An Exact Algorithm for the Maximal Sharing of Partial Terms in Multiple Constant Multiplications
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
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
Date: 2005-Nov-04 Time: 14:30:00 Room: 336
For more information:
Workshop “Metabolism and mathematical models: Two for a tango” – 2nd Edition
Title: Workshop Metabolism and mathematical models: Two for a tango – 2nd Edition
Dates: October 25-26, 2022
Location: This workshop will be held in a virtual way
The topic of this workshop is metabolism in general, with a special focus, although not exclusive, on parasitology. Besides an exploration of the biological, biochemical and biomedical aspects, the workshop will also aim at presenting some of the mathematical modelling, algorithmic theory and software development that have become crucial to explore such aspects.
This workshop is being organised in the context of two projects, both with the Inria European Team Erable. One of the projects involves a partnership with the University of São Paulo (USP), in São Paulo, Brazil, more specifically the Institute of Mathematics and Statistics (IME) and the Institute of Biomedical Sciences – Inria Associated Team Capoeira – and the other involves the Inesc-ID/IST in Portugal, ETH in Zürich and EMBL in Heidelberg – H2020 Twinning Project Olissipo.
The workshop is open to all members of these two projects but also, importantly, to the community in general.
The program and more details are available here.