..............................Figure 4. the BDDs for the function ( f + g ) and ( f . g )

Figure 4 shows the BDDs for the function ( f + g ) and ( f . g ). The construction procedure is simple. In case of “f + g ” if f is 1, then we need not evaluate g and if fis “0” the final result depends only on g . This is reflected in the final BDD for “ f + g ”. Similarly, in case of “ f. g ” if f is 0, then we need not evaluate g and if fis “1” the final result depends only on g . This is reflected in the final BDD for “f . g ”. The way we are constructing the BDDs for OR and AND operation, it is clear that the resultant BDDs are not ordered or reduced.

So, we need a method where the resultant BDD (after Boolean operations on ROBDDs) is also a ROBDD.

2.1   Binary Operations on ROBDD to procedue ROBBD

To perform the binary operation on two ROBDD's B_fand B_g , corresponding to the functions f and g respectively, we use the algorithm apply( op, B_f, B_g ). The two ROBDDs B_f and B_g have compatible ordering.

Application of apply( op, B_f, B_g ) will give a OBDD. The ordering of the resultant BDD is same as B_f or B_g but it may not be the reduced one. After constructing the resultant BDD, we may apply the reduce algorithm to get the ROBDD.

 

The function apply is based on the Shannon's expansion for f and g :

From the Shannon's expansion of f and g :

 

This is used as a control structure of apply which proceeds from the roots of B_f and B_gdownwards to construct nodes of the OBDD B_f op B_g. Let r f be the root node of B_f and r g be the root node of B_g.

apply( op, B_f, B_g ) Algorithm :

1.  If both r_fand r _g are terminal nodes with labels l_f and l_g, respectively

compute the value l_f op l_g and the resulting OBDD is B 0 if the value is 0 and B 1 otherwise.

In the remaining cases, at least one of the root nodes is a non-terminal.

2.  If both nodes are x_i -nodes (i.e., non-terminal of same order),

create an x_i -node n (called r_f, r_g) with a dashed line to apply ( op , lo( r_f ) , lo( r_g ) ) and a solid line to apply ( op , hi( r_f ) , hi( r_g) ).

3.  If r f is an x_i -node, but r_g is a terminal node or an x_j -node with j > i ,

create an x_i -node n (called r_f, r_g ) with a dashed line to apply ( op , lo( r_f ) , r_g ) and a solid line to apply (op , hi( r_f ) , r_g).

4.  The case in which r_gis a non-terminal node, but r_fis a terminal or an x_j -node with

j > i , is handled symmetrically to case 3.