Exact and Approximate Algorithms for the Optimization of Area and Delay in Multiple Constant Multiplications

• Published in: IEEE transactions on computer-aided design of integrated circuits and systems Vol. 27; no. 6; pp. 1013 - 1026
• Main Authors: Aksoy, L., da Costa, E., Flores, P., Monteiro, J.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.06.2008; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithm design and analysis; Algorithms; Approximation; Approximation algorithms; Canonical signed digit (CSD); common subexpression elimination (CSE); Constraint optimization; Delay; delay constraints; Digital filters; Digital signal processing; Exact solutions; Fast Fourier transforms; Finite impulse response filter; Integer linear programming; Mathematical analysis; Mathematical models; minimal signed digit (MSD); multiple constant multiplications (MCMs); Networks; pseudo-Boolean optimization (PBO); Signal processing algorithms; Solvers; Studies
• ISSN: 0278-0070; 1937-4151
• DOI: 10.1109/TCAD.2008.923242

The main contribution of this paper is an exact common subexpression elimination algorithm for the optimum sharing of partial terms in multiple constant multiplications (MCMs). We model this problem as a Boolean network that covers all possible partial terms that may be used to generate the set of coefficients in the MCM instance. We cast this problem into a 0-1 integer linear programming (ILP) by requiring that the single output of this network is asserted while minimizing the number of gates representing operations in the MCM implementation that evaluate to one. A satisfiability (SAT)-based 0-1 ILP 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. Because performance is often a primary design parameter, we describe how this algorithm can be modified to target the minimum area solution under a user-specified delay constraint. Additionally, we propose an approximate algorithm based on the exact approach with extremely competitive results. We have applied these algorithms on the design of digital filters and present a comprehensive set of results that evaluate ours and existing approximation schemes against exact solutions under different number representations and using different SAT solvers.