nexmon – Rev 1
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/*
* Copyright 2008-2009 Katholieke Universiteit Leuven
*
* Use of this software is governed by the GNU LGPLv2.1 license
*
* Written by Sven Verdoolaege, K.U.Leuven, Departement
* Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
*/
#include <isl_ctx_private.h>
#include <isl_map_private.h>
#include "isl_sample.h"
#include "isl_sample_piplib.h"
#include <isl/vec.h>
#include <isl/mat.h>
#include <isl/seq.h>
#include "isl_equalities.h"
#include "isl_tab.h"
#include "isl_basis_reduction.h"
#include <isl_factorization.h>
#include <isl_point_private.h>
#include <isl_options_private.h>
static struct isl_vec *empty_sample(struct isl_basic_set *bset)
{
struct isl_vec *vec;
vec = isl_vec_alloc(bset->ctx, 0);
isl_basic_set_free(bset);
return vec;
}
/* Construct a zero sample of the same dimension as bset.
* As a special case, if bset is zero-dimensional, this
* function creates a zero-dimensional sample point.
*/
static struct isl_vec *zero_sample(struct isl_basic_set *bset)
{
unsigned dim;
struct isl_vec *sample;
dim = isl_basic_set_total_dim(bset);
sample = isl_vec_alloc(bset->ctx, 1 + dim);
if (sample) {
isl_int_set_si(sample->el[0], 1);
isl_seq_clr(sample->el + 1, dim);
}
isl_basic_set_free(bset);
return sample;
}
static struct isl_vec *interval_sample(struct isl_basic_set *bset)
{
int i;
isl_int t;
struct isl_vec *sample;
bset = isl_basic_set_simplify(bset);
if (!bset)
return NULL;
if (isl_basic_set_plain_is_empty(bset))
return empty_sample(bset);
if (bset->n_eq == 0 && bset->n_ineq == 0)
return zero_sample(bset);
sample = isl_vec_alloc(bset->ctx, 2);
if (!sample)
goto error;
if (!bset)
return NULL;
isl_int_set_si(sample->block.data[0], 1);
if (bset->n_eq > 0) {
isl_assert(bset->ctx, bset->n_eq == 1, goto error);
isl_assert(bset->ctx, bset->n_ineq == 0, goto error);
if (isl_int_is_one(bset->eq[0][1]))
isl_int_neg(sample->el[1], bset->eq[0][0]);
else {
isl_assert(bset->ctx, isl_int_is_negone(bset->eq[0][1]),
goto error);
isl_int_set(sample->el[1], bset->eq[0][0]);
}
isl_basic_set_free(bset);
return sample;
}
isl_int_init(t);
if (isl_int_is_one(bset->ineq[0][1]))
isl_int_neg(sample->block.data[1], bset->ineq[0][0]);
else
isl_int_set(sample->block.data[1], bset->ineq[0][0]);
for (i = 1; i < bset->n_ineq; ++i) {
isl_seq_inner_product(sample->block.data,
bset->ineq[i], 2, &t);
if (isl_int_is_neg(t))
break;
}
isl_int_clear(t);
if (i < bset->n_ineq) {
isl_vec_free(sample);
return empty_sample(bset);
}
isl_basic_set_free(bset);
return sample;
error:
isl_basic_set_free(bset);
isl_vec_free(sample);
return NULL;
}
static struct isl_mat *independent_bounds(struct isl_basic_set *bset)
{
int i, j, n;
struct isl_mat *dirs = NULL;
struct isl_mat *bounds = NULL;
unsigned dim;
if (!bset)
return NULL;
dim = isl_basic_set_n_dim(bset);
bounds = isl_mat_alloc(bset->ctx, 1+dim, 1+dim);
if (!bounds)
return NULL;
isl_int_set_si(bounds->row[0][0], 1);
isl_seq_clr(bounds->row[0]+1, dim);
bounds->n_row = 1;
if (bset->n_ineq == 0)
return bounds;
dirs = isl_mat_alloc(bset->ctx, dim, dim);
if (!dirs) {
isl_mat_free(bounds);
return NULL;
}
isl_seq_cpy(dirs->row[0], bset->ineq[0]+1, dirs->n_col);
isl_seq_cpy(bounds->row[1], bset->ineq[0], bounds->n_col);
for (j = 1, n = 1; n < dim && j < bset->n_ineq; ++j) {
int pos;
isl_seq_cpy(dirs->row[n], bset->ineq[j]+1, dirs->n_col);
pos = isl_seq_first_non_zero(dirs->row[n], dirs->n_col);
if (pos < 0)
continue;
for (i = 0; i < n; ++i) {
int pos_i;
pos_i = isl_seq_first_non_zero(dirs->row[i], dirs->n_col);
if (pos_i < pos)
continue;
if (pos_i > pos)
break;
isl_seq_elim(dirs->row[n], dirs->row[i], pos,
dirs->n_col, NULL);
pos = isl_seq_first_non_zero(dirs->row[n], dirs->n_col);
if (pos < 0)
break;
}
if (pos < 0)
continue;
if (i < n) {
int k;
isl_int *t = dirs->row[n];
for (k = n; k > i; --k)
dirs->row[k] = dirs->row[k-1];
dirs->row[i] = t;
}
++n;
isl_seq_cpy(bounds->row[n], bset->ineq[j], bounds->n_col);
}
isl_mat_free(dirs);
bounds->n_row = 1+n;
return bounds;
}
static void swap_inequality(struct isl_basic_set *bset, int a, int b)
{
isl_int *t = bset->ineq[a];
bset->ineq[a] = bset->ineq[b];
bset->ineq[b] = t;
}
/* Skew into positive orthant and project out lineality space.
*
* We perform a unimodular transformation that turns a selected
* maximal set of linearly independent bounds into constraints
* on the first dimensions that impose that these first dimensions
* are non-negative. In particular, the constraint matrix is lower
* triangular with positive entries on the diagonal and negative
* entries below.
* If "bset" has a lineality space then these constraints (and therefore
* all constraints in bset) only involve the first dimensions.
* The remaining dimensions then do not appear in any constraints and
* we can select any value for them, say zero. We therefore project
* out this final dimensions and plug in the value zero later. This
* is accomplished by simply dropping the final columns of
* the unimodular transformation.
*/
static struct isl_basic_set *isl_basic_set_skew_to_positive_orthant(
struct isl_basic_set *bset, struct isl_mat **T)
{
struct isl_mat *U = NULL;
struct isl_mat *bounds = NULL;
int i, j;
unsigned old_dim, new_dim;
*T = NULL;
if (!bset)
return NULL;
isl_assert(bset->ctx, isl_basic_set_n_param(bset) == 0, goto error);
isl_assert(bset->ctx, bset->n_div == 0, goto error);
isl_assert(bset->ctx, bset->n_eq == 0, goto error);
old_dim = isl_basic_set_n_dim(bset);
/* Try to move (multiples of) unit rows up. */
for (i = 0, j = 0; i < bset->n_ineq; ++i) {
int pos = isl_seq_first_non_zero(bset->ineq[i]+1, old_dim);
if (pos < 0)
continue;
if (isl_seq_first_non_zero(bset->ineq[i]+1+pos+1,
old_dim-pos-1) >= 0)
continue;
if (i != j)
swap_inequality(bset, i, j);
++j;
}
bounds = independent_bounds(bset);
if (!bounds)
goto error;
new_dim = bounds->n_row - 1;
bounds = isl_mat_left_hermite(bounds, 1, &U, NULL);
if (!bounds)
goto error;
U = isl_mat_drop_cols(U, 1 + new_dim, old_dim - new_dim);
bset = isl_basic_set_preimage(bset, isl_mat_copy(U));
if (!bset)
goto error;
*T = U;
isl_mat_free(bounds);
return bset;
error:
isl_mat_free(bounds);
isl_mat_free(U);
isl_basic_set_free(bset);
return NULL;
}
/* Find a sample integer point, if any, in bset, which is known
* to have equalities. If bset contains no integer points, then
* return a zero-length vector.
* We simply remove the known equalities, compute a sample
* in the resulting bset, using the specified recurse function,
* and then transform the sample back to the original space.
*/
static struct isl_vec *sample_eq(struct isl_basic_set *bset,
struct isl_vec *(*recurse)(struct isl_basic_set *))
{
struct isl_mat *T;
struct isl_vec *sample;
if (!bset)
return NULL;
bset = isl_basic_set_remove_equalities(bset, &T, NULL);
sample = recurse(bset);
if (!sample || sample->size == 0)
isl_mat_free(T);
else
sample = isl_mat_vec_product(T, sample);
return sample;
}
/* Return a matrix containing the equalities of the tableau
* in constraint form. The tableau is assumed to have
* an associated bset that has been kept up-to-date.
*/
static struct isl_mat *tab_equalities(struct isl_tab *tab)
{
int i, j;
int n_eq;
struct isl_mat *eq;
struct isl_basic_set *bset;
if (!tab)
return NULL;
bset = isl_tab_peek_bset(tab);
isl_assert(tab->mat->ctx, bset, return NULL);
n_eq = tab->n_var - tab->n_col + tab->n_dead;
if (tab->empty || n_eq == 0)
return isl_mat_alloc(tab->mat->ctx, 0, tab->n_var);
if (n_eq == tab->n_var)
return isl_mat_identity(tab->mat->ctx, tab->n_var);
eq = isl_mat_alloc(tab->mat->ctx, n_eq, tab->n_var);
if (!eq)
return NULL;
for (i = 0, j = 0; i < tab->n_con; ++i) {
if (tab->con[i].is_row)
continue;
if (tab->con[i].index >= 0 && tab->con[i].index >= tab->n_dead)
continue;
if (i < bset->n_eq)
isl_seq_cpy(eq->row[j], bset->eq[i] + 1, tab->n_var);
else
isl_seq_cpy(eq->row[j],
bset->ineq[i - bset->n_eq] + 1, tab->n_var);
++j;
}
isl_assert(bset->ctx, j == n_eq, goto error);
return eq;
error:
isl_mat_free(eq);
return NULL;
}
/* Compute and return an initial basis for the bounded tableau "tab".
*
* If the tableau is either full-dimensional or zero-dimensional,
* the we simply return an identity matrix.
* Otherwise, we construct a basis whose first directions correspond
* to equalities.
*/
static struct isl_mat *initial_basis(struct isl_tab *tab)
{
int n_eq;
struct isl_mat *eq;
struct isl_mat *Q;
tab->n_unbounded = 0;
tab->n_zero = n_eq = tab->n_var - tab->n_col + tab->n_dead;
if (tab->empty || n_eq == 0 || n_eq == tab->n_var)
return isl_mat_identity(tab->mat->ctx, 1 + tab->n_var);
eq = tab_equalities(tab);
eq = isl_mat_left_hermite(eq, 0, NULL, &Q);
if (!eq)
return NULL;
isl_mat_free(eq);
Q = isl_mat_lin_to_aff(Q);
return Q;
}
/* Given a tableau representing a set, find and return
* an integer point in the set, if there is any.
*
* We perform a depth first search
* for an integer point, by scanning all possible values in the range
* attained by a basis vector, where an initial basis may have been set
* by the calling function. Otherwise an initial basis that exploits
* the equalities in the tableau is created.
* tab->n_zero is currently ignored and is clobbered by this function.
*
* The tableau is allowed to have unbounded direction, but then
* the calling function needs to set an initial basis, with the
* unbounded directions last and with tab->n_unbounded set
* to the number of unbounded directions.
* Furthermore, the calling functions needs to add shifted copies
* of all constraints involving unbounded directions to ensure
* that any feasible rational value in these directions can be rounded
* up to yield a feasible integer value.
* In particular, let B define the given basis x' = B x
* and let T be the inverse of B, i.e., X = T x'.
* Let a x + c >= 0 be a constraint of the set represented by the tableau,
* or a T x' + c >= 0 in terms of the given basis. Assume that
* the bounded directions have an integer value, then we can safely
* round up the values for the unbounded directions if we make sure
* that x' not only satisfies the original constraint, but also
* the constraint "a T x' + c + s >= 0" with s the sum of all
* negative values in the last n_unbounded entries of "a T".
* The calling function therefore needs to add the constraint
* a x + c + s >= 0. The current function then scans the first
* directions for an integer value and once those have been found,
* it can compute "T ceil(B x)" to yield an integer point in the set.
* Note that during the search, the first rows of B may be changed
* by a basis reduction, but the last n_unbounded rows of B remain
* unaltered and are also not mixed into the first rows.
*
* The search is implemented iteratively. "level" identifies the current
* basis vector. "init" is true if we want the first value at the current
* level and false if we want the next value.
*
* The initial basis is the identity matrix. If the range in some direction
* contains more than one integer value, we perform basis reduction based
* on the value of ctx->opt->gbr
* - ISL_GBR_NEVER: never perform basis reduction
* - ISL_GBR_ONCE: only perform basis reduction the first
* time such a range is encountered
* - ISL_GBR_ALWAYS: always perform basis reduction when
* such a range is encountered
*
* When ctx->opt->gbr is set to ISL_GBR_ALWAYS, then we allow the basis
* reduction computation to return early. That is, as soon as it
* finds a reasonable first direction.
*/
struct isl_vec *isl_tab_sample(struct isl_tab *tab)
{
unsigned dim;
unsigned gbr;
struct isl_ctx *ctx;
struct isl_vec *sample;
struct isl_vec *min;
struct isl_vec *max;
enum isl_lp_result res;
int level;
int init;
int reduced;
struct isl_tab_undo **snap;
if (!tab)
return NULL;
if (tab->empty)
return isl_vec_alloc(tab->mat->ctx, 0);
if (!tab->basis)
tab->basis = initial_basis(tab);
if (!tab->basis)
return NULL;
isl_assert(tab->mat->ctx, tab->basis->n_row == tab->n_var + 1,
return NULL);
isl_assert(tab->mat->ctx, tab->basis->n_col == tab->n_var + 1,
return NULL);
ctx = tab->mat->ctx;
dim = tab->n_var;
gbr = ctx->opt->gbr;
if (tab->n_unbounded == tab->n_var) {
sample = isl_tab_get_sample_value(tab);
sample = isl_mat_vec_product(isl_mat_copy(tab->basis), sample);
sample = isl_vec_ceil(sample);
sample = isl_mat_vec_inverse_product(isl_mat_copy(tab->basis),
sample);
return sample;
}
if (isl_tab_extend_cons(tab, dim + 1) < 0)
return NULL;
min = isl_vec_alloc(ctx, dim);
max = isl_vec_alloc(ctx, dim);
snap = isl_alloc_array(ctx, struct isl_tab_undo *, dim);
if (!min || !max || !snap)
goto error;
level = 0;
init = 1;
reduced = 0;
while (level >= 0) {
int empty = 0;
if (init) {
res = isl_tab_min(tab, tab->basis->row[1 + level],
ctx->one, &min->el[level], NULL, 0);
if (res == isl_lp_empty)
empty = 1;
isl_assert(ctx, res != isl_lp_unbounded, goto error);
if (res == isl_lp_error)
goto error;
if (!empty && isl_tab_sample_is_integer(tab))
break;
isl_seq_neg(tab->basis->row[1 + level] + 1,
tab->basis->row[1 + level] + 1, dim);
res = isl_tab_min(tab, tab->basis->row[1 + level],
ctx->one, &max->el[level], NULL, 0);
isl_seq_neg(tab->basis->row[1 + level] + 1,
tab->basis->row[1 + level] + 1, dim);
isl_int_neg(max->el[level], max->el[level]);
if (res == isl_lp_empty)
empty = 1;
isl_assert(ctx, res != isl_lp_unbounded, goto error);
if (res == isl_lp_error)
goto error;
if (!empty && isl_tab_sample_is_integer(tab))
break;
if (!empty && !reduced &&
ctx->opt->gbr != ISL_GBR_NEVER &&
isl_int_lt(min->el[level], max->el[level])) {
unsigned gbr_only_first;
if (ctx->opt->gbr == ISL_GBR_ONCE)
ctx->opt->gbr = ISL_GBR_NEVER;
tab->n_zero = level;
gbr_only_first = ctx->opt->gbr_only_first;
ctx->opt->gbr_only_first =
ctx->opt->gbr == ISL_GBR_ALWAYS;
tab = isl_tab_compute_reduced_basis(tab);
ctx->opt->gbr_only_first = gbr_only_first;
if (!tab || !tab->basis)
goto error;
reduced = 1;
continue;
}
reduced = 0;
snap[level] = isl_tab_snap(tab);
} else
isl_int_add_ui(min->el[level], min->el[level], 1);
if (empty || isl_int_gt(min->el[level], max->el[level])) {
level--;
init = 0;
if (level >= 0)
if (isl_tab_rollback(tab, snap[level]) < 0)
goto error;
continue;
}
isl_int_neg(tab->basis->row[1 + level][0], min->el[level]);
if (isl_tab_add_valid_eq(tab, tab->basis->row[1 + level]) < 0)
goto error;
isl_int_set_si(tab->basis->row[1 + level][0], 0);
if (level + tab->n_unbounded < dim - 1) {
++level;
init = 1;
continue;
}
break;
}
if (level >= 0) {
sample = isl_tab_get_sample_value(tab);
if (!sample)
goto error;
if (tab->n_unbounded && !isl_int_is_one(sample->el[0])) {
sample = isl_mat_vec_product(isl_mat_copy(tab->basis),
sample);
sample = isl_vec_ceil(sample);
sample = isl_mat_vec_inverse_product(
isl_mat_copy(tab->basis), sample);
}
} else
sample = isl_vec_alloc(ctx, 0);
ctx->opt->gbr = gbr;
isl_vec_free(min);
isl_vec_free(max);
free(snap);
return sample;
error:
ctx->opt->gbr = gbr;
isl_vec_free(min);
isl_vec_free(max);
free(snap);
return NULL;
}
static struct isl_vec *sample_bounded(struct isl_basic_set *bset);
/* Compute a sample point of the given basic set, based on the given,
* non-trivial factorization.
*/
static __isl_give isl_vec *factored_sample(__isl_take isl_basic_set *bset,
__isl_take isl_factorizer *f)
{
int i, n;
isl_vec *sample = NULL;
isl_ctx *ctx;
unsigned nparam;
unsigned nvar;
ctx = isl_basic_set_get_ctx(bset);
if (!ctx)
goto error;
nparam = isl_basic_set_dim(bset, isl_dim_param);
nvar = isl_basic_set_dim(bset, isl_dim_set);
sample = isl_vec_alloc(ctx, 1 + isl_basic_set_total_dim(bset));
if (!sample)
goto error;
isl_int_set_si(sample->el[0], 1);
bset = isl_morph_basic_set(isl_morph_copy(f->morph), bset);
for (i = 0, n = 0; i < f->n_group; ++i) {
isl_basic_set *bset_i;
isl_vec *sample_i;
bset_i = isl_basic_set_copy(bset);
bset_i = isl_basic_set_drop_constraints_involving(bset_i,
nparam + n + f->len[i], nvar - n - f->len[i]);
bset_i = isl_basic_set_drop_constraints_involving(bset_i,
nparam, n);
bset_i = isl_basic_set_drop(bset_i, isl_dim_set,
n + f->len[i], nvar - n - f->len[i]);
bset_i = isl_basic_set_drop(bset_i, isl_dim_set, 0, n);
sample_i = sample_bounded(bset_i);
if (!sample_i)
goto error;
if (sample_i->size == 0) {
isl_basic_set_free(bset);
isl_factorizer_free(f);
isl_vec_free(sample);
return sample_i;
}
isl_seq_cpy(sample->el + 1 + nparam + n,
sample_i->el + 1, f->len[i]);
isl_vec_free(sample_i);
n += f->len[i];
}
f->morph = isl_morph_inverse(f->morph);
sample = isl_morph_vec(isl_morph_copy(f->morph), sample);
isl_basic_set_free(bset);
isl_factorizer_free(f);
return sample;
error:
isl_basic_set_free(bset);
isl_factorizer_free(f);
isl_vec_free(sample);
return NULL;
}
/* Given a basic set that is known to be bounded, find and return
* an integer point in the basic set, if there is any.
*
* After handling some trivial cases, we construct a tableau
* and then use isl_tab_sample to find a sample, passing it
* the identity matrix as initial basis.
*/
static struct isl_vec *sample_bounded(struct isl_basic_set *bset)
{
unsigned dim;
struct isl_ctx *ctx;
struct isl_vec *sample;
struct isl_tab *tab = NULL;
isl_factorizer *f;
if (!bset)
return NULL;
if (isl_basic_set_plain_is_empty(bset))
return empty_sample(bset);
dim = isl_basic_set_total_dim(bset);
if (dim == 0)
return zero_sample(bset);
if (dim == 1)
return interval_sample(bset);
if (bset->n_eq > 0)
return sample_eq(bset, sample_bounded);
f = isl_basic_set_factorizer(bset);
if (!f)
goto error;
if (f->n_group != 0)
return factored_sample(bset, f);
isl_factorizer_free(f);
ctx = bset->ctx;
tab = isl_tab_from_basic_set(bset, 1);
if (tab && tab->empty) {
isl_tab_free(tab);
ISL_F_SET(bset, ISL_BASIC_SET_EMPTY);
sample = isl_vec_alloc(bset->ctx, 0);
isl_basic_set_free(bset);
return sample;
}
if (!ISL_F_ISSET(bset, ISL_BASIC_SET_NO_IMPLICIT))
if (isl_tab_detect_implicit_equalities(tab) < 0)
goto error;
sample = isl_tab_sample(tab);
if (!sample)
goto error;
if (sample->size > 0) {
isl_vec_free(bset->sample);
bset->sample = isl_vec_copy(sample);
}
isl_basic_set_free(bset);
isl_tab_free(tab);
return sample;
error:
isl_basic_set_free(bset);
isl_tab_free(tab);
return NULL;
}
/* Given a basic set "bset" and a value "sample" for the first coordinates
* of bset, plug in these values and drop the corresponding coordinates.
*
* We do this by computing the preimage of the transformation
*
* [ 1 0 ]
* x = [ s 0 ] x'
* [ 0 I ]
*
* where [1 s] is the sample value and I is the identity matrix of the
* appropriate dimension.
*/
static struct isl_basic_set *plug_in(struct isl_basic_set *bset,
struct isl_vec *sample)
{
int i;
unsigned total;
struct isl_mat *T;
if (!bset || !sample)
goto error;
total = isl_basic_set_total_dim(bset);
T = isl_mat_alloc(bset->ctx, 1 + total, 1 + total - (sample->size - 1));
if (!T)
goto error;
for (i = 0; i < sample->size; ++i) {
isl_int_set(T->row[i][0], sample->el[i]);
isl_seq_clr(T->row[i] + 1, T->n_col - 1);
}
for (i = 0; i < T->n_col - 1; ++i) {
isl_seq_clr(T->row[sample->size + i], T->n_col);
isl_int_set_si(T->row[sample->size + i][1 + i], 1);
}
isl_vec_free(sample);
bset = isl_basic_set_preimage(bset, T);
return bset;
error:
isl_basic_set_free(bset);
isl_vec_free(sample);
return NULL;
}
/* Given a basic set "bset", return any (possibly non-integer) point
* in the basic set.
*/
static struct isl_vec *rational_sample(struct isl_basic_set *bset)
{
struct isl_tab *tab;
struct isl_vec *sample;
if (!bset)
return NULL;
tab = isl_tab_from_basic_set(bset, 0);
sample = isl_tab_get_sample_value(tab);
isl_tab_free(tab);
isl_basic_set_free(bset);
return sample;
}
/* Given a linear cone "cone" and a rational point "vec",
* construct a polyhedron with shifted copies of the constraints in "cone",
* i.e., a polyhedron with "cone" as its recession cone, such that each
* point x in this polyhedron is such that the unit box positioned at x
* lies entirely inside the affine cone 'vec + cone'.
* Any rational point in this polyhedron may therefore be rounded up
* to yield an integer point that lies inside said affine cone.
*
* Denote the constraints of cone by "<a_i, x> >= 0" and the rational
* point "vec" by v/d.
* Let b_i = <a_i, v>. Then the affine cone 'vec + cone' is given
* by <a_i, x> - b/d >= 0.
* The polyhedron <a_i, x> - ceil{b/d} >= 0 is a subset of this affine cone.
* We prefer this polyhedron over the actual affine cone because it doesn't
* require a scaling of the constraints.
* If each of the vertices of the unit cube positioned at x lies inside
* this polyhedron, then the whole unit cube at x lies inside the affine cone.
* We therefore impose that x' = x + \sum e_i, for any selection of unit
* vectors lies inside the polyhedron, i.e.,
*
* <a_i, x'> - ceil{b/d} = <a_i, x> + sum a_i - ceil{b/d} >= 0
*
* The most stringent of these constraints is the one that selects
* all negative a_i, so the polyhedron we are looking for has constraints
*
* <a_i, x> + sum_{a_i < 0} a_i - ceil{b/d} >= 0
*
* Note that if cone were known to have only non-negative rays
* (which can be accomplished by a unimodular transformation),
* then we would only have to check the points x' = x + e_i
* and we only have to add the smallest negative a_i (if any)
* instead of the sum of all negative a_i.
*/
static struct isl_basic_set *shift_cone(struct isl_basic_set *cone,
struct isl_vec *vec)
{
int i, j, k;
unsigned total;
struct isl_basic_set *shift = NULL;
if (!cone || !vec)
goto error;
isl_assert(cone->ctx, cone->n_eq == 0, goto error);
total = isl_basic_set_total_dim(cone);
shift = isl_basic_set_alloc_space(isl_basic_set_get_space(cone),
0, 0, cone->n_ineq);
for (i = 0; i < cone->n_ineq; ++i) {
k = isl_basic_set_alloc_inequality(shift);
if (k < 0)
goto error;
isl_seq_cpy(shift->ineq[k] + 1, cone->ineq[i] + 1, total);
isl_seq_inner_product(shift->ineq[k] + 1, vec->el + 1, total,
&shift->ineq[k][0]);
isl_int_cdiv_q(shift->ineq[k][0],
shift->ineq[k][0], vec->el[0]);
isl_int_neg(shift->ineq[k][0], shift->ineq[k][0]);
for (j = 0; j < total; ++j) {
if (isl_int_is_nonneg(shift->ineq[k][1 + j]))
continue;
isl_int_add(shift->ineq[k][0],
shift->ineq[k][0], shift->ineq[k][1 + j]);
}
}
isl_basic_set_free(cone);
isl_vec_free(vec);
return isl_basic_set_finalize(shift);
error:
isl_basic_set_free(shift);
isl_basic_set_free(cone);
isl_vec_free(vec);
return NULL;
}
/* Given a rational point vec in a (transformed) basic set,
* such that cone is the recession cone of the original basic set,
* "round up" the rational point to an integer point.
*
* We first check if the rational point just happens to be integer.
* If not, we transform the cone in the same way as the basic set,
* pick a point x in this cone shifted to the rational point such that
* the whole unit cube at x is also inside this affine cone.
* Then we simply round up the coordinates of x and return the
* resulting integer point.
*/
static struct isl_vec *round_up_in_cone(struct isl_vec *vec,
struct isl_basic_set *cone, struct isl_mat *U)
{
unsigned total;
if (!vec || !cone || !U)
goto error;
isl_assert(vec->ctx, vec->size != 0, goto error);
if (isl_int_is_one(vec->el[0])) {
isl_mat_free(U);
isl_basic_set_free(cone);
return vec;
}
total = isl_basic_set_total_dim(cone);
cone = isl_basic_set_preimage(cone, U);
cone = isl_basic_set_remove_dims(cone, isl_dim_set,
0, total - (vec->size - 1));
cone = shift_cone(cone, vec);
vec = rational_sample(cone);
vec = isl_vec_ceil(vec);
return vec;
error:
isl_mat_free(U);
isl_vec_free(vec);
isl_basic_set_free(cone);
return NULL;
}
/* Concatenate two integer vectors, i.e., two vectors with denominator
* (stored in element 0) equal to 1.
*/
static struct isl_vec *vec_concat(struct isl_vec *vec1, struct isl_vec *vec2)
{
struct isl_vec *vec;
if (!vec1 || !vec2)
goto error;
isl_assert(vec1->ctx, vec1->size > 0, goto error);
isl_assert(vec2->ctx, vec2->size > 0, goto error);
isl_assert(vec1->ctx, isl_int_is_one(vec1->el[0]), goto error);
isl_assert(vec2->ctx, isl_int_is_one(vec2->el[0]), goto error);
vec = isl_vec_alloc(vec1->ctx, vec1->size + vec2->size - 1);
if (!vec)
goto error;
isl_seq_cpy(vec->el, vec1->el, vec1->size);
isl_seq_cpy(vec->el + vec1->size, vec2->el + 1, vec2->size - 1);
isl_vec_free(vec1);
isl_vec_free(vec2);
return vec;
error:
isl_vec_free(vec1);
isl_vec_free(vec2);
return NULL;
}
/* Give a basic set "bset" with recession cone "cone", compute and
* return an integer point in bset, if any.
*
* If the recession cone is full-dimensional, then we know that
* bset contains an infinite number of integer points and it is
* fairly easy to pick one of them.
* If the recession cone is not full-dimensional, then we first
* transform bset such that the bounded directions appear as
* the first dimensions of the transformed basic set.
* We do this by using a unimodular transformation that transforms
* the equalities in the recession cone to equalities on the first
* dimensions.
*
* The transformed set is then projected onto its bounded dimensions.
* Note that to compute this projection, we can simply drop all constraints
* involving any of the unbounded dimensions since these constraints
* cannot be combined to produce a constraint on the bounded dimensions.
* To see this, assume that there is such a combination of constraints
* that produces a constraint on the bounded dimensions. This means
* that some combination of the unbounded dimensions has both an upper
* bound and a lower bound in terms of the bounded dimensions, but then
* this combination would be a bounded direction too and would have been
* transformed into a bounded dimensions.
*
* We then compute a sample value in the bounded dimensions.
* If no such value can be found, then the original set did not contain
* any integer points and we are done.
* Otherwise, we plug in the value we found in the bounded dimensions,
* project out these bounded dimensions and end up with a set with
* a full-dimensional recession cone.
* A sample point in this set is computed by "rounding up" any
* rational point in the set.
*
* The sample points in the bounded and unbounded dimensions are
* then combined into a single sample point and transformed back
* to the original space.
*/
__isl_give isl_vec *isl_basic_set_sample_with_cone(
__isl_take isl_basic_set *bset, __isl_take isl_basic_set *cone)
{
unsigned total;
unsigned cone_dim;
struct isl_mat *M, *U;
struct isl_vec *sample;
struct isl_vec *cone_sample;
struct isl_ctx *ctx;
struct isl_basic_set *bounded;
if (!bset || !cone)
goto error;
ctx = bset->ctx;
total = isl_basic_set_total_dim(cone);
cone_dim = total - cone->n_eq;
M = isl_mat_sub_alloc6(bset->ctx, cone->eq, 0, cone->n_eq, 1, total);
M = isl_mat_left_hermite(M, 0, &U, NULL);
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));
bounded = isl_basic_set_copy(bset);
bounded = isl_basic_set_drop_constraints_involving(bounded,
total - cone_dim, cone_dim);
bounded = isl_basic_set_drop_dims(bounded, total - cone_dim, cone_dim);
sample = sample_bounded(bounded);
if (!sample || sample->size == 0) {
isl_basic_set_free(bset);
isl_basic_set_free(cone);
isl_mat_free(U);
return sample;
}
bset = plug_in(bset, isl_vec_copy(sample));
cone_sample = rational_sample(bset);
cone_sample = round_up_in_cone(cone_sample, cone, isl_mat_copy(U));
sample = vec_concat(sample, cone_sample);
sample = isl_mat_vec_product(U, sample);
return sample;
error:
isl_basic_set_free(cone);
isl_basic_set_free(bset);
return NULL;
}
static void vec_sum_of_neg(struct isl_vec *v, isl_int *s)
{
int i;
isl_int_set_si(*s, 0);
for (i = 0; i < v->size; ++i)
if (isl_int_is_neg(v->el[i]))
isl_int_add(*s, *s, v->el[i]);
}
/* Given a tableau "tab", a tableau "tab_cone" that corresponds
* to the recession cone and the inverse of a new basis U = inv(B),
* with the unbounded directions in B last,
* add constraints to "tab" that ensure any rational value
* in the unbounded directions can be rounded up to an integer value.
*
* The new basis is given by x' = B x, i.e., x = U x'.
* For any rational value of the last tab->n_unbounded coordinates
* in the update tableau, the value that is obtained by rounding
* up this value should be contained in the original tableau.
* For any constraint "a x + c >= 0", we therefore need to add
* a constraint "a x + c + s >= 0", with s the sum of all negative
* entries in the last elements of "a U".
*
* Since we are not interested in the first entries of any of the "a U",
* we first drop the columns of U that correpond to bounded directions.
*/
static int tab_shift_cone(struct isl_tab *tab,
struct isl_tab *tab_cone, struct isl_mat *U)
{
int i;
isl_int v;
struct isl_basic_set *bset = NULL;
if (tab && tab->n_unbounded == 0) {
isl_mat_free(U);
return 0;
}
isl_int_init(v);
if (!tab || !tab_cone || !U)
goto error;
bset = isl_tab_peek_bset(tab_cone);
U = isl_mat_drop_cols(U, 0, tab->n_var - tab->n_unbounded);
for (i = 0; i < bset->n_ineq; ++i) {
int ok;
struct isl_vec *row = NULL;
if (isl_tab_is_equality(tab_cone, tab_cone->n_eq + i))
continue;
row = isl_vec_alloc(bset->ctx, tab_cone->n_var);
if (!row)
goto error;
isl_seq_cpy(row->el, bset->ineq[i] + 1, tab_cone->n_var);
row = isl_vec_mat_product(row, isl_mat_copy(U));
if (!row)
goto error;
vec_sum_of_neg(row, &v);
isl_vec_free(row);
if (isl_int_is_zero(v))
continue;
tab = isl_tab_extend(tab, 1);
isl_int_add(bset->ineq[i][0], bset->ineq[i][0], v);
ok = isl_tab_add_ineq(tab, bset->ineq[i]) >= 0;
isl_int_sub(bset->ineq[i][0], bset->ineq[i][0], v);
if (!ok)
goto error;
}
isl_mat_free(U);
isl_int_clear(v);
return 0;
error:
isl_mat_free(U);
isl_int_clear(v);
return -1;
}
/* Compute and return an initial basis for the possibly
* unbounded tableau "tab". "tab_cone" is a tableau
* for the corresponding recession cone.
* Additionally, add constraints to "tab" that ensure
* that any rational value for the unbounded directions
* can be rounded up to an integer value.
*
* If the tableau is bounded, i.e., if the recession cone
* is zero-dimensional, then we just use inital_basis.
* Otherwise, we construct a basis whose first directions
* correspond to equalities, followed by bounded directions,
* i.e., equalities in the recession cone.
* The remaining directions are then unbounded.
*/
int isl_tab_set_initial_basis_with_cone(struct isl_tab *tab,
struct isl_tab *tab_cone)
{
struct isl_mat *eq;
struct isl_mat *cone_eq;
struct isl_mat *U, *Q;
if (!tab || !tab_cone)
return -1;
if (tab_cone->n_col == tab_cone->n_dead) {
tab->basis = initial_basis(tab);
return tab->basis ? 0 : -1;
}
eq = tab_equalities(tab);
if (!eq)
return -1;
tab->n_zero = eq->n_row;
cone_eq = tab_equalities(tab_cone);
eq = isl_mat_concat(eq, cone_eq);
if (!eq)
return -1;
tab->n_unbounded = tab->n_var - (eq->n_row - tab->n_zero);
eq = isl_mat_left_hermite(eq, 0, &U, &Q);
if (!eq)
return -1;
isl_mat_free(eq);
tab->basis = isl_mat_lin_to_aff(Q);
if (tab_shift_cone(tab, tab_cone, U) < 0)
return -1;
if (!tab->basis)
return -1;
return 0;
}
/* Compute and return a sample point in bset using generalized basis
* reduction. We first check if the input set has a non-trivial
* recession cone. If so, we perform some extra preprocessing in
* sample_with_cone. Otherwise, we directly perform generalized basis
* reduction.
*/
static struct isl_vec *gbr_sample(struct isl_basic_set *bset)
{
unsigned dim;
struct isl_basic_set *cone;
dim = isl_basic_set_total_dim(bset);
cone = isl_basic_set_recession_cone(isl_basic_set_copy(bset));
if (!cone)
goto error;
if (cone->n_eq < dim)
return isl_basic_set_sample_with_cone(bset, cone);
isl_basic_set_free(cone);
return sample_bounded(bset);
error:
isl_basic_set_free(bset);
return NULL;
}
static struct isl_vec *pip_sample(struct isl_basic_set *bset)
{
struct isl_mat *T;
struct isl_ctx *ctx;
struct isl_vec *sample;
bset = isl_basic_set_skew_to_positive_orthant(bset, &T);
if (!bset)
return NULL;
ctx = bset->ctx;
sample = isl_pip_basic_set_sample(bset);
if (sample && sample->size != 0)
sample = isl_mat_vec_product(T, sample);
else
isl_mat_free(T);
return sample;
}
static struct isl_vec *basic_set_sample(struct isl_basic_set *bset, int bounded)
{
struct isl_ctx *ctx;
unsigned dim;
if (!bset)
return NULL;
ctx = bset->ctx;
if (isl_basic_set_plain_is_empty(bset))
return empty_sample(bset);
dim = isl_basic_set_n_dim(bset);
isl_assert(ctx, isl_basic_set_n_param(bset) == 0, goto error);
isl_assert(ctx, bset->n_div == 0, goto error);
if (bset->sample && bset->sample->size == 1 + dim) {
int contains = isl_basic_set_contains(bset, bset->sample);
if (contains < 0)
goto error;
if (contains) {
struct isl_vec *sample = isl_vec_copy(bset->sample);
isl_basic_set_free(bset);
return sample;
}
}
isl_vec_free(bset->sample);
bset->sample = NULL;
if (bset->n_eq > 0)
return sample_eq(bset, bounded ? isl_basic_set_sample_bounded
: isl_basic_set_sample_vec);
if (dim == 0)
return zero_sample(bset);
if (dim == 1)
return interval_sample(bset);
switch (bset->ctx->opt->ilp_solver) {
case ISL_ILP_PIP:
return pip_sample(bset);
case ISL_ILP_GBR:
return bounded ? sample_bounded(bset) : gbr_sample(bset);
}
isl_assert(bset->ctx, 0, );
error:
isl_basic_set_free(bset);
return NULL;
}
__isl_give isl_vec *isl_basic_set_sample_vec(__isl_take isl_basic_set *bset)
{
return basic_set_sample(bset, 0);
}
/* Compute an integer sample in "bset", where the caller guarantees
* that "bset" is bounded.
*/
struct isl_vec *isl_basic_set_sample_bounded(struct isl_basic_set *bset)
{
return basic_set_sample(bset, 1);
}
__isl_give isl_basic_set *isl_basic_set_from_vec(__isl_take isl_vec *vec)
{
int i;
int k;
struct isl_basic_set *bset = NULL;
struct isl_ctx *ctx;
unsigned dim;
if (!vec)
return NULL;
ctx = vec->ctx;
isl_assert(ctx, vec->size != 0, goto error);
bset = isl_basic_set_alloc(ctx, 0, vec->size - 1, 0, vec->size - 1, 0);
if (!bset)
goto error;
dim = isl_basic_set_n_dim(bset);
for (i = dim - 1; i >= 0; --i) {
k = isl_basic_set_alloc_equality(bset);
if (k < 0)
goto error;
isl_seq_clr(bset->eq[k], 1 + dim);
isl_int_neg(bset->eq[k][0], vec->el[1 + i]);
isl_int_set(bset->eq[k][1 + i], vec->el[0]);
}
bset->sample = vec;
return bset;
error:
isl_basic_set_free(bset);
isl_vec_free(vec);
return NULL;
}
__isl_give isl_basic_map *isl_basic_map_sample(__isl_take isl_basic_map *bmap)
{
struct isl_basic_set *bset;
struct isl_vec *sample_vec;
bset = isl_basic_map_underlying_set(isl_basic_map_copy(bmap));
sample_vec = isl_basic_set_sample_vec(bset);
if (!sample_vec)
goto error;
if (sample_vec->size == 0) {
struct isl_basic_map *sample;
sample = isl_basic_map_empty_like(bmap);
isl_vec_free(sample_vec);
isl_basic_map_free(bmap);
return sample;
}
bset = isl_basic_set_from_vec(sample_vec);
return isl_basic_map_overlying_set(bset, bmap);
error:
isl_basic_map_free(bmap);
return NULL;
}
__isl_give isl_basic_set *isl_basic_set_sample(__isl_take isl_basic_set *bset)
{
return isl_basic_map_sample(bset);
}
__isl_give isl_basic_map *isl_map_sample(__isl_take isl_map *map)
{
int i;
isl_basic_map *sample = NULL;
if (!map)
goto error;
for (i = 0; i < map->n; ++i) {
sample = isl_basic_map_sample(isl_basic_map_copy(map->p[i]));
if (!sample)
goto error;
if (!ISL_F_ISSET(sample, ISL_BASIC_MAP_EMPTY))
break;
isl_basic_map_free(sample);
}
if (i == map->n)
sample = isl_basic_map_empty_like_map(map);
isl_map_free(map);
return sample;
error:
isl_map_free(map);
return NULL;
}
__isl_give isl_basic_set *isl_set_sample(__isl_take isl_set *set)
{
return (isl_basic_set *) isl_map_sample((isl_map *)set);
}
__isl_give isl_point *isl_basic_set_sample_point(__isl_take isl_basic_set *bset)
{
isl_vec *vec;
isl_space *dim;
dim = isl_basic_set_get_space(bset);
bset = isl_basic_set_underlying_set(bset);
vec = isl_basic_set_sample_vec(bset);
return isl_point_alloc(dim, vec);
}
__isl_give isl_point *isl_set_sample_point(__isl_take isl_set *set)
{
int i;
isl_point *pnt;
if (!set)
return NULL;
for (i = 0; i < set->n; ++i) {
pnt = isl_basic_set_sample_point(isl_basic_set_copy(set->p[i]));
if (!pnt)
goto error;
if (!isl_point_is_void(pnt))
break;
isl_point_free(pnt);
}
if (i == set->n)
pnt = isl_point_void(isl_set_get_space(set));
isl_set_free(set);
return pnt;
error:
isl_set_free(set);
return NULL;
}