An efficient variable partitioning approach for functional decomposition of circuits

Functional decomposition is a process of splitting a complex circuit into smaller sub-circuits. There exist two major strategies in decomposition, namely, serial and parallel decomposition. In serial decomposition the problem the complex function represented as a truth table with support set variables and partitioned into free and bout set variables. The minterms corresponding to the bound set variables are represented as an equivalent function called the predecessor function. Equivalent minterms of the bound set variables are assigned an output code. The assigned output codes and the free set variable minterms are represented as the successor function. Serial decomposition is further categorized into disjoint and non-disjoint decomposition, when the free and bound set variables are disjoint and non-disjoint respectively. This paper deals with the problem of determining the set of best free and bound variables (variable partitioning problem) for disjoint serial decomposition. Variable partitioning is the first step in decomposition process. An efficient variable partition algorithm is one that determines the set of all free and bound set variables that satisfy the decomposition theorem in minimal time and by exploring the search space effectively. This will allow the decomposition algorithm to determine the best variable partition of a function that results in smaller decomposed functions and with maximum number of do not cares in these functions. Classical approaches to determine the best free and bound set use exhaustive search methods. The time and memory requirements for such approaches are exponential or super exponential.A novel heuristic search approach is proposed to determine the set of good variable partitions in minimal time by minimally exploring the search space. There are two heuristics employed in the proposed search approach, (1) r-admissibility based heuristic or pruned breadth first search (PBFS) approach and (2) Information relation based heuristic or improved pruned breadth first search (IPBFS) approach. The r-admissibility based heuristic is based on r-partition characteristics of the free and bound set variables. The information relation and measure based heuristic is based on information relationship of free and bound set variables that are expressed as r-partition heuristics. The proposed variable partition search approach has been successfully implemented and test with MCNC and Espresso benchmarks and the results indicate that the time complexity is comparable to r-admissible heuristic algorithm and the quality of solution is comparable to exact variable partitioning algorithm. A comparison of PBFS and IPBFS heuristics for certain benchmarks are also discussed in this paper.