nexmon – Rev 1
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/*
* Copyright 2010 INRIA Saclay
*
* Use of this software is governed by the GNU LGPLv2.1 license
*
* Written by Sven Verdoolaege, INRIA Saclay - Ile-de-France,
* Parc Club Orsay Universite, ZAC des vignes, 4 rue Jacques Monod,
* 91893 Orsay, France
*/
#include <isl_ctx_private.h>
#include <isl_map_private.h>
#include <isl/map.h>
#include <isl/seq.h>
#include <isl_space_private.h>
#include <isl/lp.h>
#include <isl/union_map.h>
#include <isl_mat_private.h>
#include <isl_options_private.h>
int isl_map_is_transitively_closed(__isl_keep isl_map *map)
{
isl_map *map2;
int closed;
map2 = isl_map_apply_range(isl_map_copy(map), isl_map_copy(map));
closed = isl_map_is_subset(map2, map);
isl_map_free(map2);
return closed;
}
int isl_union_map_is_transitively_closed(__isl_keep isl_union_map *umap)
{
isl_union_map *umap2;
int closed;
umap2 = isl_union_map_apply_range(isl_union_map_copy(umap),
isl_union_map_copy(umap));
closed = isl_union_map_is_subset(umap2, umap);
isl_union_map_free(umap2);
return closed;
}
/* Given a map that represents a path with the length of the path
* encoded as the difference between the last output coordindate
* and the last input coordinate, set this length to either
* exactly "length" (if "exactly" is set) or at least "length"
* (if "exactly" is not set).
*/
static __isl_give isl_map *set_path_length(__isl_take isl_map *map,
int exactly, int length)
{
isl_space *dim;
struct isl_basic_map *bmap;
unsigned d;
unsigned nparam;
int k;
isl_int *c;
if (!map)
return NULL;
dim = isl_map_get_space(map);
d = isl_space_dim(dim, isl_dim_in);
nparam = isl_space_dim(dim, isl_dim_param);
bmap = isl_basic_map_alloc_space(dim, 0, 1, 1);
if (exactly) {
k = isl_basic_map_alloc_equality(bmap);
c = bmap->eq[k];
} else {
k = isl_basic_map_alloc_inequality(bmap);
c = bmap->ineq[k];
}
if (k < 0)
goto error;
isl_seq_clr(c, 1 + isl_basic_map_total_dim(bmap));
isl_int_set_si(c[0], -length);
isl_int_set_si(c[1 + nparam + d - 1], -1);
isl_int_set_si(c[1 + nparam + d + d - 1], 1);
bmap = isl_basic_map_finalize(bmap);
map = isl_map_intersect(map, isl_map_from_basic_map(bmap));
return map;
error:
isl_basic_map_free(bmap);
isl_map_free(map);
return NULL;
}
/* Check whether the overapproximation of the power of "map" is exactly
* the power of "map". Let R be "map" and A_k the overapproximation.
* The approximation is exact if
*
* A_1 = R
* A_k = A_{k-1} \circ R k >= 2
*
* Since A_k is known to be an overapproximation, we only need to check
*
* A_1 \subset R
* A_k \subset A_{k-1} \circ R k >= 2
*
* In practice, "app" has an extra input and output coordinate
* to encode the length of the path. So, we first need to add
* this coordinate to "map" and set the length of the path to
* one.
*/
static int check_power_exactness(__isl_take isl_map *map,
__isl_take isl_map *app)
{
int exact;
isl_map *app_1;
isl_map *app_2;
map = isl_map_add_dims(map, isl_dim_in, 1);
map = isl_map_add_dims(map, isl_dim_out, 1);
map = set_path_length(map, 1, 1);
app_1 = set_path_length(isl_map_copy(app), 1, 1);
exact = isl_map_is_subset(app_1, map);
isl_map_free(app_1);
if (!exact || exact < 0) {
isl_map_free(app);
isl_map_free(map);
return exact;
}
app_1 = set_path_length(isl_map_copy(app), 0, 1);
app_2 = set_path_length(app, 0, 2);
app_1 = isl_map_apply_range(map, app_1);
exact = isl_map_is_subset(app_2, app_1);
isl_map_free(app_1);
isl_map_free(app_2);
return exact;
}
/* Check whether the overapproximation of the power of "map" is exactly
* the power of "map", possibly after projecting out the power (if "project"
* is set).
*
* If "project" is set and if "steps" can only result in acyclic paths,
* then we check
*
* A = R \cup (A \circ R)
*
* where A is the overapproximation with the power projected out, i.e.,
* an overapproximation of the transitive closure.
* More specifically, since A is known to be an overapproximation, we check
*
* A \subset R \cup (A \circ R)
*
* Otherwise, we check if the power is exact.
*
* Note that "app" has an extra input and output coordinate to encode
* the length of the part. If we are only interested in the transitive
* closure, then we can simply project out these coordinates first.
*/
static int check_exactness(__isl_take isl_map *map, __isl_take isl_map *app,
int project)
{
isl_map *test;
int exact;
unsigned d;
if (!project)
return check_power_exactness(map, app);
d = isl_map_dim(map, isl_dim_in);
app = set_path_length(app, 0, 1);
app = isl_map_project_out(app, isl_dim_in, d, 1);
app = isl_map_project_out(app, isl_dim_out, d, 1);
app = isl_map_reset_space(app, isl_map_get_space(map));
test = isl_map_apply_range(isl_map_copy(map), isl_map_copy(app));
test = isl_map_union(test, isl_map_copy(map));
exact = isl_map_is_subset(app, test);
isl_map_free(app);
isl_map_free(test);
isl_map_free(map);
return exact;
}
/*
* The transitive closure implementation is based on the paper
* "Computing the Transitive Closure of a Union of Affine Integer
* Tuple Relations" by Anna Beletska, Denis Barthou, Wlodzimierz Bielecki and
* Albert Cohen.
*/
/* Given a set of n offsets v_i (the rows of "steps"), construct a relation
* 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 }
*
* 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_space *dim,
__isl_keep isl_mat *steps)
{
int i, j, k;
struct isl_basic_map *path = NULL;
unsigned d;
unsigned n;
unsigned nparam;
if (!dim || !steps)
goto error;
d = isl_space_dim(dim, isl_dim_in);
n = steps->n_row;
nparam = isl_space_dim(dim, isl_dim_param);
path = isl_basic_map_alloc_space(isl_space_copy(dim), n, d, n);
for (i = 0; i < n; ++i) {
k = isl_basic_map_alloc_div(path);
if (k < 0)
goto error;
isl_assert(steps->ctx, i == k, goto error);
isl_int_set_si(path->div[k][0], 0);
}
for (i = 0; i < d; ++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 + i], -1);
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]);
}
for (i = 0; i < n; ++i) {
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 + nparam + 2 * d + i], 1);
}
isl_space_free(dim);
path = isl_basic_map_simplify(path);
path = isl_basic_map_finalize(path);
return isl_map_from_basic_map(path);
error:
isl_space_free(dim);
isl_basic_map_free(path);
return NULL;
}
#define IMPURE 0
#define PURE_PARAM 1
#define PURE_VAR 2
#define MIXED 3
/* Check whether the parametric constant term of constraint c is never
* positive in "bset".
*/
static int parametric_constant_never_positive(__isl_keep isl_basic_set *bset,
isl_int *c, int *div_purity)
{
unsigned d;
unsigned n_div;
unsigned nparam;
int i;
int k;
int empty;
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);
bset = isl_basic_set_copy(bset);
bset = isl_basic_set_cow(bset);
bset = isl_basic_set_extend_constraints(bset, 0, 1);
k = isl_basic_set_alloc_inequality(bset);
if (k < 0)
goto error;
isl_seq_clr(bset->ineq[k], 1 + isl_basic_set_total_dim(bset));
isl_seq_cpy(bset->ineq[k], c, 1 + nparam);
for (i = 0; i < n_div; ++i) {
if (div_purity[i] != PURE_PARAM)
continue;
isl_int_set(bset->ineq[k][1 + nparam + d + i],
c[1 + nparam + d + i]);
}
isl_int_sub_ui(bset->ineq[k][0], bset->ineq[k][0], 1);
empty = isl_basic_set_is_empty(bset);
isl_basic_set_free(bset);
return empty;
error:
isl_basic_set_free(bset);
return -1;
}
/* 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 MIXED if only the coefficients of the parameters and the set
* variables are non-zero and if moreover the parametric constant
* can never attain positive values.
* Return IMPURE otherwise.
*
* If div_purity is NULL then we are dealing with a non-parametric set
* and so the constraint is obviously PURE_VAR.
*/
static int purity(__isl_keep isl_basic_set *bset, isl_int *c, int *div_purity,
int eq)
{
unsigned d;
unsigned n_div;
unsigned nparam;
int empty;
int i;
int p = 0, v = 0;
if (!div_purity)
return PURE_VAR;
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);
for (i = 0; i < n_div; ++i) {
if (isl_int_is_zero(c[1 + nparam + d + i]))
continue;
switch (div_purity[i]) {
case PURE_PARAM: p = 1; break;
case PURE_VAR: v = 1; break;
default: return IMPURE;
}
}
if (!p && isl_seq_first_non_zero(c + 1, nparam) == -1)
return PURE_VAR;
if (!v && isl_seq_first_non_zero(c + 1 + nparam, d) == -1)
return PURE_PARAM;
empty = parametric_constant_never_positive(bset, c, div_purity);
if (eq && empty >= 0 && !empty) {
isl_seq_neg(c, c, 1 + nparam + d + n_div);
empty = parametric_constant_never_positive(bset, c, div_purity);
}
return empty < 0 ? -1 : empty ? MIXED : IMPURE;
}
/* Return an array of integers indicating the type of each div in bset.
* If the div is (recursively) defined in terms of only the parameters,
* then the type is PURE_PARAM.
* If the div is (recursively) defined in terms of only the set variables,
* then the type is PURE_VAR.
* Otherwise, the type is IMPURE.
*/
static __isl_give int *get_div_purity(__isl_keep isl_basic_set *bset)
{
int i, j;
int *div_purity;
unsigned d;
unsigned n_div;
unsigned nparam;
if (!bset)
return NULL;
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);
div_purity = isl_alloc_array(bset->ctx, int, n_div);
if (!div_purity)
return NULL;
for (i = 0; i < bset->n_div; ++i) {
int p = 0, v = 0;
if (isl_int_is_zero(bset->div[i][0])) {
div_purity[i] = IMPURE;
continue;
}
if (isl_seq_first_non_zero(bset->div[i] + 2, nparam) != -1)
p = 1;
if (isl_seq_first_non_zero(bset->div[i] + 2 + nparam, d) != -1)
v = 1;
for (j = 0; j < i; ++j) {
if (isl_int_is_zero(bset->div[i][2 + nparam + d + j]))
continue;
switch (div_purity[j]) {
case PURE_PARAM: p = 1; break;
case PURE_VAR: v = 1; break;
default: p = v = 1; break;
}
}
div_purity[i] = v ? p ? IMPURE : PURE_VAR : PURE_PARAM;
}
return div_purity;
}
/* Given a path with the as yet unconstrained length at position "pos",
* check if setting the length to zero results in only the identity
* mapping.
*/
static int empty_path_is_identity(__isl_keep isl_basic_map *path, unsigned pos)
{
isl_basic_map *test = NULL;
isl_basic_map *id = NULL;
int k;
int is_id;
test = isl_basic_map_copy(path);
test = isl_basic_map_extend_constraints(test, 1, 0);
k = isl_basic_map_alloc_equality(test);
if (k < 0)
goto error;
isl_seq_clr(test->eq[k], 1 + isl_basic_map_total_dim(test));
isl_int_set_si(test->eq[k][pos], 1);
id = isl_basic_map_identity(isl_basic_map_get_space(path));
is_id = isl_basic_map_is_equal(test, id);
isl_basic_map_free(test);
isl_basic_map_free(id);
return is_id;
error:
isl_basic_map_free(test);
return -1;
}
/* If any of the constraints is found to be impure then this function
* sets *impurity to 1.
*/
static __isl_give isl_basic_map *add_delta_constraints(
__isl_take isl_basic_map *path,
__isl_keep isl_basic_set *delta, unsigned off, unsigned nparam,
unsigned d, int *div_purity, int eq, int *impurity)
{
int i, k;
int n = eq ? delta->n_eq : delta->n_ineq;
isl_int **delta_c = eq ? delta->eq : delta->ineq;
unsigned n_div;
n_div = isl_basic_set_dim(delta, isl_dim_div);
for (i = 0; i < n; ++i) {
isl_int *path_c;
int p = purity(delta, delta_c[i], div_purity, eq);
if (p < 0)
goto error;
if (p != PURE_VAR && p != PURE_PARAM && !*impurity)
*impurity = 1;
if (p == IMPURE)
continue;
if (eq && p != MIXED) {
k = isl_basic_map_alloc_equality(path);
path_c = path->eq[k];
} else {
k = isl_basic_map_alloc_inequality(path);
path_c = path->ineq[k];
}
if (k < 0)
goto error;
isl_seq_clr(path_c, 1 + isl_basic_map_total_dim(path));
if (p == PURE_VAR) {
isl_seq_cpy(path_c + off,
delta_c[i] + 1 + nparam, d);
isl_int_set(path_c[off + d], delta_c[i][0]);
} else if (p == PURE_PARAM) {
isl_seq_cpy(path_c, delta_c[i], 1 + nparam);
} else {
isl_seq_cpy(path_c + off,
delta_c[i] + 1 + nparam, d);
isl_seq_cpy(path_c, delta_c[i], 1 + nparam);
}
isl_seq_cpy(path_c + off - n_div,
delta_c[i] + 1 + nparam + d, n_div);
}
return path;
error:
isl_basic_map_free(path);
return NULL;
}
/* 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 >= 0 and B p + b >= 0 and
* C x + C'p + c >= 0 and
* D x + D'p + d >= 0 }
*
* where the constraints C x + C'p + c >= 0 are such that the parametric
* constant term of each constraint j, "C_j x + C'_j p + c_j",
* can never attain positive values, 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
* C f + C'p + c >= 0 and k >= 1 }
* union { [x] -> [x] }
*
* If the zero-length paths happen to correspond exactly to the identity
* mapping, then we return
*
* { [x] -> [y] : exists [f, k] \in Z^{n+1} : y = x + f and
* A f + k a >= 0 and B p + b >= 0 and
* C f + C'p + c >= 0 and k >= 0 }
*
* instead.
*
* Existentially quantified variables in \delta are handled by
* classifying them as independent of the parameters, purely
* parameter dependent and others. Constraints containing
* any of the other existentially quantified variables are removed.
* This is safe, but leads to an additional overapproximation.
*
* If there are any impure constraints, then we also eliminate
* the parameters from \delta, resulting in a set
*
* \delta' = { [x] : E x + e >= 0 }
*
* and add the constraints
*
* E f + k e >= 0
*
* to the constructed relation.
*/
static __isl_give isl_map *path_along_delta(__isl_take isl_space *dim,
__isl_take isl_basic_set *delta)
{
isl_basic_map *path = NULL;
unsigned d;
unsigned n_div;
unsigned nparam;
unsigned off;
int i, k;
int is_id;
int *div_purity = NULL;
int impurity = 0;
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_space(isl_space_copy(dim), n_div + d + 1,
d + 1 + delta->n_eq, 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);
}
div_purity = get_div_purity(delta);
if (!div_purity)
goto error;
path = add_delta_constraints(path, delta, off, nparam, d,
div_purity, 1, &impurity);
path = add_delta_constraints(path, delta, off, nparam, d,
div_purity, 0, &impurity);
if (impurity) {
isl_space *dim = isl_basic_set_get_space(delta);
delta = isl_basic_set_project_out(delta,
isl_dim_param, 0, nparam);
delta = isl_basic_set_add(delta, isl_dim_param, nparam);
delta = isl_basic_set_reset_space(delta, dim);
if (!delta)
goto error;
path = isl_basic_map_extend_constraints(path, delta->n_eq,
delta->n_ineq + 1);
path = add_delta_constraints(path, delta, off, nparam, d,
NULL, 1, &impurity);
path = add_delta_constraints(path, delta, off, nparam, d,
NULL, 0, &impurity);
path = isl_basic_map_gauss(path, NULL);
}
is_id = empty_path_is_identity(path, off + d);
if (is_id < 0)
goto error;
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 (!is_id)
isl_int_set_si(path->ineq[k][0], -1);
isl_int_set_si(path->ineq[k][off + d], 1);
free(div_purity);
isl_basic_set_free(delta);
path = isl_basic_map_finalize(path);
if (is_id) {
isl_space_free(dim);
return isl_map_from_basic_map(path);
}
return isl_basic_map_union(path, isl_basic_map_identity(dim));
error:
free(div_purity);
isl_space_free(dim);
isl_basic_set_free(delta);
isl_basic_map_free(path);
return NULL;
}
/* Given a dimension 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_space *dim,
unsigned param)
{
struct isl_basic_map *bmap;
unsigned d;
unsigned nparam;
int k;
d = isl_space_dim(dim, isl_dim_in);
nparam = isl_space_dim(dim, isl_dim_param);
bmap = isl_basic_map_alloc_space(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 any element with positive last coordinate (positive length)
* and zero remaining coordinates (cycle).
*/
static int is_acyclic(__isl_take isl_map *path)
{
int i;
int acyclic;
unsigned dim;
struct isl_set *delta;
delta = isl_map_deltas(path);
dim = isl_set_dim(delta, isl_dim_set);
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_space *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_space_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_plain_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_space_copy(dim), delta));
path = isl_map_coalesce(path);
} else {
isl_basic_set_free(delta);
++n;
}
}
if (n > 0) {
steps->n_row = n;
path = isl_map_apply_range(path,
path_along_steps(isl_space_copy(dim), steps));
}
if (project && *project) {
*project = is_acyclic(isl_map_copy(path));
if (*project < 0)
goto error;
}
isl_space_free(dim);
isl_mat_free(steps);
return path;
error:
isl_space_free(dim);
isl_mat_free(steps);
isl_map_free(path);
return NULL;
}
static int isl_set_overlaps(__isl_keep isl_set *set1, __isl_keep isl_set *set2)
{
isl_set *i;
int no_overlap;
if (!isl_space_tuple_match(set1->dim, isl_dim_set, set2->dim, isl_dim_set))
return 0;
i = isl_set_intersect(isl_set_copy(set1), isl_set_copy(set2));
no_overlap = isl_set_is_empty(i);
isl_set_free(i);
return no_overlap < 0 ? -1 : !no_overlap;
}
/* 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 and
* \sum_i k_i >= 1 }
*/
static __isl_give isl_map *construct_component(__isl_take isl_space *dim,
__isl_keep isl_map *map, int *exact, 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);
if (!isl_set_overlaps(domain, range)) {
isl_set_free(domain);
isl_set_free(range);
isl_space_free(dim);
map = isl_map_copy(map);
map = isl_map_add_dims(map, isl_dim_in, 1);
map = isl_map_add_dims(map, isl_dim_out, 1);
map = set_path_length(map, 1, 1);
return map;
}
app = isl_map_from_domain_and_range(domain, range);
app = isl_map_add_dims(app, isl_dim_in, 1);
app = isl_map_add_dims(app, isl_dim_out, 1);
path = construct_extended_path(isl_space_copy(dim), map,
exact && *exact ? &project : NULL);
app = isl_map_intersect(app, path);
if (exact && *exact &&
(*exact = check_exactness(isl_map_copy(map), isl_map_copy(app),
project)) < 0)
goto error;
isl_space_free(dim);
app = set_path_length(app, 0, 1);
return app;
error:
isl_space_free(dim);
isl_map_free(app);
return NULL;
}
/* Call construct_component and, if "project" is set, project out
* the final coordinates.
*/
static __isl_give isl_map *construct_projected_component(
__isl_take isl_space *dim,
__isl_keep isl_map *map, int *exact, int project)
{
isl_map *app;
unsigned d;
if (!dim)
return NULL;
d = isl_space_dim(dim, isl_dim_in);
app = construct_component(dim, map, exact, project);
if (project) {
app = isl_map_project_out(app, isl_dim_in, d - 1, 1);
app = isl_map_project_out(app, isl_dim_out, d - 1, 1);
}
return app;
}
/* Compute an extended version, i.e., with path lengths, of
* an overapproximation of the transitive closure of "bmap"
* with path lengths greater than or equal to zero and with
* domain and range equal to "dom".
*/
static __isl_give isl_map *q_closure(__isl_take isl_space *dim,
__isl_take isl_set *dom, __isl_keep isl_basic_map *bmap, int *exact)
{
int project = 1;
isl_map *path;
isl_map *map;
isl_map *app;
dom = isl_set_add_dims(dom, isl_dim_set, 1);
app = isl_map_from_domain_and_range(dom, isl_set_copy(dom));
map = isl_map_from_basic_map(isl_basic_map_copy(bmap));
path = construct_extended_path(dim, map, &project);
app = isl_map_intersect(app, path);
if ((*exact = check_exactness(map, isl_map_copy(app), project)) < 0)
goto error;
return app;
error:
isl_map_free(app);
return NULL;
}
/* Check whether qc has any elements of length at least one
* with domain and/or range outside of dom and ran.
*/
static int has_spurious_elements(__isl_keep isl_map *qc,
__isl_keep isl_set *dom, __isl_keep isl_set *ran)
{
isl_set *s;
int subset;
unsigned d;
if (!qc || !dom || !ran)
return -1;
d = isl_map_dim(qc, isl_dim_in);
qc = isl_map_copy(qc);
qc = set_path_length(qc, 0, 1);
qc = isl_map_project_out(qc, isl_dim_in, d - 1, 1);
qc = isl_map_project_out(qc, isl_dim_out, d - 1, 1);
s = isl_map_domain(isl_map_copy(qc));
subset = isl_set_is_subset(s, dom);
isl_set_free(s);
if (subset < 0)
goto error;
if (!subset) {
isl_map_free(qc);
return 1;
}
s = isl_map_range(qc);
subset = isl_set_is_subset(s, ran);
isl_set_free(s);
return subset < 0 ? -1 : !subset;
error:
isl_map_free(qc);
return -1;
}
#define LEFT 2
#define RIGHT 1
/* For each basic map in "map", except i, check whether it combines
* with the transitive closure that is reflexive on C combines
* to the left and to the right.
*
* In particular, if
*
* dom map_j \subseteq C
*
* then right[j] is set to 1. Otherwise, if
*
* ran map_i \cap dom map_j = \emptyset
*
* then right[j] is set to 0. Otherwise, composing to the right
* is impossible.
*
* Similar, for composing to the left, we have if
*
* ran map_j \subseteq C
*
* then left[j] is set to 1. Otherwise, if
*
* dom map_i \cap ran map_j = \emptyset
*
* then left[j] is set to 0. Otherwise, composing to the left
* is impossible.
*
* The return value is or'd with LEFT if composing to the left
* is possible and with RIGHT if composing to the right is possible.
*/
static int composability(__isl_keep isl_set *C, int i,
isl_set **dom, isl_set **ran, int *left, int *right,
__isl_keep isl_map *map)
{
int j;
int ok;
ok = LEFT | RIGHT;
for (j = 0; j < map->n && ok; ++j) {
int overlaps, subset;
if (j == i)
continue;
if (ok & RIGHT) {
if (!dom[j])
dom[j] = isl_set_from_basic_set(
isl_basic_map_domain(
isl_basic_map_copy(map->p[j])));
if (!dom[j])
return -1;
overlaps = isl_set_overlaps(ran[i], dom[j]);
if (overlaps < 0)
return -1;
if (!overlaps)
right[j] = 0;
else {
subset = isl_set_is_subset(dom[j], C);
if (subset < 0)
return -1;
if (subset)
right[j] = 1;
else
ok &= ~RIGHT;
}
}
if (ok & LEFT) {
if (!ran[j])
ran[j] = isl_set_from_basic_set(
isl_basic_map_range(
isl_basic_map_copy(map->p[j])));
if (!ran[j])
return -1;
overlaps = isl_set_overlaps(dom[i], ran[j]);
if (overlaps < 0)
return -1;
if (!overlaps)
left[j] = 0;
else {
subset = isl_set_is_subset(ran[j], C);
if (subset < 0)
return -1;
if (subset)
left[j] = 1;
else
ok &= ~LEFT;
}
}
}
return ok;
}
static __isl_give isl_map *anonymize(__isl_take isl_map *map)
{
map = isl_map_reset(map, isl_dim_in);
map = isl_map_reset(map, isl_dim_out);
return map;
}
/* Return a map that is a union of the basic maps in "map", except i,
* composed to left and right with qc based on the entries of "left"
* and "right".
*/
static __isl_give isl_map *compose(__isl_keep isl_map *map, int i,
__isl_take isl_map *qc, int *left, int *right)
{
int j;
isl_map *comp;
comp = isl_map_empty(isl_map_get_space(map));
for (j = 0; j < map->n; ++j) {
isl_map *map_j;
if (j == i)
continue;
map_j = isl_map_from_basic_map(isl_basic_map_copy(map->p[j]));
map_j = anonymize(map_j);
if (left && left[j])
map_j = isl_map_apply_range(map_j, isl_map_copy(qc));
if (right && right[j])
map_j = isl_map_apply_range(isl_map_copy(qc), map_j);
comp = isl_map_union(comp, map_j);
}
comp = isl_map_compute_divs(comp);
comp = isl_map_coalesce(comp);
isl_map_free(qc);
return comp;
}
/* Compute the transitive closure of "map" incrementally by
* computing
*
* map_i^+ \cup qc^+
*
* or
*
* map_i^+ \cup ((id \cup map_i^) \circ qc^+)
*
* or
*
* map_i^+ \cup (qc^+ \circ (id \cup map_i^))
*
* depending on whether left or right are NULL.
*/
static __isl_give isl_map *compute_incremental(
__isl_take isl_space *dim, __isl_keep isl_map *map,
int i, __isl_take isl_map *qc, int *left, int *right, int *exact)
{
isl_map *map_i;
isl_map *tc;
isl_map *rtc = NULL;
if (!map)
goto error;
isl_assert(map->ctx, left || right, goto error);
map_i = isl_map_from_basic_map(isl_basic_map_copy(map->p[i]));
tc = construct_projected_component(isl_space_copy(dim), map_i,
exact, 1);
isl_map_free(map_i);
if (*exact)
qc = isl_map_transitive_closure(qc, exact);
if (!*exact) {
isl_space_free(dim);
isl_map_free(tc);
isl_map_free(qc);
return isl_map_universe(isl_map_get_space(map));
}
if (!left || !right)
rtc = isl_map_union(isl_map_copy(tc),
isl_map_identity(isl_map_get_space(tc)));
if (!right)
qc = isl_map_apply_range(rtc, qc);
if (!left)
qc = isl_map_apply_range(qc, rtc);
qc = isl_map_union(tc, qc);
isl_space_free(dim);
return qc;
error:
isl_space_free(dim);
isl_map_free(qc);
return NULL;
}
/* Given a map "map", try to find a basic map such that
* map^+ can be computed as
*
* map^+ = map_i^+ \cup
* \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
*
* with C the simple hull of the domain and range of the input map.
* map_i^ \cup Id_C is computed by allowing the path lengths to be zero
* and by intersecting domain and range with C.
* Of course, we need to check that this is actually equal to map_i^ \cup Id_C.
* Also, we only use the incremental computation if all the transitive
* closures are exact and if the number of basic maps in the union,
* after computing the integer divisions, is smaller than the number
* of basic maps in the input map.
*/
static int incemental_on_entire_domain(__isl_keep isl_space *dim,
__isl_keep isl_map *map,
isl_set **dom, isl_set **ran, int *left, int *right,
__isl_give isl_map **res)
{
int i;
isl_set *C;
unsigned d;
*res = NULL;
C = isl_set_union(isl_map_domain(isl_map_copy(map)),
isl_map_range(isl_map_copy(map)));
C = isl_set_from_basic_set(isl_set_simple_hull(C));
if (!C)
return -1;
if (C->n != 1) {
isl_set_free(C);
return 0;
}
d = isl_map_dim(map, isl_dim_in);
for (i = 0; i < map->n; ++i) {
isl_map *qc;
int exact_i, spurious;
int j;
dom[i] = isl_set_from_basic_set(isl_basic_map_domain(
isl_basic_map_copy(map->p[i])));
ran[i] = isl_set_from_basic_set(isl_basic_map_range(
isl_basic_map_copy(map->p[i])));
qc = q_closure(isl_space_copy(dim), isl_set_copy(C),
map->p[i], &exact_i);
if (!qc)
goto error;
if (!exact_i) {
isl_map_free(qc);
continue;
}
spurious = has_spurious_elements(qc, dom[i], ran[i]);
if (spurious) {
isl_map_free(qc);
if (spurious < 0)
goto error;
continue;
}
qc = isl_map_project_out(qc, isl_dim_in, d, 1);
qc = isl_map_project_out(qc, isl_dim_out, d, 1);
qc = isl_map_compute_divs(qc);
for (j = 0; j < map->n; ++j)
left[j] = right[j] = 1;
qc = compose(map, i, qc, left, right);
if (!qc)
goto error;
if (qc->n >= map->n) {
isl_map_free(qc);
continue;
}
*res = compute_incremental(isl_space_copy(dim), map, i, qc,
left, right, &exact_i);
if (!*res)
goto error;
if (exact_i)
break;
isl_map_free(*res);
*res = NULL;
}
isl_set_free(C);
return *res != NULL;
error:
isl_set_free(C);
return -1;
}
/* Try and compute the transitive closure of "map" as
*
* map^+ = map_i^+ \cup
* \bigcup_j ((map_i^+ \cup Id_C)^+ \circ map_j \circ (map_i^+ \cup Id_C))^+
*
* with C either the simple hull of the domain and range of the entire
* map or the simple hull of domain and range of map_i.
*/
static __isl_give isl_map *incremental_closure(__isl_take isl_space *dim,
__isl_keep isl_map *map, int *exact, int project)
{
int i;
isl_set **dom = NULL;
isl_set **ran = NULL;
int *left = NULL;
int *right = NULL;
isl_set *C;
unsigned d;
isl_map *res = NULL;
if (!project)
return construct_projected_component(dim, map, exact, project);
if (!map)
goto error;
if (map->n <= 1)
return construct_projected_component(dim, map, exact, project);
d = isl_map_dim(map, isl_dim_in);
dom = isl_calloc_array(map->ctx, isl_set *, map->n);
ran = isl_calloc_array(map->ctx, isl_set *, map->n);
left = isl_calloc_array(map->ctx, int, map->n);
right = isl_calloc_array(map->ctx, int, map->n);
if (!ran || !dom || !left || !right)
goto error;
if (incemental_on_entire_domain(dim, map, dom, ran, left, right, &res) < 0)
goto error;
for (i = 0; !res && i < map->n; ++i) {
isl_map *qc;
int exact_i, spurious, comp;
if (!dom[i])
dom[i] = isl_set_from_basic_set(
isl_basic_map_domain(
isl_basic_map_copy(map->p[i])));
if (!dom[i])
goto error;
if (!ran[i])
ran[i] = isl_set_from_basic_set(
isl_basic_map_range(
isl_basic_map_copy(map->p[i])));
if (!ran[i])
goto error;
C = isl_set_union(isl_set_copy(dom[i]),
isl_set_copy(ran[i]));
C = isl_set_from_basic_set(isl_set_simple_hull(C));
if (!C)
goto error;
if (C->n != 1) {
isl_set_free(C);
continue;
}
comp = composability(C, i, dom, ran, left, right, map);
if (!comp || comp < 0) {
isl_set_free(C);
if (comp < 0)
goto error;
continue;
}
qc = q_closure(isl_space_copy(dim), C, map->p[i], &exact_i);
if (!qc)
goto error;
if (!exact_i) {
isl_map_free(qc);
continue;
}
spurious = has_spurious_elements(qc, dom[i], ran[i]);
if (spurious) {
isl_map_free(qc);
if (spurious < 0)
goto error;
continue;
}
qc = isl_map_project_out(qc, isl_dim_in, d, 1);
qc = isl_map_project_out(qc, isl_dim_out, d, 1);
qc = isl_map_compute_divs(qc);
qc = compose(map, i, qc, (comp & LEFT) ? left : NULL,
(comp & RIGHT) ? right : NULL);
if (!qc)
goto error;
if (qc->n >= map->n) {
isl_map_free(qc);
continue;
}
res = compute_incremental(isl_space_copy(dim), map, i, qc,
(comp & LEFT) ? left : NULL,
(comp & RIGHT) ? right : NULL, &exact_i);
if (!res)
goto error;
if (exact_i)
break;
isl_map_free(res);
res = NULL;
}
for (i = 0; i < map->n; ++i) {
isl_set_free(dom[i]);
isl_set_free(ran[i]);
}
free(dom);
free(ran);
free(left);
free(right);
if (res) {
isl_space_free(dim);
return res;
}
return construct_projected_component(dim, map, exact, project);
error:
if (dom)
for (i = 0; i < map->n; ++i)
isl_set_free(dom[i]);
free(dom);
if (ran)
for (i = 0; i < map->n; ++i)
isl_set_free(ran[i]);
free(ran);
free(left);
free(right);
isl_space_free(dim);
return NULL;
}
/* Given an array of sets "set", add "dom" at position "pos"
* and search for elements at earlier positions that overlap with "dom".
* If any can be found, then merge all of them, together with "dom", into
* a single set and assign the union to the first in the array,
* which becomes the new group leader for all groups involved in the merge.
* During the search, we only consider group leaders, i.e., those with
* group[i] = i, as the other sets have already been combined
* with one of the group leaders.
*/
static int merge(isl_set **set, int *group, __isl_take isl_set *dom, int pos)
{
int i;
group[pos] = pos;
set[pos] = isl_set_copy(dom);
for (i = pos - 1; i >= 0; --i) {
int o;
if (group[i] != i)
continue;
o = isl_set_overlaps(set[i], dom);
if (o < 0)
goto error;
if (!o)
continue;
set[i] = isl_set_union(set[i], set[group[pos]]);
set[group[pos]] = NULL;
if (!set[i])
goto error;
group[group[pos]] = i;
group[pos] = i;
}
isl_set_free(dom);
return 0;
error:
isl_set_free(dom);
return -1;
}
/* Replace each entry in the n by n grid of maps by the cross product
* with the relation { [i] -> [i + 1] }.
*/
static int add_length(__isl_keep isl_map *map, isl_map ***grid, int n)
{
int i, j, k;
isl_space *dim;
isl_basic_map *bstep;
isl_map *step;
unsigned nparam;
if (!map)
return -1;
dim = isl_map_get_space(map);
nparam = isl_space_dim(dim, isl_dim_param);
dim = isl_space_drop_dims(dim, isl_dim_in, 0, isl_space_dim(dim, isl_dim_in));
dim = isl_space_drop_dims(dim, isl_dim_out, 0, isl_space_dim(dim, isl_dim_out));
dim = isl_space_add_dims(dim, isl_dim_in, 1);
dim = isl_space_add_dims(dim, isl_dim_out, 1);
bstep = isl_basic_map_alloc_space(dim, 0, 1, 0);
k = isl_basic_map_alloc_equality(bstep);
if (k < 0) {
isl_basic_map_free(bstep);
return -1;
}
isl_seq_clr(bstep->eq[k], 1 + isl_basic_map_total_dim(bstep));
isl_int_set_si(bstep->eq[k][0], 1);
isl_int_set_si(bstep->eq[k][1 + nparam], 1);
isl_int_set_si(bstep->eq[k][1 + nparam + 1], -1);
bstep = isl_basic_map_finalize(bstep);
step = isl_map_from_basic_map(bstep);
for (i = 0; i < n; ++i)
for (j = 0; j < n; ++j)
grid[i][j] = isl_map_product(grid[i][j],
isl_map_copy(step));
isl_map_free(step);
return 0;
}
/* The core of the Floyd-Warshall algorithm.
* Updates the given n x x matrix of relations in place.
*
* The algorithm iterates over all vertices. In each step, the whole
* matrix is updated to include all paths that go to the current vertex,
* possibly stay there a while (including passing through earlier vertices)
* and then come back. At the start of each iteration, the diagonal
* element corresponding to the current vertex is replaced by its
* transitive closure to account for all indirect paths that stay
* in the current vertex.
*/
static void floyd_warshall_iterate(isl_map ***grid, int n, int *exact)
{
int r, p, q;
for (r = 0; r < n; ++r) {
int r_exact;
grid[r][r] = isl_map_transitive_closure(grid[r][r],
(exact && *exact) ? &r_exact : NULL);
if (exact && *exact && !r_exact)
*exact = 0;
for (p = 0; p < n; ++p)
for (q = 0; q < n; ++q) {
isl_map *loop;
if (p == r && q == r)
continue;
loop = isl_map_apply_range(
isl_map_copy(grid[p][r]),
isl_map_copy(grid[r][q]));
grid[p][q] = isl_map_union(grid[p][q], loop);
loop = isl_map_apply_range(
isl_map_copy(grid[p][r]),
isl_map_apply_range(
isl_map_copy(grid[r][r]),
isl_map_copy(grid[r][q])));
grid[p][q] = isl_map_union(grid[p][q], loop);
grid[p][q] = isl_map_coalesce(grid[p][q]);
}
}
}
/* Given a partition of the domains and ranges of the basic maps in "map",
* apply the Floyd-Warshall algorithm with the elements in the partition
* as vertices.
*
* In particular, there are "n" elements in the partition and "group" is
* an array of length 2 * map->n with entries in [0,n-1].
*
* We first construct a matrix of relations based on the partition information,
* apply Floyd-Warshall on this matrix of relations and then take the
* union of all entries in the matrix as the final result.
*
* If we are actually computing the power instead of the transitive closure,
* i.e., when "project" is not set, then the result should have the
* path lengths encoded as the difference between an extra pair of
* coordinates. We therefore apply the nested transitive closures
* to relations that include these lengths. In particular, we replace
* the input relation by the cross product with the unit length relation
* { [i] -> [i + 1] }.
*/
static __isl_give isl_map *floyd_warshall_with_groups(__isl_take isl_space *dim,
__isl_keep isl_map *map, int *exact, int project, int *group, int n)
{
int i, j, k;
isl_map ***grid = NULL;
isl_map *app;
if (!map)
goto error;
if (n == 1) {
free(group);
return incremental_closure(dim, map, exact, project);
}
grid = isl_calloc_array(map->ctx, isl_map **, n);
if (!grid)
goto error;
for (i = 0; i < n; ++i) {
grid[i] = isl_calloc_array(map->ctx, isl_map *, n);
if (!grid[i])
goto error;
for (j = 0; j < n; ++j)
grid[i][j] = isl_map_empty(isl_map_get_space(map));
}
for (k = 0; k < map->n; ++k) {
i = group[2 * k];
j = group[2 * k + 1];
grid[i][j] = isl_map_union(grid[i][j],
isl_map_from_basic_map(
isl_basic_map_copy(map->p[k])));
}
if (!project && add_length(map, grid, n) < 0)
goto error;
floyd_warshall_iterate(grid, n, exact);
app = isl_map_empty(isl_map_get_space(map));
for (i = 0; i < n; ++i) {
for (j = 0; j < n; ++j)
app = isl_map_union(app, grid[i][j]);
free(grid[i]);
}
free(grid);
free(group);
isl_space_free(dim);
return app;
error:
if (grid)
for (i = 0; i < n; ++i) {
if (!grid[i])
continue;
for (j = 0; j < n; ++j)
isl_map_free(grid[i][j]);
free(grid[i]);
}
free(grid);
free(group);
isl_space_free(dim);
return NULL;
}
/* Partition the domains and ranges of the n basic relations in list
* into disjoint cells.
*
* To find the partition, we simply consider all of the domains
* and ranges in turn and combine those that overlap.
* "set" contains the partition elements and "group" indicates
* to which partition element a given domain or range belongs.
* The domain of basic map i corresponds to element 2 * i in these arrays,
* while the domain corresponds to element 2 * i + 1.
* During the construction group[k] is either equal to k,
* in which case set[k] contains the union of all the domains and
* ranges in the corresponding group, or is equal to some l < k,
* with l another domain or range in the same group.
*/
static int *setup_groups(isl_ctx *ctx, __isl_keep isl_basic_map **list, int n,
isl_set ***set, int *n_group)
{
int i;
int *group = NULL;
int g;
*set = isl_calloc_array(ctx, isl_set *, 2 * n);
group = isl_alloc_array(ctx, int, 2 * n);
if (!*set || !group)
goto error;
for (i = 0; i < n; ++i) {
isl_set *dom;
dom = isl_set_from_basic_set(isl_basic_map_domain(
isl_basic_map_copy(list[i])));
if (merge(*set, group, dom, 2 * i) < 0)
goto error;
dom = isl_set_from_basic_set(isl_basic_map_range(
isl_basic_map_copy(list[i])));
if (merge(*set, group, dom, 2 * i + 1) < 0)
goto error;
}
g = 0;
for (i = 0; i < 2 * n; ++i)
if (group[i] == i) {
if (g != i) {
(*set)[g] = (*set)[i];
(*set)[i] = NULL;
}
group[i] = g++;
} else
group[i] = group[group[i]];
*n_group = g;
return group;
error:
if (*set) {
for (i = 0; i < 2 * n; ++i)
isl_set_free((*set)[i]);
free(*set);
*set = NULL;
}
free(group);
return NULL;
}
/* Check if the domains and ranges of the basic maps in "map" can
* be partitioned, and if so, apply Floyd-Warshall on the elements
* of the partition. Note that we also apply this algorithm
* if we want to compute the power, i.e., when "project" is not set.
* However, the results are unlikely to be exact since the recursive
* calls inside the Floyd-Warshall algorithm typically result in
* non-linear path lengths quite quickly.
*/
static __isl_give isl_map *floyd_warshall(__isl_take isl_space *dim,
__isl_keep isl_map *map, int *exact, int project)
{
int i;
isl_set **set = NULL;
int *group = NULL;
int n;
if (!map)
goto error;
if (map->n <= 1)
return incremental_closure(dim, map, exact, project);
group = setup_groups(map->ctx, map->p, map->n, &set, &n);
if (!group)
goto error;
for (i = 0; i < 2 * map->n; ++i)
isl_set_free(set[i]);
free(set);
return floyd_warshall_with_groups(dim, map, exact, project, group, n);
error:
isl_space_free(dim);
return NULL;
}
/* Structure for representing the nodes in the graph being traversed
* using Tarjan's algorithm.
* index represents the order in which nodes are visited.
* min_index is the index of the root of a (sub)component.
* on_stack indicates whether the node is currently on the stack.
*/
struct basic_map_sort_node {
int index;
int min_index;
int on_stack;
};
/* Structure for representing the graph being traversed
* using Tarjan's algorithm.
* len is the number of nodes
* node is an array of nodes
* stack contains the nodes on the path from the root to the current node
* sp is the stack pointer
* index is the index of the last node visited
* order contains the elements of the components separated by -1
* op represents the current position in order
*
* check_closed is set if we may have used the fact that
* a pair of basic maps can be interchanged
*/
struct basic_map_sort {
int len;
struct basic_map_sort_node *node;
int *stack;
int sp;
int index;
int *order;
int op;
int check_closed;
};
static void basic_map_sort_free(struct basic_map_sort *s)
{
if (!s)
return;
free(s->node);
free(s->stack);
free(s->order);
free(s);
}
static struct basic_map_sort *basic_map_sort_alloc(struct isl_ctx *ctx, int len)
{
struct basic_map_sort *s;
int i;
s = isl_calloc_type(ctx, struct basic_map_sort);
if (!s)
return NULL;
s->len = len;
s->node = isl_alloc_array(ctx, struct basic_map_sort_node, len);
if (!s->node)
goto error;
for (i = 0; i < len; ++i)
s->node[i].index = -1;
s->stack = isl_alloc_array(ctx, int, len);
if (!s->stack)
goto error;
s->order = isl_alloc_array(ctx, int, 2 * len);
if (!s->order)
goto error;
s->sp = 0;
s->index = 0;
s->op = 0;
s->check_closed = 0;
return s;
error:
basic_map_sort_free(s);
return NULL;
}
/* Check whether in the computation of the transitive closure
* "bmap1" (R_1) should follow (or be part of the same component as)
* "bmap2" (R_2).
*
* That is check whether
*
* R_1 \circ R_2
*
* is a subset of
*
* R_2 \circ R_1
*
* If so, then there is no reason for R_1 to immediately follow R_2
* in any path.
*
* *check_closed is set if the subset relation holds while
* R_1 \circ R_2 is not empty.
*/
static int basic_map_follows(__isl_keep isl_basic_map *bmap1,
__isl_keep isl_basic_map *bmap2, int *check_closed)
{
struct isl_map *map12 = NULL;
struct isl_map *map21 = NULL;
int subset;
if (!isl_space_tuple_match(bmap1->dim, isl_dim_in, bmap2->dim, isl_dim_out))
return 0;
map21 = isl_map_from_basic_map(
isl_basic_map_apply_range(
isl_basic_map_copy(bmap2),
isl_basic_map_copy(bmap1)));
subset = isl_map_is_empty(map21);
if (subset < 0)
goto error;
if (subset) {
isl_map_free(map21);
return 0;
}
if (!isl_space_tuple_match(bmap1->dim, isl_dim_in, bmap1->dim, isl_dim_out) ||
!isl_space_tuple_match(bmap2->dim, isl_dim_in, bmap2->dim, isl_dim_out)) {
isl_map_free(map21);
return 1;
}
map12 = isl_map_from_basic_map(
isl_basic_map_apply_range(
isl_basic_map_copy(bmap1),
isl_basic_map_copy(bmap2)));
subset = isl_map_is_subset(map21, map12);
isl_map_free(map12);
isl_map_free(map21);
if (subset)
*check_closed = 1;
return subset < 0 ? -1 : !subset;
error:
isl_map_free(map21);
return -1;
}
/* Perform Tarjan's algorithm for computing the strongly connected components
* in the graph with the disjuncts of "map" as vertices and with an
* edge between any pair of disjuncts such that the first has
* to be applied after the second.
*/
static int power_components_tarjan(struct basic_map_sort *s,
__isl_keep isl_basic_map **list, int i)
{
int j;
s->node[i].index = s->index;
s->node[i].min_index = s->index;
s->node[i].on_stack = 1;
s->index++;
s->stack[s->sp++] = i;
for (j = s->len - 1; j >= 0; --j) {
int f;
if (j == i)
continue;
if (s->node[j].index >= 0 &&
(!s->node[j].on_stack ||
s->node[j].index > s->node[i].min_index))
continue;
f = basic_map_follows(list[i], list[j], &s->check_closed);
if (f < 0)
return -1;
if (!f)
continue;
if (s->node[j].index < 0) {
power_components_tarjan(s, list, j);
if (s->node[j].min_index < s->node[i].min_index)
s->node[i].min_index = s->node[j].min_index;
} else if (s->node[j].index < s->node[i].min_index)
s->node[i].min_index = s->node[j].index;
}
if (s->node[i].index != s->node[i].min_index)
return 0;
do {
j = s->stack[--s->sp];
s->node[j].on_stack = 0;
s->order[s->op++] = j;
} while (j != i);
s->order[s->op++] = -1;
return 0;
}
/* Decompose the "len" basic relations in "list" into strongly connected
* components.
*/
static struct basic_map_sort *basic_map_sort_init(isl_ctx *ctx, int len,
__isl_keep isl_basic_map **list)
{
int i;
struct basic_map_sort *s = NULL;
s = basic_map_sort_alloc(ctx, len);
if (!s)
return NULL;
for (i = len - 1; i >= 0; --i) {
if (s->node[i].index >= 0)
continue;
if (power_components_tarjan(s, list, i) < 0)
goto error;
}
return s;
error:
basic_map_sort_free(s);
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.
* If "project" is set, then these final coordinates are not included,
* i.e., a relation of type Z^n -> Z^n is returned.
* 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 }
*
* or
*
* { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
* d = (\sum_i k_i \delta_i) and
* x in dom R and x + d in ran R }
*
* if "project" is set.
*
* We first split the map into strongly connected components, perform
* the above on each component and then join the results in the correct
* order, at each join also taking in the union of both arguments
* to allow for paths that do not go through one of the two arguments.
*/
static __isl_give isl_map *construct_power_components(__isl_take isl_space *dim,
__isl_keep isl_map *map, int *exact, int project)
{
int i, n, c;
struct isl_map *path = NULL;
struct basic_map_sort *s = NULL;
int *orig_exact;
int local_exact;
if (!map)
goto error;
if (map->n <= 1)
return floyd_warshall(dim, map, exact, project);
s = basic_map_sort_init(map->ctx, map->n, map->p);
if (!s)
goto error;
orig_exact = exact;
if (s->check_closed && !exact)
exact = &local_exact;
c = 0;
i = 0;
n = map->n;
if (project)
path = isl_map_empty(isl_map_get_space(map));
else
path = isl_map_empty(isl_space_copy(dim));
path = anonymize(path);
while (n) {
struct isl_map *comp;
isl_map *path_comp, *path_comb;
comp = isl_map_alloc_space(isl_map_get_space(map), n, 0);
while (s->order[i] != -1) {
comp = isl_map_add_basic_map(comp,
isl_basic_map_copy(map->p[s->order[i]]));
--n;
++i;
}
path_comp = floyd_warshall(isl_space_copy(dim),
comp, exact, project);
path_comp = anonymize(path_comp);
path_comb = isl_map_apply_range(isl_map_copy(path),
isl_map_copy(path_comp));
path = isl_map_union(path, path_comp);
path = isl_map_union(path, path_comb);
isl_map_free(comp);
++i;
++c;
}
if (c > 1 && s->check_closed && !*exact) {
int closed;
closed = isl_map_is_transitively_closed(path);
if (closed < 0)
goto error;
if (!closed) {
basic_map_sort_free(s);
isl_map_free(path);
return floyd_warshall(dim, map, orig_exact, project);
}
}
basic_map_sort_free(s);
isl_space_free(dim);
return path;
error:
basic_map_sort_free(s);
isl_space_free(dim);
isl_map_free(path);
return NULL;
}
/* 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 }
* or
*
* { (x) -> (x + d) | \exists k_i >= 0, \delta_i \in \Delta_i :
* d = \sum_i k_i \delta_i and \sum_i k_i > 0 }
*
* if "project" is set.
*
* If "project" is not set, then
* we 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).
*/
static __isl_give isl_map *construct_power(__isl_keep isl_map *map,
int *exact, int project)
{
struct isl_map *app = NULL;
isl_space *dim = NULL;
unsigned d;
if (!map)
return NULL;
dim = isl_map_get_space(map);
d = isl_space_dim(dim, isl_dim_in);
dim = isl_space_add_dims(dim, isl_dim_in, 1);
dim = isl_space_add_dims(dim, isl_dim_out, 1);
app = construct_power_components(isl_space_copy(dim), map,
exact, project);
isl_space_free(dim);
return app;
}
/* Compute the positive powers of "map", or an overapproximation.
* If the result is exact, then *exact is set to 1.
*
* If project is set, then we are actually interested in the transitive
* closure, so we can use a more relaxed exactness check.
* The lengths of the paths are also projected out instead of being
* encoded as the difference between an extra pair of final coordinates.
*/
static __isl_give isl_map *map_power(__isl_take isl_map *map,
int *exact, int project)
{
struct isl_map *app = NULL;
if (exact)
*exact = 1;
if (!map)
return NULL;
isl_assert(map->ctx,
isl_map_dim(map, isl_dim_in) == isl_map_dim(map, isl_dim_out),
goto error);
app = construct_power(map, exact, project);
isl_map_free(map);
return app;
error:
isl_map_free(map);
isl_map_free(app);
return NULL;
}
/* Compute the positive powers of "map", or an overapproximation.
* The result maps the exponent to a nested copy of the corresponding power.
* If the result is exact, then *exact is set to 1.
* map_power constructs an extended relation with the path lengths
* encoded as the difference between the final coordinates.
* In the final step, this difference is equated to an extra parameter
* and made positive. The extra coordinates are subsequently projected out
* and the parameter is turned into the domain of the result.
*/
__isl_give isl_map *isl_map_power(__isl_take isl_map *map, int *exact)
{
isl_space *target_dim;
isl_space *dim;
isl_map *diff;
unsigned d;
unsigned param;
if (!map)
return NULL;
d = isl_map_dim(map, isl_dim_in);
param = isl_map_dim(map, isl_dim_param);
map = isl_map_compute_divs(map);
map = isl_map_coalesce(map);
if (isl_map_plain_is_empty(map)) {
map = isl_map_from_range(isl_map_wrap(map));
map = isl_map_add_dims(map, isl_dim_in, 1);
map = isl_map_set_dim_name(map, isl_dim_in, 0, "k");
return map;
}
target_dim = isl_map_get_space(map);
target_dim = isl_space_from_range(isl_space_wrap(target_dim));
target_dim = isl_space_add_dims(target_dim, isl_dim_in, 1);
target_dim = isl_space_set_dim_name(target_dim, isl_dim_in, 0, "k");
map = map_power(map, exact, 0);
map = isl_map_add_dims(map, isl_dim_param, 1);
dim = isl_map_get_space(map);
diff = equate_parameter_to_length(dim, param);
map = isl_map_intersect(map, diff);
map = isl_map_project_out(map, isl_dim_in, d, 1);
map = isl_map_project_out(map, isl_dim_out, d, 1);
map = isl_map_from_range(isl_map_wrap(map));
map = isl_map_move_dims(map, isl_dim_in, 0, isl_dim_param, param, 1);
map = isl_map_reset_space(map, target_dim);
return map;
}
/* Compute a relation that maps each element in the range of the input
* relation to the lengths of all paths composed of edges in the input
* relation that end up in the given range element.
* The result may be an overapproximation, in which case *exact is set to 0.
* The resulting relation is very similar to the power relation.
* The difference are that the domain has been projected out, the
* range has become the domain and the exponent is the range instead
* of a parameter.
*/
__isl_give isl_map *isl_map_reaching_path_lengths(__isl_take isl_map *map,
int *exact)
{
isl_space *dim;
isl_map *diff;
unsigned d;
unsigned param;
if (!map)
return NULL;
d = isl_map_dim(map, isl_dim_in);
param = isl_map_dim(map, isl_dim_param);
map = isl_map_compute_divs(map);
map = isl_map_coalesce(map);
if (isl_map_plain_is_empty(map)) {
if (exact)
*exact = 1;
map = isl_map_project_out(map, isl_dim_out, 0, d);
map = isl_map_add_dims(map, isl_dim_out, 1);
return map;
}
map = map_power(map, exact, 0);
map = isl_map_add_dims(map, isl_dim_param, 1);
dim = isl_map_get_space(map);
diff = equate_parameter_to_length(dim, param);
map = isl_map_intersect(map, diff);
map = isl_map_project_out(map, isl_dim_in, 0, d + 1);
map = isl_map_project_out(map, isl_dim_out, d, 1);
map = isl_map_reverse(map);
map = isl_map_move_dims(map, isl_dim_out, 0, isl_dim_param, param, 1);
return map;
}
/* Check whether equality i of bset is a pure stride constraint
* on a single dimensions, i.e., of the form
*
* v = k e
*
* with k a constant and e an existentially quantified variable.
*/
static int is_eq_stride(__isl_keep isl_basic_set *bset, int i)
{
unsigned nparam;
unsigned d;
unsigned n_div;
int pos1;
int pos2;
if (!bset)
return -1;
if (!isl_int_is_zero(bset->eq[i][0]))
return 0;
nparam = isl_basic_set_dim(bset, isl_dim_param);
d = isl_basic_set_dim(bset, isl_dim_set);
n_div = isl_basic_set_dim(bset, isl_dim_div);
if (isl_seq_first_non_zero(bset->eq[i] + 1, nparam) != -1)
return 0;
pos1 = isl_seq_first_non_zero(bset->eq[i] + 1 + nparam, d);
if (pos1 == -1)
return 0;
if (isl_seq_first_non_zero(bset->eq[i] + 1 + nparam + pos1 + 1,
d - pos1 - 1) != -1)
return 0;
pos2 = isl_seq_first_non_zero(bset->eq[i] + 1 + nparam + d, n_div);
if (pos2 == -1)
return 0;
if (isl_seq_first_non_zero(bset->eq[i] + 1 + nparam + d + pos2 + 1,
n_div - pos2 - 1) != -1)
return 0;
if (!isl_int_is_one(bset->eq[i][1 + nparam + pos1]) &&
!isl_int_is_negone(bset->eq[i][1 + nparam + pos1]))
return 0;
return 1;
}
/* Given a map, compute the smallest superset of this map that is of the form
*
* { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
*
* (where p ranges over the (non-parametric) dimensions),
* compute the transitive closure of this map, i.e.,
*
* { i -> j : exists k > 0:
* k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
*
* and intersect domain and range of this transitive closure with
* the given domain and range.
*
* If with_id is set, then try to include as much of the identity mapping
* as possible, by computing
*
* { i -> j : exists k >= 0:
* k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
*
* instead (i.e., allow k = 0).
*
* In practice, we compute the difference set
*
* delta = { j - i | i -> j in map },
*
* look for stride constraint on the individual dimensions and compute
* (constant) lower and upper bounds for each individual dimension,
* adding a constraint for each bound not equal to infinity.
*/
static __isl_give isl_map *box_closure_on_domain(__isl_take isl_map *map,
__isl_take isl_set *dom, __isl_take isl_set *ran, int with_id)
{
int i;
int k;
unsigned d;
unsigned nparam;
unsigned total;
isl_space *dim;
isl_set *delta;
isl_map *app = NULL;
isl_basic_set *aff = NULL;
isl_basic_map *bmap = NULL;
isl_vec *obj = NULL;
isl_int opt;
isl_int_init(opt);
delta = isl_map_deltas(isl_map_copy(map));
aff = isl_set_affine_hull(isl_set_copy(delta));
if (!aff)
goto error;
dim = isl_map_get_space(map);
d = isl_space_dim(dim, isl_dim_in);
nparam = isl_space_dim(dim, isl_dim_param);
total = isl_space_dim(dim, isl_dim_all);
bmap = isl_basic_map_alloc_space(dim,
aff->n_div + 1, aff->n_div, 2 * d + 1);
for (i = 0; i < aff->n_div + 1; ++i) {
k = isl_basic_map_alloc_div(bmap);
if (k < 0)
goto error;
isl_int_set_si(bmap->div[k][0], 0);
}
for (i = 0; i < aff->n_eq; ++i) {
if (!is_eq_stride(aff, i))
continue;
k = isl_basic_map_alloc_equality(bmap);
if (k < 0)
goto error;
isl_seq_clr(bmap->eq[k], 1 + nparam);
isl_seq_cpy(bmap->eq[k] + 1 + nparam + d,
aff->eq[i] + 1 + nparam, d);
isl_seq_neg(bmap->eq[k] + 1 + nparam,
aff->eq[i] + 1 + nparam, d);
isl_seq_cpy(bmap->eq[k] + 1 + nparam + 2 * d,
aff->eq[i] + 1 + nparam + d, aff->n_div);
isl_int_set_si(bmap->eq[k][1 + total + aff->n_div], 0);
}
obj = isl_vec_alloc(map->ctx, 1 + nparam + d);
if (!obj)
goto error;
isl_seq_clr(obj->el, 1 + nparam + d);
for (i = 0; i < d; ++ i) {
enum isl_lp_result res;
isl_int_set_si(obj->el[1 + nparam + i], 1);
res = isl_set_solve_lp(delta, 0, obj->el, map->ctx->one, &opt,
NULL, NULL);
if (res == isl_lp_error)
goto error;
if (res == isl_lp_ok) {
k = isl_basic_map_alloc_inequality(bmap);
if (k < 0)
goto error;
isl_seq_clr(bmap->ineq[k],
1 + nparam + 2 * d + bmap->n_div);
isl_int_set_si(bmap->ineq[k][1 + nparam + i], -1);
isl_int_set_si(bmap->ineq[k][1 + nparam + d + i], 1);
isl_int_neg(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], opt);
}
res = isl_set_solve_lp(delta, 1, obj->el, map->ctx->one, &opt,
NULL, NULL);
if (res == isl_lp_error)
goto error;
if (res == isl_lp_ok) {
k = isl_basic_map_alloc_inequality(bmap);
if (k < 0)
goto error;
isl_seq_clr(bmap->ineq[k],
1 + nparam + 2 * d + bmap->n_div);
isl_int_set_si(bmap->ineq[k][1 + nparam + i], 1);
isl_int_set_si(bmap->ineq[k][1 + nparam + d + i], -1);
isl_int_set(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], opt);
}
isl_int_set_si(obj->el[1 + nparam + i], 0);
}
k = isl_basic_map_alloc_inequality(bmap);
if (k < 0)
goto error;
isl_seq_clr(bmap->ineq[k],
1 + nparam + 2 * d + bmap->n_div);
if (!with_id)
isl_int_set_si(bmap->ineq[k][0], -1);
isl_int_set_si(bmap->ineq[k][1 + nparam + 2 * d + aff->n_div], 1);
app = isl_map_from_domain_and_range(dom, ran);
isl_vec_free(obj);
isl_basic_set_free(aff);
isl_map_free(map);
bmap = isl_basic_map_finalize(bmap);
isl_set_free(delta);
isl_int_clear(opt);
map = isl_map_from_basic_map(bmap);
map = isl_map_intersect(map, app);
return map;
error:
isl_vec_free(obj);
isl_basic_map_free(bmap);
isl_basic_set_free(aff);
isl_set_free(dom);
isl_set_free(ran);
isl_map_free(map);
isl_set_free(delta);
isl_int_clear(opt);
return NULL;
}
/* Given a map, compute the smallest superset of this map that is of the form
*
* { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
*
* (where p ranges over the (non-parametric) dimensions),
* compute the transitive closure of this map, i.e.,
*
* { i -> j : exists k > 0:
* k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
*
* and intersect domain and range of this transitive closure with
* domain and range of the original map.
*/
static __isl_give isl_map *box_closure(__isl_take isl_map *map)
{
isl_set *domain;
isl_set *range;
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);
return box_closure_on_domain(map, domain, range, 0);
}
/* Given a map, compute the smallest superset of this map that is of the form
*
* { i -> j : L <= j - i <= U and exists a_p: j_p - i_p = M_p a_p }
*
* (where p ranges over the (non-parametric) dimensions),
* compute the transitive and partially reflexive closure of this map, i.e.,
*
* { i -> j : exists k >= 0:
* k L <= j - i <= k U and exists a: j_p - i_p = M_p a_p }
*
* and intersect domain and range of this transitive closure with
* the given domain.
*/
static __isl_give isl_map *box_closure_with_identity(__isl_take isl_map *map,
__isl_take isl_set *dom)
{
return box_closure_on_domain(map, dom, isl_set_copy(dom), 1);
}
/* Check whether app is the transitive closure of map.
* In particular, check that app is acyclic and, if so,
* check that
*
* app \subset (map \cup (map \circ app))
*/
static int check_exactness_omega(__isl_keep isl_map *map,
__isl_keep isl_map *app)
{
isl_set *delta;
int i;
int is_empty, is_exact;
unsigned d;
isl_map *test;
delta = isl_map_deltas(isl_map_copy(app));
d = isl_set_dim(delta, isl_dim_set);
for (i = 0; i < d; ++i)
delta = isl_set_fix_si(delta, isl_dim_set, i, 0);
is_empty = isl_set_is_empty(delta);
isl_set_free(delta);
if (is_empty < 0)
return -1;
if (!is_empty)
return 0;
test = isl_map_apply_range(isl_map_copy(app), isl_map_copy(map));
test = isl_map_union(test, isl_map_copy(map));
is_exact = isl_map_is_subset(app, test);
isl_map_free(test);
return is_exact;
}
/* Check if basic map M_i can be combined with all the other
* basic maps such that
*
* (\cup_j M_j)^+
*
* can be computed as
*
* M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
*
* In particular, check if we can compute a compact representation
* of
*
* M_i^* \circ M_j \circ M_i^*
*
* for each j != i.
* Let M_i^? be an extension of M_i^+ that allows paths
* of length zero, i.e., the result of box_closure(., 1).
* The criterion, as proposed by Kelly et al., is that
* id = M_i^? - M_i^+ can be represented as a basic map
* and that
*
* id \circ M_j \circ id = M_j
*
* for each j != i.
*
* If this function returns 1, then tc and qc are set to
* M_i^+ and M_i^?, respectively.
*/
static int can_be_split_off(__isl_keep isl_map *map, int i,
__isl_give isl_map **tc, __isl_give isl_map **qc)
{
isl_map *map_i, *id = NULL;
int j = -1;
isl_set *C;
*tc = NULL;
*qc = NULL;
C = isl_set_union(isl_map_domain(isl_map_copy(map)),
isl_map_range(isl_map_copy(map)));
C = isl_set_from_basic_set(isl_set_simple_hull(C));
if (!C)
goto error;
map_i = isl_map_from_basic_map(isl_basic_map_copy(map->p[i]));
*tc = box_closure(isl_map_copy(map_i));
*qc = box_closure_with_identity(map_i, C);
id = isl_map_subtract(isl_map_copy(*qc), isl_map_copy(*tc));
if (!id || !*qc)
goto error;
if (id->n != 1 || (*qc)->n != 1)
goto done;
for (j = 0; j < map->n; ++j) {
isl_map *map_j, *test;
int is_ok;
if (i == j)
continue;
map_j = isl_map_from_basic_map(
isl_basic_map_copy(map->p[j]));
test = isl_map_apply_range(isl_map_copy(id),
isl_map_copy(map_j));
test = isl_map_apply_range(test, isl_map_copy(id));
is_ok = isl_map_is_equal(test, map_j);
isl_map_free(map_j);
isl_map_free(test);
if (is_ok < 0)
goto error;
if (!is_ok)
break;
}
done:
isl_map_free(id);
if (j == map->n)
return 1;
isl_map_free(*qc);
isl_map_free(*tc);
*qc = NULL;
*tc = NULL;
return 0;
error:
isl_map_free(id);
isl_map_free(*qc);
isl_map_free(*tc);
*qc = NULL;
*tc = NULL;
return -1;
}
static __isl_give isl_map *box_closure_with_check(__isl_take isl_map *map,
int *exact)
{
isl_map *app;
app = box_closure(isl_map_copy(map));
if (exact)
*exact = check_exactness_omega(map, app);
isl_map_free(map);
return app;
}
/* Compute an overapproximation of the transitive closure of "map"
* using a variation of the algorithm from
* "Transitive Closure of Infinite Graphs and its Applications"
* by Kelly et al.
*
* We first check whether we can can split of any basic map M_i and
* compute
*
* (\cup_j M_j)^+
*
* as
*
* M_i \cup (\cup_{j \ne i} M_i^* \circ M_j \circ M_i^*)^+
*
* using a recursive call on the remaining map.
*
* If not, we simply call box_closure on the whole map.
*/
static __isl_give isl_map *transitive_closure_omega(__isl_take isl_map *map,
int *exact)
{
int i, j;
int exact_i;
isl_map *app;
if (!map)
return NULL;
if (map->n == 1)
return box_closure_with_check(map, exact);
for (i = 0; i < map->n; ++i) {
int ok;
isl_map *qc, *tc;
ok = can_be_split_off(map, i, &tc, &qc);
if (ok < 0)
goto error;
if (!ok)
continue;
app = isl_map_alloc_space(isl_map_get_space(map), map->n - 1, 0);
for (j = 0; j < map->n; ++j) {
if (j == i)
continue;
app = isl_map_add_basic_map(app,
isl_basic_map_copy(map->p[j]));
}
app = isl_map_apply_range(isl_map_copy(qc), app);
app = isl_map_apply_range(app, qc);
app = isl_map_union(tc, transitive_closure_omega(app, NULL));
exact_i = check_exactness_omega(map, app);
if (exact_i == 1) {
if (exact)
*exact = exact_i;
isl_map_free(map);
return app;
}
isl_map_free(app);
if (exact_i < 0)
goto error;
}
return box_closure_with_check(map, exact);
error:
isl_map_free(map);
return NULL;
}
/* Compute the transitive closure of "map", or an overapproximation.
* If the result is exact, then *exact is set to 1.
* Simply use map_power to compute the powers of map, but tell
* it to project out the lengths of the paths instead of equating
* the length to a parameter.
*/
__isl_give isl_map *isl_map_transitive_closure(__isl_take isl_map *map,
int *exact)
{
isl_space *target_dim;
int closed;
if (!map)
goto error;
if (map->ctx->opt->closure == ISL_CLOSURE_BOX)
return transitive_closure_omega(map, exact);
map = isl_map_compute_divs(map);
map = isl_map_coalesce(map);
closed = isl_map_is_transitively_closed(map);
if (closed < 0)
goto error;
if (closed) {
if (exact)
*exact = 1;
return map;
}
target_dim = isl_map_get_space(map);
map = map_power(map, exact, 1);
map = isl_map_reset_space(map, target_dim);
return map;
error:
isl_map_free(map);
return NULL;
}
static int inc_count(__isl_take isl_map *map, void *user)
{
int *n = user;
*n += map->n;
isl_map_free(map);
return 0;
}
static int collect_basic_map(__isl_take isl_map *map, void *user)
{
int i;
isl_basic_map ***next = user;
for (i = 0; i < map->n; ++i) {
**next = isl_basic_map_copy(map->p[i]);
if (!**next)
goto error;
(*next)++;
}
isl_map_free(map);
return 0;
error:
isl_map_free(map);
return -1;
}
/* Perform Floyd-Warshall on the given list of basic relations.
* The basic relations may live in different dimensions,
* but basic relations that get assigned to the diagonal of the
* grid have domains and ranges of the same dimension and so
* the standard algorithm can be used because the nested transitive
* closures are only applied to diagonal elements and because all
* compositions are peformed on relations with compatible domains and ranges.
*/
static __isl_give isl_union_map *union_floyd_warshall_on_list(isl_ctx *ctx,
__isl_keep isl_basic_map **list, int n, int *exact)
{
int i, j, k;
int n_group;
int *group = NULL;
isl_set **set = NULL;
isl_map ***grid = NULL;
isl_union_map *app;
group = setup_groups(ctx, list, n, &set, &n_group);
if (!group)
goto error;
grid = isl_calloc_array(ctx, isl_map **, n_group);
if (!grid)
goto error;
for (i = 0; i < n_group; ++i) {
grid[i] = isl_calloc_array(ctx, isl_map *, n_group);
if (!grid[i])
goto error;
for (j = 0; j < n_group; ++j) {
isl_space *dim1, *dim2, *dim;
dim1 = isl_space_reverse(isl_set_get_space(set[i]));
dim2 = isl_set_get_space(set[j]);
dim = isl_space_join(dim1, dim2);
grid[i][j] = isl_map_empty(dim);
}
}
for (k = 0; k < n; ++k) {
i = group[2 * k];
j = group[2 * k + 1];
grid[i][j] = isl_map_union(grid[i][j],
isl_map_from_basic_map(
isl_basic_map_copy(list[k])));
}
floyd_warshall_iterate(grid, n_group, exact);
app = isl_union_map_empty(isl_map_get_space(grid[0][0]));
for (i = 0; i < n_group; ++i) {
for (j = 0; j < n_group; ++j)
app = isl_union_map_add_map(app, grid[i][j]);
free(grid[i]);
}
free(grid);
for (i = 0; i < 2 * n; ++i)
isl_set_free(set[i]);
free(set);
free(group);
return app;
error:
if (grid)
for (i = 0; i < n_group; ++i) {
if (!grid[i])
continue;
for (j = 0; j < n_group; ++j)
isl_map_free(grid[i][j]);
free(grid[i]);
}
free(grid);
if (set) {
for (i = 0; i < 2 * n; ++i)
isl_set_free(set[i]);
free(set);
}
free(group);
return NULL;
}
/* Perform Floyd-Warshall on the given union relation.
* The implementation is very similar to that for non-unions.
* The main difference is that it is applied unconditionally.
* We first extract a list of basic maps from the union map
* and then perform the algorithm on this list.
*/
static __isl_give isl_union_map *union_floyd_warshall(
__isl_take isl_union_map *umap, int *exact)
{
int i, n;
isl_ctx *ctx;
isl_basic_map **list = NULL;
isl_basic_map **next;
isl_union_map *res;
n = 0;
if (isl_union_map_foreach_map(umap, inc_count, &n) < 0)
goto error;
ctx = isl_union_map_get_ctx(umap);
list = isl_calloc_array(ctx, isl_basic_map *, n);
if (!list)
goto error;
next = list;
if (isl_union_map_foreach_map(umap, collect_basic_map, &next) < 0)
goto error;
res = union_floyd_warshall_on_list(ctx, list, n, exact);
if (list) {
for (i = 0; i < n; ++i)
isl_basic_map_free(list[i]);
free(list);
}
isl_union_map_free(umap);
return res;
error:
if (list) {
for (i = 0; i < n; ++i)
isl_basic_map_free(list[i]);
free(list);
}
isl_union_map_free(umap);
return NULL;
}
/* Decompose the give union relation into strongly connected components.
* The implementation is essentially the same as that of
* construct_power_components with the major difference that all
* operations are performed on union maps.
*/
static __isl_give isl_union_map *union_components(
__isl_take isl_union_map *umap, int *exact)
{
int i;
int n;
isl_ctx *ctx;
isl_basic_map **list;
isl_basic_map **next;
isl_union_map *path = NULL;
struct basic_map_sort *s = NULL;
int c, l;
int recheck = 0;
n = 0;
if (isl_union_map_foreach_map(umap, inc_count, &n) < 0)
goto error;
if (n <= 1)
return union_floyd_warshall(umap, exact);
ctx = isl_union_map_get_ctx(umap);
list = isl_calloc_array(ctx, isl_basic_map *, n);
if (!list)
goto error;
next = list;
if (isl_union_map_foreach_map(umap, collect_basic_map, &next) < 0)
goto error;
s = basic_map_sort_init(ctx, n, list);
if (!s)
goto error;
c = 0;
i = 0;
l = n;
path = isl_union_map_empty(isl_union_map_get_space(umap));
while (l) {
isl_union_map *comp;
isl_union_map *path_comp, *path_comb;
comp = isl_union_map_empty(isl_union_map_get_space(umap));
while (s->order[i] != -1) {
comp = isl_union_map_add_map(comp,
isl_map_from_basic_map(
isl_basic_map_copy(list[s->order[i]])));
--l;
++i;
}
path_comp = union_floyd_warshall(comp, exact);
path_comb = isl_union_map_apply_range(isl_union_map_copy(path),
isl_union_map_copy(path_comp));
path = isl_union_map_union(path, path_comp);
path = isl_union_map_union(path, path_comb);
++i;
++c;
}
if (c > 1 && s->check_closed && !*exact) {
int closed;
closed = isl_union_map_is_transitively_closed(path);
if (closed < 0)
goto error;
recheck = !closed;
}
basic_map_sort_free(s);
for (i = 0; i < n; ++i)
isl_basic_map_free(list[i]);
free(list);
if (recheck) {
isl_union_map_free(path);
return union_floyd_warshall(umap, exact);
}
isl_union_map_free(umap);
return path;
error:
basic_map_sort_free(s);
if (list) {
for (i = 0; i < n; ++i)
isl_basic_map_free(list[i]);
free(list);
}
isl_union_map_free(umap);
isl_union_map_free(path);
return NULL;
}
/* Compute the transitive closure of "umap", or an overapproximation.
* If the result is exact, then *exact is set to 1.
*/
__isl_give isl_union_map *isl_union_map_transitive_closure(
__isl_take isl_union_map *umap, int *exact)
{
int closed;
if (!umap)
return NULL;
if (exact)
*exact = 1;
umap = isl_union_map_compute_divs(umap);
umap = isl_union_map_coalesce(umap);
closed = isl_union_map_is_transitively_closed(umap);
if (closed < 0)
goto error;
if (closed)
return umap;
umap = union_components(umap, exact);
return umap;
error:
isl_union_map_free(umap);
return NULL;
}
struct isl_union_power {
isl_union_map *pow;
int *exact;
};
static int power(__isl_take isl_map *map, void *user)
{
struct isl_union_power *up = user;
map = isl_map_power(map, up->exact);
up->pow = isl_union_map_from_map(map);
return -1;
}
/* Construct a map [x] -> [x+1], with parameters prescribed by "dim".
*/
static __isl_give isl_union_map *increment(__isl_take isl_space *dim)
{
int k;
isl_basic_map *bmap;
dim = isl_space_add_dims(dim, isl_dim_in, 1);
dim = isl_space_add_dims(dim, isl_dim_out, 1);
bmap = isl_basic_map_alloc_space(dim, 0, 1, 0);
k = isl_basic_map_alloc_equality(bmap);
if (k < 0)
goto error;
isl_seq_clr(bmap->eq[k], isl_basic_map_total_dim(bmap));
isl_int_set_si(bmap->eq[k][0], 1);
isl_int_set_si(bmap->eq[k][isl_basic_map_offset(bmap, isl_dim_in)], 1);
isl_int_set_si(bmap->eq[k][isl_basic_map_offset(bmap, isl_dim_out)], -1);
return isl_union_map_from_map(isl_map_from_basic_map(bmap));
error:
isl_basic_map_free(bmap);
return NULL;
}
/* Construct a map [[x]->[y]] -> [y-x], with parameters prescribed by "dim".
*/
static __isl_give isl_union_map *deltas_map(__isl_take isl_space *dim)
{
isl_basic_map *bmap;
dim = isl_space_add_dims(dim, isl_dim_in, 1);
dim = isl_space_add_dims(dim, isl_dim_out, 1);
bmap = isl_basic_map_universe(dim);
bmap = isl_basic_map_deltas_map(bmap);
return isl_union_map_from_map(isl_map_from_basic_map(bmap));
}
/* Compute the positive powers of "map", or an overapproximation.
* The result maps the exponent to a nested copy of the corresponding power.
* If the result is exact, then *exact is set to 1.
*/
__isl_give isl_union_map *isl_union_map_power(__isl_take isl_union_map *umap,
int *exact)
{
int n;
isl_union_map *inc;
isl_union_map *dm;
if (!umap)
return NULL;
n = isl_union_map_n_map(umap);
if (n == 0)
return umap;
if (n == 1) {
struct isl_union_power up = { NULL, exact };
isl_union_map_foreach_map(umap, &power, &up);
isl_union_map_free(umap);
return up.pow;
}
inc = increment(isl_union_map_get_space(umap));
umap = isl_union_map_product(inc, umap);
umap = isl_union_map_transitive_closure(umap, exact);
umap = isl_union_map_zip(umap);
dm = deltas_map(isl_union_map_get_space(umap));
umap = isl_union_map_apply_domain(umap, dm);
return umap;
}
#undef TYPE
#define TYPE isl_map
#include "isl_power_templ.c"
#undef TYPE
#define TYPE isl_union_map
#include "isl_power_templ.c"