Abstract
Chain reduction enables reduced ordered binary decision diagrams (BDDs) and zero-suppressed binary decision diagrams (ZDDs) to each take advantage of the others’ ability to symbolically represent Boolean functions in compact form. For any Boolean function, its chain-reduced ZDD (CZDD) representation will be no larger than its ZDD representation, and at most twice the size of its BDD representation. The chain-reduced BDD (CBDD) of a function will be no larger than its BDD representation, and at most three times the size of its CZDD representation. Extensions to the standard algorithms for operating on BDDs and ZDDs enable them to operate on the chain-reduced versions. Experimental evaluations on representative benchmarks for encoding word lists, solving combinatorial problems, and operating on digital circuits indicate that chain reduction can provide significant benefits in terms of both memory and execution time.
Highlights
Decision diagrams (DDs) encode sets of values in compact forms, such that operations on the sets can be performed on the encoded representation, without expanding the sets into their individual elements
We introduce two new representations: chain-reduced ordered binary decision diagrams (CBDDs), and chain-reduced zero-suppressed binary decision diagrams (CZDDs)
We show bounds on the relative sizes of the representations as: Rf (CBDD, CZDD) ≤ 3 (5)
Summary
Decision diagrams (DDs) encode sets of values in compact forms, such that operations on the sets can be performed on the encoded representation, without expanding the sets into their individual elements. We present extensions to both representations, such that BDDs can take advantage of the source of compaction provided by ZDDs, and vice-versa Both BDDs and ZDDs encode sets of binary sequences of some fixed length n, defining a Boolean function over n variables. This paper defines the two compressed representations, derives the bounds indicated in (5) and (6) and presents extensions of the core BDD and ZDD algorithms to their chained versions. It describes an implementation based on modifications of the CUDD BDD package [14]. It presents some experimental results and concludes with a discussion of the merits of chaining and possible extensions
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