Explicit and implicit algorithms for binate covering problems

Abstract

We survey techniques for solving binate covering problems, an optimization step often occurring in logic synthesis applications. Standard exact solutions are found with a branchand-bound exhaustive search, made more efficient by bounding away regions of the search space. Standard approaches are said to be explicit because they work on a direct representation of the binate table, usually as a matrix. Recently, covering problems involving large tables have been attacked with implicit techniques. They are based on the representation by reducedordered binary decision diagrams of an encoding of the binate table. We show how table reductions, computation of a lower bound, and of a branching column can be performed on the table so represented. We report experiments for two different applications that demonstrate that implicit techniques handle instances beyond the reach of explicit techniques. Various aspects of our original research are presented for the first time, together with a selection of the most important old and new results scattered in many sources. Tiziano Villa, for a photograph and biography, see this issue, p. 675. Timothy Kam (M'87), for a photograph and biography, see this issue, p. 675.

Dynamic template generation for resource sharing in control and data flow graphs

“…However, this problem is NP-Complete. Villa et al [8] proposed acceleration techniques based on cost bounding. However, even at its best, binate covering can be slow for large CDFGs.…”

“…During this procedure it is necessary to remove any overlapping nodesets that arise locally within each set of template matches. Villa et al [8] describe a method of obtaining a close approximation of the maximal independent set using an adjacency matrix, in which the number of nodesets is maximized by first selecting those that conflict with the least number of nodesets, and then eliminating any rows/columns that intersect with a '1'. Expressions with less than two matches are pruned, since there is no benefit to instantiating single-match templates.…”

Satisfiability-Based Algorithms for Boolean Optimization

“…(Details of the work on BCP can be found in [4,12,20].) Despite these improvements, and as with other NP-hard problems, additional search pruning ability allows in general very significant gains, both in the amount of search and in the run times.…”

“…BCP can be formulated as the problem of finding a satisfying assignment for a given Conjunctive Normal Form (CNF) formula subject to minimizing a given cost function. As with generic Boolean Optimization, BCP also finds many applications, including the computation of minimum-size prime implicants, of interest in Automated Reasoning and Non-Monotonic Reasoning [18], and as a modeling tool in Electronic Design Automation (EDA) [4,20].…”

Integrating CNF and BDD based SAT solvers

“…Unfortunately, their use is rather limited due to the size explosion problem. It has been observed that so long as the BDD size can be contained, solutions can be found quickly, often with predictable run-times [6] [7] [23]. If, for relatively large hard-to-solve instances, CNF tools invest a large amount of time, and BDDs run into memory explosion, how can their respective capabilities and limitations be leveraged to efficiently search for a SAT solution?…”