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