A detailed complexity model for multiple constant multiplication and an algorithm to minimize the complexity

Abstract

Multiple constant multiplication (MCM) has been an active research area for the last decade. Most work so far have only considered the number of additions to realize a number of constant multiplications with the same input. In this work we consider the number of full and half adder cells required to realize those additions and a novel complexity measure is proposed. The proposed complexity measure can be utilized for all types of constant operations based on shifts, additions and subtractions. Based on the proposed complexity measure a novel MCM algorithm is presented. Simulations show that compared with previous algorithms, the proposed MCM algorithm have a similar number of additions while the number of full adder cells are significantly reduced.

Optimization of Area and Delay at Gate-Level in Multiple Constant Multiplications

“…The GB algorithm [11], that is based on RAG-n [8], initially takes an unimplemented constant that requires the smallest number of full adders (FAs), and then synthesizes it using a single operation including one or two intermediate constants that lead to the smallest number of FAs overhead.…”

“…Although there exist a large number of algorithms designed for the MCM problem, there are only a few algorithms [10], [11] that target directly on the reduction of area in the MCM design at gate-level. However, the exact CSE algorithm of [10] considers a restricted number of possible implementations of the constant multiplications with respect to a GB algorithm and the GB algorithm of [11] is not equipped with the recently proposed efficient heuristics such as [5], [9].…”

“…However, the exact CSE algorithm of [10] considers a restricted number of possible implementations of the constant multiplications with respect to a GB algorithm and the GB algorithm of [11] is not equipped with the recently proposed efficient heuristics such as [5], [9].…”

“…In this paper, we start by introducing area efficient addition and subtraction architectures used in the MCM design, since the architectures proposed in [10] and [11] do not cover all possible addition and subtraction operations in MCM and do not consider additional simplifications respectively. Then, we propose a GB algorithm, called MINAS (MINimum Area Search algorithm), that considers more possible implementations of the constant multiplications than the exact CSE algorithm [10] and uses a better heuristic than the GB algorithm [11].…”

“…Then, we propose a GB algorithm, called MINAS (MINimum Area Search algorithm), that considers more possible implementations of the constant multiplications than the exact CSE algorithm [10] and uses a better heuristic than the GB algorithm [11]. Also, based on the MINAS algorithm, we introduce the MINAS-DC algorithm that searches for a solution with the smallest area of the MCM design under a delay constraint that is defined as the maximal number of operations in series, generally known as the number of adder-steps.…”