Performance Analysis of the Parallel Karatsuba Multiplication Algorithm for Distributed Memory Architectures

We present three parallel implementations of the Karatsuba algorithm for long integer multiplication on a distributed memory architecture and discuss the experimental results obtained on a Paragon computer. The first two implementations have both time complexityO(n) onnlog23processors, but present different behavior for inputs of practical size. The third algorithm has complexityO(nlog2nonnprocessors, offering therefore better asymptotic efficiency. A refinement of the asymptotic analysis for the important case of a constant number of processors takes into account sequential parts of the algorithm and communications overhead. It is shown that the theoretically best speed-up and efficiency can be obtained with two of the algorithms for sufficient problem size. The experimental results confirm the analysis.