source: vis_dev/glu-2.3/src/cuBdd/cuddGenCof.c @ 86

/**CFile***********************************************************************

  FileName    [cuddGenCof.c]

  PackageName [cudd]

  Synopsis    [Generalized cofactors for BDDs and ADDs.]

  Description [External procedures included in this module:
                <ul>
                <li> Cudd_bddConstrain()
                <li> Cudd_bddRestrict()
                <li> Cudd_bddNPAnd()
                <li> Cudd_addConstrain()
                <li> Cudd_bddConstrainDecomp()
                <li> Cudd_addRestrict()
                <li> Cudd_bddCharToVect()
                <li> Cudd_bddLICompaction()
                <li> Cudd_bddSqueeze()
                <li> Cudd_SubsetCompress()
                <li> Cudd_SupersetCompress()
                </ul>
            Internal procedures included in this module:
                <ul>
                <li> cuddBddConstrainRecur()
                <li> cuddBddRestrictRecur()
                <li> cuddBddNPAndRecur()
                <li> cuddAddConstrainRecur()
                <li> cuddAddRestrictRecur()
                <li> cuddBddLICompaction()
                </ul>
            Static procedures included in this module:
                <ul>
                <li> cuddBddConstrainDecomp()
                <li> cuddBddCharToVect()
                <li> cuddBddLICMarkEdges()
                <li> cuddBddLICBuildResult()
                <li> cuddBddSqueeze()
                </ul>
                ]

  Author      [Fabio Somenzi]

  Copyright   [Copyright (c) 1995-2004, Regents of the University of Colorado

  All rights reserved.

  Redistribution and use in source and binary forms, with or without
  modification, are permitted provided that the following conditions
  are met:

  Redistributions of source code must retain the above copyright
  notice, this list of conditions and the following disclaimer.

  Redistributions in binary form must reproduce the above copyright
  notice, this list of conditions and the following disclaimer in the
  documentation and/or other materials provided with the distribution.

  Neither the name of the University of Colorado nor the names of its
  contributors may be used to endorse or promote products derived from
  this software without specific prior written permission.

  THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
  "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
  LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
  FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
  COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
  INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
  BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
  LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
  CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
  ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
  POSSIBILITY OF SUCH DAMAGE.]

******************************************************************************/

#include "util.h"
#include "cuddInt.h"


/*---------------------------------------------------------------------------*/
/* Constant declarations                                                     */
/*---------------------------------------------------------------------------*/

/* Codes for edge markings in Cudd_bddLICompaction.  The codes are defined
** so that they can be bitwise ORed to implement the code priority scheme.
*/
#define DD_LIC_DC 0
#define DD_LIC_1  1
#define DD_LIC_0  2
#define DD_LIC_NL 3

/*---------------------------------------------------------------------------*/
/* Stucture declarations                                                     */
/*---------------------------------------------------------------------------*/


/*---------------------------------------------------------------------------*/
/* Type declarations                                                         */
/*---------------------------------------------------------------------------*/

/* Key for the cache used in the edge marking phase. */
typedef struct MarkCacheKey {
    DdNode *f;
    DdNode *c;
} MarkCacheKey;

/*---------------------------------------------------------------------------*/
/* Variable declarations                                                     */
/*---------------------------------------------------------------------------*/

#ifndef lint
static char rcsid[] DD_UNUSED = "$Id: cuddGenCof.c,v 1.38 2005/05/14 17:27:11 fabio Exp $";
#endif

/*---------------------------------------------------------------------------*/
/* Macro declarations                                                        */
/*---------------------------------------------------------------------------*/

#ifdef __cplusplus
extern "C" {
#endif

/**AutomaticStart*************************************************************/

/*---------------------------------------------------------------------------*/
/* Static function prototypes                                                */
/*---------------------------------------------------------------------------*/

static int cuddBddConstrainDecomp (DdManager *dd, DdNode *f, DdNode **decomp);
static DdNode * cuddBddCharToVect (DdManager *dd, DdNode *f, DdNode *x);
static int cuddBddLICMarkEdges (DdManager *dd, DdNode *f, DdNode *c, st_table *table, st_table *cache);
static DdNode * cuddBddLICBuildResult (DdManager *dd, DdNode *f, st_table *cache, st_table *table);
static int MarkCacheHash (char *ptr, int modulus);
static int MarkCacheCompare (const char *ptr1, const char *ptr2);
static enum st_retval MarkCacheCleanUp (char *key, char *value, char *arg);
static DdNode * cuddBddSqueeze (DdManager *dd, DdNode *l, DdNode *u);

/**AutomaticEnd***************************************************************/

#ifdef __cplusplus
}
#endif

/*---------------------------------------------------------------------------*/
/* Definition of exported functions                                          */
/*---------------------------------------------------------------------------*/


/**Function********************************************************************

  Synopsis    [Computes f constrain c.]

  Description [Computes f constrain c (f @ c).
  Uses a canonical form: (f' @ c) = ( f @ c)'.  (Note: this is not true
  for c.)  List of special cases:
    <ul>
    <li> f @ 0 = 0
    <li> f @ 1 = f
    <li> 0 @ c = 0
    <li> 1 @ c = 1
    <li> f @ f = 1
    <li> f @ f'= 0
    </ul>
  Returns a pointer to the result if successful; NULL otherwise. Note that if
  F=(f1,...,fn) and reordering takes place while computing F @ c, then the
  image restriction property (Img(F,c) = Img(F @ c)) is lost.]

  SideEffects [None]

  SeeAlso     [Cudd_bddRestrict Cudd_addConstrain]

******************************************************************************/
DdNode *
Cudd_bddConstrain(
  DdManager * dd,
  DdNode * f,
  DdNode * c)
{
    DdNode *res;

    do {
        dd->reordered = 0;
        res = cuddBddConstrainRecur(dd,f,c);
    } while (dd->reordered == 1);
    return(res);

} /* end of Cudd_bddConstrain */


/**Function********************************************************************

  Synopsis [BDD restrict according to Coudert and Madre's algorithm
  (ICCAD90).]

  Description [BDD restrict according to Coudert and Madre's algorithm
  (ICCAD90). Returns the restricted BDD if successful; otherwise NULL.
  If application of restrict results in a BDD larger than the input
  BDD, the input BDD is returned.]

  SideEffects [None]

  SeeAlso     [Cudd_bddConstrain Cudd_addRestrict]

******************************************************************************/
DdNode *
Cudd_bddRestrict(
  DdManager * dd,
  DdNode * f,
  DdNode * c)
{
    DdNode *suppF, *suppC, *commonSupport;
    DdNode *cplus, *res;
    int retval;
    int sizeF, sizeRes;

    /* Check terminal cases here to avoid computing supports in trivial cases.
    ** This also allows us notto check later for the case c == 0, in which
    ** there is no common support. */
    if (c == Cudd_Not(DD_ONE(dd))) return(Cudd_Not(DD_ONE(dd)));
    if (Cudd_IsConstant(f)) return(f);
    if (f == c) return(DD_ONE(dd));
    if (f == Cudd_Not(c)) return(Cudd_Not(DD_ONE(dd)));

    /* Check if supports intersect. */
    retval = Cudd_ClassifySupport(dd,f,c,&commonSupport,&suppF,&suppC);
    if (retval == 0) {
        return(NULL);
    }
    cuddRef(commonSupport); cuddRef(suppF); cuddRef(suppC);
    Cudd_IterDerefBdd(dd,suppF);

    if (commonSupport == DD_ONE(dd)) {
        Cudd_IterDerefBdd(dd,commonSupport);
        Cudd_IterDerefBdd(dd,suppC);
        return(f);
    }
    Cudd_IterDerefBdd(dd,commonSupport);

    /* Abstract from c the variables that do not appear in f. */
    cplus = Cudd_bddExistAbstract(dd, c, suppC);
    if (cplus == NULL) {
        Cudd_IterDerefBdd(dd,suppC);
        return(NULL);
    }
    cuddRef(cplus);
    Cudd_IterDerefBdd(dd,suppC);

    do {
        dd->reordered = 0;
        res = cuddBddRestrictRecur(dd, f, cplus);
    } while (dd->reordered == 1);
    if (res == NULL) {
        Cudd_IterDerefBdd(dd,cplus);
        return(NULL);
    }
    cuddRef(res);
    Cudd_IterDerefBdd(dd,cplus);
    /* Make restric safe by returning the smaller of the input and the
    ** result. */
    sizeF = Cudd_DagSize(f);
    sizeRes = Cudd_DagSize(res);
    if (sizeF <= sizeRes) {
        Cudd_IterDerefBdd(dd, res);
        return(f);
    } else {
        cuddDeref(res);
        return(res);
    }

} /* end of Cudd_bddRestrict */


/**Function********************************************************************

  Synopsis    [Computes f non-polluting-and g.]

  Description [Computes f non-polluting-and g.  The non-polluting AND
  of f and g is a hybrid of AND and Restrict.  From Restrict, this
  operation takes the idea of existentially quantifying the top
  variable of the second operand if it does not appear in the first.
  Therefore, the variables that appear in the result also appear in f.
  For the rest, the function behaves like AND.  Since the two operands
  play different roles, non-polluting AND is not commutative.

  Returns a pointer to the result if successful; NULL otherwise.]

  SideEffects [None]

  SeeAlso     [Cudd_bddConstrain Cudd_bddRestrict]

******************************************************************************/
DdNode *
Cudd_bddNPAnd(
  DdManager * dd,
  DdNode * f,
  DdNode * g)
{
    DdNode *res;

    do {
        dd->reordered = 0;
        res = cuddBddNPAndRecur(dd,f,g);
    } while (dd->reordered == 1);
    return(res);

} /* end of Cudd_bddNPAnd */


/**Function********************************************************************

  Synopsis    [Computes f constrain c for ADDs.]

  Description [Computes f constrain c (f @ c), for f an ADD and c a 0-1
  ADD.  List of special cases:
    <ul>
    <li> F @ 0 = 0
    <li> F @ 1 = F
    <li> 0 @ c = 0
    <li> 1 @ c = 1
    <li> F @ F = 1
    </ul>
  Returns a pointer to the result if successful; NULL otherwise.]

  SideEffects [None]

  SeeAlso     [Cudd_bddConstrain]

******************************************************************************/
DdNode *
Cudd_addConstrain(
  DdManager * dd,
  DdNode * f,
  DdNode * c)
{
    DdNode *res;

    do {
        dd->reordered = 0;
        res = cuddAddConstrainRecur(dd,f,c);
    } while (dd->reordered == 1);
    return(res);

} /* end of Cudd_addConstrain */


/**Function********************************************************************

  Synopsis [BDD conjunctive decomposition as in McMillan's CAV96 paper.]

  Description [BDD conjunctive decomposition as in McMillan's CAV96
  paper.  The decomposition is canonical only for a given variable
  order. If canonicity is required, variable ordering must be disabled
  after the decomposition has been computed. Returns an array with one
  entry for each BDD variable in the manager if successful; otherwise
  NULL. The components of the solution have their reference counts
  already incremented (unlike the results of most other functions in
  the package.]

  SideEffects [None]

  SeeAlso     [Cudd_bddConstrain Cudd_bddExistAbstract]

******************************************************************************/
DdNode **
Cudd_bddConstrainDecomp(
  DdManager * dd,
  DdNode * f)
{
    DdNode **decomp;
    int res;
    int i;

    /* Create an initialize decomposition array. */
    decomp = ALLOC(DdNode *,dd->size);
    if (decomp == NULL) {
        dd->errorCode = CUDD_MEMORY_OUT;
        return(NULL);
    }
    for (i = 0; i < dd->size; i++) {
        decomp[i] = NULL;
    }
    do {
        dd->reordered = 0;
        /* Clean up the decomposition array in case reordering took place. */
        for (i = 0; i < dd->size; i++) {
            if (decomp[i] != NULL) {
                Cudd_IterDerefBdd(dd, decomp[i]);
                decomp[i] = NULL;
            }
        }
        res = cuddBddConstrainDecomp(dd,f,decomp);
    } while (dd->reordered == 1);
    if (res == 0) {
        FREE(decomp);
        return(NULL);
    }
    /* Missing components are constant ones. */
    for (i = 0; i < dd->size; i++) {
        if (decomp[i] == NULL) {
            decomp[i] = DD_ONE(dd);
            cuddRef(decomp[i]);
        }
    }
    return(decomp);

} /* end of Cudd_bddConstrainDecomp */


/**Function********************************************************************

  Synopsis [ADD restrict according to Coudert and Madre's algorithm
  (ICCAD90).]

  Description [ADD restrict according to Coudert and Madre's algorithm
  (ICCAD90). Returns the restricted ADD if successful; otherwise NULL.
  If application of restrict results in an ADD larger than the input
  ADD, the input ADD is returned.]

  SideEffects [None]

  SeeAlso     [Cudd_addConstrain Cudd_bddRestrict]

******************************************************************************/
DdNode *
Cudd_addRestrict(
  DdManager * dd,
  DdNode * f,
  DdNode * c)
{
    DdNode *supp_f, *supp_c;
    DdNode *res, *commonSupport;
    int intersection;
    int sizeF, sizeRes;

    /* Check if supports intersect. */
    supp_f = Cudd_Support(dd, f);
    if (supp_f == NULL) {
        return(NULL);
    }
    cuddRef(supp_f);
    supp_c = Cudd_Support(dd, c);
    if (supp_c == NULL) {
        Cudd_RecursiveDeref(dd,supp_f);
        return(NULL);
    }
    cuddRef(supp_c);
    commonSupport = Cudd_bddLiteralSetIntersection(dd, supp_f, supp_c);
    if (commonSupport == NULL) {
        Cudd_RecursiveDeref(dd,supp_f);
        Cudd_RecursiveDeref(dd,supp_c);
        return(NULL);
    }
    cuddRef(commonSupport);
    Cudd_RecursiveDeref(dd,supp_f);
    Cudd_RecursiveDeref(dd,supp_c);
    intersection = commonSupport != DD_ONE(dd);
    Cudd_RecursiveDeref(dd,commonSupport);

    if (intersection) {
        do {
            dd->reordered = 0;
            res = cuddAddRestrictRecur(dd, f, c);
        } while (dd->reordered == 1);
        sizeF = Cudd_DagSize(f);
        sizeRes = Cudd_DagSize(res);
        if (sizeF <= sizeRes) {
            cuddRef(res);
            Cudd_RecursiveDeref(dd, res);
            return(f);
        } else {
            return(res);
        }
    } else {
        return(f);
    }

} /* end of Cudd_addRestrict */


/**Function********************************************************************

  Synopsis    [Computes a vector whose image equals a non-zero function.]

  Description [Computes a vector of BDDs whose image equals a non-zero
  function.
  The result depends on the variable order. The i-th component of the vector
  depends only on the first i variables in the order.  Each BDD in the vector
  is not larger than the BDD of the given characteristic function.  This
  function is based on the description of char-to-vect in "Verification of
  Sequential Machines Using Boolean Functional Vectors" by O. Coudert, C.
  Berthet and J. C. Madre.
  Returns a pointer to an array containing the result if successful; NULL
  otherwise. The size of the array equals the number of variables in the
  manager. The components of the solution have their reference counts 
  already incremented (unlike the results of most other functions in 
  the package).]

  SideEffects [None]

  SeeAlso     [Cudd_bddConstrain]

******************************************************************************/
DdNode **
Cudd_bddCharToVect(
  DdManager * dd,
  DdNode * f)
{
    int i, j;
    DdNode **vect;
    DdNode *res = NULL;

    if (f == Cudd_Not(DD_ONE(dd))) return(NULL);

    vect = ALLOC(DdNode *, dd->size);
    if (vect == NULL) {
        dd->errorCode = CUDD_MEMORY_OUT;
        return(NULL);
    }

    do {
        dd->reordered = 0;
        for (i = 0; i < dd->size; i++) {
            res = cuddBddCharToVect(dd,f,dd->vars[dd->invperm[i]]);
            if (res == NULL) {
                /* Clean up the vector array in case reordering took place. */
                for (j = 0; j < i; j++) {
                    Cudd_IterDerefBdd(dd, vect[dd->invperm[j]]);
                }
                break;
            }
            cuddRef(res);
            vect[dd->invperm[i]] = res;
        }
    } while (dd->reordered == 1);
    if (res == NULL) {
        FREE(vect);
        return(NULL);
    }
    return(vect);

} /* end of Cudd_bddCharToVect */


/**Function********************************************************************

  Synopsis    [Performs safe minimization of a BDD.]

  Description [Performs safe minimization of a BDD. Given the BDD
  <code>f</code> of a function to be minimized and a BDD
  <code>c</code> representing the care set, Cudd_bddLICompaction
  produces the BDD of a function that agrees with <code>f</code>
  wherever <code>c</code> is 1.  Safe minimization means that the size
  of the result is guaranteed not to exceed the size of
  <code>f</code>. This function is based on the DAC97 paper by Hong et
  al..  Returns a pointer to the result if successful; NULL
  otherwise.]

  SideEffects [None]

  SeeAlso     [Cudd_bddRestrict]

******************************************************************************/
DdNode *
Cudd_bddLICompaction(
  DdManager * dd /* manager */,
  DdNode * f /* function to be minimized */,
  DdNode * c /* constraint (care set) */)
{
    DdNode *res;

    do {
        dd->reordered = 0;
        res = cuddBddLICompaction(dd,f,c);
    } while (dd->reordered == 1);
    return(res);

} /* end of Cudd_bddLICompaction */


/**Function********************************************************************

  Synopsis    [Finds a small BDD in a function interval.]

  Description [Finds a small BDD in a function interval. Given BDDs
  <code>l</code> and <code>u</code>, representing the lower bound and
  upper bound of a function interval, Cudd_bddSqueeze produces the BDD
  of a function within the interval with a small BDD.  Returns a
  pointer to the result if successful; NULL otherwise.]

  SideEffects [None]

  SeeAlso     [Cudd_bddRestrict Cudd_bddLICompaction]

******************************************************************************/
DdNode *
Cudd_bddSqueeze(
  DdManager * dd /* manager */,
  DdNode * l /* lower bound */,
  DdNode * u /* upper bound */)
{
    DdNode *res;
    int sizeRes, sizeL, sizeU;

    do {
        dd->reordered = 0;
        res = cuddBddSqueeze(dd,l,u);
    } while (dd->reordered == 1);
    if (res == NULL) return(NULL);
    /* We now compare the result with the bounds and return the smallest.
    ** We first compare to u, so that in case l == 0 and u == 1, we return
    ** 0 as in other minimization algorithms. */
    sizeRes = Cudd_DagSize(res);
    sizeU = Cudd_DagSize(u);
    if (sizeU <= sizeRes) {
        cuddRef(res);
        Cudd_IterDerefBdd(dd,res);
        res = u;
        sizeRes = sizeU;
    }
    sizeL = Cudd_DagSize(l);
    if (sizeL <= sizeRes) {
        cuddRef(res);
        Cudd_IterDerefBdd(dd,res);
        res = l;
        sizeRes = sizeL;
    }
    return(res);

} /* end of Cudd_bddSqueeze */


/**Function********************************************************************

  Synopsis    [Finds a small BDD that agrees with <code>f</code> over
  <code>c</code>.]

  Description [Finds a small BDD that agrees with <code>f</code> over
  <code>c</code>.  Returns a pointer to the result if successful; NULL
  otherwise.]

  SideEffects [None]

  SeeAlso     [Cudd_bddRestrict Cudd_bddLICompaction Cudd_bddSqueeze]

******************************************************************************/
DdNode *
Cudd_bddMinimize(
  DdManager * dd,
  DdNode * f,
  DdNode * c)
{
    DdNode *cplus, *res;

    if (c == Cudd_Not(DD_ONE(dd))) return(c);
    if (Cudd_IsConstant(f)) return(f);
    if (f == c) return(DD_ONE(dd));
    if (f == Cudd_Not(c)) return(Cudd_Not(DD_ONE(dd)));

    cplus = Cudd_RemapOverApprox(dd,c,0,0,1.0);
    if (cplus == NULL) return(NULL);
    cuddRef(cplus);
    res = Cudd_bddLICompaction(dd,f,cplus);
    if (res == NULL) {
        Cudd_IterDerefBdd(dd,cplus);
        return(NULL);
    }
    cuddRef(res);
    Cudd_IterDerefBdd(dd,cplus);
    cuddDeref(res);
    return(res);

} /* end of Cudd_bddMinimize */


/**Function********************************************************************

  Synopsis    [Find a dense subset of BDD <code>f</code>.]

  Description [Finds a dense subset of BDD <code>f</code>. Density is
  the ratio of number of minterms to number of nodes.  Uses several
  techniques in series. It is more expensive than other subsetting
  procedures, but often produces better results. See
  Cudd_SubsetShortPaths for a description of the threshold and nvars
  parameters.  Returns a pointer to the result if successful; NULL
  otherwise.]

  SideEffects [None]

  SeeAlso     [Cudd_SubsetRemap Cudd_SubsetShortPaths Cudd_SubsetHeavyBranch
  Cudd_bddSqueeze]

******************************************************************************/
DdNode *
Cudd_SubsetCompress(
  DdManager * dd /* manager */,
  DdNode * f /* BDD whose subset is sought */,
  int  nvars /* number of variables in the support of f */,
  int  threshold /* maximum number of nodes in the subset */)
{
    DdNode *res, *tmp1, *tmp2;

    tmp1 = Cudd_SubsetShortPaths(dd, f, nvars, threshold, 0);
    if (tmp1 == NULL) return(NULL);
    cuddRef(tmp1);
    tmp2 = Cudd_RemapUnderApprox(dd,tmp1,nvars,0,1.0);
    if (tmp2 == NULL) {
        Cudd_IterDerefBdd(dd,tmp1);
        return(NULL);
    }
    cuddRef(tmp2);
    Cudd_IterDerefBdd(dd,tmp1);
    res = Cudd_bddSqueeze(dd,tmp2,f);
    if (res == NULL) {
        Cudd_IterDerefBdd(dd,tmp2);
        return(NULL);
    }
    cuddRef(res);
    Cudd_IterDerefBdd(dd,tmp2);
    cuddDeref(res);
    return(res);

} /* end of Cudd_SubsetCompress */


/**Function********************************************************************

  Synopsis    [Find a dense superset of BDD <code>f</code>.]

  Description [Finds a dense superset of BDD <code>f</code>. Density is
  the ratio of number of minterms to number of nodes.  Uses several
  techniques in series. It is more expensive than other supersetting
  procedures, but often produces better results. See
  Cudd_SupersetShortPaths for a description of the threshold and nvars
  parameters.  Returns a pointer to the result if successful; NULL
  otherwise.]

  SideEffects [None]

  SeeAlso     [Cudd_SubsetCompress Cudd_SupersetRemap Cudd_SupersetShortPaths
  Cudd_SupersetHeavyBranch Cudd_bddSqueeze]

******************************************************************************/
DdNode *
Cudd_SupersetCompress(
  DdManager * dd /* manager */,
  DdNode * f /* BDD whose superset is sought */,
  int  nvars /* number of variables in the support of f */,
  int  threshold /* maximum number of nodes in the superset */)
{
    DdNode *subset;

    subset = Cudd_SubsetCompress(dd, Cudd_Not(f),nvars,threshold);

    return(Cudd_NotCond(subset, (subset != NULL)));

} /* end of Cudd_SupersetCompress */


/*---------------------------------------------------------------------------*/
/* Definition of internal functions                                          */
/*---------------------------------------------------------------------------*/


/**Function********************************************************************

  Synopsis    [Performs the recursive step of Cudd_bddConstrain.]

  Description [Performs the recursive step of Cudd_bddConstrain.
  Returns a pointer to the result if successful; NULL otherwise.]

  SideEffects [None]

  SeeAlso     [Cudd_bddConstrain]

******************************************************************************/
DdNode *
cuddBddConstrainRecur(
  DdManager * dd,
  DdNode * f,
  DdNode * c)
{
    DdNode       *Fv, *Fnv, *Cv, *Cnv, *t, *e, *r;
    DdNode       *one, *zero;
    unsigned int topf, topc;
    int          index;
    int          comple = 0;

    statLine(dd);
    one = DD_ONE(dd);
    zero = Cudd_Not(one);

    /* Trivial cases. */
    if (c == one)               return(f);
    if (c == zero)              return(zero);
    if (Cudd_IsConstant(f))     return(f);
    if (f == c)                 return(one);
    if (f == Cudd_Not(c))       return(zero);

    /* Make canonical to increase the utilization of the cache. */
    if (Cudd_IsComplement(f)) {
        f = Cudd_Not(f);
        comple = 1;
    }
    /* Now f is a regular pointer to a non-constant node; c is also
    ** non-constant, but may be complemented.
    */

    /* Check the cache. */
    r = cuddCacheLookup2(dd, Cudd_bddConstrain, f, c);
    if (r != NULL) {
        return(Cudd_NotCond(r,comple));
    }
    
    /* Recursive step. */
    topf = dd->perm[f->index];
    topc = dd->perm[Cudd_Regular(c)->index];
    if (topf <= topc) {
        index = f->index;
        Fv = cuddT(f); Fnv = cuddE(f);
    } else {
        index = Cudd_Regular(c)->index;
        Fv = Fnv = f;
    }
    if (topc <= topf) {
        Cv = cuddT(Cudd_Regular(c)); Cnv = cuddE(Cudd_Regular(c));
        if (Cudd_IsComplement(c)) {
            Cv = Cudd_Not(Cv);
            Cnv = Cudd_Not(Cnv);
        }
    } else {
        Cv = Cnv = c;
    }

    if (!Cudd_IsConstant(Cv)) {
        t = cuddBddConstrainRecur(dd, Fv, Cv);
        if (t == NULL)
            return(NULL);
    } else if (Cv == one) {
        t = Fv;
    } else {            /* Cv == zero: return Fnv @ Cnv */
        if (Cnv == one) {
            r = Fnv;
        } else {
            r = cuddBddConstrainRecur(dd, Fnv, Cnv);
            if (r == NULL)
                return(NULL);
        }
        return(Cudd_NotCond(r,comple));
    }
    cuddRef(t);

    if (!Cudd_IsConstant(Cnv)) {
        e = cuddBddConstrainRecur(dd, Fnv, Cnv);
        if (e == NULL) {
            Cudd_IterDerefBdd(dd, t);
            return(NULL);
        }
    } else if (Cnv == one) {
        e = Fnv;
    } else {            /* Cnv == zero: return Fv @ Cv previously computed */
        cuddDeref(t);
        return(Cudd_NotCond(t,comple));
    }
    cuddRef(e);

    if (Cudd_IsComplement(t)) {
        t = Cudd_Not(t);
        e = Cudd_Not(e);
        r = (t == e) ? t : cuddUniqueInter(dd, index, t, e);
        if (r == NULL) {
            Cudd_IterDerefBdd(dd, e);
            Cudd_IterDerefBdd(dd, t);
            return(NULL);
        }
        r = Cudd_Not(r);
    } else {
        r = (t == e) ? t : cuddUniqueInter(dd, index, t, e);
        if (r == NULL) {
            Cudd_IterDerefBdd(dd, e);
            Cudd_IterDerefBdd(dd, t);
            return(NULL);
        }
    }
    cuddDeref(t);
    cuddDeref(e);

    cuddCacheInsert2(dd, Cudd_bddConstrain, f, c, r);
    return(Cudd_NotCond(r,comple));

} /* end of cuddBddConstrainRecur */


/**Function********************************************************************

  Synopsis    [Performs the recursive step of Cudd_bddRestrict.]

  Description [Performs the recursive step of Cudd_bddRestrict.
  Returns the restricted BDD if successful; otherwise NULL.]

  SideEffects [None]

  SeeAlso     [Cudd_bddRestrict]

******************************************************************************/
DdNode *
cuddBddRestrictRecur(
  DdManager * dd,
  DdNode * f,
  DdNode * c)
{
    DdNode       *Fv, *Fnv, *Cv, *Cnv, *t, *e, *r, *one, *zero;
    unsigned int topf, topc;
    int          index;
    int          comple = 0;

    statLine(dd);
    one = DD_ONE(dd);
    zero = Cudd_Not(one);

    /* Trivial cases */
    if (c == one)               return(f);
    if (c == zero)              return(zero);
    if (Cudd_IsConstant(f))     return(f);
    if (f == c)                 return(one);
    if (f == Cudd_Not(c))       return(zero);

    /* Make canonical to increase the utilization of the cache. */
    if (Cudd_IsComplement(f)) {
        f = Cudd_Not(f);
        comple = 1;
    }
    /* Now f is a regular pointer to a non-constant node; c is also
    ** non-constant, but may be complemented.
    */

    /* Check the cache. */
    r = cuddCacheLookup2(dd, Cudd_bddRestrict, f, c);
    if (r != NULL) {
        return(Cudd_NotCond(r,comple));
    }

    topf = dd->perm[f->index];
    topc = dd->perm[Cudd_Regular(c)->index];

    if (topc < topf) {  /* abstract top variable from c */
        DdNode *d, *s1, *s2;

        /* Find complements of cofactors of c. */
        if (Cudd_IsComplement(c)) {
            s1 = cuddT(Cudd_Regular(c));
            s2 = cuddE(Cudd_Regular(c));
        } else {
            s1 = Cudd_Not(cuddT(c));
            s2 = Cudd_Not(cuddE(c));
        }
        /* Take the OR by applying DeMorgan. */
        d = cuddBddAndRecur(dd, s1, s2);
        if (d == NULL) return(NULL);
        d = Cudd_Not(d);
        cuddRef(d);
        r = cuddBddRestrictRecur(dd, f, d);
        if (r == NULL) {
            Cudd_IterDerefBdd(dd, d);
            return(NULL);
        }
        cuddRef(r);
        Cudd_IterDerefBdd(dd, d);
        cuddCacheInsert2(dd, Cudd_bddRestrict, f, c, r);
        cuddDeref(r);
        return(Cudd_NotCond(r,comple));
    }

    /* Recursive step. Here topf <= topc. */
    index = f->index;
    Fv = cuddT(f); Fnv = cuddE(f);
    if (topc == topf) {
        Cv = cuddT(Cudd_Regular(c)); Cnv = cuddE(Cudd_Regular(c));
        if (Cudd_IsComplement(c)) {
            Cv = Cudd_Not(Cv);
            Cnv = Cudd_Not(Cnv);
        }
    } else {
        Cv = Cnv = c;
    }

    if (!Cudd_IsConstant(Cv)) {
        t = cuddBddRestrictRecur(dd, Fv, Cv);
        if (t == NULL) return(NULL);
    } else if (Cv == one) {
        t = Fv;
    } else {            /* Cv == zero: return(Fnv @ Cnv) */
        if (Cnv == one) {
            r = Fnv;
        } else {
            r = cuddBddRestrictRecur(dd, Fnv, Cnv);
            if (r == NULL) return(NULL);
        }
        return(Cudd_NotCond(r,comple));
    }
    cuddRef(t);

    if (!Cudd_IsConstant(Cnv)) {
        e = cuddBddRestrictRecur(dd, Fnv, Cnv);
        if (e == NULL) {
            Cudd_IterDerefBdd(dd, t);
            return(NULL);
        }
    } else if (Cnv == one) {
        e = Fnv;
    } else {            /* Cnv == zero: return (Fv @ Cv) previously computed */
        cuddDeref(t);
        return(Cudd_NotCond(t,comple));
    }
    cuddRef(e);

    if (Cudd_IsComplement(t)) {
        t = Cudd_Not(t);
        e = Cudd_Not(e);
        r = (t == e) ? t : cuddUniqueInter(dd, index, t, e);
        if (r == NULL) {
            Cudd_IterDerefBdd(dd, e);
            Cudd_IterDerefBdd(dd, t);
            return(NULL);
        }
        r = Cudd_Not(r);
    } else {
        r = (t == e) ? t : cuddUniqueInter(dd, index, t, e);
        if (r == NULL) {
            Cudd_IterDerefBdd(dd, e);
            Cudd_IterDerefBdd(dd, t);
            return(NULL);
        }
    }
    cuddDeref(t);
    cuddDeref(e);

    cuddCacheInsert2(dd, Cudd_bddRestrict, f, c, r);
    return(Cudd_NotCond(r,comple));

} /* end of cuddBddRestrictRecur */


/**Function********************************************************************

  Synopsis [Implements the recursive step of Cudd_bddAnd.]

  Description [Implements the recursive step of Cudd_bddNPAnd.
  Returns a pointer to the result is successful; NULL otherwise.]

  SideEffects [None]

  SeeAlso     [Cudd_bddNPAnd]

******************************************************************************/
DdNode *
cuddBddNPAndRecur(
  DdManager * manager,
  DdNode * f,
  DdNode * g)
{
    DdNode *F, *ft, *fe, *G, *gt, *ge;
    DdNode *one, *r, *t, *e;
    unsigned int topf, topg, index;

    statLine(manager);
    one = DD_ONE(manager);

    /* Terminal cases. */
    F = Cudd_Regular(f);
    G = Cudd_Regular(g);
    if (F == G) {
        if (f == g) return(one);
        else return(Cudd_Not(one));
    }
    if (G == one) {
        if (g == one) return(f);
        else return(g);
    }
    if (F == one) {
        return(f);
    }

    /* At this point f and g are not constant. */
    /* Check cache. */
    if (F->ref != 1 || G->ref != 1) {
        r = cuddCacheLookup2(manager, Cudd_bddNPAnd, f, g);
        if (r != NULL) return(r);
    }

    /* Here we can skip the use of cuddI, because the operands are known
    ** to be non-constant.
    */
    topf = manager->perm[F->index];
    topg = manager->perm[G->index];

    if (topg < topf) {  /* abstract top variable from g */
        DdNode *d;

        /* Find complements of cofactors of g. */
        if (Cudd_IsComplement(g)) {
            gt = cuddT(G);
            ge = cuddE(G);
        } else {
            gt = Cudd_Not(cuddT(g));
            ge = Cudd_Not(cuddE(g));
        }
        /* Take the OR by applying DeMorgan. */
        d = cuddBddAndRecur(manager, gt, ge);
        if (d == NULL) return(NULL);
        d = Cudd_Not(d);
        cuddRef(d);
        r = cuddBddNPAndRecur(manager, f, d);
        if (r == NULL) {
            Cudd_IterDerefBdd(manager, d);
            return(NULL);
        }
        cuddRef(r);
        Cudd_IterDerefBdd(manager, d);
        cuddCacheInsert2(manager, Cudd_bddNPAnd, f, g, r);
        cuddDeref(r);
        return(r);
    }

    /* Compute cofactors. */
    index = F->index;
    ft = cuddT(F);
    fe = cuddE(F);
    if (Cudd_IsComplement(f)) {
      ft = Cudd_Not(ft);
      fe = Cudd_Not(fe);
    }

    if (topg == topf) {
        gt = cuddT(G);
        ge = cuddE(G);
        if (Cudd_IsComplement(g)) {
            gt = Cudd_Not(gt);
            ge = Cudd_Not(ge);
        }
    } else {
        gt = ge = g;
    }

    t = cuddBddAndRecur(manager, ft, gt);
    if (t == NULL) return(NULL);
    cuddRef(t);

    e = cuddBddAndRecur(manager, fe, ge);
    if (e == NULL) {
        Cudd_IterDerefBdd(manager, t);
        return(NULL);
    }
    cuddRef(e);

    if (t == e) {
        r = t;
    } else {
        if (Cudd_IsComplement(t)) {
            r = cuddUniqueInter(manager,(int)index,Cudd_Not(t),Cudd_Not(e));
            if (r == NULL) {
                Cudd_IterDerefBdd(manager, t);
                Cudd_IterDerefBdd(manager, e);
                return(NULL);
            }
            r = Cudd_Not(r);
        } else {
            r = cuddUniqueInter(manager,(int)index,t,e);
            if (r == NULL) {
                Cudd_IterDerefBdd(manager, t);
                Cudd_IterDerefBdd(manager, e);
                return(NULL);
            }
        }
    }
    cuddDeref(e);
    cuddDeref(t);
    if (F->ref != 1 || G->ref != 1)
        cuddCacheInsert2(manager, Cudd_bddNPAnd, f, g, r);
    return(r);

} /* end of cuddBddNPAndRecur */


/**Function********************************************************************

  Synopsis    [Performs the recursive step of Cudd_addConstrain.]

  Description [Performs the recursive step of Cudd_addConstrain.
  Returns a pointer to the result if successful; NULL otherwise.]

  SideEffects [None]

  SeeAlso     [Cudd_addConstrain]

******************************************************************************/
DdNode *
cuddAddConstrainRecur(
  DdManager * dd,
  DdNode * f,
  DdNode * c)
{
    DdNode       *Fv, *Fnv, *Cv, *Cnv, *t, *e, *r;
    DdNode       *one, *zero;
    unsigned int topf, topc;
    int          index;

    statLine(dd);
    one = DD_ONE(dd);
    zero = DD_ZERO(dd);

    /* Trivial cases. */
    if (c == one)               return(f);
    if (c == zero)              return(zero);
    if (Cudd_IsConstant(f))     return(f);
    if (f == c)                 return(one);

    /* Now f and c are non-constant. */

    /* Check the cache. */
    r = cuddCacheLookup2(dd, Cudd_addConstrain, f, c);
    if (r != NULL) {
        return(r);
    }
    
    /* Recursive step. */
    topf = dd->perm[f->index];
    topc = dd->perm[c->index];
    if (topf <= topc) {
        index = f->index;
        Fv = cuddT(f); Fnv = cuddE(f);
    } else {
        index = c->index;
        Fv = Fnv = f;
    }
    if (topc <= topf) {
        Cv = cuddT(c); Cnv = cuddE(c);
    } else {
        Cv = Cnv = c;
    }

    if (!Cudd_IsConstant(Cv)) {
        t = cuddAddConstrainRecur(dd, Fv, Cv);
        if (t == NULL)
            return(NULL);
    } else if (Cv == one) {
        t = Fv;
    } else {            /* Cv == zero: return Fnv @ Cnv */
        if (Cnv == one) {
            r = Fnv;
        } else {
            r = cuddAddConstrainRecur(dd, Fnv, Cnv);
            if (r == NULL)
                return(NULL);
        }
        return(r);
    }
    cuddRef(t);

    if (!Cudd_IsConstant(Cnv)) {
        e = cuddAddConstrainRecur(dd, Fnv, Cnv);
        if (e == NULL) {
            Cudd_RecursiveDeref(dd, t);
            return(NULL);
        }
    } else if (Cnv == one) {
        e = Fnv;
    } else {            /* Cnv == zero: return Fv @ Cv previously computed */
        cuddDeref(t);
        return(t);
    }
    cuddRef(e);

    r = (t == e) ? t : cuddUniqueInter(dd, index, t, e);
    if (r == NULL) {
        Cudd_RecursiveDeref(dd, e);
        Cudd_RecursiveDeref(dd, t);
        return(NULL);
    }
    cuddDeref(t);
    cuddDeref(e);

    cuddCacheInsert2(dd, Cudd_addConstrain, f, c, r);
    return(r);

} /* end of cuddAddConstrainRecur */


/**Function********************************************************************

  Synopsis    [Performs the recursive step of Cudd_addRestrict.]

  Description [Performs the recursive step of Cudd_addRestrict.
  Returns the restricted ADD if successful; otherwise NULL.]

  SideEffects [None]

  SeeAlso     [Cudd_addRestrict]

******************************************************************************/
DdNode *
cuddAddRestrictRecur(
  DdManager * dd,
  DdNode * f,
  DdNode * c)
{
    DdNode       *Fv, *Fnv, *Cv, *Cnv, *t, *e, *r, *one, *zero;
    unsigned int topf, topc;
    int          index;

    statLine(dd);
    one = DD_ONE(dd);
    zero = DD_ZERO(dd);

    /* Trivial cases */
    if (c == one)               return(f);
    if (c == zero)              return(zero);
    if (Cudd_IsConstant(f))     return(f);
    if (f == c)                 return(one);

    /* Now f and c are non-constant. */

    /* Check the cache. */
    r = cuddCacheLookup2(dd, Cudd_addRestrict, f, c);
    if (r != NULL) {
        return(r);
    }

    topf = dd->perm[f->index];
    topc = dd->perm[c->index];

    if (topc < topf) {  /* abstract top variable from c */
        DdNode *d, *s1, *s2;

        /* Find cofactors of c. */
        s1 = cuddT(c);
        s2 = cuddE(c);
        /* Take the OR by applying DeMorgan. */
        d = cuddAddApplyRecur(dd, Cudd_addOr, s1, s2);
        if (d == NULL) return(NULL);
        cuddRef(d);
        r = cuddAddRestrictRecur(dd, f, d);
        if (r == NULL) {
            Cudd_RecursiveDeref(dd, d);
            return(NULL);
        }
        cuddRef(r);
        Cudd_RecursiveDeref(dd, d);
        cuddCacheInsert2(dd, Cudd_addRestrict, f, c, r);
        cuddDeref(r);
        return(r);
    }

    /* Recursive step. Here topf <= topc. */
    index = f->index;
    Fv = cuddT(f); Fnv = cuddE(f);
    if (topc == topf) {
        Cv = cuddT(c); Cnv = cuddE(c);
    } else {
        Cv = Cnv = c;
    }

    if (!Cudd_IsConstant(Cv)) {
        t = cuddAddRestrictRecur(dd, Fv, Cv);
        if (t == NULL) return(NULL);
    } else if (Cv == one) {
        t = Fv;
    } else {            /* Cv == zero: return(Fnv @ Cnv) */
        if (Cnv == one) {
            r = Fnv;
        } else {
            r = cuddAddRestrictRecur(dd, Fnv, Cnv);
            if (r == NULL) return(NULL);
        }
        return(r);
    }
    cuddRef(t);

    if (!Cudd_IsConstant(Cnv)) {
        e = cuddAddRestrictRecur(dd, Fnv, Cnv);
        if (e == NULL) {
            Cudd_RecursiveDeref(dd, t);
            return(NULL);
        }
    } else if (Cnv == one) {
        e = Fnv;
    } else {            /* Cnv == zero: return (Fv @ Cv) previously computed */
        cuddDeref(t);
        return(t);
    }
    cuddRef(e);

    r = (t == e) ? t : cuddUniqueInter(dd, index, t, e);
    if (r == NULL) {
        Cudd_RecursiveDeref(dd, e);
        Cudd_RecursiveDeref(dd, t);
        return(NULL);
    }
    cuddDeref(t);
    cuddDeref(e);

    cuddCacheInsert2(dd, Cudd_addRestrict, f, c, r);
    return(r);

} /* end of cuddAddRestrictRecur */



/**Function********************************************************************

  Synopsis    [Performs safe minimization of a BDD.]

  Description [Performs safe minimization of a BDD. Given the BDD
  <code>f</code> of a function to be minimized and a BDD
  <code>c</code> representing the care set, Cudd_bddLICompaction
  produces the BDD of a function that agrees with <code>f</code>
  wherever <code>c</code> is 1.  Safe minimization means that the size
  of the result is guaranteed not to exceed the size of
  <code>f</code>. This function is based on the DAC97 paper by Hong et
  al..  Returns a pointer to the result if successful; NULL
  otherwise.]

  SideEffects [None]

  SeeAlso     [Cudd_bddLICompaction]

******************************************************************************/
DdNode *
cuddBddLICompaction(
  DdManager * dd /* manager */,
  DdNode * f /* function to be minimized */,
  DdNode * c /* constraint (care set) */)
{
    st_table *marktable, *markcache, *buildcache;
    DdNode *res, *zero;

    zero = Cudd_Not(DD_ONE(dd));
    if (c == zero) return(zero);

    /* We need to use local caches for both steps of this operation.
    ** The results of the edge marking step are only valid as long as the
    ** edge markings themselves are available. However, the edge markings
    ** are lost at the end of one invocation of Cudd_bddLICompaction.
    ** Hence, the cache entries for the edge marking step must be
    ** invalidated at the end of this function.
    ** For the result of the building step we argue as follows. The result
    ** for a node and a given constrain depends on the BDD in which the node
    ** appears. Hence, the same node and constrain may give different results
    ** in successive invocations.
    */
    marktable = st_init_table(st_ptrcmp,st_ptrhash);
    if (marktable == NULL) {
        return(NULL);
    }
    markcache = st_init_table(MarkCacheCompare,MarkCacheHash);
    if (markcache == NULL) {
        st_free_table(marktable);
        return(NULL);
    }
    if (cuddBddLICMarkEdges(dd,f,c,marktable,markcache) == CUDD_OUT_OF_MEM) {
        st_foreach(markcache, MarkCacheCleanUp, NULL);
        st_free_table(marktable);
        st_free_table(markcache);
        return(NULL);
    }
    st_foreach(markcache, MarkCacheCleanUp, NULL);
    st_free_table(markcache);
    buildcache = st_init_table(st_ptrcmp,st_ptrhash);
    if (buildcache == NULL) {
        st_free_table(marktable);
        return(NULL);
    }
    res = cuddBddLICBuildResult(dd,f,buildcache,marktable);
    st_free_table(buildcache);
    st_free_table(marktable);
    return(res);

} /* end of cuddBddLICompaction */


/*---------------------------------------------------------------------------*/
/* Definition of static functions                                            */
/*---------------------------------------------------------------------------*/


/**Function********************************************************************

  Synopsis    [Performs the recursive step of Cudd_bddConstrainDecomp.]

  Description [Performs the recursive step of Cudd_bddConstrainDecomp.
  Returns f super (i) if successful; otherwise NULL.]

  SideEffects [None]

  SeeAlso     [Cudd_bddConstrainDecomp]

******************************************************************************/
static int
cuddBddConstrainDecomp(
  DdManager * dd,
  DdNode * f,
  DdNode ** decomp)
{
    DdNode *F, *fv, *fvn;
    DdNode *fAbs;
    DdNode *result;
    int ok;

    if (Cudd_IsConstant(f)) return(1);
    /* Compute complements of cofactors. */
    F = Cudd_Regular(f);
    fv = cuddT(F);
    fvn = cuddE(F);
    if (F == f) {
        fv = Cudd_Not(fv);
        fvn = Cudd_Not(fvn);
    }
    /* Compute abstraction of top variable. */
    fAbs = cuddBddAndRecur(dd, fv, fvn);
    if (fAbs == NULL) {
        return(0);
    }
    cuddRef(fAbs);
    fAbs = Cudd_Not(fAbs);
    /* Recursively find the next abstraction and the components of the
    ** decomposition. */
    ok = cuddBddConstrainDecomp(dd, fAbs, decomp);
    if (ok == 0) {
        Cudd_IterDerefBdd(dd,fAbs);
        return(0);
    }
    /* Compute the component of the decomposition corresponding to the
    ** top variable and store it in the decomposition array. */
    result = cuddBddConstrainRecur(dd, f, fAbs);
    if (result == NULL) {
        Cudd_IterDerefBdd(dd,fAbs);
        return(0);
    }
    cuddRef(result);
    decomp[F->index] = result;
    Cudd_IterDerefBdd(dd, fAbs);
    return(1);

} /* end of cuddBddConstrainDecomp */


/**Function********************************************************************

  Synopsis    [Performs the recursive step of Cudd_bddCharToVect.]

  Description [Performs the recursive step of Cudd_bddCharToVect.
  This function maintains the invariant that f is non-zero.
  Returns the i-th component of the vector if successful; otherwise NULL.]

  SideEffects [None]

  SeeAlso     [Cudd_bddCharToVect]

******************************************************************************/
static DdNode *
cuddBddCharToVect(
  DdManager * dd,
  DdNode * f,
  DdNode * x)
{
    unsigned int topf;
    unsigned int level;
    int comple;

    DdNode *one, *zero, *res, *F, *fT, *fE, *T, *E;

    statLine(dd);
    /* Check the cache. */
    res = cuddCacheLookup2(dd, cuddBddCharToVect, f, x);
    if (res != NULL) {
        return(res);
    }

    F = Cudd_Regular(f);

    topf = cuddI(dd,F->index);
    level = dd->perm[x->index];

    if (topf > level) return(x);

    one = DD_ONE(dd);
    zero = Cudd_Not(one);

    comple = F != f;
    fT = Cudd_NotCond(cuddT(F),comple);
    fE = Cudd_NotCond(cuddE(F),comple);

    if (topf == level) {
        if (fT == zero) return(zero);
        if (fE == zero) return(one);
        return(x);
    }

    /* Here topf < level. */
    if (fT == zero) return(cuddBddCharToVect(dd, fE, x));
    if (fE == zero) return(cuddBddCharToVect(dd, fT, x));

    T = cuddBddCharToVect(dd, fT, x);
    if (T == NULL) {
        return(NULL);
    }
    cuddRef(T);
    E = cuddBddCharToVect(dd, fE, x);
    if (E == NULL) {
        Cudd_IterDerefBdd(dd,T);
        return(NULL);
    }
    cuddRef(E);
    res = cuddBddIteRecur(dd, dd->vars[F->index], T, E);
    if (res == NULL) {
        Cudd_IterDerefBdd(dd,T);
        Cudd_IterDerefBdd(dd,E);
        return(NULL);
    }
    cuddDeref(T);
    cuddDeref(E);
    cuddCacheInsert2(dd, cuddBddCharToVect, f, x, res);
    return(res);

} /* end of cuddBddCharToVect */


/**Function********************************************************************

  Synopsis    [Performs the edge marking step of Cudd_bddLICompaction.]

  Description [Performs the edge marking step of Cudd_bddLICompaction.
  Returns the LUB of the markings of the two outgoing edges of <code>f</code>
  if successful; otherwise CUDD_OUT_OF_MEM.]

  SideEffects [None]

  SeeAlso     [Cudd_bddLICompaction cuddBddLICBuildResult]

******************************************************************************/
static int
cuddBddLICMarkEdges(
  DdManager * dd,
  DdNode * f,
  DdNode * c,
  st_table * table,
  st_table * cache)
{
    DdNode *Fv, *Fnv, *Cv, *Cnv;
    DdNode *one, *zero;
    unsigned int topf, topc;
    int comple;
    int resT, resE, res, retval;
    char **slot;
    MarkCacheKey *key;

    one = DD_ONE(dd);
    zero = Cudd_Not(one);

    /* Terminal cases. */
    if (c == zero) return(DD_LIC_DC);
    if (f == one)  return(DD_LIC_1);
    if (f == zero) return(DD_LIC_0);

    /* Make canonical to increase the utilization of the cache. */
    comple = Cudd_IsComplement(f);
    f = Cudd_Regular(f);
    /* Now f is a regular pointer to a non-constant node; c may be
    ** constant, or it may be complemented.
    */

    /* Check the cache. */
    key = ALLOC(MarkCacheKey, 1);
    if (key == NULL) {
        dd->errorCode = CUDD_MEMORY_OUT;
        return(CUDD_OUT_OF_MEM);
    }
    key->f = f; key->c = c;
    if (st_lookup_int(cache, (char *)key, &res)) {
        FREE(key);
        if (comple) {
            if (res == DD_LIC_0) res = DD_LIC_1;
            else if (res == DD_LIC_1) res = DD_LIC_0;
        }
        return(res);
    }

    /* Recursive step. */
    topf = dd->perm[f->index];
    topc = cuddI(dd,Cudd_Regular(c)->index);
    if (topf <= topc) {
        Fv = cuddT(f); Fnv = cuddE(f);
    } else {
        Fv = Fnv = f;
    }
    if (topc <= topf) {
        /* We know that c is not constant because f is not. */
        Cv = cuddT(Cudd_Regular(c)); Cnv = cuddE(Cudd_Regular(c));
        if (Cudd_IsComplement(c)) {
            Cv = Cudd_Not(Cv);
            Cnv = Cudd_Not(Cnv);
        }
    } else {
        Cv = Cnv = c;
    }

    resT = cuddBddLICMarkEdges(dd, Fv, Cv, table, cache);
    if (resT == CUDD_OUT_OF_MEM) {
        FREE(key);
        return(CUDD_OUT_OF_MEM);
    }
    resE = cuddBddLICMarkEdges(dd, Fnv, Cnv, table, cache);
    if (resE == CUDD_OUT_OF_MEM) {
        FREE(key);
        return(CUDD_OUT_OF_MEM);
    }

    /* Update edge markings. */
    if (topf <= topc) {
        retval = st_find_or_add(table, (char *)f, (char ***)&slot);
        if (retval == 0) {
            *slot = (char *) (ptrint)((resT << 2) | resE);
        } else if (retval == 1) {
            *slot = (char *) (ptrint)((int)((ptrint) *slot) | (resT << 2) | resE);
        } else {
            FREE(key);
            return(CUDD_OUT_OF_MEM);
        }
    }

    /* Cache result. */
    res = resT | resE;
    if (st_insert(cache, (char *)key, (char *)(ptrint)res) == ST_OUT_OF_MEM) {
        FREE(key);
        return(CUDD_OUT_OF_MEM);
    }

    /* Take into account possible complementation. */
    if (comple) {
        if (res == DD_LIC_0) res = DD_LIC_1;
        else if (res == DD_LIC_1) res = DD_LIC_0;
    }
    return(res);

} /* end of cuddBddLICMarkEdges */


/**Function********************************************************************

  Synopsis    [Builds the result of Cudd_bddLICompaction.]

  Description [Builds the results of Cudd_bddLICompaction.
  Returns a pointer to the minimized BDD if successful; otherwise NULL.]

  SideEffects [None]

  SeeAlso     [Cudd_bddLICompaction cuddBddLICMarkEdges]

******************************************************************************/
static DdNode *
cuddBddLICBuildResult(
  DdManager * dd,
  DdNode * f,
  st_table * cache,
  st_table * table)
{
    DdNode *Fv, *Fnv, *r, *t, *e;
    DdNode *one, *zero;
    int index;
    int comple;
    int markT, markE, markings;

    one = DD_ONE(dd);
    zero = Cudd_Not(one);

    if (Cudd_IsConstant(f)) return(f);
    /* Make canonical to increase the utilization of the cache. */
    comple = Cudd_IsComplement(f);
    f = Cudd_Regular(f);

    /* Check the cache. */
    if (st_lookup(cache, f, &r)) {
        return(Cudd_NotCond(r,comple));
    }

    /* Retrieve the edge markings. */
    if (st_lookup_int(table, (char *)f, &markings) == 0)
        return(NULL);
    markT = markings >> 2;
    markE = markings & 3;

    index = f->index;
    Fv = cuddT(f); Fnv = cuddE(f);

    if (markT == DD_LIC_NL) {
        t = cuddBddLICBuildResult(dd,Fv,cache,table);
        if (t == NULL) {
            return(NULL);
        }
    } else if (markT == DD_LIC_1) {
        t = one;
    } else {
        t = zero;
    }
    cuddRef(t);
    if (markE == DD_LIC_NL) {
        e = cuddBddLICBuildResult(dd,Fnv,cache,table);
        if (e == NULL) {
            Cudd_IterDerefBdd(dd,t);
            return(NULL);
        }
    } else if (markE == DD_LIC_1) {
        e = one;
    } else {
        e = zero;
    }
    cuddRef(e);

    if (markT == DD_LIC_DC && markE != DD_LIC_DC) {
        r = e;
    } else if (markT != DD_LIC_DC && markE == DD_LIC_DC) {
        r = t;
    } else {
        if (Cudd_IsComplement(t)) {
            t = Cudd_Not(t);
            e = Cudd_Not(e);
            r = (t == e) ? t : cuddUniqueInter(dd, index, t, e);
            if (r == NULL) {
                Cudd_IterDerefBdd(dd, e);
                Cudd_IterDerefBdd(dd, t);
                return(NULL);
            }
            r = Cudd_Not(r);
        } else {
            r = (t == e) ? t : cuddUniqueInter(dd, index, t, e);
            if (r == NULL) {
                Cudd_IterDerefBdd(dd, e);
                Cudd_IterDerefBdd(dd, t);
                return(NULL);
            }
        }
    }
    cuddDeref(t);
    cuddDeref(e);

    if (st_insert(cache, (char *)f, (char *)r) == ST_OUT_OF_MEM) {
        cuddRef(r);
        Cudd_IterDerefBdd(dd,r);
        return(NULL);
    }

    return(Cudd_NotCond(r,comple));

} /* end of cuddBddLICBuildResult */


/**Function********************************************************************

  Synopsis    [Hash function for the computed table of cuddBddLICMarkEdges.]

  Description [Hash function for the computed table of
  cuddBddLICMarkEdges.  Returns the bucket number.]

  SideEffects [None]

  SeeAlso     [Cudd_bddLICompaction]

******************************************************************************/
static int
MarkCacheHash(
  char * ptr,
  int  modulus)
{
    int val = 0;
    MarkCacheKey *entry;

    entry = (MarkCacheKey *) ptr;

    val = (int) (ptrint) entry->f;
    val = val * 997 + (int) (ptrint) entry->c;

    return ((val < 0) ? -val : val) % modulus;

} /* end of MarkCacheHash */


/**Function********************************************************************

  Synopsis    [Comparison function for the computed table of
  cuddBddLICMarkEdges.]

  Description [Comparison function for the computed table of
  cuddBddLICMarkEdges. Returns 0 if the two nodes of the key are equal; 1
  otherwise.]

  SideEffects [None]

  SeeAlso     [Cudd_bddLICompaction]

******************************************************************************/
static int
MarkCacheCompare(
  const char * ptr1,
  const char * ptr2)
{
    MarkCacheKey *entry1, *entry2;

    entry1 = (MarkCacheKey *) ptr1;
    entry2 = (MarkCacheKey *) ptr2;
    
    return((entry1->f != entry2->f) || (entry1->c != entry2->c));

} /* end of MarkCacheCompare */



/**Function********************************************************************

  Synopsis    [Frees memory associated with computed table of
  cuddBddLICMarkEdges.]

  Description [Frees memory associated with computed table of
  cuddBddLICMarkEdges. Returns ST_CONTINUE.]

  SideEffects [None]

  SeeAlso     [Cudd_bddLICompaction]

******************************************************************************/
static enum st_retval
MarkCacheCleanUp(
  char * key,
  char * value,
  char * arg)
{
    MarkCacheKey *entry;

    entry = (MarkCacheKey *) key;
    FREE(entry);
    return ST_CONTINUE;

} /* end of MarkCacheCleanUp */


/**Function********************************************************************

  Synopsis    [Performs the recursive step of Cudd_bddSqueeze.]

  Description [Performs the recursive step of Cudd_bddSqueeze.  This
  procedure exploits the fact that if we complement and swap the
  bounds of the interval we obtain a valid solution by taking the
  complement of the solution to the original problem. Therefore, we
  can enforce the condition that the upper bound is always regular.
  Returns a pointer to the result if successful; NULL otherwise.]

  SideEffects [None]

  SeeAlso     [Cudd_bddSqueeze]

******************************************************************************/
static DdNode *
cuddBddSqueeze(
  DdManager * dd,
  DdNode * l,
  DdNode * u)
{
    DdNode *one, *zero, *r, *lt, *le, *ut, *ue, *t, *e;
#if 0
    DdNode *ar;
#endif
    int comple = 0;
    unsigned int topu, topl;
    int index;

    statLine(dd);
    if (l == u) {
        return(l);
    }
    one = DD_ONE(dd);
    zero = Cudd_Not(one);
    /* The only case when l == zero && u == one is at the top level,
    ** where returning either one or zero is OK. In all other cases
    ** the procedure will detect such a case and will perform
    ** remapping. Therefore the order in which we test l and u at this
    ** point is immaterial. */
    if (l == zero) return(l);
    if (u == one)  return(u);

    /* Make canonical to increase the utilization of the cache. */
    if (Cudd_IsComplement(u)) {
        DdNode *temp;
        temp = Cudd_Not(l);
        l = Cudd_Not(u);
        u = temp;
        comple = 1;
    }
    /* At this point u is regular and non-constant; l is non-constant, but
    ** may be complemented. */

    /* Here we could check the relative sizes. */

    /* Check the cache. */
    r = cuddCacheLookup2(dd, Cudd_bddSqueeze, l, u);
    if (r != NULL) {
        return(Cudd_NotCond(r,comple));
    }

    /* Recursive step. */
    topu = dd->perm[u->index];
    topl = dd->perm[Cudd_Regular(l)->index];
    if (topu <= topl) {
        index = u->index;
        ut = cuddT(u); ue = cuddE(u);
    } else {
        index = Cudd_Regular(l)->index;
        ut = ue = u;
    }
    if (topl <= topu) {
        lt = cuddT(Cudd_Regular(l)); le = cuddE(Cudd_Regular(l));
        if (Cudd_IsComplement(l)) {
            lt = Cudd_Not(lt);
            le = Cudd_Not(le);
        }
    } else {
        lt = le = l;
    }

    /* If one interval is contained in the other, use the smaller
    ** interval. This corresponds to one-sided matching. */
    if ((lt == zero || Cudd_bddLeq(dd,lt,le)) &&
        (ut == one  || Cudd_bddLeq(dd,ue,ut))) { /* remap */
        r = cuddBddSqueeze(dd, le, ue);
        if (r == NULL)
            return(NULL);
        return(Cudd_NotCond(r,comple));
    } else if ((le == zero || Cudd_bddLeq(dd,le,lt)) &&
               (ue == one  || Cudd_bddLeq(dd,ut,ue))) { /* remap */
        r = cuddBddSqueeze(dd, lt, ut);
        if (r == NULL)
            return(NULL);
        return(Cudd_NotCond(r,comple));
    } else if ((le == zero || Cudd_bddLeq(dd,le,Cudd_Not(ut))) &&
               (ue == one  || Cudd_bddLeq(dd,Cudd_Not(lt),ue))) { /* c-remap */
        t = cuddBddSqueeze(dd, lt, ut);
        cuddRef(t);
        if (Cudd_IsComplement(t)) {
            r = cuddUniqueInter(dd, index, Cudd_Not(t), t);
            if (r == NULL) {
                Cudd_IterDerefBdd(dd, t);
                return(NULL);
            }
            r = Cudd_Not(r);
        } else {
            r = cuddUniqueInter(dd, index, t, Cudd_Not(t));
            if (r == NULL) {
                Cudd_IterDerefBdd(dd, t);
                return(NULL);
            }
        }
        cuddDeref(t);
        if (r == NULL)
            return(NULL);
        cuddCacheInsert2(dd, Cudd_bddSqueeze, l, u, r);
        return(Cudd_NotCond(r,comple));
    } else if ((lt == zero || Cudd_bddLeq(dd,lt,Cudd_Not(ue))) &&
               (ut == one  || Cudd_bddLeq(dd,Cudd_Not(le),ut))) { /* c-remap */
        e = cuddBddSqueeze(dd, le, ue);
        cuddRef(e);
        if (Cudd_IsComplement(e)) {
            r = cuddUniqueInter(dd, index, Cudd_Not(e), e);
            if (r == NULL) {
                Cudd_IterDerefBdd(dd, e);
                return(NULL);
            }
        } else {
            r = cuddUniqueInter(dd, index, e, Cudd_Not(e));
            if (r == NULL) {
                Cudd_IterDerefBdd(dd, e);
                return(NULL);
            }
            r = Cudd_Not(r);
        }
        cuddDeref(e);
        if (r == NULL)
            return(NULL);
        cuddCacheInsert2(dd, Cudd_bddSqueeze, l, u, r);
        return(Cudd_NotCond(r,comple));
    }

#if 0
    /* If the two intervals intersect, take a solution from
    ** the intersection of the intervals. This guarantees that the
    ** splitting variable will not appear in the result.
    ** This approach corresponds to two-sided matching, and is very
    ** expensive. */
    if (Cudd_bddLeq(dd,lt,ue) && Cudd_bddLeq(dd,le,ut)) {
        DdNode *au, *al;
        au = cuddBddAndRecur(dd,ut,ue);
        if (au == NULL)
            return(NULL);
        cuddRef(au);
        al = cuddBddAndRecur(dd,Cudd_Not(lt),Cudd_Not(le));
        if (al == NULL) {
            Cudd_IterDerefBdd(dd,au);
            return(NULL);
        }
        cuddRef(al);
        al = Cudd_Not(al);
        ar = cuddBddSqueeze(dd, al, au);
        if (ar == NULL) {
            Cudd_IterDerefBdd(dd,au);
            Cudd_IterDerefBdd(dd,al);
            return(NULL);
        }
        cuddRef(ar);
        Cudd_IterDerefBdd(dd,au);
        Cudd_IterDerefBdd(dd,al);
    } else {
        ar = NULL;
    }
#endif

    t = cuddBddSqueeze(dd, lt, ut);
    if (t == NULL) {
        return(NULL);
    }
    cuddRef(t);
    e = cuddBddSqueeze(dd, le, ue);
    if (e == NULL) {
        Cudd_IterDerefBdd(dd,t);
        return(NULL);
    }
    cuddRef(e);

    if (Cudd_IsComplement(t)) {
        t = Cudd_Not(t);
        e = Cudd_Not(e);
        r = (t == e) ? t : cuddUniqueInter(dd, index, t, e);
        if (r == NULL) {
            Cudd_IterDerefBdd(dd, e);
            Cudd_IterDerefBdd(dd, t);
            return(NULL);
        }
        r = Cudd_Not(r);
    } else {
        r = (t == e) ? t : cuddUniqueInter(dd, index, t, e);
        if (r == NULL) {
            Cudd_IterDerefBdd(dd, e);
            Cudd_IterDerefBdd(dd, t);
            return(NULL);
        }
    }
    cuddDeref(t);
    cuddDeref(e);

#if 0
    /* Check whether there is a result obtained by abstraction and whether
    ** it is better than the one obtained by recursion. */
    cuddRef(r);
    if (ar != NULL) {
        if (Cudd_DagSize(ar) <= Cudd_DagSize(r)) {
            Cudd_IterDerefBdd(dd, r);
            r = ar;
        } else {
            Cudd_IterDerefBdd(dd, ar);
        }
    }
    cuddDeref(r);
#endif

    cuddCacheInsert2(dd, Cudd_bddSqueeze, l, u, r);
    return(Cudd_NotCond(r,comple));

} /* end of cuddBddSqueeze */