#include "isl_map.h"
#include "isl_map_private.h"
+#include "isl_seq.h"
/*
* The transitive closure implementation is based on the paper
* Albert Cohen.
*/
-/* Given a union of translations (uniform dependences), return a matrix
- * with as many rows as there are disjuncts in the union and as many
- * columns as there are input dimensions (which should be equal to
- * the number of output dimensions).
- * Each row contains the translation performed by the corresponding disjunct.
- * If "map" turns out not to be a union of translations, then the contents
- * of the returned matrix are undefined and *ok is set to 0.
- */
-static __isl_give isl_mat *extract_steps(__isl_keep isl_map *map, int *ok)
-{
- int i, j;
- struct isl_mat *steps;
- unsigned dim = isl_map_dim(map, isl_dim_in);
-
- *ok = 1;
-
- steps = isl_mat_alloc(map->ctx, map->n, dim);
- if (!steps)
- return NULL;
-
- for (i = 0; i < map->n; ++i) {
- struct isl_basic_set *delta;
-
- delta = isl_basic_map_deltas(isl_basic_map_copy(map->p[i]));
-
- for (j = 0; j < dim; ++j) {
- int fixed;
-
- fixed = isl_basic_set_fast_dim_is_fixed(delta, j,
- &steps->row[i][j]);
- if (fixed < 0) {
- isl_basic_set_free(delta);
- goto error;
- }
- if (!fixed)
- break;
- }
-
- isl_basic_set_free(delta);
-
- if (j < dim)
- break;
- }
-
- if (i < map->n)
- *ok = 0;
-
- return steps;
-error:
- isl_mat_free(steps);
- return NULL;
-}
-
/* Given a set of n offsets v_i (the rows of "steps"), construct a relation
- * of the given dimension specification that maps a element x to any
- * element that can be reached by taking a positive number of steps
- * along any of the offsets, where the number of steps k is equal to
- * parameter "param". That is, construct
+ * of the given dimension specification (Z^{n+1} -> Z^{n+1})
+ * that maps an element x to any element that can be reached
+ * by taking a non-negative number of steps along any of
+ * the extended offsets v'_i = [v_i 1].
+ * That is, construct
*
- * { [x] -> [y] : exists k_i >= 0, y = x + \sum_i k_i v_i, k = \sum_i k_i > 0 }
+ * { [x] -> [y] : exists k_i >= 0, y = x + \sum_i k_i v'_i }
*
+ * For any element in this relation, the number of steps taken
+ * is equal to the difference in the final coordinates.
*/
static __isl_give isl_map *path_along_steps(__isl_take isl_dim *dim,
- __isl_keep isl_mat *steps, unsigned param)
+ __isl_keep isl_mat *steps)
{
int i, j, k;
struct isl_basic_map *path = NULL;
n = steps->n_row;
nparam = isl_dim_size(dim, isl_dim_param);
- path = isl_basic_map_alloc_dim(isl_dim_copy(dim), n, d + 1, n + 1);
+ path = isl_basic_map_alloc_dim(isl_dim_copy(dim), n, d, n);
for (i = 0; i < n; ++i) {
k = isl_basic_map_alloc_div(path);
isl_seq_clr(path->eq[k], 1 + isl_basic_map_total_dim(path));
isl_int_set_si(path->eq[k][1 + nparam + i], 1);
isl_int_set_si(path->eq[k][1 + nparam + d + i], -1);
- for (j = 0; j < n; ++j)
- isl_int_set(path->eq[k][1 + nparam + 2 * d + j],
- steps->row[j][i]);
+ if (i == d - 1)
+ for (j = 0; j < n; ++j)
+ isl_int_set_si(path->eq[k][1 + nparam + 2 * d + j], 1);
+ else
+ for (j = 0; j < n; ++j)
+ isl_int_set(path->eq[k][1 + nparam + 2 * d + j],
+ steps->row[j][i]);
}
- k = isl_basic_map_alloc_equality(path);
- if (k < 0)
- goto error;
- isl_seq_clr(path->eq[k], 1 + isl_basic_map_total_dim(path));
- isl_int_set_si(path->eq[k][1 + param], -1);
- for (j = 0; j < n; ++j)
- isl_int_set_si(path->eq[k][1 + nparam + 2 * d + j], 1);
-
for (i = 0; i < n; ++i) {
k = isl_basic_map_alloc_inequality(path);
if (k < 0)
isl_int_set_si(path->ineq[k][1 + nparam + 2 * d + i], 1);
}
+ isl_dim_free(dim);
+
+ path = isl_basic_map_simplify(path);
+ path = isl_basic_map_finalize(path);
+ return isl_map_from_basic_map(path);
+error:
+ isl_dim_free(dim);
+ isl_basic_map_free(path);
+ return NULL;
+}
+
+#define IMPURE 0
+#define PURE_PARAM 1
+#define PURE_VAR 2
+
+/* Return PURE_PARAM if only the coefficients of the parameters are non-zero.
+ * Return PURE_VAR if only the coefficients of the set variables are non-zero.
+ * Return IMPURE otherwise.
+ */
+static int purity(__isl_keep isl_basic_set *bset, isl_int *c)
+{
+ unsigned d;
+ unsigned n_div;
+ unsigned nparam;
+
+ n_div = isl_basic_set_dim(bset, isl_dim_div);
+ d = isl_basic_set_dim(bset, isl_dim_set);
+ nparam = isl_basic_set_dim(bset, isl_dim_param);
+
+ if (isl_seq_first_non_zero(c + 1 + nparam + d, n_div) != -1)
+ return IMPURE;
+ if (isl_seq_first_non_zero(c + 1, nparam) == -1)
+ return PURE_VAR;
+ if (isl_seq_first_non_zero(c + 1 + nparam, d) == -1)
+ return PURE_PARAM;
+ return IMPURE;
+}
+
+/* Given a set of offsets "delta", construct a relation of the
+ * given dimension specification (Z^{n+1} -> Z^{n+1}) that
+ * is an overapproximation of the relations that
+ * maps an element x to any element that can be reached
+ * by taking a non-negative number of steps along any of
+ * the elements in "delta".
+ * That is, construct an approximation of
+ *
+ * { [x] -> [y] : exists f \in \delta, k \in Z :
+ * y = x + k [f, 1] and k >= 0 }
+ *
+ * For any element in this relation, the number of steps taken
+ * is equal to the difference in the final coordinates.
+ *
+ * In particular, let delta be defined as
+ *
+ * \delta = [p] -> { [x] : A x + a >= and B p + b >= 0 and
+ * C x + C'p + c >= 0 }
+ *
+ * then the relation is constructed as
+ *
+ * { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
+ * A f + k a >= 0 and B p + b >= 0 and k >= 1 }
+ * union { [x] -> [x] }
+ *
+ * Existentially quantified variables in \delta are currently ignored.
+ * This is safe, but leads to an additional overapproximation.
+ */
+static __isl_give isl_map *path_along_delta(__isl_take isl_dim *dim,
+ __isl_take isl_basic_set *delta)
+{
+ isl_basic_map *path = NULL;
+ unsigned d;
+ unsigned n_div;
+ unsigned nparam;
+ unsigned off;
+ int i, k;
+
+ if (!delta)
+ goto error;
+ n_div = isl_basic_set_dim(delta, isl_dim_div);
+ d = isl_basic_set_dim(delta, isl_dim_set);
+ nparam = isl_basic_set_dim(delta, isl_dim_param);
+ path = isl_basic_map_alloc_dim(isl_dim_copy(dim), n_div + d + 1,
+ d + 1 + delta->n_eq, delta->n_ineq + 1);
+ off = 1 + nparam + 2 * (d + 1) + n_div;
+
+ for (i = 0; i < n_div + d + 1; ++i) {
+ k = isl_basic_map_alloc_div(path);
+ if (k < 0)
+ goto error;
+ isl_int_set_si(path->div[k][0], 0);
+ }
+
+ for (i = 0; i < d + 1; ++i) {
+ k = isl_basic_map_alloc_equality(path);
+ if (k < 0)
+ goto error;
+ isl_seq_clr(path->eq[k], 1 + isl_basic_map_total_dim(path));
+ isl_int_set_si(path->eq[k][1 + nparam + i], 1);
+ isl_int_set_si(path->eq[k][1 + nparam + d + 1 + i], -1);
+ isl_int_set_si(path->eq[k][off + i], 1);
+ }
+
+ for (i = 0; i < delta->n_eq; ++i) {
+ int p = purity(delta, delta->eq[i]);
+ if (p == IMPURE)
+ continue;
+ k = isl_basic_map_alloc_equality(path);
+ if (k < 0)
+ goto error;
+ isl_seq_clr(path->eq[k], 1 + isl_basic_map_total_dim(path));
+ if (p == PURE_VAR) {
+ isl_seq_cpy(path->eq[k] + off,
+ delta->eq[i] + 1 + nparam, d);
+ isl_int_set(path->eq[k][off + d], delta->eq[i][0]);
+ } else
+ isl_seq_cpy(path->eq[k], delta->eq[i], 1 + nparam);
+ }
+
+ for (i = 0; i < delta->n_ineq; ++i) {
+ int p = purity(delta, delta->ineq[i]);
+ if (p == IMPURE)
+ continue;
+ k = isl_basic_map_alloc_inequality(path);
+ if (k < 0)
+ goto error;
+ isl_seq_clr(path->ineq[k], 1 + isl_basic_map_total_dim(path));
+ if (p == PURE_VAR) {
+ isl_seq_cpy(path->ineq[k] + off,
+ delta->ineq[i] + 1 + nparam, d);
+ isl_int_set(path->ineq[k][off + d], delta->ineq[i][0]);
+ } else
+ isl_seq_cpy(path->ineq[k], delta->ineq[i], 1 + nparam);
+ }
+
k = isl_basic_map_alloc_inequality(path);
if (k < 0)
goto error;
isl_seq_clr(path->ineq[k], 1 + isl_basic_map_total_dim(path));
- isl_int_set_si(path->ineq[k][1 + param], 1);
isl_int_set_si(path->ineq[k][0], -1);
-
- isl_dim_free(dim);
-
- path = isl_basic_map_simplify(path);
+ isl_int_set_si(path->ineq[k][off + d], 1);
+
+ isl_basic_set_free(delta);
path = isl_basic_map_finalize(path);
- return isl_map_from_basic_map(path);
+ return isl_basic_map_union(path,
+ isl_basic_map_identity(isl_dim_domain(dim)));
error:
isl_dim_free(dim);
+ isl_basic_set_free(delta);
isl_basic_map_free(path);
return NULL;
}
-/* Check whether "path" is acyclic.
+/* Given a dimenion specification Z^{n+1} -> Z^{n+1} and a parameter "param",
+ * construct a map that equates the parameter to the difference
+ * in the final coordinates and imposes that this difference is positive.
+ * That is, construct
+ *
+ * { [x,x_s] -> [y,y_s] : k = y_s - x_s > 0 }
+ */
+static __isl_give isl_map *equate_parameter_to_length(__isl_take isl_dim *dim,
+ unsigned param)
+{
+ struct isl_basic_map *bmap;
+ unsigned d;
+ unsigned nparam;
+ int k;
+
+ d = isl_dim_size(dim, isl_dim_in);
+ nparam = isl_dim_size(dim, isl_dim_param);
+ bmap = isl_basic_map_alloc_dim(dim, 0, 1, 1);
+ k = isl_basic_map_alloc_equality(bmap);
+ if (k < 0)
+ goto error;
+ isl_seq_clr(bmap->eq[k], 1 + isl_basic_map_total_dim(bmap));
+ isl_int_set_si(bmap->eq[k][1 + param], -1);
+ isl_int_set_si(bmap->eq[k][1 + nparam + d - 1], -1);
+ isl_int_set_si(bmap->eq[k][1 + nparam + d + d - 1], 1);
+
+ k = isl_basic_map_alloc_inequality(bmap);
+ if (k < 0)
+ goto error;
+ isl_seq_clr(bmap->ineq[k], 1 + isl_basic_map_total_dim(bmap));
+ isl_int_set_si(bmap->ineq[k][1 + param], 1);
+ isl_int_set_si(bmap->ineq[k][0], -1);
+
+ bmap = isl_basic_map_finalize(bmap);
+ return isl_map_from_basic_map(bmap);
+error:
+ isl_basic_map_free(bmap);
+ return NULL;
+}
+
+/* Check whether "path" is acyclic, where the last coordinates of domain
+ * and range of path encode the number of steps taken.
* That is, check whether
*
* { d | d = y - x and (x,y) in path }
*
- * does not contain the origin.
+ * does not contain any element with positive last coordinate (positive length)
+ * and zero remaining coordinates (cycle).
*/
static int is_acyclic(__isl_take isl_map *path)
{
delta = isl_map_deltas(path);
dim = isl_set_dim(delta, isl_dim_set);
- for (i = 0; i < dim; ++i)
- delta = isl_set_fix_si(delta, isl_dim_set, i, 0);
+ for (i = 0; i < dim; ++i) {
+ if (i == dim -1)
+ delta = isl_set_lower_bound_si(delta, isl_dim_set, i, 1);
+ else
+ delta = isl_set_fix_si(delta, isl_dim_set, i, 0);
+ }
acyclic = isl_set_is_empty(delta);
isl_set_free(delta);
return acyclic;
}
+/* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
+ * and a dimension specification (Z^{n+1} -> Z^{n+1}),
+ * construct a map that is an overapproximation of the map
+ * that takes an element from the space D \times Z to another
+ * element from the same space, such that the first n coordinates of the
+ * difference between them is a sum of differences between images
+ * and pre-images in one of the R_i and such that the last coordinate
+ * is equal to the number of steps taken.
+ * That is, let
+ *
+ * \Delta_i = { y - x | (x, y) in R_i }
+ *
+ * then the constructed map is an overapproximation of
+ *
+ * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
+ * d = (\sum_i k_i \delta_i, \sum_i k_i) }
+ *
+ * The elements of the singleton \Delta_i's are collected as the
+ * rows of the steps matrix. For all these \Delta_i's together,
+ * a single path is constructed.
+ * For each of the other \Delta_i's, we compute an overapproximation
+ * of the paths along elements of \Delta_i.
+ * Since each of these paths performs an addition, composition is
+ * symmetric and we can simply compose all resulting paths in any order.
+ */
+static __isl_give isl_map *construct_extended_path(__isl_take isl_dim *dim,
+ __isl_keep isl_map *map, int *project)
+{
+ struct isl_mat *steps = NULL;
+ struct isl_map *path = NULL;
+ unsigned d;
+ int i, j, n;
+
+ d = isl_map_dim(map, isl_dim_in);
+
+ path = isl_map_identity(isl_dim_domain(isl_dim_copy(dim)));
+
+ steps = isl_mat_alloc(map->ctx, map->n, d);
+ if (!steps)
+ goto error;
+
+ n = 0;
+ for (i = 0; i < map->n; ++i) {
+ struct isl_basic_set *delta;
+
+ delta = isl_basic_map_deltas(isl_basic_map_copy(map->p[i]));
+
+ for (j = 0; j < d; ++j) {
+ int fixed;
+
+ fixed = isl_basic_set_fast_dim_is_fixed(delta, j,
+ &steps->row[n][j]);
+ if (fixed < 0) {
+ isl_basic_set_free(delta);
+ goto error;
+ }
+ if (!fixed)
+ break;
+ }
+
+
+ if (j < d) {
+ path = isl_map_apply_range(path,
+ path_along_delta(isl_dim_copy(dim), delta));
+ } else {
+ isl_basic_set_free(delta);
+ ++n;
+ }
+ }
+
+ if (n > 0) {
+ steps->n_row = n;
+ path = isl_map_apply_range(path,
+ path_along_steps(isl_dim_copy(dim), steps));
+ }
+
+ if (project && *project) {
+ *project = is_acyclic(isl_map_copy(path));
+ if (*project < 0)
+ goto error;
+ }
+
+ isl_dim_free(dim);
+ isl_mat_free(steps);
+ return path;
+error:
+ isl_dim_free(dim);
+ isl_mat_free(steps);
+ isl_map_free(path);
+ return NULL;
+}
+
+/* Given a union of basic maps R = \cup_i R_i \subseteq D \times D
+ * and a dimension specification (Z^{n+1} -> Z^{n+1}),
+ * construct a map that is an overapproximation of the map
+ * that takes an element from the dom R \times Z to an
+ * element from ran R \times Z, such that the first n coordinates of the
+ * difference between them is a sum of differences between images
+ * and pre-images in one of the R_i and such that the last coordinate
+ * is equal to the number of steps taken.
+ * That is, let
+ *
+ * \Delta_i = { y - x | (x, y) in R_i }
+ *
+ * then the constructed map is an overapproximation of
+ *
+ * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
+ * d = (\sum_i k_i \delta_i, \sum_i k_i) and
+ * x in dom R and x + d in ran R}
+ */
+static __isl_give isl_map *construct_extended_power(__isl_take isl_dim *dim,
+ __isl_keep isl_map *map, int *project)
+{
+ struct isl_set *domain = NULL;
+ struct isl_set *range = NULL;
+ struct isl_map *app = NULL;
+ struct isl_map *path = NULL;
+
+ domain = isl_map_domain(isl_map_copy(map));
+ domain = isl_set_coalesce(domain);
+ range = isl_map_range(isl_map_copy(map));
+ range = isl_set_coalesce(range);
+ app = isl_map_from_domain_and_range(domain, range);
+ app = isl_map_add(app, isl_dim_in, 1);
+ app = isl_map_add(app, isl_dim_out, 1);
+
+ path = construct_extended_path(dim, map, project);
+ app = isl_map_intersect(app, path);
+
+ return app;
+}
+
+/* Given a union of basic maps R = \cup_i R_i \subseteq D \times D,
+ * construct a map that is an overapproximation of the map
+ * that takes an element from the space D to another
+ * element from the same space, such that the difference between
+ * them is a strictly positive sum of differences between images
+ * and pre-images in one of the R_i.
+ * The number of differences in the sum is equated to parameter "param".
+ * That is, let
+ *
+ * \Delta_i = { y - x | (x, y) in R_i }
+ *
+ * then the constructed map is an overapproximation of
+ *
+ * { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
+ * d = \sum_i k_i \delta_i and k = \sum_i k_i > 0 }
+ *
+ * We first construct an extended mapping with an extra coordinate
+ * that indicates the number of steps taken. In particular,
+ * the difference in the last coordinate is equal to the number
+ * of steps taken to move from a domain element to the corresponding
+ * image element(s).
+ * In the final step, this difference is equated to the parameter "param"
+ * and made positive. The extra coordinates are subsequently projected out.
+ */
+static __isl_give isl_map *construct_power(__isl_keep isl_map *map,
+ unsigned param, int *project)
+{
+ struct isl_map *app = NULL;
+ struct isl_map *diff;
+ struct isl_dim *dim = NULL;
+ unsigned d;
+
+ if (!map)
+ return NULL;
+
+ dim = isl_map_get_dim(map);
+
+ d = isl_dim_size(dim, isl_dim_in);
+ dim = isl_dim_add(dim, isl_dim_in, 1);
+ dim = isl_dim_add(dim, isl_dim_out, 1);
+
+ app = construct_extended_power(isl_dim_copy(dim), map, project);
+
+ diff = equate_parameter_to_length(dim, param);
+ app = isl_map_intersect(app, diff);
+ app = isl_map_project_out(app, isl_dim_in, d, 1);
+ app = isl_map_project_out(app, isl_dim_out, d, 1);
+
+ return app;
+}
+
/* Shift variable at position "pos" up by one.
* That is, replace the corresponding variable v by v - 1.
*/
* Otherwise, we check if the power is exact.
*/
static int check_exactness(__isl_take isl_map *map, __isl_take isl_map *app,
- __isl_take isl_map *path, unsigned param, int project)
+ unsigned param, int project)
{
isl_map *test;
int exact;
- if (project) {
- project = is_acyclic(path);
- if (project < 0)
- goto error;
- } else
- isl_map_free(path);
-
if (!project)
return check_power_exactness(map, app, param);
static __isl_give isl_map *map_power(__isl_take isl_map *map, unsigned param,
int *exact, int project)
{
- struct isl_mat *steps = NULL;
- struct isl_set *domain = NULL;
- struct isl_set *range = NULL;
struct isl_map *app = NULL;
- struct isl_map *path = NULL;
- int ok;
if (exact)
*exact = 1;
isl_map_dim(map, isl_dim_in) == isl_map_dim(map, isl_dim_out),
goto error);
- domain = isl_map_domain(isl_map_copy(map));
- domain = isl_set_coalesce(domain);
- range = isl_map_range(isl_map_copy(map));
- range = isl_set_coalesce(range);
- app = isl_map_from_domain_and_range(isl_set_copy(domain),
- isl_set_copy(range));
-
- steps = extract_steps(map, &ok);
- if (!ok)
- goto not_exact;
-
- path = path_along_steps(isl_map_get_dim(map), steps, param);
- app = isl_map_intersect(app, isl_map_copy(path));
+ app = construct_power(map, param, exact ? &project : NULL);
if (exact &&
(*exact = check_exactness(isl_map_copy(map), isl_map_copy(app),
- isl_map_copy(path), param, project)) < 0)
+ param, project)) < 0)
goto error;
- if (0) {
-not_exact:
- if (exact)
- *exact = 0;
- }
- isl_set_free(domain);
- isl_set_free(range);
- isl_mat_free(steps);
- isl_map_free(path);
isl_map_free(map);
return app;
error:
- isl_set_free(domain);
- isl_set_free(range);
- isl_mat_free(steps);
- isl_map_free(path);
isl_map_free(map);
isl_map_free(app);
return NULL;
projects / platform / upstream / isl.git / blobdiff