Anna Slobodová

Speeding up variable reordering of OBDDs

The use of Ordered Binary Decision Diagrams (OB-DDs) as a representation of Boolean functions brought essential progress in many different applications. The optimization of the OBDD-size by the choice of the variable ordering is known to be NP-hard. The known heuristics for finding an initial ordering and for reordering still have insufficient performance. Rudells sifting is one of the most successful reordering algorithms that is application independent and can be used dynamically. In this paper, we propose a method based on some communication complexity considerations that improves the time performance of the sifting. The main idea is to restrict the reordering of variables to blocks that are determined according to readily computable OBDD-measure. Experimental comparison of the proposed block-restricted sifting with the original algorithm shows a speed-up by factor two without any loss in the final size.

Efficient OBDD-Based Boolean Manipulation in CAD Beyond Current Limits

We present the concept of TBDDs which considerably enlarges the class of Boolean functions that can be efficiently manipulated in terms of OBDDs. It extends the idea of using domain transformations, which is well-known in many areas of mathematics, physics, and technical sciences, to the context of OBDD-based Boolean function manipulation in CAD: Instead of working with the OBDD-representation of a function f, TBDDs allow working with an OBDD-representation of a suited cube transformed version of f. Besides of giving some theoretical insights into the new concept, we investigate in some detail cube transformations which are based on complete types. We - show that such TBDD-representations can be derived similarly as OBDD-representations, - give evidence of the practical importance of such TBDDs by presenting very small-size TBDD-representations of the hidden weighted bit functions HWBn which were proved to have only very large OBDD-representations, and - report some promising experimental results with some ISCAS benchmark circuits including the multiplier circuit C6288.

Some heuristics for generating tree-like FBDD types

Reduced ordered binary decision diagrams (OBDDs) are nowadays the state-of-the-art representation scheme for Boolean functions in Boolean manipulation. Recent results have shown that it is possible to use the more general concept of free binary decision diagrams (FBDDs) without giving up most of the useful computational properties of OBDDs, but possibly reducing the space requirements considerably. The amount of space reduction depends essentially on the shape of so-called FBDD-types the Boolean manipulation in terms of FBDDs is based on. Here, we propose some heuristics for deriving tree-like FBDD-types from given circuit descriptions. The experimental results we obtained clearly demonstrate that the FBDD-approach is not only of theoretical interest, but also of practical usefulness even in the ease of using merely such simple-structured tree-based FBDD-types as produced by the investigated heuristics.

On the Complexity of Constructing Optimal Ordered Binary Decision Diagrams

In the following we prove the widely assumed conjecture that the construction of an optimal OBDD-representation of a Boolean function f is intractable if f is given in terms of DNFs, CNFs, logical circuits, or Ω-BDDs. The problem remains NP-hard if one wants to construct an optimal OBDD-representation merely in such cases its size is bounded by a given constant. Both results remain true if the variable ordering of an optimal OBDD-representation is known in advance.

Boolean manipulation with free BDD's. First experimental results

It is shown that Free Binary Decision Diagrams (FBDDs), with respect to a predefined type, provide a canonical representation and allow efficient solutions of the basic tasks in Boolean manipulation in a similar manner to the well-known OBDDs. However, in contrast to OBDDs, the FBDDs allow more succinct representations of Boolean functions. For experimentation we have used an FBDD-package and the types worked with are tree-based. Using different type-creating heuristics, we compare the size of FBDD-representations of some ISCAS benchmarks with the size of their OBDD-representations.<<ETX>>

Global rebuilding of OBDDs Avoiding Memory Requirement Maxima

It is well-known that the size of an ordered binary decision diagram (OBDD) may depend crucially on the order in which the variables occur. In the paper, we describe an implementation of an outputefficient algorithm that transforms an OBDD P representing a Boolean function f with respect to one variable ordering π into an OBDD Q that represents f with respect to another variable ordering σ. The algorithm runs in average time O(¦P∥Q¦) and requires O(¦P¦+n¦Q¦) space.

Application driven variable reordering and an example implementation in reachability analysis

Variable reordering is the main approach to minimize the size of Ordered Binary Decision Diagrams. But despite the huge effort spent, up to now, to design different reordering heuristics, their performance often does not meet the needs of the applications. In many OBDD-based computations, the time cost for reordering dominates the time spent by the computation itself. There are some known approaches for accelerating the reordering by taking advantage of structural properties of OBDDs and functions represented. In this paper, we propose a reordering method that exploits application specific information. The main idea is to drive the reordering process by the computation. This effects an acceleration of the whole computation rather than of the reordering only. The power of the approach is illustrated by speeding up forward traversal of finite state machines.

A Unifying Theoretical Background for Some Bdd-based Data Structures

In the paper, we propose a general concept (denoted by TBDD) for Boolean functions manipulation that is based on cube transformations. The basic idea is to manipulate a Boolean function by converting it by means of a cube transformation into a function that can be efficiently represented and manipulated in terms of ordered binary decision diagrams (OBDDs). We show that the new concept unifies several BDD–based data structures considerably, and simplifies their manipulation to work with the simple and well–understood data struture of OBDDs. This is especially important for practical applications.Further, to give an example of how TBDDs open new ways in the search for efficient data structures for Boolean functions, we discuss the data structure of typed kFBDDs.

Global rebuilding of OBDD's avoiding memory requirement maxima

It is well-known that the size of an ordered binary decision diagram (OBDD) may depend crucially on the order in which the variables occur. In the paper, we describe an implementation of an output-efficient algorithm that transforms an OBDD P representing a Boolean function f with respect to one variable ordering /spl pi/ into an OBDD Q that represents f with respect to another variable ordering /spl sigma/. The algorithm runs in average time O(|P/spl par/Q|) and requires O(|P|+|Q|) space. The importance of the algorithm is demonstrated by means of experimental results on basically two different applications. In one of them, the algorithm is used merely once. Such transformations are needed to test equivalence or to perform synthesis on OBDDs in which variables appear in different orders. The other application shows a way how to decrease the size of intermediate OBDD representations of a given circuit in the course of its symbolic simulation. Here, the algorithm is used dynamically, whenever the size of the manipulated OBDDs becomes too large.

On the Composition Problem for OBDDs with Multiple Variable Orders

Ordered Binary Decision Diagram (OBDD) is a favorite data structure used for representation Boolean functions in computer-aided synthesis and verification of digital systems. The secret of its success is the efficiency of the algorithms for Boolean operations, satisfiability and equivalence check. However, the algorithms work well under condition only that the variable order of considered OBDDs is the same.