tempest/resources/3rdparty/cudd-3.0.0/cudd/cuddSymmetry.c

/**
  @file

  @ingroup cudd

  @brief Functions for symmetry-based variable reordering.

  @author Shipra Panda, Fabio Somenzi

  @copyright@parblock
  Copyright (c) 1995-2015, 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.
  @endparblock

*/

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

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

#define MV_OOM (Move *)1

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

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

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

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

/** \cond */

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

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

/** \endcond */


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


/**
  @brief Prints statistics on symmetric variables.

  @details The information is accurate only if this function is called
  right after reordering with methods CUDD_REORDER_SYMM_SIFT or
  CUDD_REORDER_SYMM_SIFT_CONV.

  @sideeffect None

*/
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                                          */
/*---------------------------------------------------------------------------*/


/**
  @brief Checks for symmetry of x and y.

  @details Ignores projection functions, unless they are isolated.

  @return 1 in case of symmetry; 0 otherwise.

  @sideeffect None

*/
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 */


/**
  @brief Symmetric sifting algorithm.

  @details Assumes that no dead nodes are present.
    <ol>
    <li> Order all the variables according to the number of entries in
    each unique subtable.
    <li> Sift the variable up and down, remembering each time the total
    size of the DD heap and grouping variables that are symmetric.
    <li> Select the best permutation.
    <li> Repeat 3 and 4 for all variables.
    </ol>

  @return 1 plus the number of symmetric variables if successful; 0
  otherwise.

  @sideeffect None

  @see cuddSymmSiftingConv

*/
int
cuddSymmSifting(
  DdManager * table,
  int  lower,
  int  upper)
{
    int		i;
    IndexKey	*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 = ALLOC(IndexKey,size);
    if (var == NULL) {
	table->errorCode = CUDD_MEMORY_OUT;
	goto ddSymmSiftingOutOfMem;
    }

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

    util_qsort(var,size,sizeof(IndexKey),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 (table->ddTotalNumberSwapping >= table->siftMaxSwap)
	    break;
        if (util_cpu_time() - table->startTime > table->timeLimit) {
            table->autoDyn = 0; /* prevent further reordering */
            break;
        }
        if (table->terminationCallback != NULL &&
            table->terminationCallback(table->tcbArg)) {
            table->autoDyn = 0; /* prevent further reordering */
            break;
        }
	x = table->perm[var[i].index];
#ifdef DD_STATS
	previousSize = (int) (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);

    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 (var != NULL) FREE(var);

    return(0);

} /* end of cuddSymmSifting */


/**
  @brief Symmetric sifting to convergence algorithm.

  @details Assumes that no dead nodes are present.
    <ol>
    <li> Order all the variables according to the number of entries in
    each unique subtable.
    <li> Sift the variable up and down, remembering each time the total
    size of the %DD heap and grouping variables that are symmetric.
    <li> Select the best permutation.
    <li> Repeat 3 and 4 for all variables.
    <li> Repeat 1-4 until no further improvement.
    </ol>

  @return 1 plus the number of symmetric variables if successful; 0
  otherwise.

  @sideeffect None

  @see cuddSymmSifting

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

    initialSize = (int) (table->keys - table->isolated);

    size = table->size;

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

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

    util_qsort(var,size,sizeof(IndexKey),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 (table->ddTotalNumberSwapping >= table->siftMaxSwap)
	    break;
        if (util_cpu_time() - table->startTime > table->timeLimit) {
            table->autoDyn = 0; /* prevent further reordering */
            break;
        }
        if (table->terminationCallback != NULL &&
            table->terminationCallback(table->tcbArg)) {
            table->autoDyn = 0; /* prevent further reordering */
            break;
        }
	x = table->perm[var[i].index];
	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 = (int) (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 = (int) (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];
	    var[classes].keys = table->subtables[x].keys;
	    var[classes].index = i;
	}

	util_qsort(var,classes,sizeof(IndexKey),ddSymmUniqueCompare);

	/* Now sift. */
	for (i = 0; i < ddMin(table->siftMaxVar,classes); i++) {
	    if (table->ddTotalNumberSwapping >= table->siftMaxSwap)
		break;
            if (util_cpu_time() - table->startTime > table->timeLimit) {
              table->autoDyn = 0; /* prevent further reordering */
              break;
            }
            if (table->terminationCallback != NULL &&
                table->terminationCallback(table->tcbArg)) {
                table->autoDyn = 0; /* prevent further reordering */
                break;
            }
	    x = table->perm[var[i].index];
	    if ((unsigned) x >= table->subtables[x].next) {
#ifdef DD_STATS
              previousSize = (int) (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);

    return(1+symvars);

ddSymmSiftingConvOutOfMem:

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

    return(0);

} /* end of cuddSymmSiftingConv */


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


/**
  @brief Comparison function used by qsort.

  @details Used to order the variables according to the number of keys
  in the subtables.

  @return the difference in number of keys between the two variables
  being compared.

  @sideeffect None

*/
static int
ddSymmUniqueCompare(
  void const * ptrX,
  void const * ptrY)
{
    IndexKey const * pX = (IndexKey const *) ptrX;
    IndexKey const * pY = (IndexKey const *) ptrY;
#if 0
    if (pY->keys == pX->keys) {
	return(pX->index - pY->index);
    }
#endif
    return(pY->keys - pX->keys);

} /* end of ddSymmUniqueCompare */


/**
  @brief Given xLow <= x <= xHigh moves x up and down between the
  boundaries.

  @details Finds the best position and does the required changes.
  Assumes that x is not part of a symmetry group.

  @return 1 if successful; 0 otherwise.

  @sideeffect None

*/
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 = (int) (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 = (int) (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 = (int) (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;
		cuddDeallocMove(table, moveDown);
		moveDown = move;
	    }
	    initialSize = (int) (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;
		cuddDeallocMove(table, moveUp);
		moveUp = move;
	    }
	    initialSize = (int) (table->keys - table->isolated);
	    moveUp = ddSymmSiftingUp(table,x,xLow);
	    result = ddSymmSiftingBackward(table,moveUp,initialSize);
	}
	if (!result) goto ddSymmSiftingAuxOutOfMem;
    }

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

    return(1);

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

    return(0);

} /* end of ddSymmSiftingAux */


/**
  @brief Given xLow <= x <= xHigh moves x up and down between the
  boundaries.

  @details Finds the best position and does the required changes.
  Assumes that x is either an isolated variable, or it is the bottom of
  a symmetry group. All symmetries may not have been found, because of
  exceeded growth limit.

  @return 1 if successful; 0 otherwise.

  @sideeffect None

*/
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 = (int) (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 = (int) (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 = (int) (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;
		cuddDeallocMove(table, moveDown);
		moveDown = move;
	    }
	    initialSize = (int) (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;
		cuddDeallocMove(table, moveUp);
		moveUp = move;
	    }
	    initialSize = (int) (table->keys - table->isolated);
	    moveUp = ddSymmSiftingUp(table,x,xLow);
	    result = ddSymmSiftingBackward(table,moveUp,initialSize);
	}
	if (!result) goto ddSymmSiftingConvAuxOutOfMem;
    }

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

    return(1);

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

    return(0);

} /* end of ddSymmSiftingConvAux */


/**
  @brief Moves x up until either it reaches the bound (xLow) or
  the size of the %DD heap increases too much.

  @details Assumes that x is the top of a symmetry group.  Checks x
  for symmetry to the adjacent variables. If symmetry is found, the
  symmetry group of x is merged with the symmetry group of the other
  variable.

  @return the set of moves in case of success; MV_OOM if memory is
  full.

  @sideeffect None

*/
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 = (int) (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 -= (int) 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 = (int) (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 -= (int) 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 += (int) 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 += (int) 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;
	cuddDeallocMove(table, moves);
	moves = move;
    }
    return(MV_OOM);

} /* end of ddSymmSiftingUp */


/**
  @brief Moves x down until either it reaches the bound (xHigh) or
  the size of the %DD heap increases too much.

  @details Assumes that x is the bottom of a symmetry group. Checks x
  for symmetry to the adjacent variables. If symmetry is found, the
  symmetry group of x is merged with the symmetry group of the other
  variable.

  @return the set of moves in case of success; MV_OOM if memory is
  full.

  @sideeffect None

*/
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 = (int) (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 += (int) 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 += (int) 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 -= (int) 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 -= (int) 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 += (int) table->subtables[z].keys - isolated;
		}
	    }
	}
	x = gybot;
	y = cuddNextHigh(table,x);
    }

    return(moves);

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

} /* end of ddSymmSiftingDown */


/**
  @brief Swaps two groups.

  @details x is assumed to be the bottom variable of the first
  group. y is assumed to be the top variable of the second group.
  Updates the list of moves.

  @return the number of keys in the table if successful; 0 otherwise.

  @sideeffect None

*/
static int
ddSymmGroupMove(
  DdManager * table,
  int  x,
  int  y,
  Move ** moves)
{
    Move *move;
    int	 size = 0;
    int  i,j;
    int  xtop,xbot,xsize,ytop,ybot,ysize,newxtop;
    int  swapx = 0, swapy = 0;

#ifdef 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 */


/**
  @brief Undoes the swap of two groups.

  @details x is assumed to be the bottom variable of the first
  group. y is assumed to be the top variable of the second group.

  @return the number of keys in the table if successful; 0 otherwise.

  @sideeffect None

*/
static int
ddSymmGroupMoveBackward(
  DdManager * table,
  int  x,
  int  y)
{
  int	size = (int) (table->keys - table->isolated);
    int i,j;
    int	xtop,xbot,xsize,ytop,ybot,ysize,newxtop;

#ifdef 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;
#ifdef DD_DEBUG
    assert(xsize > 0);
    assert(ysize > 0);
#endif

    /* 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 */


/**
  @brief Given a set of moves, returns the %DD heap to the position
  giving the minimum size.

  @details In case of ties, returns to the closest position giving the
  minimum size.

  @return 1 in case of success; 0 otherwise.

  @sideeffect None

*/
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 */


/**
  @brief Counts numbers of symmetric variables and symmetry groups.

  @sideeffect None

*/
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 */