_constraints( __isl_keep isl_basic_map *bmap, int bmap_n_eq, __isl_keep isl_basic_map *context, int context_n_eq) { isl_size total, n_div; isl_ctx *ctx; isl_mat *A, *B, *T, *V; total = isl_basic_map_dim(context, isl_dim_all); n_div = isl_basic_map_dim(context, isl_dim_div); if (total < 0 || n_div < 0) return NULL; total -= n_div; ctx = isl_basic_map_get_ctx(bmap); V = combined_variable_compression(bmap, bmap_n_eq, context, context_n_eq, total); A = isl_mat_sub_alloc6(ctx, context->eq, 0, context_n_eq, 0, 1 + total); B = isl_mat_sub_alloc6(ctx, context->eq, 0, context_n_eq, 1 + total, n_div); A = isl_mat_product(A, isl_mat_copy(V)); T = isl_mat_parameter_compression_ext(A, B); T = isl_mat_product(V, T); n_div = isl_basic_map_dim(bmap, isl_dim_div); if (n_div < 0) T = isl_mat_free(T); else T = isl_mat_diagonal(T, isl_mat_identity(ctx, n_div)); A = isl_mat_sub_alloc6(ctx, bmap->eq, 0, bmap_n_eq, 0, 1 + total + n_div); A = isl_mat_product(A, T); return A; } /* Remove the prime factors from *g that have an exponent that * is strictly smaller than the exponent in "c". * All exponents in *g are known to be smaller than or equal * to those in "c". * * That is, if *g is equal to * * p_1^{e_1} p_2^{e_2} ... p_n^{e_n} * * and "c" is equal to * * p_1^{f_1} p_2^{f_2} ... p_n^{f_n} * * then update *g to * * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ... * p_n^{e_n * (e_n = f_n)} * * If e_i = f_i, then c / *g does not have any p_i factors and therefore * neither does the gcd of *g and c / *g. * If e_i < f_i, then the gcd of *g and c / *g has a positive * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors. * Dividing *g by this gcd therefore strictly reduces the exponent * of the prime factors that need to be removed, while leaving the * other prime factors untouched. * Repeating this process until gcd(*g, c / *g) = 1 therefore * removes all undesired factors, without removing any others. */ static void remove_incomplete_powers(isl_int *g, isl_int c) { isl_int t; isl_int_init(t); for (;;) { isl_int_divexact(t, c, *g); isl_int_gcd(t, t, *g); if (isl_int_is_one(t)) break; isl_int_divexact(*g, *g, t); } isl_int_clear(t); } /* Reduce the "n" stride constraints in "bmap" based on a copy "A" * of the same stride constraints in a compressed space that exploits * all equalities in the context and the other equalities in "bmap". * * If the stride constraints of "bmap" are of the form * * C(x) + D(y) = 0 * * then A is of the form * * B(x') + D(y) = 0 * * If any of these constraints involves only a single local variable y, * then the constraint appears as * * f(x) + m y_i = 0 * * in "bmap" and as * * h(x') + m y_i = 0 * * in "A". * * Let g be the gcd of m and the coefficients of h. * Then, in particular, g is a divisor of the coefficients of h and * * f(x) = h(x') * * is known to be a multiple of g. * If some prime factor in m appears with the same exponent in g, * then it can be removed from m because f(x) is already known * to be a multiple of g and therefore in particular of this power * of the prime factors. * Prime factors that appear with a smaller exponent in g cannot * be removed from m. * Let g' be the divisor of g containing all prime factors that * appear with the same exponent in m and g, then * * f(x) + m y_i = 0 * * can be replaced by * * f(x) + m/g' y_i' = 0 * * Note that (if g' != 1) this changes the explicit representation * of y_i to that of y_i', so the integer division at position i * is marked unknown and later recomputed by a call to * isl_basic_map_gauss. */ static __isl_give isl_basic_map *reduce_stride_constraints( __isl_take isl_basic_map *bmap, int n, __isl_keep isl_mat *A) { int i; isl_size total, n_div; int any = 0; isl_int gcd; total = isl_basic_map_dim(bmap, isl_dim_all); n_div = isl_basic_map_dim(bmap, isl_dim_div); if (total < 0 || n_div < 0 || !A) return isl_basic_map_free(bmap); total -= n_div; isl_int_init(gcd); for (i = 0; i < n; ++i) { int div; div = isl_seq_first_non_zero(bmap->eq[i] + 1 + total, n_div); if (div < 0) isl_die(isl_basic_map_get_ctx(bmap), isl_error_internal, "equality constraints modified unexpectedly", goto error); if (isl_seq_first_non_zero(bmap->eq[i] + 1 + total + div + 1, n_div - div - 1) != -1) continue; if (isl_mat_row_gcd(A, i, &gcd) < 0) goto error; if (isl_int_is_one(gcd)) continue; remove_incomplete_powers(&gcd, bmap->eq[i][1 + total + div]); if (isl_int_is_one(gcd)) continue; isl_int_divexact(bmap->eq[i][1 + total + div], bmap->eq[i][1 + total + div], gcd); bmap = isl_basic_map_mark_div_unknown(bmap, div); if (!bmap) goto error; any = 1; } isl_int_clear(gcd); if (any) bmap = isl_basic_map_gauss(bmap, NULL); return bmap; error: isl_int_clear(gcd); isl_basic_map_free(bmap); return NULL; } /* Simplify the stride constraints in "bmap" based on * the remaining equality constraints in "bmap" and all equality * constraints in "context". * Only do this if both "bmap" and "context" have stride constraints. * * First extract a copy of the stride constraints in "bmap" in a compressed * space exploiting all the other equality constraints and then * use this compressed copy to simplify the original stride constraints. */ static __isl_give isl_basic_map *gist_strides(__isl_take isl_basic_map *bmap, __isl_keep isl_basic_map *context) { int bmap_n_eq, context_n_eq; isl_mat *A; if (!bmap || !context) return isl_basic_map_free(bmap); bmap_n_eq = n_div_eq(bmap); context_n_eq = n_div_eq(context); if (bmap_n_eq < 0 || context_n_eq < 0) return isl_basic_map_free(bmap); if (bmap_n_eq == 0 || context_n_eq == 0) return bmap; A = extract_compressed_stride_constraints(bmap, bmap_n_eq, context, context_n_eq); bmap = reduce_stride_constraints(bmap, bmap_n_eq, A); isl_mat_free(A); return bmap; } /* Return a basic map that has the same intersection with "context" as "bmap" * and that is as "simple" as possible. * * The core computation is performed on the pure constraints. * When we add back the meaning of the integer divisions, we need * to (re)introduce the div constraints. If we happen to have * discovered that some of these integer divisions are equal to * some affine combination of other variables, then these div * constraints may end up getting simplified in terms of the equalities, * resulting in extra inequalities on the other variables that * may have been removed already or that may not even have been * part of the input. We try and remove those constraints of * this form that are most obviously redundant with respect to * the context. We also remove those div constraints that are * redundant with respect to the other constraints in the result. * * The stride constraints among the equality constraints in "bmap" are * also simplified with respecting to the other equality constraints * in "bmap" and with respect to all equality constraints in "context". */ __isl_give isl_basic_map *isl_basic_map_gist(__isl_take isl_basic_map *bmap, __isl_take isl_basic_map *context) { isl_basic_set *bset, *eq; isl_basic_map *eq_bmap; isl_size total, n_div, n_div_bmap; unsigned extra, n_eq, n_ineq; if (!bmap || !context) goto error; if (isl_basic_map_plain_is_universe(bmap)) { isl_basic_map_free(context); return bmap; } if (isl_basic_map_plain_is_empty(context)) { isl_space *space = isl_basic_map_get_space(bmap); isl_basic_map_free(bmap); isl_basic_map_free(context); return isl_basic_map_universe(space); } if (isl_basic_map_plain_is_empty(bmap)) { isl_basic_map_free(context); return bmap; } bmap = isl_basic_map_remove_redundancies(bmap); context = isl_basic_map_remove_redundancies(context); bmap = isl_basic_map_order_divs(bmap); context = isl_basic_map_align_divs(context, bmap); n_div = isl_basic_map_dim(context, isl_dim_div); total = isl_basic_map_dim(bmap, isl_dim_all); n_div_bmap = isl_basic_map_dim(bmap, isl_dim_div); if (n_div < 0 || total < 0 || n_div_bmap < 0) goto error; extra = n_div - n_div_bmap; bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap)); bset = isl_basic_set_add_dims(bset, isl_dim_set, extra); bset = uset_gist(bset, isl_basic_map_underlying_set(isl_basic_map_copy(context))); bset = isl_basic_set_project_out(bset, isl_dim_set, total, extra); if (!bset || bset->n_eq == 0 || n_div == 0 || isl_basic_set_plain_is_empty(bset)) { isl_basic_map_free(context); return isl_basic_map_overlying_set(bset, bmap); } n_eq = bset->n_eq; n_ineq = bset->n_ineq; eq = isl_basic_set_copy(bset); eq = isl_basic_set_cow(eq); eq = isl_basic_set_free_inequality(eq, n_ineq); bset = isl_basic_set_free_equality(bset, n_eq); eq_bmap = isl_basic_map_overlying_set(eq, isl_basic_map_copy(bmap)); eq_bmap = gist_strides(eq_bmap, context); eq_bmap = isl_basic_map_remove_shifted_constraints(eq_bmap, context); bmap = isl_basic_map_overlying_set(bset, bmap); bmap = isl_basic_map_intersect(bmap, eq_bmap); bmap = isl_basic_map_remove_redundancies(bmap); return bmap; error: isl_basic_map_free(bmap); isl_basic_map_free(context); return NULL; } /* * Assumes context has no implicit divs. */ __isl_give isl_map *isl_map_gist_basic_map(__isl_take isl_map *map, __isl_take isl_basic_map *context) { int i; if (!map || !context) goto error; if (isl_basic_map_plain_is_empty(context)) { isl_space *space = isl_map_get_space(map); isl_map_free(map); isl_basic_map_free(context); return isl_map_universe(space); } context = isl_basic_map_remove_redundancies(context); map = isl_map_cow(map); if (isl_map_basic_map_check_equal_space(map, context) < 0) goto error; map = isl_map_compute_divs(map); if (!map) goto error; for (i = map->n - 1; i >= 0; --i) { map->p[i] = isl_basic_map_gist(map->p[i], isl_basic_map_copy(context)); if (!map->p[i]) goto error; if (isl_basic_map_plain_is_empty(map->p[i])) { isl_basic_map_free(map->p[i]); if (i != map->n - 1) map->p[i] = map->p[map->n - 1]; map->n--; } } isl_basic_map_free(context); ISL_F_CLR(map, ISL_MAP_NORMALIZED); return map; error: isl_map_free(map); isl_basic_map_free(context); return NULL; } /* Drop all inequalities from "bmap" that also appear in "context". * "context" is assumed to have only known local variables and * the initial local variables of "bmap" are assumed to be the same * as those of "context". * The constraints of both "bmap" and "context" are assumed * to have been sorted using isl_basic_map_sort_constraints. * * Run through the inequality constraints of "bmap" and "context" * in sorted order. * If a constraint of "bmap" involves variables not in "context", * then it cannot appear in "context". * If a matching constraint is found, it is removed from "bmap". */ static __isl_give isl_basic_map *drop_inequalities( __isl_take isl_basic_map *bmap, __isl_keep isl_basic_map *context) { int i1, i2; isl_size total, bmap_total; unsigned extra; total = isl_basic_map_dim(context, isl_dim_all); bmap_total = isl_basic_map_dim(bmap, isl_dim_all); if (total < 0 || bmap_total < 0) return isl_basic_map_free(bmap); extra = bmap_total - total; i1 = bmap->n_ineq - 1; i2 = context->n_ineq - 1; while (bmap && i1 >= 0 && i2 >= 0) { int cmp; if (isl_seq_first_non_zero(bmap->ineq[i1] + 1 + total, extra) != -1) { --i1; continue; } cmp = isl_basic_map_constraint_cmp(context, bmap->ineq[i1], context->ineq[i2]); if (cmp < 0) { --i2; continue; } if (cmp > 0) { --i1; continue; } if (isl_int_eq(bmap->ineq[i1][0], context->ineq[i2][0])) { bmap = isl_basic_map_cow(bmap); if (isl_basic_map_drop_inequality(bmap, i1) < 0) bmap = isl_basic_map_free(bmap); } --i1; --i2; } return bmap; } /* Drop all equalities from "bmap" that also appear in "context". * "context" is assumed to have only known local variables and * the initial local variables of "bmap" are assumed to be the same * as those of "context". * * Run through the equality constraints of "bmap" and "context" * in sorted order. * If a constraint of "bmap" involves variables not in "context", * then it cannot appear in "context". * If a matching constraint is found, it is removed from "bmap". */ static __isl_give isl_basic_map *drop_equalities( __isl_take isl_basic_map *bmap, __isl_keep isl_basic_map *context) { int i1, i2; isl_size total, bmap_total; unsigned extra; total = isl_basic_map_dim(context, isl_dim_all); bmap_total = isl_basic_map_dim(bmap, isl_dim_all); if (total < 0 || bmap_total < 0) return isl_basic_map_free(bmap); extra = bmap_total - total; i1 = bmap->n_eq - 1; i2 = context->n_eq - 1; while (bmap && i1 >= 0 && i2 >= 0) { int last1, last2; if (isl_seq_first_non_zero(bmap->eq[i1] + 1 + total, extra) != -1) break; last1 = isl_seq_last_non_zero(bmap->eq[i1] + 1, total); last2 = isl_seq_last_non_zero(context->eq[i2] + 1, total); if (last1 > last2) { --i2; continue; } if (last1 < last2) { --i1; continue; } if (isl_seq_eq(bmap->eq[i1], context->eq[i2], 1 + total)) { bmap = isl_basic_map_cow(bmap); if (isl_basic_map_drop_equality(bmap, i1) < 0) bmap = isl_basic_map_free(bmap); } --i1; --i2; } return bmap; } /* Remove the constraints in "context" from "bmap". * "context" is assumed to have explicit representations * for all local variables. * * First align the divs of "bmap" to those of "context" and * sort the constraints. Then drop all constraints from "bmap" * that appear in "context". */ __isl_give isl_basic_map *isl_basic_map_plain_gist( __isl_take isl_basic_map *bmap, __isl_take isl_basic_map *context) { isl_bool done, known; done = isl_basic_map_plain_is_universe(context); if (done == isl_bool_false) done = isl_basic_map_plain_is_universe(bmap); if (done == isl_bool_false) done = isl_basic_map_plain_is_empty(context); if (done == isl_bool_false) done = isl_basic_map_plain_is_empty(bmap); if (done < 0) goto error; if (done) { isl_basic_map_free(context); return bmap; } known = isl_basic_map_divs_known(context); if (known < 0) goto error; if (!known) isl_die(isl_basic_map_get_ctx(bmap), isl_error_invalid, "context has unknown divs", goto error); context = isl_basic_map_order_divs(context); bmap = isl_basic_map_align_divs(bmap, context); bmap = isl_basic_map_gauss(bmap, NULL); bmap = isl_basic_map_sort_constraints(bmap); context = isl_basic_map_sort_constraints(context); bmap = drop_inequalities(bmap, context); bmap = drop_equalities(bmap, context); isl_basic_map_free(context); bmap = isl_basic_map_finalize(bmap); return bmap; error: isl_basic_map_free(bmap); isl_basic_map_free(context); return NULL; } /* Replace "map" by the disjunct at position "pos" and free "context". */ static __isl_give isl_map *replace_by_disjunct(__isl_take isl_map *map, int pos, __isl_take isl_basic_map *context) { isl_basic_map *bmap; bmap = isl_basic_map_copy(map->p[pos]); isl_map_free(map); isl_basic_map_free(context); return isl_map_from_basic_map(bmap); } /* Remove the constraints in "context" from "map". * If any of the disjuncts in the result turns out to be the universe, * then return this universe. * "context" is assumed to have explicit representations * for all local variables. */ __isl_give isl_map *isl_map_plain_gist_basic_map(__isl_take isl_map *map, __isl_take isl_basic_map *context) { int i; isl_bool univ, known; univ = isl_basic_map_plain_is_universe(context); if (univ < 0) goto error; if (univ) { isl_basic_map_free(context); return map; } known = isl_basic_map_divs_known(context); if (known < 0) goto error; if (!known) isl_die(isl_map_get_ctx(map), isl_error_invalid, "context has unknown divs", goto error); map = isl_map_cow(map); if (!map) goto error; for (i = 0; i < map->n; ++i) { map->p[i] = isl_basic_map_plain_gist(map->p[i], isl_basic_map_copy(context)); univ = isl_basic_map_plain_is_universe(map->p[i]); if (univ < 0) goto error; if (univ && map->n > 1) return replace_by_disjunct(map, i, context); } isl_basic_map_free(context); ISL_F_CLR(map, ISL_MAP_NORMALIZED); if (map->n > 1) ISL_F_CLR(map, ISL_MAP_DISJOINT); return map; error: isl_map_free(map); isl_basic_map_free(context); return NULL; } /* Remove the constraints in "context" from "set". * If any of the disjuncts in the result turns out to be the universe, * then return this universe. * "context" is assumed to have explicit representations * for all local variables. */ __isl_give isl_set *isl_set_plain_gist_basic_set(__isl_take isl_set *set, __isl_take isl_basic_set *context) { return set_from_map(isl_map_plain_gist_basic_map(set_to_map(set), bset_to_bmap(context))); } /* Remove the constraints in "context" from "map". * If any of the disjuncts in the result turns out to be the universe, * then return this universe. * "context" is assumed to consist of a single disjunct and * to have explicit representations for all local variables. */ __isl_give isl_map *isl_map_plain_gist(__isl_take isl_map *map, __isl_take isl_map *context) { isl_basic_map *hull; hull = isl_map_unshifted_simple_hull(context); return isl_map_plain_gist_basic_map(map, hull); } /* Replace "map" by a universe map in the same space and free "drop". */ static __isl_give isl_map *replace_by_universe(__isl_take isl_map *map, __isl_take isl_map *drop) { isl_map *res; res = isl_map_universe(isl_map_get_space(map)); isl_map_free(map); isl_map_free(drop); return res; } /* Return a map that has the same intersection with "context" as "map" * and that is as "simple" as possible. * * If "map" is already the universe, then we cannot make it any simpler. * Similarly, if "context" is the universe, then we cannot exploit it * to simplify "map" * If "map" and "context" are identical to each other, then we can * return the corresponding universe. * * If either "map" or "context" consists of multiple disjuncts, * then check if "context" happens to be a subset of "map", * in which case all constraints can be removed. * In case of multiple disjuncts, the standard procedure * may not be able to detect that all constraints can be removed. * * If none of these cases apply, we have to work a bit harder. * During this computation, we make use of a single disjunct context, * so if the original context consists of more than one disjunct * then we need to approximate the context by a single disjunct set. * Simply taking the simple hull may drop constraints that are * only implicitly available in each disjunct. We therefore also * look for constraints among those defining "map" that are valid * for the context. These can then be used to simplify away * the corresponding constraints in "map". */ __isl_give isl_map *isl_map_gist(__isl_take isl_map *map, __isl_take isl_map *context) { int equal; int is_universe; isl_size n_disjunct_map, n_disjunct_context; isl_bool subset; isl_basic_map *hull; is_universe = isl_map_plain_is_universe(map); if (is_universe >= 0 && !is_universe) is_universe = isl_map_plain_is_universe(context); if (is_universe < 0) goto error; if (is_universe) { isl_map_free(context); return map; } isl_map_align_params_bin(&map, &context); equal = isl_map_plain_is_equal(map, context); if (equal < 0) goto error; if (equal) return replace_by_universe(map, context); n_disjunct_map = isl_map_n_basic_map(map); n_disjunct_context = isl_map_n_basic_map(context); if (n_disjunct_map < 0 || n_disjunct_context < 0) goto error; if (n_disjunct_map != 1 || n_disjunct_context != 1) { subset = isl_map_is_subset(context, map); if (subset < 0) goto error; if (subset) return replace_by_universe(map, context); } context = isl_map_compute_divs(context); if (!context) goto error; if (n_disjunct_context == 1) { hull = isl_map_simple_hull(context); } else { isl_ctx *ctx; isl_map_list *list; ctx = isl_map_get_ctx(map); list = isl_map_list_alloc(ctx, 2); list = isl_map_list_add(list, isl_map_copy(context)); list = isl_map_list_add(list, isl_map_copy(map)); hull = isl_map_unshifted_simple_hull_from_map_list(context, list); } return isl_map_gist_basic_map(map, hull); error: isl_map_free(map); isl_map_free(context); return NULL; } __isl_give isl_basic_set *isl_basic_set_gist(__isl_take isl_basic_set *bset, __isl_take isl_basic_set *context) { return bset_from_bmap(isl_basic_map_gist(bset_to_bmap(bset), bset_to_bmap(context))); } __isl_give isl_set *isl_set_gist_basic_set(__isl_take isl_set *set, __isl_take isl_basic_set *context) { return set_from_map(isl_map_gist_basic_map(set_to_map(set), bset_to_bmap(context))); } __isl_give isl_set *isl_set_gist_params_basic_set(__isl_take isl_set *set, __isl_take isl_basic_set *context) { isl_space *space = isl_set_get_space(set); isl_basic_set *dom_context = isl_basic_set_universe(space); dom_context = isl_basic_set_intersect_params(dom_context, context); return isl_set_gist_basic_set(set, dom_context); } __isl_give isl_set *isl_set_gist(__isl_take isl_set *set, __isl_take isl_set *context) { return set_from_map(isl_map_gist(set_to_map(set), set_to_map(context))); } /* Compute the gist of "bmap" with respect to the constraints "context" * on the domain. */ __isl_give isl_basic_map *isl_basic_map_gist_domain( __isl_take isl_basic_map *bmap, __isl_take isl_basic_set *context) { isl_space *space = isl_basic_map_get_space(bmap); isl_basic_map *bmap_context = isl_basic_map_universe(space); bmap_context = isl_basic_map_intersect_domain(bmap_context, context); return isl_basic_map_gist(bmap, bmap_context); } __isl_give isl_map *isl_map_gist_domain(__isl_take isl_map *map, __isl_take isl_set *context) { isl_map *map_context = isl_map_universe(isl_map_get_space(map)); map_context = isl_map_intersect_domain(map_context, context); return isl_map_gist(map, map_context); } __isl_give isl_map *isl_map_gist_range(__isl_take isl_map *map, __isl_take isl_set *context) { isl_map *map_context = isl_map_universe(isl_map_get_space(map)); map_context = isl_map_intersect_range(map_context, context); return isl_map_gist(map, map_context); } __isl_give isl_map *isl_map_gist_params(__isl_take isl_map *map, __isl_take isl_set *context) { isl_map *map_context = isl_map_universe(isl_map_get_space(map)); map_context = isl_map_intersect_params(map_context, context); return isl_map_gist(map, map_context); } __isl_give isl_set *isl_set_gist_params(__isl_take isl_set *set, __isl_take isl_set *context) { return isl_map_gist_params(set, context); } /* Quick check to see if two basic maps are disjoint. * In particular, we reduce the equalities and inequalities of * one basic map in the context of the equalities of the other * basic map and check if we get a contradiction. */ isl_bool isl_basic_map_plain_is_disjoint(__isl_keep isl_basic_map *bmap1, __isl_keep isl_basic_map *bmap2) { struct isl_vec *v = NULL; int *elim = NULL; isl_size total; int i; if (isl_basic_map_check_equal_space(bmap1, bmap2) < 0) return isl_bool_error; if (bmap1->n_div || bmap2->n_div) return isl_bool_false; if (!bmap1->n_eq && !bmap2->n_eq) return isl_bool_false; total = isl_space_dim(bmap1->dim, isl_dim_all); if (total < 0) return isl_bool_error; if (total == 0) return isl_bool_false; v = isl_vec_alloc(bmap1->ctx, 1 + total); if (!v) goto error; elim = isl_alloc_array(bmap1->ctx, int, total); if (!elim) goto error; compute_elimination_index(bmap1, elim, total); for (i = 0; i < bmap2->n_eq; ++i) { int reduced; reduced = reduced_using_equalities(v->block.data, bmap2->eq[i], bmap1, elim, total); if (reduced && !isl_int_is_zero(v->block.data[0]) && isl_seq_first_non_zero(v->block.data + 1, total) == -1) goto disjoint; } for (i = 0; i < bmap2->n_ineq; ++i) { int reduced; reduced = reduced_using_equalities(v->block.data, bmap2->ineq[i], bmap1, elim, total); if (reduced && isl_int_is_neg(v->block.data[0]) && isl_seq_first_non_zero(v->block.data + 1, total) == -1) goto disjoint; } compute_elimination_index(bmap2, elim, total); for (i = 0; i < bmap1->n_ineq; ++i) { int reduced; reduced = reduced_using_equalities(v->block.data, bmap1->ineq[i], bmap2, elim, total); if (reduced && isl_int_is_neg(v->block.data[0]) && isl_seq_first_non_zero(v->block.data + 1, total) == -1) goto disjoint; } isl_vec_free(v); free(elim); return isl_bool_false; disjoint: isl_vec_free(v); free(elim); return isl_bool_true; error: isl_vec_free(v); free(elim); return isl_bool_error; } int isl_basic_set_plain_is_disjoint(__isl_keep isl_basic_set *bset1, __isl_keep isl_basic_set *bset2) { return isl_basic_map_plain_is_disjoint(bset_to_bmap(bset1), bset_to_bmap(bset2)); } /* Does "test" hold for all pairs of basic maps in "map1" and "map2"? */ static isl_bool all_pairs(__isl_keep isl_map *map1, __isl_keep isl_map *map2, isl_bool (*test)(__isl_keep isl_basic_map *bmap1, __isl_keep isl_basic_map *bmap2)) { int i, j; if (!map1 || !map2) return isl_bool_error; for (i = 0; i < map1->n; ++i) { for (j = 0; j < map2->n; ++j) { isl_bool d = test(map1->p[i], map2->p[j]); if (d != isl_bool_true) return d; } } return isl_bool_true; } /* Are "map1" and "map2" obviously disjoint, based on information * that can be derived without looking at the individual basic maps? * * In particular, if one of them is empty or if they live in different spaces * (ignoring parameters), then they are clearly disjoint. */ static isl_bool isl_map_plain_is_disjoint_global(__isl_keep isl_map *map1, __isl_keep isl_map *map2) { isl_bool disjoint; isl_bool match; if (!map1 || !map2) return isl_bool_error; disjoint = isl_map_plain_is_empty(map1); if (disjoint < 0 || disjoint) return disjoint; disjoint = isl_map_plain_is_empty(map2); if (disjoint < 0 || disjoint) return disjoint; match = isl_map_tuple_is_equal(map1, isl_dim_in, map2, isl_dim_in); if (match < 0 || !match) return match < 0 ? isl_bool_error : isl_bool_true; match = isl_map_tuple_is_equal(map1, isl_dim_out, map2, isl_dim_out); if (match < 0 || !match) return match < 0 ? isl_bool_error : isl_bool_true; return isl_bool_false; } /* Are "map1" and "map2" obviously disjoint? * * If one of them is empty or if they live in different spaces (ignoring * parameters), then they are clearly disjoint. * This is checked by isl_map_plain_is_disjoint_global. * * If they have different parameters, then we skip any further tests. * * If they are obviously equal, but not obviously empty, then we will * not be able to detect if they are disjoint. * * Otherwise we check if each basic map in "map1" is obviously disjoint * from each basic map in "map2". */ isl_bool isl_map_plain_is_disjoint(__isl_keep isl_map *map1, __isl_keep isl_map *map2) { isl_bool disjoint; isl_bool intersect; isl_bool match; disjoint = isl_map_plain_is_disjoint_global(map1, map2); if (disjoint < 0 || disjoint) return disjoint; match = isl_map_has_equal_params(map1, map2); if (match < 0 || !match) return match < 0 ? isl_bool_error : isl_bool_false; intersect = isl_map_plain_is_equal(map1, map2); if (intersect < 0 || intersect) return intersect < 0 ? isl_bool_error : isl_bool_false; return all_pairs(map1, map2, &isl_basic_map_plain_is_disjoint); } /* Are "map1" and "map2" disjoint? * The parameters are assumed to have been aligned. * * In particular, check whether all pairs of basic maps are disjoint. */ static isl_bool isl_map_is_disjoint_aligned(__isl_keep isl_map *map1, __isl_keep isl_map *map2) { return all_pairs(map1, map2, &isl_basic_map_is_disjoint); } /* Are "map1" and "map2" disjoint? * * They are disjoint if they are "obviously disjoint" or if one of them * is empty. Otherwise, they are not disjoint if one of them is universal. * If the two inputs are (obviously) equal and not empty, then they are * not disjoint. * If none of these cases apply, then check if all pairs of basic maps * are disjoint after aligning the parameters. */ isl_bool isl_map_is_disjoint(__isl_keep isl_map *map1, __isl_keep isl_map *map2) { isl_bool disjoint; isl_bool intersect; disjoint = isl_map_plain_is_disjoint_global(map1, map2); if (disjoint < 0 || disjoint) return disjoint; disjoint = isl_map_is_empty(map1); if (disjoint < 0 || disjoint) return disjoint; disjoint = isl_map_is_empty(map2); if (disjoint < 0 || disjoint) return disjoint; intersect = isl_map_plain_is_universe(map1); if (intersect < 0 || intersect) return isl_bool_not(intersect); intersect = isl_map_plain_is_universe(map2); if (intersect < 0 || intersect) return isl_bool_not(intersect); intersect = isl_map_plain_is_equal(map1, map2); if (intersect < 0 || intersect) return isl_bool_not(intersect); return isl_map_align_params_map_map_and_test(map1, map2, &isl_map_is_disjoint_aligned); } /* Are "bmap1" and "bmap2" disjoint? * * They are disjoint if they are "obviously disjoint" or if one of them * is empty. Otherwise, they are not disjoint if one of them is universal. * If none of these cases apply, we compute the intersection and see if * the result is empty. */ isl_bool isl_basic_map_is_disjoint(__isl_keep isl_basic_map *bmap1, __isl_keep isl_basic_map *bmap2) { isl_bool disjoint; isl_bool intersect; isl_basic_map *test; disjoint = isl_basic_map_plain_is_disjoint(bmap1, bmap2); if (disjoint < 0 || disjoint) return disjoint; disjoint = isl_basic_map_is_empty(bmap1); if (disjoint < 0 || disjoint) return disjoint; disjoint = isl_basic_map_is_empty(bmap2); if (disjoint < 0 || disjoint) return disjoint; intersect = isl_basic_map_plain_is_universe(bmap1); if (intersect < 0 || intersect) return isl_bool_not(intersect); intersect = isl_basic_map_plain_is_universe(bmap2); if (intersect < 0 || intersect) return isl_bool_not(intersect); test = isl_basic_map_intersect(isl_basic_map_copy(bmap1), isl_basic_map_copy(bmap2)); disjoint = isl_basic_map_is_empty(test); isl_basic_map_free(test); return disjoint; } /* Are "bset1" and "bset2" disjoint? */ isl_bool isl_basic_set_is_disjoint(__isl_keep isl_basic_set *bset1, __isl_keep isl_basic_set *bset2) { return isl_basic_map_is_disjoint(bset1, bset2); } isl_bool isl_set_plain_is_disjoint(__isl_keep isl_set *set1, __isl_keep isl_set *set2) { return isl_map_plain_is_disjoint(set_to_map(set1), set_to_map(set2)); } /* Are "set1" and "set2" disjoint? */ isl_bool isl_set_is_disjoint(__isl_keep isl_set *set1, __isl_keep isl_set *set2) { return isl_map_is_disjoint(set1, set2); } /* Is "v" equal to 0, 1 or -1? */ static int is_zero_or_one(isl_int v) { return isl_int_is_zero(v) || isl_int_is_one(v) || isl_int_is_negone(v); } /* Are the "n" coefficients starting at "first" of inequality constraints * "i" and "j" of "bmap" opposite to each other? */ static int is_opposite_part(__isl_keep isl_basic_map *bmap, int i, int j, int first, int n) { return isl_seq_is_neg(bmap->ineq[i] + first, bmap->ineq[j] + first, n); } /* Are inequality constraints "i" and "j" of "bmap" opposite to each other, * apart from the constant term? */ static isl_bool is_opposite(__isl_keep isl_basic_map *bmap, int i, int j) { isl_size total; total = isl_basic_map_dim(bmap, isl_dim_all); if (total < 0) return isl_bool_error; return is_opposite_part(bmap, i, j, 1, total); } /* Check if we can combine a given div with lower bound l and upper * bound u with some other div and if so return that other div. * Otherwise, return a position beyond the integer divisions. * Return -1 on error. * * We first check that * - the bounds are opposites of each other (except for the constant * term) * - the bounds do not reference any other div * - no div is defined in terms of this div * * Let m be the size of the range allowed on the div by the bounds. * That is, the bounds are of the form * * e <= a <= e + m - 1 * * with e some expression in the other variables. * We look for another div b such that no third div is defined in terms * of this second div b and such that in any constraint that contains * a (except for the given lower and upper bound), also contains b * with a coefficient that is m times that of b. * That is, all constraints (except for the lower and upper bound) * are of the form * * e + f (a + m b) >= 0 * * Furthermore, in the constraints that only contain b, the coefficient * of b should be equal to 1 or -1. * If so, we return b so that "a + m b" can be replaced by * a single div "c = a + m b". */ static int div_find_coalesce(__isl_keep isl_basic_map *bmap, int *pairs, unsigned div, unsigned l, unsigned u) { int i, j; unsigned n_div; isl_size v_div; int coalesce; isl_bool opp; n_div = isl_basic_map_dim(bmap, isl_dim_div); if (n_div <= 1) return n_div; v_div = isl_basic_map_var_offset(bmap, isl_dim_div); if (v_div < 0) return -1; if (isl_seq_first_non_zero(bmap->ineq[l] + 1 + v_div, div) != -1) return n_div; if (isl_seq_first_non_zero(bmap->ineq[l] + 1 + v_div + div + 1, n_div - div - 1) != -1) return n_div; opp = is_opposite(bmap, l, u); if (opp < 0 || !opp) return opp < 0 ? -1 : n_div; for (i = 0; i < n_div; ++i) { if (isl_int_is_zero(bmap->div[i][0])) continue; if (!isl_int_is_zero(bmap->div[i][1 + 1 + v_div + div])) return n_div; } isl_int_add(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]); if (isl_int_is_neg(bmap->ineq[l][0])) { isl_int_sub(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]); bmap = isl_basic_map_copy(bmap); bmap = isl_basic_map_set_to_empty(bmap); isl_basic_map_free(bmap); return n_div; } isl_int_add_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1); coalesce = n_div; for (i = 0; i < n_div; ++i) { if (i == div) continue; if (!pairs[i]) continue; for (j = 0; j < n_div; ++j) { if (isl_int_is_zero(bmap->div[j][0])) continue; if (!isl_int_is_zero(bmap->div[j][1 + 1 + v_div + i])) break; } if (j < n_div) continue; for (j = 0; j < bmap->n_ineq; ++j) { int valid; if (j == l || j == u) continue; if (isl_int_is_zero(bmap->ineq[j][1 + v_div + div])) { if (is_zero_or_one(bmap->ineq[j][1 + v_div + i])) continue; break; } if (isl_int_is_zero(bmap->ineq[j][1 + v_div + i])) break; isl_int_mul(bmap->ineq[j][1 + v_div + div], bmap->ineq[j][1 + v_div + div], bmap->ineq[l][0]); valid = isl_int_eq(bmap->ineq[j][1 + v_div + div], bmap->ineq[j][1 + v_div + i]); isl_int_divexact(bmap->ineq[j][1 + v_div + div], bmap->ineq[j][1 + v_div + div], bmap->ineq[l][0]); if (!valid) break; } if (j < bmap->n_ineq) continue; coalesce = i; break; } isl_int_sub_ui(bmap->ineq[l][0], bmap->ineq[l][0], 1); isl_int_sub(bmap->ineq[l][0], bmap->ineq[l][0], bmap->ineq[u][0]); return coalesce; } /* Internal data structure used during the construction and/or evaluation of * an inequality that ensures that a pair of bounds always allows * for an integer value. * * "tab" is the tableau in which the inequality is evaluated. It may * be NULL until it is actually needed. * "v" contains the inequality coefficients. * "g", "fl" and "fu" are temporary scalars used during the construction and * evaluation. */ struct test_ineq_data { struct isl_tab *tab; isl_vec *v; isl_int g; isl_int fl; isl_int fu; }; /* Free all the memory allocated by the fields of "data". */ static void test_ineq_data_clear(struct test_ineq_data *data) { isl_tab_free(data->tab); isl_vec_free(data->v); isl_int_clear(data->g); isl_int_clear(data->fl); isl_int_clear(data->fu); } /* Is the inequality stored in data->v satisfied by "bmap"? * That is, does it only attain non-negative values? * data->tab is a tableau corresponding to "bmap". */ static isl_bool test_ineq_is_satisfied(__isl_keep isl_basic_map *bmap, struct test_ineq_data *data) { isl_ctx *ctx; enum isl_lp_result res; ctx = isl_basic_map_get_ctx(bmap); if (!data->tab) data->tab = isl_tab_from_basic_map(bmap, 0); res = isl_tab_min(data->tab, data->v->el, ctx->one, &data->g, NULL, 0); if (res == isl_lp_error) return isl_bool_error; return res == isl_lp_ok && isl_int_is_nonneg(data->g); } /* Given a lower and an upper bound on div i, do they always allow * for an integer value of the given div? * Determine this property by constructing an inequality * such that the property is guaranteed when the inequality is nonnegative. * The lower bound is inequality l, while the upper bound is inequality u. * The constructed inequality is stored in data->v. * * Let the upper bound be * * -n_u a + e_u >= 0 * * and the lower bound * * n_l a + e_l >= 0 * * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l). * We have * * - f_u e_l <= f_u f_l g a <= f_l e_u * * Since all variables are integer valued, this is equivalent to * * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1) * * If this interval is at least f_u f_l g, then it contains at least * one integer value for a. * That is, the test constraint is * * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g * * or * * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0 * * If the coefficients of f_l e_u + f_u e_l have a common divisor g', * then the constraint can be scaled down by a factor g', * with the constant term replaced by * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g'). * Note that the result of applying Fourier-Motzkin to this pair * of constraints is * * f_l e_u + f_u e_l >= 0 * * If the constant term of the scaled down version of this constraint, * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant * term of the scaled down test constraint, then the test constraint * is known to hold and no explicit evaluation is required. * This is essentially the Omega test. * * If the test constraint consists of only a constant term, then * it is sufficient to look at the sign of this constant term. */ static isl_bool int_between_bounds(__isl_keep isl_basic_map *bmap, int i, int l, int u, struct test_ineq_data *data) { unsigned offset; isl_size n_div; offset = isl_basic_map_offset(bmap, isl_dim_div); n_div = isl_basic_map_dim(bmap, isl_dim_div); if (n_div < 0) return isl_bool_error; isl_int_gcd(data->g, bmap->ineq[l][offset + i], bmap->ineq[u][offset + i]); isl_int_divexact(data->fl, bmap->ineq[l][offset + i], data->g); isl_int_divexact(data->fu, bmap->ineq[u][offset + i], data->g); isl_int_neg(data->fu, data->fu); isl_seq_combine(data->v->el, data->fl, bmap->ineq[u], data->fu, bmap->ineq[l], offset + n_div); isl_int_mul(data->g, data->g, data->fl); isl_int_mul(data->g, data->g, data->fu); isl_int_sub(data->g, data->g, data->fl); isl_int_sub(data->g, data->g, data->fu); isl_int_add_ui(data->g, data->g, 1); isl_int_sub(data->fl, data->v->el[0], data->g); isl_seq_gcd(data->v->el + 1, offset - 1 + n_div, &data->g); if (isl_int_is_zero(data->g)) return isl_int_is_nonneg(data->fl); if (isl_int_is_one(data->g)) { isl_int_set(data->v->el[0], data->fl); return test_ineq_is_satisfied(bmap, data); } isl_int_fdiv_q(data->fl, data->fl, data->g); isl_int_fdiv_q(data->v->el[0], data->v->el[0], data->g); if (isl_int_eq(data->fl, data->v->el[0])) return isl_bool_true; isl_int_set(data->v->el[0], data->fl); isl_seq_scale_down(data->v->el + 1, data->v->el + 1, data->g, offset - 1 + n_div); return test_ineq_is_satisfied(bmap, data); } /* Remove more kinds of divs that are not strictly needed. * In particular, if all pairs of lower and upper bounds on a div * are such that they allow at least one integer value of the div, * then we can eliminate the div using Fourier-Motzkin without * introducing any spurious solutions. * * If at least one of the two constraints has a unit coefficient for the div, * then the presence of such a value is guaranteed so there is no need to check. * In particular, the value attained by the bound with unit coefficient * can serve as this intermediate value. */ static __isl_give isl_basic_map *drop_more_redundant_divs( __isl_take isl_basic_map *bmap, __isl_take int *pairs, int n) { isl_ctx *ctx; struct test_ineq_data data = { NULL, NULL }; unsigned off; isl_size n_div; int remove = -1; isl_int_init(data.g); isl_int_init(data.fl); isl_int_init(data.fu); n_div = isl_basic_map_dim(bmap, isl_dim_div); if (n_div < 0) goto error; ctx = isl_basic_map_get_ctx(bmap); off = isl_basic_map_offset(bmap, isl_dim_div); data.v = isl_vec_alloc(ctx, off + n_div); if (!data.v) goto error; while (n > 0) { int i, l, u; int best = -1; isl_bool has_int; for (i = 0; i < n_div; ++i) { if (!pairs[i]) continue; if (best >= 0 && pairs[best] <= pairs[i]) continue; best = i; } i = best; for (l = 0; l < bmap->n_ineq; ++l) { if (!isl_int_is_pos(bmap->ineq[l][off + i])) continue; if (isl_int_is_one(bmap->ineq[l][off + i])) continue; for (u = 0; u < bmap->n_ineq; ++u) { if (!isl_int_is_neg(bmap->ineq[u][off + i])) continue; if (isl_int_is_negone(bmap->ineq[u][off + i])) continue; has_int = int_between_bounds(bmap, i, l, u, &data); if (has_int < 0) goto error; if (data.tab && data.tab->empty) break; if (!has_int) break; } if (u < bmap->n_ineq) break; } if (data.tab && data.tab->empty) { bmap = isl_basic_map_set_to_empty(bmap); break; } if (l == bmap->n_ineq) { remove = i; break; } pairs[i] = 0; --n; } test_ineq_data_clear(&data); free(pairs); if (remove < 0) return bmap; bmap = isl_basic_map_remove_dims(bmap, isl_dim_div, remove, 1); return isl_basic_map_drop_redundant_divs(bmap); error: free(pairs); isl_basic_map_free(bmap); test_ineq_data_clear(&data); return NULL; } /* Given a pair of divs div1 and div2 such that, except for the lower bound l * and the upper bound u, div1 always occurs together with div2 in the form * (div1 + m div2), where m is the constant range on the variable div1 * allowed by l and u, replace the pair div1 and div2 by a single * div that is equal to div1 + m div2. * * The new div will appear in the location that contains div2. * We need to modify all constraints that contain * div2 = (div - div1) / m * The coefficient of div2 is known to be equal to 1 or -1. * (If a constraint does not contain div2, it will also not contain div1.) * If the constraint also contains div1, then we know they appear * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div, * i.e., the coefficient of div is f. * * Otherwise, we first need to introduce div1 into the constraint. * Let l be * * div1 + f >=0 * * and u * * -div1 + f' >= 0 * * A lower bound on div2 * * div2 + t >= 0 * * can be replaced by * * m div2 + div1 + m t + f >= 0 * * An upper bound * * -div2 + t >= 0 * * can be replaced by * * -(m div2 + div1) + m t + f' >= 0 * * These constraint are those that we would obtain from eliminating * div1 using Fourier-Motzkin. * * After all constraints have been modified, we drop the lower and upper * bound and then drop div1. * Since the new div is only placed in the same location that used * to store div2, but otherwise has a different meaning, any possible * explicit representation of the original div2 is removed. */ static __isl_give isl_basic_map *coalesce_divs(__isl_take isl_basic_map *bmap, unsigned div1, unsigned div2, unsigned l, unsigned u) { isl_ctx *ctx; isl_int m; isl_size v_div; unsigned total; int i; ctx = isl_basic_map_get_ctx(bmap); v_div = isl_basic_map_var_offset(bmap, isl_dim_div); if (v_div < 0) return isl_basic_map_free(bmap); total = 1 + v_div + bmap->n_div; isl_int_init(m); isl_int_add(m, bmap->ineq[l][0], bmap->ineq[u][0]); isl_int_add_ui(m, m, 1); for (i = 0; i < bmap->n_ineq; ++i) { if (i == l || i == u) continue; if (isl_int_is_zero(bmap->ineq[i][1 + v_div + div2])) continue; if (isl_int_is_zero(bmap->ineq[i][1 + v_div + div1])) { if (isl_int_is_pos(bmap->ineq[i][1 + v_div + div2])) isl_seq_combine(bmap->ineq[i], m, bmap->ineq[i], ctx->one, bmap->ineq[l], total); else isl_seq_combine(bmap->ineq[i], m, bmap->ineq[i], ctx->one, bmap->ineq[u], total); } isl_int_set(bmap->ineq[i][1 + v_div + div2], bmap->ineq[i][1 + v_div + div1]); isl_int_set_si(bmap->ineq[i][1 + v_div + div1], 0); } isl_int_clear(m); if (l > u) { isl_basic_map_drop_inequality(bmap, l); isl_basic_map_drop_inequality(bmap, u); } else { isl_basic_map_drop_inequality(bmap, u); isl_basic_map_drop_inequality(bmap, l); } bmap = isl_basic_map_mark_div_unknown(bmap, div2); bmap = isl_basic_map_drop_div(bmap, div1); return bmap; } /* First check if we can coalesce any pair of divs and * then continue with dropping more redundant divs. * * We loop over all pairs of lower and upper bounds on a div * with coefficient 1 and -1, respectively, check if there * is any other div "c" with which we can coalesce the div * and if so, perform the coalescing. */ static __isl_give isl_basic_map *coalesce_or_drop_more_redundant_divs( __isl_take isl_basic_map *bmap, int *pairs, int n) { int i, l, u; isl_size v_div; isl_size n_div; v_div = isl_basic_map_var_offset(bmap, isl_dim_div); n_div = isl_basic_map_dim(bmap, isl_dim_div); if (v_div < 0 || n_div < 0) return isl_basic_map_free(bmap); for (i = 0; i < n_div; ++i) { if (!pairs[i]) continue; for (l = 0; l < bmap->n_ineq; ++l) { if (!isl_int_is_one(bmap->ineq[l][1 + v_div + i])) continue; for (u = 0; u < bmap->n_ineq; ++u) { int c; if (!isl_int_is_negone(bmap->ineq[u][1+v_div+i])) continue; c = div_find_coalesce(bmap, pairs, i, l, u); if (c < 0) goto error; if (c >= n_div) continue; free(pairs); bmap = coalesce_divs(bmap, i, c, l, u); return isl_basic_map_drop_redundant_divs(bmap); } } } if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY)) { free(pairs); return bmap; } return drop_more_redundant_divs(bmap, pairs, n); error: free(pairs); isl_basic_map_free(bmap); return NULL; } /* Are the "n" coefficients starting at "first" of inequality constraints * "i" and "j" of "bmap" equal to each other? */ static int is_parallel_part(__isl_keep isl_basic_map *bmap, int i, int j, int first, int n) { return isl_seq_eq(bmap->ineq[i] + first, bmap->ineq[j] + first, n); } /* Are inequality constraints "i" and "j" of "bmap" equal to each other, * apart from the constant term and the coefficient at position "pos"? */ static isl_bool is_parallel_except(__isl_keep isl_basic_map *bmap, int i, int j, int pos) { isl_size total; total = isl_basic_map_dim(bmap, isl_dim_all); if (total < 0) return isl_bool_error; return is_parallel_part(bmap, i, j, 1, pos - 1) && is_parallel_part(bmap, i, j, pos + 1, total - pos); } /* Are inequality constraints "i" and "j" of "bmap" opposite to each other, * apart from the constant term and the coefficient at position "pos"? */ static isl_bool is_opposite_except(__isl_keep isl_basic_map *bmap, int i, int j, int pos) { isl_size total; total = isl_basic_map_dim(bmap, isl_dim_all); if (total < 0) return isl_bool_error; return is_opposite_part(bmap, i, j, 1, pos - 1) && is_opposite_part(bmap, i, j, pos + 1, total - pos); } /* Restart isl_basic_map_drop_redundant_divs after "bmap" has * been modified, simplying it if "simplify" is set. * Free the temporary data structure "pairs" that was associated * to the old version of "bmap". */ static __isl_give isl_basic_map *drop_redundant_divs_again( __isl_take isl_basic_map *bmap, __isl_take int *pairs, int simplify) { if (simplify) bmap = isl_basic_map_simplify(bmap); free(pairs); return isl_basic_map_drop_redundant_divs(bmap); } /* Is "div" the single unknown existentially quantified variable * in inequality constraint "ineq" of "bmap"? * "div" is known to have a non-zero coefficient in "ineq". */ static isl_bool single_unknown(__isl_keep isl_basic_map *bmap, int ineq, int div) { int i; isl_size n_div; unsigned o_div; isl_bool known; known = isl_basic_map_div_is_known(bmap, div); if (known < 0 || known) return isl_bool_not(known); n_div = isl_basic_map_dim(bmap, isl_dim_div); if (n_div < 0) return isl_bool_error; if (n_div == 1) return isl_bool_true; o_div = isl_basic_map_offset(bmap, isl_dim_div); for (i = 0; i < n_div; ++i) { isl_bool known; if (i == div) continue; if (isl_int_is_zero(bmap->ineq[ineq][o_div + i])) continue; known = isl_basic_map_div_is_known(bmap, i); if (known < 0 || !known) return known; } return isl_bool_true; } /* Does integer division "div" have coefficient 1 in inequality constraint * "ineq" of "map"? */ static isl_bool has_coef_one(__isl_keep isl_basic_map *bmap, int div, int ineq) { unsigned o_div; o_div = isl_basic_map_offset(bmap, isl_dim_div); if (isl_int_is_one(bmap->ineq[ineq][o_div + div])) return isl_bool_true; return isl_bool_false; } /* Turn inequality constraint "ineq" of "bmap" into an equality and * then try and drop redundant divs again, * freeing the temporary data structure "pairs" that was associated * to the old version of "bmap". */ static __isl_give isl_basic_map *set_eq_and_try_again( __isl_take isl_basic_map *bmap, int ineq, __isl_take int *pairs) { bmap = isl_basic_map_cow(bmap); isl_basic_map_inequality_to_equality(bmap, ineq); return drop_redundant_divs_again(bmap, pairs, 1); } /* Drop the integer division at position "div", along with the two * inequality constraints "ineq1" and "ineq2" in which it appears * from "bmap" and then try and drop redundant divs again, * freeing the temporary data structure "pairs" that was associated * to the old version of "bmap". */ static __isl_give isl_basic_map *drop_div_and_try_again( __isl_take isl_basic_map *bmap, int div, int ineq1, int ineq2, __isl_take int *pairs) { if (ineq1 > ineq2) { isl_basic_map_drop_inequality(bmap, ineq1); isl_basic_map_drop_inequality(bmap, ineq2); } else { isl_basic_map_drop_inequality(bmap, ineq2); isl_basic_map_drop_inequality(bmap, ineq1); } bmap = isl_basic_map_drop_div(bmap, div); return drop_redundant_divs_again(bmap, pairs, 0); } /* Given two inequality constraints * * f(x) + n d + c >= 0, (ineq) * * with d the variable at position "pos", and * * f(x) + c0 >= 0, (lower) * * compute the maximal value of the lower bound ceil((-f(x) - c)/n) * determined by the first constraint. * That is, store * * ceil((c0 - c)/n) * * in *l. */ static void lower_bound_from_parallel(__isl_keep isl_basic_map *bmap, int ineq, int lower, int pos, isl_int *l) { isl_int_neg(*l, bmap->ineq[ineq][0]); isl_int_add(*l, *l, bmap->ineq[lower][0]); isl_int_cdiv_q(*l, *l, bmap->ineq[ineq][pos]); } /* Given two inequality constraints * * f(x) + n d + c >= 0, (ineq) * * with d the variable at position "pos", and * * -f(x) - c0 >= 0, (upper) * * compute the minimal value of the lower bound ceil((-f(x) - c)/n) * determined by the first constraint. * That is, store * * ceil((-c1 - c)/n) * * in *u. */ static void lower_bound_from_opposite(__isl_keep isl_basic_map *bmap, int ineq, int upper, int pos, isl_int *u) { isl_int_neg(*u, bmap->ineq[ineq][0]); isl_int_sub(*u, *u, bmap->ineq[upper][0]); isl_int_cdiv_q(*u, *u, bmap->ineq[ineq][pos]); } /* Given a lower bound constraint "ineq" on "div" in "bmap", * does the corresponding lower bound have a fixed value in "bmap"? * * In particular, "ineq" is of the form * * f(x) + n d + c >= 0 * * with n > 0, c the constant term and * d the existentially quantified variable "div". * That is, the lower bound is * * ceil((-f(x) - c)/n) * * Look for a pair of constraints * * f(x) + c0 >= 0 * -f(x) + c1 >= 0 * * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value. * That is, check that * * ceil((-c1 - c)/n) = ceil((c0 - c)/n) * * If so, return the index of inequality f(x) + c0 >= 0. * Otherwise, return bmap->n_ineq. * Return -1 on error. */ static int lower_bound_is_cst(__isl_keep isl_basic_map *bmap, int div, int ineq) { int i; int lower = -1, upper = -1; unsigned o_div; isl_int l, u; int equal; o_div = isl_basic_map_offset(bmap, isl_dim_div); for (i = 0; i < bmap->n_ineq && (lower < 0 || upper < 0); ++i) { isl_bool par, opp; if (i == ineq) continue; if (!isl_int_is_zero(bmap->ineq[i][o_div + div])) continue; par = isl_bool_false; if (lower < 0) par = is_parallel_except(bmap, ineq, i, o_div + div); if (par < 0) return -1; if (par) { lower = i; continue; } opp = isl_bool_false; if (upper < 0) opp = is_opposite_except(bmap, ineq, i, o_div + div); if (opp < 0) return -1; if (opp) upper = i; } if (lower < 0 || upper < 0) return bmap->n_ineq; isl_int_init(l); isl_int_init(u); lower_bound_from_parallel(bmap, ineq, lower, o_div + div, &l); lower_bound_from_opposite(bmap, ineq, upper, o_div + div, &u); equal = isl_int_eq(l, u); isl_int_clear(l); isl_int_clear(u); return equal ? lower : bmap->n_ineq; } /* Given a lower bound constraint "ineq" on the existentially quantified * variable "div", such that the corresponding lower bound has * a fixed value in "bmap", assign this fixed value to the variable and * then try and drop redundant divs again, * freeing the temporary data structure "pairs" that was associated * to the old version of "bmap". * "lower" determines the constant value for the lower bound. * * In particular, "ineq" is of the form * * f(x) + n d + c >= 0, * * while "lower" is of the form * * f(x) + c0 >= 0 * * The lower bound is ceil((-f(x) - c)/n) and its constant value * is ceil((c0 - c)/n). */ static __isl_give isl_basic_map *fix_cst_lower(__isl_take isl_basic_map *bmap, int div, int ineq, int lower, int *pairs) { isl_int c; unsigned o_div; isl_int_init(c); o_div = isl_basic_map_offset(bmap, isl_dim_div); lower_bound_from_parallel(bmap, ineq, lower, o_div + div, &c); bmap = isl_basic_map_fix(bmap, isl_dim_div, div, c); free(pairs); isl_int_clear(c); return isl_basic_map_drop_redundant_divs(bmap); } /* Do any of the integer divisions of "bmap" involve integer division "div"? * * The integer division "div" could only ever appear in any later * integer division (with an explicit representation). */ static isl_bool any_div_involves_div(__isl_keep isl_basic_map *bmap, int div) { int i; isl_size v_div, n_div; v_div = isl_basic_map_var_offset(bmap, isl_dim_div); n_div = isl_basic_map_dim(bmap, isl_dim_div); if (v_div < 0 || n_div < 0) return isl_bool_error; for (i = div + 1; i < n_div; ++i) { isl_bool unknown; unknown = isl_basic_map_div_is_marked_unknown(bmap, i); if (unknown < 0) return isl_bool_error; if (unknown) continue; if (!isl_int_is_zero(bmap->div[i][1 + 1 + v_div + div])) return isl_bool_true; } return isl_bool_false; } /* Remove divs that are not strictly needed based on the inequality * constraints. * In particular, if a div only occurs positively (or negatively) * in constraints, then it can simply be dropped. * Also, if a div occurs in only two constraints and if moreover * those two constraints are opposite to each other, except for the constant * term and if the sum of the constant terms is such that for any value * of the other values, there is always at least one integer value of the * div, i.e., if one plus this sum is greater than or equal to * the (absolute value) of the coefficient of the div in the constraints, * then we can also simply drop the div. * * If an existentially quantified variable does not have an explicit * representation, appears in only a single lower bound that does not * involve any other such existentially quantified variables and appears * in this lower bound with coefficient 1, * then fix the variable to the value of the lower bound. That is, * turn the inequality into an equality. * If for any value of the other variables, there is any value * for the existentially quantified variable satisfying the constraints, * then this lower bound also satisfies the constraints. * It is therefore safe to pick this lower bound. * * The same reasoning holds even if the coefficient is not one. * However, fixing the variable to the value of the lower bound may * in general introduce an extra integer division, in which case * it may be better to pick another value. * If this integer division has a known constant value, then plugging * in this constant value removes the existentially quantified variable * completely. In particular, if the lower bound is of the form * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n), * then the existentially quantified variable can be assigned this * shared value. * * We skip divs that appear in equalities or in the definition of other divs. * Divs that appear in the definition of other divs usually occur in at least * 4 constraints, but the constraints may have been simplified. * * If any divs are left after these simple checks then we move on * to more complicated cases in drop_more_redundant_divs. */ static __isl_give isl_basic_map *isl_basic_map_drop_redundant_divs_ineq( __isl_take isl_basic_map *bmap) { int i, j; isl_size off; int *pairs = NULL; int n = 0; isl_size n_ineq; if (!bmap) goto error; if (bmap->n_div == 0) return bmap; off = isl_basic_map_var_offset(bmap, isl_dim_div); if (off < 0) return isl_basic_map_free(bmap); pairs = isl_calloc_array(bmap->ctx, int, bmap->n_div); if (!pairs) goto error; n_ineq = isl_basic_map_n_inequality(bmap); if (n_ineq < 0) goto error; for (i = 0; i < bmap->n_div; ++i) { int pos, neg; int last_pos, last_neg; int redundant; int defined; isl_bool involves, opp, set_div; defined = !isl_int_is_zero(bmap->div[i][0]); involves = any_div_involves_div(bmap, i); if (involves < 0) goto error; if (involves) continue; for (j = 0; j < bmap->n_eq; ++j) if (!isl_int_is_zero(bmap->eq[j][1 + off + i])) break; if (j < bmap->n_eq) continue; ++n; pos = neg = 0; for (j = 0; j < bmap->n_ineq; ++j) { if (isl_int_is_pos(bmap->ineq[j][1 + off + i])) { last_pos = j; ++pos; } if (isl_int_is_neg(bmap->ineq[j][1 + off + i])) { last_neg = j; ++neg; } } pairs[i] = pos * neg; if (pairs[i] == 0) { for (j = bmap->n_ineq - 1; j >= 0; --j) if (!isl_int_is_zero(bmap->ineq[j][1+off+i])) isl_basic_map_drop_inequality(bmap, j); bmap = isl_basic_map_drop_div(bmap, i); return drop_redundant_divs_again(bmap, pairs, 0); } if (pairs[i] != 1) opp = isl_bool_false; else opp = is_opposite(bmap, last_pos, last_neg); if (opp < 0) goto error; if (!opp) { int lower; isl_bool single, one; if (pos != 1) continue; single = single_unknown(bmap, last_pos, i); if (single < 0) goto error; if (!single) continue; one = has_coef_one(bmap, i, last_pos); if (one < 0) goto error; if (one) return set_eq_and_try_again(bmap, last_pos, pairs); lower = lower_bound_is_cst(bmap, i, last_pos); if (lower < 0) goto error; if (lower < n_ineq) return fix_cst_lower(bmap, i, last_pos, lower, pairs); continue; } isl_int_add(bmap->ineq[last_pos][0], bmap->ineq[last_pos][0], bmap->ineq[last_neg][0]); isl_int_add_ui(bmap->ineq[last_pos][0], bmap->ineq[last_pos][0], 1); redundant = isl_int_ge(bmap->ineq[last_pos][0], bmap->ineq[last_pos][1+off+i]); isl_int_sub_ui(bmap->ineq[last_pos][0], bmap->ineq[last_pos][0], 1); isl_int_sub(bmap->ineq[last_pos][0], bmap->ineq[last_pos][0], bmap->ineq[last_neg][0]); if (redundant) return drop_div_and_try_again(bmap, i, last_pos, last_neg, pairs); if (defined) set_div = isl_bool_false; else set_div = ok_to_set_div_from_bound(bmap, i, last_pos); if (set_div < 0) return isl_basic_map_free(bmap); if (set_div) { bmap = set_div_from_lower_bound(bmap, i, last_pos); return drop_redundant_divs_again(bmap, pairs, 1); } pairs[i] = 0; --n; } if (n > 0) return coalesce_or_drop_more_redundant_divs(bmap, pairs, n); free(pairs); return bmap; error: free(pairs); isl_basic_map_free(bmap); return NULL; } /* Consider the coefficients at "c" as a row vector and replace * them with their product with "T". "T" is assumed to be a square matrix. */ static isl_stat preimage(isl_int *c, __isl_keep isl_mat *T) { isl_size n; isl_ctx *ctx; isl_vec *v; n = isl_mat_rows(T); if (n < 0) return isl_stat_error; if (isl_seq_first_non_zero(c, n) == -1) return isl_stat_ok; ctx = isl_mat_get_ctx(T); v = isl_vec_alloc(ctx, n); if (!v) return isl_stat_error; isl_seq_swp_or_cpy(v->el, c, n); v = isl_vec_mat_product(v, isl_mat_copy(T)); if (!v) return isl_stat_error; isl_seq_swp_or_cpy(c, v->el, n); isl_vec_free(v); return isl_stat_ok; } /* Plug in T for the variables in "bmap" starting at "pos". * T is a linear unimodular matrix, i.e., without constant term. */ static __isl_give isl_basic_map *isl_basic_map_preimage_vars( __isl_take isl_basic_map *bmap, unsigned pos, __isl_take isl_mat *T) { int i; isl_size n_row, n_col; bmap = isl_basic_map_cow(bmap); n_row = isl_mat_rows(T); n_col = isl_mat_cols(T); if (!bmap || n_row < 0 || n_col < 0) goto error; if (n_col != n_row) isl_die(isl_mat_get_ctx(T), isl_error_invalid, "expecting square matrix", goto error); if (isl_basic_map_check_range(bmap, isl_dim_all, pos, n_col) < 0) goto error; for (i = 0; i < bmap->n_eq; ++i) if (preimage(bmap->eq[i] + 1 + pos, T) < 0) goto error; for (i = 0; i < bmap->n_ineq; ++i) if (preimage(bmap->ineq[i] + 1 + pos, T) < 0) goto error; for (i = 0; i < bmap->n_div; ++i) { if (isl_basic_map_div_is_marked_unknown(bmap, i)) continue; if (preimage(bmap->div[i] + 1 + 1 + pos, T) < 0) goto error; } isl_mat_free(T); return bmap; error: isl_basic_map_free(bmap); isl_mat_free(T); return NULL; } /* Remove divs that are not strictly needed. * * First look for an equality constraint involving two or more * existentially quantified variables without an explicit * representation. Replace the combination that appears * in the equality constraint by a single existentially quantified * variable such that the equality can be used to derive * an explicit representation for the variable. * If there are no more such equality constraints, then continue * with isl_basic_map_drop_redundant_divs_ineq. * * In particular, if the equality constraint is of the form * * f(x) + \sum_i c_i a_i = 0 * * with a_i existentially quantified variable without explicit * representation, then apply a transformation on the existentially * quantified variables to turn the constraint into * * f(x) + g a_1' = 0 * * with g the gcd of the c_i. * In order to easily identify which existentially quantified variables * have a complete explicit representation, i.e., without being defined * in terms of other existentially quantified variables without * an explicit representation, the existentially quantified variables * are first sorted. * * The variable transformation is computed by extending the row * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation * * [a_1'] [c_1/g ... c_n/g] [ a_1 ] * [a_2'] [ a_2 ] * ... = U .... * [a_n'] [ a_n ] * * with [c_1/g ... c_n/g] representing the first row of U. * The inverse of U is then plugged into the original constraints. * The call to isl_basic_map_simplify makes sure the explicit * representation for a_1' is extracted from the equality constraint. */ __isl_give isl_basic_map *isl_basic_map_drop_redundant_divs( __isl_take isl_basic_map *bmap) { int first; int i; unsigned o_div; isl_size n_div; int l; isl_ctx *ctx; isl_mat *T; if (!bmap) return NULL; if (isl_basic_map_divs_known(bmap)) return isl_basic_map_drop_redundant_divs_ineq(bmap); if (bmap->n_eq == 0) return isl_basic_map_drop_redundant_divs_ineq(bmap); bmap = isl_basic_map_sort_divs(bmap); if (!bmap) return NULL; first = isl_basic_map_first_unknown_div(bmap); if (first < 0) return isl_basic_map_free(bmap); o_div = isl_basic_map_offset(bmap, isl_dim_div); n_div = isl_basic_map_dim(bmap, isl_dim_div); if (n_div < 0) return isl_basic_map_free(bmap); for (i = 0; i < bmap->n_eq; ++i) { l = isl_seq_first_non_zero(bmap->eq[i] + o_div + first, n_div - (first)); if (l < 0) continue; l += first; if (isl_seq_first_non_zero(bmap->eq[i] + o_div + l + 1, n_div - (l + 1)) == -1) continue; break; } if (i >= bmap->n_eq) return isl_basic_map_drop_redundant_divs_ineq(bmap); ctx = isl_basic_map_get_ctx(bmap); T = isl_mat_alloc(ctx, n_div - l, n_div - l); if (!T) return isl_basic_map_free(bmap); isl_seq_cpy(T->row[0], bmap->eq[i] + o_div + l, n_div - l); T = isl_mat_normalize_row(T, 0); T = isl_mat_unimodular_complete(T, 1); T = isl_mat_right_inverse(T); for (i = l; i < n_div; ++i) bmap = isl_basic_map_mark_div_unknown(bmap, i); bmap = isl_basic_map_preimage_vars(bmap, o_div - 1 + l, T); bmap = isl_basic_map_simplify(bmap); return isl_basic_map_drop_redundant_divs(bmap); } /* Does "bmap" satisfy any equality that involves more than 2 variables * and/or has coefficients different from -1 and 1? */ static isl_bool has_multiple_var_equality(__isl_keep isl_basic_map *bmap) { int i; isl_size total; total = isl_basic_map_dim(bmap, isl_dim_all); if (total < 0) return isl_bool_error; for (i = 0; i < bmap->n_eq; ++i) { int j, k; j = isl_seq_first_non_zero(bmap->eq[i] + 1, total); if (j < 0) continue; if (!isl_int_is_one(bmap->eq[i][1 + j]) && !isl_int_is_negone(bmap->eq[i][1 + j])) return isl_bool_true; j += 1; k = isl_seq_first_non_zero(bmap->eq[i] + 1 + j, total - j); if (k < 0) continue; j += k; if (!isl_int_is_one(bmap->eq[i][1 + j]) && !isl_int_is_negone(bmap->eq[i][1 + j])) return isl_bool_true; j += 1; k = isl_seq_first_non_zero(bmap->eq[i] + 1 + j, total - j); if (k >= 0) return isl_bool_true; } return isl_bool_false; } /* Remove any common factor g from the constraint coefficients in "v". * The constant term is stored in the first position and is replaced * by floor(c/g). If any common factor is removed and if this results * in a tightening of the constraint, then set *tightened. */ static __isl_give isl_vec *normalize_constraint(__isl_take isl_vec *v, int *tightened) { isl_ctx *ctx; if (!v) return NULL; ctx = isl_vec_get_ctx(v); isl_seq_gcd(v->el + 1, v->size - 1, &ctx->normalize_gcd); if (isl_int_is_zero(ctx->normalize_gcd)) return v; if (isl_int_is_one(ctx->normalize_gcd)) return v; v = isl_vec_cow(v); if (!v) return NULL; if (tightened && !isl_int_is_divisible_by(v->el[0], ctx->normalize_gcd)) *tightened = 1; isl_int_fdiv_q(v->el[0], v->el[0], ctx->normalize_gcd); isl_seq_scale_down(v->el + 1, v->el + 1, ctx->normalize_gcd, v->size - 1); return v; } /* If "bmap" is an integer set that satisfies any equality involving * more than 2 variables and/or has coefficients different from -1 and 1, * then use variable compression to reduce the coefficients by removing * any (hidden) common factor. * In particular, apply the variable compression to each constraint, * factor out any common factor in the non-constant coefficients and * then apply the inverse of the compression. * At the end, we mark the basic map as having reduced constants. * If this flag is still set on the next invocation of this function, * then we skip the computation. * * Removing a common factor may result in a tightening of some of * the constraints. If this happens, then we may end up with two * opposite inequalities that can be replaced by an equality. * We therefore call isl_basic_map_detect_inequality_pairs, * which checks for such pairs of inequalities as well as eliminate_divs_eq * and isl_basic_map_gauss if such a pair was found. * * Tightening may also result in some other constraints becoming * (rationally) redundant with respect to the tightened constraint * (in combination with other constraints). The basic map may * therefore no longer be assumed to have no redundant constraints. * * Note that this function may leave the result in an inconsistent state. * In particular, the constraints may not be gaussed. * Unfortunately, isl_map_coalesce actually depends on this inconsistent state * for some of the test cases to pass successfully. * Any potential modification of the representation is therefore only * performed on a single copy of the basic map. */ __isl_give isl_basic_map *isl_basic_map_reduce_coefficients( __isl_take isl_basic_map *bmap) { isl_size total; isl_bool multi; isl_ctx *ctx; isl_vec *v; isl_mat *eq, *T, *T2; int i; int tightened; if (!bmap) return NULL; if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_REDUCED_COEFFICIENTS)) return bmap; if (isl_basic_map_is_rational(bmap)) return bmap; if (bmap->n_eq == 0) return bmap; multi = has_multiple_var_equality(bmap); if (multi < 0) return isl_basic_map_free(bmap); if (!multi) return bmap; total = isl_basic_map_dim(bmap, isl_dim_all); if (total < 0) return isl_basic_map_free(bmap); ctx = isl_basic_map_get_ctx(bmap); v = isl_vec_alloc(ctx, 1 + total); if (!v) return isl_basic_map_free(bmap); eq = isl_mat_sub_alloc6(ctx, bmap->eq, 0, bmap->n_eq, 0, 1 + total); T = isl_mat_variable_compression(eq, &T2); if (!T || !T2) goto error; if (T->n_col == 0) { isl_mat_free(T); isl_mat_free(T2); isl_vec_free(v); return isl_basic_map_set_to_empty(bmap); } bmap = isl_basic_map_cow(bmap); if (!bmap) goto error; tightened = 0; for (i = 0; i < bmap->n_ineq; ++i) { isl_seq_cpy(v->el, bmap->ineq[i], 1 + total); v = isl_vec_mat_product(v, isl_mat_copy(T)); v = normalize_constraint(v, &tightened); v = isl_vec_mat_product(v, isl_mat_copy(T2)); if (!v) goto error; isl_seq_cpy(bmap->ineq[i], v->el, 1 + total); } isl_mat_free(T); isl_mat_free(T2); isl_vec_free(v); ISL_F_SET(bmap, ISL_BASIC_MAP_REDUCED_COEFFICIENTS); if (tightened) { int progress = 0; ISL_F_CLR(bmap, ISL_BASIC_MAP_NO_REDUNDANT); bmap = isl_basic_map_detect_inequality_pairs(bmap, &progress); if (progress) { bmap = eliminate_divs_eq(bmap, &progress); bmap = isl_basic_map_gauss(bmap, NULL); } } return bmap; error: isl_mat_free(T); isl_mat_free(T2); isl_vec_free(v); return isl_basic_map_free(bmap); } /* Shift the integer division at position "div" of "bmap" * by "shift" times the variable at position "pos". * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0 * corresponds to the constant term. * * That is, if the integer division has the form * * floor(f(x)/d) * * then replace it by * * floor((f(x) + shift * d * x_pos)/d) - shift * x_pos */ __isl_give isl_basic_map *isl_basic_map_shift_div( __isl_take isl_basic_map *bmap, int div, int pos, isl_int shift) { int i; isl_size total, n_div; if (isl_int_is_zero(shift)) return bmap; total = isl_basic_map_dim(bmap, isl_dim_all); n_div = isl_basic_map_dim(bmap, isl_dim_div); total -= n_div; if (total < 0 || n_div < 0) return isl_basic_map_free(bmap); isl_int_addmul(bmap->div[div][1 + pos], shift, bmap->div[div][0]); for (i = 0; i < bmap->n_eq; ++i) { if (isl_int_is_zero(bmap->eq[i][1 + total + div])) continue; isl_int_submul(bmap->eq[i][pos], shift, bmap->eq[i][1 + total + div]); } for (i = 0; i < bmap->n_ineq; ++i) { if (isl_int_is_zero(bmap->ineq[i][1 + total + div])) continue; isl_int_submul(bmap->ineq[i][pos], shift, bmap->ineq[i][1 + total + div]); } for (i = 0; i < bmap->n_div; ++i) { if (isl_int_is_zero(bmap->div[i][0])) continue; if (isl_int_is_zero(bmap->div[i][1 + 1 + total + div])) continue; isl_int_submul(bmap->div[i][1 + pos], shift, bmap->div[i][1 + 1 + total + div]); } return bmap; }