Using Transposition to Efficiently Solve Constant Matrix-Vector Multiplication and Sum of Product Problems

Abstract

In this work, we present an approach to alleviate the potential benefit of adder graph algorithms by solving the transposed form of the problem and then transposing the solution. The key contribution is a systematic way to obtain the transposed realization with a minimum number of cascaded adders subject to the input realization. In this way, wide and low constant matrix multiplication problems, with sum of products as a special case, which are normally exceptionally time consuming to solve using adder graph algorithms, can be solved by first transposing the matrix and then transposing the solution. Examples show that while the relation between the adder depth of the solution to the transposed problem and the original problem is not straightforward, there are many cases where the reduction in adder cost will more than compensate for the potential increase in adder depth and result in implementations with reduced power consumption compared to using sub-expression sharing algorithms, which can both solve the original problem directly in reasonable time and guarantee a minimum adder depth. Keywords Constant matrix multiplication (CMM) • Multiple constant multiplication (MCM) • Shift-and-add • Sum of products (SOP) • Minimum depth expansion algorithm.