Obtaining Minimum Depth Sum of Products from Multiple Constant Multiplication

Abstract

In this work, an approach for transposing solutions to the multiple constant multiplication (MCM) problem to obtain a sum of product (SOP) computation with minimum depth is proposed. The reason for doing this is that solving the SOP problem directly is highly computationally intensive when adder graph algorithms are used. Compared to using sub-expression sharing algorithms, which has a lower computational complexity, directly for the SOP problem, it is shown that the proposed approach, as expected, results in lower complexity for the SOP. It is also shown that there is no obvious way to construct the MCM solution in such a way that the SOP solution has the minimum theoretical depth. However, the proposed approach guarantees minimum depth subject to the MCM solution given as input.Index Terms-multiple constant multiplication (MCM), shiftand-add, Sum of Product (SOP), minimum depth expansion algorithm, and Vertex reduced SOP adder Graph (VSG).

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

“…However, to the best knowledge of the authors, there has been no earlier work that shows how to perform the transposition in practice, less considers the adder depth while doing it. A preliminary version of this work was introduced in [18] for the SOP problem only.…”