source: vis_dev/glu-2.3/src/cuBdd/cuddSat.c

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

  FileName    [cuddSat.c]

  PackageName [cudd]

  Synopsis    [Functions for the solution of satisfiability related
  problems.]

  Description [External procedures included in this file:
                <ul>
                <li> Cudd_Eval()
                <li> Cudd_ShortestPath()
                <li> Cudd_LargestCube()
                <li> Cudd_ShortestLength()
                <li> Cudd_Decreasing()
                <li> Cudd_Increasing()
                <li> Cudd_EquivDC()
                <li> Cudd_bddLeqUnless()
                <li> Cudd_EqualSupNorm()
                <li> Cudd_bddMakePrime()
                </ul>
        Internal procedures included in this module:
                <ul>
                <li> cuddBddMakePrime()
                </ul>
        Static procedures included in this module:
                <ul>
                <li> freePathPair()
                <li> getShortest()
                <li> getPath()
                <li> getLargest()
                <li> getCube()
                </ul>]

  Author      [Seh-Woong Jeong, 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                                                     */
/*---------------------------------------------------------------------------*/

#define DD_BIGGY        1000000

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

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

typedef struct cuddPathPair {
    int pos;
    int neg;
} cuddPathPair;

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

#ifndef lint
static char rcsid[] DD_UNUSED = "$Id: cuddSat.c,v 1.36 2009/03/08 02:49:02 fabio Exp $";
#endif

static  DdNode  *one, *zero;

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

#define WEIGHT(weight, col)     ((weight) == NULL ? 1 : weight[col])

#ifdef __cplusplus
extern "C" {
#endif

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

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

static enum st_retval freePathPair (char *key, char *value, char *arg);
static cuddPathPair getShortest (DdNode *root, int *cost, int *support, st_table *visited);
static DdNode * getPath (DdManager *manager, st_table *visited, DdNode *f, int *weight, int cost);
static cuddPathPair getLargest (DdNode *root, st_table *visited);
static DdNode * getCube (DdManager *manager, st_table *visited, DdNode *f, int cost);

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

#ifdef __cplusplus
}
#endif

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


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

  Synopsis    [Returns the value of a DD for a given variable assignment.]

  Description [Finds the value of a DD for a given variable
  assignment. The variable assignment is passed in an array of int's,
  that should specify a zero or a one for each variable in the support
  of the function. Returns a pointer to a constant node. No new nodes
  are produced.]

  SideEffects [None]

  SeeAlso     [Cudd_bddLeq Cudd_addEvalConst]

******************************************************************************/
DdNode *
Cudd_Eval(
  DdManager * dd,
  DdNode * f,
  int * inputs)
{
    int comple;
    DdNode *ptr;

    comple = Cudd_IsComplement(f);
    ptr = Cudd_Regular(f);

    while (!cuddIsConstant(ptr)) {
        if (inputs[ptr->index] == 1) {
            ptr = cuddT(ptr);
        } else {
            comple ^= Cudd_IsComplement(cuddE(ptr));
            ptr = Cudd_Regular(cuddE(ptr));
        }
    }
    return(Cudd_NotCond(ptr,comple));

} /* end of Cudd_Eval */


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

  Synopsis    [Finds a shortest path in a DD.]

  Description [Finds a shortest path in a DD. f is the DD we want to
  get the shortest path for; weight\[i\] is the weight of the THEN arc
  coming from the node whose index is i. If weight is NULL, then unit
  weights are assumed for all THEN arcs. All ELSE arcs have 0 weight.
  If non-NULL, both weight and support should point to arrays with at
  least as many entries as there are variables in the manager.
  Returns the shortest path as the BDD of a cube.]

  SideEffects [support contains on return the true support of f.
  If support is NULL on entry, then Cudd_ShortestPath does not compute
  the true support info. length contains the length of the path.]

  SeeAlso     [Cudd_ShortestLength Cudd_LargestCube]

******************************************************************************/
DdNode *
Cudd_ShortestPath(
  DdManager * manager,
  DdNode * f,
  int * weight,
  int * support,
  int * length)
{
    DdNode      *F;
    st_table    *visited;
    DdNode      *sol;
    cuddPathPair *rootPair;
    int         complement, cost;
    int         i;

    one = DD_ONE(manager);
    zero = DD_ZERO(manager);

    /* Initialize support. Support does not depend on variable order.
    ** Hence, it does not need to be reinitialized if reordering occurs.
    */
    if (support) {
      for (i = 0; i < manager->size; i++) {
        support[i] = 0;
      }
    }

    if (f == Cudd_Not(one) || f == zero) {
      *length = DD_BIGGY;
      return(Cudd_Not(one));
    }
    /* From this point on, a path exists. */

    do {
        manager->reordered = 0;

        /* Initialize visited table. */
        visited = st_init_table(st_ptrcmp, st_ptrhash);

        /* Now get the length of the shortest path(s) from f to 1. */
        (void) getShortest(f, weight, support, visited);

        complement = Cudd_IsComplement(f);

        F = Cudd_Regular(f);

        if (!st_lookup(visited, F, &rootPair)) return(NULL);

        if (complement) {
          cost = rootPair->neg;
        } else {
          cost = rootPair->pos;
        }

        /* Recover an actual shortest path. */
        sol = getPath(manager,visited,f,weight,cost);

        st_foreach(visited, freePathPair, NULL);
        st_free_table(visited);

    } while (manager->reordered == 1);

    *length = cost;
    return(sol);

} /* end of Cudd_ShortestPath */


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

  Synopsis    [Finds a largest cube in a DD.]

  Description [Finds a largest cube in a DD. f is the DD we want to
  get the largest cube for. The problem is translated into the one of
  finding a shortest path in f, when both THEN and ELSE arcs are assumed to
  have unit length. This yields a largest cube in the disjoint cover
  corresponding to the DD. Therefore, it is not necessarily the largest
  implicant of f.  Returns the largest cube as a BDD.]

  SideEffects [The number of literals of the cube is returned in length.]

  SeeAlso     [Cudd_ShortestPath]

******************************************************************************/
DdNode *
Cudd_LargestCube(
  DdManager * manager,
  DdNode * f,
  int * length)
{
    register    DdNode  *F;
    st_table    *visited;
    DdNode      *sol;
    cuddPathPair *rootPair;
    int         complement, cost;

    one = DD_ONE(manager);
    zero = DD_ZERO(manager);

    if (f == Cudd_Not(one) || f == zero) {
        *length = DD_BIGGY;
        return(Cudd_Not(one));
    }
    /* From this point on, a path exists. */

    do {
        manager->reordered = 0;

        /* Initialize visited table. */
        visited = st_init_table(st_ptrcmp, st_ptrhash);

        /* Now get the length of the shortest path(s) from f to 1. */
        (void) getLargest(f, visited);

        complement = Cudd_IsComplement(f);

        F = Cudd_Regular(f);

        if (!st_lookup(visited, F, &rootPair)) return(NULL);

        if (complement) {
          cost = rootPair->neg;
        } else {
          cost = rootPair->pos;
        }

        /* Recover an actual shortest path. */
        sol = getCube(manager,visited,f,cost);

        st_foreach(visited, freePathPair, NULL);
        st_free_table(visited);

    } while (manager->reordered == 1);

    *length = cost;
    return(sol);

} /* end of Cudd_LargestCube */


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

  Synopsis    [Find the length of the shortest path(s) in a DD.]

  Description [Find the length of the shortest path(s) in a DD. f is
  the DD we want to get the shortest path for; weight\[i\] is the
  weight of the THEN edge coming from the node whose index is i. All
  ELSE edges have 0 weight. Returns the length of the shortest
  path(s) if such a path is found; a large number if the function is
  identically 0, and CUDD_OUT_OF_MEM in case of failure.]

  SideEffects [None]

  SeeAlso     [Cudd_ShortestPath]

******************************************************************************/
int
Cudd_ShortestLength(
  DdManager * manager,
  DdNode * f,
  int * weight)
{
    register    DdNode  *F;
    st_table    *visited;
    cuddPathPair *my_pair;
    int         complement, cost;

    one = DD_ONE(manager);
    zero = DD_ZERO(manager);

    if (f == Cudd_Not(one) || f == zero) {
        return(DD_BIGGY);
    }

    /* From this point on, a path exists. */
    /* Initialize visited table and support. */
    visited = st_init_table(st_ptrcmp, st_ptrhash);

    /* Now get the length of the shortest path(s) from f to 1. */
    (void) getShortest(f, weight, NULL, visited);

    complement = Cudd_IsComplement(f);

    F = Cudd_Regular(f);

    if (!st_lookup(visited, F, &my_pair)) return(CUDD_OUT_OF_MEM);
    
    if (complement) {
        cost = my_pair->neg;
    } else {
        cost = my_pair->pos;
    }

    st_foreach(visited, freePathPair, NULL);
    st_free_table(visited);

    return(cost);

} /* end of Cudd_ShortestLength */


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

  Synopsis    [Determines whether a BDD is negative unate in a
  variable.]

  Description [Determines whether the function represented by BDD f is
  negative unate (monotonic decreasing) in variable i. Returns the
  constant one is f is unate and the (logical) constant zero if it is not.
  This function does not generate any new nodes.]

  SideEffects [None]

  SeeAlso     [Cudd_Increasing]

******************************************************************************/
DdNode *
Cudd_Decreasing(
  DdManager * dd,
  DdNode * f,
  int  i)
{
    unsigned int topf, level;
    DdNode *F, *fv, *fvn, *res;
    DD_CTFP cacheOp;

    statLine(dd);
#ifdef DD_DEBUG
    assert(0 <= i && i < dd->size);
#endif

    F = Cudd_Regular(f);
    topf = cuddI(dd,F->index);

    /* Check terminal case. If topf > i, f does not depend on var.
    ** Therefore, f is unate in i.
    */
    level = (unsigned) dd->perm[i];
    if (topf > level) {
        return(DD_ONE(dd));
    }

    /* From now on, f is not constant. */

    /* Check cache. */
    cacheOp = (DD_CTFP) Cudd_Decreasing;
    res = cuddCacheLookup2(dd,cacheOp,f,dd->vars[i]);
    if (res != NULL) {
        return(res);
    }

    /* Compute cofactors. */
    fv = cuddT(F); fvn = cuddE(F);
    if (F != f) {
        fv = Cudd_Not(fv);
        fvn = Cudd_Not(fvn);
    }

    if (topf == (unsigned) level) {
        /* Special case: if fv is regular, fv(1,...,1) = 1;
        ** If in addition fvn is complemented, fvn(1,...,1) = 0.
        ** But then f(1,1,...,1) > f(0,1,...,1). Hence f is not
        ** monotonic decreasing in i.
        */
        if (!Cudd_IsComplement(fv) && Cudd_IsComplement(fvn)) {
            return(Cudd_Not(DD_ONE(dd)));
        }
        res = Cudd_bddLeq(dd,fv,fvn) ? DD_ONE(dd) : Cudd_Not(DD_ONE(dd));
    } else {
        res = Cudd_Decreasing(dd,fv,i);
        if (res == DD_ONE(dd)) {
            res = Cudd_Decreasing(dd,fvn,i);
        }
    }

    cuddCacheInsert2(dd,cacheOp,f,dd->vars[i],res);
    return(res);

} /* end of Cudd_Decreasing */


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

  Synopsis    [Determines whether a BDD is positive unate in a
  variable.]

  Description [Determines whether the function represented by BDD f is
  positive unate (monotonic increasing) in variable i. It is based on
  Cudd_Decreasing and the fact that f is monotonic increasing in i if
  and only if its complement is monotonic decreasing in i.]

  SideEffects [None]

  SeeAlso     [Cudd_Decreasing]

******************************************************************************/
DdNode *
Cudd_Increasing(
  DdManager * dd,
  DdNode * f,
  int  i)
{
    return(Cudd_Decreasing(dd,Cudd_Not(f),i));

} /* end of Cudd_Increasing */


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

  Synopsis    [Tells whether F and G are identical wherever D is 0.]

  Description [Tells whether F and G are identical wherever D is 0.  F
  and G are either two ADDs or two BDDs.  D is either a 0-1 ADD or a
  BDD.  The function returns 1 if F and G are equivalent, and 0
  otherwise.  No new nodes are created.]

  SideEffects [None]

  SeeAlso     [Cudd_bddLeqUnless]

******************************************************************************/
int
Cudd_EquivDC(
  DdManager * dd,
  DdNode * F,
  DdNode * G,
  DdNode * D)
{
    DdNode *tmp, *One, *Gr, *Dr;
    DdNode *Fv, *Fvn, *Gv, *Gvn, *Dv, *Dvn;
    int res;
    unsigned int flevel, glevel, dlevel, top;

    One = DD_ONE(dd);

    statLine(dd);
    /* Check terminal cases. */
    if (D == One || F == G) return(1);
    if (D == Cudd_Not(One) || D == DD_ZERO(dd) || F == Cudd_Not(G)) return(0);

    /* From now on, D is non-constant. */

    /* Normalize call to increase cache efficiency. */
    if (F > G) {
        tmp = F;
        F = G;
        G = tmp;
    }
    if (Cudd_IsComplement(F)) {
        F = Cudd_Not(F);
        G = Cudd_Not(G);
    }

    /* From now on, F is regular. */

    /* Check cache. */
    tmp = cuddCacheLookup(dd,DD_EQUIV_DC_TAG,F,G,D);
    if (tmp != NULL) return(tmp == One);

    /* Find splitting variable. */
    flevel = cuddI(dd,F->index);
    Gr = Cudd_Regular(G);
    glevel = cuddI(dd,Gr->index);
    top = ddMin(flevel,glevel);
    Dr = Cudd_Regular(D);
    dlevel = dd->perm[Dr->index];
    top = ddMin(top,dlevel);

    /* Compute cofactors. */
    if (top == flevel) {
        Fv = cuddT(F);
        Fvn = cuddE(F);
    } else {
        Fv = Fvn = F;
    }
    if (top == glevel) {
        Gv = cuddT(Gr);
        Gvn = cuddE(Gr);
        if (G != Gr) {
            Gv = Cudd_Not(Gv);
            Gvn = Cudd_Not(Gvn);
        }
    } else {
        Gv = Gvn = G;
    }
    if (top == dlevel) {
        Dv = cuddT(Dr);
        Dvn = cuddE(Dr);
        if (D != Dr) {
            Dv = Cudd_Not(Dv);
            Dvn = Cudd_Not(Dvn);
        }
    } else {
        Dv = Dvn = D;
    }

    /* Solve recursively. */
    res = Cudd_EquivDC(dd,Fv,Gv,Dv);
    if (res != 0) {
        res = Cudd_EquivDC(dd,Fvn,Gvn,Dvn);
    }
    cuddCacheInsert(dd,DD_EQUIV_DC_TAG,F,G,D,(res) ? One : Cudd_Not(One));

    return(res);

} /* end of Cudd_EquivDC */


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

  Synopsis    [Tells whether f is less than of equal to G unless D is 1.]

  Description [Tells whether f is less than of equal to G unless D is
  1.  f, g, and D are BDDs.  The function returns 1 if f is less than
  of equal to G, and 0 otherwise.  No new nodes are created.]

  SideEffects [None]

  SeeAlso     [Cudd_EquivDC Cudd_bddLeq Cudd_bddIteConstant]

******************************************************************************/
int
Cudd_bddLeqUnless(
  DdManager *dd,
  DdNode *f,
  DdNode *g,
  DdNode *D)
{
    DdNode *tmp, *One, *F, *G;
    DdNode *Ft, *Fe, *Gt, *Ge, *Dt, *De;
    int res;
    unsigned int flevel, glevel, dlevel, top;

    statLine(dd);

    One = DD_ONE(dd);

    /* Check terminal cases. */
    if (f == g || g == One || f == Cudd_Not(One) || D == One ||
        D == f || D == Cudd_Not(g)) return(1);
    /* Check for two-operand cases. */
    if (D == Cudd_Not(One) || D == g || D == Cudd_Not(f))
        return(Cudd_bddLeq(dd,f,g));
    if (g == Cudd_Not(One) || g == Cudd_Not(f)) return(Cudd_bddLeq(dd,f,D));
    if (f == One) return(Cudd_bddLeq(dd,Cudd_Not(g),D));

    /* From now on, f, g, and D are non-constant, distinct, and
    ** non-complementary. */

    /* Normalize call to increase cache efficiency.  We rely on the
    ** fact that f <= g unless D is equivalent to not(g) <= not(f)
    ** unless D and to f <= D unless g.  We make sure that D is
    ** regular, and that at most one of f and g is complemented.  We also
    ** ensure that when two operands can be swapped, the one with the
    ** lowest address comes first. */

    if (Cudd_IsComplement(D)) {
        if (Cudd_IsComplement(g)) {
            /* Special case: if f is regular and g is complemented,
            ** f(1,...,1) = 1 > 0 = g(1,...,1).  If D(1,...,1) = 0, return 0.
            */
            if (!Cudd_IsComplement(f)) return(0);
            /* !g <= D unless !f  or  !D <= g unless !f */
            tmp = D;
            D = Cudd_Not(f);
            if (g < tmp) {
                f = Cudd_Not(g);
                g = tmp;
            } else {
                f = Cudd_Not(tmp);
            }
        } else {
            if (Cudd_IsComplement(f)) {
                /* !D <= !f unless g  or  !D <= g unless !f */
                tmp = f;
                f = Cudd_Not(D);
                if (tmp < g) {
                    D = g;
                    g = Cudd_Not(tmp);
                } else {
                    D = Cudd_Not(tmp);
                }
            } else {
                /* f <= D unless g  or  !D <= !f unless g */
                tmp = D;
                D = g;
                if (tmp < f) {
                    g = Cudd_Not(f);
                    f = Cudd_Not(tmp);
                } else {
                    g = tmp;
                }
            }
        }
    } else {
        if (Cudd_IsComplement(g)) {
            if (Cudd_IsComplement(f)) {
                /* !g <= !f unless D  or  !g <= D unless !f */
                tmp = f;
                f = Cudd_Not(g);
                if (D < tmp) {
                    g = D;
                    D = Cudd_Not(tmp);
                } else {
                    g = Cudd_Not(tmp);
                }
            } else {
                /* f <= g unless D  or  !g <= !f unless D */
                if (g < f) {
                    tmp = g;
                    g = Cudd_Not(f);
                    f = Cudd_Not(tmp);
                }
            }
        } else {
            /* f <= g unless D  or  f <= D unless g */
            if (D < g) {
                tmp = D;
                D = g;
                g = tmp;
            }
        }
    }

    /* From now on, D is regular. */

    /* Check cache. */
    tmp = cuddCacheLookup(dd,DD_BDD_LEQ_UNLESS_TAG,f,g,D);
    if (tmp != NULL) return(tmp == One);

    /* Find splitting variable. */
    F = Cudd_Regular(f);
    flevel = dd->perm[F->index];
    G = Cudd_Regular(g);
    glevel = dd->perm[G->index];
    top = ddMin(flevel,glevel);
    dlevel = dd->perm[D->index];
    top = ddMin(top,dlevel);

    /* Compute cofactors. */
    if (top == flevel) {
        Ft = cuddT(F);
        Fe = cuddE(F);
        if (F != f) {
            Ft = Cudd_Not(Ft);
            Fe = Cudd_Not(Fe);
        }
    } else {
        Ft = Fe = f;
    }
    if (top == glevel) {
        Gt = cuddT(G);
        Ge = cuddE(G);
        if (G != g) {
            Gt = Cudd_Not(Gt);
            Ge = Cudd_Not(Ge);
        }
    } else {
        Gt = Ge = g;
    }
    if (top == dlevel) {
        Dt = cuddT(D);
        De = cuddE(D);
    } else {
        Dt = De = D;
    }

    /* Solve recursively. */
    res = Cudd_bddLeqUnless(dd,Ft,Gt,Dt);
    if (res != 0) {
        res = Cudd_bddLeqUnless(dd,Fe,Ge,De);
    }
    cuddCacheInsert(dd,DD_BDD_LEQ_UNLESS_TAG,f,g,D,Cudd_NotCond(One,!res));

    return(res);

} /* end of Cudd_bddLeqUnless */


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

  Synopsis    [Compares two ADDs for equality within tolerance.]

  Description [Compares two ADDs for equality within tolerance. Two
  ADDs are reported to be equal if the maximum difference between them
  (the sup norm of their difference) is less than or equal to the
  tolerance parameter. Returns 1 if the two ADDs are equal (within
  tolerance); 0 otherwise. If parameter <code>pr</code> is positive
  the first failure is reported to the standard output.]

  SideEffects [None]

  SeeAlso     []

******************************************************************************/
int
Cudd_EqualSupNorm(
  DdManager * dd /* manager */,
  DdNode * f /* first ADD */,
  DdNode * g /* second ADD */,
  CUDD_VALUE_TYPE  tolerance /* maximum allowed difference */,
  int  pr /* verbosity level */)
{
    DdNode *fv, *fvn, *gv, *gvn, *r;
    unsigned int topf, topg;

    statLine(dd);
    /* Check terminal cases. */
    if (f == g) return(1);
    if (Cudd_IsConstant(f) && Cudd_IsConstant(g)) {
        if (ddEqualVal(cuddV(f),cuddV(g),tolerance)) {
            return(1);
        } else {
            if (pr>0) {
                (void) fprintf(dd->out,"Offending nodes:\n");
                (void) fprintf(dd->out,
                               "f: address = %p\t value = %40.30f\n",
                               (void *) f, cuddV(f));
                (void) fprintf(dd->out,
                               "g: address = %p\t value = %40.30f\n",
                               (void *) g, cuddV(g));
            }
            return(0);
        }
    }

    /* We only insert the result in the cache if the comparison is
    ** successful. Therefore, if we hit we return 1. */
    r = cuddCacheLookup2(dd,(DD_CTFP)Cudd_EqualSupNorm,f,g);
    if (r != NULL) {
        return(1);
    }

    /* Compute the cofactors and solve the recursive subproblems. */
    topf = cuddI(dd,f->index);
    topg = cuddI(dd,g->index);

    if (topf <= topg) {fv = cuddT(f); fvn = cuddE(f);} else {fv = fvn = f;}
    if (topg <= topf) {gv = cuddT(g); gvn = cuddE(g);} else {gv = gvn = g;}

    if (!Cudd_EqualSupNorm(dd,fv,gv,tolerance,pr)) return(0);
    if (!Cudd_EqualSupNorm(dd,fvn,gvn,tolerance,pr)) return(0);

    cuddCacheInsert2(dd,(DD_CTFP)Cudd_EqualSupNorm,f,g,DD_ONE(dd));

    return(1);

} /* end of Cudd_EqualSupNorm */


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

  Synopsis    [Expands cube to a prime implicant of f.]

  Description [Expands cube to a prime implicant of f. Returns the prime
  if successful; NULL otherwise.  In particular, NULL is returned if cube
  is not a real cube or is not an implicant of f.]

  SideEffects [None]

  SeeAlso     []

******************************************************************************/
DdNode *
Cudd_bddMakePrime(
  DdManager *dd /* manager */,
  DdNode *cube /* cube to be expanded */,
  DdNode *f /* function of which the cube is to be made a prime */)
{
    DdNode *res;

    if (!Cudd_bddLeq(dd,cube,f)) return(NULL);

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

} /* end of Cudd_bddMakePrime */


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


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

  Synopsis    [Performs the recursive step of Cudd_bddMakePrime.]

  Description [Performs the recursive step of Cudd_bddMakePrime.
  Returns the prime if successful; NULL otherwise.]

  SideEffects [None]

  SeeAlso     []

******************************************************************************/
DdNode *
cuddBddMakePrime(
  DdManager *dd /* manager */,
  DdNode *cube /* cube to be expanded */,
  DdNode *f /* function of which the cube is to be made a prime */)
{
    DdNode *scan;
    DdNode *t, *e;
    DdNode *res = cube;
    DdNode *zero = Cudd_Not(DD_ONE(dd));

    Cudd_Ref(res);
    scan = cube;
    while (!Cudd_IsConstant(scan)) {
        DdNode *reg = Cudd_Regular(scan);
        DdNode *var = dd->vars[reg->index];
        DdNode *expanded = Cudd_bddExistAbstract(dd,res,var);
        if (expanded == NULL) {
            return(NULL);
        }
        Cudd_Ref(expanded);
        if (Cudd_bddLeq(dd,expanded,f)) {
            Cudd_RecursiveDeref(dd,res);
            res = expanded;
        } else {
            Cudd_RecursiveDeref(dd,expanded);
        }
        cuddGetBranches(scan,&t,&e);
        if (t == zero) {
            scan = e;
        } else if (e == zero) {
            scan = t;
        } else {
            Cudd_RecursiveDeref(dd,res);
            return(NULL);       /* cube is not a cube */
        }
    }

    if (scan == DD_ONE(dd)) {
        Cudd_Deref(res);
        return(res);
    } else {
        Cudd_RecursiveDeref(dd,res);
        return(NULL);
    }

} /* end of cuddBddMakePrime */


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


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

  Synopsis    [Frees the entries of the visited symbol table.]

  Description [Frees the entries of the visited symbol table. Returns
  ST_CONTINUE.]

  SideEffects [None]

******************************************************************************/
static enum st_retval
freePathPair(
  char * key,
  char * value,
  char * arg)
{
    cuddPathPair *pair;

    pair = (cuddPathPair *) value;
        FREE(pair);
    return(ST_CONTINUE);

} /* end of freePathPair */


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

  Synopsis    [Finds the length of the shortest path(s) in a DD.]

  Description [Finds the length of the shortest path(s) in a DD.
  Uses a local symbol table to store the lengths for each
  node. Only the lengths for the regular nodes are entered in the table,
  because those for the complement nodes are simply obtained by swapping
  the two lenghts.
  Returns a pair of lengths: the length of the shortest path to 1;
  and the length of the shortest path to 0. This is done so as to take
  complement arcs into account.]

  SideEffects [Accumulates the support of the DD in support.]

  SeeAlso     []

******************************************************************************/
static cuddPathPair
getShortest(
  DdNode * root,
  int * cost,
  int * support,
  st_table * visited)
{
    cuddPathPair *my_pair, res_pair, pair_T, pair_E;
    DdNode      *my_root, *T, *E;
    int         weight;

    my_root = Cudd_Regular(root);

    if (st_lookup(visited, my_root, &my_pair)) {
        if (Cudd_IsComplement(root)) {
            res_pair.pos = my_pair->neg;
            res_pair.neg = my_pair->pos;
        } else {
            res_pair.pos = my_pair->pos;
            res_pair.neg = my_pair->neg;
        }
        return(res_pair);
    }

    /* In the case of a BDD the following test is equivalent to
    ** testing whether the BDD is the constant 1. This formulation,
    ** however, works for ADDs as well, by assuming the usual
    ** dichotomy of 0 and != 0.
    */
    if (cuddIsConstant(my_root)) {
        if (my_root != zero) {
            res_pair.pos = 0;
            res_pair.neg = DD_BIGGY;
        } else {
            res_pair.pos = DD_BIGGY;
            res_pair.neg = 0;
        }
    } else {
        T = cuddT(my_root);
        E = cuddE(my_root);

        pair_T = getShortest(T, cost, support, visited);
        pair_E = getShortest(E, cost, support, visited);
        weight = WEIGHT(cost, my_root->index);
        res_pair.pos = ddMin(pair_T.pos+weight, pair_E.pos);
        res_pair.neg = ddMin(pair_T.neg+weight, pair_E.neg);

        /* Update support. */
        if (support != NULL) {
            support[my_root->index] = 1;
        }
    }

    my_pair = ALLOC(cuddPathPair, 1);
    if (my_pair == NULL) {
        if (Cudd_IsComplement(root)) {
            int tmp = res_pair.pos;
            res_pair.pos = res_pair.neg;
            res_pair.neg = tmp;
        }
        return(res_pair);
    }
    my_pair->pos = res_pair.pos;
    my_pair->neg = res_pair.neg;

    st_insert(visited, (char *)my_root, (char *)my_pair);
    if (Cudd_IsComplement(root)) {
        res_pair.pos = my_pair->neg;
        res_pair.neg = my_pair->pos;
    } else {
        res_pair.pos = my_pair->pos;
        res_pair.neg = my_pair->neg;
    }
    return(res_pair);

} /* end of getShortest */


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

  Synopsis    [Build a BDD for a shortest path of f.]

  Description [Build a BDD for a shortest path of f.
  Given the minimum length from the root, and the minimum
  lengths for each node (in visited), apply triangulation at each node.
  Of the two children of each node on a shortest path, at least one is
  on a shortest path. In case of ties the procedure chooses the THEN
  children.
  Returns a pointer to the cube BDD representing the path if
  successful; NULL otherwise.]

  SideEffects [None]

  SeeAlso     []

******************************************************************************/
static DdNode *
getPath(
  DdManager * manager,
  st_table * visited,
  DdNode * f,
  int * weight,
  int  cost)
{
    DdNode      *sol, *tmp;
    DdNode      *my_dd, *T, *E;
    cuddPathPair *T_pair, *E_pair;
    int         Tcost, Ecost;
    int         complement;

    my_dd = Cudd_Regular(f);
    complement = Cudd_IsComplement(f);

    sol = one;
    cuddRef(sol);

    while (!cuddIsConstant(my_dd)) {
        Tcost = cost - WEIGHT(weight, my_dd->index);
        Ecost = cost;

        T = cuddT(my_dd);
        E = cuddE(my_dd);

        if (complement) {T = Cudd_Not(T); E = Cudd_Not(E);}

        st_lookup(visited, Cudd_Regular(T), &T_pair);
        if ((Cudd_IsComplement(T) && T_pair->neg == Tcost) ||
        (!Cudd_IsComplement(T) && T_pair->pos == Tcost)) {
            tmp = cuddBddAndRecur(manager,manager->vars[my_dd->index],sol);
            if (tmp == NULL) {
                Cudd_RecursiveDeref(manager,sol);
                return(NULL);
            }
            cuddRef(tmp);
            Cudd_RecursiveDeref(manager,sol);
            sol = tmp;

            complement =  Cudd_IsComplement(T);
            my_dd = Cudd_Regular(T);
            cost = Tcost;
            continue;
        }
        st_lookup(visited, Cudd_Regular(E), &E_pair);
        if ((Cudd_IsComplement(E) && E_pair->neg == Ecost) ||
        (!Cudd_IsComplement(E) && E_pair->pos == Ecost)) {
            tmp = cuddBddAndRecur(manager,Cudd_Not(manager->vars[my_dd->index]),sol);
            if (tmp == NULL) {
                Cudd_RecursiveDeref(manager,sol);
                return(NULL);
            }
            cuddRef(tmp);
            Cudd_RecursiveDeref(manager,sol);
            sol = tmp;
            complement = Cudd_IsComplement(E);
            my_dd = Cudd_Regular(E);
            cost = Ecost;
            continue;
        }
        (void) fprintf(manager->err,"We shouldn't be here!!\n");
        manager->errorCode = CUDD_INTERNAL_ERROR;
        return(NULL);
    }

    cuddDeref(sol);
    return(sol);

} /* end of getPath */


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

  Synopsis    [Finds the size of the largest cube(s) in a DD.]

  Description [Finds the size of the largest cube(s) in a DD.
  This problem is translated into finding the shortest paths from a node
  when both THEN and ELSE arcs have unit lengths.
  Uses a local symbol table to store the lengths for each
  node. Only the lengths for the regular nodes are entered in the table,
  because those for the complement nodes are simply obtained by swapping
  the two lenghts.
  Returns a pair of lengths: the length of the shortest path to 1;
  and the length of the shortest path to 0. This is done so as to take
  complement arcs into account.]

  SideEffects [none]

  SeeAlso     []

******************************************************************************/
static cuddPathPair
getLargest(
  DdNode * root,
  st_table * visited)
{
    cuddPathPair *my_pair, res_pair, pair_T, pair_E;
    DdNode      *my_root, *T, *E;

    my_root = Cudd_Regular(root);

    if (st_lookup(visited, my_root, &my_pair)) {
        if (Cudd_IsComplement(root)) {
            res_pair.pos = my_pair->neg;
            res_pair.neg = my_pair->pos;
        } else {
            res_pair.pos = my_pair->pos;
            res_pair.neg = my_pair->neg;
        }
        return(res_pair);
    }

    /* In the case of a BDD the following test is equivalent to
    ** testing whether the BDD is the constant 1. This formulation,
    ** however, works for ADDs as well, by assuming the usual
    ** dichotomy of 0 and != 0.
    */
    if (cuddIsConstant(my_root)) {
        if (my_root != zero) {
            res_pair.pos = 0;
            res_pair.neg = DD_BIGGY;
        } else {
            res_pair.pos = DD_BIGGY;
            res_pair.neg = 0;
        }
    } else {
        T = cuddT(my_root);
        E = cuddE(my_root);

        pair_T = getLargest(T, visited);
        pair_E = getLargest(E, visited);
        res_pair.pos = ddMin(pair_T.pos, pair_E.pos) + 1;
        res_pair.neg = ddMin(pair_T.neg, pair_E.neg) + 1;
    }

    my_pair = ALLOC(cuddPathPair, 1);
    if (my_pair == NULL) {      /* simply do not cache this result */
        if (Cudd_IsComplement(root)) {
            int tmp = res_pair.pos;
            res_pair.pos = res_pair.neg;
            res_pair.neg = tmp;
        }
        return(res_pair);
    }
    my_pair->pos = res_pair.pos;
    my_pair->neg = res_pair.neg;

    /* Caching may fail without affecting correctness. */
    st_insert(visited, (char *)my_root, (char *)my_pair);
    if (Cudd_IsComplement(root)) {
        res_pair.pos = my_pair->neg;
        res_pair.neg = my_pair->pos;
    } else {
        res_pair.pos = my_pair->pos;
        res_pair.neg = my_pair->neg;
    }
    return(res_pair);

} /* end of getLargest */


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

  Synopsis    [Build a BDD for a largest cube of f.]

  Description [Build a BDD for a largest cube of f.
  Given the minimum length from the root, and the minimum
  lengths for each node (in visited), apply triangulation at each node.
  Of the two children of each node on a shortest path, at least one is
  on a shortest path. In case of ties the procedure chooses the THEN
  children.
  Returns a pointer to the cube BDD representing the path if
  successful; NULL otherwise.]

  SideEffects [None]

  SeeAlso     []

******************************************************************************/
static DdNode *
getCube(
  DdManager * manager,
  st_table * visited,
  DdNode * f,
  int  cost)
{
    DdNode      *sol, *tmp;
    DdNode      *my_dd, *T, *E;
    cuddPathPair *T_pair, *E_pair;
    int         Tcost, Ecost;
    int         complement;

    my_dd = Cudd_Regular(f);
    complement = Cudd_IsComplement(f);

    sol = one;
    cuddRef(sol);

    while (!cuddIsConstant(my_dd)) {
        Tcost = cost - 1;
        Ecost = cost - 1;

        T = cuddT(my_dd);
        E = cuddE(my_dd);

        if (complement) {T = Cudd_Not(T); E = Cudd_Not(E);}

        if (!st_lookup(visited, Cudd_Regular(T), &T_pair)) return(NULL);
        if ((Cudd_IsComplement(T) && T_pair->neg == Tcost) ||
        (!Cudd_IsComplement(T) && T_pair->pos == Tcost)) {
            tmp = cuddBddAndRecur(manager,manager->vars[my_dd->index],sol);
            if (tmp == NULL) {
                Cudd_RecursiveDeref(manager,sol);
                return(NULL);
            }
            cuddRef(tmp);
            Cudd_RecursiveDeref(manager,sol);
            sol = tmp;

            complement =  Cudd_IsComplement(T);
            my_dd = Cudd_Regular(T);
            cost = Tcost;
            continue;
        }
        if (!st_lookup(visited, Cudd_Regular(E), &E_pair)) return(NULL);
        if ((Cudd_IsComplement(E) && E_pair->neg == Ecost) ||
        (!Cudd_IsComplement(E) && E_pair->pos == Ecost)) {
            tmp = cuddBddAndRecur(manager,Cudd_Not(manager->vars[my_dd->index]),sol);
            if (tmp == NULL) {
                Cudd_RecursiveDeref(manager,sol);
                return(NULL);
            }
            cuddRef(tmp);
            Cudd_RecursiveDeref(manager,sol);
            sol = tmp;
            complement = Cudd_IsComplement(E);
            my_dd = Cudd_Regular(E);
            cost = Ecost;
            continue;
        }
        (void) fprintf(manager->err,"We shouldn't be here!\n");
        manager->errorCode = CUDD_INTERNAL_ERROR;
        return(NULL);
    }

    cuddDeref(sol);
    return(sol);

} /* end of getCube */