Adaptive Booth Algorithm for Three-integers Multiplication for Reconfigurable Mesh

Abstract

This paper presents a three-integers multiplication algorithm R = A * X * Y for Reconfigurable Mesh (RM). It is based on a three-integer multiplication algorithm for faster FPGA implementations. We show that multiplying three integers of n bits can be performed on a 3D RM of size (3n + log⁡ n + 1) × (2n+1 +3) × n+1 using 44 + 18 · log log MNO steps, where MNO is a bound which is related to the number of sequences of ‘1’s in the multiplied numbers. The value of MNO is bounded by n but experimentally we show that on the average it is n. Two algorithms for solving multiplication on a RM exists and their techniques are asymptotically better time wise, O(1) and O(log⁡* n), but they suffer from large hidden constants and slow data insertion time (O(n)) respectively. The proposed algorithm is relatively simple and faster on the average (via sampling input values) then the previous two algorithms thus contributes in making the RM a practical and feasible model. Our experiments show a significant improvement in the expected number of elementary operations for the proposed algorithm.

Field Programmable Reconfigurable Mesh (FPRM)

“…The or-term of these and-terms is computed in another broadcast along the last column. Obviously, the FPRM can be used to execute the O 1 ðÞRM algorithms such as summing of n numbers, multiplication [6,28], sorting, convex hull [2], graph algorithms [9] and image processing [11]). However, here we consider the problem of parallel evaluation of circuits with large depths for which no previous RM algorithm exists.…”