/**
  @file

  @ingroup cudd

  @brief Functions for exact variable reordering.

  @author Cheng Hua, 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                                                     */
/*---------------------------------------------------------------------------*/

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

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

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

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

/** \cond */

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

static int getMaxBinomial (int n);
static DdHalfWord ** getMatrix (int rows, int cols);
static void freeMatrix (DdHalfWord **matrix);
static int getLevelKeys (DdManager *table, int l);
static int ddShuffle (DdManager *table, DdHalfWord *permutation, int lower, int upper);
static int ddSiftUp (DdManager *table, int x, int xLow);
static int updateUB (DdManager *table, int oldBound, DdHalfWord *bestOrder, int lower, int upper);
static int ddCountRoots (DdManager *table, int lower, int upper);
static void ddClearGlobal (DdManager *table, int lower, int maxlevel);
static int computeLB (DdManager *table, DdHalfWord *order, int roots, int cost, int lower, int upper, int level);
static int updateEntry (DdManager *table, DdHalfWord *order, int level, int cost, DdHalfWord **orders, int *costs, int subsets, char *mask, int lower, int upper);
static void pushDown (DdHalfWord *order, int j, int level);
static DdHalfWord * initSymmInfo (DdManager *table, int lower, int upper);
static int checkSymmInfo (DdManager *table, DdHalfWord *symmInfo, int index, int level);

/** \endcond */


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

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


/**
  @brief Exact variable ordering algorithm.

  @details Finds an optimum order for the variables between lower and
  upper.

  @return 1 if successful; 0 otherwise.

  @sideeffect None

*/
int
cuddExact(
  DdManager * table,
  int  lower,
  int  upper)
{
    int k, i, j;
    int maxBinomial, oldSubsets, newSubsets;
    int subsetCost;
    int size;			/* number of variables to be reordered */
    int unused, nvars, level, result;
    int upperBound, lowerBound, cost;
    int roots;
    char *mask = NULL;
    DdHalfWord  *symmInfo = NULL;
    DdHalfWord **newOrder = NULL;
    DdHalfWord **oldOrder = NULL;
    int *newCost = NULL;
    int *oldCost = NULL;
    DdHalfWord **tmpOrder;
    int *tmpCost;
    DdHalfWord *bestOrder = NULL;
    DdHalfWord *order;
#ifdef DD_STATS
    int  ddTotalSubsets;
#endif

    /* Restrict the range to be reordered by excluding unused variables
    ** at the two ends. */
    while (table->subtables[lower].keys == 1 &&
	   table->vars[table->invperm[lower]]->ref == 1 &&
	   lower < upper)
	lower++;
    while (table->subtables[upper].keys == 1 &&
	   table->vars[table->invperm[upper]]->ref == 1 &&
	   lower < upper)
	upper--;
    if (lower == upper) return(1); /* trivial problem */

    /* Apply symmetric sifting to get a good upper bound and to extract
    ** symmetry information. */
    result = cuddSymmSiftingConv(table,lower,upper);
    if (result == 0) goto cuddExactOutOfMem;

#ifdef DD_STATS
    (void) fprintf(table->out,"\n");
    table->totalShuffles = 0;
    ddTotalSubsets = 0;
#endif

    /* Initialization. */
    nvars = table->size;
    size = upper - lower + 1;
    /* Count unused variable among those to be reordered.  This is only
    ** used to compute maxBinomial. */
    unused = 0;
    for (i = lower + 1; i < upper; i++) {
	if (table->subtables[i].keys == 1 &&
	    table->vars[table->invperm[i]]->ref == 1)
	    unused++;
    }

    /* Find the maximum number of subsets we may have to store. */
    maxBinomial = getMaxBinomial(size - unused);
    if (maxBinomial == -1) goto cuddExactOutOfMem;

    newOrder = getMatrix(maxBinomial, size);
    if (newOrder == NULL) goto cuddExactOutOfMem;

    newCost = ALLOC(int, maxBinomial);
    if (newCost == NULL) goto cuddExactOutOfMem;

    oldOrder = getMatrix(maxBinomial, size);
    if (oldOrder == NULL) goto cuddExactOutOfMem;

    oldCost = ALLOC(int, maxBinomial);
    if (oldCost == NULL) goto cuddExactOutOfMem;

    bestOrder = ALLOC(DdHalfWord, size);
    if (bestOrder == NULL) goto cuddExactOutOfMem;

    mask = ALLOC(char, nvars);
    if (mask == NULL) goto cuddExactOutOfMem;

    symmInfo = initSymmInfo(table, lower, upper);
    if (symmInfo == NULL) goto cuddExactOutOfMem;

    roots = ddCountRoots(table, lower, upper);

    /* Initialize the old order matrix for the empty subset and the best
    ** order to the current order. The cost for the empty subset includes
    ** the cost of the levels between upper and the constants. These levels
    ** are not going to change. Hence, we count them only once.
    */
    oldSubsets = 1;
    for (i = 0; i < size; i++) {
	oldOrder[0][i] = bestOrder[i] = (DdHalfWord) table->invperm[i+lower];
    }
    subsetCost = (int) table->constants.keys;
    for (i = upper + 1; i < nvars; i++)
	subsetCost += getLevelKeys(table,i);
    oldCost[0] = subsetCost;
    /* The upper bound is initialized to the current size of the BDDs. */
    upperBound = (int) (table->keys - table->isolated);

    /* Now consider subsets of increasing size. */
    for (k = 1; k <= size; k++) {
#ifdef DD_STATS
	(void) fprintf(table->out,"Processing subsets of size %d\n", k);
	fflush(table->out);
#endif
	newSubsets = 0;
	level = size - k;		/* offset of first bottom variable */

	for (i = 0; i < oldSubsets; i++) { /* for each subset of size k-1 */
	    order = oldOrder[i];
	    cost = oldCost[i];
	    lowerBound = computeLB(table, order, roots, cost, lower, upper,
				   level);
	    if (lowerBound >= upperBound)
		continue;
	    /* Impose new order. */
	    result = ddShuffle(table, order, lower, upper);
	    if (result == 0) goto cuddExactOutOfMem;
	    upperBound = updateUB(table,upperBound,bestOrder,lower,upper);
	    /* For each top bottom variable. */
	    for (j = level; j >= 0; j--) {
		/* Skip unused variables. */
		if (table->subtables[j+lower-1].keys == 1 &&
		    table->vars[table->invperm[j+lower-1]]->ref == 1) continue;
		/* Find cost under this order. */
		subsetCost = cost + getLevelKeys(table, lower + level);
		newSubsets = updateEntry(table, order, level, subsetCost,
					 newOrder, newCost, newSubsets, mask,
					 lower, upper);
		if (j == 0)
		    break;
		if (checkSymmInfo(table, symmInfo, (int) order[j-1], level) == 0)
		    continue;
		pushDown(order,j-1,level);
		/* Impose new order. */
		result = ddShuffle(table, order, lower, upper);
		if (result == 0) goto cuddExactOutOfMem;
		upperBound = updateUB(table,upperBound,bestOrder,lower,upper);
	    } /* for each bottom variable */
	} /* for each subset of size k */

	/* New orders become old orders in preparation for next iteration. */
	tmpOrder = oldOrder; tmpCost = oldCost;
	oldOrder = newOrder; oldCost = newCost;
	newOrder = tmpOrder; newCost = tmpCost;
#ifdef DD_STATS
	ddTotalSubsets += newSubsets;
#endif
	oldSubsets = newSubsets;
    }
    result = ddShuffle(table, bestOrder, lower, upper);
    if (result == 0) goto cuddExactOutOfMem;
#ifdef DD_STATS
#ifdef DD_VERBOSE
    (void) fprintf(table->out,"\n");
#endif
    (void) fprintf(table->out,"#:S_EXACT   %8d: total subsets\n",
		   ddTotalSubsets);
    (void) fprintf(table->out,"#:H_EXACT   %8d: total shuffles",
		   table->totalShuffles);
#endif

    freeMatrix(newOrder);
    freeMatrix(oldOrder);
    FREE(bestOrder);
    FREE(oldCost);
    FREE(newCost);
    FREE(symmInfo);
    FREE(mask);
    return(1);

cuddExactOutOfMem:

    if (newOrder != NULL) freeMatrix(newOrder);
    if (oldOrder != NULL) freeMatrix(oldOrder);
    if (bestOrder != NULL) FREE(bestOrder);
    if (oldCost != NULL) FREE(oldCost);
    if (newCost != NULL) FREE(newCost);
    if (symmInfo != NULL) FREE(symmInfo);
    if (mask != NULL) FREE(mask);
    table->errorCode = CUDD_MEMORY_OUT;
    return(0);

} /* end of cuddExact */


/**
  @brief Returns the maximum value of `(n choose k)` for a given `n`.

  @details Computes the maximum value of `(n choose k)` for a given
  `n`.  The maximum value occurs for `k = n/2` when `n` is even, or `k =
  (n-1)/2` when `n` is odd.  The algorithm used in this procedure avoids
  intermediate overflow problems.  It is based on the identity

      binomial(n,k) = n/k * binomial(n-1,k-1).

  @return the computed value if successful; -1 if out of range.

  @sideeffect None

*/
static int
getMaxBinomial(
  int n)
{
    double i, j, result;

    if (n < 0 || n > 33) return(-1); /* error */
    if (n < 2) return(1);

    for (result = (double)((n+3)/2), i = result+1, j=2; i <= n; i++, j++) {
	result *= i;
	result /= j;
    }

    return((int)result);

} /* end of getMaxBinomial */


#if 0
/**
  @brief Returns the gcd of two integers.

  @details Uses the binary GCD algorithm described in Cormen,
  Leiserson, and Rivest.

  @sideeffect None

*/
static int
gcd(
  int  x,
  int  y)
{
    int a;
    int b;
    int lsbMask;

    /* GCD(n,0) = n. */
    if (x == 0) return(y);
    if (y == 0) return(x);

    a = x; b = y; lsbMask = 1;

    /* Here both a and b are != 0. The iteration maintains this invariant.
    ** Hence, we only need to check for when they become equal.
    */
    while (a != b) {
	if (a & lsbMask) {
	    if (b & lsbMask) {	/* both odd */
		if (a < b) {
		    b = (b - a) >> 1;
		} else {
		    a = (a - b) >> 1;
		}
	    } else {		/* a odd, b even */
		b >>= 1;
	    }
	} else {
	    if (b & lsbMask) {	/* a even, b odd */
		a >>= 1;
	    } else {		/* both even */
		lsbMask <<= 1;
	    }
	}
    }

    return(a);

} /* end of gcd */
#endif


/**
  @brief Allocates a two-dimensional matrix of ints.

  @return the pointer to the matrix if successful; NULL otherwise.

  @sideeffect None

  @see freeMatrix

*/
static DdHalfWord **
getMatrix(
  int  rows /* number of rows */,
  int  cols /* number of columns */)
{
    DdHalfWord **matrix;
    int i;

    if (cols*rows == 0) return(NULL);
    matrix = ALLOC(DdHalfWord *, rows);
    if (matrix == NULL) return(NULL);
    matrix[0] = ALLOC(DdHalfWord, cols*rows);
    if (matrix[0] == NULL) {
	FREE(matrix);
	return(NULL);
    }
    for (i = 1; i < rows; i++) {
	matrix[i] = matrix[i-1] + cols;
    }
    return(matrix);

} /* end of getMatrix */


/**
  @brief Frees a two-dimensional matrix allocated by getMatrix.

  @sideeffect None

  @see getMatrix

*/
static void
freeMatrix(
  DdHalfWord ** matrix)
{
    FREE(matrix[0]);
    FREE(matrix);
    return;

} /* end of freeMatrix */


/**
  @brief Returns the number of nodes at one level of a unique table.

  @details The projection function, if isolated, is not counted.

  @sideeffect None

*/
static int
getLevelKeys(
  DdManager * table,
  int  l)
{
    int isolated;
    int x;        /* x is an index */

    x = table->invperm[l];
    isolated = table->vars[x]->ref == 1;

    return((int) table->subtables[l].keys - isolated);

} /* end of getLevelKeys */


/**
  @brief Reorders variables according to a given permutation.

  @details The i-th permutation array contains the index of the
  variable that should be brought to the i-th level. ddShuffle assumes
  that no dead nodes are present and that the interaction matrix is
  properly initialized.  The reordering is achieved by a series of
  upward sifts.

  @return 1 if successful; 0 otherwise.

  @sideeffect None

*/
static int
ddShuffle(
  DdManager * table,
  DdHalfWord * permutation,
  int  lower,
  int  upper)
{
    DdHalfWord	index;
    int		level;
    int		position;
#if 0
    int		numvars;
#endif
    int		result;
#if defined(DD_STATS) && defined(DD_VERBOSE)
    int		initialSize;
    int		finalSize;
#endif

#if defined(DD_STATS) && defined(DD_VERBOSE)
    initialSize = (int) (table->keys - table->isolated);
#endif

#if 0
    numvars = table->size;

    (void) fprintf(table->out,"%d:", table->totalShuffles);
    for (level = 0; level < numvars; level++) {
	(void) fprintf(table->out," %d", table->invperm[level]);
    }
    (void) fprintf(table->out,"\n");
#endif

    for (level = 0; level <= upper - lower; level++) {
	index = permutation[level];
	position = table->perm[index];
	result = ddSiftUp(table,position,level+lower);
	if (!result) return(0);
    }

#ifdef DD_STATS
    table->totalShuffles++;
#ifdef DD_VERBOSE
    finalSize = (int) (table->keys - table->isolated);
    if (finalSize < initialSize) {
	(void) fprintf(table->out,"-");
    } else if (finalSize > initialSize) {
	(void) fprintf(table->out,"+");
    } else {
	(void) fprintf(table->out,"=");
    }
    if ((table->totalShuffles & 63) == 0) (void) fprintf(table->out,"\n");
    fflush(table->out);
#endif
#endif

    return(1);

} /* end of ddShuffle */


/**
  @brief Moves one variable up.

  @details Takes a variable from position x and sifts it up to
  position xLow;  xLow should be less than or equal to x.

  @return 1 if successful; 0 otherwise

  @sideeffect None

*/
static int
ddSiftUp(
  DdManager * table,
  int  x,
  int  xLow)
{
    int        y;
    int        size;

    y = cuddNextLow(table,x);
    while (y >= xLow) {
	size = cuddSwapInPlace(table,y,x);
	if (size == 0) {
	    return(0);
	}
	x = y;
	y = cuddNextLow(table,x);
    }
    return(1);

} /* end of ddSiftUp */


/**
  @brief Updates the upper bound and saves the best order seen so far.

  @return the current value of the upper bound.

  @sideeffect None

*/
static int
updateUB(
  DdManager * table,
  int  oldBound,
  DdHalfWord * bestOrder,
  int  lower,
  int  upper)
{
    int i;
    int newBound = (int) (table->keys - table->isolated);

    if (newBound < oldBound) {
#ifdef DD_STATS
	(void) fprintf(table->out,"New upper bound = %d\n", newBound);
	fflush(table->out);
#endif
	for (i = lower; i <= upper; i++)
	    bestOrder[i-lower] = (DdHalfWord) table->invperm[i];
	return(newBound);
    } else {
	return(oldBound);
    }

} /* end of updateUB */


/**
  @brief Counts the number of roots.

  @details Counts the number of roots at the levels between lower and
  upper.  The computation is based on breadth-first search.  A node is
  a root if it is not reachable from any previously visited node.
  (All the nodes at level lower are therefore considered roots.)  The
  roots that are constant nodes are always ignored.  The visited flag
  uses the LSB of the next pointer.

  @return the root count.

  @sideeffect None

  @see ddClearGlobal

*/
static int
ddCountRoots(
  DdManager * table,
  int  lower,
  int  upper)
{
    int i,j;
    DdNode *f;
    DdNodePtr *nodelist;
    DdNode *sentinel = &(table->sentinel);
    int slots;
    int roots = 0;
    int maxlevel = lower;

    for (i = lower; i <= upper; i++) {
	nodelist = table->subtables[i].nodelist;
	slots = (int) table->subtables[i].slots;
	for (j = 0; j < slots; j++) {
	    f = nodelist[j];
	    while (f != sentinel) {
		/* A node is a root of the DAG if it cannot be
		** reached by nodes above it. If a node was never
		** reached during the previous depth-first searches,
		** then it is a root, and we start a new depth-first
		** search from it.
		*/
		if (!Cudd_IsComplement(f->next)) {
		    if (f != table->vars[f->index]) {
			roots++;
		    }
		}
		if (!cuddIsConstant(cuddT(f))) {
		    cuddT(f)->next = Cudd_Complement(cuddT(f)->next);
		    if (table->perm[cuddT(f)->index] > maxlevel)
			maxlevel = table->perm[cuddT(f)->index];
		}
		if (!Cudd_IsConstantInt(cuddE(f))) {
		    Cudd_Regular(cuddE(f))->next =
			Cudd_Complement(Cudd_Regular(cuddE(f))->next);
		    if (table->perm[Cudd_Regular(cuddE(f))->index] > maxlevel)
			maxlevel = table->perm[Cudd_Regular(cuddE(f))->index];
		}
		f = Cudd_Regular(f->next);
	    }
	}
    }
    ddClearGlobal(table, lower, maxlevel);

    return(roots);

} /* end of ddCountRoots */


/**
  @brief Scans the %DD and clears the LSB of the next pointers.

  @details The LSB of the next pointers are used as markers to tell
  whether a node was reached. Once the roots are counted, these flags
  are reset.

  @sideeffect None

  @see ddCountRoots

*/
static void
ddClearGlobal(
  DdManager * table,
  int  lower,
  int  maxlevel)
{
    int i,j;
    DdNode *f;
    DdNodePtr *nodelist;
    DdNode *sentinel = &(table->sentinel);
    int slots;

    for (i = lower; i <= maxlevel; i++) {
	nodelist = table->subtables[i].nodelist;
	slots = (int) table->subtables[i].slots;
	for (j = 0; j < slots; j++) {
	    f = nodelist[j];
	    while (f != sentinel) {
		f->next = Cudd_Regular(f->next);
		f = f->next;
	    }
	}
    }

} /* end of ddClearGlobal */


/**
  @brief Computes a lower bound on the size of a %BDD.

  @details The lower bound depends on the following factors:
  <ul>
  <li> size of the lower part of it;
  <li> size of the part of the upper part not subjected to reordering;
  <li> number of roots in the part of the %BDD subjected to reordering;
  <li> variable in the support of the roots in the upper part of the
       %BDD subjected to reordering.
  </ul>

  @sideeffect None

*/
static int
computeLB(
  DdManager * table		/**< manager */,
  DdHalfWord * order		/**< optimal order for the subset */,
  int  roots			/**< roots between lower and upper */,
  int  cost			/**< minimum cost for the subset */,
  int  lower			/**< lower level to be reordered */,
  int  upper			/**< upper level to be reordered */,
  int  level			/**< offset for the current top bottom var */
  )
{
    int i;
    int lb = cost;
    int lb1 = 0;
    int lb2;
    int support;
    DdHalfWord ref;

    /* The levels not involved in reordering are not going to change.
    ** Add their sizes to the lower bound.
    */
    for (i = 0; i < lower; i++) {
	lb += getLevelKeys(table,i);
    }
    /* If a variable is in the support, then there is going
    ** to be at least one node labeled by that variable.
    */
    for (i = lower; i <= lower+level; i++) {
	support = table->subtables[i].keys > 1 ||
	    table->vars[order[i-lower]]->ref > 1;
	lb1 += support;
    }

    /* Estimate the number of nodes required to connect the roots to
    ** the nodes in the bottom part. */
    if (lower+level+1 < table->size) {
	if (lower+level < upper)
	    ref = table->vars[order[level+1]]->ref;
	else
	    ref = table->vars[table->invperm[upper+1]]->ref;
	lb2 = (int) table->subtables[lower+level+1].keys -
            (ref > (DdHalfWord) 1) - roots;
    } else {
	lb2 = 0;
    }

    lb += lb1 > lb2 ? lb1 : lb2;

    return(lb);

} /* end of computeLB */


/**
  @brief Updates entry for a subset.

  @details Finds the subset, if it exists.  If the new order for the
  subset has lower cost, or if the subset did not exist, it stores the
  new order and cost.

  @return the number of subsets currently in the table.

  @sideeffect None

*/
static int
updateEntry(
  DdManager * table,
  DdHalfWord * order,
  int  level,
  int  cost,
  DdHalfWord ** orders,
  int * costs,
  int  subsets,
  char * mask,
  int  lower,
  int  upper)
{
    int i, j;
    int size = upper - lower + 1;

    /* Build a mask that says what variables are in this subset. */
    for (i = lower; i <= upper; i++)
	mask[table->invperm[i]] = 0;
    for (i = level; i < size; i++)
	mask[order[i]] = 1;

    /* Check each subset until a match is found or all subsets are examined. */
    for (i = 0; i < subsets; i++) {
	DdHalfWord *subset = orders[i];
	for (j = level; j < size; j++) {
	    if (mask[subset[j]] == 0)
		break;
	}
	if (j == size)		/* no mismatches: success */
	    break;
    }
    if (i == subsets || cost < costs[i]) {		/* add or replace */
	for (j = 0; j < size; j++)
	    orders[i][j] = order[j];
	costs[i] = cost;
	subsets += (i == subsets);
    }
    return(subsets);

} /* end of updateEntry */


/**
  @brief Pushes a variable in the order down to position "level."

  @sideeffect None

*/
static void
pushDown(
  DdHalfWord * order,
  int  j,
  int  level)
{
    int i;
    DdHalfWord tmp;

    tmp = order[j];
    for (i = j; i < level; i++) {
	order[i] = order[i+1];
    }
    order[level] = tmp;
    return;

} /* end of pushDown */


/**
  @brief Gathers symmetry information.

  @details Translates the symmetry information stored in the next
  field of each subtable from level to indices. This procedure is called
  immediately after symmetric sifting, so that the next fields are correct.
  By translating this informaton in terms of indices, we make it independent
  of subsequent reorderings. The format used is that of the next fields:
  a circular list where each variable points to the next variable in the
  same symmetry group. Only the entries between lower and upper are
  considered.  The procedure returns a pointer to an array
  holding the symmetry information if successful; NULL otherwise.

  @sideeffect None

  @see checkSymmInfo

*/
static DdHalfWord *
initSymmInfo(
  DdManager * table,
  int  lower,
  int  upper)
{
    int level, index, next, nextindex;
    DdHalfWord *symmInfo;

    symmInfo =  ALLOC(DdHalfWord, table->size);
    if (symmInfo == NULL) return(NULL);

    for (level = lower; level <= upper; level++) {
	index = table->invperm[level];
	next =  (int) table->subtables[level].next;
	nextindex = table->invperm[next];
	symmInfo[index] = (DdHalfWord) nextindex;
    }
    return(symmInfo);

} /* end of initSymmInfo */


/**
  @brief Check symmetry condition.

  @details Returns 1 if a variable is the one with the highest index
  among those belonging to a symmetry group that are in the top part of
  the %BDD.  The top part is given by level.

  @sideeffect None

  @see initSymmInfo

*/
static int
checkSymmInfo(
  DdManager * table,
  DdHalfWord * symmInfo,
  int  index,
  int  level)
{
    int i;

    i = (int) symmInfo[index];
    while (i != index) {
	if (index < i && table->perm[i] <= level)
	    return(0);
	i = (int) symmInfo[i];
    }
    return(1);

} /* end of checkSymmInfo */