| /* |
| * Copyright 2008-2009 Katholieke Universiteit Leuven |
| * Copyright 2010 INRIA Saclay |
| * Copyright 2012 Ecole Normale Superieure |
| * |
| * Use of this software is governed by the MIT license |
| * |
| * Written by Sven Verdoolaege, K.U.Leuven, Departement |
| * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium |
| * and INRIA Saclay - Ile-de-France, Parc Club Orsay Universite, |
| * ZAC des vignes, 4 rue Jacques Monod, 91893 Orsay, France |
| * and Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris, France |
| */ |
| |
| #include <isl_ctx_private.h> |
| #include <isl_map_private.h> |
| #include <isl_seq.h> |
| #include <isl/set.h> |
| #include <isl/lp.h> |
| #include <isl/map.h> |
| #include "isl_equalities.h" |
| #include "isl_sample.h" |
| #include "isl_tab.h" |
| #include <isl_mat_private.h> |
| #include <isl_vec_private.h> |
| |
| #include <bset_to_bmap.c> |
| #include <bset_from_bmap.c> |
| #include <set_to_map.c> |
| #include <set_from_map.c> |
| |
| __isl_give isl_basic_map *isl_basic_map_implicit_equalities( |
| __isl_take isl_basic_map *bmap) |
| { |
| struct isl_tab *tab; |
| |
| if (!bmap) |
| return bmap; |
| |
| bmap = isl_basic_map_gauss(bmap, NULL); |
| if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY)) |
| return bmap; |
| if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_NO_IMPLICIT)) |
| return bmap; |
| if (bmap->n_ineq <= 1) |
| return bmap; |
| |
| tab = isl_tab_from_basic_map(bmap, 0); |
| if (isl_tab_detect_implicit_equalities(tab) < 0) |
| goto error; |
| bmap = isl_basic_map_update_from_tab(bmap, tab); |
| isl_tab_free(tab); |
| bmap = isl_basic_map_gauss(bmap, NULL); |
| ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT); |
| return bmap; |
| error: |
| isl_tab_free(tab); |
| isl_basic_map_free(bmap); |
| return NULL; |
| } |
| |
| struct isl_basic_set *isl_basic_set_implicit_equalities( |
| struct isl_basic_set *bset) |
| { |
| return bset_from_bmap( |
| isl_basic_map_implicit_equalities(bset_to_bmap(bset))); |
| } |
| |
| /* Make eq[row][col] of both bmaps equal so we can add the row |
| * add the column to the common matrix. |
| * Note that because of the echelon form, the columns of row row |
| * after column col are zero. |
| */ |
| static void set_common_multiple( |
| struct isl_basic_set *bset1, struct isl_basic_set *bset2, |
| unsigned row, unsigned col) |
| { |
| isl_int m, c; |
| |
| if (isl_int_eq(bset1->eq[row][col], bset2->eq[row][col])) |
| return; |
| |
| isl_int_init(c); |
| isl_int_init(m); |
| isl_int_lcm(m, bset1->eq[row][col], bset2->eq[row][col]); |
| isl_int_divexact(c, m, bset1->eq[row][col]); |
| isl_seq_scale(bset1->eq[row], bset1->eq[row], c, col+1); |
| isl_int_divexact(c, m, bset2->eq[row][col]); |
| isl_seq_scale(bset2->eq[row], bset2->eq[row], c, col+1); |
| isl_int_clear(c); |
| isl_int_clear(m); |
| } |
| |
| /* Delete a given equality, moving all the following equalities one up. |
| */ |
| static void delete_row(struct isl_basic_set *bset, unsigned row) |
| { |
| isl_int *t; |
| int r; |
| |
| t = bset->eq[row]; |
| bset->n_eq--; |
| for (r = row; r < bset->n_eq; ++r) |
| bset->eq[r] = bset->eq[r+1]; |
| bset->eq[bset->n_eq] = t; |
| } |
| |
| /* Make first row entries in column col of bset1 identical to |
| * those of bset2, using the fact that entry bset1->eq[row][col]=a |
| * is non-zero. Initially, these elements of bset1 are all zero. |
| * For each row i < row, we set |
| * A[i] = a * A[i] + B[i][col] * A[row] |
| * B[i] = a * B[i] |
| * so that |
| * A[i][col] = B[i][col] = a * old(B[i][col]) |
| */ |
| static void construct_column( |
| struct isl_basic_set *bset1, struct isl_basic_set *bset2, |
| unsigned row, unsigned col) |
| { |
| int r; |
| isl_int a; |
| isl_int b; |
| unsigned total; |
| |
| isl_int_init(a); |
| isl_int_init(b); |
| total = 1 + isl_basic_set_n_dim(bset1); |
| for (r = 0; r < row; ++r) { |
| if (isl_int_is_zero(bset2->eq[r][col])) |
| continue; |
| isl_int_gcd(b, bset2->eq[r][col], bset1->eq[row][col]); |
| isl_int_divexact(a, bset1->eq[row][col], b); |
| isl_int_divexact(b, bset2->eq[r][col], b); |
| isl_seq_combine(bset1->eq[r], a, bset1->eq[r], |
| b, bset1->eq[row], total); |
| isl_seq_scale(bset2->eq[r], bset2->eq[r], a, total); |
| } |
| isl_int_clear(a); |
| isl_int_clear(b); |
| delete_row(bset1, row); |
| } |
| |
| /* Make first row entries in column col of bset1 identical to |
| * those of bset2, using only these entries of the two matrices. |
| * Let t be the last row with different entries. |
| * For each row i < t, we set |
| * A[i] = (A[t][col]-B[t][col]) * A[i] + (B[i][col]-A[i][col) * A[t] |
| * B[i] = (A[t][col]-B[t][col]) * B[i] + (B[i][col]-A[i][col) * B[t] |
| * so that |
| * A[i][col] = B[i][col] = old(A[t][col]*B[i][col]-A[i][col]*B[t][col]) |
| */ |
| static int transform_column( |
| struct isl_basic_set *bset1, struct isl_basic_set *bset2, |
| unsigned row, unsigned col) |
| { |
| int i, t; |
| isl_int a, b, g; |
| unsigned total; |
| |
| for (t = row-1; t >= 0; --t) |
| if (isl_int_ne(bset1->eq[t][col], bset2->eq[t][col])) |
| break; |
| if (t < 0) |
| return 0; |
| |
| total = 1 + isl_basic_set_n_dim(bset1); |
| isl_int_init(a); |
| isl_int_init(b); |
| isl_int_init(g); |
| isl_int_sub(b, bset1->eq[t][col], bset2->eq[t][col]); |
| for (i = 0; i < t; ++i) { |
| isl_int_sub(a, bset2->eq[i][col], bset1->eq[i][col]); |
| isl_int_gcd(g, a, b); |
| isl_int_divexact(a, a, g); |
| isl_int_divexact(g, b, g); |
| isl_seq_combine(bset1->eq[i], g, bset1->eq[i], a, bset1->eq[t], |
| total); |
| isl_seq_combine(bset2->eq[i], g, bset2->eq[i], a, bset2->eq[t], |
| total); |
| } |
| isl_int_clear(a); |
| isl_int_clear(b); |
| isl_int_clear(g); |
| delete_row(bset1, t); |
| delete_row(bset2, t); |
| return 1; |
| } |
| |
| /* The implementation is based on Section 5.2 of Michael Karr, |
| * "Affine Relationships Among Variables of a Program", |
| * except that the echelon form we use starts from the last column |
| * and that we are dealing with integer coefficients. |
| */ |
| static struct isl_basic_set *affine_hull( |
| struct isl_basic_set *bset1, struct isl_basic_set *bset2) |
| { |
| unsigned total; |
| int col; |
| int row; |
| |
| if (!bset1 || !bset2) |
| goto error; |
| |
| total = 1 + isl_basic_set_n_dim(bset1); |
| |
| row = 0; |
| for (col = total-1; col >= 0; --col) { |
| int is_zero1 = row >= bset1->n_eq || |
| isl_int_is_zero(bset1->eq[row][col]); |
| int is_zero2 = row >= bset2->n_eq || |
| isl_int_is_zero(bset2->eq[row][col]); |
| if (!is_zero1 && !is_zero2) { |
| set_common_multiple(bset1, bset2, row, col); |
| ++row; |
| } else if (!is_zero1 && is_zero2) { |
| construct_column(bset1, bset2, row, col); |
| } else if (is_zero1 && !is_zero2) { |
| construct_column(bset2, bset1, row, col); |
| } else { |
| if (transform_column(bset1, bset2, row, col)) |
| --row; |
| } |
| } |
| isl_assert(bset1->ctx, row == bset1->n_eq, goto error); |
| isl_basic_set_free(bset2); |
| bset1 = isl_basic_set_normalize_constraints(bset1); |
| return bset1; |
| error: |
| isl_basic_set_free(bset1); |
| isl_basic_set_free(bset2); |
| return NULL; |
| } |
| |
| /* Find an integer point in the set represented by "tab" |
| * that lies outside of the equality "eq" e(x) = 0. |
| * If "up" is true, look for a point satisfying e(x) - 1 >= 0. |
| * Otherwise, look for a point satisfying -e(x) - 1 >= 0 (i.e., e(x) <= -1). |
| * The point, if found, is returned. |
| * If no point can be found, a zero-length vector is returned. |
| * |
| * Before solving an ILP problem, we first check if simply |
| * adding the normal of the constraint to one of the known |
| * integer points in the basic set represented by "tab" |
| * yields another point inside the basic set. |
| * |
| * The caller of this function ensures that the tableau is bounded or |
| * that tab->basis and tab->n_unbounded have been set appropriately. |
| */ |
| static struct isl_vec *outside_point(struct isl_tab *tab, isl_int *eq, int up) |
| { |
| struct isl_ctx *ctx; |
| struct isl_vec *sample = NULL; |
| struct isl_tab_undo *snap; |
| unsigned dim; |
| |
| if (!tab) |
| return NULL; |
| ctx = tab->mat->ctx; |
| |
| dim = tab->n_var; |
| sample = isl_vec_alloc(ctx, 1 + dim); |
| if (!sample) |
| return NULL; |
| isl_int_set_si(sample->el[0], 1); |
| isl_seq_combine(sample->el + 1, |
| ctx->one, tab->bmap->sample->el + 1, |
| up ? ctx->one : ctx->negone, eq + 1, dim); |
| if (isl_basic_map_contains(tab->bmap, sample)) |
| return sample; |
| isl_vec_free(sample); |
| sample = NULL; |
| |
| snap = isl_tab_snap(tab); |
| |
| if (!up) |
| isl_seq_neg(eq, eq, 1 + dim); |
| isl_int_sub_ui(eq[0], eq[0], 1); |
| |
| if (isl_tab_extend_cons(tab, 1) < 0) |
| goto error; |
| if (isl_tab_add_ineq(tab, eq) < 0) |
| goto error; |
| |
| sample = isl_tab_sample(tab); |
| |
| isl_int_add_ui(eq[0], eq[0], 1); |
| if (!up) |
| isl_seq_neg(eq, eq, 1 + dim); |
| |
| if (sample && isl_tab_rollback(tab, snap) < 0) |
| goto error; |
| |
| return sample; |
| error: |
| isl_vec_free(sample); |
| return NULL; |
| } |
| |
| __isl_give isl_basic_set *isl_basic_set_recession_cone( |
| __isl_take isl_basic_set *bset) |
| { |
| int i; |
| |
| bset = isl_basic_set_cow(bset); |
| if (!bset) |
| return NULL; |
| isl_assert(bset->ctx, bset->n_div == 0, goto error); |
| |
| for (i = 0; i < bset->n_eq; ++i) |
| isl_int_set_si(bset->eq[i][0], 0); |
| |
| for (i = 0; i < bset->n_ineq; ++i) |
| isl_int_set_si(bset->ineq[i][0], 0); |
| |
| ISL_F_CLR(bset, ISL_BASIC_SET_NO_IMPLICIT); |
| return isl_basic_set_implicit_equalities(bset); |
| error: |
| isl_basic_set_free(bset); |
| return NULL; |
| } |
| |
| /* Move "sample" to a point that is one up (or down) from the original |
| * point in dimension "pos". |
| */ |
| static void adjacent_point(__isl_keep isl_vec *sample, int pos, int up) |
| { |
| if (up) |
| isl_int_add_ui(sample->el[1 + pos], sample->el[1 + pos], 1); |
| else |
| isl_int_sub_ui(sample->el[1 + pos], sample->el[1 + pos], 1); |
| } |
| |
| /* Check if any points that are adjacent to "sample" also belong to "bset". |
| * If so, add them to "hull" and return the updated hull. |
| * |
| * Before checking whether and adjacent point belongs to "bset", we first |
| * check whether it already belongs to "hull" as this test is typically |
| * much cheaper. |
| */ |
| static __isl_give isl_basic_set *add_adjacent_points( |
| __isl_take isl_basic_set *hull, __isl_take isl_vec *sample, |
| __isl_keep isl_basic_set *bset) |
| { |
| int i, up; |
| int dim; |
| |
| if (!sample) |
| goto error; |
| |
| dim = isl_basic_set_dim(hull, isl_dim_set); |
| |
| for (i = 0; i < dim; ++i) { |
| for (up = 0; up <= 1; ++up) { |
| int contains; |
| isl_basic_set *point; |
| |
| adjacent_point(sample, i, up); |
| contains = isl_basic_set_contains(hull, sample); |
| if (contains < 0) |
| goto error; |
| if (contains) { |
| adjacent_point(sample, i, !up); |
| continue; |
| } |
| contains = isl_basic_set_contains(bset, sample); |
| if (contains < 0) |
| goto error; |
| if (contains) { |
| point = isl_basic_set_from_vec( |
| isl_vec_copy(sample)); |
| hull = affine_hull(hull, point); |
| } |
| adjacent_point(sample, i, !up); |
| if (contains) |
| break; |
| } |
| } |
| |
| isl_vec_free(sample); |
| |
| return hull; |
| error: |
| isl_vec_free(sample); |
| isl_basic_set_free(hull); |
| return NULL; |
| } |
| |
| /* Extend an initial (under-)approximation of the affine hull of basic |
| * set represented by the tableau "tab" |
| * by looking for points that do not satisfy one of the equalities |
| * in the current approximation and adding them to that approximation |
| * until no such points can be found any more. |
| * |
| * The caller of this function ensures that "tab" is bounded or |
| * that tab->basis and tab->n_unbounded have been set appropriately. |
| * |
| * "bset" may be either NULL or the basic set represented by "tab". |
| * If "bset" is not NULL, we check for any point we find if any |
| * of its adjacent points also belong to "bset". |
| */ |
| static __isl_give isl_basic_set *extend_affine_hull(struct isl_tab *tab, |
| __isl_take isl_basic_set *hull, __isl_keep isl_basic_set *bset) |
| { |
| int i, j; |
| unsigned dim; |
| |
| if (!tab || !hull) |
| goto error; |
| |
| dim = tab->n_var; |
| |
| if (isl_tab_extend_cons(tab, 2 * dim + 1) < 0) |
| goto error; |
| |
| for (i = 0; i < dim; ++i) { |
| struct isl_vec *sample; |
| struct isl_basic_set *point; |
| for (j = 0; j < hull->n_eq; ++j) { |
| sample = outside_point(tab, hull->eq[j], 1); |
| if (!sample) |
| goto error; |
| if (sample->size > 0) |
| break; |
| isl_vec_free(sample); |
| sample = outside_point(tab, hull->eq[j], 0); |
| if (!sample) |
| goto error; |
| if (sample->size > 0) |
| break; |
| isl_vec_free(sample); |
| |
| if (isl_tab_add_eq(tab, hull->eq[j]) < 0) |
| goto error; |
| } |
| if (j == hull->n_eq) |
| break; |
| if (tab->samples && |
| isl_tab_add_sample(tab, isl_vec_copy(sample)) < 0) |
| hull = isl_basic_set_free(hull); |
| if (bset) |
| hull = add_adjacent_points(hull, isl_vec_copy(sample), |
| bset); |
| point = isl_basic_set_from_vec(sample); |
| hull = affine_hull(hull, point); |
| if (!hull) |
| return NULL; |
| } |
| |
| return hull; |
| error: |
| isl_basic_set_free(hull); |
| return NULL; |
| } |
| |
| /* Construct an initial underapproximation of the hull of "bset" |
| * from "sample" and any of its adjacent points that also belong to "bset". |
| */ |
| static __isl_give isl_basic_set *initialize_hull(__isl_keep isl_basic_set *bset, |
| __isl_take isl_vec *sample) |
| { |
| isl_basic_set *hull; |
| |
| hull = isl_basic_set_from_vec(isl_vec_copy(sample)); |
| hull = add_adjacent_points(hull, sample, bset); |
| |
| return hull; |
| } |
| |
| /* Look for all equalities satisfied by the integer points in bset, |
| * which is assumed to be bounded. |
| * |
| * The equalities are obtained by successively looking for |
| * a point that is affinely independent of the points found so far. |
| * In particular, for each equality satisfied by the points so far, |
| * we check if there is any point on a hyperplane parallel to the |
| * corresponding hyperplane shifted by at least one (in either direction). |
| */ |
| static struct isl_basic_set *uset_affine_hull_bounded(struct isl_basic_set *bset) |
| { |
| struct isl_vec *sample = NULL; |
| struct isl_basic_set *hull; |
| struct isl_tab *tab = NULL; |
| unsigned dim; |
| |
| if (isl_basic_set_plain_is_empty(bset)) |
| return bset; |
| |
| dim = isl_basic_set_n_dim(bset); |
| |
| if (bset->sample && bset->sample->size == 1 + dim) { |
| int contains = isl_basic_set_contains(bset, bset->sample); |
| if (contains < 0) |
| goto error; |
| if (contains) { |
| if (dim == 0) |
| return bset; |
| sample = isl_vec_copy(bset->sample); |
| } else { |
| isl_vec_free(bset->sample); |
| bset->sample = NULL; |
| } |
| } |
| |
| tab = isl_tab_from_basic_set(bset, 1); |
| if (!tab) |
| goto error; |
| if (tab->empty) { |
| isl_tab_free(tab); |
| isl_vec_free(sample); |
| return isl_basic_set_set_to_empty(bset); |
| } |
| |
| if (!sample) { |
| struct isl_tab_undo *snap; |
| snap = isl_tab_snap(tab); |
| sample = isl_tab_sample(tab); |
| if (isl_tab_rollback(tab, snap) < 0) |
| goto error; |
| isl_vec_free(tab->bmap->sample); |
| tab->bmap->sample = isl_vec_copy(sample); |
| } |
| |
| if (!sample) |
| goto error; |
| if (sample->size == 0) { |
| isl_tab_free(tab); |
| isl_vec_free(sample); |
| return isl_basic_set_set_to_empty(bset); |
| } |
| |
| hull = initialize_hull(bset, sample); |
| |
| hull = extend_affine_hull(tab, hull, bset); |
| isl_basic_set_free(bset); |
| isl_tab_free(tab); |
| |
| return hull; |
| error: |
| isl_vec_free(sample); |
| isl_tab_free(tab); |
| isl_basic_set_free(bset); |
| return NULL; |
| } |
| |
| /* Given an unbounded tableau and an integer point satisfying the tableau, |
| * construct an initial affine hull containing the recession cone |
| * shifted to the given point. |
| * |
| * The unbounded directions are taken from the last rows of the basis, |
| * which is assumed to have been initialized appropriately. |
| */ |
| static __isl_give isl_basic_set *initial_hull(struct isl_tab *tab, |
| __isl_take isl_vec *vec) |
| { |
| int i; |
| int k; |
| struct isl_basic_set *bset = NULL; |
| struct isl_ctx *ctx; |
| unsigned dim; |
| |
| if (!vec || !tab) |
| return NULL; |
| ctx = vec->ctx; |
| isl_assert(ctx, vec->size != 0, goto error); |
| |
| bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0); |
| if (!bset) |
| goto error; |
| dim = isl_basic_set_n_dim(bset) - tab->n_unbounded; |
| for (i = 0; i < dim; ++i) { |
| k = isl_basic_set_alloc_equality(bset); |
| if (k < 0) |
| goto error; |
| isl_seq_cpy(bset->eq[k] + 1, tab->basis->row[1 + i] + 1, |
| vec->size - 1); |
| isl_seq_inner_product(bset->eq[k] + 1, vec->el +1, |
| vec->size - 1, &bset->eq[k][0]); |
| isl_int_neg(bset->eq[k][0], bset->eq[k][0]); |
| } |
| bset->sample = vec; |
| bset = isl_basic_set_gauss(bset, NULL); |
| |
| return bset; |
| error: |
| isl_basic_set_free(bset); |
| isl_vec_free(vec); |
| return NULL; |
| } |
| |
| /* Given a tableau of a set and a tableau of the corresponding |
| * recession cone, detect and add all equalities to the tableau. |
| * If the tableau is bounded, then we can simply keep the |
| * tableau in its state after the return from extend_affine_hull. |
| * However, if the tableau is unbounded, then |
| * isl_tab_set_initial_basis_with_cone will add some additional |
| * constraints to the tableau that have to be removed again. |
| * In this case, we therefore rollback to the state before |
| * any constraints were added and then add the equalities back in. |
| */ |
| struct isl_tab *isl_tab_detect_equalities(struct isl_tab *tab, |
| struct isl_tab *tab_cone) |
| { |
| int j; |
| struct isl_vec *sample; |
| struct isl_basic_set *hull = NULL; |
| struct isl_tab_undo *snap; |
| |
| if (!tab || !tab_cone) |
| goto error; |
| |
| snap = isl_tab_snap(tab); |
| |
| isl_mat_free(tab->basis); |
| tab->basis = NULL; |
| |
| isl_assert(tab->mat->ctx, tab->bmap, goto error); |
| isl_assert(tab->mat->ctx, tab->samples, goto error); |
| isl_assert(tab->mat->ctx, tab->samples->n_col == 1 + tab->n_var, goto error); |
| isl_assert(tab->mat->ctx, tab->n_sample > tab->n_outside, goto error); |
| |
| if (isl_tab_set_initial_basis_with_cone(tab, tab_cone) < 0) |
| goto error; |
| |
| sample = isl_vec_alloc(tab->mat->ctx, 1 + tab->n_var); |
| if (!sample) |
| goto error; |
| |
| isl_seq_cpy(sample->el, tab->samples->row[tab->n_outside], sample->size); |
| |
| isl_vec_free(tab->bmap->sample); |
| tab->bmap->sample = isl_vec_copy(sample); |
| |
| if (tab->n_unbounded == 0) |
| hull = isl_basic_set_from_vec(isl_vec_copy(sample)); |
| else |
| hull = initial_hull(tab, isl_vec_copy(sample)); |
| |
| for (j = tab->n_outside + 1; j < tab->n_sample; ++j) { |
| isl_seq_cpy(sample->el, tab->samples->row[j], sample->size); |
| hull = affine_hull(hull, |
| isl_basic_set_from_vec(isl_vec_copy(sample))); |
| } |
| |
| isl_vec_free(sample); |
| |
| hull = extend_affine_hull(tab, hull, NULL); |
| if (!hull) |
| goto error; |
| |
| if (tab->n_unbounded == 0) { |
| isl_basic_set_free(hull); |
| return tab; |
| } |
| |
| if (isl_tab_rollback(tab, snap) < 0) |
| goto error; |
| |
| if (hull->n_eq > tab->n_zero) { |
| for (j = 0; j < hull->n_eq; ++j) { |
| isl_seq_normalize(tab->mat->ctx, hull->eq[j], 1 + tab->n_var); |
| if (isl_tab_add_eq(tab, hull->eq[j]) < 0) |
| goto error; |
| } |
| } |
| |
| isl_basic_set_free(hull); |
| |
| return tab; |
| error: |
| isl_basic_set_free(hull); |
| isl_tab_free(tab); |
| return NULL; |
| } |
| |
| /* Compute the affine hull of "bset", where "cone" is the recession cone |
| * of "bset". |
| * |
| * We first compute a unimodular transformation that puts the unbounded |
| * directions in the last dimensions. In particular, we take a transformation |
| * that maps all equalities to equalities (in HNF) on the first dimensions. |
| * Let x be the original dimensions and y the transformed, with y_1 bounded |
| * and y_2 unbounded. |
| * |
| * [ y_1 ] [ y_1 ] [ Q_1 ] |
| * x = U [ y_2 ] [ y_2 ] = [ Q_2 ] x |
| * |
| * Let's call the input basic set S. We compute S' = preimage(S, U) |
| * and drop the final dimensions including any constraints involving them. |
| * This results in set S''. |
| * Then we compute the affine hull A'' of S''. |
| * Let F y_1 >= g be the constraint system of A''. In the transformed |
| * space the y_2 are unbounded, so we can add them back without any constraints, |
| * resulting in |
| * |
| * [ y_1 ] |
| * [ F 0 ] [ y_2 ] >= g |
| * or |
| * [ Q_1 ] |
| * [ F 0 ] [ Q_2 ] x >= g |
| * or |
| * F Q_1 x >= g |
| * |
| * The affine hull in the original space is then obtained as |
| * A = preimage(A'', Q_1). |
| */ |
| static struct isl_basic_set *affine_hull_with_cone(struct isl_basic_set *bset, |
| struct isl_basic_set *cone) |
| { |
| unsigned total; |
| unsigned cone_dim; |
| struct isl_basic_set *hull; |
| struct isl_mat *M, *U, *Q; |
| |
| if (!bset || !cone) |
| goto error; |
| |
| total = isl_basic_set_total_dim(cone); |
| cone_dim = total - cone->n_eq; |
| |
| M = isl_mat_sub_alloc6(bset->ctx, cone->eq, 0, cone->n_eq, 1, total); |
| M = isl_mat_left_hermite(M, 0, &U, &Q); |
| if (!M) |
| goto error; |
| isl_mat_free(M); |
| |
| U = isl_mat_lin_to_aff(U); |
| bset = isl_basic_set_preimage(bset, isl_mat_copy(U)); |
| |
| bset = isl_basic_set_drop_constraints_involving(bset, total - cone_dim, |
| cone_dim); |
| bset = isl_basic_set_drop_dims(bset, total - cone_dim, cone_dim); |
| |
| Q = isl_mat_lin_to_aff(Q); |
| Q = isl_mat_drop_rows(Q, 1 + total - cone_dim, cone_dim); |
| |
| if (bset && bset->sample && bset->sample->size == 1 + total) |
| bset->sample = isl_mat_vec_product(isl_mat_copy(Q), bset->sample); |
| |
| hull = uset_affine_hull_bounded(bset); |
| |
| if (!hull) { |
| isl_mat_free(Q); |
| isl_mat_free(U); |
| } else { |
| struct isl_vec *sample = isl_vec_copy(hull->sample); |
| U = isl_mat_drop_cols(U, 1 + total - cone_dim, cone_dim); |
| if (sample && sample->size > 0) |
| sample = isl_mat_vec_product(U, sample); |
| else |
| isl_mat_free(U); |
| hull = isl_basic_set_preimage(hull, Q); |
| if (hull) { |
| isl_vec_free(hull->sample); |
| hull->sample = sample; |
| } else |
| isl_vec_free(sample); |
| } |
| |
| isl_basic_set_free(cone); |
| |
| return hull; |
| error: |
| isl_basic_set_free(bset); |
| isl_basic_set_free(cone); |
| return NULL; |
| } |
| |
| /* Look for all equalities satisfied by the integer points in bset, |
| * which is assumed not to have any explicit equalities. |
| * |
| * The equalities are obtained by successively looking for |
| * a point that is affinely independent of the points found so far. |
| * In particular, for each equality satisfied by the points so far, |
| * we check if there is any point on a hyperplane parallel to the |
| * corresponding hyperplane shifted by at least one (in either direction). |
| * |
| * Before looking for any outside points, we first compute the recession |
| * cone. The directions of this recession cone will always be part |
| * of the affine hull, so there is no need for looking for any points |
| * in these directions. |
| * In particular, if the recession cone is full-dimensional, then |
| * the affine hull is simply the whole universe. |
| */ |
| static struct isl_basic_set *uset_affine_hull(struct isl_basic_set *bset) |
| { |
| struct isl_basic_set *cone; |
| |
| if (isl_basic_set_plain_is_empty(bset)) |
| return bset; |
| |
| cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset)); |
| if (!cone) |
| goto error; |
| if (cone->n_eq == 0) { |
| isl_space *space; |
| space = isl_basic_set_get_space(bset); |
| isl_basic_set_free(cone); |
| isl_basic_set_free(bset); |
| return isl_basic_set_universe(space); |
| } |
| |
| if (cone->n_eq < isl_basic_set_total_dim(cone)) |
| return affine_hull_with_cone(bset, cone); |
| |
| isl_basic_set_free(cone); |
| return uset_affine_hull_bounded(bset); |
| error: |
| isl_basic_set_free(bset); |
| return NULL; |
| } |
| |
| /* Look for all equalities satisfied by the integer points in bmap |
| * that are independent of the equalities already explicitly available |
| * in bmap. |
| * |
| * We first remove all equalities already explicitly available, |
| * then look for additional equalities in the reduced space |
| * and then transform the result to the original space. |
| * The original equalities are _not_ added to this set. This is |
| * the responsibility of the calling function. |
| * The resulting basic set has all meaning about the dimensions removed. |
| * In particular, dimensions that correspond to existential variables |
| * in bmap and that are found to be fixed are not removed. |
| */ |
| static struct isl_basic_set *equalities_in_underlying_set( |
| struct isl_basic_map *bmap) |
| { |
| struct isl_mat *T1 = NULL; |
| struct isl_mat *T2 = NULL; |
| struct isl_basic_set *bset = NULL; |
| struct isl_basic_set *hull = NULL; |
| |
| bset = isl_basic_map_underlying_set(bmap); |
| if (!bset) |
| return NULL; |
| if (bset->n_eq) |
| bset = isl_basic_set_remove_equalities(bset, &T1, &T2); |
| if (!bset) |
| goto error; |
| |
| hull = uset_affine_hull(bset); |
| if (!T2) |
| return hull; |
| |
| if (!hull) { |
| isl_mat_free(T1); |
| isl_mat_free(T2); |
| } else { |
| struct isl_vec *sample = isl_vec_copy(hull->sample); |
| if (sample && sample->size > 0) |
| sample = isl_mat_vec_product(T1, sample); |
| else |
| isl_mat_free(T1); |
| hull = isl_basic_set_preimage(hull, T2); |
| if (hull) { |
| isl_vec_free(hull->sample); |
| hull->sample = sample; |
| } else |
| isl_vec_free(sample); |
| } |
| |
| return hull; |
| error: |
| isl_mat_free(T1); |
| isl_mat_free(T2); |
| isl_basic_set_free(bset); |
| isl_basic_set_free(hull); |
| return NULL; |
| } |
| |
| /* Detect and make explicit all equalities satisfied by the (integer) |
| * points in bmap. |
| */ |
| __isl_give isl_basic_map *isl_basic_map_detect_equalities( |
| __isl_take isl_basic_map *bmap) |
| { |
| int i, j; |
| struct isl_basic_set *hull = NULL; |
| |
| if (!bmap) |
| return NULL; |
| if (bmap->n_ineq == 0) |
| return bmap; |
| if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_EMPTY)) |
| return bmap; |
| if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_ALL_EQUALITIES)) |
| return bmap; |
| if (ISL_F_ISSET(bmap, ISL_BASIC_MAP_RATIONAL)) |
| return isl_basic_map_implicit_equalities(bmap); |
| |
| hull = equalities_in_underlying_set(isl_basic_map_copy(bmap)); |
| if (!hull) |
| goto error; |
| if (ISL_F_ISSET(hull, ISL_BASIC_SET_EMPTY)) { |
| isl_basic_set_free(hull); |
| return isl_basic_map_set_to_empty(bmap); |
| } |
| bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim), 0, |
| hull->n_eq, 0); |
| for (i = 0; i < hull->n_eq; ++i) { |
| j = isl_basic_map_alloc_equality(bmap); |
| if (j < 0) |
| goto error; |
| isl_seq_cpy(bmap->eq[j], hull->eq[i], |
| 1 + isl_basic_set_total_dim(hull)); |
| } |
| isl_vec_free(bmap->sample); |
| bmap->sample = isl_vec_copy(hull->sample); |
| isl_basic_set_free(hull); |
| ISL_F_SET(bmap, ISL_BASIC_MAP_NO_IMPLICIT | ISL_BASIC_MAP_ALL_EQUALITIES); |
| bmap = isl_basic_map_simplify(bmap); |
| return isl_basic_map_finalize(bmap); |
| error: |
| isl_basic_set_free(hull); |
| isl_basic_map_free(bmap); |
| return NULL; |
| } |
| |
| __isl_give isl_basic_set *isl_basic_set_detect_equalities( |
| __isl_take isl_basic_set *bset) |
| { |
| return bset_from_bmap( |
| isl_basic_map_detect_equalities(bset_to_bmap(bset))); |
| } |
| |
| __isl_give isl_map *isl_map_detect_equalities(__isl_take isl_map *map) |
| { |
| return isl_map_inline_foreach_basic_map(map, |
| &isl_basic_map_detect_equalities); |
| } |
| |
| __isl_give isl_set *isl_set_detect_equalities(__isl_take isl_set *set) |
| { |
| return set_from_map(isl_map_detect_equalities(set_to_map(set))); |
| } |
| |
| /* Return the superset of "bmap" described by the equalities |
| * satisfied by "bmap" that are already known. |
| */ |
| __isl_give isl_basic_map *isl_basic_map_plain_affine_hull( |
| __isl_take isl_basic_map *bmap) |
| { |
| bmap = isl_basic_map_cow(bmap); |
| if (bmap) |
| isl_basic_map_free_inequality(bmap, bmap->n_ineq); |
| bmap = isl_basic_map_finalize(bmap); |
| return bmap; |
| } |
| |
| /* Return the superset of "bset" described by the equalities |
| * satisfied by "bset" that are already known. |
| */ |
| __isl_give isl_basic_set *isl_basic_set_plain_affine_hull( |
| __isl_take isl_basic_set *bset) |
| { |
| return isl_basic_map_plain_affine_hull(bset); |
| } |
| |
| /* After computing the rational affine hull (by detecting the implicit |
| * equalities), we compute the additional equalities satisfied by |
| * the integer points (if any) and add the original equalities back in. |
| */ |
| __isl_give isl_basic_map *isl_basic_map_affine_hull( |
| __isl_take isl_basic_map *bmap) |
| { |
| bmap = isl_basic_map_detect_equalities(bmap); |
| bmap = isl_basic_map_plain_affine_hull(bmap); |
| return bmap; |
| } |
| |
| struct isl_basic_set *isl_basic_set_affine_hull(struct isl_basic_set *bset) |
| { |
| return bset_from_bmap(isl_basic_map_affine_hull(bset_to_bmap(bset))); |
| } |
| |
| /* Given a rational affine matrix "M", add stride constraints to "bmap" |
| * that ensure that |
| * |
| * M(x) |
| * |
| * is an integer vector. The variables x include all the variables |
| * of "bmap" except the unknown divs. |
| * |
| * If d is the common denominator of M, then we need to impose that |
| * |
| * d M(x) = 0 mod d |
| * |
| * or |
| * |
| * exists alpha : d M(x) = d alpha |
| * |
| * This function is similar to add_strides in isl_morph.c |
| */ |
| static __isl_give isl_basic_map *add_strides(__isl_take isl_basic_map *bmap, |
| __isl_keep isl_mat *M, int n_known) |
| { |
| int i, div, k; |
| isl_int gcd; |
| |
| if (isl_int_is_one(M->row[0][0])) |
| return bmap; |
| |
| bmap = isl_basic_map_extend_space(bmap, isl_space_copy(bmap->dim), |
| M->n_row - 1, M->n_row - 1, 0); |
| |
| isl_int_init(gcd); |
| for (i = 1; i < M->n_row; ++i) { |
| isl_seq_gcd(M->row[i], M->n_col, &gcd); |
| if (isl_int_is_divisible_by(gcd, M->row[0][0])) |
| continue; |
| div = isl_basic_map_alloc_div(bmap); |
| if (div < 0) |
| goto error; |
| isl_int_set_si(bmap->div[div][0], 0); |
| k = isl_basic_map_alloc_equality(bmap); |
| if (k < 0) |
| goto error; |
| isl_seq_cpy(bmap->eq[k], M->row[i], M->n_col); |
| isl_seq_clr(bmap->eq[k] + M->n_col, bmap->n_div - n_known); |
| isl_int_set(bmap->eq[k][M->n_col - n_known + div], |
| M->row[0][0]); |
| } |
| isl_int_clear(gcd); |
| |
| return bmap; |
| error: |
| isl_int_clear(gcd); |
| isl_basic_map_free(bmap); |
| return NULL; |
| } |
| |
| /* If there are any equalities that involve (multiple) unknown divs, |
| * then extract the stride information encoded by those equalities |
| * and make it explicitly available in "bmap". |
| * |
| * We first sort the divs so that the unknown divs appear last and |
| * then we count how many equalities involve these divs. |
| * |
| * Let these equalities be of the form |
| * |
| * A(x) + B y = 0 |
| * |
| * where y represents the unknown divs and x the remaining variables. |
| * Let [H 0] be the Hermite Normal Form of B, i.e., |
| * |
| * B = [H 0] Q |
| * |
| * Then x is a solution of the equalities iff |
| * |
| * H^-1 A(x) (= - [I 0] Q y) |
| * |
| * is an integer vector. Let d be the common denominator of H^-1. |
| * We impose |
| * |
| * d H^-1 A(x) = d alpha |
| * |
| * in add_strides, with alpha fresh existentially quantified variables. |
| */ |
| static __isl_give isl_basic_map *isl_basic_map_make_strides_explicit( |
| __isl_take isl_basic_map *bmap) |
| { |
| int known; |
| int n_known; |
| int n, n_col; |
| int total; |
| isl_ctx *ctx; |
| isl_mat *A, *B, *M; |
| |
| known = isl_basic_map_divs_known(bmap); |
| if (known < 0) |
| return isl_basic_map_free(bmap); |
| if (known) |
| return bmap; |
| bmap = isl_basic_map_sort_divs(bmap); |
| bmap = isl_basic_map_gauss(bmap, NULL); |
| if (!bmap) |
| return NULL; |
| |
| for (n_known = 0; n_known < bmap->n_div; ++n_known) |
| if (isl_int_is_zero(bmap->div[n_known][0])) |
| break; |
| ctx = isl_basic_map_get_ctx(bmap); |
| total = isl_space_dim(bmap->dim, isl_dim_all); |
| for (n = 0; n < bmap->n_eq; ++n) |
| if (isl_seq_first_non_zero(bmap->eq[n] + 1 + total + n_known, |
| bmap->n_div - n_known) == -1) |
| break; |
| if (n == 0) |
| return bmap; |
| B = isl_mat_sub_alloc6(ctx, bmap->eq, 0, n, 0, 1 + total + n_known); |
| n_col = bmap->n_div - n_known; |
| A = isl_mat_sub_alloc6(ctx, bmap->eq, 0, n, 1 + total + n_known, n_col); |
| A = isl_mat_left_hermite(A, 0, NULL, NULL); |
| A = isl_mat_drop_cols(A, n, n_col - n); |
| A = isl_mat_lin_to_aff(A); |
| A = isl_mat_right_inverse(A); |
| B = isl_mat_insert_zero_rows(B, 0, 1); |
| B = isl_mat_set_element_si(B, 0, 0, 1); |
| M = isl_mat_product(A, B); |
| if (!M) |
| return isl_basic_map_free(bmap); |
| bmap = add_strides(bmap, M, n_known); |
| bmap = isl_basic_map_gauss(bmap, NULL); |
| isl_mat_free(M); |
| |
| return bmap; |
| } |
| |
| /* Compute the affine hull of each basic map in "map" separately |
| * and make all stride information explicit so that we can remove |
| * all unknown divs without losing this information. |
| * The result is also guaranteed to be gaussed. |
| * |
| * In simple cases where a div is determined by an equality, |
| * calling isl_basic_map_gauss is enough to make the stride information |
| * explicit, as it will derive an explicit representation for the div |
| * from the equality. If, however, the stride information |
| * is encoded through multiple unknown divs then we need to make |
| * some extra effort in isl_basic_map_make_strides_explicit. |
| */ |
| static __isl_give isl_map *isl_map_local_affine_hull(__isl_take isl_map *map) |
| { |
| int i; |
| |
| map = isl_map_cow(map); |
| if (!map) |
| return NULL; |
| |
| for (i = 0; i < map->n; ++i) { |
| map->p[i] = isl_basic_map_affine_hull(map->p[i]); |
| map->p[i] = isl_basic_map_gauss(map->p[i], NULL); |
| map->p[i] = isl_basic_map_make_strides_explicit(map->p[i]); |
| if (!map->p[i]) |
| return isl_map_free(map); |
| } |
| |
| return map; |
| } |
| |
| static __isl_give isl_set *isl_set_local_affine_hull(__isl_take isl_set *set) |
| { |
| return isl_map_local_affine_hull(set); |
| } |
| |
| /* Return an empty basic map living in the same space as "map". |
| */ |
| static __isl_give isl_basic_map *replace_map_by_empty_basic_map( |
| __isl_take isl_map *map) |
| { |
| isl_space *space; |
| |
| space = isl_map_get_space(map); |
| isl_map_free(map); |
| return isl_basic_map_empty(space); |
| } |
| |
| /* Compute the affine hull of "map". |
| * |
| * We first compute the affine hull of each basic map separately. |
| * Then we align the divs and recompute the affine hulls of the basic |
| * maps since some of them may now have extra divs. |
| * In order to avoid performing parametric integer programming to |
| * compute explicit expressions for the divs, possible leading to |
| * an explosion in the number of basic maps, we first drop all unknown |
| * divs before aligning the divs. Note that isl_map_local_affine_hull tries |
| * to make sure that all stride information is explicitly available |
| * in terms of known divs. This involves calling isl_basic_set_gauss, |
| * which is also needed because affine_hull assumes its input has been gaussed, |
| * while isl_map_affine_hull may be called on input that has not been gaussed, |
| * in particular from initial_facet_constraint. |
| * Similarly, align_divs may reorder some divs so that we need to |
| * gauss the result again. |
| * Finally, we combine the individual affine hulls into a single |
| * affine hull. |
| */ |
| __isl_give isl_basic_map *isl_map_affine_hull(__isl_take isl_map *map) |
| { |
| struct isl_basic_map *model = NULL; |
| struct isl_basic_map *hull = NULL; |
| struct isl_set *set; |
| isl_basic_set *bset; |
| |
| map = isl_map_detect_equalities(map); |
| map = isl_map_local_affine_hull(map); |
| map = isl_map_remove_empty_parts(map); |
| map = isl_map_remove_unknown_divs(map); |
| map = isl_map_align_divs_internal(map); |
| |
| if (!map) |
| return NULL; |
| |
| if (map->n == 0) |
| return replace_map_by_empty_basic_map(map); |
| |
| model = isl_basic_map_copy(map->p[0]); |
| set = isl_map_underlying_set(map); |
| set = isl_set_cow(set); |
| set = isl_set_local_affine_hull(set); |
| if (!set) |
| goto error; |
| |
| while (set->n > 1) |
| set->p[0] = affine_hull(set->p[0], set->p[--set->n]); |
| |
| bset = isl_basic_set_copy(set->p[0]); |
| hull = isl_basic_map_overlying_set(bset, model); |
| isl_set_free(set); |
| hull = isl_basic_map_simplify(hull); |
| return isl_basic_map_finalize(hull); |
| error: |
| isl_basic_map_free(model); |
| isl_set_free(set); |
| return NULL; |
| } |
| |
| struct isl_basic_set *isl_set_affine_hull(struct isl_set *set) |
| { |
| return bset_from_bmap(isl_map_affine_hull(set_to_map(set))); |
| } |