Hypergraph Based Minimum Arborescence Algorithm for the Optimization and Reoptimization of Multiple Constant Multiplications

Graph-based heuristic for Multiple Constant Multiplication (MCM) has been an area of intensive research over the last two decades. To overcome the pitfalls of hidden partial sums and local minima inherent in trivial digraph formulation, this paper introduces an elementary digraph-to-hypergraph transformation to enable a new minimum arborescence (MA) formulation of MCM problem. Our proposed algorithm iteratively solves the MA problem from a strongly connected graph formed by adding new vertex from the least cost surviving hypergraph in the preceding iteration until no new vertex is found. As the visible and hidden edges of an elementary hypergraph (EHG) are interchangeable by swapping their values encoded in its hyperedge, previously removed vertex can be revived if it is a predecessor of the least cost surviving hyperedge as the hypergraph evolves. A by-product of this iterative algorithm is that its underlying EHG minimization can also be used for quality improvement of existing MCM solutions generated by other design algorithms or for engineering changes. Experimental results on benchmark filters show that our proposed algorithm can reduce their multiplier block areas by 2.9% to 14.9% and power consumptions by 2.8% to 20.1% over other advanced algorithms. A further ~20% of area can be saved from the less area-efficient solutions of other design algorithms by the proposed EHG reoptimization.