isl_scheduler.c: non_neg_lexmin: improve documentation

[isl.git] / isl_farkas.c

/*
 * Copyright 2010      INRIA Saclay
 *
 * Use of this software is governed by the MIT 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_map_private.h>
#include <isl/set.h>
#include <isl_space_private.h>
#include <isl_seq.h>

/*
 * Let C be a cone and define
 *
 *      C' := { y | forall x in C : y x >= 0 }
 *
 * C' contains the coefficients of all linear constraints
 * that are valid for C.
 * Furthermore, C'' = C.
 *
 * If C is defined as { x | A x >= 0 }
 * then any element in C' must be a non-negative combination
 * of the rows of A, i.e., y = t A with t >= 0.  That is,
 *
 *      C' = { y | exists t >= 0 : y = t A }
 *
 * If any of the rows in A actually represents an equality, then
 * also negative combinations of this row are allowed and so the
 * non-negativity constraint on the corresponding element of t
 * can be dropped.
 *
 * A polyhedron P = { x | b + A x >= 0 } can be represented
 * in homogeneous coordinates by the cone
 * C = { [z,x] | b z + A x >= and z >= 0 }
 * The valid linear constraints on C correspond to the valid affine
 * constraints on P.
 * This is essentially Farkas' lemma.
 *
 * Since
 *                                [ 1 0 ]
 *              [ w y ] = [t_0 t] [ b A ]
 *
 * we have
 *
 *      C' = { w, y | exists t_0, t >= 0 : y = t A and w = t_0 + t b }
 * or
 *
 *      C' = { w, y | exists t >= 0 : y = t A and w - t b >= 0 }
 *
 * In practice, we introduce an extra variable (w), shifting all
 * other variables to the right, and an extra inequality
 * (w - t b >= 0) corresponding to the positivity constraint on
 * the homogeneous coordinate.
 *
 * When going back from coefficients to solutions, we immediately
 * plug in 1 for z, which corresponds to shifting all variables
 * to the left, with the leftmost ending up in the constant position.
 */

/* Add the given prefix to all named isl_dim_set dimensions in "dim".
 */
static __isl_give isl_space *isl_space_prefix(__isl_take isl_space *dim,
        const char *prefix)
{
        int i;
        isl_ctx *ctx;
        unsigned nvar;
        size_t prefix_len = strlen(prefix);

        if (!dim)
                return NULL;

        ctx = isl_space_get_ctx(dim);
        nvar = isl_space_dim(dim, isl_dim_set);

        for (i = 0; i < nvar; ++i) {
                const char *name;
                char *prefix_name;

                name = isl_space_get_dim_name(dim, isl_dim_set, i);
                if (!name)
                        continue;

                prefix_name = isl_alloc_array(ctx, char,
                                              prefix_len + strlen(name) + 1);
                if (!prefix_name)
                        goto error;
                memcpy(prefix_name, prefix, prefix_len);
                strcpy(prefix_name + prefix_len, name);

                dim = isl_space_set_dim_name(dim, isl_dim_set, i, prefix_name);
                free(prefix_name);
        }

        return dim;
error:
        isl_space_free(dim);
        return NULL;
}

/* Given a dimension specification of the solutions space, construct
 * a dimension specification for the space of coefficients.
 *
 * In particular transform
 *
 *      [params] -> { S }
 *
 * to
 *
 *      { coefficients[[cst, params] -> S] }
 *
 * and prefix each dimension name with "c_".
 */
static __isl_give isl_space *isl_space_coefficients(__isl_take isl_space *dim)
{
        isl_space *dim_param;
        unsigned nvar;
        unsigned nparam;

        nvar = isl_space_dim(dim, isl_dim_set);
        nparam = isl_space_dim(dim, isl_dim_param);
        dim_param = isl_space_copy(dim);
        dim_param = isl_space_drop_dims(dim_param, isl_dim_set, 0, nvar);
        dim_param = isl_space_move_dims(dim_param, isl_dim_set, 0,
                                 isl_dim_param, 0, nparam);
        dim_param = isl_space_prefix(dim_param, "c_");
        dim_param = isl_space_insert_dims(dim_param, isl_dim_set, 0, 1);
        dim_param = isl_space_set_dim_name(dim_param, isl_dim_set, 0, "c_cst");
        dim = isl_space_drop_dims(dim, isl_dim_param, 0, nparam);
        dim = isl_space_prefix(dim, "c_");
        dim = isl_space_join(isl_space_from_domain(dim_param),
                           isl_space_from_range(dim));
        dim = isl_space_wrap(dim);
        dim = isl_space_set_tuple_name(dim, isl_dim_set, "coefficients");

        return dim;
}

/* Drop the given prefix from all named dimensions of type "type" in "dim".
 */
static __isl_give isl_space *isl_space_unprefix(__isl_take isl_space *dim,
        enum isl_dim_type type, const char *prefix)
{
        int i;
        unsigned n;
        size_t prefix_len = strlen(prefix);

        n = isl_space_dim(dim, type);

        for (i = 0; i < n; ++i) {
                const char *name;

                name = isl_space_get_dim_name(dim, type, i);
                if (!name)
                        continue;
                if (strncmp(name, prefix, prefix_len))
                        continue;

                dim = isl_space_set_dim_name(dim, type, i, name + prefix_len);
        }

        return dim;
}

/* Given a dimension specification of the space of coefficients, construct
 * a dimension specification for the space of solutions.
 *
 * In particular transform
 *
 *      { coefficients[[cst, params] -> S] }
 *
 * to
 *
 *      [params] -> { S }
 *
 * and drop the "c_" prefix from the dimension names.
 */
static __isl_give isl_space *isl_space_solutions(__isl_take isl_space *dim)
{
        unsigned nparam;

        dim = isl_space_unwrap(dim);
        dim = isl_space_drop_dims(dim, isl_dim_in, 0, 1);
        dim = isl_space_unprefix(dim, isl_dim_in, "c_");
        dim = isl_space_unprefix(dim, isl_dim_out, "c_");
        nparam = isl_space_dim(dim, isl_dim_in);
        dim = isl_space_move_dims(dim, isl_dim_param, 0, isl_dim_in, 0, nparam);
        dim = isl_space_range(dim);

        return dim;
}

/* Return the rational universe basic set in the given space.
 */
static __isl_give isl_basic_set *rational_universe(__isl_take isl_space *space)
{
        isl_basic_set *bset;

        bset = isl_basic_set_universe(space);
        bset = isl_basic_set_set_rational(bset);

        return bset;
}

/* Compute the dual of "bset" by applying Farkas' lemma.
 * As explained above, we add an extra dimension to represent
 * the coefficient of the constant term when going from solutions
 * to coefficients (shift == 1) and we drop the extra dimension when going
 * in the opposite direction (shift == -1).  "dim" is the space in which
 * the dual should be created.
 *
 * If "bset" is (obviously) empty, then the way this emptiness
 * is represented by the constraints does not allow for the application
 * of the standard farkas algorithm.  We therefore handle this case
 * specifically and return the universe basic set.
 */
static __isl_give isl_basic_set *farkas(__isl_take isl_space *space,
        __isl_take isl_basic_set *bset, int shift)
{
        int i, j, k;
        isl_basic_set *dual = NULL;
        unsigned total;

        if (isl_basic_set_plain_is_empty(bset)) {
                isl_basic_set_free(bset);
                return rational_universe(space);
        }

        total = isl_basic_set_total_dim(bset);

        dual = isl_basic_set_alloc_space(space, bset->n_eq + bset->n_ineq,
                                        total, bset->n_ineq + (shift > 0));
        dual = isl_basic_set_set_rational(dual);

        for (i = 0; i < bset->n_eq + bset->n_ineq; ++i) {
                k = isl_basic_set_alloc_div(dual);
                if (k < 0)
                        goto error;
                isl_int_set_si(dual->div[k][0], 0);
        }

        for (i = 0; i < total; ++i) {
                k = isl_basic_set_alloc_equality(dual);
                if (k < 0)
                        goto error;
                isl_seq_clr(dual->eq[k], 1 + shift + total);
                isl_int_set_si(dual->eq[k][1 + shift + i], -1);
                for (j = 0; j < bset->n_eq; ++j)
                        isl_int_set(dual->eq[k][1 + shift + total + j],
                                    bset->eq[j][1 + i]);
                for (j = 0; j < bset->n_ineq; ++j)
                        isl_int_set(dual->eq[k][1 + shift + total + bset->n_eq + j],
                                    bset->ineq[j][1 + i]);
        }

        for (i = 0; i < bset->n_ineq; ++i) {
                k = isl_basic_set_alloc_inequality(dual);
                if (k < 0)
                        goto error;
                isl_seq_clr(dual->ineq[k],
                            1 + shift + total + bset->n_eq + bset->n_ineq);
                isl_int_set_si(dual->ineq[k][1 + shift + total + bset->n_eq + i], 1);
        }

        if (shift > 0) {
                k = isl_basic_set_alloc_inequality(dual);
                if (k < 0)
                        goto error;
                isl_seq_clr(dual->ineq[k], 2 + total);
                isl_int_set_si(dual->ineq[k][1], 1);
                for (j = 0; j < bset->n_eq; ++j)
                        isl_int_neg(dual->ineq[k][2 + total + j],
                                    bset->eq[j][0]);
                for (j = 0; j < bset->n_ineq; ++j)
                        isl_int_neg(dual->ineq[k][2 + total + bset->n_eq + j],
                                    bset->ineq[j][0]);
        }

        dual = isl_basic_set_remove_divs(dual);
        dual = isl_basic_set_simplify(dual);
        dual = isl_basic_set_finalize(dual);

        isl_basic_set_free(bset);
        return dual;
error:
        isl_basic_set_free(bset);
        isl_basic_set_free(dual);
        return NULL;
}

/* Construct a basic set containing the tuples of coefficients of all
 * valid affine constraints on the given basic set.
 */
__isl_give isl_basic_set *isl_basic_set_coefficients(
        __isl_take isl_basic_set *bset)
{
        isl_space *dim;

        if (!bset)
                return NULL;
        if (bset->n_div)
                isl_die(bset->ctx, isl_error_invalid,
                        "input set not allowed to have local variables",
                        goto error);

        dim = isl_basic_set_get_space(bset);
        dim = isl_space_coefficients(dim);

        return farkas(dim, bset, 1);
error:
        isl_basic_set_free(bset);
        return NULL;
}

/* Construct a basic set containing the elements that satisfy all
 * affine constraints whose coefficient tuples are
 * contained in the given basic set.
 */
__isl_give isl_basic_set *isl_basic_set_solutions(
        __isl_take isl_basic_set *bset)
{
        isl_space *dim;

        if (!bset)
                return NULL;
        if (bset->n_div)
                isl_die(bset->ctx, isl_error_invalid,
                        "input set not allowed to have local variables",
                        goto error);

        dim = isl_basic_set_get_space(bset);
        dim = isl_space_solutions(dim);

        return farkas(dim, bset, -1);
error:
        isl_basic_set_free(bset);
        return NULL;
}

/* Construct a basic set containing the tuples of coefficients of all
 * valid affine constraints on the given set.
 */
__isl_give isl_basic_set *isl_set_coefficients(__isl_take isl_set *set)
{
        int i;
        isl_basic_set *coeff;

        if (!set)
                return NULL;
        if (set->n == 0) {
                isl_space *space = isl_set_get_space(set);
                space = isl_space_coefficients(space);
                isl_set_free(set);
                return rational_universe(space);
        }

        coeff = isl_basic_set_coefficients(isl_basic_set_copy(set->p[0]));

        for (i = 1; i < set->n; ++i) {
                isl_basic_set *bset, *coeff_i;
                bset = isl_basic_set_copy(set->p[i]);
                coeff_i = isl_basic_set_coefficients(bset);
                coeff = isl_basic_set_intersect(coeff, coeff_i);
        }

        isl_set_free(set);
        return coeff;
}

/* Construct a basic set containing the elements that satisfy all
 * affine constraints whose coefficient tuples are
 * contained in the given set.
 */
__isl_give isl_basic_set *isl_set_solutions(__isl_take isl_set *set)
{
        int i;
        isl_basic_set *sol;

        if (!set)
                return NULL;
        if (set->n == 0) {
                isl_space *space = isl_set_get_space(set);
                space = isl_space_solutions(space);
                isl_set_free(set);
                return rational_universe(space);
        }

        sol = isl_basic_set_solutions(isl_basic_set_copy(set->p[0]));

        for (i = 1; i < set->n; ++i) {
                isl_basic_set *bset, *sol_i;
                bset = isl_basic_set_copy(set->p[i]);
                sol_i = isl_basic_set_solutions(bset);
                sol = isl_basic_set_intersect(sol, sol_i);
        }

        isl_set_free(set);
        return sol;
}