Bdd Minimization by Linear Transformations

Binary Decision Diagrams (BDDs) are a powerful tool and are frequently used in many applications in VLSI CAD, like synthesis and veriication. Unfortunately, BDDs are very sensitive to the variable ordering and their size often becomes infeasible. Recently, a new approach for BDD minimization based on linear transformations, i.e. a special type of spectral techniques, has been proposed. In this paper we study this minimization method in more detail. While so far only experimental results are known, we prove for a family of Boolean functions that by linear transformations an exponential blow up of the BDD size can be prevented. Furthermore, we present a heuristic method that allows to signiicantly reduce the BDD sizes. By experiments we give a comparison to the state-of-the-art method to demonstrate the advantages of our approach. Binary Decision Diagrams (BDDs) are the state-of-the-art data structure in VLSI CAD and in the meantime are integrated in many commercial tools. They are widely used in applications, like synthesis 11, 5] and veriication 1, 4, 6]. The major drawback of all BDD based systems is that the underlying data structure is very sensitive to a chosen variable ordering, and for some functions even no eecient representation exists, like for the Boolean multiplication 3]. For this, several extensions of the basic BDD structure have been proposed in the last few years. A very promising extension is based on spectral techniques, since the BDD data structure itself remains unchanged, and only an encoding of the inputs is carried out: Instead of transforming the function table, the input variables are transformed by linear transformations. In contrast to spectral methods that operate with 2 n 2 n-matrices, for linear transformations n n-matrices are suucient. An eecient heuristic for BDD minimization under linear transformations has been presented in 12], which is called \linear sifting". The widely used sifting algorithm 14] for dynamic BDD minimization has been extended by an eeective linear operator which combines neighboring variables of the BDD. The algorithm runs very fast, since the basic operation is (very similar to variable reordering 8]) a local operation. By this, it is possible to obtain much smaller BDD sizes than with sifting alone. Recently, an exact algorithm for BDD minimization under linear transformations has been proposed 9], by which the size of the BDDs could be reduced signiicantly in comparison to linear sifting. But although an eecient branch&bound technique known from BDD minimization …