doc: add a note about isl_*_intersect and parametric sets

[isl.git] / isl_bernstein.c

/*
 * Copyright 2006-2007 Universiteit Leiden
 * Copyright 2008-2009 Katholieke Universiteit Leuven
 * Copyright 2010      INRIA Saclay
 *
 * Use of this software is governed by the GNU LGPLv2.1 license
 *
 * Written by Sven Verdoolaege, Leiden Institute of Advanced Computer Science,
 * Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
 * and 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
 */

#include <isl_ctx_private.h>
#include <isl_map_private.h>
#include <isl/set.h>
#include <isl/seq.h>
#include <isl_morph.h>
#include <isl_factorization.h>
#include <isl_vertices_private.h>
#include <isl_polynomial_private.h>
#include <isl_options_private.h>
#include <isl_bernstein.h>

struct bernstein_data {
        enum isl_fold type;
        isl_qpolynomial *poly;
        int check_tight;

        isl_cell *cell;

        isl_qpolynomial_fold *fold;
        isl_qpolynomial_fold *fold_tight;
        isl_pw_qpolynomial_fold *pwf;
        isl_pw_qpolynomial_fold *pwf_tight;
};

static int vertex_is_integral(__isl_keep isl_basic_set *vertex)
{
        unsigned nvar;
        unsigned nparam;
        int i;

        nvar = isl_basic_set_dim(vertex, isl_dim_set);
        nparam = isl_basic_set_dim(vertex, isl_dim_param);
        for (i = 0; i < nvar; ++i) {
                int r = nvar - 1 - i;
                if (!isl_int_is_one(vertex->eq[r][1 + nparam + i]) &&
                    !isl_int_is_negone(vertex->eq[r][1 + nparam + i]))
                        return 0;
        }

        return 1;
}

static __isl_give isl_qpolynomial *vertex_coordinate(
        __isl_keep isl_basic_set *vertex, int i, __isl_take isl_space *dim)
{
        unsigned nvar;
        unsigned nparam;
        int r;
        isl_int denom;
        isl_qpolynomial *v;

        nvar = isl_basic_set_dim(vertex, isl_dim_set);
        nparam = isl_basic_set_dim(vertex, isl_dim_param);
        r = nvar - 1 - i;

        isl_int_init(denom);
        isl_int_set(denom, vertex->eq[r][1 + nparam + i]);
        isl_assert(vertex->ctx, !isl_int_is_zero(denom), goto error);

        if (isl_int_is_pos(denom))
                isl_seq_neg(vertex->eq[r], vertex->eq[r],
                                1 + isl_basic_set_total_dim(vertex));
        else
                isl_int_neg(denom, denom);

        v = isl_qpolynomial_from_affine(dim, vertex->eq[r], denom);
        isl_int_clear(denom);

        return v;
error:
        isl_space_free(dim);
        isl_int_clear(denom);
        return NULL;
}

/* Check whether the bound associated to the selection "k" is tight,
 * which is the case if we select exactly one vertex and if that vertex
 * is integral for all values of the parameters.
 */
static int is_tight(int *k, int n, int d, isl_cell *cell)
{
        int i;

        for (i = 0; i < n; ++i) {
                int v;
                if (k[i] != d) {
                        if (k[i])
                                return 0;
                        continue;
                }
                v = cell->ids[n - 1 - i];
                return vertex_is_integral(cell->vertices->v[v].vertex);
        }

        return 0;
}

static void add_fold(__isl_take isl_qpolynomial *b, __isl_keep isl_set *dom,
        int *k, int n, int d, struct bernstein_data *data)
{
        isl_qpolynomial_fold *fold;

        fold = isl_qpolynomial_fold_alloc(data->type, b);

        if (data->check_tight && is_tight(k, n, d, data->cell))
                data->fold_tight = isl_qpolynomial_fold_fold_on_domain(dom,
                                                        data->fold_tight, fold);
        else
                data->fold = isl_qpolynomial_fold_fold_on_domain(dom,
                                                        data->fold, fold);
}

/* Extract the coefficients of the Bernstein base polynomials and store
 * them in data->fold and data->fold_tight.
 *
 * In particular, the coefficient of each monomial
 * of multi-degree (k[0], k[1], ..., k[n-1]) is divided by the corresponding
 * multinomial coefficient d!/k[0]! k[1]! ... k[n-1]!
 *
 * c[i] contains the coefficient of the selected powers of the first i+1 vars.
 * multinom[i] contains the partial multinomial coefficient.
 */
static void extract_coefficients(isl_qpolynomial *poly,
        __isl_keep isl_set *dom, struct bernstein_data *data)
{
        int i;
        int d;
        int n;
        isl_ctx *ctx;
        isl_qpolynomial **c = NULL;
        int *k = NULL;
        int *left = NULL;
        isl_vec *multinom = NULL;

        if (!poly)
                return;

        ctx = isl_qpolynomial_get_ctx(poly);
        n = isl_qpolynomial_dim(poly, isl_dim_in);
        d = isl_qpolynomial_degree(poly);
        isl_assert(ctx, n >= 2, return);

        c = isl_calloc_array(ctx, isl_qpolynomial *, n);
        k = isl_alloc_array(ctx, int, n);
        left = isl_alloc_array(ctx, int, n);
        multinom = isl_vec_alloc(ctx, n);
        if (!c || !k || !left || !multinom)
                goto error;

        isl_int_set_si(multinom->el[0], 1);
        for (k[0] = d; k[0] >= 0; --k[0]) {
                int i = 1;
                isl_qpolynomial_free(c[0]);
                c[0] = isl_qpolynomial_coeff(poly, isl_dim_in, n - 1, k[0]);
                left[0] = d - k[0];
                k[1] = -1;
                isl_int_set(multinom->el[1], multinom->el[0]);
                while (i > 0) {
                        if (i == n - 1) {
                                int j;
                                isl_space *dim;
                                isl_qpolynomial *b;
                                isl_qpolynomial *f;
                                for (j = 2; j <= left[i - 1]; ++j)
                                        isl_int_divexact_ui(multinom->el[i],
                                                multinom->el[i], j);
                                b = isl_qpolynomial_coeff(c[i - 1], isl_dim_in,
                                        n - 1 - i, left[i - 1]);
                                b = isl_qpolynomial_project_domain_on_params(b);
                                dim = isl_qpolynomial_get_domain_space(b);
                                f = isl_qpolynomial_rat_cst_on_domain(dim, ctx->one,
                                        multinom->el[i]);
                                b = isl_qpolynomial_mul(b, f);
                                k[n - 1] = left[n - 2];
                                add_fold(b, dom, k, n, d, data);
                                --i;
                                continue;
                        }
                        if (k[i] >= left[i - 1]) {
                                --i;
                                continue;
                        }
                        ++k[i];
                        if (k[i])
                                isl_int_divexact_ui(multinom->el[i],
                                        multinom->el[i], k[i]);
                        isl_qpolynomial_free(c[i]);
                        c[i] = isl_qpolynomial_coeff(c[i - 1], isl_dim_in,
                                        n - 1 - i, k[i]);
                        left[i] = left[i - 1] - k[i];
                        k[i + 1] = -1;
                        isl_int_set(multinom->el[i + 1], multinom->el[i]);
                        ++i;
                }
                isl_int_mul_ui(multinom->el[0], multinom->el[0], k[0]);
        }

        for (i = 0; i < n; ++i)
                isl_qpolynomial_free(c[i]);

        isl_vec_free(multinom);
        free(left);
        free(k);
        free(c);
        return;
error:
        isl_vec_free(multinom);
        free(left);
        free(k);
        if (c)
                for (i = 0; i < n; ++i)
                        isl_qpolynomial_free(c[i]);
        free(c);
        return;
}

/* Perform bernstein expansion on the parametric vertices that are active
 * on "cell".
 *
 * data->poly has been homogenized in the calling function.
 *
 * We plug in the barycentric coordinates for the set variables
 *
 *              \vec x = \sum_i \alpha_i v_i(\vec p)
 *
 * and the constant "1 = \sum_i \alpha_i" for the homogeneous dimension.
 * Next, we extract the coefficients of the Bernstein base polynomials.
 */
static int bernstein_coefficients_cell(__isl_take isl_cell *cell, void *user)
{
        int i, j;
        struct bernstein_data *data = (struct bernstein_data *)user;
        isl_space *dim_param;
        isl_space *dim_dst;
        isl_qpolynomial *poly = data->poly;
        unsigned nvar;
        int n_vertices;
        isl_qpolynomial **subs;
        isl_pw_qpolynomial_fold *pwf;
        isl_set *dom;
        isl_ctx *ctx;

        if (!poly)
                goto error;

        nvar = isl_qpolynomial_dim(poly, isl_dim_in) - 1;
        n_vertices = cell->n_vertices;

        ctx = isl_qpolynomial_get_ctx(poly);
        if (n_vertices > nvar + 1 && ctx->opt->bernstein_triangulate)
                return isl_cell_foreach_simplex(cell,
                                            &bernstein_coefficients_cell, user);

        subs = isl_alloc_array(ctx, isl_qpolynomial *, 1 + nvar);
        if (!subs)
                goto error;

        dim_param = isl_basic_set_get_space(cell->dom);
        dim_dst = isl_qpolynomial_get_domain_space(poly);
        dim_dst = isl_space_add_dims(dim_dst, isl_dim_set, n_vertices);

        for (i = 0; i < 1 + nvar; ++i)
                subs[i] = isl_qpolynomial_zero_on_domain(isl_space_copy(dim_dst));

        for (i = 0; i < n_vertices; ++i) {
                isl_qpolynomial *c;
                c = isl_qpolynomial_var_on_domain(isl_space_copy(dim_dst), isl_dim_set,
                                        1 + nvar + i);
                for (j = 0; j < nvar; ++j) {
                        int k = cell->ids[i];
                        isl_qpolynomial *v;
                        v = vertex_coordinate(cell->vertices->v[k].vertex, j,
                                                isl_space_copy(dim_param));
                        v = isl_qpolynomial_add_dims(v, isl_dim_in,
                                                        1 + nvar + n_vertices);
                        v = isl_qpolynomial_mul(v, isl_qpolynomial_copy(c));
                        subs[1 + j] = isl_qpolynomial_add(subs[1 + j], v);
                }
                subs[0] = isl_qpolynomial_add(subs[0], c);
        }
        isl_space_free(dim_dst);

        poly = isl_qpolynomial_copy(poly);

        poly = isl_qpolynomial_add_dims(poly, isl_dim_in, n_vertices);
        poly = isl_qpolynomial_substitute(poly, isl_dim_in, 0, 1 + nvar, subs);
        poly = isl_qpolynomial_drop_dims(poly, isl_dim_in, 0, 1 + nvar);

        data->cell = cell;
        dom = isl_set_from_basic_set(isl_basic_set_copy(cell->dom));
        data->fold = isl_qpolynomial_fold_empty(data->type, isl_space_copy(dim_param));
        data->fold_tight = isl_qpolynomial_fold_empty(data->type, dim_param);
        extract_coefficients(poly, dom, data);

        pwf = isl_pw_qpolynomial_fold_alloc(data->type, isl_set_copy(dom),
                                            data->fold);
        data->pwf = isl_pw_qpolynomial_fold_fold(data->pwf, pwf);
        pwf = isl_pw_qpolynomial_fold_alloc(data->type, dom, data->fold_tight);
        data->pwf_tight = isl_pw_qpolynomial_fold_fold(data->pwf_tight, pwf);

        isl_qpolynomial_free(poly);
        isl_cell_free(cell);
        for (i = 0; i < 1 + nvar; ++i)
                isl_qpolynomial_free(subs[i]);
        free(subs);
        return 0;
error:
        isl_cell_free(cell);
        return -1;
}

/* Base case of applying bernstein expansion.
 *
 * We compute the chamber decomposition of the parametric polytope "bset"
 * and then perform bernstein expansion on the parametric vertices
 * that are active on each chamber.
 */
static __isl_give isl_pw_qpolynomial_fold *bernstein_coefficients_base(
        __isl_take isl_basic_set *bset,
        __isl_take isl_qpolynomial *poly, struct bernstein_data *data, int *tight)
{
        unsigned nvar;
        isl_space *dim;
        isl_pw_qpolynomial_fold *pwf;
        isl_vertices *vertices;
        int covers;

        nvar = isl_basic_set_dim(bset, isl_dim_set);
        if (nvar == 0) {
                isl_set *dom;
                isl_qpolynomial_fold *fold;

                fold = isl_qpolynomial_fold_alloc(data->type, poly);
                dom = isl_set_from_basic_set(bset);
                if (tight)
                        *tight = 1;
                pwf = isl_pw_qpolynomial_fold_alloc(data->type, dom, fold);
                return isl_pw_qpolynomial_fold_project_domain_on_params(pwf);
        }

        if (isl_qpolynomial_is_zero(poly)) {
                isl_set *dom;
                isl_qpolynomial_fold *fold;
                fold = isl_qpolynomial_fold_alloc(data->type, poly);
                dom = isl_set_from_basic_set(bset);
                pwf = isl_pw_qpolynomial_fold_alloc(data->type, dom, fold);
                if (tight)
                        *tight = 1;
                return isl_pw_qpolynomial_fold_project_domain_on_params(pwf);
        }

        dim = isl_basic_set_get_space(bset);
        dim = isl_space_params(dim);
        dim = isl_space_from_domain(dim);
        dim = isl_space_add_dims(dim, isl_dim_set, 1);
        data->pwf = isl_pw_qpolynomial_fold_zero(isl_space_copy(dim), data->type);
        data->pwf_tight = isl_pw_qpolynomial_fold_zero(dim, data->type);
        data->poly = isl_qpolynomial_homogenize(isl_qpolynomial_copy(poly));
        vertices = isl_basic_set_compute_vertices(bset);
        isl_vertices_foreach_disjoint_cell(vertices,
                &bernstein_coefficients_cell, data);
        isl_vertices_free(vertices);
        isl_qpolynomial_free(data->poly);

        isl_basic_set_free(bset);
        isl_qpolynomial_free(poly);

        covers = isl_pw_qpolynomial_fold_covers(data->pwf_tight, data->pwf);
        if (covers < 0)
                goto error;

        if (tight)
                *tight = covers;

        if (covers) {
                isl_pw_qpolynomial_fold_free(data->pwf);
                return data->pwf_tight;
        }

        data->pwf = isl_pw_qpolynomial_fold_fold(data->pwf, data->pwf_tight);

        return data->pwf;
error:
        isl_pw_qpolynomial_fold_free(data->pwf_tight);
        isl_pw_qpolynomial_fold_free(data->pwf);
        return NULL;
}

/* Apply bernstein expansion recursively by working in on len[i]
 * set variables at a time, with i ranging from n_group - 1 to 0.
 */
static __isl_give isl_pw_qpolynomial_fold *bernstein_coefficients_recursive(
        __isl_take isl_pw_qpolynomial *pwqp,
        int n_group, int *len, struct bernstein_data *data, int *tight)
{
        int i;
        unsigned nparam;
        unsigned nvar;
        isl_pw_qpolynomial_fold *pwf;

        if (!pwqp)
                return NULL;

        nparam = isl_pw_qpolynomial_dim(pwqp, isl_dim_param);
        nvar = isl_pw_qpolynomial_dim(pwqp, isl_dim_in);

        pwqp = isl_pw_qpolynomial_move_dims(pwqp, isl_dim_param, nparam,
                                        isl_dim_in, 0, nvar - len[n_group - 1]);
        pwf = isl_pw_qpolynomial_bound(pwqp, data->type, tight);

        for (i = n_group - 2; i >= 0; --i) {
                nparam = isl_pw_qpolynomial_fold_dim(pwf, isl_dim_param);
                pwf = isl_pw_qpolynomial_fold_move_dims(pwf, isl_dim_in, 0,
                                isl_dim_param, nparam - len[i], len[i]);
                if (tight && !*tight)
                        tight = NULL;
                pwf = isl_pw_qpolynomial_fold_bound(pwf, tight);
        }

        return pwf;
}

static __isl_give isl_pw_qpolynomial_fold *bernstein_coefficients_factors(
        __isl_take isl_basic_set *bset,
        __isl_take isl_qpolynomial *poly, struct bernstein_data *data, int *tight)
{
        isl_factorizer *f;
        isl_set *set;
        isl_pw_qpolynomial *pwqp;
        isl_pw_qpolynomial_fold *pwf;

        f = isl_basic_set_factorizer(bset);
        if (!f)
                goto error;
        if (f->n_group == 0) {
                isl_factorizer_free(f);
                return  bernstein_coefficients_base(bset, poly, data, tight);
        }

        set = isl_set_from_basic_set(bset);
        pwqp = isl_pw_qpolynomial_alloc(set, poly);
        pwqp = isl_pw_qpolynomial_morph_domain(pwqp, isl_morph_copy(f->morph));

        pwf = bernstein_coefficients_recursive(pwqp, f->n_group, f->len, data,
                                                tight);

        isl_factorizer_free(f);

        return pwf;
error:
        isl_basic_set_free(bset);
        isl_qpolynomial_free(poly);
        return NULL;
}

static __isl_give isl_pw_qpolynomial_fold *bernstein_coefficients_full_recursive(
        __isl_take isl_basic_set *bset,
        __isl_take isl_qpolynomial *poly, struct bernstein_data *data, int *tight)
{
        int i;
        int *len;
        unsigned nvar;
        isl_pw_qpolynomial_fold *pwf;
        isl_set *set;
        isl_pw_qpolynomial *pwqp;

        if (!bset || !poly)
                goto error;

        nvar = isl_basic_set_dim(bset, isl_dim_set);
        
        len = isl_alloc_array(bset->ctx, int, nvar);
        if (!len)
                goto error;

        for (i = 0; i < nvar; ++i)
                len[i] = 1;

        set = isl_set_from_basic_set(bset);
        pwqp = isl_pw_qpolynomial_alloc(set, poly);

        pwf = bernstein_coefficients_recursive(pwqp, nvar, len, data, tight);

        free(len);

        return pwf;
error:
        isl_basic_set_free(bset);
        isl_qpolynomial_free(poly);
        return NULL;
}

/* Compute a bound on the polynomial defined over the parametric polytope
 * using bernstein expansion and store the result
 * in bound->pwf and bound->pwf_tight.
 *
 * If bernstein_recurse is set to ISL_BERNSTEIN_FACTORS, we check if
 * the polytope can be factorized and apply bernstein expansion recursively
 * on the factors.
 * If bernstein_recurse is set to ISL_BERNSTEIN_INTERVALS, we apply
 * bernstein expansion recursively on each dimension.
 * Otherwise, we apply bernstein expansion on the entire polytope.
 */
int isl_qpolynomial_bound_on_domain_bernstein(__isl_take isl_basic_set *bset,
        __isl_take isl_qpolynomial *poly, struct isl_bound *bound)
{
        struct bernstein_data data;
        isl_pw_qpolynomial_fold *pwf;
        unsigned nvar;
        int tight = 0;
        int *tp = bound->check_tight ? &tight : NULL;

        if (!bset || !poly)
                goto error;

        data.type = bound->type;
        data.check_tight = bound->check_tight;

        nvar = isl_basic_set_dim(bset, isl_dim_set);

        if (bset->ctx->opt->bernstein_recurse & ISL_BERNSTEIN_FACTORS)
                pwf = bernstein_coefficients_factors(bset, poly, &data, tp);
        else if (nvar > 1 &&
            (bset->ctx->opt->bernstein_recurse & ISL_BERNSTEIN_INTERVALS))
                pwf = bernstein_coefficients_full_recursive(bset, poly, &data, tp);
        else
                pwf = bernstein_coefficients_base(bset, poly, &data, tp);

        if (tight)
                bound->pwf_tight = isl_pw_qpolynomial_fold_fold(bound->pwf_tight, pwf);
        else
                bound->pwf = isl_pw_qpolynomial_fold_fold(bound->pwf, pwf);

        return 0;
error:
        isl_basic_set_free(bset);
        isl_qpolynomial_free(poly);
        return -1;
}