glpk_polyhedron_affine_sign: special case 0D polyhedra

[barvinok.git] / summate.c

#include <barvinok/options.h>
#include <barvinok/util.h>
#include "bernoulli.h"
#include "euler.h"
#include "laurent.h"
#include "summate.h"
#include "section_array.h"
#include "remove_equalities.h"

extern evalue *evalue_outer_floor(evalue *e);
extern int evalue_replace_floor(evalue *e, const evalue *floor, int var);
extern void evalue_drop_floor(evalue *e, const evalue *floor);

#define ALLOC(type) (type*)malloc(sizeof(type))
#define ALLOCN(type,n) (type*)malloc((n) * sizeof(type))

/* Apply the variable transformation specified by T and CP on
 * the polynomial e.  T expresses the old variables in terms
 * of the new variables (and optionally also the new parameters),
 * while CP expresses the old parameters in terms of the new
 * parameters.
 */
static void transform_polynomial(evalue *E, Matrix *T, Matrix *CP,
                                 unsigned nvar, unsigned nparam,
                                 unsigned new_nvar, unsigned new_nparam)
{
    int j;
    evalue **subs;

    subs = ALLOCN(evalue *, nvar+nparam);

    for (j = 0; j < nvar; ++j) {
        if (T)
            subs[j] = affine2evalue(T->p[j], T->p[T->NbRows-1][T->NbColumns-1],
                                    T->NbColumns-1);
        else
            subs[j] = evalue_var(j);
    }
    for (j = 0; j < nparam; ++j) {
        if (CP)
            subs[nvar+j] = affine2evalue(CP->p[j], CP->p[nparam][new_nparam],
                                         new_nparam);
        else
            subs[nvar+j] = evalue_var(j);
        evalue_shift_variables(subs[nvar+j], 0, new_nvar);
    }

    evalue_substitute(E, subs);
    reduce_evalue(E);

    for (j = 0; j < nvar+nparam; ++j)
        evalue_free(subs[j]);
    free(subs);
}

static evalue *sum_over_polytope_with_equalities(Polyhedron *P, evalue *E,
                                 unsigned nvar,
                                 struct evalue_section_array *sections,
                                 struct barvinok_options *options)
{
    unsigned dim = P->Dimension;
    unsigned new_dim, new_nparam;
    Matrix *T = NULL, *CP = NULL;
    evalue *sum;

    if (emptyQ(P))
        return evalue_zero();

    assert(P->NbEq > 0);

    remove_all_equalities(&P, NULL, &CP, &T, dim-nvar, options->MaxRays);

    if (emptyQ(P)) {
        Polyhedron_Free(P);
        return evalue_zero();
    }

    new_nparam = CP ? CP->NbColumns-1 : dim - nvar;
    new_dim = T ? T->NbColumns-1 : nvar + new_nparam;

    /* We can avoid these substitutions if E is a constant */
    E = evalue_dup(E);
    transform_polynomial(E, T, CP, nvar, dim-nvar,
                         new_dim-new_nparam, new_nparam);

    if (new_dim-new_nparam > 0) {
        sum = barvinok_sum_over_polytope(P, E, new_dim-new_nparam,
                                         sections, options);
        evalue_free(E);
        Polyhedron_Free(P);
    } else {
        sum = ALLOC(evalue);
        value_init(sum->d);
        sum->x.p = new_enode(partition, 2, new_dim);
        EVALUE_SET_DOMAIN(sum->x.p->arr[0], P);
        value_clear(sum->x.p->arr[1].d);
        sum->x.p->arr[1] = *E;
        free(E);
    }

    if (CP) {
        evalue_backsubstitute(sum, CP, options->MaxRays);
        Matrix_Free(CP);
    }

    if (T)
        Matrix_Free(T);

    return sum;
}

static evalue *sum_base(Polyhedron *P, evalue *E, unsigned nvar,
                                     struct barvinok_options *options)
{
    if (options->summation == BV_SUM_EULER)
        return euler_summate(P, E, nvar, options);
    else if (options->summation == BV_SUM_LAURENT)
        return laurent_summate(P, E, nvar, options);
    else if (options->summation == BV_SUM_LAURENT_OLD)
        return laurent_summate_old(P, E, nvar, options);
    assert(0);
}

/* Count the number of non-zero terms in e when viewed as a polynomial
 * in only the first nvar variables.  "count" is the number counted
 * so far.
 */
static int evalue_count_terms(const evalue *e, unsigned nvar, int count)
{
    int i;

    if (EVALUE_IS_ZERO(*e))
        return count;

    if (value_zero_p(e->d))
        assert(e->x.p->type == polynomial);
    if (value_notzero_p(e->d) || e->x.p->pos >= nvar+1)
        return count+1;

    for (i = 0; i < e->x.p->size; ++i)
        count = evalue_count_terms(&e->x.p->arr[i], nvar, count);

    return count;
}

/* Create placeholder structure for unzipping.
 * A "polynomial" is created with size terms in variable pos,
 * with each term having itself as coefficient.
 */
static evalue *create_placeholder(int size, int pos)
{
    int i, j;
    evalue *E = ALLOC(evalue);
    value_init(E->d);
    E->x.p = new_enode(polynomial, size, pos+1);
    for (i = 0; i < size; ++i) {
        E->x.p->arr[i].x.p = new_enode(polynomial, i+1, pos+1);
        for (j = 0; j < i; ++j)
            evalue_set_si(&E->x.p->arr[i].x.p->arr[j], 0, 1);
        evalue_set_si(&E->x.p->arr[i].x.p->arr[i], 1, 1);
    }
    return E;
}

/* Interchange each non-zero term in e (when viewed as a polynomial
 * in only the first nvar variables) with a placeholder in ph (created
 * by create_placeholder), resulting in two polynomials in the
 * placeholder variable such that for each non-zero term in e
 * there is a power of the placeholder variable such that the factors
 * in the first nvar variables form the coefficient of that power in
 * the first polynomial (e) and the factors in the remaining variables
 * form the coefficient of that power in the second polynomial (ph).
 */
static int evalue_unzip_terms(evalue *e, evalue *ph, unsigned nvar, int count)
{
    int i;

    if (EVALUE_IS_ZERO(*e))
        return count;

    if (value_zero_p(e->d))
        assert(e->x.p->type == polynomial);
    if (value_notzero_p(e->d) || e->x.p->pos >= nvar+1) {
        evalue t = *e;
        *e = ph->x.p->arr[count];
        ph->x.p->arr[count] = t;
        return count+1;
    }

    for (i = 0; i < e->x.p->size; ++i)
        count = evalue_unzip_terms(&e->x.p->arr[i], ph, nvar, count);

    return count;
}

/* Remove n variables at pos (0-based) from the polyhedron P.
 * Each of these variables is assumed to be completely free,
 * i.e., there is a line in the polyhedron corresponding to
 * each of these variables.
 */
static Polyhedron *Polyhedron_Remove_Columns(Polyhedron *P, unsigned pos,
                                        unsigned n)
{
    int i, j;
    unsigned NbConstraints = 0;
    unsigned NbRays = 0;
    Polyhedron *Q;

    if (n == 0)
        return P;

    assert(pos <= P->Dimension);

    if (POL_HAS(P, POL_INEQUALITIES))
        NbConstraints = P->NbConstraints;
    if (POL_HAS(P, POL_POINTS))
        NbRays = P->NbRays - n;

    Q = Polyhedron_Alloc(P->Dimension - n, NbConstraints, NbRays);
    if (POL_HAS(P, POL_INEQUALITIES)) {
        Q->NbEq = P->NbEq;
        for (i = 0; i < P->NbConstraints; ++i) {
            Vector_Copy(P->Constraint[i], Q->Constraint[i], 1+pos);
            Vector_Copy(P->Constraint[i]+1+pos+n, Q->Constraint[i]+1+pos,
                        Q->Dimension-pos+1);
        }
    }
    if (POL_HAS(P, POL_POINTS)) {
        Q->NbBid = P->NbBid - n;
        for (i = 0; i < n; ++i)
            value_set_si(Q->Ray[i][1+pos+i], 1);
        for (i = 0, j = 0; i < P->NbRays; ++i) {
            int line = First_Non_Zero(P->Ray[i], 1+P->Dimension+1);
            assert(line != -1);
            if (line-1 >= pos && line-1 < pos+n) {
                ++j;
                assert(j <= n);
                continue;
            }
            assert(i-j < Q->NbRays);
            Vector_Copy(P->Ray[i], Q->Ray[i-j], 1+pos);
            Vector_Copy(P->Ray[i]+1+pos+n, Q->Ray[i-j]+1+pos,
                        Q->Dimension-pos+1);
        }
    }
    POL_SET(Q, POL_VALID);
    if (POL_HAS(P, POL_INEQUALITIES))
        POL_SET(Q, POL_INEQUALITIES);
    if (POL_HAS(P, POL_POINTS))
        POL_SET(Q, POL_POINTS);
    if (POL_HAS(P, POL_VERTICES))
        POL_SET(Q, POL_VERTICES);
    return Q;
}

/* Remove n variables at pos (0-based) from the union of polyhedra P.
 * Each of these variables is assumed to be completely free,
 * i.e., there is a line in the polyhedron corresponding to
 * each of these variables.
 */
static Polyhedron *Domain_Remove_Columns(Polyhedron *P, unsigned pos,
                                        unsigned n)
{
    Polyhedron *R;
    Polyhedron **next = &R;

    for (; P; P = P->next) {
        *next = Polyhedron_Remove_Columns(P, pos, n);
        next = &(*next)->next;
    }
    return R;
}

/* Drop n parameters starting at first from partition evalue e */
static void drop_parameters(evalue *e, int first, int n)
{
    int i;

    if (EVALUE_IS_ZERO(*e))
        return;

    assert(value_zero_p(e->d) && e->x.p->type == partition);
    for (i = 0; i < e->x.p->size/2; ++i) {
        Polyhedron *P = EVALUE_DOMAIN(e->x.p->arr[2*i]);
        Polyhedron *Q = Domain_Remove_Columns(P, first, n);
        EVALUE_SET_DOMAIN(e->x.p->arr[2*i], Q);
        Domain_Free(P);
        evalue_shift_variables(&e->x.p->arr[2*i+1], first, -n);
    }
    e->x.p->pos -= n;
}

static void extract_term_into(const evalue *src, int var, int exp, evalue *dst)
{
    int i;

    if (value_notzero_p(src->d) ||
            src->x.p->type != polynomial ||
            src->x.p->pos > var+1) {
        if (exp == 0)
            evalue_copy(dst, src);
        else
            evalue_set_si(dst, 0, 1);
        return;
    }

    if (src->x.p->pos == var+1) {
        if (src->x.p->size > exp)
            evalue_copy(dst, &src->x.p->arr[exp]);
        else
            evalue_set_si(dst, 0, 1);
        return;
    }

    dst->x.p = new_enode(polynomial, src->x.p->size, src->x.p->pos);
    for (i = 0; i < src->x.p->size; ++i)
        extract_term_into(&src->x.p->arr[i], var, exp,
                            &dst->x.p->arr[i]);
}

/* Extract the coefficient of var^exp.
 */
static evalue *extract_term(const evalue *e, int var, int exp)
{
    int i;
    evalue *res;

    if (EVALUE_IS_ZERO(*e))
        return evalue_zero();

    assert(value_zero_p(e->d) && e->x.p->type == partition);
    res = ALLOC(evalue);
    value_init(res->d);
    res->x.p = new_enode(partition, e->x.p->size, e->x.p->pos);
    for (i = 0; i < e->x.p->size/2; ++i) {
        EVALUE_SET_DOMAIN(res->x.p->arr[2*i],
                          Domain_Copy(EVALUE_DOMAIN(e->x.p->arr[2*i])));
        extract_term_into(&e->x.p->arr[2*i+1], var, exp,
                            &res->x.p->arr[2*i+1]);
        reduce_evalue(&res->x.p->arr[2*i+1]);
    }
    return res;
}

/* Insert n free variables at pos (0-based) in the polyhedron P.
 */
static Polyhedron *Polyhedron_Insert_Columns(Polyhedron *P, unsigned pos,
                                        unsigned n)
{
    int i;
    unsigned NbConstraints = 0;
    unsigned NbRays = 0;
    Polyhedron *Q;

    if (!P)
        return P;
    if (n == 0)
        return P;

    assert(pos <= P->Dimension);

    if (POL_HAS(P, POL_INEQUALITIES))
        NbConstraints = P->NbConstraints;
    if (POL_HAS(P, POL_POINTS))
        NbRays = P->NbRays + n;

    Q = Polyhedron_Alloc(P->Dimension+n, NbConstraints, NbRays);
    if (POL_HAS(P, POL_INEQUALITIES)) {
        Q->NbEq = P->NbEq;
        for (i = 0; i < P->NbConstraints; ++i) {
            Vector_Copy(P->Constraint[i], Q->Constraint[i], 1+pos);
            Vector_Copy(P->Constraint[i]+1+pos, Q->Constraint[i]+1+pos+n,
                        P->Dimension-pos+1);
        }
    }
    if (POL_HAS(P, POL_POINTS)) {
        Q->NbBid = P->NbBid + n;
        for (i = 0; i < n; ++i)
            value_set_si(Q->Ray[i][1+pos+i], 1);
        for (i = 0; i < P->NbRays; ++i) {
            Vector_Copy(P->Ray[i], Q->Ray[n+i], 1+pos);
            Vector_Copy(P->Ray[i]+1+pos, Q->Ray[n+i]+1+pos+n,
                        P->Dimension-pos+1);
        }
    }
    POL_SET(Q, POL_VALID);
    if (POL_HAS(P, POL_INEQUALITIES))
        POL_SET(Q, POL_INEQUALITIES);
    if (POL_HAS(P, POL_POINTS))
        POL_SET(Q, POL_POINTS);
    if (POL_HAS(P, POL_VERTICES))
        POL_SET(Q, POL_VERTICES);
    return Q;
}

/* Perform summation of e over a list of 1 or more factors F, with context C.
 * nvar is the total number of variables in the remaining factors.
 * extra is the number of placeholder parameters introduced in e,
 * but not (yet) in F or C.
 *
 * If there is only one factor left, F is intersected with the
 * context C, the placeholder variables are added, and then
 * e is summed over the resulting parametric polytope.
 *
 * If there is more than one factor left, we create two polynomials
 * in a new placeholder variable (which is placed after the regular
 * parameters, but before any previously introduced placeholder
 * variables) that has the factors of the variables in the first
 * factor of F and the factor of the remaining variables of
 * each term as its coefficients.
 * These two polynomials are then summed over their domains
 * and afterwards the results are combined and the placeholder
 * variable is removed again.
 */
static evalue *sum_factors(Polyhedron *F, Polyhedron *C, evalue *e,
                                     unsigned nvar, unsigned extra,
                                     struct barvinok_options *options)
{
    Polyhedron *P = F;
    unsigned nparam = C->Dimension;
    unsigned F_var = F->Dimension - C->Dimension;
    int i, n;
    evalue *s1, *s2, *s;
    evalue *ph;

    if (!F->next) {
        Polyhedron *CA = align_context(C, nvar+nparam, options->MaxRays);
        Polyhedron *P = DomainIntersection(F, CA, options->MaxRays);
        Polyhedron *Q = Polyhedron_Insert_Columns(P, nvar+nparam, extra);
        Polyhedron_Free(CA);
        Polyhedron_Free(F);
        Polyhedron_Free(P);
        evalue *sum = sum_base(Q, e, nvar, options);
        Polyhedron_Free(Q);
        return sum;
    }

    n = evalue_count_terms(e, F_var, 0);
    ph = create_placeholder(n, nvar+nparam);
    evalue_shift_variables(e, nvar+nparam, 1);
    evalue_unzip_terms(e, ph, F_var, 0);
    evalue_shift_variables(e, nvar, -(nvar-F_var));
    evalue_reorder_terms(ph);
    evalue_shift_variables(ph, 0, -F_var);

    s2 = sum_factors(F->next, C, ph, nvar-F_var, extra+1, options);
    evalue_free(ph);
    F->next = NULL;
    s1 = sum_factors(F, C, e, F_var, extra+1, options);

    if (n == 1) {
        /* remove placeholder "polynomial" */
        reduce_evalue(s1);
        emul(s1, s2);
        evalue_free(s1);
        drop_parameters(s2, nparam, 1);
        return s2;
    }

    s = evalue_zero();
    for (i = 0; i < n; ++i) {
        evalue *t1, *t2;
        t1 = extract_term(s1, nparam, i);
        t2 = extract_term(s2, nparam, i);
        emul(t1, t2);
        eadd(t2, s);
        evalue_free(t1);
        evalue_free(t2);
    }
    evalue_free(s1);
    evalue_free(s2);

    drop_parameters(s, nparam, 1);
    return s;
}

/* Perform summation over a product of factors F, obtained using
 * variable transformation T from the original problem specification.
 *
 * We first perform the corresponding transformation on the polynomial E,
 * compute the common context over all factors and then perform
 * the actual summation over the factors.
 */
static evalue *sum_product(Polyhedron *F, evalue *E, Matrix *T, unsigned nparam,
                                     struct barvinok_options *options)
{
    int i;
    Matrix *T2;
    unsigned nvar = T->NbRows;
    Polyhedron *C;
    evalue *sum;

    assert(nvar == T->NbColumns);
    T2 = Matrix_Alloc(nvar+1, nvar+1);
    for (i = 0; i < nvar; ++i)
        Vector_Copy(T->p[i], T2->p[i], nvar);
    value_set_si(T2->p[nvar][nvar], 1);

    transform_polynomial(E, T2, NULL, nvar, nparam, nvar, nparam);

    C = Factor_Context(F, nparam, options->MaxRays);
    if (F->Dimension == nparam) {
        Polyhedron *T = F;
        F = F->next;
        T->next = NULL;
        Polyhedron_Free(T);
    }
    sum = sum_factors(F, C, E, nvar, 0, options);

    Polyhedron_Free(C);
    Matrix_Free(T2);
    Matrix_Free(T);
    return sum;
}

/* Add two constraints corresponding to floor = floor(e/d),
 *
 *       e - d t       >= 0
 *      -e + d t + d-1 >= 0
 *
 * e is assumed to be an affine expression.
 */
Polyhedron *add_floor_var(Polyhedron *P, unsigned nvar, const evalue *floor,
                                     struct barvinok_options *options)
{
    int i;
    unsigned dim = P->Dimension+1;
    Matrix *M = Matrix_Alloc(P->NbConstraints+2, 2+dim);
    Polyhedron *CP;
    Value *d = &M->p[0][1+nvar];
    evalue_extract_affine(floor, M->p[0]+1, M->p[0]+1+dim, d);
    value_oppose(*d, *d);
    value_set_si(M->p[0][0], 1);
    value_set_si(M->p[1][0], 1);
    Vector_Oppose(M->p[0]+1, M->p[1]+1, M->NbColumns-1);
    value_subtract(M->p[1][1+dim], M->p[1][1+dim], *d);
    value_decrement(M->p[1][1+dim], M->p[1][1+dim]);

    for (i = 0; i < P->NbConstraints; ++i) {
        Vector_Copy(P->Constraint[i], M->p[i+2], 1+nvar);
        Vector_Copy(P->Constraint[i]+1+nvar, M->p[i+2]+1+nvar+1, dim-nvar-1+1);
    }

    CP = Constraints2Polyhedron(M, options->MaxRays);
    Matrix_Free(M);
    return CP;
}

static evalue *evalue_add(evalue *a, evalue *b)
{
    if (!a)
        return b;
    if (!b)
        return a;
    eadd(a, b);
    evalue_free(a);
    return b;
}

/* Compute sum of a step-polynomial over a polytope by grouping
 * terms containing the same floor-expressions and introducing
 * new variables for each such expression.
 * In particular, while there is any floor-expression left,
 * the step-polynomial is split into a polynomial containing
 * the expression, which is then converted to a new variable,
 * and a polynomial not containing the expression.
 */
static evalue *sum_step_polynomial(Polyhedron *P, evalue *E, unsigned nvar,
                                     struct barvinok_options *options)
{
    evalue *floor;
    evalue *cur = E;
    evalue *sum = NULL;
    evalue *t;

    while ((floor = evalue_outer_floor(cur))) {
        Polyhedron *CP;
        evalue *converted;
        evalue *converted_floor;

        /* Ignore floors that do not depend on variables. */
        if (value_notzero_p(floor->d) || floor->x.p->pos >= nvar+1)
            break;

        converted = evalue_dup(cur);
        converted_floor = evalue_dup(floor);
        evalue_shift_variables(converted, nvar, 1);
        evalue_shift_variables(converted_floor, nvar, 1);
        evalue_replace_floor(converted, converted_floor, nvar);
        CP = add_floor_var(P, nvar, converted_floor, options);
        evalue_free(converted_floor);
        t = sum_step_polynomial(CP, converted, nvar+1, options);
        evalue_free(converted);
        Polyhedron_Free(CP);
        sum = evalue_add(t, sum);

        if (cur == E)
            cur = evalue_dup(cur);
        evalue_drop_floor(cur, floor);
        evalue_free(floor);
    }
    if (floor) {
        evalue_floor2frac(cur);
        evalue_free(floor);
    }

    if (EVALUE_IS_ZERO(*cur))
        t = NULL;
    else {
        Matrix *T;
        unsigned nparam = P->Dimension - nvar;
        Polyhedron *F = Polyhedron_Factor(P, nparam, &T, options->MaxRays);
        if (!F)
            t = sum_base(P, cur, nvar, options);
        else {
            if (cur == E)
                cur = evalue_dup(cur);
            t = sum_product(F, cur, T, nparam, options);
        }
    }

    if (E != cur)
        evalue_free(cur);

    return evalue_add(t, sum);
}

evalue *barvinok_sum_over_polytope(Polyhedron *P, evalue *E, unsigned nvar,
                                     struct evalue_section_array *sections,
                                     struct barvinok_options *options)
{
    if (P->NbEq)
        return sum_over_polytope_with_equalities(P, E, nvar, sections, options);

    if (options->summation == BV_SUM_BERNOULLI)
        return bernoulli_summate(P, E, nvar, sections, options);
    else if (options->summation == BV_SUM_BOX)
        return box_summate(P, E, nvar, options->MaxRays);

    evalue_frac2floor2(E, 0);

    return sum_step_polynomial(P, E, nvar, options);
}

evalue *barvinok_summate(evalue *e, int nvar, struct barvinok_options *options)
{
    int i;
    struct evalue_section_array sections;
    evalue *sum;

    assert(nvar >= 0);
    if (nvar == 0 || EVALUE_IS_ZERO(*e))
        return evalue_dup(e);

    assert(value_zero_p(e->d));
    assert(e->x.p->type == partition);

    evalue_section_array_init(&sections);
    sum = evalue_zero();

    for (i = 0; i < e->x.p->size/2; ++i) {
        Polyhedron *D;
        for (D = EVALUE_DOMAIN(e->x.p->arr[2*i]); D; D = D->next) {
            Polyhedron *next = D->next;
            evalue *tmp;
            D->next = NULL;

            tmp = barvinok_sum_over_polytope(D, &e->x.p->arr[2*i+1], nvar,
                                             &sections, options);
            assert(tmp);
            eadd(tmp, sum);
            evalue_free(tmp);

            D->next = next;
        }
    }

    free(sections.s);

    reduce_evalue(sum);
    return sum;
}

evalue *evalue_sum(evalue *E, int nvar, unsigned MaxRays)
{
    evalue *sum;
    struct barvinok_options *options = barvinok_options_new_with_defaults();
    options->MaxRays = MaxRays;
    sum = barvinok_summate(E, nvar, options);
    barvinok_options_free(options);
    return sum;
}

evalue *esum(evalue *e, int nvar)
{
    evalue *sum;
    struct barvinok_options *options = barvinok_options_new_with_defaults();
    sum = barvinok_summate(e, nvar, options);
    barvinok_options_free(options);
    return sum;
}

/* Turn unweighted counting problem into "weighted" counting problem
 * with weight equal to 1 and call barvinok_summate on this weighted problem.
 */
evalue *barvinok_summate_unweighted(Polyhedron *P, Polyhedron *C,
                                    struct barvinok_options *options)
{
    Polyhedron *CA, *D;
    evalue e;
    evalue *sum;

    if (emptyQ(P) || emptyQ(C))
        return evalue_zero();

    CA = align_context(C, P->Dimension, options->MaxRays);
    D = DomainIntersection(P, CA, options->MaxRays);
    Domain_Free(CA);

    if (emptyQ(D)) {
        Domain_Free(D);
        return evalue_zero();
    }

    value_init(e.d);
    e.x.p = new_enode(partition, 2, P->Dimension);
    EVALUE_SET_DOMAIN(e.x.p->arr[0], D);
    evalue_set_si(&e.x.p->arr[1], 1, 1);
    sum = barvinok_summate(&e, P->Dimension - C->Dimension, options);
    free_evalue_refs(&e);
    return sum;
}