# An Exact Algorithm for the Maximal Sharing of Partial Terms in Multiple Constant Multiplications

**Paulo Flores**,

**Inesc-ID** –

## Abstract:

Several computationally intensive operations, such as, Finite Impulse

Response (FIR) filters and Fast Fourier Transforms (FFT), involve a

sequence of Multiply-Accumulate (MAC) operations with constant

coefficients. Hardwired dedicated architectures are the best option

for maximum performance and minimum power consumption.

Constant coefficients allow for a great simplification of the

multipliers, which can be reduced to shift-adders. Shifts are free in

terms of hardware, hence the hardware required for a multiplication

with a constant with N bits set to 1 is simply N-1 adders. In many MAC

operations, the same input is to be multiplied by a set of

coefficients, an operation known as Multiple Constant Multiplications

(MCM). An example of this is the transposed form architecture of a FIR

filter. In this situation, significant reductions in hardware, and

consequently power, can be obtained by sharing the partial products of

the input.

In this talk I propose an exact algorithm that maximizes the sharing

of partial terms in MCM operations. We model this problem as a Boolean

network that covers all possible partial terms which may be used to

generate the set of coefficients in the MCM instance. The PIs to this

network are shifted versions of the MCM input. An AND gate represents

an adder or a subtracter, ie, an AND gate generates a new partial

term. All partial terms that have the same numerical value are ORed

together. There is a single output which is an AND over all the

coefficients in the MCM. We cast this problem into a 0-1 Integer

Linear Programming (ILP) problem by requiring that the output is

asserted while minimizing the total number of AND gates that evaluate

to one. A SAT-based solver is used to obtain the exact solution. We

argue that for many real problems the size of the problem is within

the capabilities of current SAT solvers.

We present results using binary, CSD and MSD representations. Two main

conclusions can be drawn from the results. One is that, in many cases,

existing heuristics perform well, computing the best solution, or one

close to it. The other is that the flexibility of the MSD

representation does not have a significant impact in the solution

obtained.

Date: 2005-Nov-04 Time: 14:30:00 Room: 336

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**Title: ** Workshop Metabolism and mathematical models: Two for a tango – 2nd Edition

**Dates:** October 25-26, 2022

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