path: root/src/misc/extra/extraBddCas.c

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

  FileName    [extraBddCas.c]

  PackageName [extra]

  Synopsis    [Procedures related to LUT cascade synthesis.]

  Author      [Alan Mishchenko]

  Affiliation [UC Berkeley]

  Date        [Ver. 2.0. Started - September 1, 2003.]

  Revision    [$Id: extraBddCas.c,v 1.0 2003/05/21 18:03:50 alanmi Exp $]

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

#include "extra.h"

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

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

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

// the table to store cofactor operations
#define _TABLESIZE_COF  51113
typedef struct 
{
    unsigned Sign;  
    DdNode * Arg1;  
} _HashEntry_cof;
_HashEntry_cof HHTable1[_TABLESIZE_COF];

// the table to store the result of computation of the number of minterms
#define _TABLESIZE_MINT  15113
typedef struct 
{
    DdNode * Arg1;  
    unsigned Arg2;  
    unsigned Res;  
} _HashEntry_mint;
_HashEntry_mint HHTable2[_TABLESIZE_MINT];

typedef struct 
{
    int      nEdges;  // the number of in-coming edges of the node
    DdNode * bSum;    // the sum of paths of the incoming edges
} traventry;

// the signature used for hashing
static unsigned s_Signature = 1;

static int s_CutLevel = 0;

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

// because the proposed solution to the optimal encoding problem has exponential complexity
// we limit the depth of the branch and bound procedure to 5 levels
static int s_MaxDepth = 5;      

static int s_nVarsBest;          // the number of vars in the best ordering
static int s_VarOrderBest[32];   // storing the best ordering of vars in the "simple encoding"
static int s_VarOrderCur[32];    // storing the current ordering of vars
 
// the place to store the supports of the encoded function
static DdNode * s_Field[8][256]; // the size should be K, 2^K, where K is no less than MaxDepth
static DdNode * s_Encoded;       // this is the original function
static DdNode * s_VarAll;        // the set of all column variables
static int s_MultiStart;         // the total number of encoding variables used
// the array field now stores the supports

static DdNode ** s_pbTemp;       // the temporary storage for the columns

static int s_BackTracks;
static int s_BackTrackLimit = 100;

static DdNode * s_Terminal;      // the terminal value for counting minterms


static int s_EncodingVarsLevel;


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


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

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

static DdNode * CreateTheCodes_rec( DdManager * dd, DdNode * bEncoded, int Level, DdNode ** pCVars );
static void EvaluateEncodings_rec( DdManager * dd, DdNode * bVarsCol, int nVarsCol, int nMulti, int Level );
// functions called from EvaluateEncodings_rec()
static DdNode * ComputeVarSetAndCountMinterms( DdManager * dd, DdNode * bVars, DdNode * bVarTop, unsigned * Cost );
static DdNode * ComputeVarSetAndCountMinterms2( DdManager * dd, DdNode * bVars, DdNode * bVarTop, unsigned * Cost );
unsigned Extra_CountCofactorMinterms( DdManager * dd, DdNode * bFunc, DdNode * bVarsCof, DdNode * bVarsAll );
static unsigned Extra_CountMintermsSimple( DdNode * bFunc, unsigned max );

static void CountNodeVisits_rec( DdManager * dd, DdNode * aFunc, st_table * Visited );
static void CollectNodesAndComputePaths_rec( DdManager * dd, DdNode * aFunc, DdNode * bCube, st_table * Visited, st_table * CutNodes );

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


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

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

  Synopsis    [Performs the binary encoding of the set of function using the given vars.]

  Description [Performs a straight binary encoding of the set of functions using 
  the variable cubes formed from the given set of variables. ]

  SideEffects []

  SeeAlso     []

******************************************************************************/
DdNode * 
Extra_bddEncodingBinary( 
  DdManager * dd, 
  DdNode ** pbFuncs,    // pbFuncs is the array of columns to be encoded
  int nFuncs,           // nFuncs is the number of columns in the array
  DdNode ** pbVars,     // pbVars is the array of variables to use for the codes
  int nVars )           // nVars is the column multiplicity, [log2(nFuncs)]
{
    int i;
    DdNode * bResult;
    DdNode * bCube, * bTemp, * bProd;

    assert( nVars >= Extra_Base2Log(nFuncs) );

    bResult = b0;   Cudd_Ref( bResult );
    for ( i = 0; i < nFuncs; i++ )
    {
        bCube   = Extra_bddBitsToCube( dd, i, nVars, pbVars, 1 );   Cudd_Ref( bCube );
        bProd   = Cudd_bddAnd( dd, bCube, pbFuncs[i] );         Cudd_Ref( bProd );
        Cudd_RecursiveDeref( dd, bCube );

        bResult = Cudd_bddOr( dd, bProd, bTemp = bResult );     Cudd_Ref( bResult );
        Cudd_RecursiveDeref( dd, bTemp );
        Cudd_RecursiveDeref( dd, bProd );
    }

    Cudd_Deref( bResult );
    return bResult;
} /* end of Extra_bddEncodingBinary */


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

  Synopsis    [Solves the column encoding problem using a sophisticated method.]

  Description [The encoding is based on the idea of deriving functions which 
  depend on only one variable, which corresponds to the case of non-disjoint 
  decompostion. It is assumed that the variables pCVars are ordered below the variables
  representing the solumns, and the first variable pCVars[0] is the topmost one.]

  SideEffects []

  SeeAlso     [Extra_bddEncodingBinary]

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

DdNode * 
Extra_bddEncodingNonStrict( 
  DdManager * dd, 
  DdNode ** pbColumns,   // pbColumns is the array of columns to be encoded;
  int nColumns,          // nColumns is the number of columns in the array
  DdNode * bVarsCol,     // bVarsCol is the cube of variables on which the columns depend
  DdNode ** pCVars,      // pCVars is the array of variables to use for the codes
  int nMulti,            // nMulti is the column multiplicity, [log2(nColumns)]
  int * pSimple )        // pSimple gets the number of code variables taken from the input varibles without change
{
    DdNode * bEncoded, * bResult;
    int nVarsCol = Cudd_SupportSize(dd,bVarsCol);
    long clk;

    // cannot work with more that 32-bit codes
    assert( nMulti < 32 );

    // perform the preliminary encoding using the straight binary code
    bEncoded = Extra_bddEncodingBinary( dd, pbColumns, nColumns, pCVars, nMulti );   Cudd_Ref( bEncoded );
    //printf( "Node count = %d", Cudd_DagSize(bEncoded) );

    // set the backgroup value for counting minterms
    s_Terminal = b0;
    // set the level of the encoding variables
    s_EncodingVarsLevel = dd->invperm[pCVars[0]->index];

    // the current number of backtracks
    s_BackTracks = 0;
    // the variables that are cofactored on the topmost level where everything starts (no vars)
    s_Field[0][0] = b1;   
    // the size of the best set of "simple" encoding variables found so far
    s_nVarsBest   = 0;

    // set the relation to be accessible to traversal procedures
    s_Encoded = bEncoded;
    // the set of all vars to be accessible to traversal procedures
    s_VarAll  = bVarsCol;
    // the column multiplicity
    s_MultiStart  = nMulti;


    clk = clock();
    // find the simplest encoding
    if ( nColumns > 2 )
    EvaluateEncodings_rec( dd, bVarsCol, nVarsCol, nMulti, 1 );
//    printf( "The number of backtracks = %d\n", s_BackTracks );
//    s_EncSearchTime += clock() - clk;

    // allocate the temporary storage for the columns
    s_pbTemp = (DdNode **) malloc( nColumns * sizeof(DdNode *) );

//    clk = clock();
    bResult = CreateTheCodes_rec( dd, bEncoded, 0, pCVars );   Cudd_Ref( bResult );
//    s_EncComputeTime += clock() - clk;
    
    // delocate the preliminarily encoded set
    Cudd_RecursiveDeref( dd, bEncoded );
//    Cudd_RecursiveDeref( dd, aEncoded );

    free( s_pbTemp );

    *pSimple = s_nVarsBest;
    Cudd_Deref( bResult );
    return bResult;
}
 
/**Function********************************************************************

  Synopsis    [Collects the nodes under the cut and, for each node, computes the sum of paths leading to it from the root.]

  Description [The table returned contains the set of BDD nodes pointed to under the cut
  and, for each node, the BDD of the sum of paths leading to this node from the root
  The sums of paths in the table are referenced. CutLevel is the first DD level 
  considered to be under the cut.]

  SideEffects []

  SeeAlso     [Extra_bddNodePaths]

******************************************************************************/
st_table * Extra_bddNodePathsUnderCut( DdManager * dd, DdNode * bFunc, int CutLevel )
{
    st_table * Visited;  // temporary table to remember the visited nodes
    st_table * CutNodes; // the result goes here
    st_table * Result; // the result goes here
    DdNode * aFunc;

    s_CutLevel = CutLevel;

    Result  = st_init_table(st_ptrcmp,st_ptrhash);
    // the terminal cases
    if ( Cudd_IsConstant( bFunc ) )
    {
        if ( bFunc == b1 )
        {
            st_insert( Result, (char*)b1, (char*)b1 );
            Cudd_Ref( b1 );
            Cudd_Ref( b1 );
        }
        else
        {
            st_insert( Result, (char*)b0, (char*)b0 );
            Cudd_Ref( b0 );
            Cudd_Ref( b0 );
        }
        return Result;
    }

    // create the ADD to simplify processing (no complemented edges)
    aFunc = Cudd_BddToAdd( dd, bFunc );   Cudd_Ref( aFunc );

    // Step 1: Start the tables and collect information about the nodes above the cut 
    // this information tells how many edges point to each node
    Visited  = st_init_table(st_ptrcmp,st_ptrhash);
    CutNodes = st_init_table(st_ptrcmp,st_ptrhash);

    CountNodeVisits_rec( dd, aFunc, Visited );

    // Step 2: Traverse the BDD using the visited table and compute the sum of paths
    CollectNodesAndComputePaths_rec( dd, aFunc, b1, Visited, CutNodes );

    // at this point the table of cut nodes is ready and the table of visited is useless
    {
        st_generator * gen;
        DdNode * aNode;
        traventry * p;
        st_foreach_item( Visited, gen, (char**)&aNode, (char**)&p ) 
        {
            Cudd_RecursiveDeref( dd, p->bSum );
            free( p );
        }
        st_free_table( Visited );
    }

    // go through the table CutNodes and create the BDD and the path to be returned
    {
        st_generator * gen;
        DdNode * aNode, * bNode, * bSum;
        st_foreach_item( CutNodes, gen, (char**)&aNode, (char**)&bSum) 
        {
            // aNode is not referenced, because aFunc is holding it
            bNode = Cudd_addBddPattern( dd, aNode );  Cudd_Ref( bNode ); 
            st_insert( Result, (char*)bNode, (char*)bSum );
            // the new table takes both refs
        }
        st_free_table( CutNodes );
    }

    // dereference the ADD
    Cudd_RecursiveDeref( dd, aFunc );

    // return the table
    return Result;

} /* end of Extra_bddNodePathsUnderCut */

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

  Synopsis    [Collects the nodes under the cut in the ADD starting from the given set of ADD nodes.]

  Description [Takes the array, paNodes, of ADD nodes to start the traversal,
  the array, pbCubes, of BDD cubes to start the traversal with in each node, 
  and the number, nNodes, of ADD nodes and BDD cubes in paNodes and pbCubes. 
  Returns the number of columns found. Fills in paNodesRes (pbCubesRes) 
  with the set of ADD columns (BDD paths). These arrays should be allocated 
  by the user.]

  SideEffects []

  SeeAlso     [Extra_bddNodePaths]

******************************************************************************/
int Extra_bddNodePathsUnderCutArray( DdManager * dd, DdNode ** paNodes, DdNode ** pbCubes, int nNodes, DdNode ** paNodesRes, DdNode ** pbCubesRes, int CutLevel )
{
    st_table * Visited;  // temporary table to remember the visited nodes
    st_table * CutNodes; // the nodes under the cut go here
    int i, Counter;

    s_CutLevel = CutLevel;

    // there should be some nodes
    assert( nNodes > 0 );
    if ( nNodes == 1 && Cudd_IsConstant( paNodes[0] ) )
    {
        if ( paNodes[0] == a1 )
        {
            paNodesRes[0] = a1;          Cudd_Ref( a1 );
            pbCubesRes[0] = pbCubes[0];  Cudd_Ref( pbCubes[0] );
        }
        else
        {
            paNodesRes[0] = a0;          Cudd_Ref( a0 );
            pbCubesRes[0] = pbCubes[0];  Cudd_Ref( pbCubes[0] );
        }
        return 1;
    }

    // Step 1: Start the table and collect information about the nodes above the cut 
    // this information tells how many edges point to each node
    CutNodes = st_init_table(st_ptrcmp,st_ptrhash);
    Visited  = st_init_table(st_ptrcmp,st_ptrhash);

    for ( i = 0; i < nNodes; i++ )
        CountNodeVisits_rec( dd, paNodes[i], Visited );

    // Step 2: Traverse the BDD using the visited table and compute the sum of paths
    for ( i = 0; i < nNodes; i++ )
        CollectNodesAndComputePaths_rec( dd, paNodes[i], pbCubes[i], Visited, CutNodes );

    // at this point, the table of cut nodes is ready and the table of visited is useless
    {
        st_generator * gen;
        DdNode * aNode;
        traventry * p;
        st_foreach_item( Visited, gen, (char**)&aNode, (char**)&p ) 
        {
            Cudd_RecursiveDeref( dd, p->bSum );
            free( p );
        }
        st_free_table( Visited );
    }

    // go through the table CutNodes and create the BDD and the path to be returned
    {
        st_generator * gen;
        DdNode * aNode, * bSum;
        Counter = 0;
        st_foreach_item( CutNodes, gen, (char**)&aNode, (char**)&bSum) 
        {
            paNodesRes[Counter] = aNode;   Cudd_Ref( aNode ); 
            pbCubesRes[Counter] = bSum;
            Counter++;
        }
        st_free_table( CutNodes );
    }

    // return the number of cofactors found
    return Counter;

} /* end of Extra_bddNodePathsUnderCutArray */

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

  Synopsis    [Collects all the BDD nodes into the table.]

  Description []

  SideEffects []

  SeeAlso     []

***********************************************************************/
void extraCollectNodes( DdNode * Func, st_table * tNodes )
{
    DdNode * FuncR;
    FuncR = Cudd_Regular(Func);
    if ( st_find_or_add( tNodes, (char*)FuncR, NULL ) )
        return;
    if ( cuddIsConstant(FuncR) )
        return;
    extraCollectNodes( cuddE(FuncR), tNodes );
    extraCollectNodes( cuddT(FuncR), tNodes );
}

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

  Synopsis    [Collects all the nodes of one DD into the table.]

  Description []

  SideEffects []

  SeeAlso     []

***********************************************************************/
st_table * Extra_CollectNodes( DdNode * Func )
{
    st_table * tNodes;
    tNodes = st_init_table( st_ptrcmp, st_ptrhash );
    extraCollectNodes( Func, tNodes );
    return tNodes;
}

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

  Synopsis    [Updates the topmost level from which the given node is referenced.]

  Description [Takes the table which maps each BDD nodes (including the constants)
  into the topmost level on which this node counts as a cofactor. Takes the topmost
  level, on which this node counts as a cofactor (see Extra_ProfileWidthFast(). 
  Takes the node, for which the table entry should be updated.]

  SideEffects []

  SeeAlso     []

***********************************************************************/
void extraProfileUpdateTopLevel( st_table * tNodeTopRef, int TopLevelNew, DdNode * node )
{
    int * pTopLevel;

    if ( st_find_or_add( tNodeTopRef, (char*)node, (char***)&pTopLevel ) )
    { // the node is already referenced
        // the current top level should be updated if it is larger than the new level
        if ( *pTopLevel > TopLevelNew ) 
             *pTopLevel = TopLevelNew;
    }
    else
    { // the node is not referenced
        // its level should be set to the current new level
        *pTopLevel = TopLevelNew;
    }
}
/**Function*************************************************************

  Synopsis    [Fast computation of the BDD profile.]

  Description [The array to store the profile is given by the user and should
  contain at least as many entries as there is the maximum of the BDD/ZDD
  size of the manager PLUS ONE.
  When we say that the widths of the DD on level L is W, we mean the following.
  Let us create the cut between the level L-1 and the level L and count the number
  of different DD nodes pointed to across the cut. This number is the width W.
  From this it follows the on level 0, the width is equal to the number of external
  pointers to the considered DDs. If there is only one DD, then the profile on 
  level 0 is always 1. If this DD is rooted in the topmost variable, then the width
  on level 1 is always 2, etc. The width at the level equal to dd->size is the
  number of terminal nodes in the DD. (Because we consider the first level #0
  and the last level #dd->size, the profile array should contain dd->size+1 entries.)
  ]

  SideEffects [This procedure will not work for BDDs w/ complement edges, only for ADDs and ZDDs]

  SeeAlso     []

***********************************************************************/
int Extra_ProfileWidth( DdManager * dd, DdNode * Func, int * pProfile, int CutLevel )
{
    st_generator * gen;
    st_table * tNodeTopRef; // this table stores the top level from which this node is pointed to
    st_table * tNodes;
    DdNode * node;
    DdNode * nodeR;
    int LevelStart, Limit;
    int    i, size;
    int WidthMax;
  
    // start the mapping table
    tNodeTopRef = st_init_table(st_ptrcmp,st_ptrhash);
    // add the topmost node to the profile
    extraProfileUpdateTopLevel( tNodeTopRef, 0, Func );

    // collect all nodes
    tNodes = Extra_CollectNodes( Func );
    // go though all the nodes and set the top level the cofactors are pointed from
//    Cudd_ForeachNode( dd, Func, genDD, node )
    st_foreach_item( tNodes, gen, (char**)&node, NULL )
    {
//        assert( Cudd_Regular(node) );  // this procedure works only with ADD/ZDD (not BDD w/ compl.edges)
        nodeR = Cudd_Regular(node);
        if ( cuddIsConstant(nodeR) )
            continue;
        // this node is not a constant - consider its cofactors
        extraProfileUpdateTopLevel( tNodeTopRef, dd->perm[node->index]+1, cuddE(nodeR) );
        extraProfileUpdateTopLevel( tNodeTopRef, dd->perm[node->index]+1, cuddT(nodeR) );
    }
    st_free_table( tNodes );

    // clean the profile
    size = ddMax(dd->size, dd->sizeZ) + 1;
    for ( i = 0; i < size; i++ ) 
        pProfile[i] = 0;

    // create the profile
    st_foreach_item( tNodeTopRef, gen, (char**)&node, (char**)&LevelStart )
    {
        nodeR = Cudd_Regular(node);
        Limit = (cuddIsConstant(nodeR))? dd->size: dd->perm[nodeR->index];
        for ( i = LevelStart; i <= Limit; i++ )
            pProfile[i]++;
    }

    if ( CutLevel != -1 && CutLevel != 0  )
        size = CutLevel;

    // get the max width
    WidthMax = 0;
    for ( i = 0; i < size; i++ ) 
        if ( WidthMax < pProfile[i] )
            WidthMax = pProfile[i];

    // deref the table
    st_free_table( tNodeTopRef );

    return WidthMax;
} /* end of Extra_ProfileWidth */


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

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

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

  Synopsis    [Computes the non-strict codes when evaluation is finished.]

  Description [The information about the best code is stored in s_VarOrderBest,
  which has s_nVarsBest entries.]

  SideEffects [None]

******************************************************************************/
DdNode * CreateTheCodes_rec( DdManager * dd, DdNode * bEncoded, int Level, DdNode ** pCVars )
// bEncoded is the preliminarily encoded set of columns
// Level is the current level in the recursion
// pCVars are the variables to be used for encoding
{
    DdNode * bRes;
    if ( Level == s_nVarsBest )
    { // the terminal case, when we need to remap the encoded function 
        // from the preliminary encoded variables to the new ones
        st_table * CutNodes;
        int nCols;
//        double nMints;
/*
#ifdef _DEBUG

        {
        DdNode * bTemp;
        // make sure that the given number of variables is enough
        bTemp  = Cudd_bddExistAbstract( dd, bEncoded, s_VarAll );           Cudd_Ref( bTemp );
//        nMints = Cudd_CountMinterm( dd, bTemp, s_MultiStart );
        nMints = Extra_CountMintermsSimple( bTemp, (1<<s_MultiStart) );
        if ( nMints > Extra_Power2( s_MultiStart-Level ) ) 
        {  // the number of minterms is too large to encode the columns 
           // using the given minimum number of encoding variables
            assert( 0 );
        }
        Cudd_RecursiveDeref( dd, bTemp );
        }
#endif
*/
        // get the columns to be re-encoded
        CutNodes = Extra_bddNodePathsUnderCut( dd, bEncoded, s_EncodingVarsLevel );
        // LUT size is the cut level because because the temporary encoding variables 
        // are above the functional variables - this is not true!!!
        // the temporary variables are below!

        // put the entries from the table into the temporary array
        { 
            st_generator * gen;
            DdNode * bColumn, * bCode;
            nCols = 0;
            st_foreach_item( CutNodes, gen, (char**)&bCode, (char**)&bColumn ) 
            {
                if ( bCode == b0 )
                { // the unused part of the columns
                    Cudd_RecursiveDeref( dd, bColumn );
                    Cudd_RecursiveDeref( dd, bCode );
                    continue;
                }
                else
                {
                    s_pbTemp[ nCols ] = bColumn; // takes ref
                    Cudd_RecursiveDeref( dd, bCode );
                    nCols++;
                }
            }
            st_free_table( CutNodes );
//            assert( nCols == (int)nMints );
        }

        // encode the columns
        if ( s_MultiStart-Level == 0 ) // we reached the bottom level of recursion
        {
            assert( nCols       == 1 );
//            assert( (int)nMints == 1 );
            bRes = s_pbTemp[0];     Cudd_Ref( bRes );
        }
        else
        {
            bRes = Extra_bddEncodingBinary( dd, s_pbTemp, nCols, pCVars+Level, s_MultiStart-Level ); Cudd_Ref( bRes );
        }

        // deref the columns
        {
            int i;
            for ( i = 0; i < nCols; i++ )
                Cudd_RecursiveDeref( dd, s_pbTemp[i] );
        }
    }
    else
    {
        // cofactor the problem as specified in the best solution
        DdNode * bCof0,  * bCof1;
        DdNode * bRes0,  * bRes1;
        DdNode * bProd0, * bProd1;
        DdNode * bTemp;
        DdNode * bVarNext = dd->vars[ s_VarOrderBest[Level] ];

        bCof0  = Cudd_Cofactor( dd, bEncoded,  Cudd_Not( bVarNext ) );   Cudd_Ref( bCof0 );
        bCof1  = Cudd_Cofactor( dd, bEncoded,            bVarNext   );   Cudd_Ref( bCof1 );

        // call recursively
        bRes0 = CreateTheCodes_rec( dd, bCof0, Level+1, pCVars );  Cudd_Ref( bRes0 );
        bRes1 = CreateTheCodes_rec( dd, bCof1, Level+1, pCVars );  Cudd_Ref( bRes1 );

        Cudd_RecursiveDeref( dd, bCof0 );
        Cudd_RecursiveDeref( dd, bCof1 );

        // compose the result using the identity (bVarNext <=> pCVars[Level])  - this is wrong!
        // compose the result as follows: x'y'F0 + xyF1
        bProd0 = Cudd_bddAnd( dd, Cudd_Not(bVarNext), Cudd_Not(pCVars[Level]) );   Cudd_Ref( bProd0 );
        bProd1 = Cudd_bddAnd( dd,          bVarNext ,          pCVars[Level]  );   Cudd_Ref( bProd1 );

        bProd0 = Cudd_bddAnd( dd, bTemp = bProd0, bRes0 );   Cudd_Ref( bProd0 );
        Cudd_RecursiveDeref( dd, bTemp );
        Cudd_RecursiveDeref( dd, bRes0 );

        bProd1 = Cudd_bddAnd( dd, bTemp = bProd1, bRes1 );   Cudd_Ref( bProd1 );
        Cudd_RecursiveDeref( dd, bTemp );
        Cudd_RecursiveDeref( dd, bRes1 );

        bRes = Cudd_bddOr( dd, bProd0, bProd1 );             Cudd_Ref( bRes );

        Cudd_RecursiveDeref( dd, bProd0 );
        Cudd_RecursiveDeref( dd, bProd1 );
    }
    Cudd_Deref( bRes );
    return bRes;
}

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

  Synopsis    [Computes the current set of variables and counts the number of minterms.]

  Description [Old implementation.]

  SideEffects []

  SeeAlso     []

******************************************************************************/
void EvaluateEncodings_rec( DdManager * dd, DdNode * bVarsCol, int nVarsCol, int nMulti, int Level )
// bVarsCol is the set of remaining variables
// nVarsCol is the number of remaining variables
// nMulti is the number of encoding variables to be used
// Level is the level of recursion, from which this function is called
// if we successfully finish this procedure, Level also stands for how many encoding variabled we saved
{
    int i, k;
    int nEntries = (1<<(Level-1)); // the number of entries in the field of the previous level
    DdNode * bVars0, * bVars1; // the cofactors
    unsigned nMint0, nMint1;   // the number of minterms
    DdNode * bTempV;
    DdNode * bVarTop;
    int fBreak;


    // there is no need to search above this level
    if ( Level > s_MaxDepth )
        return;

    // if there are no variables left, quit the research
    if ( bVarsCol == b1 )
        return;

    if ( s_BackTracks > s_BackTrackLimit )
        return;

    s_BackTracks++;

    // otherwise, go through the remaining variables
    for ( bTempV = bVarsCol; bTempV != b1; bTempV = cuddT(bTempV) )
    {
        // the currently tested variable
        bVarTop = dd->vars[bTempV->index];

        // put it into the array
        s_VarOrderCur[Level-1] = bTempV->index;

        // go through the entries and fill them out by cofactoring
        fBreak = 0;
        for ( i = 0; i < nEntries; i++ )
        {
            bVars0 = ComputeVarSetAndCountMinterms( dd, s_Field[Level-1][i], Cudd_Not(bVarTop), &nMint0 );
            Cudd_Ref( bVars0 );

            if ( nMint0 > Extra_Power2( nMulti-1 ) ) 
            {
                // there is no way to encode - dereference and return
                Cudd_RecursiveDeref( dd, bVars0 );
                fBreak = 1;
                break;
            }

            bVars1 = ComputeVarSetAndCountMinterms( dd, s_Field[Level-1][i], bVarTop, &nMint1 );
            Cudd_Ref( bVars1 );

            if ( nMint1 > Extra_Power2( nMulti-1 ) ) 
            {
                // there is no way to encode - dereference and return
                Cudd_RecursiveDeref( dd, bVars0 );
                Cudd_RecursiveDeref( dd, bVars1 );
                fBreak = 1;
                break;
            }

            // otherwise, add these two cofactors
            s_Field[Level][2*i + 0] = bVars0; // takes ref
            s_Field[Level][2*i + 1] = bVars1; // takes ref
        } 

        if ( !fBreak )
        {
            DdNode * bVarsRem;
            // if we ended up here, it means that the cofactors w.r.t. variable bVarTop satisfy the condition
            // save this situation
            if ( s_nVarsBest < Level )
            {
                s_nVarsBest = Level;
                // copy the variable assignment
                for ( k = 0; k < Level; k++ )
                    s_VarOrderBest[k] = s_VarOrderCur[k];
            }

            // call recursively
            // get the new variable set
            if ( nMulti-1 > 0 )
            {
                bVarsRem = Cudd_bddExistAbstract( dd, bVarsCol, bVarTop );   Cudd_Ref( bVarsRem );
                EvaluateEncodings_rec( dd, bVarsRem, nVarsCol-1, nMulti-1, Level+1 );
                Cudd_RecursiveDeref( dd, bVarsRem );
            }
        }

        // deref the contents of the array
        for ( k = 0; k < i; k++ )
        {
            Cudd_RecursiveDeref( dd, s_Field[Level][2*k + 0] );
            Cudd_RecursiveDeref( dd, s_Field[Level][2*k + 1] );
        }

        // if the solution is found, there is no need to continue
        if ( s_nVarsBest == s_MaxDepth )
            return;

        // if the solution is found, there is no need to continue
        if ( s_nVarsBest == s_MultiStart )
            return;
    }
    // at this point, we have tried all possible directions in the space of variables
}

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

  Synopsis    [Computes the current set of variables and counts the number of minterms.]

  Description []

  SideEffects []

  SeeAlso     []

******************************************************************************/
DdNode * ComputeVarSetAndCountMinterms( DdManager * dd, DdNode * bVars, DdNode * bVarTop, unsigned * Cost )
// takes bVars - the variables cofactored so far (some of them may be in negative polarity)
// bVarTop - the topmost variable w.r.t. which to cofactor (may be in negative polarity)
// returns the cost and the new set of variables (bVars & bVarTop)
{
    DdNode * bVarsRes;

    // get the resulting set of variables
    bVarsRes = Cudd_bddAnd( dd, bVars, bVarTop );  Cudd_Ref( bVarsRes );

    // increment signature before calling Cudd_CountCofactorMinterms()
    s_Signature++;
    *Cost = Extra_CountCofactorMinterms( dd, s_Encoded, bVarsRes, s_VarAll );

    Cudd_Deref( bVarsRes );
//    s_CountCalls++;
    return bVarsRes;
}

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

  Synopsis    [Computes the current set of variables and counts the number of minterms.]

  Description [The old implementation, which is approximately 4 times slower.]

  SideEffects []

  SeeAlso     []

******************************************************************************/
DdNode * ComputeVarSetAndCountMinterms2( DdManager * dd, DdNode * bVars, DdNode * bVarTop, unsigned * Cost )
{
    DdNode * bVarsRes;
    DdNode * bCof, * bFun;

    bVarsRes = Cudd_bddAnd( dd, bVars, bVarTop );             Cudd_Ref( bVarsRes );

    bCof  = Cudd_Cofactor( dd, s_Encoded,  bVarsRes );        Cudd_Ref( bCof );
    bFun  = Cudd_bddExistAbstract( dd, bCof, s_VarAll );      Cudd_Ref( bFun );
    *Cost = (unsigned)Cudd_CountMinterm( dd, bFun, s_MultiStart );
    Cudd_RecursiveDeref( dd, bFun );
    Cudd_RecursiveDeref( dd, bCof );

    Cudd_Deref( bVarsRes );
//    s_CountCalls++;
    return bVarsRes;
}


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

  Synopsis    [Counts the number of encoding minterms pointed to by the cofactor of the function.]

  Description []

  SideEffects [None]

******************************************************************************/
unsigned Extra_CountCofactorMinterms( DdManager * dd, DdNode * bFunc, DdNode * bVarsCof, DdNode * bVarsAll )
// this function computes how many minterms depending on the encoding variables
// are there in the cofactor of bFunc w.r.t. variables bVarsCof
// bFunc is assumed to depend on variables s_VarsAll
// the variables s_VarsAll should be ordered above the encoding variables
{
    unsigned HKey;
    DdNode * bFuncR;

    // if the function is zero, there are no minterms
//    if ( bFunc == b0 )
//        return 0;

//    if ( st_lookup(Visited, (char*)bFunc, NULL) )
//        return 0;

//    HKey = hashKey2c( s_Signature, bFuncR );
//    if ( HHTable1[HKey].Sign == s_Signature && HHTable1[HKey].Arg1 == bFuncR ) // this node is visited
//        return 0;


    // check the hash-table 
    bFuncR = Cudd_Regular(bFunc);
//    HKey = hashKey2( s_Signature, bFuncR, _TABLESIZE_COF );
    HKey = hashKey2( s_Signature, bFunc, _TABLESIZE_COF );
    for ( ;  HHTable1[HKey].Sign == s_Signature; HKey = (HKey+1) % _TABLESIZE_COF )
//        if ( HHTable1[HKey].Arg1 == bFuncR ) // this node is visited
        if ( HHTable1[HKey].Arg1 == bFunc ) // this node is visited
            return 0;


    // if the function is already the code
    if ( dd->perm[bFuncR->index] >= s_EncodingVarsLevel )
    {
//        st_insert(Visited, (char*)bFunc, NULL);

//        HHTable1[HKey].Sign = s_Signature;
//        HHTable1[HKey].Arg1 = bFuncR;

        assert( HHTable1[HKey].Sign != s_Signature );
        HHTable1[HKey].Sign = s_Signature;
//        HHTable1[HKey].Arg1 = bFuncR;
        HHTable1[HKey].Arg1 = bFunc;

        return Extra_CountMintermsSimple( bFunc, (1<<s_MultiStart) );
    }
    else
    {
        DdNode * bFunc0,    * bFunc1;
        DdNode * bVarsCof0, * bVarsCof1;
        DdNode * bVarsCofR = Cudd_Regular(bVarsCof);
        unsigned Res;

        // get the levels
        int LevelF = dd->perm[bFuncR->index];
        int LevelC = cuddI(dd,bVarsCofR->index);
        int LevelA = dd->perm[bVarsAll->index];

        int LevelTop = LevelF;

        if ( LevelTop > LevelC )
             LevelTop = LevelC;

        if ( LevelTop > LevelA )
             LevelTop = LevelA;

        // the top var in the function or in cofactoring vars always belongs to the set of all vars
        assert( !( LevelTop == LevelF || LevelTop == LevelC ) || LevelTop == LevelA );

        // cofactor the function
        if ( LevelTop == LevelF )
        {
            if ( bFuncR != bFunc ) // bFunc is complemented 
            {
                bFunc0 = Cudd_Not( cuddE(bFuncR) );
                bFunc1 = Cudd_Not( cuddT(bFuncR) );
            }
            else
            {
                bFunc0 = cuddE(bFuncR);
                bFunc1 = cuddT(bFuncR);
            }
        }
        else // bVars is higher in the variable order 
            bFunc0 = bFunc1 = bFunc;

        // cofactor the cube
        if ( LevelTop == LevelC )
        {
            if ( bVarsCofR != bVarsCof ) // bFunc is complemented 
            {
                bVarsCof0 = Cudd_Not( cuddE(bVarsCofR) );
                bVarsCof1 = Cudd_Not( cuddT(bVarsCofR) );
            }
            else
            {
                bVarsCof0 = cuddE(bVarsCofR);
                bVarsCof1 = cuddT(bVarsCofR);
            }
        }
        else // bVars is higher in the variable order 
            bVarsCof0 = bVarsCof1 = bVarsCof;

        // there are two cases: 
        // (1) the top variable belongs to the cofactoring variables
        // (2) the top variable does not belong to the cofactoring variables

        // (1) the top variable belongs to the cofactoring variables
        Res = 0;
        if ( LevelTop == LevelC )
        {
            if ( bVarsCof1 == b0 ) // this is a negative cofactor
            {
                if ( bFunc0 != b0 )
                Res = Extra_CountCofactorMinterms( dd, bFunc0, bVarsCof0, cuddT(bVarsAll) );
            }
            else                        // this is a positive cofactor
            {
                if ( bFunc1 != b0 )
                Res = Extra_CountCofactorMinterms( dd, bFunc1, bVarsCof1, cuddT(bVarsAll) );
            }
        }
        else
        {
            if ( bFunc0 != b0 )
            Res += Extra_CountCofactorMinterms( dd, bFunc0, bVarsCof0, cuddT(bVarsAll) );
            
            if ( bFunc1 != b0 )
            Res += Extra_CountCofactorMinterms( dd, bFunc1, bVarsCof1, cuddT(bVarsAll) );
        }

//        st_insert(Visited, (char*)bFunc, NULL);

//        HHTable1[HKey].Sign = s_Signature;
//        HHTable1[HKey].Arg1 = bFuncR;

        // skip through the entries with the same signatures 
        // (these might have been created at the time of recursive calls)
        for ( ; HHTable1[HKey].Sign == s_Signature; HKey = (HKey+1) % _TABLESIZE_COF );
        assert( HHTable1[HKey].Sign != s_Signature );
        HHTable1[HKey].Sign = s_Signature;
//        HHTable1[HKey].Arg1 = bFuncR;
        HHTable1[HKey].Arg1 = bFunc;

        return Res;
    }
}

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

  Synopsis    [Counts the number of minterms.]

  Description [This function counts minterms for functions up to 32 variables
  using a local cache. The terminal value (s_Termina) should be adjusted for
  BDDs and ADDs.]

  SideEffects [None]

******************************************************************************/
unsigned Extra_CountMintermsSimple( DdNode * bFunc, unsigned max )
{
    unsigned HKey;

    // normalize
    if ( Cudd_IsComplement(bFunc) )
        return max - Extra_CountMintermsSimple( Cudd_Not(bFunc), max );

    // now it is known that the function is not complemented
    if ( cuddIsConstant(bFunc) )
        return ((bFunc==s_Terminal)? 0: max);

    // check cache
    HKey = hashKey2( bFunc, max, _TABLESIZE_MINT );
    if ( HHTable2[HKey].Arg1 == bFunc && HHTable2[HKey].Arg2 == max )
        return HHTable2[HKey].Res;
    else
    {
        // min = min0/2 + min1/2;
        unsigned min = (Extra_CountMintermsSimple( cuddE(bFunc), max ) >> 1) +
                       (Extra_CountMintermsSimple( cuddT(bFunc), max ) >> 1);

        HHTable2[HKey].Arg1 = bFunc;
        HHTable2[HKey].Arg2 = max;
        HHTable2[HKey].Res  = min;

        return min;
    }
}    /* end of Extra_CountMintermsSimple */


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

  Synopsis    [Visits the nodes.]

  Description [Visits the nodes above the cut and the nodes pointed to below the cut;
  collects the visited nodes, counts how many times each node is visited, and sets 
  the path-sum to be the constant zero BDD.]

  SideEffects []

  SeeAlso     []

******************************************************************************/
void CountNodeVisits_rec( DdManager * dd, DdNode * aFunc, st_table * Visited )

{
    traventry * p;
    char **slot;
    if ( st_find_or_add(Visited, (char*)aFunc, &slot) )
    { // the entry already exists
        p = (traventry*) *slot;
        // increment the counter of incoming edges
        p->nEdges++;
        return; 
    }
    // this node has not been visited
    assert( !Cudd_IsComplement(aFunc) );

    // create the new traversal entry
    p = (traventry *) malloc( sizeof(traventry) );
    // set the initial sum of edges to zero BDD
    p->bSum = b0;   Cudd_Ref( b0 );
    // set the starting number of incoming edges
    p->nEdges = 1;
    // set this entry into the slot
    *slot = (char*)p;

    // recur if the node is above the cut
    if ( cuddI(dd,aFunc->index) < s_CutLevel )
    {
        CountNodeVisits_rec( dd, cuddE(aFunc), Visited );
        CountNodeVisits_rec( dd, cuddT(aFunc), Visited );
    }
} /* end of CountNodeVisits_rec */


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

  Synopsis    [Revisits the nodes and computes the paths.]

  Description [This function visits the nodes above the cut having the goal of 
  summing all the incomming BDD edges; when this function comes across the node 
  below the cut, it saves this node in the CutNode table.]

  SideEffects []

  SeeAlso     []

******************************************************************************/
void CollectNodesAndComputePaths_rec( DdManager * dd, DdNode * aFunc, DdNode * bCube, st_table * Visited, st_table * CutNodes )
{    
    // find the node in the visited table
    DdNode * bTemp;
    traventry * p;
    char **slot;
    if ( st_find_or_add(Visited, (char*)aFunc, &slot) )
    { // the node is found
        // get the pointer to the traversal entry
        p = (traventry*) *slot;

        // make sure that the counter of incoming edges is positive
        assert( p->nEdges > 0 );

        // add the cube to the currently accumulated cubes
        p->bSum = Cudd_bddOr( dd, bTemp = p->bSum, bCube );  Cudd_Ref( p->bSum );
        Cudd_RecursiveDeref( dd, bTemp );

        // decrement the number of visits
        p->nEdges--;

        // if more visits to this node are expected, return
        if ( p->nEdges ) 
            return;
        else // if ( p->nEdges == 0 )
        { // this is the last visit - propagate the cube

            // check where this node is
            if ( cuddI(dd,aFunc->index) < s_CutLevel )
            { // the node is above the cut
                DdNode * bCube0, * bCube1;

                // get the top-most variable
                DdNode * bVarTop = dd->vars[aFunc->index];

                // compute the propagated cubes
                bCube0 = Cudd_bddAnd( dd, p->bSum, Cudd_Not( bVarTop ) );   Cudd_Ref( bCube0 );
                bCube1 = Cudd_bddAnd( dd, p->bSum,           bVarTop   );   Cudd_Ref( bCube1 );

                // call recursively
                CollectNodesAndComputePaths_rec( dd, cuddE(aFunc), bCube0, Visited, CutNodes );
                CollectNodesAndComputePaths_rec( dd, cuddT(aFunc), bCube1, Visited, CutNodes );

                // dereference the cubes
                Cudd_RecursiveDeref( dd, bCube0 );
                Cudd_RecursiveDeref( dd, bCube1 );
                return;
            }
            else
            { // the node is below the cut
                // add this node to the cut node table, if it is not yet there

//                DdNode * bNode;
//                bNode = Cudd_addBddPattern( dd, aFunc );  Cudd_Ref( bNode );
                if ( st_find_or_add(CutNodes, (char*)aFunc, &slot) )
                { // the node exists - should never happen
                    assert( 0 );
                }
                *slot = (char*) p->bSum;   Cudd_Ref( p->bSum );
                // the table takes the reference of bNode
                return;
            }
        }
    }

    // the node does not exist in the visited table - should never happen
    assert(0);

} /* end of CollectNodesAndComputePaths_rec */



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