src/bdd/cudd/cuddSymmetry.c

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#include "util_hack.h"
#include "cuddInt.h"

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

#define MV_OOM (Move *)1

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

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

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

#ifndef lint
static char rcsid[] DD_UNUSED = "$Id: cuddSymmetry.c,v 1.1.1.1 2003/02/24 22:23:53 wjiang Exp $";
#endif

static  int     *entry;

extern  int     ddTotalNumberSwapping;
#ifdef DD_STATS
extern  int     ddTotalNISwaps;
#endif

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

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

static int ddSymmUniqueCompare ARGS((int *ptrX, int *ptrY));
static int ddSymmSiftingAux ARGS((DdManager *table, int x, int xLow, int xHigh));
static int ddSymmSiftingConvAux ARGS((DdManager *table, int x, int xLow, int xHigh));
static Move * ddSymmSiftingUp ARGS((DdManager *table, int y, int xLow));
static Move * ddSymmSiftingDown ARGS((DdManager *table, int x, int xHigh));
static int ddSymmGroupMove ARGS((DdManager *table, int x, int y, Move **moves));
static int ddSymmGroupMoveBackward ARGS((DdManager *table, int x, int y));
static int ddSymmSiftingBackward ARGS((DdManager *table, Move *moves, int size));
static void ddSymmSummary ARGS((DdManager *table, int lower, int upper, int *symvars, int *symgroups));

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


void
Cudd_SymmProfile(
  DdManager * table,
  int  lower,
  int  upper)
{
    int i,x,gbot;
    int TotalSymm = 0;
    int TotalSymmGroups = 0;

    for (i = lower; i <= upper; i++) {
        if (table->subtables[i].next != (unsigned) i) {
            x = i;
            (void) fprintf(table->out,"Group:");
            do {
                (void) fprintf(table->out,"  %d",table->invperm[x]);
                TotalSymm++;
                gbot = x;
                x = table->subtables[x].next;
            } while (x != i);
            TotalSymmGroups++;
#ifdef DD_DEBUG
            assert(table->subtables[gbot].next == (unsigned) i);
#endif
            i = gbot;
            (void) fprintf(table->out,"\n");
        }
    }
    (void) fprintf(table->out,"Total Symmetric = %d\n",TotalSymm);
    (void) fprintf(table->out,"Total Groups = %d\n",TotalSymmGroups);

} /* end of Cudd_SymmProfile */


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


int
cuddSymmCheck(
  DdManager * table,
  int  x,
  int  y)
{
    DdNode *f,*f0,*f1,*f01,*f00,*f11,*f10;
    int comple;         /* f0 is complemented */
    int xsymmy;         /* x and y may be positively symmetric */
    int xsymmyp;        /* x and y may be negatively symmetric */
    int arccount;       /* number of arcs from layer x to layer y */
    int TotalRefCount;  /* total reference count of layer y minus 1 */
    int yindex;
    int i;
    DdNodePtr *list;
    int slots;
    DdNode *sentinel = &(table->sentinel);
#ifdef DD_DEBUG
    int xindex;
#endif

    /* Checks that x and y are not the projection functions.
    ** For x it is sufficient to check whether there is only one
    ** node; indeed, if there is one node, it is the projection function
    ** and it cannot point to y. Hence, if y isn't just the projection
    ** function, it has one arc coming from a layer different from x.
    */
    if (table->subtables[x].keys == 1) {
        return(0);
    }
    yindex = table->invperm[y];
    if (table->subtables[y].keys == 1) {
        if (table->vars[yindex]->ref == 1)
            return(0);
    }

    xsymmy = xsymmyp = 1;
    arccount = 0;
    slots = table->subtables[x].slots;
    list = table->subtables[x].nodelist;
    for (i = 0; i < slots; i++) {
        f = list[i];
        while (f != sentinel) {
            /* Find f1, f0, f11, f10, f01, f00. */
            f1 = cuddT(f);
            f0 = Cudd_Regular(cuddE(f));
            comple = Cudd_IsComplement(cuddE(f));
            if ((int) f1->index == yindex) {
                arccount++;
                f11 = cuddT(f1); f10 = cuddE(f1);
            } else {
                if ((int) f0->index != yindex) {
                    /* If f is an isolated projection function it is
                    ** allowed to bypass layer y.
                    */
                    if (f1 != DD_ONE(table) || f0 != DD_ONE(table) || f->ref != 1)
                        return(0); /* f bypasses layer y */
                }
                f11 = f10 = f1;
            }
            if ((int) f0->index == yindex) {
                arccount++;
                f01 = cuddT(f0); f00 = cuddE(f0);
            } else {
                f01 = f00 = f0;
            }
            if (comple) {
                f01 = Cudd_Not(f01);
                f00 = Cudd_Not(f00);
            }

            if (f1 != DD_ONE(table) || f0 != DD_ONE(table) || f->ref != 1) {
                xsymmy &= f01 == f10;
                xsymmyp &= f11 == f00;
                if ((xsymmy == 0) && (xsymmyp == 0))
                    return(0);
            }

            f = f->next;
        } /* while */
    } /* for */

    /* Calculate the total reference counts of y */
    TotalRefCount = -1; /* -1 for projection function */
    slots = table->subtables[y].slots;
    list = table->subtables[y].nodelist;
    for (i = 0; i < slots; i++) {
        f = list[i];
        while (f != sentinel) {
            TotalRefCount += f->ref;
            f = f->next;
        }
    }

#if defined(DD_DEBUG) && defined(DD_VERBOSE)
    if (arccount == TotalRefCount) {
        xindex = table->invperm[x];
        (void) fprintf(table->out,
                       "Found symmetry! x =%d\ty = %d\tPos(%d,%d)\n",
                       xindex,yindex,x,y);
    }
#endif

    return(arccount == TotalRefCount);

} /* end of cuddSymmCheck */


int
cuddSymmSifting(
  DdManager * table,
  int  lower,
  int  upper)
{
    int         i;
    int         *var;
    int         size;
    int         x;
    int         result;
    int         symvars;
    int         symgroups;
#ifdef DD_STATS
    int         previousSize;
#endif

    size = table->size;

    /* Find order in which to sift variables. */
    var = NULL;
    entry = ALLOC(int,size);
    if (entry == NULL) {
        table->errorCode = CUDD_MEMORY_OUT;
        goto ddSymmSiftingOutOfMem;
    }
    var = ALLOC(int,size);
    if (var == NULL) {
        table->errorCode = CUDD_MEMORY_OUT;
        goto ddSymmSiftingOutOfMem;
    }

    for (i = 0; i < size; i++) {
        x = table->perm[i];
        entry[i] = table->subtables[x].keys;
        var[i] = i;
    }

    qsort((void *)var,size,sizeof(int),(int (*)(const void *, const void *))ddSymmUniqueCompare);

    /* Initialize the symmetry of each subtable to itself. */
    for (i = lower; i <= upper; i++) {
        table->subtables[i].next = i;
    }

    for (i = 0; i < ddMin(table->siftMaxVar,size); i++) {
        if (ddTotalNumberSwapping >= table->siftMaxSwap)
            break;
        x = table->perm[var[i]];
#ifdef DD_STATS
        previousSize = table->keys - table->isolated;
#endif
        if (x < lower || x > upper) continue;
        if (table->subtables[x].next == (unsigned) x) {
            result = ddSymmSiftingAux(table,x,lower,upper);
            if (!result) goto ddSymmSiftingOutOfMem;
#ifdef DD_STATS
            if (table->keys < (unsigned) previousSize + table->isolated) {
                (void) fprintf(table->out,"-");
            } else if (table->keys > (unsigned) previousSize +
                       table->isolated) {
                (void) fprintf(table->out,"+"); /* should never happen */
            } else {
                (void) fprintf(table->out,"=");
            }
            fflush(table->out);
#endif
        }
    }

    FREE(var);
    FREE(entry);

    ddSymmSummary(table, lower, upper, &symvars, &symgroups);

#ifdef DD_STATS
    (void) fprintf(table->out, "\n#:S_SIFTING %8d: symmetric variables\n",
                   symvars);
    (void) fprintf(table->out, "#:G_SIFTING %8d: symmetric groups",
                   symgroups);
#endif

    return(1+symvars);

ddSymmSiftingOutOfMem:

    if (entry != NULL) FREE(entry);
    if (var != NULL) FREE(var);

    return(0);

} /* end of cuddSymmSifting */


int
cuddSymmSiftingConv(
  DdManager * table,
  int  lower,
  int  upper)
{
    int         i;
    int         *var;
    int         size;
    int         x;
    int         result;
    int         symvars;
    int         symgroups;
    int         classes;
    int         initialSize;
#ifdef DD_STATS
    int         previousSize;
#endif

    initialSize = table->keys - table->isolated;

    size = table->size;

    /* Find order in which to sift variables. */
    var = NULL;
    entry = ALLOC(int,size);
    if (entry == NULL) {
        table->errorCode = CUDD_MEMORY_OUT;
        goto ddSymmSiftingConvOutOfMem;
    }
    var = ALLOC(int,size);
    if (var == NULL) {
        table->errorCode = CUDD_MEMORY_OUT;
        goto ddSymmSiftingConvOutOfMem;
    }

    for (i = 0; i < size; i++) {
        x = table->perm[i];
        entry[i] = table->subtables[x].keys;
        var[i] = i;
    }

    qsort((void *)var,size,sizeof(int),(int (*)(const void *, const void *))ddSymmUniqueCompare);

    /* Initialize the symmetry of each subtable to itself
    ** for first pass of converging symmetric sifting.
    */
    for (i = lower; i <= upper; i++) {
        table->subtables[i].next = i;
    }

    for (i = 0; i < ddMin(table->siftMaxVar, table->size); i++) {
        if (ddTotalNumberSwapping >= table->siftMaxSwap)
            break;
        x = table->perm[var[i]];
        if (x < lower || x > upper) continue;
        /* Only sift if not in symmetry group already. */
        if (table->subtables[x].next == (unsigned) x) {
#ifdef DD_STATS
            previousSize = table->keys - table->isolated;
#endif
            result = ddSymmSiftingAux(table,x,lower,upper);
            if (!result) goto ddSymmSiftingConvOutOfMem;
#ifdef DD_STATS
            if (table->keys < (unsigned) previousSize + table->isolated) {
                (void) fprintf(table->out,"-");
            } else if (table->keys > (unsigned) previousSize +
                       table->isolated) {
                (void) fprintf(table->out,"+");
            } else {
                (void) fprintf(table->out,"=");
            }
            fflush(table->out);
#endif
        }
    }

    /* Sifting now until convergence. */
    while ((unsigned) initialSize > table->keys - table->isolated) {
        initialSize = table->keys - table->isolated;
#ifdef DD_STATS
        (void) fprintf(table->out,"\n");
#endif
        /* Here we consider only one representative for each symmetry class. */
        for (x = lower, classes = 0; x <= upper; x++, classes++) {
            while ((unsigned) x < table->subtables[x].next) {
                x = table->subtables[x].next;
            }
            /* Here x is the largest index in a group.
            ** Groups consist of adjacent variables.
            ** Hence, the next increment of x will move it to a new group.
            */
            i = table->invperm[x];
            entry[i] = table->subtables[x].keys;
            var[classes] = i;
        }

        qsort((void *)var,classes,sizeof(int),(int (*)(const void *, const void *))ddSymmUniqueCompare);

        /* Now sift. */
        for (i = 0; i < ddMin(table->siftMaxVar,classes); i++) {
            if (ddTotalNumberSwapping >= table->siftMaxSwap)
                break;
            x = table->perm[var[i]];
            if ((unsigned) x >= table->subtables[x].next) {
#ifdef DD_STATS
                previousSize = table->keys - table->isolated;
#endif
                result = ddSymmSiftingConvAux(table,x,lower,upper);
                if (!result ) goto ddSymmSiftingConvOutOfMem;
#ifdef DD_STATS
                if (table->keys < (unsigned) previousSize + table->isolated) {
                    (void) fprintf(table->out,"-");
                } else if (table->keys > (unsigned) previousSize +
                           table->isolated) {
                    (void) fprintf(table->out,"+");
                } else {
                    (void) fprintf(table->out,"=");
                }
                fflush(table->out);
#endif
            }
        } /* for */
    }

    ddSymmSummary(table, lower, upper, &symvars, &symgroups);

#ifdef DD_STATS
    (void) fprintf(table->out, "\n#:S_SIFTING %8d: symmetric variables\n",
                   symvars);
    (void) fprintf(table->out, "#:G_SIFTING %8d: symmetric groups",
                   symgroups);
#endif

    FREE(var);
    FREE(entry);

    return(1+symvars);

ddSymmSiftingConvOutOfMem:

    if (entry != NULL) FREE(entry);
    if (var != NULL) FREE(var);

    return(0);

} /* end of cuddSymmSiftingConv */


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


static int
ddSymmUniqueCompare(
  int * ptrX,
  int * ptrY)
{
#if 0
    if (entry[*ptrY] == entry[*ptrX]) {
        return((*ptrX) - (*ptrY));
    }
#endif
    return(entry[*ptrY] - entry[*ptrX]);

} /* end of ddSymmUniqueCompare */


static int
ddSymmSiftingAux(
  DdManager * table,
  int  x,
  int  xLow,
  int  xHigh)
{
    Move *move;
    Move *moveUp;       /* list of up moves */
    Move *moveDown;     /* list of down moves */
    int  initialSize;
    int  result;
    int  i;
    int  topbot;        /* index to either top or bottom of symmetry group */
    int  initGroupSize, finalGroupSize;


#ifdef DD_DEBUG
    /* check for previously detected symmetry */
    assert(table->subtables[x].next == (unsigned) x);
#endif

    initialSize = table->keys - table->isolated;

    moveDown = NULL;
    moveUp = NULL;

    if ((x - xLow) > (xHigh - x)) {
        /* Will go down first, unless x == xHigh:
        ** Look for consecutive symmetries above x.
        */
        for (i = x; i > xLow; i--) {
            if (!cuddSymmCheck(table,i-1,i))
                break;
            topbot = table->subtables[i-1].next; /* find top of i-1's group */
            table->subtables[i-1].next = i;
            table->subtables[x].next = topbot; /* x is bottom of group so its */
                                               /* next is top of i-1's group */
            i = topbot + 1; /* add 1 for i--; new i is top of symm group */
        }
    } else {
        /* Will go up first unless x == xlow:
        ** Look for consecutive symmetries below x.
        */
        for (i = x; i < xHigh; i++) {
            if (!cuddSymmCheck(table,i,i+1))
                break;
            /* find bottom of i+1's symm group */
            topbot = i + 1;
            while ((unsigned) topbot < table->subtables[topbot].next) {
                topbot = table->subtables[topbot].next;
            }
            table->subtables[topbot].next = table->subtables[i].next;
            table->subtables[i].next = i + 1;
            i = topbot - 1; /* subtract 1 for i++; new i is bottom of group */
        }
    }

    /* Now x may be in the middle of a symmetry group.
    ** Find bottom of x's symm group.
    */
    while ((unsigned) x < table->subtables[x].next)
        x = table->subtables[x].next;

    if (x == xLow) { /* Sift down */

#ifdef DD_DEBUG
        /* x must be a singleton */
        assert((unsigned) x == table->subtables[x].next);
#endif
        if (x == xHigh) return(1);      /* just one variable */

        initGroupSize = 1;

        moveDown = ddSymmSiftingDown(table,x,xHigh);
            /* after this point x --> xHigh, unless early term */
        if (moveDown == MV_OOM) goto ddSymmSiftingAuxOutOfMem;
        if (moveDown == NULL) return(1);

        x = moveDown->y;
        /* Find bottom of x's group */
        i = x;
        while ((unsigned) i < table->subtables[i].next) {
            i = table->subtables[i].next;
        }
#ifdef DD_DEBUG
        /* x should be the top of the symmetry group and i the bottom */
        assert((unsigned) i >= table->subtables[i].next);
        assert((unsigned) x == table->subtables[i].next);
#endif
        finalGroupSize = i - x + 1;

        if (initGroupSize == finalGroupSize) {
            /* No new symmetry groups detected, return to best position */
            result = ddSymmSiftingBackward(table,moveDown,initialSize);
        } else {
            initialSize = table->keys - table->isolated;
            moveUp = ddSymmSiftingUp(table,x,xLow);
            result = ddSymmSiftingBackward(table,moveUp,initialSize);
        }
        if (!result) goto ddSymmSiftingAuxOutOfMem;

    } else if (cuddNextHigh(table,x) > xHigh) { /* Sift up */
        /* Find top of x's symm group */
        i = x;                          /* bottom */
        x = table->subtables[x].next;   /* top */

        if (x == xLow) return(1); /* just one big group */

        initGroupSize = i - x + 1;

        moveUp = ddSymmSiftingUp(table,x,xLow);
            /* after this point x --> xLow, unless early term */
        if (moveUp == MV_OOM) goto ddSymmSiftingAuxOutOfMem;
        if (moveUp == NULL) return(1);

        x = moveUp->x;
        /* Find top of x's group */
        i = table->subtables[x].next;
#ifdef DD_DEBUG
        /* x should be the bottom of the symmetry group and i the top */
        assert((unsigned) x >= table->subtables[x].next);
        assert((unsigned) i == table->subtables[x].next);
#endif
        finalGroupSize = x - i + 1;

        if (initGroupSize == finalGroupSize) {
            /* No new symmetry groups detected, return to best position */
            result = ddSymmSiftingBackward(table,moveUp,initialSize);
        } else {
            initialSize = table->keys - table->isolated;
            moveDown = ddSymmSiftingDown(table,x,xHigh);
            result = ddSymmSiftingBackward(table,moveDown,initialSize);
        }
        if (!result) goto ddSymmSiftingAuxOutOfMem;

    } else if ((x - xLow) > (xHigh - x)) { /* must go down first: shorter */

        moveDown = ddSymmSiftingDown(table,x,xHigh);
        /* at this point x == xHigh, unless early term */
        if (moveDown == MV_OOM) goto ddSymmSiftingAuxOutOfMem;

        if (moveDown != NULL) {
            x = moveDown->y;    /* x is top here */
            i = x;
            while ((unsigned) i < table->subtables[i].next) {
                i = table->subtables[i].next;
            }
        } else {
            i = x;
            while ((unsigned) i < table->subtables[i].next) {
                i = table->subtables[i].next;
            }
            x = table->subtables[i].next;
        }
#ifdef DD_DEBUG
        /* x should be the top of the symmetry group and i the bottom */
        assert((unsigned) i >= table->subtables[i].next);
        assert((unsigned) x == table->subtables[i].next);
#endif
        initGroupSize = i - x + 1;

        moveUp = ddSymmSiftingUp(table,x,xLow);
        if (moveUp == MV_OOM) goto ddSymmSiftingAuxOutOfMem;

        if (moveUp != NULL) {
            x = moveUp->x;
            i = table->subtables[x].next;
        } else {
            i = x;
            while ((unsigned) x < table->subtables[x].next)
                x = table->subtables[x].next;
        }
#ifdef DD_DEBUG
        /* x should be the bottom of the symmetry group and i the top */
        assert((unsigned) x >= table->subtables[x].next);
        assert((unsigned) i == table->subtables[x].next);
#endif
        finalGroupSize = x - i + 1;

        if (initGroupSize == finalGroupSize) {
            /* No new symmetry groups detected, return to best position */
            result = ddSymmSiftingBackward(table,moveUp,initialSize);
        } else {
            while (moveDown != NULL) {
                move = moveDown->next;
                cuddDeallocNode(table, (DdNode *) moveDown);
                moveDown = move;
            }
            initialSize = table->keys - table->isolated;
            moveDown = ddSymmSiftingDown(table,x,xHigh);
            result = ddSymmSiftingBackward(table,moveDown,initialSize);
        }
        if (!result) goto ddSymmSiftingAuxOutOfMem;

    } else { /* moving up first: shorter */
        /* Find top of x's symmetry group */
        x = table->subtables[x].next;

        moveUp = ddSymmSiftingUp(table,x,xLow);
        /* at this point x == xHigh, unless early term */
        if (moveUp == MV_OOM) goto ddSymmSiftingAuxOutOfMem;

        if (moveUp != NULL) {
            x = moveUp->x;
            i = table->subtables[x].next;
        } else {
            while ((unsigned) x < table->subtables[x].next)
                x = table->subtables[x].next;
            i = table->subtables[x].next;
        }
#ifdef DD_DEBUG
        /* x is bottom of the symmetry group and i is top */
        assert((unsigned) x >= table->subtables[x].next);
        assert((unsigned) i == table->subtables[x].next);
#endif
        initGroupSize = x - i + 1;

        moveDown = ddSymmSiftingDown(table,x,xHigh);
        if (moveDown == MV_OOM) goto ddSymmSiftingAuxOutOfMem;

        if (moveDown != NULL) {
            x = moveDown->y;
            i = x;
            while ((unsigned) i < table->subtables[i].next) {
                i = table->subtables[i].next;
            }
        } else {
            i = x;
            x = table->subtables[x].next;
        }
#ifdef DD_DEBUG
        /* x should be the top of the symmetry group and i the bottom */
        assert((unsigned) i >= table->subtables[i].next);
        assert((unsigned) x == table->subtables[i].next);
#endif
        finalGroupSize = i - x + 1;

        if (initGroupSize == finalGroupSize) {
            /* No new symmetries detected, go back to best position */
            result = ddSymmSiftingBackward(table,moveDown,initialSize);
        } else {
            while (moveUp != NULL) {
                move = moveUp->next;
                cuddDeallocNode(table, (DdNode *) moveUp);
                moveUp = move;
            }
            initialSize = table->keys - table->isolated;
            moveUp = ddSymmSiftingUp(table,x,xLow);
            result = ddSymmSiftingBackward(table,moveUp,initialSize);
        }
        if (!result) goto ddSymmSiftingAuxOutOfMem;
    }

    while (moveDown != NULL) {
        move = moveDown->next;
        cuddDeallocNode(table, (DdNode *) moveDown);
        moveDown = move;
    }
    while (moveUp != NULL) {
        move = moveUp->next;
        cuddDeallocNode(table, (DdNode *) moveUp);
        moveUp = move;
    }

    return(1);

ddSymmSiftingAuxOutOfMem:
    if (moveDown != MV_OOM) {
        while (moveDown != NULL) {
            move = moveDown->next;
            cuddDeallocNode(table, (DdNode *) moveDown);
            moveDown = move;
        }
    }
    if (moveUp != MV_OOM) {
        while (moveUp != NULL) {
            move = moveUp->next;
            cuddDeallocNode(table, (DdNode *) moveUp);
            moveUp = move;
        }
    }

    return(0);

} /* end of ddSymmSiftingAux */


static int
ddSymmSiftingConvAux(
  DdManager * table,
  int  x,
  int  xLow,
  int  xHigh)
{
    Move *move;
    Move *moveUp;       /* list of up moves */
    Move *moveDown;     /* list of down moves */
    int  initialSize;
    int  result;
    int  i;
    int  initGroupSize, finalGroupSize;


    initialSize = table->keys - table->isolated;

    moveDown = NULL;
    moveUp = NULL;

    if (x == xLow) { /* Sift down */
#ifdef DD_DEBUG
        /* x is bottom of symmetry group */
        assert((unsigned) x >= table->subtables[x].next);
#endif
        i = table->subtables[x].next;
        initGroupSize = x - i + 1;

        moveDown = ddSymmSiftingDown(table,x,xHigh);
        /* at this point x == xHigh, unless early term */
        if (moveDown == MV_OOM) goto ddSymmSiftingConvAuxOutOfMem;
        if (moveDown == NULL) return(1);

        x = moveDown->y;
        i = x;
        while ((unsigned) i < table->subtables[i].next) {
            i = table->subtables[i].next;
        }
#ifdef DD_DEBUG
        /* x should be the top of the symmetric group and i the bottom */
        assert((unsigned) i >= table->subtables[i].next);
        assert((unsigned) x == table->subtables[i].next);
#endif
        finalGroupSize = i - x + 1;

        if (initGroupSize == finalGroupSize) {
            /* No new symmetries detected, go back to best position */
            result = ddSymmSiftingBackward(table,moveDown,initialSize);
        } else {
            initialSize = table->keys - table->isolated;
            moveUp = ddSymmSiftingUp(table,x,xLow);
            result = ddSymmSiftingBackward(table,moveUp,initialSize);
        }
        if (!result) goto ddSymmSiftingConvAuxOutOfMem;

    } else if (cuddNextHigh(table,x) > xHigh) { /* Sift up */
        /* Find top of x's symm group */
        while ((unsigned) x < table->subtables[x].next)
            x = table->subtables[x].next;
        i = x;                          /* bottom */
        x = table->subtables[x].next;   /* top */

        if (x == xLow) return(1);

        initGroupSize = i - x + 1;

        moveUp = ddSymmSiftingUp(table,x,xLow);
            /* at this point x == xLow, unless early term */
        if (moveUp == MV_OOM) goto ddSymmSiftingConvAuxOutOfMem;
        if (moveUp == NULL) return(1);

        x = moveUp->x;
        i = table->subtables[x].next;
#ifdef DD_DEBUG
        /* x should be the bottom of the symmetry group and i the top */
        assert((unsigned) x >= table->subtables[x].next);
        assert((unsigned) i == table->subtables[x].next);
#endif
        finalGroupSize = x - i + 1;

        if (initGroupSize == finalGroupSize) {
            /* No new symmetry groups detected, return to best position */
            result = ddSymmSiftingBackward(table,moveUp,initialSize);
        } else {
            initialSize = table->keys - table->isolated;
            moveDown = ddSymmSiftingDown(table,x,xHigh);
            result = ddSymmSiftingBackward(table,moveDown,initialSize);
        }
        if (!result)
            goto ddSymmSiftingConvAuxOutOfMem;

    } else if ((x - xLow) > (xHigh - x)) { /* must go down first: shorter */
        moveDown = ddSymmSiftingDown(table,x,xHigh);
            /* at this point x == xHigh, unless early term */
        if (moveDown == MV_OOM) goto ddSymmSiftingConvAuxOutOfMem;

        if (moveDown != NULL) {
            x = moveDown->y;
            i = x;
            while ((unsigned) i < table->subtables[i].next) {
                i = table->subtables[i].next;
            }
        } else {
            while ((unsigned) x < table->subtables[x].next)
                x = table->subtables[x].next;
            i = x;
            x = table->subtables[x].next;
        }
#ifdef DD_DEBUG
        /* x should be the top of the symmetry group and i the bottom */
        assert((unsigned) i >= table->subtables[i].next);
        assert((unsigned) x == table->subtables[i].next);
#endif
        initGroupSize = i - x + 1;

        moveUp = ddSymmSiftingUp(table,x,xLow);
        if (moveUp == MV_OOM) goto ddSymmSiftingConvAuxOutOfMem;

        if (moveUp != NULL) {
            x = moveUp->x;
            i = table->subtables[x].next;
        } else {
            i = x;
            while ((unsigned) x < table->subtables[x].next)
                x = table->subtables[x].next;
        }
#ifdef DD_DEBUG
        /* x should be the bottom of the symmetry group and i the top */
        assert((unsigned) x >= table->subtables[x].next);
        assert((unsigned) i == table->subtables[x].next);
#endif
        finalGroupSize = x - i + 1;

        if (initGroupSize == finalGroupSize) {
            /* No new symmetry groups detected, return to best position */
            result = ddSymmSiftingBackward(table,moveUp,initialSize);
        } else {
            while (moveDown != NULL) {
                move = moveDown->next;
                cuddDeallocNode(table, (DdNode *) moveDown);
                moveDown = move;
            }
            initialSize = table->keys - table->isolated;
            moveDown = ddSymmSiftingDown(table,x,xHigh);
            result = ddSymmSiftingBackward(table,moveDown,initialSize);
        }
        if (!result) goto ddSymmSiftingConvAuxOutOfMem;

    } else { /* moving up first: shorter */
        /* Find top of x's symmetry group */
        x = table->subtables[x].next;

        moveUp = ddSymmSiftingUp(table,x,xLow);
        /* at this point x == xHigh, unless early term */
        if (moveUp == MV_OOM) goto ddSymmSiftingConvAuxOutOfMem;

        if (moveUp != NULL) {
            x = moveUp->x;
            i = table->subtables[x].next;
        } else {
            i = x;
            while ((unsigned) x < table->subtables[x].next)
                x = table->subtables[x].next;
        }
#ifdef DD_DEBUG
        /* x is bottom of the symmetry group and i is top */
        assert((unsigned) x >= table->subtables[x].next);
        assert((unsigned) i == table->subtables[x].next);
#endif
        initGroupSize = x - i + 1;

        moveDown = ddSymmSiftingDown(table,x,xHigh);
        if (moveDown == MV_OOM) goto ddSymmSiftingConvAuxOutOfMem;

        if (moveDown != NULL) {
            x = moveDown->y;
            i = x;
            while ((unsigned) i < table->subtables[i].next) {
                i = table->subtables[i].next;
            }
        } else {
            i = x;
            x = table->subtables[x].next;
        }
#ifdef DD_DEBUG
        /* x should be the top of the symmetry group and i the bottom */
        assert((unsigned) i >= table->subtables[i].next);
        assert((unsigned) x == table->subtables[i].next);
#endif
        finalGroupSize = i - x + 1;

        if (initGroupSize == finalGroupSize) {
            /* No new symmetries detected, go back to best position */
            result = ddSymmSiftingBackward(table,moveDown,initialSize);
        } else {
            while (moveUp != NULL) {
                move = moveUp->next;
                cuddDeallocNode(table, (DdNode *) moveUp);
                moveUp = move;
            }
            initialSize = table->keys - table->isolated;
            moveUp = ddSymmSiftingUp(table,x,xLow);
            result = ddSymmSiftingBackward(table,moveUp,initialSize);
        }
        if (!result) goto ddSymmSiftingConvAuxOutOfMem;
    }

    while (moveDown != NULL) {
        move = moveDown->next;
        cuddDeallocNode(table, (DdNode *) moveDown);
        moveDown = move;
    }
    while (moveUp != NULL) {
        move = moveUp->next;
        cuddDeallocNode(table, (DdNode *) moveUp);
        moveUp = move;
    }

    return(1);

ddSymmSiftingConvAuxOutOfMem:
    if (moveDown != MV_OOM) {
        while (moveDown != NULL) {
            move = moveDown->next;
            cuddDeallocNode(table, (DdNode *) moveDown);
            moveDown = move;
        }
    }
    if (moveUp != MV_OOM) {
        while (moveUp != NULL) {
            move = moveUp->next;
            cuddDeallocNode(table, (DdNode *) moveUp);
            moveUp = move;
        }
    }

    return(0);

} /* end of ddSymmSiftingConvAux */


static Move *
ddSymmSiftingUp(
  DdManager * table,
  int  y,
  int  xLow)
{
    Move *moves;
    Move *move;
    int  x;
    int  size;
    int  i;
    int  gxtop,gybot;
    int  limitSize;
    int  xindex, yindex;
    int  zindex;
    int  z;
    int  isolated;
    int  L;     /* lower bound on DD size */
#ifdef DD_DEBUG
    int  checkL;
#endif


    moves = NULL;
    yindex = table->invperm[y];

    /* Initialize the lower bound.
    ** The part of the DD below the bottom of y' group will not change.
    ** The part of the DD above y that does not interact with y will not
    ** change. The rest may vanish in the best case, except for
    ** the nodes at level xLow, which will not vanish, regardless.
    */
    limitSize = L = table->keys - table->isolated;
    gybot = y;
    while ((unsigned) gybot < table->subtables[gybot].next)
        gybot = table->subtables[gybot].next;
    for (z = xLow + 1; z <= gybot; z++) {
        zindex = table->invperm[z];
        if (zindex == yindex || cuddTestInteract(table,zindex,yindex)) {
            isolated = table->vars[zindex]->ref == 1;
            L -= table->subtables[z].keys - isolated;
        }
    }

    x = cuddNextLow(table,y);
    while (x >= xLow && L <= limitSize) {
#ifdef DD_DEBUG
        gybot = y;
        while ((unsigned) gybot < table->subtables[gybot].next)
            gybot = table->subtables[gybot].next;
        checkL = table->keys - table->isolated;
        for (z = xLow + 1; z <= gybot; z++) {
            zindex = table->invperm[z];
            if (zindex == yindex || cuddTestInteract(table,zindex,yindex)) {
                isolated = table->vars[zindex]->ref == 1;
                checkL -= table->subtables[z].keys - isolated;
            }
        }
        assert(L == checkL);
#endif
        gxtop = table->subtables[x].next;
        if (cuddSymmCheck(table,x,y)) {
            /* Symmetry found, attach symm groups */
            table->subtables[x].next = y;
            i = table->subtables[y].next;
            while (table->subtables[i].next != (unsigned) y)
                i = table->subtables[i].next;
            table->subtables[i].next = gxtop;
        } else if (table->subtables[x].next == (unsigned) x &&
                   table->subtables[y].next == (unsigned) y) {
            /* x and y have self symmetry */
            xindex = table->invperm[x];
            size = cuddSwapInPlace(table,x,y);
#ifdef DD_DEBUG
            assert(table->subtables[x].next == (unsigned) x);
            assert(table->subtables[y].next == (unsigned) y);
#endif
            if (size == 0) goto ddSymmSiftingUpOutOfMem;
            /* Update the lower bound. */
            if (cuddTestInteract(table,xindex,yindex)) {
                isolated = table->vars[xindex]->ref == 1;
                L += table->subtables[y].keys - isolated;
            }
            move = (Move *) cuddDynamicAllocNode(table);
            if (move == NULL) goto ddSymmSiftingUpOutOfMem;
            move->x = x;
            move->y = y;
            move->size = size;
            move->next = moves;
            moves = move;
            if ((double) size > (double) limitSize * table->maxGrowth)
                return(moves);
            if (size < limitSize) limitSize = size;
        } else { /* Group move */
            size = ddSymmGroupMove(table,x,y,&moves);
            if (size == 0) goto ddSymmSiftingUpOutOfMem;
            /* Update the lower bound. */
            z = moves->y;
            do {
                zindex = table->invperm[z];
                if (cuddTestInteract(table,zindex,yindex)) {
                    isolated = table->vars[zindex]->ref == 1;
                    L += table->subtables[z].keys - isolated;
                }
                z = table->subtables[z].next;
            } while (z != (int) moves->y);
            if ((double) size > (double) limitSize * table->maxGrowth)
                return(moves);
            if (size < limitSize) limitSize = size;
        }
        y = gxtop;
        x = cuddNextLow(table,y);
    }

    return(moves);

ddSymmSiftingUpOutOfMem:
    while (moves != NULL) {
        move = moves->next;
        cuddDeallocNode(table, (DdNode *) moves);
        moves = move;
    }
    return(MV_OOM);

} /* end of ddSymmSiftingUp */


static Move *
ddSymmSiftingDown(
  DdManager * table,
  int  x,
  int  xHigh)
{
    Move *moves;
    Move *move;
    int  y;
    int  size;
    int  limitSize;
    int  gxtop,gybot;
    int  R;     /* upper bound on node decrease */
    int  xindex, yindex;
    int  isolated;
    int  z;
    int  zindex;
#ifdef DD_DEBUG
    int  checkR;
#endif

    moves = NULL;
    /* Initialize R */
    xindex = table->invperm[x];
    gxtop = table->subtables[x].next;
    limitSize = size = table->keys - table->isolated;
    R = 0;
    for (z = xHigh; z > gxtop; z--) {
        zindex = table->invperm[z];
        if (zindex == xindex || cuddTestInteract(table,xindex,zindex)) {
            isolated = table->vars[zindex]->ref == 1;
            R += table->subtables[z].keys - isolated;
        }
    }

    y = cuddNextHigh(table,x);
    while (y <= xHigh && size - R < limitSize) {
#ifdef DD_DEBUG
        gxtop = table->subtables[x].next;
        checkR = 0;
        for (z = xHigh; z > gxtop; z--) {
            zindex = table->invperm[z];
            if (zindex == xindex || cuddTestInteract(table,xindex,zindex)) {
                isolated = table->vars[zindex]->ref == 1;
                checkR += table->subtables[z].keys - isolated;
            }
        }
        assert(R == checkR);
#endif
        gybot = table->subtables[y].next;
        while (table->subtables[gybot].next != (unsigned) y)
            gybot = table->subtables[gybot].next;
        if (cuddSymmCheck(table,x,y)) {
            /* Symmetry found, attach symm groups */
            gxtop = table->subtables[x].next;
            table->subtables[x].next = y;
            table->subtables[gybot].next = gxtop;
        } else if (table->subtables[x].next == (unsigned) x &&
                   table->subtables[y].next == (unsigned) y) {
            /* x and y have self symmetry */
            /* Update upper bound on node decrease. */
            yindex = table->invperm[y];
            if (cuddTestInteract(table,xindex,yindex)) {
                isolated = table->vars[yindex]->ref == 1;
                R -= table->subtables[y].keys - isolated;
            }
            size = cuddSwapInPlace(table,x,y);
#ifdef DD_DEBUG
            assert(table->subtables[x].next == (unsigned) x);
            assert(table->subtables[y].next == (unsigned) y);
#endif
            if (size == 0) goto ddSymmSiftingDownOutOfMem;
            move = (Move *) cuddDynamicAllocNode(table);
            if (move == NULL) goto ddSymmSiftingDownOutOfMem;
            move->x = x;
            move->y = y;
            move->size = size;
            move->next = moves;
            moves = move;
            if ((double) size > (double) limitSize * table->maxGrowth)
                return(moves);
            if (size < limitSize) limitSize = size;
        } else { /* Group move */
            /* Update upper bound on node decrease: first phase. */
            gxtop = table->subtables[x].next;
            z = gxtop + 1;
            do {
                zindex = table->invperm[z];
                if (zindex == xindex || cuddTestInteract(table,xindex,zindex)) {
                    isolated = table->vars[zindex]->ref == 1;
                    R -= table->subtables[z].keys - isolated;
                }
                z++;
            } while (z <= gybot);
            size = ddSymmGroupMove(table,x,y,&moves);
            if (size == 0) goto ddSymmSiftingDownOutOfMem;
            if ((double) size > (double) limitSize * table->maxGrowth)
                return(moves);
            if (size < limitSize) limitSize = size;
            /* Update upper bound on node decrease: second phase. */
            gxtop = table->subtables[gybot].next;
            for (z = gxtop + 1; z <= gybot; z++) {
                zindex = table->invperm[z];
                if (zindex == xindex || cuddTestInteract(table,xindex,zindex)) {
                    isolated = table->vars[zindex]->ref == 1;
                    R += table->subtables[z].keys - isolated;
                }
            }
        }
        x = gybot;
        y = cuddNextHigh(table,x);
    }

    return(moves);

ddSymmSiftingDownOutOfMem:
    while (moves != NULL) {
        move = moves->next;
        cuddDeallocNode(table, (DdNode *) moves);
        moves = move;
    }
    return(MV_OOM);

} /* end of ddSymmSiftingDown */


static int
ddSymmGroupMove(
  DdManager * table,
  int  x,
  int  y,
  Move ** moves)
{
    Move *move;
    int  size;
    int  i,j;
    int  xtop,xbot,xsize,ytop,ybot,ysize,newxtop;
    int  swapx,swapy;

#if DD_DEBUG
    assert(x < y);      /* we assume that x < y */
#endif
    /* Find top, bottom, and size for the two groups. */
    xbot = x;
    xtop = table->subtables[x].next;
    xsize = xbot - xtop + 1;
    ybot = y;
    while ((unsigned) ybot < table->subtables[ybot].next)
        ybot = table->subtables[ybot].next;
    ytop = y;
    ysize = ybot - ytop + 1;

    /* Sift the variables of the second group up through the first group. */
    for (i = 1; i <= ysize; i++) {
        for (j = 1; j <= xsize; j++) {
            size = cuddSwapInPlace(table,x,y);
            if (size == 0) return(0);
            swapx = x; swapy = y;
            y = x;
            x = y - 1;
        }
        y = ytop + i;
        x = y - 1;
    }

    /* fix symmetries */
    y = xtop; /* ytop is now where xtop used to be */
    for (i = 0; i < ysize-1 ; i++) {
        table->subtables[y].next = y + 1;
        y = y + 1;
    }
    table->subtables[y].next = xtop; /* y is bottom of its group, join */
                                     /* its symmetry to top of its group */
    x = y + 1;
    newxtop = x;
    for (i = 0; i < xsize - 1 ; i++) {
        table->subtables[x].next = x + 1;
        x = x + 1;
    }
    table->subtables[x].next = newxtop; /* x is bottom of its group, join */
                                        /* its symmetry to top of its group */
    /* Store group move */
    move = (Move *) cuddDynamicAllocNode(table);
    if (move == NULL) return(0);
    move->x = swapx;
    move->y = swapy;
    move->size = size;
    move->next = *moves;
    *moves = move;

    return(size);

} /* end of ddSymmGroupMove */


static int
ddSymmGroupMoveBackward(
  DdManager * table,
  int  x,
  int  y)
{
    int size;
    int i,j;
    int xtop,xbot,xsize,ytop,ybot,ysize,newxtop;

#if DD_DEBUG
    assert(x < y); /* We assume that x < y */
#endif

    /* Find top, bottom, and size for the two groups. */
    xbot = x;
    xtop = table->subtables[x].next;
    xsize = xbot - xtop + 1;
    ybot = y;
    while ((unsigned) ybot < table->subtables[ybot].next)
        ybot = table->subtables[ybot].next;
    ytop = y;
    ysize = ybot - ytop + 1;

    /* Sift the variables of the second group up through the first group. */
    for (i = 1; i <= ysize; i++) {
        for (j = 1; j <= xsize; j++) {
            size = cuddSwapInPlace(table,x,y);
            if (size == 0) return(0);
            y = x;
            x = cuddNextLow(table,y);
        }
        y = ytop + i;
        x = y - 1;
    }

    /* Fix symmetries. */
    y = xtop;
    for (i = 0; i < ysize-1 ; i++) {
        table->subtables[y].next = y + 1;
        y = y + 1;
    }
    table->subtables[y].next = xtop; /* y is bottom of its group, join */
                                     /* its symmetry to top of its group */
    x = y + 1;
    newxtop = x;
    for (i = 0; i < xsize-1 ; i++) {
        table->subtables[x].next = x + 1;
        x = x + 1;
    }
    table->subtables[x].next = newxtop; /* x is bottom of its group, join */
                                        /* its symmetry to top of its group */

    return(size);

} /* end of ddSymmGroupMoveBackward */


static int
ddSymmSiftingBackward(
  DdManager * table,
  Move * moves,
  int  size)
{
    Move *move;
    int  res;

    for (move = moves; move != NULL; move = move->next) {
        if (move->size < size) {
            size = move->size;
        }
    }

    for (move = moves; move != NULL; move = move->next) {
        if (move->size == size) return(1);
        if (table->subtables[move->x].next == move->x && table->subtables[move->y].next == move->y) {
            res = cuddSwapInPlace(table,(int)move->x,(int)move->y);
#ifdef DD_DEBUG
            assert(table->subtables[move->x].next == move->x);
            assert(table->subtables[move->y].next == move->y);
#endif
        } else { /* Group move necessary */
            res = ddSymmGroupMoveBackward(table,(int)move->x,(int)move->y);
        }
        if (!res) return(0);
    }

    return(1);

} /* end of ddSymmSiftingBackward */


static void
ddSymmSummary(
  DdManager * table,
  int  lower,
  int  upper,
  int * symvars,
  int * symgroups)
{
    int i,x,gbot;
    int TotalSymm = 0;
    int TotalSymmGroups = 0;

    for (i = lower; i <= upper; i++) {
        if (table->subtables[i].next != (unsigned) i) {
            TotalSymmGroups++;
            x = i;
            do {
                TotalSymm++;
                gbot = x;
                x = table->subtables[x].next;
            } while (x != i);
#ifdef DD_DEBUG
            assert(table->subtables[gbot].next == (unsigned) i);
#endif
            i = gbot;
        }
    }
    *symvars = TotalSymm;
    *symgroups = TotalSymmGroups;

    return;

} /* end of ddSymmSummary */

---