Detlef Sieling

Graph driven BDDs—a new data structure for Boolean functions

Abstract (Ordered) binary decision diagrams (OBDDs) are used as a data structure for Boolean functions in the logical synthesis process, for verification and test pattern generation, and as part of CAD tools. For several important functions like arithmetical and logical units with quite different functions, the indirect storage access function or the hidden weighted bit function OBDDs have exponential size for any ordering of the variables. Since an ordering of the variables may be stored as a list, OBDDs may also be called list driven BDDs. Two new generalized models of graph driven BDDs are presented. The above mentioned and many other functions can be represented in small polynomial size in this model and the usual operations on OBDDs can be performed efficiently also for graph driven BDDs.

Reduction of OBDDs in linear time

Ordered binary decision diagrams (OBDDs) play an important role as data structure for Boolean functions. They are used, e.g., in the logical synthesis process, for verification and test pattern generation, and as part of CAD tools. For a given ordering of the variables and a given Boolean function f the reduced OBDD, i.e. the OBDD of minimal size, is unique (up to isomorphisms) and can be computed from any OBDD G for f of size |G| in time O(¦G¦log¦G¦). A new reduction algorithm which works in optimal linear time O(|G|) is presented.

NC-ALGORITHMS FOR OPERATIONS ON BINARY DECISION DIAGRAMS

(Ordered) binary decision diagrams are a powerful representation for Boolean functions and are widely used in logical synthesis, verification, test pattern generation or as part of CAD tools. NC-al...

Quantum branching programs and space-bounded nonuniform quantum complexity

In this paper, the space complexity of non-uniform quantum algorithms is investigated using the model of quantum branching programs (QBPs). In order to clarify the relationship between QBPs and non-uniform quantum Turing machines, simulations between these two models are presented which allow to transfer upper and lower bound results. Exploiting additional insights about the connection between the running time and the precision of amplitudes, it is shown that non-uniform quantum Turing machines with algebraic amplitudes and QBPs with a suitable analogous set of amplitudes are equivalent in computational power if both models work with bounded or unbounded error. Furthermore, quantum ordered binary decision diagrams (QOBDDs) are considered, which are restricted QBPs that can be regarded as a non-uniform analog of one-way quantum finite automata. Upper and lower bounds are proved that allow a classification of the computational power of QOBDDs in comparison to usual deterministic and randomized variants of the model. Finally, an extension of QBPs is proposed where the performed unitary operation may depend on the result of a previous measurement. A simulation of randomized BPs by this generalized QBP model as well as exponential lower bounds for its ordered variant are presented.

Hierarchy theorems for k OBDDs and k IBDDs

Almost the same types of restricted branching programs (or binary decision diagrams BDDs) are considered in complexity theory and in applications like hardware verification. These models are read-once branching programs (free BDDs) and certain types of oblivious branching programs (ordered and indexed BDDs with k layers). The complexity of the satisfiability problem for these restricted branching programs is investigated and tight hierarchy results are proved for the classes of functions representable by k layers of ordered or indexed BDDs of polynomial size.

New Lower Bounds and Hierarchy Results for Restricted Branching Programs

In unrestricted branching programs all variables may be tested arbitrarily often on each path. But exponential lower bounds are only known if on each path the number of tests of each variable is bounded. We examine branching programs in which for each path the number of variables that are tested more than once is bounded bykbut we do not bound the number of tests of those variables. Using a new lower bound method we can prove that such branching programs become more powerful by increasingkonly by 1: Fork?(1??)(n/3)(1/3)/log2/3n, where?>0, we exhibit Boolean functions that can be represented in polynomial size ifkvariables may be tested more than once on each path, but only in exponential size ifk?1 variables may be tested more than once on each path. Therefore, we obtain a tight hierarchy.

Minimization of decision trees is hard to approximate

Decision trees are representations of discrete functions with widespread applications in, e.g., complexity theory and data mining and exploration. In these areas it is important to obtain decision trees of small size. The minimization problem for decision trees is known to be NP-hard. In this paper the problem is shown to be even hard to approximate up to any constant factor under the assumption P NP.

A Comparison of Free BDDs and Transformed BDDs

Ordered binary decision diagrams (OBDDs) introduced by Bryant (IEEE Trans. on Computers, Vol. 35, pp. 677–691, 1986) have found a lot of applications in verification and CAD. Their use is limited if the OBDD size of the considered functions is too large. Therefore, a variety of generalized BDD models has been presented, among them FBDDs (free BDDs) and TBDDs (transformed BDDs). Here the quite tight relations between these models are revealed and their advantages and disadvantages are discussed.

Parity OBDDs cannot be handled efficiently enough

Parity Ordered Binary Decision Diagrams (POBDDs) seem to be a promising alternative to OBDDs because they admit a more compact representation of Boolean functions and polynomial time algorithms for all important operations on Boolean functions. However, for the reduction of POBDDs only an algorithm based on Gaussian elimination is known. Since the reduction is used quite frequently, a linear time algorithm would be desirable. In this note it is proved that it is unlikely that such an algorithm exists because a linear time algorithm for POBDD reduction would imply a linear time algorithm for matrix rank computation.

A hierarchy result for read-once branching programs with restricted parity nondeterminism

Restricted branching programs are considered in complexity theory in order to study the space complexity of sequential computations and in applications as a data structure for Boolean functions. In this paper (○+, k)-branching programs and (V, k)-branching programs are considered, i.e., branching programs starting with a ○+- (or V-)node with a fan-out of k whose successors are k read-once branching programs. This model is motivated by the investigation of the power of nondeterminism in branching programs and of similar variants that have been considered as a data structure. Lower bound methods for these variants of branching programs are presented, which allow to prove even hierarchy results for polynomial size (○+, k)- and (V, k)-branching programs with respect to k.