/*

 * 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);

	isl_basic_set_simplify(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;

}